COM BINATORIAL, ALGEBRAIC A N D TOPOLOGICAL R E P R E S E NTATION S 0F G ROU PS, S E M I G R O U P S A N D CATEGORIES
North-Holland Mathematical Library Board of Advisory Editors : M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring, H. Halberstam, L. V. Hormander, M. Kac, J. H. B. Kemperman, H. A. Lauwerier, W. A. J. Luxemburg, P. F. Peterson, I. M. Singer and A. C . Zaanen
VOLUME 22
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COMBINATORIAL, ALGEBRAIC AND TOPOLOGICAL REPRESENTATIONS OF GROUPS, SEMIGROUPS AND CATEGORIES ALES PULTR TRNKOVA
VERA
198C
NORTH-HOLLAND PUBLISHING COMPANY
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@ Ale: Pultr, Vera Trnkova, Prague 1980 Translation @ Ale2 Pultr, Vera T'rnko\,i. Prague 1980 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers
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PREFACE The theory of representations of groups, semigroups and categories as groups of symmetries, semigroups of endomorphisms and concrete categories, respectively, underwent a rapid development in the sixties and at the beginning of the seventies. Before that time, only a few basic results had been obtained on representing groups as symmetry groups of graphs, lattices and topological spaces, and on rigidity (see the Introduction). At the present moment, writing a monography on the subject one is even forced to omit several interesting results to avoid covering much more than half a thousand pages. A very important moment was the one at which, besides the automorphism groups, the investigation of the endomorphism semigroups and concrete categories was started. However, it would be totally misleading to consider the subsequent development as a mere generalization of the known results on groups. The vast majority of the facts concerning the representation of groups as automorphism groups known in, say, 1973 have appeared later than 1963. This can also be said, perhaps even more pointedly, about another ,,classical" area in the field, namely, about the questions concerning rigidity. The interest in representing categories, above all, introduced the categorial language and techniques into the field, and this proved to be very beneficiary also for those questions that do not concern categories at all. What could be better evidence for the usefulness of a theory? In this book much is said about categories and the categorial language is cften used in places where one might avoid it if one so wished. Howevef, this is not a book on category theory. No previous knowledge of category theory is assumed at all. All the necessary terminology and technique (the
VI amount of which is not extensive) are introduced in the text. What we would like to assume from reader is interest, on his part, in combinatorics, or universal algebra, or topology, or general questions on structures. We wish to express our thanks to Z. Hedrlin and L. KuEera for allowing us to reproduce here some of their results (Chapter 111), which have only been partly published and without which this book would be very incomplete. We are also very grateful to M. KatEtov and J. Novak for their encouragement during the work on the manuscript. Further, we are indebted to L. Beran, J. Hejcman, J. J. Charatonik, V. Koubek and D. Zaremba for careful reading and remarks on the text, to J. Fried, J. Janda, J. Mach, J. Pavelka, J. Scholz, J. SiSka and J. Vinarek for considerable help with the final preparation of the manuscript, to V1. Mach for linguistic advice and, last but not least, to Mrs. A. FuEikova for careful typing.
ALESPULTR,VERA TRNKOVA
CONTENTS PREFACE
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INTRODUCTION . .
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CHAPTER I . PRELIMINARIES . .
8
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6 8
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. . . . . . . 1 . Relations. algebraic operations. homomorphisms . 2. Monoids and concrete monoids . . . . . 3. Categories. functors. transformations . . . . 4. Concrete categories . . . . . . . . . 5. Some important concrete catcgoi-ies . . . . 6. Embeddings . . . . . . : . . . . 7 . Two easy but important embeddings . . . . 8 . The representation problems . . . . . . 9. Bibliographical remarks . . . . . . . .
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CHAPTER I1. BASICEMBEDDINGS . . . . . . . . . . . 0 1. Three obvious realizations . . . . . . . . . . $ 2. Two important extensions . . . . . . . . . . $ 3 . Rigid binary relations . . . . . . . . . . . $ 4. Rigid symmetric binary relations . . . . . . . . $ 5. Graph is alg-universal. Consequences . . . . . . . 5 6. Assumption (M) and strong embedding of S ( P - ) into Graph $ 7. Strong embedding of S ( P + )into S ( P - ) . . . . . . $ 8. Bibliographical remarks . . . . . . . . . . .
v 1 21 21 24 28 38 40 42 50 52 56 57 57 59
63 66 71 77 81 86
CHAPTER 111. UNIVERSALITY OF S ( P + ) .
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. . . . . . Q 1. Strong embedding of S(P+...., P’) into S(P+) . . . . Q 2. Representations of thin categories . . . . . . . . 4 3. Categories 9 ( F ; (T, 5 ) )and realizations of concrete categories . . . . . . . . . . . . . . . . 9 4. s ( P + )is universal . . . . . . . . . . . . . Q 5 . Bibliographical remarks . . . . . . . . . . .
CHAPTER IV . COMBINATORICS . .
. . . . . Q 1. Graphs, symmetric graphs. undirected graphs . 3 2. The “arrow construction” in its simplest form . Q 3. Two applications of the arrow construction: graphs and acyclic graphs . . . . . . 9 4. More about undirected graphs . . . . . Q 5 . Partially ordered sets . . . . . . . 3 6. Graphs with strong homomorphisms . . . 4 7. Graphswithloops . . . . . . . . 9 8. Sets with two equivalences . . . . . . Q 9. A technicallemma . . . . . . . . 9: 10. On a problem by S. Ulam . . . . . . 9 11. Bibliographical remarks . . . . . . .
CHAPTER V . ALGEBRA . .
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Symmetric
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Q 2. Embeddings into the categories of semigroups and monoids
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0 1. An elementary result on To-spaces . . . 9 2. Some special mappings . Quontients and 9 9 0
99
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9 3. Categories of rings . . . . . . . . . . . .
CHAPTER VI . TOPOLOGY . .
94 36
. . . 101 . . . . 101 . . . . 105
9 1. Some easy results . . . . . . . . . . . . . Q 4. Categories of lattices . . . . Q 5 . Unary algebras . . . . . 9 6. Categories of small categories . 9 7. Remarks on categories of functors $ 8. Bibliographical remarks . . .
87 87 89
. . . . . sums of metric
spaces . . . . . . . . . . . . . . . . 3. The functors A, Mo. 2,do.Asand Au. . . . . . 4. Some full embeddings into categories of metric spaces . . 5 . Labelled topologizedgraphs. The functor 9 . . . . .
142 143 145 154 160 173 185 196 204 206 207 208 213 218 220
9 6. Construction of sufficiently rigid basic and fundamental classes .
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. 222
9 7. Some strong embeddings into categories of metric spaces . 227 9 8. Some universal categories of metric spaces . . . . . 230
ij 9. Negative results on open and locally one-to-one mappings
233
9 10. Techniques for Tl-spaces . . . . . . . . . . 236 9 11. Alg-universal categories of TI-spaces . . . . . . . . 239 3 12. Strong embeddings into categories of Tl-spaces . . . . 241 9 13. The category of TI-spaces and their open continuous mappings is universal .
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8 14. Rigid spaces and stiff classes of spaces . . . 8 15. The category of paracompact spaces is almost 4 16. Compact Hausdorff spaces . . . . . . 9 17. Some negative results . . . . . . . 9 18. Bibliographictil remarks . . . . . . .
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. 244 . 246
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universal
249 253 258 264
CHAPTER VlI . STRONGEMBEDDINGS AND STRONGLY ALGEBRAIC CATEGORIES
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. 266
9 1. Strong embeddings of categories S(F) into S ( P f )and Graph 266 9 2. Which concrete categories are strongly embeddable into S(P+) .
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. 268
9 3 . Strong universality . . . . . . . . . . . . 273 9 4. Examples of strongly hniversal categories . . . . . . 274 ij 5. Strong embeddings of Alg(d) into Alg(d’) .
8 6. Q 7. 3 8. 9 9. 8 10.
. . . . 274 Some negative results and the notion of strong algebraicity 278 Categories A((Fi, di)iEJ). . . . . . . . . . . 280 A criterion of strong algebraicity . . . . . . . . 283 Applications . . . . . . . . . . . . . . 287 Bibliographical remarks . . . . . . . . . . . 290
APPENDIX A . COOKCONTINUA
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ij 1. Continua and their basic properties
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. 300
291 291
9 2. Indecomposable continua . . . . . . . . . . 293 9 3. Limits of diagrams in Top . . . . . . . . . . 294 8 4. Snake-like and circle-like continua . Solenoids . . . . . 296 § 5. Crookedmappings
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9 6. Hereditarily indecomposable continua . . . . . . . 304 9 7. Monotone, atomicandconfluent mappings . . . . . 307 9 8. Upper semicontinuous mappings . . . . . . . . 309
9 9. Mappings of a circle into itself . . . . . . . . . 31 1 9 10. Upper semicontinuous mappings berween circle-like continua . Non-homeomorphic solenoids .
. . . . . . 315
9 11. Cook continua . . . . . . . . . . . . . . 5 12. The double diagram D . . . . . . . . . . . 0 13. Construction lemmas . . . . . . . . . . . Q 14. The d-process . . . . . . . . . 9 15. Processes 98,%,' 9, 6 and construction of D . APPENDIXB . MEASURABLE CARDINALS
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. .
. .
AND NON-ALGEBRAIC CATE-
. . . . . . . . . . . . . . . . . 1. @-additivemeasure . . . . . . . . . . . . 2. If an a-additive measure exists. it is on a rather large set 3. F-measures . . . . . . . . . . . . . 4. Existence of non-algebraic categories under non(M) . .
GORIES
9 5 9 9
RESEARCH PROBLEMS .
319 321 324 . 330 . 334
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342 342 345 346 348
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BIBLIOGRAPHY . .
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INDEX . . . . INDEXOF SYMBOLS .
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INTRODUCTION 1.
Every group is isomorphic to a group of permutations of a set. Actually, it is isomorphic to a group of permutations of its own underlying set: it suffices to represent each element a by the so-called left translation by a-the permutation sending x to ax. Although this is an important observation, it does not really say all that much. After all, group theory started by investigating groups of permutations, and all that we have here is an affirmation that there are no others. Now, suppose we have a group of permutations and wish to have a way of distinguishing the wanted from the unwanted ones more comfortable than just listing the former. This can hardly be achieved on an unstructured set. Given, however, a structure on a set, we have a distinction between those permutations which behave and those which do not. The first subsystem, the permutations preserving the structure in question (the symmetries, the automorphisms or whatever name one uses for them), forms itself a group. So, e.g., consider the set with a binary relation shown in Fig. 0.1 (where the nodes indicate the elements of the set, an arrow x --t y indicates that (x,y ) is in the relation).
The relation-preserving permutations form a group here, isomorphic to the additive group Z3of integers modulo 3. Similarly one can represent Z,, the additive group of integers modulo n, for an arbitrary n. Or it is
2
INTRCDUCTION 1
easy to see that the symmetries of the set with the binary relation as shown in Fig. 0.2 represent the group Z2 Z,. Using the well known fact that
+
a finite abelian group is a product of cyclic groups, the reader may try to represent an arbitrary finite abelian group in this way. (A little trick is needed: one has to prevent interchanging equally long cycles, if there are any. Generalize, e.g., the representation of Z , + Z3as shown in Fig. 0.3.)
Actually a far more general statement, and a much stronger one as well, was proved by Frucht ([Fr]) for the finite case and by Sabidussi ([Sa,]) for the infinite case: Every group is isomorphic to the group of all automorphisms of a symmetric graph (i.e., a set with a symmetric binary relation). To give examples of other structures forcing automorphisms to behave after a prescribed pattern, let us point to the following theorems: Every group is isomorphic to the automorphism group of a distributive lattice (G. Birkhoff [Bff]). Every group is isomorphic to the autohomeomorphism group ofa topological space (J. de Groot [dG]). The latter result was proved by using the above result on binary relation and a construction which is fairly instructive. Therefore, we will sketch it here, without going into much detail. It goes as follows: Firstly, take a set with binary relation representing our group as the automorphism group. Consider it a set of nodes joined by arrows at the places where the corresponding points are
3
INTRODUCTION 1
in relation. Thus, our group is represented by the permutations sending arrows into arrows. If, moreover, we assume every point to be joined with some other one, we see that the group is actually the group of those permutations of the set of arrows which do not tear them apart. Secondly, choose a topological space A and two distinct points i and t in it. Now, take as many copies of it as we have arrows in the pattern, and glue them, i in the initial points, t in the terminal ones, into it instead of the arrows. For example, if A is the space of Fig. 0.4(a), then the triangle (b) becomes the space shown in (c). Denote the obtained space by X . Now, if A can be constructed so that it is rigid in the sense that there is no homeomorphism of A into itself but the identity mapping and, in addition, that there is no embedding of A into X except onto one of its copies, then the autohomeomorphism group of X consists precisely of those mappings replacing the “arrows” A by other copies, and is, therefore, isomorphic to the original group.
A
Fig. 0.4
De Groot and Wille ([dGW]) constructed a rigid space as follows: Take a closed disc D in the euclidean plane, and a sequence of interior points {x,, x2, x3, . . .} which is dense in D. Now, put y , = x,, and remove the interior of a two-bladed “propeller” with centre in y1 and with diameter small enough not to touch the boundary of D. Denote by A , the space thus obtained. Suppose we have constructed A n - , . Denote by y. the first so far unused x, lying in the interior of A n - l and remove the interior of an (n 1)-bladed “propeller” with centre in y, and with diameter small enough not to touch the boundary of A,- I . Denote by A , the space obtained. Thus, e.g., A , will look, say, as in Fig. 0.5. In the
+
0.5
4 intersection A
INTRODUCTION 1 , 2
nAi,the neighborhood systems of the y’s are different from each other a
=
i= 1
and from the neighborhood systems of the other points to such an extent that. under a homeomorphism, y. has to be sent to itself. Consequently, since { y , , j 2 ...} , is dense in A , the homeomorphism is the identity on the whole of A .
We dwelt this long at the construction of X above because we wanted to stress two of its features. One is the idea of replacing arrows by some object. It proved a very fruitful basis of more general-and sometimes considerably more complicated-constructions. The other, here related closely to the first one, is the idea of rigid object. The rigidity (i.e., the absence of proper symmetries, or a stronger property such as the absence of non-identical structure-preserving mappings into itself-which was not the case for the propeller spaces above, the constants being certainly very legitimate mappings of a topological space into itself) also plays a useful role in constructions of different nature, some of which will be described in the text. And the question of the existence of nontrivial rigid objects with some structure or other is certainly an interesting problem in itself. 2.
Before returning to the original question of representing groups, let us make a provisory agreement concerning some notions of which the reader certainly has an intuitive idea at least (this is meant, by no means, to make up for definitions !). A structure type Y consists of two kinds of data: I. A pattern for additional information added to sets: Mostly it consists of a construction pattern concerning the sets, and of a system of axioms to be satisfied. Table 0.1 gives a few examples. Particular data satisfying the pattern, added to a set X , will be called a structure of type Y on X . 11. A rule for deciding which mappings between sets with structures are well-behaved : That is, it has to be understood in which way the structures are supposed to be “preserved by mappings. Very often, the rule is, or seems to be, obvious from the construction pattern of the structure type. This is why, in an intuitive reasoning about structures, so often only part I is considered and part I1 forgotten. But it is also often the case that one deals with identical information patterns and very different rules for well-behaved mappings.
5
INTRODUCTION 2
A particularly colorful variety may be met in topology: one considers continuous, open, closed, perfect mappings and so on, local homeomorphisms, etc.
For our purposes, we will assume that the rule is such that a composition of well-behaved mappings is well-behaved (more precisely, if f: X 4 Y and g: Y - t Z with is well-behaved with respect to r on X and s on respect to s on Y and t on 2 then g o f is well-behaved with respect to r and t ) and that the identity mapping is well-behaved (with respect to one and the same structure). Some more conventions: A set X together with a structure s of type 9, often referred to as ( X , s), is called an object of the structure type 9. A well-behaved mapping is called (not quite accurately, cf. Ch. I, 5 3) a morphism, and a well-behaved mapping from (X,s) into itself an endomorphism of (X, s). A well-behaved mapping with a well-behaved inverse is called an isomorphism, and an isomorphism from ( X , s ) into itself an automorphism of (x, s). The automorphisms of ( X , s ) form a group which is referred to as the autornorphism group of ( X ,s). Table 0.1 additional information pattern construction pattern
axioms
Binary operation
a mapping w of X x X into X
none
Commutative binary operation
a mapping w of X x X into X
w(x,y ) = w(y, x)
Binary relation
a subset R of X x X
none
Transitive binary relation
a subset R of X x X
Topology
a subset T of expX
6
INTRODUCTION 2
Using the terminology introduced, we can formulate a class of probl in the paradigm (Abstract Group Problem): Given a structure type 9, can every group be represented as the L morphism group of an object? We have already mentioned that the answer is positive if the struc type is that of a symmetric binary relation or that of a topology. Ir text, the answer is shown to be positive for many structures-most’ a particular case of a considerably stronger result. We will not, there give any examples here. Some will be given after formulating more gel problems. Under the assumptions on composition of well-behaved mappings on the existence of the identity we have above, we see that the syste all endomorphisms of an object (with composition as the operation an identity isomorphism as the unit) forms a monoid*) which will be ref to as the endomorphism monoid of the object. Since every monoid c: represented as a monoid of mappings of a set into itse1f-c.g. usin same trick with left translations as for groups above (the Cayley I sentation, see also Ch. I, 6 2 F t h e following class of problems nati arise (the Abstract Monoid Problem): Given a structure type 9, can every monoid be represented as the morphism monoid of an object ? Below we give several examples of cases which have been solved posi’ All of them-and many more-will be proved and discussed in thr in particular in Chapters IV, V, and VI. And all of them have alsc stantially stronger properties, which will be pointed out in the foll section. Of the “combinatorial” structures types, let us name the binary re and relation-preserving mappings. One can require (e.g.) the relat be symmetric*), or transitive. Of the algebraic cases let us mention e.g. commutative binary ope (“commutative groupoids”), semigroup, ring with unit, two unary tions (with homomorphisms in all of these cases). Of the topological ones, let us mention the structure types with to] (possibly a topology satisfying special requirements) for the “add *) A monoid, sometimes called semigroupwith unit, is a set with an associative binary o and a neutral element with respect to this operation.
*) Here one has what is often called (symmetric) graphs and graph homomorphi also the formulation of Frucht’s result above.
INTRODUCTION 2 , 3
7
information pattern” and with the well-behaved mappings e.g. local homeomorphisms, or open continuous mappings; and metric spaces with e.g. open contractions, locally one-to-one contr$ctions, or open uniformly continuous mappings. An interesting case is also that of topological spaces and non-constant continuous mappings. This goes beyond our convention on structure types, because a composition of non-constant mappings can be constant. The result here, properly formulated, says that for every monoid M there is a topological space (and, if there are no measurable cardinals, even a compact Hausdorff one) such that its non-constant continuous mappings into itself form a monoid, namely one that is isomorphic to M . Most of the constructions preserve finiteness, i.e. one obtains finite monoids represented by finite objects. In cases where the nature of the structure evidently excludes finiteness, one often has, for the finite monoids, the results at least strengthened: e g , every finite monoid can be represented as the monoid of all non-constant continuous mappings of a compact metric space into itself.
The word “abstract” in Abstract Monoid (Group) Problem, as the reader has certainly guessed, did not refer to leaving the structure type unspecified. The point was in disregarding the possible concrete forms of the groups or monoids. Suppose we are given, say, a monoid M of mappings of a set X into itself (with composition for the operation and the identity mapping for the unit). The positive solution of the Abstract Monoid Problem for Y only says that M is isomorphic to the endomorphism monoid of an object (I:s) of the structure type 9’. A considerably finer question (and a harder one-mostly too hard to be answered) goes as follows (Concrete Monoid Problem): I s there a structure s of the tj‘pc .Y on X such that th<,~ , i l i l o l l ~ ~ monoid oj ( X ,s) coincides with M ? Let US look at an easy example where the Concrete Monoid Problem is not solvable, although the Abstract Monoid Problem is; namely the binary relation:
Example. Take a three-point set X and consider the monoid M consisting of the identity mapping of X and all those mappings which are not one-to-one. Suppose M is the endomorphism monoid of ( X , R ) where R is a binary relation on X. R is non-void, the void relation being preserved by all permutations. For the same reason, R cannot consist only
~ ~ ~ ~ ~ l ;
8
INTRODUCTION 3,4
of couples of the form (x, x): Because of the constant mappings, it would consist of all (x,x), and the relation is preserved by all permutations. So let (xo,yo), xo $; yo, be in R. Let (x, y ) be arbitrary. Define f: X + X putting f ( x o ) = x, f ( z ) = y otherwise. We have ,fc M and hence (x, y) = ( f ( x o ) ,f(yo))e R. Thus, R = X x X and here are all the permutations again.
The analogy of the Abstract Monoid Problem, the question as to whether, in a given structure type, every monoid of mappings is representable as indicated above, has for reasonable structure types a negative answer. A more proper problem, the problem of characterizing those monoids of mappings which have such representations in some structure type or other, is so far unsolved even in the simplest cases. We can, however, offer a weakened type of concrete case problem, where the answers are more satisfactory, namely the Concrete Monoid Extension Problem (a concrete monoid ( M , X ) consists of a set X and a monoid M of mappings of X into itself-f course with the composition as the operation and with the identity mapping as the unit): Given a structure type 9, can every concrete monoid ( M , X ) be represented by the endomorphism monoid of an object (I:s) such that (1) x = I: (2) the mappings f E M are represented by their, uniquely defined, extensions Y-+ Y ? The answers are positive in all the cases given as examples of positive solutions of the Abstract Monoid Problem.
4. A category ‘i!I consists of two classes, obja, called the class of objects of 2I, and morpha, called the class of morphisms, with the following further data: morpha is presented as a disjoint union of sets ‘%(a,b) of “morphisms from a to b”, determined for all couples of objects; a composition a o p of morphisms is given, defined if and only if a starts where terminates; this composition is associative, i.e. . O ( B O Y )
=(
a 0 P ) O Y ;
in every E(a,a) a morphism 1, is pointed out, such that a0 1, and 1 , o p = p whenever defined. (A more pedantic definition will be given in Chapter I, 6 3). Let us take notice of the following two facts:
= ci
INTRODUCTION 4
9
(1) Every %(a, a) is a monoid: CI 0 is defined for any two of its elements, and 0 is associative; we have the 1, as unit. On the other hand, a monoid can be viewed as a category with only one object. Thus, formaly, we can regard the notion of category as generalization of the notion of monoid (sometimes, this formal standpoint can be rather misleading). (2) A structure type we spoke about in Section 2 gives obviously rise to a category. One just has to be a little more careful about morphisms: one mapping may behave well with respect to different pairs of structures. Hence, reforming a structure type to a category, we should add to the mapping itself in some way or other the information on the objects between which it is considered at the very moment. So we can speak, e.g., of the category of sets and all mappings (resulting from the structure type with no additional information on sets), the category of abelian groups and homomorphisms, the category of topological spaces and continuous mappings etc. In view of the observation (2) one may reformulate the problems of Section 2. So we have, e.g., the Abstract Monoid Problem for a category \LI: I s every monoid isomorphic to an '%(a, a)? But the real aim of this section goes rather in the line of the observation (1). We are going to extend the problems from monoids to categories. It was mentioned above that regarding categories as generalized monoids is often not an advantageous point of view. It should be said perhaps a bit more about it. Category theory (from its very start in the famous Eilenberg and MacLane article [EM]) is by no means a generalization of the theory of monoids. Most important notions and most non-trivial and well applicable theorems of category theory prove to be uninteresting or trivial if applied to the case of one object. On the other hand, they are often applied fruitfully e.g. to another (slightly less obvious - see Chapter I, 3.6) particular case, namely that of partially ordered sets. Still, a narrow consideration of a category as a generalization of a partially ordered set is hardly a good idea. Thus, the reader may wonder whether making the step from monoids to categories now is a reasonable thing to do. It is. Looking back at the problems, we see that they are, in the part where one represents, already problems on categories anyway and that limiting the part of what one represents to special cases would be artificial-which proved to be the case also in the results: actually it is true that, for most everyday-life structure types, if monoids are representable, categories are representable as well. Moreover, there are results on representation of monoids that
10
INTRODUCTION 4
follow from results on categories with more than one object and that would be hard to prove within just monoid reasoning. NOW,let us start to formulate the problem of representation of a category W using a structure type Y , say, as follows: Can the category 9l be represented so that its objects are replaced by objects of the structure type Y in such a way that the morphisms of W get represented exactly by the well-behaved mappings between them-of course with the composition preserved? We get a more lucid formulation after introducing a very natural notion. A category W is called a full subcategory*) of a category 23 if it is obtained by taking some objects of 23 and all morphisms between them, i.e. if it is the whole part of 23 spanned over some subclass of objB. If we now use, instead of the structure type 9, the resulting category 23 (in the sense of the'observation (2) above), we obtain a much simpler formulation of the problem : I s 2l isomorphic to a full subcategory of 23? Now, a careless attempt to extend the Abstract Monoid Problem might go as follows: Is every category W isomorphic to a full subcategory of the category resulting from the structure type in question? Giving it a second thought, we see, however, that we have overlooked an important question. Before formulating the Group and the Monoid problems we had already known that it was possible to represent groups and monoids by systems of mappings. Is this the case with categories? That is, can every category be represented by replacing objects by sets and morphisms by some mappipgs between them? Obviously only such categories have a chance to be isomorphic to subcategories, not to speak about full subcategories, of those resulting from structure types. For the case of the categories with only a set of morphisms terminating (or starting) at every object, in particular, if the whole category is a set, one obtains easily the affirmative answer by an extension of the trick used for monoids ([EM], here see Chapter I, 7.1). For the general case, the problem was unsolved for years. Its full solution is due to P. Freyd and J. R. Isbell. J. R. Isbell formulated in [I2] a condition (we will present it below) which he proved to be necessary for the concretizability (i.e. representability by sets and mappings) of a category. This enabled him to construct a counterexample to concretizability simply by describing a category *) If one does not assume that all the morphisms are taken, one speaks just about a subcategory.
11
INTRODUCTION 4
which fails to satisfy that condition. In [F], P. Freyd proved that the condition was also sufficient for concretizability. Before presenting Isbell’s condition, let us first prove the following easy Lemma. For mappings a: Z
+X
,
8: 2 --* Y
S(a, 8) = {(a(.),
put
8(z))I z E z}= x
x y
S(a, 8) = S(a’, p). Then for an arbitrary couple Let for some further a’: 2’ + X, p’: Z --* v : Y + W we have p a a = v o 8 iff pea'= v o p . of mappings p: X --t
Proof. Let p a J
=
v 0 8. For a ~ ’ E find Z a Z E Z such that
(@’(a B’(4) (44B(4) =
’
We have p(a’(z’)) = p(a(z))= v(p(z))= v(p(z‘)).
Now let % be a category, X , Y some of its objects. For couples of morphisms
Z
X
v 3
of % put (a, 8)
-
-
Z‘ Y
Y
Y
X
Y
(a’, B’) if for arbitrary morphisms p, v of rU
iff
pc”a=vop
poa‘=vo
B’.
Obviously, is an equivalence relation on the class T[X,Y ] of all the couples ( a : Z + X , 8: Z + Y ) . Let T(a, p) designate the equivalence class containing (a, @).If % is a subcategory of the category of all sets and mappings, we have, by Lemma, the implication
T(a,8)
+ T(a’,B’)
*
S(a, 8) =+ S(a‘, 8’)
Since we have only a set of possible S(a, P), a necessary condition (this is the one given by Isbell) is the following: For every couple of objects X, Y of rU there is a subset M of T [ X , Y ] containing for every (a, 8)E T [ x , Y ] an equivalent one. Now one can easily construct a category which does not satisfy this condition e.g. in the following way (it is worth noting that P. Freyd showed also in [ F ] that some everyday-life categories, as e.g. the category of topological spaces and the homotopy classes of continuous mappings, are not concretizable): Take a proper class d and two elements X , Y which are not in d x 2 and consider the category % with objA = (d x 2) u { X , Y } , %(n, n) consisting only of the identity for any n, %((a,0), X ) , %((a, 0), Y), % ( X , (a, l)), %( Y, (a, 1)) and %((a,O), (b, 1)) for a 9 b consisting of one morphism each, say 5.0, V ~ O 5.1. , Vol, %(u, v ) = 0 otherwise, the composition and respectively, %((a,0),(a, 1)) = { (,: C,},’
cob
defined in the only non-trivial cases by 5.1
0
5.0
=
c
7
Vn1
O
‘loo =
i,‘
12
INTRODUCTION 4 , 5
(in the other cases one has no choice). It is indicated in Fig. 0.6. If a =+ b, then T(l,,, vo0) + + T(tbO,qbO),for t b l too= q b l qo0 while tbl t b r J =+ % I %O. Hence, each couple from T [ X , Y ] is equivalent only with itself, and T [ X , Y ] is a proper class. rn 0
0
0
I
X
I
0
I
I
I
E-*
/
(a
I
'a"
\
I I 1
Fig. 0.6
Thus, instead of the question above, one has to extend the Abstract Monoid Problem to Abstract Concretizable Category Problem : For a given structure type 9, denote by 9 the resulting category. I s every concretizabie category isomorphic to a full subcategory of p? To present some examples of positive answers we just advise the reader to turn the leaves back to the end of Section 2. All the examples mentioned there are of that kind. (Mostly, there is a set-theoretic assumption to be required. This will be discussed in Section 7 below.)
5. Now, what about the "concrete problems"? To formulate them we have to elucidate another notion, the notion of concrete category. After all, it will not be all that new (also, almost all the examples of categories we have mentioned so far were actually concrete categories). While a concretizable category was a category which could be represented as one with sets for objects and mappings for morphisms, a concrete category is a category which is represented so (i.e. such a representation is chosen and fixed). Thus, one and the same category together with two different setmapping representations forms two different concrete categories.
INTRODUCTION 5
13
Describing a concrete category, we will mostly use a symbol like ('21, U ) where '% is a category, U is a fixed set-mapping representation. The set U ( U )representing an object a will be called the underlying set of a, the mapping U(a) will be called the underlying mapping of the morphism a. Looking back at the notion of a concrete monoid, we see (unless we are too pedantic about the distinction in taking in a concrete monoid the morphisms directly as the mappings representing them) that it is a particular case of concrete category in the same sense as a monoid was a particular case of category. It is very important to realize-and it is what we meant in stating that the notion of a concrete category would not be quite new for the readerthat a concrete category is almost the same as the notion (somewhat nebulous) of a structure type we have used in formulating the problems till now. If one turns the leaves back to the description (not a definitionthere was not any) of the structure type, one sees that the difference is only in two, unessential for our purposes, details: in a structure type, the underlying sets and mappings were explicitly parts of the objects and morphisms, and we were not concerned with distinguishing morphisms from their underlying mappings there*). From now on, we will use, in formulation of problems, concrete categories also in places of structures types. To simplify further formulations, let us now agree about the use of two expressions (though we feel that the reader will guess their meaning without explicit explanation). 1. A concrete (full) subcategory of a concrete category ('%,U) is a concrete category consisting of a (full) subcategory of '% and the restriction of the rule U on it. 2. Speaking about two concrete categories as isomorphic, we mean that there is a one-to-one correspondence which preserves not only the composition and identities, but also the ,underlying sets and mappings. (Here we differ from the definition of isomorphism in the text, which is more general. This restricted form, however, will do well now.) So the problem corresponding to the Concrete Monoid Problem may be formulated as follows: I s (2l,U ) isomorphic to a concrete full subcategory of ('23, V ) ? *) On the level of intuitive feeling about the notions, when speaking about structure types, we wanted the reader to have very concrete constructions on sets and very concrete specifications of well-behaved mappings in mind. That is why we avoided a definitionit would have been too general for a start.
14
INTRODUCTION 5
Answers to this question will play auxiliary, but often very important roles in the text. Putting for an everyday-life (23, V )the question as to whether every (%, U ) is isomorphic to a concrete subcategory of it does not give much sense. It has been already mentioned when treating the Concrete Monoid Problems that answers were too often negative there. Here, the answer is negative in reasonable cases always, and for a very obvious reason: The concrete categories (a, U ) one usually encounters have the property that there is a cardinal 3c = x(%, U ) such that every system of non-isomorphic objects with one-point underlying set has cardinality less than K. Obviously. a concrete category with objects, all of them one-point sets, and only their identity mappings for morphisms, cannot be isomorphic to a concrete U). full subcategory of (a, One has, however, satisfactory answers to the Concrete Category Extension Problems. We say that (%, U ) has an extension in (%, V ) if we can replace the rule U by a U' such that (1) U'(a) = u(a); (2) on the original sets, U ' ( Mcoincide ) with U(cr); (3) (a, U') is isomorphic to a concrete full subcategory of (23,V). (Roughly speaking, (a, U ) can be made to a concrete full subcategory of (23,V ) merely by expanding the underlying sets and mappings.) Now the Concrete Category Extension Problem for (23, V ) can be formulated as follows : Has every concrete category an extension in (23,V )? The answer will be shown to be positive e.g. for all the cases given as examples after the Monoid Problem. Moreover, there is a theorem (perhaps somewhat surprising, but easy to prove) that for reasonable concrete categories (23, V )a positive solution of the Abstract Concretizable Category Problem readily implies a positive solution of the Concrete Category Extension Problem (see Chapter I, 6.10). Let us, in this place, touch a problem, which is closely related to the extension one (and which does not differ from it in the case of monoids). The expanded underlying sets and mappings in an extension could (and often have to) depend upon the whole objccrs 4 much slronger problem is the question as to whether one can have the extension with the new underlying sets and mappings depending only on the original ones. In other words-if, for a moment, we adopt the language of structure types again-the question as to whether one canfirst expand sets and mappings canonically, and then obtain a representation of the first structure type
INTRODUCTION 5 , 6
15
in the other one by associating structures of the second type on the expanded sets with the structures of the first type on the original ones. Such a representation is called strong embedding (see Chapter I, 6.11). We will say a little more about it in Section 7.
6. The Abstract Group Problems, Abstract Monoid Problems, etc., were formulated in the rather strong form requiring representability of all groups, all monoids, etc. This was so mainly because, as will be shown in the text, one really does obtain very often a positive answer in that strong form. Sometimes, however, one is concerned with representing some particular group, monoid, etc., only. Among the tasks of this kind, representing trivial groups and monoids plays a very particular role. A structure of a given type Y is said to be automorphism rigid (endomorphism rigid, resp.) if the only automorphism (endomorphism, resp.) is identity. The endomorphism rigid ones are mostly called rigid (in fact, automorphism rigid is a special case of rigid: it is obtained after modifying the given structure to a new one, in which for the well-behaved mappings only the isomorphisms of the original one are taken). The corresponding object is called rigid object. Thus, let a structure type Y be given. The “abstract” form of the problem discussed is the question as to whether there exists a non-trivial rigid object (we have some obvious trivial ones mostly), the “concrete” form asks for a rigid structure on a given set, the “extensional” form asks for a rigid object at least so large as the given set. Some questions of that type had been put and solved before for various types of structures. So one has the above-mentioned de Groot’s construction of an automorphisrh rigid topological space (it is, by the way, also rigid if, e g , we take all local homomorphisms for well-behaved mappings). Let us name here also Cook’s continuum ([C,]), which is rigid even with respect to all non-constant continuous mappings (and which will play a considerable role in various constructions in Chapter VI). Other examples of a solved rigidity problem are the rigid Boolean algebras ([Ka], [Ri]), and the constructions of rigid semigroups in [DN] and [HL]. In the constructions in this book, a particular role will be played by rigid binary relations (and by some rigid binary symmetric relations). A rigid binary relation exists on any set.
’
16
INTRODUCTION 6 , 7 (The difficulty arises with infinite sets; on a finite set one can take
for example. Stop at this place and prove that it is really rigid, and try to find a countable rigid object, which is still easy.)
This problem was first solved inductively for accessible cardinals ([PHI). Thereafter, P. VopEnka found an explicit construction which did not require the accessibility ([VPH]). This construction will be used in the text (see Chapter 11, 0 3). A rigid symmetric relation exists on sets with either cardinality one, or cardinality at least eight. Here, the trouble was with the finite sets, not with the infinite ones (once the result on the non-symmetric case was already known, of course). One of the important corollaries of a positive answer to the Abstract Concretizable Category Problem for a structure type is that one has arbitrarily many non-isomorphic rigid objects there. Moreover, one has a proper class of rigid objects with no well-behaved mappings between any two of them. So, e.g., one has a proper class of topological spaces with only constants and identities as continuous mappitngs between them (Chapter VI).
7. There were two or three topics in the previous sections we did not want to go into in the places they were touched. We shall spend a little time on them now. Naming examples of categories containing isomorphic copies of all concretizable categories as full subcategories, it was remarked that mostly an assumption on the set theory one works in has to be done. It is the assumption that there are not too many measurable cardinals, more exactly, (M) There is a cardinal LY such that every a-additive two-valued measure is trivial. Let us say a little more about it. There is a notion of an algebraic category defined by J. R. Isbell (in [I3]) as a category which is isomorphic to a category of some algebras and all homomorphisms between them. It is easy to prove (though not immediately obvious) that every small category is algebraic. Thus, we have the following picture small c algebraic c concretizable .
INTRODUCTION 7
17
In the first years of discussing algebraic categories the interest was focused to questions as to whether that or other category which is not explicitly defined by algebraic means, is algebraic (the first result of this kind was the Isbell’s result on the category of compact Hausdorff spaces and continuous mappings). In 1966, it was already known that every “constructive” structure type yields an algebraic category ([HPT]). In 1969, KuEera and Hedrlin proved that ([H5], [K,]) under (M) every concretizable category is algebraic
(both results mentioned above were proved also under (M)). In 1971 it was proved ([KP,]) that (M) was necessary and that under non(M) the algebraicity was actually a rare property ( e g the category of compact spaces mentioned above is then nonalgebraic). Now, what one actually proves instead of proving that every concrettizable category is isomorphic to a full subcategory of the 23 in question is that every algebraic category is isomorphic to a full subcategory of 23,
which under (M) means the same. On the other hand, if 23 itself is algebraic, one cannot prove any more. According to a result we mentioned in another place, one has the following: under (M), every concrete category has an extension in a concrete category of algebras. But not every concrete category is strongly embeddable (see the last line of Section 5) into one. Those which are we call strongly algebraic. This notion allows to point out those categories which (though not defined by algebraic means) behave more algebraically than others. E.g. the category of compact Hausdorff spaces and continuous mappings is strongly algebraic while the category of all topological spaces is not. Seeing that concrete categories of algebras are not, in the strong embedding sense, “universal”, although many of them are in the extension sense, it is only too natural to put the question as to whether there are, among the everyday-life categories, such ones that every concrete category is strongly embeddable into. Putting the question in this generality, one obtains negative answers simply because, so to say, “only a reasonable concrete category can be strongly embedded into a reasonable one” (it will be put more correctly in a proper place, let us here remark only that the
18
INTRODUCTION 7
property of having only a set of non-isomorphic objects with equally large underlying sets is inherited by a strongly embedded category). Confining ourselves only to “reasonable” categories we obtain positive results e.g. at the category of graphs and graph homomorphisms, at its subcategories of symmetric or transitive graphs, at partial algebras, etc. How to read this book:
First, let us summarize the contents. The book is divided into Introduction, seven chapters and two appendices. The first chapter contains the necessary definitions and some easy facts. In the second there are the proofs of the alg-universality of the category of graphs and the categories of algebras of non-trivial type. This is the basis of the alg-universality results throughout the book. Chapter 111 deals with the universality of some naturally defined categories. The kernel of the book is in Chapters IV, V and VI containing the representations by combinatorial, algebraic, and topological means. In Chapter VII, a particular kind of embedding is discussed. . Appendix A is devoted to a technical (but also very important in itself) fact, important in the topological representations, in appendix B the role of a set theoretical assumption is discussed. A reader interested in all the questions we are presenting is a highly improbable phenomenon. Therefore, we would like to give here a few hints of how to read the book in order to reach the subject one is interested in promtly. There are two basic points: what one wants to represent, and what kind of structure one wants to use. If one is interested in representing just groups, monoids, small Categories o r generally categories which are, for some reason, obviously algebraic (in the sense mentioned above) one can omit Chapter I11 fully (in fact, also in the opposite case, we think that at the first reading it is not necessary to read all the proofs here), all sections dealing with universality (in contrast with the mere alg-universality see 1.8.6) and certainly appendix B. Now to the point of the structures one wants to use: How to reach Chapter IV (V, VI, respectively) soon if one is interested exclusively, or primarily, in the combinatorial (algebraic, topological7 respectively) representations ? One can get the necessary information reading carefully this Introduction, and further, for more exact formulation of the problems and the terminology used, $5 6 and 8 of Chapter I. Then, one should take from Chapter I1 at least the facts on the alg-universality of the categories
INTRODUCTION
19
of graphs and algebras of non-trivial types (11, 9 5) including, perhaps, the basic idea of the proofs. After this one can start reading the one chosen from among the Chapters IV, V, VI. There are a few places in Chapters V and VI based on special results of Chapter IV, but one can leaf back at the moment one really needs it. In case of a particularly detailed interest in the topological questions, Chapter VI will be supplemented with Appendix A.
No knowledge of category theory and very little set theory is assumed. Almost throughout the book, all that one needs, except the so-called naive set theory, is to understand, vaguely at least, the distinction between sets and classes (in the Godel-Bernays sense). The few places where one needs more than that can be, at the first reading, done without. All the conventions and notations used, except the most standard ones, are introduced very explicitly in proper places. Here, we will just mention the convention used when dealing with mappings. After a long hesitation we decided to write the composition of mappings in a conservative way, from the right to the left. The identity mapping of a set X onto itself is usually denoted by 1, (or just 1, if the index is obvious) nowadays. Sometimes we used this convention, too, but avoid it in the case when 1 is used in the vicinity for another purpose (say, a unit of an algebra). In such a case we prefer denoting the identity mapping by id, or id. The expression f’=const (or const,) is an abbreviation for stating that f is a constant mapping (or the constant mapping with the common value a for all the arguments). The cardinality of a set X is denoted by card X or simply by 1x1. Quotations: The books and papers (summarized in References at the end of the book) are quoted, as usual, using square brackets. The items from the text of the book are quoted as follows: First, the chapter is indicated by Roman numerals (if inside the same chapter, this is omitted), then the numbers of the section and its subsection follow. When quoting exercises, we add the letter E in front of the quotation. Thus, when quoting subsection 4 of 9 5 of Chapter 11, we write 11.5.4, if we are in Chapter 11, just 5.4. The quotation E V.2.1. means Exercise 1 to 9 2 in Chapter V.
This page intentionally left blank
Chapter I PRELIMINARIES
$ 1 of this chapter has a character of agreement on terminology; the notions themselves are hardly unfamiliar to the reader. $ 2 deals with monoids and concrete monoids. In $ 3 , the most necessary notions from category theory are introduced. Such a small torso of the theory may very easily give rise to a misunderstanding as to what category theory is about. To reduce this, we present more notions and theorems in Exercises associated to this section. These will not be used in the main text; however, some of them appear in further exercises. $9 4 and 5 deal with concrete categories. The substance of this chapter is in $56 and 8 analyzing and formulating the representation problems.
0 1.
Relations, algebraic operations, homomorphisms
The purpose of this section is to make an agreement on some definitions and notation. The reader has certainly met the notions before in some form or other. 1.1. A binary relation on X is usually defined as a subset R of X x X, a ternary one is defined as R c X x X x X ; more generally, an n-ary relation on X is a subset R c X x X x ... x X (n times). For relations with infinite arities one would have to use infinite Cartesian powers of X . For many purposes, however, it is more convenient to use the following description of relations:
If A , X are sets, denote by X A the set of all mappings of A into X and define :
(*) an A-ary relation on X is a subset r of X A .
22
Ch. I, PRELIMINARIES
Regarding, as it is done mostly, the natural number n as the set n
=
= (0, 1, .. ., n - 1) of all smaller natural numbers (hence 0 is synonymous
with Q)), we obtain an obvious translation of the first mentioned n-ary relations into the (*)-form viewing an n-tuple (xo,xl, ..., x,- E X x X x x ... x X as a “table” of the mapping n X sending i to xi. When working with unary (1-ary) and binary (2-ary) relations, we will prefer the relations represented as subsets of X and X x X mostly. -+
1.2. If Y is an A-ary relation on X and s an A-ary relation on ping f : X -+ Y is said to be an rs-homomorphism if
for every a from r, the composition
fo
a map-
a is in s.
It is also said that the mapping f is rs-compatible. If not necessary, the relations are not explicitly mentioned and we speak only about a homomorphism or a compatible mapping. If we take binary relations in the form R c X x X , S c Y x E; we see immediately that the condition for an RS-homomorphism f translates into for unary relations R c X, S c Y we obtain simply
f ( R )= S ‘ 1.3. Remark. For binary relations, (x, y ) E R is mostly written a5 X Ry (in the case of partial orderings almost exclusively so). Thus, e.g. for partially ordered sets ( X , <), (E; <) the homomorphisms are exactly the monotone mappings (defined by x < y f ( x )
1.4. An A-ary algebraic operation on a set X is a mapping a : X A + X . For small non-zero arities, one uses, of course, rather the classical description of an operation as a mapping X x X x ... x X 3 X (the transX. lation as in 1.1). Thus, unary operations are simply mappings X Nullary (0-ary) operations (often also called the constants) are usually represented as elements of X ( X o is a one-point set and a mapping a : X o 3 X is determined by the value in the only element). -+
1.5. If a (b, resp.) is an A-ary (algebraic) operation on X (E; resp.), a mapping f : X + Y is said to be ap-homomorphism (or simply homomorphism, if for every ( E X A , b(f 5) = f ( a ( 5 ) ) . 0
23
$1. Relations, operations, homomorphisms
Taking algebraic operations in the form X x X x . . . x X dition translates into the usual one: for every ( x o , ..., x n - 1), P(f(xo),...>f ( X n - 1)) = f(a(x0, ...>xn-
-+ X
, this con-
1)).
For unary operations, in particular, we obtain the commutativity condition = f o a. In the case of nullary operations represented as x o € X and y o E Y we obtain the condition f ( x o ) = yo. P a f
1.6. Taking an element a =I= A and putting B = A u (a}, we can represent A-ary operations as B-ary relations putting, for a : X A + X ,
z = {< E xB1 t ( a ) = a(< 1 A ) ) .
This representation preserves the choice of homomorphisms: Really, if f: X Y is an ag-homomorphism and < E Z , we have -+
(f 4 (4 = f ( W )= f ( 4 4 1 A ) ) = P ( f 0 (i" I 4)= P((f0 4;) I 4
<
a.
7
so that f o E If f is an $-homomorphism and 5: A + X , define q : B -+ X by putting q A = 4 and ~ ( a=) a(<). We have then q E Z and hence = f(M)= P ( ( f 0 A ) = P ( f c (v A ) ) = B(fJ *
f(45))
1
1
1
0
1.7. The identity mapping is always an rr-homomorphism. The composition of homomorphisms is a homomorphism. More precisely if f : X -+ Y is an rs-homomorphism and g: Y + Z is an st-homomorphism, then g o f is an rt-homomorphism. Indeed, if r, s, t are relations, we have J".E s for E r, and hence (g of) t = g o (fo t)E t . If r, s, t are operations, the statement now follows by the fact just proved and by 1.6, but it is also obvious directly by the definition.
<
<
0
EXERCISES
1. The projections X x ... x X X sending ( x l ,..., xm) to x i (i fixed) are n-ary operations such that every mapping f : X -+X is a homomorphism with respect to them. Are there any other operations with this property? 2. What n-ary relations on X are preserved by all f : X -+ X ? -+
24
5 2.
Ch. I, PRELIMINARIES
Monoids and concrete monoids
2.1. A semigroup is a set X with a binary associative operation, i.e. with an operation a : X x X -+ X such that (A) a(cr(x,y), z ) = a(x, a(y, 2 ) ) for any x, y , z in X . Instead of the clumsy notation a(x, y), the value of the operation is usually denoted by means of a symbol between the arguments, e.g. as x . y , x o y, x y etc., or simply by juxtaposition as xy. The associativity condition (A) then takes the usual form (xy) z = x(yz). A monoid is a semigroup with a nullary operation e, called unit, such that for every x, xe = ex = x.
+
2.2. Referring to a semigroup (monoid, resp.) in a precise way, we should use a notation mentioning the operation (operations, resp.), say ( X , a) ( ( X ,a, e), resp.). This will, however, be done if really necessary. If there is no danger of confusion, we will usually use the same symbol for the semigroup (monoid) and for its underlying set (the X above).
2.3. Let X be a set, let M c X x be such that for any two mappings f, g from M the composition f o g is in M . Then M with the composition (as binary operation) is obviously a semigroup. If, moreover, M contains the identity mapping, it becomes a monoid (taking the identity mapping for the unit). 2.4. Example. The system of all rr-homomorphisms for a relation or an operation r on X forms a monoid (see 1.7). More generally, let X , J be sets, ri (i E J) relations or operations on X . Then the system of all those mappings f: X +. X which are simultaneously riri-homomorphisms for all i e J forms a monoid. 2.5. The monoids from 2.3 and 2.4 are examples of the so called concrete monoids. A concrete monoid is a couple ( M ,X ) where M c X x contains the identity mapping and is closed under composition. The monoid M (with the composition and the identity) is called the abstractization of ( M , X ) , X is called its carrier. Remarks. (1) The requirement for the identity to play the role of a unit is substantial. M c X x consisting of one constant mapping is a monoid under composition, but ( M , X) is not a concrete monoid, unless = 1. (2) Usually, one uses the expression “transformation monoid”. We decided to use the expression concrete monoid for two reasons: First, it is
1x1
$2. Monoids and concrete monoids
25
essentially a particular case of a concrete category (see $4) to which it relates as a monoid to a category. Second, we want to avoid the confusion with a most important notion of (natural) transformation (see $ 3 ) with which it is in no relation at all. 2.6. So far, the concrete monoid has differed from its abstractization in explicit mentioning the carrier only-which information is implicitly hidden in the M itself as well. The real difference occurs in the notion of isomorphism. Monoids M and N are said to be isomorphic if there exists a one-to-one mapping cp of M onto N such that cp(x.y) = cp(x).cp(y), i.e. such that it is a homomorphism (this cp is called isomorphism). Two concrete monoids ( M ,X ) and ( N , Y ) are said to be isomorphic if there is an isomorphism cp of M onto N ond a one-to-one mapping f of X onto Y such that always
4)( f ( x ) )= f ( t ( x ) ) . Thus, ( M ,X ) and ( N , Y ) are isomorphic Iff one obtains ( N , Y ) from ( M , X ) replacing the elements of X by new ones and then letting them move after while isomorphic M the original mappings (we have cp(5) = f o 5 o f - ' ) , and N may have no similarity in the concrete nature of their elements. To avoid confusion, we will sometimes refer to isomorphic concrete monoids as concretely isomorphic. If solely M and N are isomorphic we will say that ( M ,X ) and ( N , Y ) are abstractly isomorphic. To elucidate the difference, consider the following trivial example: Let M x consist of the identity mapping on X . Any ( M , , X ) and ( M y , Y ) are abstractly isomorphic while they are concretely isomosphic (if and) only if cardX = card I: Another example is the following: Let X = Y = 4 = = (0, 1,2,3}, let M consist of the identity and the mapping interchanging simultaneously 0 with 1 and 2 with 3, and let N consist of the identity and the mapping interchanging 0 with 1 and leaving 2 and 3 fixed (see Fig. 1.1).
The two concrete monoids are abstractly, but not concretely, isomorphic.
26
Ch. 1, PRELIMINARIES
2.7. Now, let M be an arbitrary monoid. Let us denote the operation by juxtaposition and the unit by e. For a E M define a mapping La: M + M
by putting L,(x) = ax. This mapping is called the left translation of M by a. Take notice of the following facts: I. The left translation by the unit is the identity mapping of M . 11. L,(Lb(x)) = L,(bx) = a(bx) = (ab)x = Lab(X) and hence we have La Lb = L a b . Thus, in particular, a composition of two left translations is again a left translation. 111. For a P b, La += L b since L,(e) = a =k b = Lb(e). Denote by A?’ the set of all left translations. By I and 11, (&, M ) IS ’ a conCrete monoid. Define q : M + A by q(a) = La. By the definition of At, q is onto, by I11 it is one-to-one. Thus, by 11, it is an isomorphism and we have 0
Theorem. Every monoid is isomorphic to the abstractization of a concrete monoid. m (A, M ) is called the Cayley representation of M . 2.8. The procedure from 2.7 allows us to go a step further. Define right translations R,: M -+ M putting R,(x) = xa. Consider the R,‘s as unary operations on M . Take notice of the following facts: I. For any a and any b, La is an R,R,-homomorphism (see 1.5). Indeed, we have Rb(L,(x))= (ax)b = a(xb) = L,(Rb(x)). 11. On the other hand, let f: M M be an R,R,-homomorphism for every a, i.e., for every a, R, o f = f o R,. Hence, we have f ( x ) = f(R,(e)) = = R,(f(e)) = f ( e )x. Thus, f = Lfc,).Combining I and I1 we obtain --f
Theorem. For every monoid M there is a set X and a collection (c&, of unary operations such that the monoid of all mappings which are simultaneously ciicci-homomorphismsfor all i is isomorphic to M . 2.9. This was a statement on an abstract isomorphism. Given a concrete monoid ( M , X), a more restrictive question arises as to whether it is concretely isomorphic to a monoid of such homomorphisms. This results in a question as to whether there is a collection of unary operations on X such that M coincides with the set of all the simultaneous homomorphisms (cf. Concrete Monoid Problem in Introduction, Section 2). Since unary operations are binary relations (see 1.6), the answer is negative according to the first part of the following
42. Monoids and concrete monoids
27
Statement. (1) For a set X denote by N ( X ) the monoid consisting of the identity mapping and all those mqppings f : X -+ X for which card f ( X ) < card X , I f N ( X ) coincides with the set of all the mappings which are ) ~ ~ ~ ri Ai-ary, simultuneously riri-homomorphismjor a collection ( Y ~ of'relations, then for some i card A i 2 card X . (2) On the other hand, let card A 2 card X . Then for any concrete monoid ( M ,X ) there is an A-ary relation r on X such that M coincides with the set of rr-homomorphisms.
Proof. (1) Suppose that card Ai < card X . Let every f~ N ( X ) be an riri-homomorphism. Then, if a E Ti, every p : Ai X such that .(a) = a(b) iff p(a) = p(b) is also in ri, because putting f ( a ( a ) )= b(a) and f ( x ) = a(ao) for x 4 a(Ai) we obtain an f~ N ( X ) with f o a = 8. But if this is so, every one-to-one mapping X -+ X is an riri-homomorphism (and it is not, with the exception of the identity, in N ( X ) ) . (2) If card A 2 card X take a mapping e of A onto X and put ---f
r=
{ f o e p q .
Obviously, every g E M is an rr-homomorphism. On the other hand, let g be an rr-homomorphism. Since the identity is in M , e is in r and hence g is in r. Thus, there is an f~ M with g e = f o e. Since 4 is onto, we infer g = f e M . H 0
0
2.10. The statements in 2.8 and 2.9, easy as they are, are a basis for stronger results. We shall see later, e.g., that the number of unary operations in Theorem 2.8 can be reduced to two (and therefore, particularly, made independent of the cardinality of the monoid). We shall also improve the concrete representations. Of course, nothing can be done with the negative result of 2.9. (l),but one can prove theorems on extensions (see $ 3 of the Introduction) and even something a little stronger. EXERCISES
1. Consider the set (0, 1, ..., n - 1) with the binary operation x y min (x,y). It is a monoid with il - 1 as unit. Prove that if it is represented as a concrete monoid ( M , X ) , then card X 2 n. (Hint: Let i be represented by q ~ ' ~X: -+ X . The sets cpi(X) form an increasing sequence.) 2. Let the monoid from E 2.1 be represented as ( M , X ) with card X = n. Then ( M , X ) is concretely isomorphic to the Cayley representation. =
28
Ch. I, PRELIMINARIES
3. Let Z,, the additive group of integers modulo n, be represented as concrete monoid ( M , X ) . Does it hold necessarily that card X 2 n ? (Hint: Consider Z6.)
5 3.
Categories, functors, transformations
3.1. A category 'u consists of the following:
(1) a class A, called the class of objects of 'u, (2) sets M(a, b) given for each two objects a and b in A such that, for (a, b) =+ (c, d), M(a, b) n M(c, d) = 0; the elements of M(a, b) are called morpRisms from a to b, (3) an element e, of M ( a , a) for every a E A ; e, is called the unit or the identity of the object a, (4) mappings mabc:M(b, c) x M(a, b) + M(a, c) for all the triples of objects a, b, c such that: m(m(r, D), a) = m(y, 4% a)) > (4 m(a, e,) = m(eb, B) = a ; (b) the mapping m is called the composition of 'u.
3.2. Usually, if there is no danger of confusion, the following conventions are used: Given a category 'u, the class of objects is denoted by obj'u, the set of morphisms from a to b by %(a, b). The fact that a is a morphism from a to b is denoted by a: a + b. The identity of a is often written as 1,. Composition is usually denoted by D o a, or simply by pa (instead of mabc(fi,
a) ).
(Using this notation, one obtains the (4) (a), (b) from 3.1 in the more lucid form (y/?)a= y(Pa), me, = eba = a.) If a is a morphism from a to b, then a is called the domain of a and b the range of a. 3.3. A category 'u is said to be small if obj 'i? is l a set. If obj 53 is a proper class, and we want to stress the fact, we refer to 53 as large or big. Since the M(a,b) have been assumed to be sets, the system of all morphisms of a small category is a set. A category 53 is to said be finite if it has only finitely many morphisms.
3.4. A category ?I' = ( A ' , M', e', m') (notation as in 3.1) is said to be a subcategory of 91 = ( A , M , e, m) if
53. Categories, functors, transformations
29
(1) A’ c A, ( 2 ) for any two a, b E A’, M’(n, b) c M(a, b), (3) for any a E A‘, eb = e,, (4) for any p E M’(b, c ) and LX E M’(a, b) , mhbc(p, &)
= mabc(p,
.) .
The definition of a full subcategory is obtained by replacing ( 2 ) by the stronger (2 full) for any two a, b E A‘, M‘(a, b) = M ( a , b). Remark. Note that (3) is essential in the definition of a general subcategory. It is, however, superfluous in the definition of a full subcategory.
3.5. Examples. (A) Set, the category of all sets and mappings: The objects are all sets, in other words, objSet is the universal class. The morphisms from X to Y are all the mappings from X to E: (Here it is important to make an agreement as to what is understood under a mapping from X to I: Its common definition as a subset f of X x Y such that for every x there is exactly one y such that (x,y) E J would not do for our purposes. We have to add to it the information on Y in question, which cannot be deduced from the “graph of the mapping”. Otherwise, the sets of morphisiiis would not be dicjoint. which would cause trouble far from being formal only.) The identities are the identity mappings, the composition is the usual composition In the following examples (B) - (G), the units and the composition are always the obvious ones. (B) The category of all finite sets and all mappings between them. It is a full subcategory of Set. (C) The category of all sets and all one-to-one mappings. This is also a subcategory of Set, but not a full one. (D) The category of groups and their homomorphisms. Its full subcategory of abelian groups. (E) The category Top of all topological spaces and their continuous mappings. Considering only special spaces, say Hausdorff, completely regular, or compact, we obtain examples of full subcategories of Top. (F) The category of topological spaces and their local homeomorphisms. It is a subcategory of Top, not a full one. (G) The category of multigraphs: The objects are quadruples G = A, i, t ) where i and c are mappings from A into I/ ( V is the set of vertices, A the set of arrows, and i and t associate its initial and terminal
(x
.
30
Ch. I, PRELIMINARIES
vertex, respectively, with an arrow), the morphisms from G into G’ = (V‘, A’, i‘, t’) are quadruples (G, fv, f A , , G’) with fv: V -+ V’ and f A : AjA‘suchthat i‘ofA=f,oiandt’of,=fVot. (H) A monoid M = ( M , o , e ) can be viewed as a category with one object, say o, putting M(o, o) = M , 1, = e, rn,,,(a, b) = a b. (I) The discrete category 2l: %(a7a) = {la},‘%(a,b) = 0 for a =k b. 0
Remark. To be more precise, the morphisms in (D), (E), (F) have to be understood as containing information on the domains and ranges. Thus, e.g. in (E), the “continuous mapping” from a topological space (X,z) to (X’, z’) is a triple ( ( X ,z),f, (X’, z‘)), where f:X -+ X‘ is a mapping continuous with respect to the topologies z and z’. 3.6. An important case of category is the so-called thin category. It is a category 2I such that card %(a, b) I 1. It is fully described by the binary relation I on obj2l consisting of all couples (a, b) with ‘%(a,b) non-void. We see easily that thus a pre-ordering (i.e., a transitive, reflexive, but not necessarily antisymmetric relation) is obtained. On the other hand, every preordered class ( A , I )can be viewed as a thin category 2l by putting obj2l = A, M(a, b) = {(b,a ) } if a I b, void otherwise, and (c, b) 0 (b,a) =
(c, a).
3.7. Let 2I = (A,M, e , m ) be a category. I t 5 dual is the category %?Iop e, mop), where MoP(b,a) = M(a, b) aiid IIP’(x, 0)= m(p, a). Thus the dual of a category is obtained by “reversing the arrows and the order of morphisms in compositions”. Obviously, (21°P)OP = 2l. = (A, M O P ,
3.8. A morphism a : a -+ b of 2I is said to be isomorphism (in 2I) if there is a *) p: b -+ a such that crp = I,, and f l c r = 1,. Two objects a and b are said to be isomorphic if there exists an isomorphism a : a -+ b. A morphism 01: a -+b is said to be monomorphism (resp. epimorphism) if crfl = cry (resp. p a = ycr) implies p = y. Obviously, every isomorphism is both a monomorphism and an epimorphism (if crp = cry, we have p = c r - ’ c r f l = cr-lcry = y, and similarly for Bcr = ycr). On the other hand, a morphism which is both a monomorphism and an epimorphism need not necessarily be an isomorphism. It is e.g. true in Set, but not in Top. *) This fl is uniquely determined (if /?’ also has the properties, ,9’ and is called the inverse of a. Often it is denoted by a - ’ .
= b ( a p ) = ( / ? ’ a ) b=
b)
31
$3. Categories, functors, transformations
3.9. Take notice of the following obvious but important facts: (1) The isomorphisms in 2l and Wp coincide, the monomorphisms (epimorphisms, resp.) in 2l coincide with the epimorphisms (monomorphisms, resp.) in %*P. Let 2l' be a subcategory of 2I: (2) If a from a'is a monomorphism (epimorphism, resp.) in 2l, it is a monomorphism (epimorphism, resp.) in a', But a can be a mono- or epimorphism in 'U' without being so in a. ( 3 ) On the other hand, an isomorphism from 2l' is an isomorphism in 2l, and a non-isomorphic a from YI' can still be an isomorphism in Z. (4) If 2l' is full, an a from a'which is an isomorphism in 2I is an isomorphism in W.
3.10. Let 2l and 23 be categories. A functor F : YI 23 consists of two mappings (which are customarily denoted by the same symbol, namely by the one used to denote the whole functor), the one mapping obj2l into obj23, the other mapping the class of morphisms of into that of 23, such that: --+
(1) (2) (3)
For a : a + b,
F(a
F(1a) 0
F(a): F(a) + F ( b ) , lF(a)3
p) = F(a) F ( B ) . 0
(Thus, actually the object part of a functor is determined by the morphism part.) Sometimes, the notion just described is named a covariant functor from CU to '23. A contravariant functor from CU to 23 is a functor from 2 l Q P to 23,
Remark. Evidently, a composition of two functors is a functor. 3.11. Examples. (A) The identity functor 1,: 2I -+ given by 1a(a)= a and la(a) = a. (B) The functor Q : Set + Set defined by Q ( X ) = X x X , Q ( f ) ( x ,y ) = (.f(x), f ( Y ) ) .
1
(C) The functor P + : Set + Set defined by P ' ( X ) = ( A A c x), P + ( , f () A ) = f(A) (the image of the subset A under f). (D) The functor U : Top --+ Set sending ( X , 7) to Y and ( ( X ,z), f, (T: 9)) to f' (for the notation see 3.5, the Remark). Similar functors from the categories of 3.5 D and 3.5 F into Set. Those functors are often called the forgetful functors since they forget about the structures. More about this will be said in $4.
32
Ch. I, PRELIMINARIES
(E) Let M , M' be monoids. Consider them categories (in the sense of 3.5 H). The functors F : M -+ M' (more precisely, the morphism parts of them) coincide with homomorphisms with respect to the binary operations and units. (F) Consider two preordered classes as thin categories (see 3.6). The functors between them (more precisely the object parts of them this time) coincide with the order-preserving (monotone) mappings. (G) The constant functors from 'u to 23 sending all the morphisms of 2I into 1, for a fixed object b E obj 23. 3.12. Let F, G: (u -+ 23 be two functors. A natural transformation G is a collection ( f ) a E o b j a of morphisms z": F(a) G(a) in 23 such that for every cp: x + y in (u the diagram z: F
-+
--f
F(x) 5G(x)
F(Y!
IY
G(Y)
commutes, i.e. zy o F ( q ) = G(q) o zx. A natural transformation z = (z8),,,bjC)t: F -+ G such that each z" is a n isomorphism is called natural equivalence. Note the fact that then ((7;') is a natural transformation G -+ F (the so-called inverse of 7). We say that F and G are naturally equivalent if there exists a natural equivalence 7: F --* G. This fact is often indicated by F r G. Mostly, we will write briefly transformation for natural transformation.
3.13. Let z = (?)a&jW: F -+ G, 9 = ( 9 ) a E o b j a : G -+ H be transformations. Then 9 z, defined as (9" z8)8Eo,,ja is a transformation F -+ H . It is called the composition of r and 9. Ifz = (za),&ja: F -+ G is a transformation, F, G : (u -+ 23 and H : 23 -+ Q (resp. H : Q -+2I) functors, then ( H ( T ~ )is) a~transformation ~ ~ ~ ~ ~ H oF -+ H G, denoted by H z ( (+c))csobj(E a transformation F o H + G o H , denoted by z H , respectively). 0
0
0
3.14. Examples. The notions of natural transformation and of natural equivalence belong to the most important in fundamental category theory. Perhaps, they are the most important notions there. The few examples we can give at this place cannot, in the least, cover even the most typical cases. The reader is advised to check explicitly the commutativity properties.
33
$3. Categories, functors, transformations
(A) The identical transformation lF = (lF(a))oEobjl,: F -+ F. Note that a natural equivalence T : F G is a transformation such that there is a transformation 3 : G -+ F with 3 z = l F and T 9 = lG. (B) The "diagonal" 6: l S e t -+ Q (see 3.11.B) defined by dX(x)= (x,x). (C) T : lSet P' (see 3.1 l.C) given by T"(x) = (x} (the one-point set containing x). Another transformation 3: lSet-,P c given by 9'(x) = 8. (D) Let C,, C,: 2I 23 be constant functors (see 3.11.G) sending the morphisms of 2I into l,, 1, respectively. Let cp: x -+ y be a morphism = cp for all a is a transformation C , -+ C,. in 23. The i7, given by (E) For a set A denote by Q A the functor Set Set defined by Q A ( X )= X A , QA(f)(5) = f o 5. We have Qz E Q ; indeed, it suffices to define T : Q 2 -+ Q by = (t(O), {(l))(cf. l.l), similarly, Q1 E l,,,. (F) Denote by Pa the functor SetoP-+ Set defined by P A ( X )= A X , PA(f)(5) = 5 o f , by P - the functor defined by P - ( X ) = ( M M c X } , P-(f) ( M ) = f-'(M) (the inverse image of M under f). We have P2 E P - ; indeed, it suffices to define T : P2 -+ P - by ~ " (=l )c-'({l}). -+
0
0
-+
-+
--$
~"(0
I
3.15. A transformation T = ( T ~ is said ) ~to be ~ monotransformation ~ ~ ~ ~ (epitransformation,resp.) if every T" is monomorphism (epimorphism, resp.). We will mostly deal with transformations between functors F , G: 2I + Set. It is easy to see that in Set the monomorphisms are exactly the one-to-one mappings, the epimorphisms the onto ones. Thus, e.g., a monotransformation here will be a T = ( T " ) with all the T"S one-to-one.
EXERCISES
A skeleton of a category 2l is its full subcategory 23 such that every u E obj 2I is isomorphic to exactly one b E obj 23. 1. Two skeletons of the same category are isomorphic. Categories CU and 23 are said to be equivalent if there are functors F:CU-+Band G : B - + % s u c h t h a t F o G ~ l ~ a G n d oF~l~(see3.12). 2. 2I and 23 are equivalent iff they have isomorphic skeletons. Let J be a set, let (aj)jE Jbe a collection of objects of 2I. The product (coproduct, resp.) of this collection consists of an object a of 2I and a collection ( a j : a -+ a j ) j e J(resp. ( a j : a j -+ a ) j E J )of morphisms such that for every collection ( p j : b -+ aj)jsJ (resp. (pj: a j -+ b)j,J)there is exactly one . morphism /? with a j 0 p = 8, (resp. / ? o ai = pj). The a in the product (resp. coproduct) is usually denoted by )(aj (resp. v a j ) or n u j (resp. u a j ) ,
34
Ch. I, PRELIMINARIES
for small collections e.g. a x b, a x b x c, a, x ... x a, (resp. a v b, a v b v c, a, v ... v a,,). Often, the term (co)product is used for the object x a j (resp. vaj). A product of a collection (aj)j,J with aj = c for all j is .I
.I
often called power of c and denoted by cJ. An a E obj CU is called singleton or the terminal object (cosingleton or initial object, resp.) of CU if for every b E obj ’u there is exactly one morphism b a (resp. a b). 3. A singleton is the product of a void collection. 4. What is a singleton in a partially ordered set (see 3.6)? -+
-+
Let a,8: a 4 b be two morphisms. An equalizer (cocqualizer. rcsp.) of a and 8 is a morphism y: c + a (resp. y : b - c ) such 1Ii:tt ( I ) Q 7 == / 3 c y (resp. y o a = yap), and (2) whenever a13 = PI3 (resp. d a = dfl) there is exactly one E with 6 = y E (resp. I3 = E y). 5. An equalizer is always a monomorphism. Prove and dualize. 0
0
Let a : a - + c and 8: b + c (resp. a : c - + a and 8: c + b) be two morphisms. The pullback (pushout, resp.) of a and p is a couple of morphisms a’:d + b and p’: d -+ a (resp. a’: b + d and p’: a + d ) such that (1) ap’ = pa’ (resp. a‘p = p’a), and (2) whenever up‘‘ = pa’’ (resp. a “ p = 8 “ a ) there is exactly one y with a’y = a“ and p ’ y = /3” (resp. ya‘ = a’’and y p ’ = B”). 6. Let c be a singleton. Then the pullback of a: a c and 8: b c coincides with the product of a and b. 7. Let a: a -+ c, ,8: b + c be morphisms, (a x b, (na,q,))a product of a and b, y an equalizer of a rc, and 8 rcb. Then, n, y and nby form a pullback of a and 8. 8. Let a: a -+ b be a morphism. Then, ( l a ,1,) is a pullback of a and a iff a is a monomorphism. 9. In Set (see 3.5 A), the product is the usual Cartesian product with J , union u M j x ( j } the natural projections. For the coproduct of ( M j ) J E the (the “disjoint union” of the collection) with the natural injectidns can be taken. The equalizer of f, g : M -+ N is the .embedding mapping of ( m f ( m ) = g(m)} into M . Construct also a coequalizer o f f , g. 10. In Top (see 3.5 E) the product coincides with the usual topological product, the coproduct with the sum of spaces. An equalizer o f f , g: X Y is the embedding mapping of (x E X f(x) = g(x)), endowed with the subspace topology, into X . 11. What is a concrete form of products, coproducts, equalizers and
-
-
I
1
-+
s3. Categories, functors, transformations
35
coequalizers in the category of abelian groups? (Attention: a v b coincides here with ~1 x b. Infinite coproduct<.however, are proper subgroup of the related products.) Let K be a small category, D : K -+ 2I a functor (such a functor starting in a small category is often called a diagram in a).A lower (upper, resp.) bound of D consists of an object a E obj ‘$1 and a collection ( C ( k ) k E o b j K of morphisms ak: a + D(k) (resp. ak:D ( k ) -+ a ) of 2I such that for every p : k -+ 1 of K , D ( p ) a, = a, (resp. a1c D(p) = a,). The lower bounds are sometimes called cones, the upper ones cocones. 12. Give a definition of the bounds using the notion of natural transformation. (Hint: See 3.11 G, 3.14 D.) 13. Consider a partially ordered set as a category (see 3.6). Compare the new definitions of the lower and upper bound with the usual synonymous notions in partially ordered sets. 0
A lower (upper, resp.) bound (a,(ak))of D is said to be a limit (colimit, resp.) of D (notation: lim D, colim D ) if for every lower (upper, resp.) bound (b,(fik))of D there is exactly one morphism p such that for every k a, fi = (resp. ctk = p k ) . 14. Reformulate the definitions in the sense of the exercise 12. 15. A limit (colimit) of a diagram is determined up to isomorphism. and (a‘, (a;)) are limits of D,there is an isomorphism More exactly, if (a, (ak)) I : a + a’ such that a; z = @k for all k . Similarly for colimits. 16. Formulate exactly the duality relations between limits and colimits (“In WP, the limits from 2I become colimits, and vice versa”.). 17. Redefine the notions of product, equalizer and pullback as limits of special diagrams. Dualize. 18. If 2I is a partially ordered class, the limit of D : K + 2I coincides with inf D(k). 0
a,
0
0
objK
A category 2I is said to be complete (cocomplete, resp.) if every diagram D : K -+ 2I ( K arbitrary small) has a limit (colimit, resp.) in 2I. 19. A partially ordered set (see 3.6) is a complete category iff it is a cocomplete category iff it is a complete lattice. 20. If a category is small and complete, it is thin. (Hint: Let a,B: a -+ b be distinct. Consider the powers bJ with J increasing and show that the number of morphisms a -+ bJ increases.) 21. Let K and L be equivalent (see exercise 2), let every D : K -+ have a limit. Then every E : L -+ 2I has a limit.
36
Ch. I, PRELIMINARIES
22. (Maranda’s completeness theorem) 2I is complete iff every collection of its objects has a product and any two morphisms with domains and ranges coinciding have an equalizer. (Hint: Let 2I have products and equalizers. Let r = ((y,, Y ; ) ) , ~be~ a collection of pairs of morphisms of 2I, y,,y:: A -+ B,, ( B ,(71,)) a product of (B,),EJ.Let y,7’: A 4 B be the morphisms with y, = n, y, yi = n, y’ for all I . Then the equalizer of y,y’ is a kind of “common equalizer” of the pairs from r. Now, let D: I( + CU be a diagram, ( P , ( p k ) ) a product of (D(k))k.Consider the r consisting of all pairs (nk,,D ( p ) nk) with p: k + k‘ a morphism in K . 23. Let CU be a category, y, y’: A -, B some of its morphisms. Let ( A x B, (n,,nb))be a product of ( A ,B), let 7,T’: A A x B be the morphisms with n, 7 = l,, nRo7 = y, n,oT‘ = l,, ngoy’ = y’. The following statements are equivalent: 0
0
--f
(i) 6 is an equalizer of y and y’ (ii) 6 is an equalizer of 7 and 7‘ (iii) (6,d) is a pullback of (7,y). 24. (Freyd’s completeness theorem). CU is complete iff it has products and pullbacks. A functor F : CU + 23 is said to preserve (co)limits if (F(a),( F ( C ( ~ ) ) ~ ~ ~ , , , , is a (co)limit of F D whenever ( ( I , (Y~))I S a limit of D. 25. If CU and 23 are complete, F : \u + 23 preserves limits iff it preserves products and equalizers. 0
Let A be a small category, 23 an arbitrary one. The category
23’
(often denoted by [ A , 2331)
is defined as follows: The objects are all the functors F : A + 23, the morphisms from F to G are the transformations z: F + G, the composition is the usual composition of transformations. We have, for every a E obj A , a functor e,: B A -, B
defined by e,(F) = F(a), e,(z) = 7,. 26. If 23 is (co)complete then every 23, is (co)complete, and every e, preserves (co)limits. (Hint: For a diagram D: K -+ BA consider the limits (L(a),(A:)) of e, 0 D, define, for a : a -, a’, L(a): L(a) -, L(a’) by G’0 L(a) = (D(k)) (a) A:. Show that thus a functor L : A -, 23 is obtained and that there is a limit (L,(x,JkEobjly) of D.) 0
37
$3. Categories, functors, transformations
If 2I, 23 are categories, the category ax23
is defined by obj(‘2I x 23) = o b j a x obj23, morph((U x 23) = m o r p h a x morph 23, l(o,a) = ( l a ,la), (a, P) (a’, P ) = (a 0 u’, P P’). 27. Show that x 23 has formally the properties of “the product of and 23” if we take functors instead of morphisms. 0
0
The evaluation functor ( A , 23 are categories, A small) e: A x
is defined by e ( u , f ) = f(a), and, for a : a + a ‘ and = ta’o f ( a ) = g(a) 0 to. 28. Check the correctness of the definition of e. Given a category
x
-+
Set
%(a, b) = the set of morphisms a
The Yoneda functor
f - 9, e(t, a)
a,let us denote by the same symbol the functor ‘?fop
defined by
t:
(%(sP))(cp) =
-+
b,
P O c p O @ .
Y: AoP+ SetA
is defined as follows: For a E obj A , Y(a): A + Set is defined by (Y(a))(b) = A(a, b), (Y(u)) (p) = A(ia, P); for a morphism I, the transformation Y(a) iS defined by Y(c()b = A(a, 1b). 29. Check the correctness of the definition of Y. 30. (The Yoneda lemma). The functors and
SetA(Y--, -): A x SetA-+ Set
e : A x SetA+Set
are naturally equivalent. (Construct transformations SetA(Y(u), f) +f(a), A ~ , f(a) ~ : + set”(Y(a),f) putting for t: ~ ( a+ ) j x(t) = .“(lo), for X E X (A(x)b)(cp)= (f(cp))(x), check the correctness and prove that x and A are mutually inverse.) 31. Y is a full embedding. 32. Reformulate the Yoneda lemma to a form including large categories ‘2I as well. How many transformations lkt -+ QA are there? How many Q-P’?
38
Ch. I. PRELIMINARIES
Functors L : 2I 23, R : 23 2I are said to be adjoint ( L is said to be a left adjoint to R , R a right one to L ) if the functors -+
-+
23(L-, -), 2 I ( - , R - ) :
WP x 23-+ Set
are naturally equivalent. 33. Functors L : 9I + 23, R : 23 -+ 2I are adjoint ( L a left adjoint to R ) iff there are natural transformations
l%-+RoL,
Q:
LoR+lB
such that e L L p = 1, and R e p R = 1,. (Let x,b: %(La,b) + %(a, Rb) be the natural equivalence. Put p, = X , , , ~ ( I ~ ~ ) ,e, = X,&(lRb). Given the natural transformations p, e, consider the correspondences 23(La,b) '%(a, Rb), %(a, Rb) %(La,b) sending cp to R cp p, and t+h to @ b L cp.) 34. Define K A : Set -+ Set putting K A ( X )= X x A, K A ( f ) ( x a) , = ( f ( x ) ,u). KA is a left adjoint to Q A . 35. Let U be the natural forgetful functor from the category of abelian groups into Set. U is a right adjoint to the functor F sending a set X to the free abelian group over X . 36. The natural forgetful functor Top -+ Set (see 3.5 E) has both a right and a left adjoints. (No separation axiom is assumed!) 37. The right adjoints preserve limits and left adjoints preserve colimits. (Let R be a right adjoint, (b,(&)) a limit of D. For a lower bound (a, ( a k ) ) of R D consider the lower bound (La, (e, 0 L(ak))k)of D with @ from exercise 33.) 38. If A , B are (small) complete lattices (cf. exercise 19), R : B + A is a right adjoint iff it preserves infima. 0
0
-+
0
0
0
A full subcategory 2I of 23 is said to be its reflective (coreflective, resp.) subcategory, if the embedding functor is a right (left, resp.) adjoint. 39. The category of abelian groups is reflective in the category of all groups, but it is not coreflective there. 40. The category of sets with symmetric relations is both reflective and coreflective in Re1 (2) (see 5.6 below).
0 4.
Concrete categories
4.1. In the examples 3.5 (A)-(F) (but not in the examples 3.5 (G), (H) and in 3.6) the categories looked as follows: the objects were sets, mostly with some extra information, the morphisms were some of the mappings between them, the identities were the identity mappings, and the morphisms
39
s4. Concrete categories
composed as mappings. Return, for a moment, to $2, and compare the definition of monoid and that of concrete monoid on one hand, with the definition of category and the just discussed form of some of them on the other hand. In this paragraph we will discuss a definition of the concrete category. 4.2. Two categories 'u and 23 are said to be isomorphic if there are functors F : 'u + 23, G: 23 + 'u with G 0 F = 1, and F 0 G = 1, (such functors are called isofunctors). A functor F : CU-23 is said to be faithful if it is one-to-one on each %(x, y ) (i.e., if, for cp: x + y and $: x + y we have cp $, then ~ ( c p ) ==! F($) - but it may happen that c p r : Y, + y,, i = 1,2 are distinct and F(cp,) = F(cp,). Of course, if F is faithful and its object part is one-to-one then the whole F is one-to-one.
+
4.3. A category 'u is said to be concretizable if there exists a faithful U ) where 'u is functor U : 'u + Set. A concrete category is a couple (a, a category and U a fixed faithful functor 2I -+ Set. U is called the forgetful functor of ('u,U ) , \u is called its abstract part. 4.4. A concrete category ('u,U ) is said to be concrete subcategory of (23, V ) if 'u is a subcategory of 23 and if, for c( in 'u, U ( a ) = V(a).
4.5. Remarks. (1) Obviously, 'u is concretizable iff it is isomorphic to a subcategory of Set (i.e., if there is a one-to-one F : 'u + Set). Indeed, take the faithful U : a + Set and define F putting, say, F(a) = U(a) x ( a ) , (F(cp))(u, a) = (u(cp)(u), b) for cp: a b. (2) Theorem in 2.7 says that every category with one object is concretizable. We shall see shortly (in § 7 ) that in fact every small one is concretizable. For the general concretizability problem see Introduction, 4. -+
4.6. If a category 'u is explicitly defined as mentioned in 4.1, i.e. with objects sets and morphisms mappings-both perhaps with some additional information-the functor U : 'u + Set defined by forgetting the additional information (see also 3.11 D) is usually called the natural forgetful functor. 4.7. A concrete category with one object is a slightly more general notion than that of a concrete monoid from 2.5. In fact, the concrete monoids coincide with the concrete one-object categories with natural forgetful functors. The widening of the notion goes no further then allowing the mappings in formulas concerning only composition to be represented by more arbitrary symbols. To be exact, viewing a concrete monoid ( M , X )
40
Ch. 1, PRELIMINARIES
as a concrete category, one replaces it by ( M , U ) where U : M + Set is given by U(f)= J: Taking, e.g., the functor P + from 3.11 C , one can define a one-object concrete category ( M , P + 0 U ) which is obtained from no ( N , Y ) by the replacement above. It is, of course, concretely isomorphic in the sense of the following definition with ((F"(f)f~ M ) , V ) where V is the natural embedding of this new concrete monoid into Set.
1
4.8. Two concrete categories (a, U) and (23,V ) are said to be (concretely) isomorphic if there is an isofunctor F : CU + B and a natural equivalence z: U + I/. F . If, moreover, U = V o F , we say that the categories are equally realized. Remark. Two concrete monoids are isomorphic iff they are isomorphic if viewed as concrete categories. 4.9. Usually, the following expressions are used: U ( a ) is called the underlying set of the object a, U ( M )the underlying mapping of the morphism a. A mapping f: U(a)-+ U(b) is said to carry a morphism from a to b if there is a q : a + b with f = U ( q ) .To avoid confusion, let us recall the use of the first expression in tj 2. We spoke there on the underlying set of a monoid in the sense of the natural forgetful functor in the category of monoids. On the other hand, the carrier of a concrete monoid is the underlying set of the object. EXERCISES
1. Prove that concrete categories (Set, ISet) and (Set, P') are not concretely isomorphic. 2. Find different forgetful functors for the ordered class of all ordinals. 3. Find a forgetful functor for the category of multigraphs (see 3.5 G). 4. If A is small and B is concretizable, then BA (see E 3.25) is concretizable.
0 5.
Some important concrete categories
5.1. A type A is a mapping of a set into the universal class*). Thus, to give a type means to give a set, say B, and for every b E B to determine, sets A h . We will usually indicate such a type by (&)beB. Types with B a natural number, a small one in particular, will often be written as se*) I t is, hence, the same as what we often refer to as a collection
41
$5. Some important concrete categories
5.2. Let A = (Ab)b,B be a type. A relational structure of the type A on a set X is a collection r = (rb)b.B where rb is an Ab-ary relation on X . An algebraic structure of the type A on X is a collection a = (ab)b.B where ub is an A,-ary operation on X . The couple (X,a) is then called algebra of the type A . 5.3. Remark. Writing small types as sequences, we see that e.g. a semigroup can be viewed as an algebra of type (2), a monoid as one of type (2,0), a group as an algebra of type (2,0,1) if we consider the unary operation of inverse, a lattice as one of type (2,2), a ring as one of type (2,0,1,2); a ring with u n i t I \ of type (2,0, 1,2,0), etc. To describe the type of vector space consider 8 = (u, u, w >u R, where R is the set of real numbers, u, z . I \ di\tinct elements outside of R, A , = 2, A , = 0, A , = 1 (for the abelian group part of the structure) and A , = 1 for x E R (for multiplying by the real numbers). 5.4. Let r = (rb)bGBbe a relational (algebraic, resp.) structure of a type A on X , s = (Sb)beB a relational (algebraic, resp.) structure of the type A on Y A mapping f’: X Y is said to be rs-homomorphism if it is an rbsbhomomorphism (see 1.2, 1.5) for all b E B . If X = Y and r = s we speak about an endomorphism, if, moreover, it has an inverse which is also endomorphism, we speak about an automorphism. In the case of relations, rs-homomorphisms are sometimes called rs-compatible mappings. --f
5.5. In this terminology, the theorem in 2.8 can be reformulated to read: Every monoid is isomorphic to the endomorphism monoid of an algebra. The algebraic structure can be chosen so that it consists of unary operations only. 5.6. The category of relational structures of a given type A , denoted by
=
Re I ( A ) > is defined as follows: The objects are couples ( X ,r), where X is a set and r is a relational structure of the type A on X . The morphisms from ( X , r ) into (I:s) are triples ((xs),f; ( X ,r ) ) where f: X + Y is an rs-homomorphism. The composition is defined by ( ( Z ,t), g, (K s)) ((I:s),L r)) = ((2,t), 9 sf, ( X , r)). 5.7. The category of algebras of the type A , denoted by 0
(x,
Alg (4 is defined as follows: The objects are algebras of the type A , the morphisms * from ( X , a) to ( y /3) are triples ((I:/3),f, ( X , a)) where f is an ~ P - h o m o morphism. The composition is defined in the obvious way. 7
42
Ch. I, PRELIMINARIES
5.8. The categories
Let J be a set, let there be given, for every i E J a functor Fi: Set -+ Set, covariant or contravariant. S((FJiEJ) is defined as follows: The objects are couples ( X , (rJieJ) where X is a set and, for every i, ri c Fi(X). Mor= (Y, ( s J i S J )are triples (EL 2)where phisms from % = (X, (r&) to f : X + Y is a mapping such that for the covariant F;’s F i ( f ) ( r i )c si, and for the contravariant ones Fi(f)(si)c ri. The composition is defined in the obvious way. For small sets J we write the functors in a sequence. So we write e.g. S(F0,F1, F2) for S((Fi)icz3)> S(F0,F,, ...>Fn- 1) for S ( ( F i ) i E n ) , S ( F ) for S((Fi)iEl) with Fo = F . 5.9. Remarks. (1) For a set A define QA Set -,Set putting Q A ( X )= X A , Q A ( f )(<) = f 4. We see that for d = (A&&, Rel(d) coincides with S((&)~GB). (2) The (concrete) category of topological spaces and their continuous mappings can be considered a full concrete subcategory of S ( P - ) (for P see 3.14F), so can the category of uniform spaces and their uniformly continuous mappings in S ( P - 0 Q). 0
5.10. Important conventions. If a category is defined so that it has the natural forgetful functor (see 4.6; this concerns particularly the categories just defined in 5.6, 5.7 and 5.8) and if we speak of it as of a concrete category, we automatically assume it, unless stated otherwise, to be endowed with the natural forgetful functor. If there is no danger of confusion, we often represent morphisms from such categories by their underlying mappings. EXERCISES
1. (See 5.9.2.) The concrete embedding of Top into S ( P - ) can be done in two different ways. 2. Categories S((Fi)iEJ)and Alg(d) are complete and cocomplete (use E 3.22).
5 6.
Embeddings
6.1. A functor F: M + B is said to be full if for any two objects x , y of M and any morphism $: F ( x ) -+ F(y) in B there is a cp: x -+ y in M such that F(cp) = $.
43
96. Embeddings
A full embedding of a category 2I into a category 23 is a full one-to-one functor F : 23. If there is a full embedding of 2I into 23, we say that 2I is fully embeddable into 23. -+
6.2. Remark. Thus, obviously, the full embeddability of 2I into 23 means that 2I is isomorphic to a full subcategory of 23. A full embedding does not differ too much from a slightly more general full faithful functor. Namely, a full faithful functor may only fail to be one-to-one by asigning common value to isomorphic objects. Indeed, if one has F ( a ) = F(b) = x, due to the fullness one has to have an M : a -+ b and a b : b -+ a such that F ( M )= F ( a ) = 1,. Thus F(@) = 1, = F(1,) and @M) = 1, = F(la),so that, due to the faithfulness, M and /3 are mutually inverse isomorphisms. It is very often the case that the category 23 in question has, for every object, a proper class of isomorphic ones. Then, a full faithful functor can easily be modified to a full embedding.
6.3. Let (a, U ) , (23,V ) be concrete categories. We say that (a, U ) has an extension in (23, V ) if there is a full embedding F : 2I + 23 and a monotransformation p: U - + V o F The functor F is then called extension of
(a, U ) into (23,V ) .
6.4. Remark. Our new definition of extension requires a confrontation with the old one from Introduction, 5, to see clearly whether, or to what extent, we have here what we wanted to have. The main point to be elucidated lies in the use of the notion of isomorphism of concrete categories. If we understand it in the sense of 4.8, the notions of extension from Introduction and from 6.3 coincide. Indeed, let us have i i n cvtension F of (91, Lr) into (a, V ) in the sense of 6.3, i.e. a monolransformation p : L' + 1' F . For u ~ o b j \ U put Lr'(a) = U ( u )u (VF(~i)\p"(U(~i))),and define c": Lj"(o) + V F ( a ) by E"(x) = pLy(x)for Y E U(a), c"(.\) = x otherwise. For an 1: a + a' put U'(r) = ( e n ) W ( r ) E". We see easily that C7' is a faithful functor, E an equivalence of U' and V F and that, for any C I : a + a' and x E U(a), (U'(a))(x) = (U(r))(x). O n the other hand, let U ' : 91 + Set be such that always U'(a) 3 U ( a ) and ( U '( a ) ) ( x )= = (U(cx))(x), and that (a, U') is isomorphic to a full concrete subcategory (0, V &) of (b,v). Hence, we have an invertible functor @: CU + b and a natural equivalence E : U' -* ( V &) :@. Put F = J u @, where J : Q + B is given by J ( y ) = y, and define a monotransformation p : L' -+ V p F by p("(x)= E"(x). m.
'
J
I
I
But, in the Introduction, isomorphic concrete categories were understood as more closely related than that, namely as what we called equally realized
44
Ch. I. PRELIMINARIES
in 4.8. If we reformulate the definition of extension from the Introduction we obtain the following requirement: (1) There is a full embedding F : 2l + 23 such that, for every object a, U ( a ) c VF(a), and, for every morphism a: a -+ a' and x E U(u), ( V W (4 = (Q)) Let us refer to the situation described in (1) stating that (a,U ) has an extension in the stronger sense in (23, V ) . Fortunately, as will be seen below (in 6.7), in reasonable cases, having an extension (or even less than that) already implies having an extension in the stronger sense.
(4.
6.5. A concrete category (a, U ) is said to be regular if the following implication holds: if a: a -+ b is an isomorphism and if U(a) = Irr(o),then a = 1,. (Very roughly speaking, the regularity means that if two isomorphic objects differ, they do so already at the set level: if the underlying sets happen to coincide, they have at least to be permuted to reform one of the objects into the other.) Obviously, regularity is carried over by concrete isomorphism, and inherited by concrete subcategories. All the examples of concrete categories introduced so far were regular. 6.6. A concrete category (a, U ) is said to have the transfer property if for every a E obj 'u and for every invertible mapping f : U ( a ) M there is an isomorphism q : a + b with f = U ( q ) . Obviously, if a concrete category has the transfer property, then each of its concrete subcategories which, with every object, contain all the isomorphic ones has; further every equally realized category has the transfer property. The examples of concrete categories mentioned so far had the transfer property. --f
6.7. Theorem. Let (a, U ) be regular and let (93, V ) have the trunsfer property. If there is afullfaithjulfunctor F : CU --* 23 and a monotransformation p : U --t V o F , then (a, U ) has an extension in the stronger sense in (23,V ) . If no p' is onto, the assumption of regularity can be omitted. Proof. For a E obj '2l put where o(a) = (a, U(a)).It is there to make sure that
45
56. Embeddings
Z(a)n U ( a ) is void,
(1) (2)
if
Define mappings
Z(u)
+ 8 + Z(b)
U(a)u Z(a)
+
+
and a b , U(b)u Z(b).
k": VF(a)-+ U ( a )u Z(a)
putting kU(pU(x)) = x, k"(u) = (u, o(u))otherwise. Since (23, V )has the transfer property, there are isomorphisms
xa: F(a) -+ G(a) with V(x") = k". Now, define a functor G : 9I + B putting, for cp: u -+ b, G(cp) = xb. . F ( c p ) . ( x " ) - ' .Evidently, G is a full faithful functor. For cp: a -+ b and x E U ( a ) we have
(vw) (4
(3) =
(4) = (4))q c p ) (x).
= k b ( ( V G 4((k")kb((VF(cp))( P L a ( X ) ) )= kb(Pb(U(cp) =
Finally, let G(u) = G(b). By 6.2, there is an isomorphism a: a b with G(a) = lG(").Hence, we see easily that either Z(a) =k (b 4 Z(b) or Z(a) = = Z(b) = 8. In the first case, u = b by (2) above, since UG(c) = U(c)u Z(c). If Z(a) = 8 = Z(b), we have U ( u ) = VG(a)= V(lG(")) = lVG(a) = lU(") by ( 3 ) and hence CL = 1, by the regularity. Thus, G is one-to-one. rn -+
6.8. Now, we are going to show that in some cases extendability follows readily from full embeddability. Namely, we have :
Lemma. Let (!! UI, ) have the following properties. (1) There is an object I with U(1) a one-point set, say {i), such that for every a E obj'2I and x E U ( a ) there exists a morpkism tax: I -+ a with U ( t : ) ( i ) = x. (2) There is an object T and two distinct u, v E U ( T ) such that for every u E obj !!Iand distinct x,y E U ( a ) there is a morphisnz if&: a + T with ~(v:.)(x) = u and ~(q:,)(y) = ZI. Then for any (23,V ) and every full faithful F : 21 -+ 23 there is a monotransformation p: U - + V o F .
Proof. First, because U is faithful, we see that the morphism determined by u and x. Consequently, for every q : u b, --f
cp
rf:= r't(,,,,,
'
5: is uniquely
46
Ch. I, PRELIMINARIES
so that p"(x)
+ p"(y).
6.9. Lemma, Every (small,finite) category is equally realized with a full concretc subcategory of a (small, finite) concrete category satisfying (1) and (2)from 6.8. Proof. For
(a,U ) construct (%, V ) as follows: obj% = (obj2I x 1) u {I, T } ,
where I, T are distinct elements which are not in obj2I x 1 (the multiplying by 1 = (0) is used to make sure that there are such I and T - the class obj 2I itself could have been the whole universal class). For a, b E obj 2l,
o), (b,0)) = %(a, b) x
'f3((a,
%(I, I) = ( 1 1 ) %(I, (a, 0)) = U(a) x
1,
3
%((a,
o), I ) = 8 ,
%(X T ) = (1,) q T , (a, 0)) = 8 %((a,o), T ) = { M M %(I, T ) = 2 x { l ) ,
{(a, 0)) 2
2
2
1
= u(a))x
((0,l))
7
B(T I ) = 8
(the multiplying by 1, ((a, 0)), ((a, I)), { 1) just provides for the sets to be disjoint).
47
$6. Embeddings
The composition rules are given by:
(a,0 ) O (P, 0 ) = (aO fi, O ) , (a,0 ) ( x ,(a, 0)) = ( U ( 4 (x),(b,0)) ( M ,(b, 1)) (a,0) = ( U ( a ) - ( M ) ,(a, I)) O
for a : a + 6 ,
0
1) for x $ M 1, 1) for X E M
( M ,(a, 1)) (x,(a, 0)) = 0
The forgetful functor V is given by V(a,0) = u(a), V ( I )= 1 ,
V ( T )= 2 ,
V(a,0) = U ( a ) ,
v ( ( x ,(a,0)))(0)= x ,
It is easy to make sure that this is a correctly defined concrete category and that it has the required properties. Since only two objects were added, it holds that if CU was small (finite), $‘3 is small (finite), too. N 6.10. Theorem. (1) Let (%, U ) be a concrete category with the transfer property. Let for every concretizable (small, finite, resp.) category 6 exist u full faithful F : K -+ %. Then every concrete (small concrete, finite concrete, resp.) category (6, V ) has an extension in the stronger sense in (%, U). ( 2 ) More generally, let afamily of concrete categories with the transfer property be such that from every concretizable (small,finite, resp.) category there is a full faithful functor into some of them. Then every concrete (small concrete, finite concrete, resp.) category has an extension in the stronger sense in some of them.
Proof. We will prove (1). (2) follows by an obvious modification of the procedure. First, define a new forgetful functor if’: 6 + Set putting V’(c) = V ( c ) u (V(c)>for c E obj6, V’(cp)( x ) = V(cp)( x ) for x E V(c) and V(q)(V(c)) = V ( d ) for a morphism cp: c -+ d. Thus, lKis an extension of (6, V ) into (6, V’). By 6.9 there is a category (23,W ) satisfying (1) and (2) from 6.8 such that (0, V’) is equally realized with its concrete subcategory. Consequently, by 6.8, there is a full faithful functor F : C i + % and a mono-. transformation p’: V’ -+ U F . Composing this with the obvious monotransformation V -+ V’, we obtain a monotransformation p: V + U F 0
0
48
Ch. I, PRELIMINARIES
such that no pc is onto. Thus, by 6.7, there is an extc-lsion in the strong U). sense of (&, V ) in (!!I, 6.11. Let (!!I, U ) , (23, V ) be concrete categories. A full embedding F : -+ 23 is said to be a strong embedding of (a, U ) into (23,V ) if there exists a faithful functor G : Set + Set such that
lu - 1" I
G
Set
Set
commutes, i.e. that G c U = V o F. If there exists a strong embedding of (a, U ) into (23,V ) , we say that (a, U ) is strongly embeddable into (23, V ) . The functor G is sometimes called the underlying functor of the strong embedding F . Remark. Obviously, if (23, V ) has the transfer property, the existence of a G such that G o U 2 V c F suffices.
6.12. Lemma. Let G : Set 4 Set be a faithful (covariant)jknctor. Then there exists a monotransjormation V : lSet +
G.
Proof. This is a consequence of 6.8. It suffices to put 2€ = '23 = Set, U = F = lSetand V = G. In (Set, lSet),one can take the one-point set for I and the two-point one for 7: d
6.13. Corollary. Every strong embedding is an extension. Proof. Take a strong embedding F of (a, U ) into (23,V ) , G its underlying functor. By 6.12 there is a monotransformation 11: lSet-+ G. Thus, we have a monotransformation p = v U : U 4 G c U = V o F (see 3.13). d
6.14. Remark. The existence of an extension of (a, U ) into (23,V )means, as it was discussed above, that 21 can be made to a full concrete subcategory of (23,V ) by extending the original underlying sets adding, if necessary, some new points to the original ones, and extending the original underlying mappings over sets thus augmented. This augmentation of the underlying sets may vary from object to object, i.e., VF(a)could differ arbitrarily from VF(b)while U ( a )is equal to U(b).
$6. Embeddings
49
The strong embeddability means that one can first expand sets canonically, regardless of the possible structures on them, and then represent the objects carried by a set by objects carried by the expanded one.
6.15. Let (%, U ) , (23, V ) be concrete categories. A full embedding F : % -+ 23 is said to be realization of (%, U ) in (23, V ) if V OF = U . If there is such an F , we say that (a, U ) is realizable in (23, V ) . 6.16. Remarks. (1) Thus, in other words, (%, U ) is realizable in (23.V ) , if it is equally realized with its full concrete subcategory. The reader may ask why we ch'ose the stronger of two possible definitions here (the other one requiring just an isomorphism) while in 6.3 the weaker form was preferred. The Peason is technical only: it is more convenient to work with the weaker form when dealing extensions, and, on the other hand, an isomorphism of a concrete category with a concrete full subcategory of another one occurs in constructions in the form of an equal realization mostly. (2) (21, U ) is strongly embeddable into (23, V ) iff there is a faithful G such that (%, G U ) is realizable in (23, V ) . (3) (%, U ) has an extension in ($' 3, V ) iff there is a monotransforrnation p : U -+ U' such tha! (%, C7') is realizable in (23, V ) . 0
6.17. Proving a realizability of (%, U ) ir, (23, V ) is usually based on the following trivial observation: (a,U ) is realizable in (23,V ) iff there is a mapping F : obj21 --* obj23 such that (1) For all a E objELI, VF(a) = U(a), (2) a mapping f : U(a)-+ U(b) carries a morphism a -+ b iff it carries a morphism F(a) F(b). --f
6.18. This is an obvious fact, actually already used implicitly above, but it should be, once at least, stated explicitly: Full embeddings (extensions, strong embeddings, realizations, resp.) compose to full embeddings (extensions, strong embeddings, realizations, resp.). In the sequel, it will be used without being mentioned, including the cases when used in combination with the facts that every realization is a strong embedding, every strong embedding is an extension, and every extension is a full embedding.
50
Ch. I, PRELIMINARIES
EXERCISES
1. Formulate and prove a statement concerning realization analogous to Theorem 6.7.
0 7.
Two easy but important embeddings
In 4.5 (2) we remarked that every small category is concretizable. We will prove it now using the so-called Cayley-MacLane representation (which is a generalization of the Cayley representation of semigroups from 2.7). In addition, we will prove generalizations of Theorem 2.8 (see also 5.5) and the positive part of the Statement in 2.9.
7.1. The Cayley-MacLane representation. Let A be a small category. For a E objA put u(a)=
u
A(x,a).
xcobj A
For cp: a
-+
b define U(cp): U(a)-+ U ( b ) putting
u(cp)(t)= cp t O
'
Obviously, U(1,) = lLl(,)and U ( $ o 'p) = U ( $ ) U(cp). Thus, a functor U : A -+ Set is obtained. If cp, $: a b are distinct, we have U(cp)(I,) = q =!= rc/ = U($)(l,). Hence, U is faithful. Actually, U is one-to-one, since U ( a )is the only U ( x )containing 1, (keep in mind that the morphism sets are disjoint). Thus, we see that Every small category is concretizable. rn Also, take notice of the fact that a finite category has a forgetful functor with finite values. 0
-+
7.2. As a generalization of 2.8 we have Theorem. For every small category A there is a full embedding of A into A l g ( A ) with A = (l)s.B where B is the set of all morphisrns o f A plus one additional element ( A I g ( A ) is hence the category of algebras with so many unary operations as is the number of morphisms of A plus one). Proof. If M is the set of all morphisms of A, put B = M u {o}, o $ M . Choose, further, two distinct elements u, u outside M and put for cp: a
--$
6, V(cp)(t)
=
U ( q )(t) if x E U(a), l f ( q ) ( <=)
<
for
< = u, z..
51
$7. Two easy embeddings
Now, define an algebraic structure mu = ( ~ i ; of) the ~ ~type ~ A on V ( a )putting:
8:
if
x + y,
1
=
(l)sGB
for 5 : y + a for <: z + a for ( E ( U , U )
t o p
o;(<) = u u
m'(5)
=
i:j
for
<EM
for
5 E (u, u> .
For any cp: a -+ b, V ( q )is an woob-homomorphism.(Really, if p : x -+ y , V ( q )(w;(t)) and o:(V(cp)(5)) have the common value cp 0 5 a, u, u resp. for 5 : y + a, 5 : z + a, 5 E (u, u ) resp. Further, we have V(p) (w;(c)) = oR(V(q)(5)) = u, u respectively if 5 E M , 5 E (u, u ) respectively.) 0
Thus. we can define a functor F : A + Alg(A) putting F ( a ) = (V(a),o"), F ( q ) = (F(b),V ( q ) ,F(a)) for cp: a -+ 6. F is obviously one-to-one. Now, let = (F(b),g, F(a)) be a morphism in Alg(A), i.e. g : V ( a )+ V ( b ) a w"ob-homomorphism.We have g(u) = g(w:(u)) = ot(g(u)) and hence (1)
g(u) = u ,
since u is the only element remaining fixed under have (take a p: x -+ y )
we
g(u) = f(wi(u)) = oi(g(u)) = wbg(z1) = u .
(2)
Further, o:(g(())
=
g(u) = u, so that
g P ( 4 = U(b)
(3) Now, put cp
0:.Consequently,
'
=
g(1J We have, for
5 E U(a),
(the only alternative for o$(cp) lying in U(b)).Thus, g Hence, F is full.
=
V(cp), @ = F(cp).
Remark. An immediate generalization of the proof of 2.8 would give for the case of more than one object in A partial operations only. That is why we paid the efforts with u and u : just to enlarge the underlying sets in such a way that the partial operations can be extended over all the underlying sets without losing the required properties.
.
52
Ch. I , PRELIMINARIES
7.3. Using 7.2 and 6.10. (2) we obtain immediately: Corollary. For every small concrete category ( A , U ) there is a set B suck that there is an extension (in the strong sense) o f ( A , U ) in Alg((lbsB).
7.4. As a generalization of 2.9 we have Theorem. For every small regular concrete category ( A , U ) there I S a realization in Rel(d) with d = (U(C)),,,,,,~.If ( A , U ) is not regular, there cannot he (see 6.5) a realization in a regular ~ u r e g o r y B u t , one has a full faithful F : A -+ Re lid) with V c F = U (where V / A rhf nariirul forqc@ul functor of Rel(d)). Proof. Put F(a) = (U(a),r') where ra = (t-:)cE,,bjA
is defined by
r:={U(t)lt: c-+a}; for cp: a = U(cp
0
b put F(cp) = (F(b),U(cp),F(a)). Since, for 5 : a -+ a, U(cp) U(5) 5) E r,b, U(cp)is a homomorphism and hence a functor -+
0
F: A
-+
Rel(d)
is obtained. Since U is faithful, F is also faithful. Now, let 9 : U ( a )-+ U(b) carry a morphism in Rel(d). We have U(1,) E r', and hence g = g U(1,)c rt. Thus, g = U(cp) for a cp: a -+ b, so that g carries a morphism in A . The statement for regular categories follows by the construction from 6.7. 0
EXERCISES
1. For every small category A there is a full embedding of SetA into a Rel(d). (Hint: For a functor f:A -+ Set consider the set f ( a )x {a)
u
and for every morphism a : a ((x,4 (if(4 a')
(4 4
0 8.
obj A
-+
a' the binary relation R, consisting of all
Tne representation problems
In this paragraph, we are going to reformulate some of the problems discussed in the Introduction using the terminology introduced in this chapter. The reader is advised to confront the new formulations with the related parts of the Introduction, consulting also the remarks from 4 6.
48. The representation problems
'
53
8.1. A category is said to be universal if every concretizable category is fully embeddable into it. A concrete category is said to be extensionally universal if every concrete category has an extension in the strong sense in it.
8.2. Thus, the universal categories are those for which the Abstract Concretizable Category Problem (see Introduction, 4) has a positive solution, the extensionally universal concrete categories are those where the Concrete Category Extension Problem (see Introduction, 5) is positively answered. 8.3. Trivially, an extensionally universal category (to be more pedantic, its abstract part) is universal. The converse, with a little restriction, is true, too. Namely, by 6.10. (1) we have immediately Theorem. Let 2I be universal. Then every (a,U ) with transfer property is extensionally universal. 8.4. A category 9l is said to be algebraic if it is fully embeddable into Alg(d) for a type A (cf. Introduction, 7). 8.5. By 7.2 we have
Proposition. Every small category is algebraic.
8.6. A category 2I is said to be algebraically universal (briefly, alguniversal) if every algebraic category is fully embeddable into it. 8.7. Let us formulate explicitly this immediate consequence of 8.5: Theorem. I f 2I is alg-universal then every small category, every monoid in particular, is fully embeddable into 2I.
8.8. By 6.10. (1) we obtain immediately Theorem. I f 9l is alg-universal and if (a,U ) has the transfer property, then every small concrete category, particularly every concrete monoid, has an extension in the strong sense in ('8,U ) . 8.9. Remarks. (1) Thus, the alg-universal categories have, in particular, positive solution of the Abstract Monoid Problem (see Introduction, 2) and of the Concrete Monoid Extension Problem (see Introduction, 3). (2) The importance of the notion of alg-universal category is by far not exhaused by the fact mentioned in (l), nor by the more general property concerning small categories. Combining the main result of Chapter 111 with some results of Chapter 11, we will see that under a reasonable set theoretic assumption, every concretizable category is algebraic.
54
Ch. 1, PRELIMINARIES
Consequently, under the set theoretic assumption, one can prove extensional universality of (3, U ) by proving that CU is alg-universal (which is done mostly by full embedding of a category about which one already knows that it is alg-universal-such suitable testing categories are discussed in Chapter 11-into 2l). In some cases, we can prove the universality without the set theoretic assumption, but the interesting cases are very often only alg-universal and, in fact, under explicit negation of the set theoretic assumption (see Chapter 11, § 6), not universal. (3) One could formulate a property to be extensionally alg-universal in the obvious way (i.e., every (23, V ) with 23 algebraic has an extension in the category in question). Here (unless the situation is such that the alguniversal coincides with the universal) there is no counterpart of Theorem 8.3. Actually, without any set theoretic assumption, every extensionally alguniversal category is extensionally universal. (4) The transfer property in 8.3 and 8.8 had to be assumed because of the extensions in the strong sense. By 6.9 and 6.8 we see that if one wants just extensions, the assumption may be omitted.
8.10. The question naturally arises as to whether that or another concrete category is universal with respect to strong embeddings. If we put it simply as to whether every concrete category is strongly embeddable into the category in question, we obtain, in case of reasonable categories, cheap negative results simply because a category strongly embeddable into a reasonable one has to be reasonable itself (e.g., regularity is preserved under a strong embedding, and, what is more substantial, so is the following property : for every cardinality a there is only a set of nonisornorphic objects with so large underlying sets, which is shared by all full concrete subcategories of categories S((Fi)i,J)- see 5.8.). In Chapter VII, “reasonable concrete categories” are specified and the problem of universality with respect to strong embeddings restricted to them is dealt with. Surprisingly enough, one sees that it is often the case with a universal everyday life category that it is also universal in the strong sense under discussion. But not always--e.g. it is never the case with an Alg(A).
8.11. To reformulate the group problems, if ‘$ is aIcategory, let us introduce, the symbol Is0 2l for the subcategory of ‘u consisting of all its objects and all its isomorphism.
$8. The representation problems
55
Further, for a concrete category (W,U ) put Zso(%,
u) = (Zso2II, u I Iso2I).
Now, the Abstract Group Problem for ‘u acquires the form: Is every group fully embeddable into Zso%? The Concrete Group Extension Problem for (%, U ) reads: Has every concrete group an extension in Zso(’u, U)? 8.12. Remark. The group problems are often ofan interest at categories ‘ill which are not alg-universal (see e.g. the de Groot’s result on topological spaces: it is certainly understood as a representation in the category of topological spaces and continuous mappings). Sometimes, the best way to solve it (though, at a first sight, it may appear as making the problem worse) is to look for an alg-universal 23 with Zso23 = Iso2I. 8.13. Sometimes, a category is not universal for an obvious reason and a question arises as to whether it is the only reason for it. In particular, we often meet the case that a concrete category is not universal since all constant mappings carry morphisms (graphs with loops, topological. spaces and continuous mappings, etc.) and hence e.g. no non-trivial group is representable there. It is only too natural to ask whether it would be universal if it were not for the constants. This induces the following definition: A concrete category (a, U ) is said to be almost (alg-)universal if for every concretizable (algebraic) category 23 there is a one-to-one functor F: 23 -+ % such that (1) U(F(P))is never a constant mapping, (2) for every CL such that U(a)is not a constant there is a P with F(8) = a. (Such a functor F is said to be almost full. In an analogous sense, the expression “almost strong” is used.) 8.14. Remark. A composition of two non-constant mappings can be a constant one. Hence, the system of the morphisms of 2l with non-constant V ( a )does not, as a rule, form a category. (21, U ) is almost (alg-)universal iff the system of its non-constant morphisms contains a full subsystem which is an (alg-)universal category. (The first statement is an immediate consequence of the second, this one follows from the first and from the existence of universal categories which will be proved in next chapters.)
56
Ch. 1, PRELIMINARIES
5 9.
Bibliographical remarks
Concrete (“transformation”) monoids were studied during the last two decades with growing intensity. Of the many papers let us mention e.g. [HI] and [GH], dealing with other questions than the representation (if we included this, we would have to mention a considerable part of the bibliography of this book). In an implicit form, the notion is as old as modern algebra (the first groups investigated being permutation groups). All the definitions of category theory introduced in 4 3 appeared as early as in Eilenberg-MacLane paper [EM], the foundation stone of the theory. The term of concrete category was originally used for what we call concretizable now. In the last decade, however, the use of the word in connection with fixed forgetful functors prevails. The Cayley-MacLane representation is used already in [EM]. The embedding notions introduced in 9 6 have developed to this explicit form during the past fifteen years, the full embedding being, of course, first (see e.g. [I3]). Implicitly, they are essentially contained in representation problems of various kinds. The extension (under a different name) and the strong embedding are defined in [P5], where one can also find Theorems 7.4, 6.10 and 6.7 in some form or another, and the realization was defined in [P2]. The term “universal category” was sometimes used in a narrower sense (e.g. in [PI] with respect to embeddings of small categories only) sometimes in a broader one (e.g. in [T,] with respect to all categories, not only to concretizable ones.) The notion of algebraic category was introduced by J. R. Isbell in [I3]. Algebraic categories were also referred to as boundable, and what we now call alg-universal was often called binding. The categories S(Fi) were introduced in [HP,], and investigated further in e.g. [P,], [P3] and [KP,].
Chapter II BASIC EMBEDDINGS
5 1.
Three obvious realizations
The proofs in this section are straightforward. The only reason for giving them so explicitly as we do is to help the reader get used to the notions (for the definitions see, if necessary, I $6 1 and 6). Otherwise, they could be left, with perhaps a few hints, as an easy exercise. 1.1. Let A = ( A b ) h e b . , A’ = ( C d ) d e D be types. Write
A
if there is a one-to-one mapping x: B card A b I card Cx(,,).
1.2. Proposition. If A
< A‘,
-+ D
such that, for each b E B,
then Rel(d) is realizable in Rel(d’).
Proof. Take A , A’ and a x from 1.1 and choose, for each b E B , a one-toone mapping &: A , -+ C x ( b ) . For a relational system r = (Yb)beB of the type A on a set X define I. = (?&ED of the type A‘ putting ?x(b)
Fd
=
I
= (‘P ‘P
ib E r b )
Y
0 for d $ x ( B ) .
Now, let f ’ : X + X ’ be rr’-compatible. If q is in I:,,b),u c have cp /+,E I;. hence j ’ 40 ibE r;,,,) and hence f -cp E ?:(b,. Obviously f’is ?,?:,-compatible for t l $ x(B).Thus, j is C’-compatible. On the other hand, let f be ??’-compatible. Take a q ~ r b Since . & is one-to-one, we have a $: C x ( b ) + X with & = cp (this equation determines the $ on & ( & , ) , outside we can define $ arbitrarily). Thus, $ E ?;@) 2
$ 0
-
58
Ch. 11, BASIC EMBEDDINGS
and hence f o t,b E Y L ( b ) . Since 3t is one-to-one, x(b) determines the b and fn'pErb. wehave f o t + b & = Hence, f is rr'-compatible iff it is +-compatible so that the mapping obj Re I ( A ) into obj Re I (A') sending ( X ,r ) to ( X , Y) gives rise to a realization (see I, 6.15). H 1.3. Proposition. I f A = ( A b ) b t B
< A'
=
and
(cd)d,D
either there is a b with A , there is no d with c d or
0 =0, =
then A l g ( A ) is realizable in A l g ( A ' ) .
Proof. Again, take one-to-one mappings &: A , 4 C x ( b ) . For an algebraic structure m = ( m b ) B of the type A on X define cl = (@& an algebraic structure of the type A' on X as follows:
(4
*
@x(b)(q)
= ab('P
ib)
9
(b) for d $ 3t(B) and C d 0 choose a Cd E C d and put @ d ( ' p ) = 'p(Cd), (c) if there are elements d 4 x(B) with C d = 0, there is, according to the assumption, an n E B with A , = 0. Take such a fixed n and put, for all the d's in question, -
cld
Now let f : X
-+
X' be an am'-homomorphism. We have
f(ix(b)('P))
for d$3t(B) with
C d
= f(clb('P
cd
3Lb)) = P b ( f n
'P
Ib)
= &(b)(f
=
@ & ( f o'PI
'P)
9
=I8 =
f(@d((P))
for d $ Z ( B ) with
= a,.
=f('P(cd)) =
( f n'P) (cd)
=0 f(@d)
=f(a,) =
= k&.
Thus, f is an @a'-homomorphism.One the other hand, let f be an @@'-homomorphism. For p: Ab -+ X take a $: C x ( b ) + X with 'p = $ &. We have 0
f('b('P))
A,)) = f(@x(b)(t,b)) = % ( b ) ( f o $) = & ( f 0 $ ' Ab) = ak(fn 'p)
= f(mb(t,b
3
so that f is an mu'-homomorphism. Thus, the correspondence ,sending ( X , m ) to (X,a') gives rise to a realization of A l g ( A ) in A l g ( A ' ) .
59
92. ‘Twoimportant extensions
1.4. Remark. The condition of the zeros in 1.3 is also necessary. If one does not have any zero in a type A , it is very easy to find an algebra ( X , a) of the type A and an aa-homomorphism cp such that ~ ( x4) x for every x. This cannot happen to an algebra ( X , E ) where cl contains a nullary operation. 1.5. As an immediate consequence of I, 1.6 we obtain: Proposition. For d = ( A b ) b e B put A + is realizable in R e I ( A ’).
=
(&
U {&,))beB
Then A i g ( ~ 1 )
EXERCISES
1. (see 1.4) Show explicitly that if A has no zero and A‘ has some, there is no realization of Aig(A) in Alg(d‘).
5 2.
Two important extensions
2.1. Among the concrete categories Rei(A), the category Rei((2)) of sets with binary relations plays a very essential role. Since it is, from a very natural viewpoint, the category of directed graphs (more will be said about it in Chapter IV, now it suffices to realize that a set X with a binary relation R can be viewed as a graph with X the set of vertices and with an arrow going from x to y iff (x,y ) E R ; we have already used this point of view when indicating relations in the Introduction), let 1.1s denote it by Graph.
Thus, the objects of Graph are couples ( X , R ) with R c X x X , the morphisms from ( X , R ) to (X’, R’) are triples ((X‘,I?), f , ( X . R)) with f: X -+ X’ such that (f(x),f ( y ) )E R’ whenever (x,y ) E R, and it is viewed as a concrete category endowed with the natural forgetful functor. 2.2. Theorem. Graph has an extension in Aig(1,l) and in Aig(1, 1, 0).
Proof. Actually, all that one has to see to is, by I, 6.10, to obtain a full embedding. One can take (1,O) for I and (2, 2 x 2) for T in Graph. But the constructed embedding will be an extension quite obviously. For an object ( X ,R ) of Graph put F ( X , R )= = x U R x { I > ~2 x (21, ( ~ ( x , R p ()x, , ~ ) ) ) ~ o b J A k ( l ~ l ) (2 is the set (0, 1)) with the operations a = a ( X , R ) and p = p ( X , R ) defined by
(x (01
OI(x,O) = (0,2),
pfx, 0) = (I, 2)
7
a(x,y, 1) = (x,o), a(i, 2) = (1,2),
P(x, y , 1) = ( y ,
o),
B(i, 2) = (0,2) 7
i =0 9 1 7 i = 091 .
60
Ch. 11, BASIC EMBEDDINGS
The role of multiplying by (0}, (l}, (2} is to make the union disjoint. To simplify the notation, we will disregard them from now on, and work with X u R u 2 as with a disjoint union. So we have F ( X , R) = ) 1, p(x) = 1, = ( X u R u 2, (CI,p)) with CI(X) = 0, a(x, y ) = x, ~ ( i = b(x, y ) = y and B(i) = 0. For a morphism f = ((X’,R’),f,( X ,R)) put
q P ) = ( F ( X ’ ,R’)?9, F ( X , R)) with g(x) = f(x) for x E X, g(x, y ) = (f(x), f ( y ) ) for (x, y ) E X x X and y(i) = i for iE 2. It is easy to make sure that F is a one-to-one functor from Graph into Alg ( A 1 ) . Now let g: X u R u 2 -+ X’ LJ R’ u 2 be both a ( X , R)a(X‘,R‘)- and p ( X , R ) p(X’, R’)-homomorphism. We have P(g(0)) = g(P(0)) = do) 4 4 1 ) ) = s(a(1)) = g(1). 2
Since 0 (1, resp.) is the only element remaining fixed under we obtain g(i) = i for i~ 2 .
(a, resp.),
Now for x E X we have cl(g(x)) = g(ct(x))= g(0) = 0 and hence g(x) E X’. Thus,
g(x)c X‘
and we can define an f : X -+ X’ by f(x) Further, for (x, y ) E R we have
>
=
g(x).
%(X, Y ) = g(+, Y ) ) = g(x) = f(x ) Pg(x, Y ) = g(P(x, Y ) ) = g ( y ) = f ( v ) . 9
This, by the definition of a and p, leaves for g(x, y ) only (f(x), f(y)). Finally, if (x, y ) E R, we have (j(x),f(y)) = g(x, y ) E R‘ and hence f is RR’-compatible. Thus, (F(X‘, R’), 9, F ( X , R )) = F(((X’,R’),f, (X, R))). The statement on Alg (1,1,0) is obtained by the same argument, just specifying, say, (0,2) in 2 x {2> as the nullary operation: it does not move anyway. 2.3. Theorem. Alg(1, 1) is strongly embeddable into Alg(2) and into A k (2,O).
61
$2. Two important extensions
Proof. Construct a functor G: Set -+Set putting G ( , Y ) u 2 x { I ) , G(f)(x,O) = ( f ( x ) ,O), G ( f ) ( i , 1) = (& 1). Now, for an object ( X ,a, p) of Alg (1, 1),put
=
X x (0) u
F ( X , a, 0) = (G(X), o ( X , a, B)) E Q A k ( 2 )
with o
=
o ( X , a, p) defined by
o((x, o), (Y, 0)) = (0,1)
>
o((x, o), (0,l)) = o((0,I), (x, 0)) = (+),
0) ,
w((x, o), (L1)) = o((L I), (40)) = (B(X)> 1) >
o((0,I), (0,l)) = ( L 1 ) 3 4 1 . I)>(L1)) = @ , I ) o((0,I), (1, 1)) = o((4I), (0,l)) 2
= (0,q 3
to write it correctly for a moment at least. From now on, since (0) and (1) just made the union disjoint, we will work with X u 2 -assuming it disjoint-and write the operation o simply by juxtaposition. Thus, we have for x, y E X , xy
0 , Ox
=
oo=
1,
= x0 = a(.),
11 = o ,
1x = x l = p(x),
01 = 1 0 = 0 ,
which certainly gives a better picture of what is going on. For a homomorphism f = ((X’, a‘ pl), fi ( X , CL,B)) put F ( f ) = = (F(X’, a’, pl), G(j”),F ( X , a, p)). Since g = G ( f ) is a homomorphism [ g(xy) = g(0) = 0 = g(x) g(y), g(0x) = .fi.(x)) = .’(f(x)) = Of(X) = g(0)g(x), similarly g( l x ) = g( 1) g(x), and finally g(ij) = i j = g(i) g(j) for i, j E 2 we see easily that F is a one-to-one functor from Alg (1, 1) to Alg (2). Let in turn @ = (F(X’,a‘, pl), g, F ( X , a, p)) be a morphism in Alg (2). We have first g(0) = g P ) = g ( l ) g ( W
1,
<
whatever g( 1) may be, because all g are in 2. If g(0) = 0, we have g( 1) = = g(O0) = 00 = 1, if g(0) = 1 we have g(1) = g(O0) = 11 = 0. Thus, g(0)
= g(O1) = g(0) g(1) =
(either 01 or 10) = 0 and hence also
g(l) = 1.
Now let g(x) = 0 for an x E X . We obtain
Hence, further,
0
=
0 = y(0)
= g(1) g(x) = g(1x) = s(P(x)). =
y(xP(.))
= y(x)g(B(x)) =
00
=
1
62
Ch. 11, BASIC EMBEDDINGS
which is a contradiction. If g(x) = 1, we have
0
=
Og(x) = g(0x) = g(or(x))
which has already been excluded. Thus, g ( X ) c X’ and we can define a mapping f:x + X’ putting f ( x ) = g(x). We have f ( a ( x ) )= g(Ox) = = Og(x) = a‘(f(x)),f ( B ( x ) )= g(1x) = lg(x) = /3’(g(x)), so that f is a homomorphism and we have lj = F((X’, a’, /3’),f,( X , a, b)). We have obviously V o F = G U with U , Vthe natural forgetful functors. The statement on Alg (2,O)is obtained similarly as the one on Alg (1,1,0) in the previous statement, putting a constant on a place fixed anyway, say (0,l). 0
Remark. The operation in F ( X , a, p) is commutative. Thus, actually an embedding into the category of commutative groupoids has been constructed. 2.4. The sum of a type A = (AbfBwritten C A is the sum of the cardinalities of A,, i.e., the cardinality of x (b}.
U&
2.5. Corollary. Graph has an extension in any Alg(A) with
CA 2 2.
Proof. If E d 2 2 and A contains a zero, we have either ( 1 , 1,O) < A
or (2,O) < A ,
otherwise, we have either (1, 1) < A or (2) < A. Applying 1.3 we see that at least one of A l g (1, I), Alg (1, 1, O), Alg(2), Alg (2,O) is realizable in Alg(A). Now it suffices to use 2.2 and 2.3.
2.6. Remark. On the other hand, there is never a strong embedding of Graph into an Alg(d). Indeed, consider a non-void set X and the morphism a = ( ( X , O), Ix, ( X , X x X ) ) in Graph. Suppose we have a strong embedding F : Graph -+ Alg(A), let G: Set -+ Set be the functor with V o F = G U ( U , I/ being the natural forgetful functors). We have V(F(ct))= G(1,) = 1,(,,, so that F(a), being a one-to-one onto homomorphism between algebras, is an isomorphism. Thus, there is a j3 = F(or)-’: F ( X , X x X ) -+ F(X,@). This is a contradiction, since F is full and there is no morphism from ( X , X x X ) to (X,O). 0
$3, Rigid binary relations
63
EXERCISES
1. If ReI(n) is realizable in ReI(rn), then n I m. 2. I f Alg ( n ) is realizable in Alg (m),then IZ I m. 3. Alg ( n ) is not realizable in Re1 (11). 4. For every covariant F : Set Set there is a cardinal number a ( F ) such that there is no strong embedding of Alg ((l)iEa(F)) in Alg (1, 1) with the underlying functor F . If F(2) is finite, there is a finite a(F). 5. If F is an underlying functor of a strong embedding of Rei(n) into Graph, and if there is a monotransformation F + Qk, then k 2 n/2. 6. Generalize the statement of 5. (Hint for all the exercises: Consider the numbers of objects with the same underlying set.) -+
9 3.
Rigid binary relations
In this paragraph we are going to prove that on every set X there is a rigid binary relation, i.e. such a relation R that there is no homomorphism (X, R ) -+ ( X ,R ) except the identity. 3.1. Before going into the construction, let us recall a remark on finite sets from the Introduction. Say on n + 1 = (0,1, ..., n } we can give a rigid relation simply by putting R = {(i, i + 1) i = 0, ..., n - l}. [If a mapping f : n 1 -+ n 1 preserves this relation, we have always ( f ( i ) ,f ( i 1))E R. Thus, for i < n, f ( i ) < n and f ( i + I) = f ( i ) + 1. From the latter, f ( i ) = = f(0) + i for i < n. Confronting it with the former we see that f ( n - 1) = = f(0) n - 1 < n, hence f(0) = 0 and f(i) = i.] On a countable set, say the set ooof all natural numbers, we can take e.g. R = {(i, i 1) i E w o } u {(0,2)}. The trouble begins when we come to uncountable sets. The reader is advised to stop here for a moment and try to find a construction of his own, to see where the trouble is. He could also find a new trick.
+
+
I
+
+
+ I
+
3.2. Construction. Let w be an infinite ordinal. Put X = o 2 and decompose X into disjoint sum X o u XI u X 2 where X o consists of those c r ~ Xwhich are cofinal with oo (oodesignates the first infinite ordinal), X I consists of the other limit ordinals a E X , X 2 consists of the non-limit ordinals cr E X .
64
Ch. 11. BASIC EMBEDDINGS
For every a E X, choose an increasing sequence a2, a39 a49
such that
...,a,, ...
(4
sup a, = a (b) a, = E, + n where En is a limit ordinal. Now, define a relation R on X as follows (we use the expressions a < 0, a 5 B in the usual sense): (i) for p E X,, (a, P)E R iff a = a, for an n 2 2, (ii) for B E X I , (a, p) E R iff a < p and a is a limit ordinal, (iii) for p € X 2 \ j 2 , Q l}, (a, /?)E R iff a 1 = p, if either a = 0 or a = 1, (iv) ( a , 2 ) R~ iff either a = w or a ~ X ~ \ { o+ 1). (v) (a, Q + 1 ) ~R : In what follows the notation of 3.2 will be used without further mentioning. 3.3. Observation. (a, p) E R implies a < B.
+ +
3.4. Observation. If B E X is a limit ordinal and a < fl, then there is a y with a < y and ( y , / 3 ) ~ R .
3.5. From now on, let f : X -+ X be RR-compatible. Lemma. a < p implies f ( a ) < f(B). Consequently, f is one-to-one. Proof. Let fl be the least ordinal such that there exists an a <
s(42 f(PI.
with
a)
We have /3 =t= a + 1 since otherwise (a, E R, hence (.f(a),f(p))E R and hence f ( a ) < f(P) (see 3.3). Thus, by 3.4 there is a y with a < y and (Y, B) E R. Hence, f(.) < f ( Y ) and ( f ( Y ) >f(B)) E R, so that f(.) < f ( Y ) < f(P), which is a contradiction.
3.6. Lemma. For every a, f ( a ) 2 a. Proof. This is a well-known fact on well-order preserving mapping. For the first a such that f ( a ) < a we obtain the immediate contradiction U < f ( a ) < a.
f(f(4)
+
+
3.7. Lemma. f ( w I) = w 1, f ( o ) = 0. Proof follows immediately by 3.5 and 3.6. 3.8. Lemma. f ( x 2 )c x,. Proof. Take an R E X 2 . If a = o + 1, we have f ( a ) = w 1 E X , . Otherwise, (a,w + 1 ) R~ and hence ( f ( a ) ,o 1) E R, so that f ( a )E X 2 by 3.2 (v).
+
+
65
$3. Rigid binary relations
3.9. Lemma. f ( x , ) = x,, f ( x 1 )= X I . Proof. Take an a € X O . We have (f(a,),f(a))~R. BY 3.8, f ( a , ) ~ X ~ . Thus, there are (see 3.5) infinitely many y E X 2 with (y,f ( a ) )E R. Hence, by the definition of R, and since, by 3.5 and 3.7, f ( a ) $. o 1, f ( a ) E X , . Take an CI E X I . We have ( y , a) for infinitely many y E X,, hence, by the first statement and 3.5, ( 6 , f ( a ) )E R for infinitely many 6 E X,. Thus,
+
f(U)E&.
+
+
3.10. Lemma. For n < oO,f ( n ) = n and f ( a n) = f ( a ) n. Proof. We have (f(O), f(l)), (f(o), f(2)), (f(l), f(2)) E R, and, by 3.8 and 3.7, f(2) 6 X,\{o + 1). Thus, by the definition of R, f(2) = 2, by 3.5, further, f ( 0 ) = 0 and f(1) = 1. Now, it suffices to prove that for every @€X f(a 1) = f(a) 1 .
+
+
+
+
For f ( a + 1) = 2 and f(a 1) = o 1 we already know that it is true. By 3.8, f(a 1 ) X ~2 . If f ( a + 1) = 2, o 1, there is only one y with (y,f ( a + 1))E R, namely that for which f ( a 1)= y 1. Since (a, a 1)E R, we have (f(a), f ( a + 1))E R and the statement follows.
+
+ +
+
3.11. Lemma. f(ct,,) = f(a),,. Proof. By 3.10, (f(a,,),f(a)) = ( f ( E , ) n, f(a)) E R 2 3 . 9 , ordinal. Thus, by 3.2 (i), f(a,,) = f ( ~ )(and , , f(%) = f(a),).
+
+
f(&) is a limit
3.12. Theorem. There exists a rigid binary relation on any set. Moreover, there exists such that is a subrelation of a well-ordering of the set. Proof. For the case of a finite set see 3.1. To prove the statement for infinite sets, it is enough to show that ( X , R ) above is rigid (the cardinality of X unrestricted). Let f : X + X be RR-compatible. Iff is not the identity mapping, denote by a the least ordinal such that f ( a ) 4= a. Thus, by 3.6, f ( a ) > a. By 3.5, a
< f ( a ) < f " a ) < ... < p ( a ) < ...
(where f' = f o f , f"" = fcf"). By 3.7, for every n fn(a) < o SO that we have a /3 = supf"(a) in X . Obviously, P E X , . For every k there is an n with B k < f f l ( a )< B . Thus, by 3.11, f(p)k = f(Bk)< f f l + ' ( a ) < p, so that f(P) = /? (see also 3.6). Hence also ,f(pn)= Pn.For a sufficiently large n, a < fin. Thus, f ( a ) < pn and by induction f ' ( a ) < fin, so that p I P,, < P.
*
66
Ch. 11, BASIC EMBEDDINGS
3.13. Before formulating a simple consequence of 3.12 let us formulate a property which is very common among current concrete categories ('3,U). For the time being, we will not give it a name. It is the following: (1) For every cardinal x there is cardinal A(%) such that there are at most A(%) non-isomorphic objects with the cardinality of the underlying set x. The reader can see easily that (1)is satisfied e.g. in every category realizable in an S((Fi)iEJ). 3.14. We have the following Theorem. Let (a, U ) be a concrete category satisfying (1). If there exists a full embedding of Graph into 'u, then there is,for every cardinal x, a rigid object a with card U(a)2 x. Proof. Let F : Graph + 'u be a full embedding. Denote the R from 3.2 by R(w). Since F is full, the objects F((w + 2, R(o)))with o cardinals form a proper class of non-isomorphic rigid objects. Now, the statement follows immediately by (1). EXERCISES
1. Without using the constructions of $2, construct a rigid binary operation and a rigid algebraic structure of the type (1, 1) on finite and countable sets. 2. How many non-isomorphic rigid binary relations are there on a three-point set?
0 4.
Rigid symmetric binary relations
In this paragraph we will show that on every infinite set there is a rigid symmetric binary relation and that for every finite set there is at least so large a finite set with a rigid symmetric relation. We situate this topic here although we will not be needing it for some time yet; whereas the result of $ 3 will be needed urgently right in the next section, the existence of a rigid symmetric relation will not be used until Chapter IV. But we think that the question is very naturally related to the non-symmetric case, in particular when we consider binary relations as graphs: symmetric graphs are of a particular interest among them. Also, it is an opportunity to present an instructive procedure.
67
$4. Rigid symmetric rclntions
4.1. To simplify the notation, we will write x R y and say that x is joined with y if ( x , y ) E R. Throughout this paragraph, a relation will always be a binary symmetric antireflexive one (ie., x R y implies y R x , and for no x x R x ) . By saying that a set of couples of elements generates a relation we will mean that the relation consists of those couples and of the reverted ones. Let ( X , R ) be a set with a relation. A sequence XO,X I ,
...)X n -
1
is said to be a cycle (in ( X , R ) ) of length n if x i R x i + l for 0 I i < n -2 and xn- R x o . It is said to be a proper cycle if n 2 3 and xi ==! x j for i =i= j. We say that a cycle xo,x l , ..., x , - ~ contains a cycle y o , y l , ..., y k - l if {y,IiEk) c {xilien}. 4.2. Lemma. Every cycle of an odd length n 2 3 contains a proper cycle. Proof by induction: if n = 3, a cycle x o , x l , x 2 is proper itself (the relation is assumed antireflexive!). Now, let the statement hold for cycles of lengths 2p 1, p < q. Let x o , x l , ..., x2q be a cycle. If xi x j for ii j it is proper itself. If there are i < j with x i = x j we have j - i 2 2 (antireflexivity!) and the following two cycles: 2q + 1 - j + i, X O , x l , ..., xi, x j +1, ..., x2q of length Xi,Xi+l,*..,Xj-l of length j - i. Since the sum of the two lengths is 2q + 1, one of them has an odd length n, 3 5 n < 2q 1 and hence contains a proper cycle.
+
+
+
4.3. Lemma. On the set 2n + 1 = (0, 1, ..., 2n) generate a relation R = R2n+l by the couples (0, l ) ,(1,2),..., (2n - 1, 2n) and (2n,0). Let f : n n be RR-compatible. Then --f
+ i,
either f (i) = f (0) or
f(i)= f ( 0 )- i .
+
(The addition and subtraction are understood modulo 2n 1.) Proof. The operations will be understood modulo 2n 1, thus e.g. 0 - 1 = 2n, 2n 2 = 1 etc. First, since i R j iff either j = i 1 or j = i - 1, we see easily that the only proper cycles are
+
+
k, k + 1, ..., k + 2n k, k - 1, ..., k - 2n
1
(k
=
0, 1, ..., 2n)
+
68
Ch. 11, BASIC EMBEDDINGS
Now, since 0, 1, ..., 2n is a cycle and since f is compatible,
f(O)> f ( $ . .>f(2n) (2) is a cycle. By 4.2 it contains a proper one. But all the proper ones have the length 2n + 1 and hence (2) itself is proper and has to coincide with some of the cycles listed under (1) which gives the statement. '
-
1.4. Corollary. Denote by cp: the maping giuen by f ( i ) = k + i, by cpk rhe one giuen by f ( i ) = k - i. Let c1: (C,S) + (2n + 1, R,,, CI': (C', S') + 1 3 1 1, R2,,+1 ) be isomorphisms. Then a compatible mapping p: (C,S) + c p l 0 CI or of the form a'-' 0 cp; 0 a. + (C', S') is either of the form a'-' I n particular, such a mapping is determined by its values in two distinct points. Proof. a' 0 fi 0 a-' is RR-compatible and hence it is either cp: or c.p;
+
0
4.5. Construction. Let a, b be two distinct elements, a , b $ 1 1 = (0, 1, ..., lo). On the set Y = 11 u (a, b } generate a relation S by the couples (0, I)>(1>2), (2,3), (3,4), (4,5), (5,6), (6, o), (5,7)?(7, 8), (8, b), (b,2), (7,9), (9, lo), (10, a), (a,0). =
The situation is indicated in Fig. 2.1. a
(the relation is indicated by segments). Further, put Co = (0, 1, 2, 3, 4, 5, 61, C1 C2 = (0, 6, 5, 7 , 9, 10, a}.
=
(2, 3, 4, 5, 7, 8, b } ,
4.6. Observation. (Y, S) contains no cycles of an odd length less than or equal to 7 except those running through a whole C i (i = 0,1,2).
$4. Rigid symmetric relations
4.7. Proposition.
69
(I:S) is rigid.
Proof. Let f : Y + Y be compatible. Hence, it sends cycles to cycles, and hence, by 4.2 and 4.6,
(1) f(ci)= Cj(i) (2) the restriction of f on Ci is always one-to-one. 7
Now, j(0) = j(1) implies f(6) = f ( 7 ) (because of the common part of C, and C,) in contradiction to (2). Similarly, j(0) = j ( 2 ) leads to the contradiction f(4) = f ( 7 ) , and j(1) = j(2) to f(4) = f(6). Thus, j : 3 + 3 is a one-to-one mapping, and hence, by (1) and (2), f is one-to-one. Now, sincef’is compatible and one-to-one, each of f ( O ) , f ( 2 ) , f ( 5 )and f ( 7 ) has to be joined with at least three elements. Thus, (3
1
f({0,2,5,7))
=
{032,537)
Of those four elements, only 5 and 7 are joined, and hence Further, only 0 and 5 are both joined with one third element. Thus, (5)
f ( { Q 5 ) ) = {0,5)
By (5) and (4), f ( 5 ) = 5; consequently, by (5), f ( 0 ) = 0 and by (4), f ( 7 ) = 7. Finally, by (3), f ( 2 ) = 2. Now, we see that f ( C i ) = Ci and we already have, in every C ithree elements fixed. Since the relation R restricted on Ci gives a graph isomorphic to (7, R7),f is the identity by 4.4. 4.8. Construction. Take a rigid (A‘,R) from 0 3 (for the finite case take the one from 3.1) and the symmetric (I:S) from 4.5. Put = X u (11 x R ) and generate a symmetric relation on by the couples ((i, x, y), ( j , x, y ) ) with i, j E 11 and (i,j) E S , ((0,.Y. y ) , X)’ ((10,.Y, y), x), ((2. %!.)-!.). ((8,%!.)> !*). (Thus, (X, S) is obtained taking for every (x, y ) E R a copy of (Y, s) and ) ~ to ~ x, the b to y.) glueing the u of the ( ~ , y one
70
Ch. 11, BASlC EMBEDDINGS
Denote, further,
co(x,Y ) = co x ( ( X ? Y ) } > Cl(X>Y ) = ( C l \ ( q ) x ((x,Y ) } u ( Y ) CZ(X,Y ) = (c2\(a)) x ( ( x ,Y ) } u c ( x ,Y ) = co(x,Y ) u Cl(X,Y ) u CZ(T Y ) . 7
(4
9
(Thus, C,(x,y ) is the C,-part of the ( x ,y)’h copy C(x,y ) of
(X,q
( Y S).)
4.9. Observation. contains no cycles of an odd length less than or equal to 7 except those running through a whole C,(x,y). (Really, by 4.2 it suffices to show that there is no other proper cycle. A proper cycle which is not contained in a C ( x , y ) and has a point in 11 x { ( x ,y ) } has to leave C(x,y ) in one of the points x , y. Since it is proper, it has to come back through the other one. Now, to get from x to y it has to run through a least three further points (see Fig. 2.1). Since this is the case in every 11 x {(x’,y’)} it meets, it has to have the length 8 at least.) W 4.10. Theorem. For everyfinite set there is a largerfinite set with a rigid symmetric relation. On every infinite set there exists a rigid symmetric relation.
Proof. It suffices to prove that (X, q is rigid. Let f : + X be compatible. Take an ( x ,y ) E R. Since f sends cycles to cycles, we see by 4.9
w
and 4.2 that
f(Ci(x9Y ) ) = Cj(r,x,y)(Xu yi) .
But since C,(x, y ) and C,(x, y ) intersect in at least two points while C,(x, y ) and Cq(x’,y’) in at most one if ( x ,y ) (x’,y’), and sincefhas to be one-to-one on every Ci(x,y), necessarily
+
(xo, Y o ) = ( X I ? Yl) = (x2,Y z ) = (g(x3 Y ) ?h(x, Y ) ) .
Thus. t(c(.,r)) = C(g(x, Y), h(x, Y)). By 4.7 (and by the obvious isomorphism of (Y, S) and (C(u, v), S n (C(u,v) x C(u, u)) ), (1)
(2)
f ( i , x , y ) = (i,g(x, y), h(x, Y ) )
for i E 11
f ( x )= g(x3 Y), f ( Y ) = h(x, Y ) .
The equations (2) used back in (1) give further
(3)
f ( k x,Y ) = (4 f ( x ) ,f(v)).
Since, in the ( X ,R ) used, for every x E X there is a y such that either ( x ,y ) E R or ( y , X ) E R, the equations (2) assert that f ( X ) = X . Define
$5. Graph IS alg-unicersal
71
f ’ : X -+ X by putting f ’ ( x )= f ( x ) .By (3), (x,y ) E R implies ( f ’ ( x ) , f ’ ( y ) ) ~ R , so that f ‘ is compatible. Since (X, R ) is rigid, f ‘ is the identity. Thus,
and by (3) also f ( i ,
X,
y)
=
for X E X ,
x
f(x)=
rn
( i , X, V ) .
4.11. Remark. Compare the construction in 4.8 with the replacing of arrows by a topological space in the de Groot’s construction mentioned in Introduction. 1. EXERCISES
In the following exercises, ( X , R ) is a fivite symmetric graph with card X > 1 and j(x) is the number of vertices that x is joined with. 1. If (X, R ) is rigid, j ( x ) > 1 for every x E X . 2. If (X, R ) is rigid, j(x) > 2 for some x E X . 3. If ( X , R ) is rigid, there is no vertex x with j(.x) = cardX - 2. 4. If ( A . R ) is rigid and i(x) = card X - 1, the full subgraph generated by X ,( A ; is also rigid. 5. There IS no rigid (X, R) with 2 I cardX 5 5. 6. Let ( X , R ) be rigid and let j(x) = cardX - 3. Let y, z be the two vertices which are not joined with x. Then y and z are joined. 7. There is no rigid (X, R ) with cardX = 6.
3 5.
Graph is alg-universal. Consequences
5.1. Proposition. For every type A = ( & ) b S B there is n set c and a strong ernbellding of Rel(d) into Re I ((2)csc),(The somewhat confusing symbol ( 2 ) c E ~ constitutes, of course, the type (A:)cecwith A: = 2 for every c E C.) Proof. First, choose a non-void set M disjoint with B, such that card& i x b : A, + M.
I card M for all b, and one-to-one mappings Put = B v M.
c
For an object (X, ( rb) B) of Re I ( A ) define an object F(X, ( Y b ) B )
of Rel((2)csc)with (LY, E i,,,for r n E M iff to X ( l ? l ) , (U,P ) € 7, for b E B
a)
=
(XM> (fc)cEc)
is the constant mapping sending the whole M
iff
L 2
Xb E r b .
.
72
Ch. 11, BASIC EMBEDDINGS
For f = ((X’, r’),f, ( X , r)) put F(P) = (F(X’, r‘), Q M ( ~ ) ,F ( X , r)) where Q M : Set -+ Set is defined by Q M ( X )= XM, Q M ( f )(l)= f~ t. Obviously, Q M ( f )is PP-compatible and we have a one-to-one functor F: Re I ( A ) + Re I((2),) satisfying Vo F = Q M 0 U where U , V are the natural forgetful functors. Thus, it suffices to prove that F is full. Let g : XM -+ X ‘ M be an PF‘compatible mapping. For convenience, denote by yx the constant mapping of M into X or X’ sending everything to x. We have (yx, y x ) E,P, hence (g(y,), g(yx))E Pk, so that g(yx) is a constant mapping. Thus, we can define an f : X -+ X ’ by
S(YX)=
YfCX).
Now take a < € X M and an mE M. We have ((.;52,,3,))~F,,, and hence ( g ( t ) ,Yf(,(,),) E so t h a t (I(<) (m) = f ( l ( m ) ) .Thus, 9 = Q.\,(,f). q, = r. We have Finally, for an X E r,, choose a 5: M --t X with (t,t) E Fb, hence (j.22. ,) <) E Pb, and hence . f o c1 = J’ :C zh E r;. Thus, f is rr’-compatible.
c,
<
0
5.2. Again (now with a general binary relation) we will write x R y for (x,y ) E R and say that a sequence XO,..,X,-l
is a cycle if x O R x l ,x 1R x Z ,..., x,- Rxo. One speaks of cycles of the length n, or simply of n-cycles. We will also write x R y R z for x R y and y R z , etc. In the sequel, the preservation of cycles by compatible mappings will play an important role again. This time, it is even a stronger tool than it is in the symmetric case, where one has the cycles arising from symmetry. Here, one can, so to say, put cycles into constructions only in places one really needs them. 5.3. Theorem. For every type A there is a strong embedding of Re I ( A ) into Graph. Proof. By 5.1, it suffices to prove that there is a strong embedding of Re I ((2),) into Graph. Define a function G: Set -+ Set putting
G(X)
= 4 x (0)uc x { l } u X x ( 2 ) u x x (recall that 4 = {0,1,2,3))
G ( f ) ( l ,i )
=
(t7i )
for i = 0, 1 ,
,G(f) (x7 2) = ( f ( x ) 2) , G(f)( X >Y , 3) = ( f ( 4f ( Y ) >3 ) . 2
xx
{3),
95. Graph is alg-universal
For an object
8 = ( X , (Rc)cGc)of
73
Re I ((2),)) put
w),
F ( 8 ) = (G(X),
where R will be described after simplifying the notation. Namely, the multiplying by (i} -as in several constructions before-just makes the summands disjoint. In the sequel we will omit it and work with 4 u C u u X u X x X as with the underlying set of F ( 2 ) (and we will take it for a disjoint union). Now, R is defined as follows: Take a rigid binary relation Q on C without cycles (by 3.12 we can even take a subrelation of a well-ordering). Put (I) OR^ iff 5 = 1 or 5 = 3 or < E X , (2) 1 R t iff 5 = 2 or ~ E C , (3) 2 R t iff 5 = 0, (4) 3 R l iff 5 = 0, (5) for C E C , c w t iff 5 = 2 or (t = ~ E and C ced), (6) for X E X , x R t iff t = 1 or l = ( x , y ) ~ Xx X, (7) for ( x , ~ ) E xx X, ( x , y ) R c iff 5 = y or (c = C E C and xR,y). This formal description becomes lucid in Fig. 2.2. It is easy to see that if a mapping f : X + X’ is (R& (R&compatible, G ( f ) is RR’-compatible.
X
C
U
Fig. 2.2
Thus, the F defined for objects so far can be made to a functor Re i((2)~)+ +Graph defining the values in morphisms according to the formula V o F = G 0 U (with U , V the natural forgetful functors). Obviously, F is one-to-one. Now, let g: G ( X ) + G(X’) be f@’-compatible. We want to prove that there is an (R,),-(R~),-compatible f : X + X’ such that g = G(f).
74
Ch. 11. BASIC EMBEDDlNGS
First, let us inventarize short cycles in ( G ( X ) , R ) .Put
1
M n ( l )= { q there exist Cyl, ..., tnp1 with
l R l 1 f?
... & - R y }
Evidently, 5 lies within a cycle of length n iff 5 E M n ( [ )iff there is an y with ( R q and t E Mnpl(q). Now, we see that: Every cycle containing an element of C has the length of at least 4. Really, it cannot be contained in C by the choice of e. It can leave C only via 2 and we have M,(2) = (0), M2(2) = (1,3} u X . 2-cycles: We have M2(1) c (0,2} u C, M 2 ( 2 )= (1.3) u X , M2((x,y))c c {l, 2) u C u { ( y ,z ) z E X } . Thus, the only candidates for elements of of 2-cycles are 0 , 3 , x and (x,x) and we see easily that the 2-cycles are exactly
I
3-cycles: We have M,(3) = (0, 1,2} u C u X x X , M3(x) c (0,1,2>u u C u (X x X ) , M3((x,y ) ) c {0,2>u C u X . Thus, the only 3-cycles are
(0, 1,2)! (1,2,0)
and (2,0, 1).
Thus, we have either g((O,3}) = {O, 3 ) or g({O, 3 ) ) = {x,(x, x)), and g((0, 1,2)) = (0, 1,2f. This leaves only one possible value for g(O), namely 0. Immediately, we obtain g(3) = 3 , g(1) = 1
and
g(2) = 2 .
For c E C, 1R c R 2 and hence 1Rrg(c)R’2, so that, by definition of g(c) E C. Thus, since Q is rigid, we have
R‘,
for every c E C , g(c) = c . Further, for x E X, O R x R 1, hence OR’g(x)R’ 1 and hence g(x)E X’. Define f : X 4 X’ by f ( x ) = g(x). We have, for (x, y) E X x X , x R ( x , y ) R y ,and hence f ( x ) R ’ g ( x , y ) R ’ f ( y ) . Thus, g(x, y ) = (f(x), f(y)). Finally, if xR,y, we have (x, y ) E I?,; hence (f(x), f ( y ) )E I?, which implies f(x)R:j(y). Thus, g = G ( f ) and f’ is (R& (R&-compatible. rn 5.4. Corollary. A category (LL is alg-universal iff there is a full embedding of Graph into (LL. Thus, e.q. Rel(d) and Alg ( A ) with E d 2 2 are alguniversal, particularlj Alg ( I , 1) and Alg(2). Proof. Follows by 5.3, 1.5.
75
$5. Graph is alg-iinivcrsal
5.5. Theorem. Alg ( A ) is alg-universal iff CA 2 2. Proof. If C A = 0, Alg(A) is obviously not alg-universal. Thus, by 5.4, it suffices to prove that Alg(A) with c d = 1 is never alg-universal. Suppose some such Alg(A) ever is. Then, in particular, every group G
is representable as an endomorphism group there, i.e., for every group G there is a set X , a subset A . c X and a mapping a: X X such that G is isomorphic to E = E ( X , A,, E) -+
=((P:X
+ X l p ( a ) = a for U E A , and
E(P=(P~)
(with composition). Put A
=
(x E X 1 ( ~ ( x= ) x for every
(1)
(P E E ) .
We have
a(A) c A .
Really, if x E A , we have for (P E E cp(a(x))= acp(x)= a(.). Further, (2) if E is a group, x $ A and ~(x) E A , then there is an X =!= x, X $ A such that ~ ( x=) ~(2). Indeed, since x $ A , there is a ( P E E with ( ~ ( x ) x. We have ap(x) = = cpa(x) = a(x), and cp(x)$A, since, otherwise, x = 'p-l'p(x) = ~ ( x ) In . fact, we have
+
if E is a group, then a(X\A)
(3)
Suppose the contrary. Fix an a E A such that a the identity mapping):
c X\A =
(ao is
Y
=
1%
{XEX
=
n(x), x " ( x ) $ A and
a ( . )
. for an x $A and put
E"+'(x) = u } .
By (l), the natural numbers n(x) from the definition of Y are uniquely determined by x. Now, either a ( X ) 3
consequently there is a sequence
xo,x1, x2, ..' such that a(xo)= a, .(xi) = xi-l for i > 0 and consequently the mapping $: X -+ X defined by $(y) = x,(~,)for Y E r, $(z) = z otherwise, commutes with E and preserves the points of A, and hence it is in E , which is a contradiction, since it is not invertible by (2) (we have $(Xo) = xg = $(xo)). Or, there is a Y E Y\a(X). For such a y define a natural number r(y) as the least k such that a"(y) = E ( Z ) for some z E r, z ak-' ( y ) (such an r(y) exists by (2) ). Now take a y with the least possible r(y), and a z E Y with + & -1( y ) and a(.) = ~ ' ( ~ ) (Then, y ) . there are elements zo, . .., zr(y) -1 =z
+
76
Ch. 11, BASIC EMBEDDINGS
such that a(zi) = zifl. Put $(ai(y))= zi for i = 0, ..., r(y) - 1, $(u) = u otherwise. We see easily that $ E E, in contradiction with the assumption that E is a group. Thus, (3) is proved. Now, let E = E ( X , An, a) be a group. Define p : X -+ X by p(x) = a(.) for x $ A , p ( x ) = x for X E A .By (1) and (3), j 3 o a = a o p and hence P E E . Furthermore, for any cp E E , cp j3 = p cp (because of q-', cp(X\A) c c (X\A). Now, take a non-trivial group G with no element but the identity commuting with all the others ( e g the group of all permutations of a threepoint set). If this is isomorphic to E, the mapping p necessarily is the identity. Thus, a(.) = x for x E X\A and hence E contains all the mappings $ such that $(X\A) c X\A and $(a) = a for a E A , which is a contradiction. 0
0
H Remark. To be more explicit, we proved that: Whenever a group G is fully embeddable into an Alg(d) with Ed = 1, there is a gnEG, distinct from the unit, such that, for all g E G, g 90 = go g. A finer analysis of the E ( X , Ao, a) in [HP,] shows that actually if it is a group, it is either the infinite cyclic group, or a product of at most countable number of finite cyclic groups with orders which mutually do not divide each other. 0
0
5.6. Corollary. A category 2I is algebraic iffthere is a full embedding of 2I into any of the following categories Graph, Alg (1, l), Alg (2).
Remark. The categories mentioned are, so to say, the simplest alguniversal categories. We shall see in Chapter 3 that under a set theoretic assumption they are even universal. Proving universality of categories goes often over an embedding of some of them, especially one of the first two, into the category in question. 5.7. In particular, we have the following statement (compare with I, 7.2). Every monoid is isomorphic to the endomorphism monoid of an algebra with two unary operations. W 5.8. As a consequknce of I, 7.4 and 5.3 we obtain immediately
Theorem. Every regular small concrete category is strongly embeddable into Graph. H
$6. S(P-)into
77
Graph
EXERCISES
1. If A is a small category, SetA is algebraic. (Hint: Embed SetA into Re1((2)iemorphA)
.)
2. Show that every small category is algebraic using 1. (Hint: see E I ,
3.31.)
5 6.
Assumption (M) and strong embedding of S ( P - ) into Graph
6.1. A filter on a set X is a set 9 of subsets of X such that (1) 8 # 9, ~ A c B c X then B E F , (4) if A E ~ (2) X E ~ ( 3, ) if A E and and BE^ then A n B E 4 . An ultrafilter on X is a filter for which, in addition (5) if A u B E 9, then either A E or ~B E E Let x be an element of X . The set ( A ~ E cAX } is obviously an ultrafilter. The ultrafilters of this form are said to be trivial. More about filters and ultrafilters will be said in Appendix B.
I
6.2. Recall the contravariant functor P - : Set .+ Set from I, 3.14. F. It was defined by P - ( X ) = { A A c X } , P - ( f ) ( A )= f - ' ( A ) .
I
6.3. Lemma. Let X , X ' be sets, g: P - ( X ' ) + P - ( X ) a mapping. Then the following two statements are equivalent: (i) There is an f: X + X ' with g = P - ( f ) . (ii) For every trivial ultrafilter 9 c P - ( x )on X , g - ' ( 8 ) is a triuial ultrafilter. Proof, We have ( P - ( f ) ) - l ( { A l x ~ c A X } ) = { B c X ' l x ~ f - ~ ( B=) ) = { B j ( x )E B c X ' } , so that (i) (ii). On the other hand, let g satisfy (ii). If X is void, P - ( X ) is a one-point set and therefore g is the constant mapping P - ( f ) obtained from the void mapping f : 8 + X'. Now let X be non-void. By ( 2 ) we have for every x E X an j ( x )E X ' such that
I
(1)
I
I
g - ' ( { A x E A c X } ) = { B f ( x )E B c X ' ) .
Thus a mapping j : X
.+
X ' is defined. By (l),we have for B c X '
f ( x ) B~
iff g ( B ) 3 x
so that g(B) = {x I f ( x ) ~B } = f - ' ( B ) = P-(f)(B). Remark. Obviously, J' is uniquely determined by P - ( f ) . 6.4. The assumption (M). By (4) in 6.1 the intersection of afinite number of elements of a filter is still in the filter. In a trivial ultrafilter 9, the inter-
78
Ch 11. BASIC EMBEDDINGS
section of any number of its elements is in F. It is very easy to see that among ultrafilters only the trivial ones have this property. Now, between “any finite” and ‘‘any“ there is a long distance and it is only too natural to put the question as to whether there is a non-trivial ultrafilter closed with respect to intersection of any a of its elements, where a is an infinite cardinal. The answer is, perhaps, a bit surprising, namely, that it is quite reasonable to assume that there is none. That is, such an assumption does not lead to a contradiction with the axioms of set theory (we have in mind the Godel-Bernays system). In Appendix B the substance of this fact will be shown, namely the fact that a set X on which a non-trivial ultrafilter already with countable intersections could be defined would have to be tremendously large: so large that it could not be successively constructed from smaller ones using powers and unions. We will use, in some places, a weakened form of the assumption of nonexistence of such ultrafilters, namely: (M) There is a cardinal number SI such that every ultrafilter closed with respect to intersections of a of its elements is trivial. To indicate that in a proof of a statement (M) is used we will use the expression “under (M)” instead of “in Godel-Bernays set theory with (M)”. 6.5. Let (a, U ) , (23, V ) be concrete categories. A strong co-embedding of (a, U ) into (23, V ) is a contraaariant full one-to-one functor F : 2I + 23 such that there exists a contravariant G: Set + Set with V o F = G U . It is an auxiliary, but a very useful notion. 0
6.6. Obviously, a composition of two strong co-embeddings is a strong embedding, a composition of a strong co-embedding with a strong embedding is a strong co-embedding. 6.7. Construction. Let X be a set, a a cardinal number. On P - ( X ) define a unary relation ro(X),a binary relation r l ( X )and an (a u (a))-ary relation r z ( X ) putting ro(X) = (0) r1(x)= ( ( A ,X\A) A = x}, r z ( ~ =) ( ( A ~ )A,~ =~X ,~ ~A , ( =~ ~ . 2
I
I
n
BEa
6.8. Lemma. Let (M)hold with the cardinul a. Thenfor g : P - ( X ’ ) + P - ( X ) the following two statements are equivalent: (i) g = p - ( f ) for an f : x + x‘. (ii) g is r,(X’)r,(X)-compatiblefor i = 0, I, 2.
79
k6, S ( P - ) into Graph
Proof. Obviously, (i) * (ii), since P - ( J ' ) preserves the void set, complement and intersections. Now let g be compatible. Hence, by yo, g(@)= 8.
(1) By r1 obviously
(2) BY
#\A)
r2,
(3)
g(
=
X\q(A).
nA,) n g(As),in particular g(A n B ) =
BET
=
g(A) n g(B).
BE7
By (2) and (3) we obtain immediately (4)
S(A u B ) = g(X'\(X'\4 n (X'\B))
By (3), since A c B iff A u B
=
A cB
(5)
=
g(A)u g(B)
'
A,
* g(A) c g(B).
Now, take a trivial ultrafilter
F ={ A I X E A E X ) .
I
We have g - ' ( F ) = {A' c X' x E g(A')}. By (l), @ $ g - ' ( FBy ) . (2), g ( X ' ) = X 3x, so that X ' € g - ' ( F ) . By (5), if A' E g- '(9) and B' 3 A', also B' E g - ' ( F ) . Let A; E g-'(F) for E a. By (3) we obtain g ( 0 A ; ) = n g ( A ; )3 X, SO that E g-'(F). Finally, a
a
a
let A' u B' be in g - ' ( 9 ) . Hence, g(A')u g(B') = g(A' u B') 3 x, so that either A' or B' is in gP1(9). Thus, g - ' ( F ) is an ultrafilter closed with respect to intersections of a elements and hence it is trivial by (M). NOW, the statement follows by 6.3.
6.9. Proposition. Under (M), there exists a strong co-embedding of Graph into Graph. Proof. By 5.3 and 6.6, it suffices to find a strong co-embedding of Graph into a R e l ( A ) . We will take A = (Ai)iE4with A. = 1, A1 = 2, A2 = M U( a } ( a is the cardinal from (M)) and A 3 = 2. For (X, R ) E obj Graph define
F ( X , R ) = ( P - ( x ) , ( R i ( X ,R))ic*) E obj R e 1 ( A ) putting Ri(X,R ) = ri(X) from 6.7 for i E 3, (A, B ) E R,(X, R )
iff
( A x B) nR
=
0.
80
Ch. 11, BASIC EMBEDDINGS
Take ( X , R),(X’, R’) and a mapping f : X + X’. It holds that (1) P-(f) is R3(X’,R’)R 3 ( X ,R)-compatible iff f is RR’-compatible. Indeed, let f be compatible, (A, B ) € R 3 ( X ’ , R ’ ) .If there is an ( x , y ) e E (f - ‘ ( A )x f - ‘(B))n R, we have a contradiction (f(x),f ( y ) )E ( A x B ) nR‘. Thus, (P-(f) ( A ) ,P-(f) ( B ) )E R , ( X , R). On the other hand, let P-(f) be compatible. Let there be an (x,y ) E R with ( f ( x ) ,f ( y ) ) $ R’. Then ( ( f ( x ) )(, f ( Y ) } )E R3, hence f-l({f(x)}) x f -‘ ( ( f ( Y ) ) ) n R = 8 which is a contradiction, since the set contains ( x ,y). Now, by definition of R i ( X ,R), by lemma 6.8 and by (1) we see immediately that the condition Vo F = = P - o U ( U , V are the natural forgetful functors) with the definition of F ( X , R ) above determines a contravariant full functor Graph -+ Re I ( A ) . Obviously, it is one-to-one. rn 6.10. Theorem. Under (M) there exists a strong embedding of S(P-) into Graph. Proof. By 6.9, 5.3 and 6.6 it suffices to find a strong co-embedding into a Rel(A). This time, we take A = (Ai)ie4 with A . = A 3 = 1, A l = 2, A , = c( u { a } .
For an object ( X , r ) of S ( P - ) put F ( X , r ) = ( P - ( X ) , (Ri(X, r))itd) with R i ( X , r ) = r i ( X ) from 6.7 again for i E 3 , and R 3 ( X , r )= r. Now R3(X’,r ’ ) R,(X, r)-compatibility of P - ( f ) means the same as F(f) (r’) c r , so that we see immediately that the condition V OF = P - U (with the natural forgetful functors U , V ) and the definition of F ( X , r ) determines a strong co-embedding F . rn 0
6.11. Remarks. Thus, under (M), S ( P - ) is algebraic, particularly its full subcategory Top (see I. 3.5E) is algebraic. The procedure used was originally a strating point for proving that “constructively defined” concrete categories are algebraic (see [HP,], [P,]). We do not need to go into it any more since (as the reader will see in Chapter 3), under (M) every concretizable category is algebraic. For the same reason we did not formulate the obvious consequence of 6.9 by saying that under (M) a dual of an algebraic category is algebraic (this fact was, using another procedure, proved by J. Isbell already in [I3]). It is worth noting that in all the facts mentioned the validity of (M) is necessary. Under non(M), e.g., already SetoPis non-algebraic ([KP,], see appendix B).
$7. S ( P + )into S ( P - )
81
EXERCISES
Let J be a set, Fi ( i E J ) functors Set + Set. Define V F , : Set -+ Set J
by ( V F i ) ( X ) = U F i ( X ) X {i), ( ( V F i ) (f)(x,j ) = (Fj(x),j). A functor F : Set + Set is Jsaid to be small if there is an epitransformation E : V Q A , -+ F. J
A functor F : Set -+ Set is said to be excessive if there is a cardinal x such that for a ! X with card X 2 x we have card F ( X ) > cardX.
1. No small functor is excessive. 2. If F is excessive and G small, then an underlying functor of a strong embedding of S ( F ) into S(G) is excessive. 3. A strong embedding of Top into any S ( F ) with a small F (particularly, into any Rel(d)) has an excessive underlying functor. (Hint: for an infinite X, there are expexpcardx topologies on X.)
8 7.
Strong embedding of S ( P + )into S ( P - )
One of the roles of the statement we are going to prove here is in the consequence that under (M) S ( P + )is algebraic (recall that P + : Set -+ Set isdefined-seeI,3.11.C-by P ' ( X ) = { A I A cX),P+(f)(A)={ f ( a ) ( a ~ A ) . The last is often denoted simply by f(A),which is sometimes confusing. We will take care to use this notation only at safe places.). Hence the universality of S(P+)which will be proved in Chapter 3 will imply that every concretizable category is algebraic. But a full embedding of S ( P + ) into Graph under (M) could be found by simpler means. An important feature of the constructions in this paragraph is that (M) is never used. Thus, the universality of S ( P + )will imply the universality of the very important category S ( P - ) . Actually, it will be proved more, namely that S(P+)is strongly embeddable into any S ( F ) with a contravariant faithful1F. 7.1. Lemma. For a subset M of a set N define A ~ k i fi Then for a mapping cp: N (i) cp(M) = M' (ii) ~ - ( c p ) (A?) c A.
-+
ki c P - ( N ) by
AnM=@.
N' the following two statements are equivalent;
82
Cli. 11, BASIC FMHEDDINGS
Proof. If q ( M ) c M‘ we have M c cp- ‘ ( M ’ ) .Thus, if A E fi‘ we have cp-’(A) n M c cp- ‘ ( A n M‘) = @ so that P-(cp) ( A )E fi. On the other hand, let (ii) hold. In particular
N\cp-’(M‘)
=
cp-’(N’\M’)
c
N\M
rn
so that M c cp- ‘ ( M ’ ) and hence cp(M) c M’.
7.2. Proposition. There exists a realization of S ( P - ) in S ( P - c P - ) . Proof. For ( X , r ) = objS(P-) put F ( X , r) = ( X , F) E objS(P- P - ) (F from7.1) Putting M = P - ( X ) , N = r and cp = F(f) we see by 7.1 that a mapping f : X --t X‘ carries a morphism ( X , r) + (X’, r’) in S(P-) iff it carries a morphism (X, 7) ( X ’ , 7’) in S(P- P - ) . -+
7.3. Theorem. There exists a strong embedding of S ( P - P - ) into S(P-). Proof. It suffices to prove that there exists a strong co-embedding F of S(P- c P - ) into S ( P - ) . In fact the composition
s(P-
0
P - ) 1,S ( F ) L s(p- P - ) 0
5 s(p-),
where J is the realization from 7.2, gives the required strong embedding by 6.6. For an ( X ,r) E obj S ( P - i P - ) put F ( X , r)
=
( P - ( X ) u A ( X ) ,1.) E obj S ( P - )
where A ( X ) = { a ( X ) ,b ( X ) ,c ( X ) ,d ( X ) ) consists of four distinct elements and is chosen so that P - ( X ) n A ( X ) = (4 and that for X += X ’ also P - ( X ) u
u A ( X ) =k P - ( X ’ ) u A(X’), 1. = nical reasons) with
4
U r, (this decomposition is done for tech-
r=O
ro = ( { a ) , ( b ) , {c}, ( d ) }
rl
2
((b,c,d}j> r 2 = { { a , b ) , (b, c}) , r3 ’= {Z u {a, b, c} Z a trivial ultrafilter on X} , r4= ( Z U A ~ Z E ~ ) . =
I
Define a contravariant functor G: Set G(X)
=
-+
Set by
p - ( x )u A ( X ) ,
G(f)(M)= P-(f)(M) for M c X , G(f)(ufx)) = ufx’) for u ( X )E A ( X ) .
83
67. S(P+)into S(P-)
For a morphism
=
((A'', r ' ) , j , ( X , 1 . ) ) put
F(P)= (f(X r'), G ( f ) >qx,r ) ) .
We see easily that P - ( G ( f ) )( F ) c F' (in fact even separately P - ( G ( f ) )(Y,) c ri) so that F is a contravariant functor S ( P - c P - ) -+ S ( P - ) , obviously one-toone. We have I/- F = G U for the natural forgetful functors U , T/: Thus, it remains to be proved that it is full. Let y: P u A ( X ' )+ P - ( x ) u A ( X ) 5
-(x')
be such that F ( y ) (1.) c 7.Consider, first, the following four disjoint sets Since they are in
r, we have gP({a))>g-Y{b))> g - ' ( { C ) ) >
s-'(@))
in F'. The preimages of disjoint sets are disjoint. Since no element of r', u r > u r ; u r ; is disjoint with more than two other elements of F', we have Consequently g(P-(x'))c p - ( x ) , g(A) = A . Hence, we can define mappings h : P - ( X ' ) + P - ( X ) , k : A + A by h ( M ) = = g ( M ) and k(u) = g(u), and we have, for Z c P - ( X ) and N c A gP1(Zu N ) = k - ' ( Z ) v k - ' ( N ) . (3) So, in particular, g-'({b, c, d}) = k - ' ( { b , c, d)). Since by (2) k is one-to-one, it has to be a three-point set and there is only one such, { b , c , d } , in F' (the 2 parts in r3 are always non-void). Thus,
k-'({b,c,d})
=
{b,c,dj,
so that, by (2), k(u) = a. We have further {a,bj in r, hence k - ' ( { a , b}) is either {u, b ) or {b,c ) , since it has to be a two-point set. Since we already know that a = k(a), it has to be { a , b } and we obtain k(b) = b. Thus, further, k - ' ( { b , c}) = {b, c} (since k is one-to-one, it cannot be {a, b } ) and we infer k(c) = c. We have k ( d ) = d since it is the only U E A left, and we have (4)
k(u) =
L1
(UEA).
84
Ch. 11, BASIC EMBEDDINGS
Thus, (3) acquires the form
g-’(Z u N ) = h-’(Z) u N (5) Taking A = {u, b,c) we see that h - ’ sends trivial ultrafilters to trivial ultrafilters, and hence, by 6.3, (6)
h =P-(f)
for an f : X
-+
X’.
Taking N = A we see that for Z E r , P - P - ( ~ ) ( z )= h - ’ ( Z ) E r ’ .
(7)
p
Hence, = ((X’, r’),f, ( X , r)) is a morphism and we have g (4) and (6). Thus, F is full.
=
G ( f ) by
7.4. Proposition. Let G, G : Set -+ Set be both covariant or both contravariant functors. Let there exist a monotransformation p: G G’. Then S(G) is realizable in S(G’). -+
Proof. For ( X , r ) E obj S(G) put
F ( X , r) = (X, px(r))E obj S(G’). It suffices to show that for an f : X + X‘, (1) G ( f ) ( r )c r‘ ( G ( f ) ( r ’ )c r in the contravariant case) holds if€ (2) G ’ ( f )(px(r))c px(r’) ( G ’ ( f )(px(r’))c ( r ) in the contravariant case). We will do the proof for the covariant case. The contravariant one differs only in the position of primes over r and X . . have Let (1) hold. Take a u = ~ ‘ ( u ) ,v ~ rWe G ( f )(u) = G!(f’)p X ( v )= px’G ( f )(u) E ~ ‘ ’ ( r ’ ) . Let (2) hold. Take a v E r. We have
PX G ( f )(u) = G ‘ ( f )(&))
.
E pX’(r‘)
Since px’ is one-to-one, we obtain G ( f )( v ) E r‘. 7.5. Theorem. There exists a strong embedding of S(P+)into S(P-). Proof. We will prove that there exists a monotransformation p: P + -+ -+ P - 0 P-. The statement then follows immediately by 7.4 and 7.3. Define p x : P + ( X )-+ P - ( P - ( X ) ) by
I
pX(A) = { B A c B c
x}.
85
For a mapping f : X
+
Y we have
p-(p-(f))(Px(A)) = P-(f)-' ( { B I A = B = x>) {C I C c Y and P-(f)(C) =f-'(C) 3 A ) =
=
=
I
{C f(A)= c
= y > = PY(P+(f)( A ) ) .
Thus, I.( = (px)is a transformation. If A = A',I either A $ p X ( A ' ) or A'#pX(A). Thus, every px is one-to-one. H 7.6. Lemma. Let F be a faithful contravariant functor. Then there exist transformat ions 1.1: P - + F and E: F +Psuch that
E o
p is the identity transformation of P - .
Proof. Let X be a set. For X E X define
t::
1+x
putting (Z(0)= x. For A c X define
x:; putting xz(x) = 1 iff x
E A.
We have obviously
for f : X
(2) (3)
x+2
for f:x
+
Y
+
Y
fc
t," = r:,,, of=
,
1;
Since F is faithful, the mappings F((F): F(2) -+ F(1) ( i = 0, 1) differ and hence there is a b E F(2) such that a0
=
F(t;)(b)
+ F(t:)(b)
= a1
.
Now, define p X : P - ( X ) -+ F ( X ) putting PX(A)= F ( X 3 ( 4 . p = (px) is a transformation P - + F since for a mapping f : X
have, by (3)
F(f)(PY(B))= F(f)(F(Xi)(4)= F(Xi o f ) (b) = =
F(X?- ' ( B J (b) = P X ( f - ' ( B ) )= P x ( P - ( f )
+
Y we
86
Ch. 11. BASlC EMBEDDINGS
Define
E
~
F. ( X ) + P - ( X ) putting EX(.)
c
=
=
(c") is a transformation F
have, by (2),
{x I F(Z)).( +
=
al)
P - since for a mapping J :
e X ( F ( f )(2.)) = .( I F(C)( F ( f )(t')) = .i> = {x 1 F ( f C)(t')= .I> = (x I F(%,)(U) = 0 1 ) = {x If(.)€EY(U)) = f-y&Y(c)) = P - ( f ) ( & y ( u ) ) .
=
x +Y
we
=
Finally, we have, by (l),
I
t X ( P X ( A= ) ) EX(F(Xf;) (b))= !x F(t,X)( F ( A 2 )(b)) = 4 ) ): (b) = a l l = {x x: = t;; = A , = {x F(X: <
I
so that ex p X is the identity mapping.
1
<:
a
7.7. Corollary. For euer): contravariant faithful F there e.xists ~irealization of S ( P - ) in S ( F ) ant1 LI strong embedding qf S ( P + ) into S(F).
x
Proof. p from 7.6 is a monotransformation since px(x) = px(y) gives c x p X ( x )= &'pX(y). Thus, the statement follows from 7.4 and 7.5.
=
8 8.
Bibliographical remarks
The embeddings of $0 1 and 2 appeared in [HP,] (the one in 2.3 is slightly modified here). The existence of rigid binary relations was first proved for accessible cardinals by induction in [PHI. Then P. VopEnka found an explicit construction ([VPH]) independent of the accessibility. This is the construction presented in (3 3. The existence of non-trivial rigid symmetric relations was first proved, we believe, by P. Erdos. An explicit example appeared in [HP,]. Then, in [HP,]. it was proved that a rigid symmetric relation on a set X exists iff cardX = 1 or X 2 8. The first constructions yielding the alg-universality of Graph appeared in [HP,], [PHI (used there explicitly for representations of semigroups only), [PI], [HP,] and [HP,]. In the text we used a more elegant construction from [HL]. The strong embedding of S(P-)and similar categories into Graph ($6) appeared in [HP,] in solving Isbell's problem on the algebraicity of the category of topological spaces. The strong embeddability of S(P+)into S(P-) without use of (M) was first communicated by L. KuEera. Here we present an independent proof.
Chapter III UNIVERSALITY OF S(P') The aim of this chapter is to present a proof of the fact that the category S ( P + )is universal. Such an example of a naturally defined universal category is important in itself, but the main reason why we place it here (though it is more difficult to read than the following two chapters and though we will not really need it in the arguments there) lies in its consequences. The most important one is that (under the assumption (M) from Chapter 11, $ 6 ) every concretizable category is algebraic. Hence, all the results on alguniversality in the following chapters turn, under (M), into results on universality. In the remaining chapters no reference will be made to the techniques of this chapter. Thus, a reader willing to take the universality of S(P+) for granted may omit this chapter at the first reading. It will cause no trouble for understanding the arguments in the sequel.
9 1.
Strong embedding of S ( P + , . . ., P') into S ( P + )
1.1. Lemma. For ( X , r ) E objS(P+) put o ( X , r ) = (I:R ) where Y = = U { A A E r, 1 card A 5 21 and R is the binary symmetric relation on Y tlejined by
I
ij7
(XJ)ER
(X,Y)El..
For eoery ( ( X ,r), j , ( X ' , r'))E morphS(P+) vt'e haoe f(Y ) c Y' and the nlapping f : Y + Y' defiiied bj- f ( j ) = f ( y ) is RR'-coinpatible.
Proof follows immediately by the fact that for {x, y ) E r (not necessarily
x
*
Y), f ( { K Y ) ) =
{ f(.+fO.')}
E 1.'.
rn
88
Ch. 111, UNIVERSALITY OF S ( P + )
1.2. Define
Po+: Set putting P;(X)
-+
P+(X)\(@f, P:(f)(A)
=
Set =
P + ( f ) ( A ) .We have
Theorem. For every natural number n, S((Fi)iGn) with Fi = P + .for all i is strongly embeddable into S(P;).
Proof. We can assume n 2 3 since S(Pt) and S(P+,P') are obviously realizable in S ( P + ,P + , P') (e.g. S(Pt, P') by sending (X, ro, r l ) to yo, r1, r1)).
(x,
Define a functor G: Set G ( f )(x, i) = (f(x), i), G ( f ) (i)
Set by G(X) = ( X x n) u (2n = i. Since for n 2 3 -+
+
l),
we can find 2n distinct four-point subsets of 2n
M o , M ~...) , Mn-
+ 1.
For ( X , with 7
u
2n+2
=
E obj S(P ',
17
N O ,N I ,...>N n - 1
. .., P') put
F ( X , (ri)ien) = (G(X),7) E obj s(po+) Ti where
i=O
Tn+i
(i}) u M i I A E ri} , = ( ( A x (i}) u Ni 1 A c X},
~ 2 n
= {(x} x n j x ~ x 9 }
for i < n Ti
T
= ((Ax
~ = ~((0,+I}, {I, ~ 2 } , ( 2 , 3 } , ..., (2n - 1, 2n}, (2n,O}} ,
72n+2
= {(0,1,3}).
We see easily that for
f = ( ( X ,(ri)),f,( X ' , (ri))) a morphism
?'I)
F ( f ) = ((G(X),f ) , G ( f ) ,(G(X'),
is a morphism of S(P:). Thus, a functor F: S ( P + ,..., P ' ) defined. It is obviously one-to-one. Now let 9: G ( X )-+ G(X')
be such that P,f(g)(i.) c 7'.
-+ S ( P , f )
is
$2. Representations of thin categories
89
+
o(G(X),i ) coincides with the (2n 1, Rz,+l) from 11.4.3. Thus, by 11.4.3 and 1.1, (1) g(2n + 1) = 2n + 1 and either g(i) = g(0) i or g(i) = g(0) - i (modulo 2n 1). Thus, further, g({O,l, 3)) has to be a three point set in 7‘ and a subset of 2n + 1. The set (0, 1,3} alone is such. This, together with the formulas in (I), gives g(i) = i for all i E 2n 1.
+
+
+
Consequently, g ( M j ) = M j and g ( N j ) = N j for all j . For (x, i ) E X x n, {(x, i)} u N iis in 7 and hence g({(x, i ) } u Ni) = {g(x, i)} u N iis in 7’. Since no sets of 7‘ but those of contain Ni,we infer that g(x, i ) E X’ x { i ) , and we can define fi: X +-X‘ by g(x, i) = (J(x), i). Further, we have {x} x n E i and hence g({x} x n) = ((fi(x), i) i E n} E 7‘. Since this set meets every X ’ x { i } , it is necessarily equal to {x’) x n for an X ’ E X . Thus, J(x) = x’ for all i. Denoting this common value by f ( x ) , we obtain a mapping f:X + X’ such that g = G(f). It remains to be shown that P + ( f )(ri) c r;. Take an A E ri. We have ( A x {i}) u M iE i and hence
c+i
I
g((A x (i})
u Mi) = ( f ( A ) x { i } ) u Mi E 7 ’ .
In i, only the sets ( B x (i}) u M i with B ~ r fintersect 2n Thus, f ( ~=) B Eri.
+ 1 in Mi.
1.3. By the construction in the proof of 1.2 we have, in fact, a stronger statement : Theorem. Denote by ‘2I the full concrete subcategory of S ( P + ) generated by the objects ( X ,r) such that 0 $ r 0. Then every S ( P + , ..., P’) is strongly embeddable into (LI.
+
5 2.
Representations of thin categories
2.1. Let us first recall some facts from the set theory we will use extensively in the sequel. We are working in the Godel-Bernays system. Every element of a set (and of a class as well) is a set. For any two proper classes, there exists a one-to-one mapping of one onto the other. In particular, every proper class is “equally large” as the class of all cardinal numbers. An ordinal CI is the set of all smaller ordinals. For an ordinal CI denote the next one, i.e. CI u {cI},by c(+.
90
Ch. Ill, UNIVERSALITY OF S ( P ' )
2.2. The category
lncl
is the subcategory of Set with obj lncl = objSet and (,f:X iff*) for every x E X , f'(x) c x.
-+
E morphlncl
X')E
2.3. Theorem. For ewry thin category with no distinct isomorphic objects
tlirrr is u firll embedding into Incl.
Proof. In other words, we have to prove that for any partially ordered (i.e. a class with a transitive and reflexive relation such that class (T, I) t 5 t' and t' I t implies t = t') there is a mapping F : T obj lncl such that (1) if t I t' there is exactly one morphism F ( t ) F( t ' ) (2) otherwise, there is none. Since all proper classes are equally large and since provided the statement holds for large categories, it holds for small ones as well, we can assume that T is the class of all ordinals. Thus, E is a well-ordering on 7: We will use the symbol a f p in the obvious sense. For an ordinal a define -+
-+
w, = { a l p 5
a 5
83
Further, define sets V, and W, by A E k,
iff ( V ~ E A wB , c A)&(n c u + ) & ( ~ E A ) ,
A
iff ( A E
E
W,
K)&(aE A
and
p
I s( only if /3 = a ) .
Obviously, W,E W,. Put, for A E V , and define
A" = { A n p' I p
such that A n F'
E
Wo),
G(a)= (A"IAEV,j. First we will prove that (1) if a 5 p, there is a morphism G(a) + C(b) in Incl. So, let a 5 p. If a = we can take the identity. Let a E b. For A" E G(a) put f(A) = B where B = A u w p (which is obviously in Vo).We have
(*I
A
since, if y E won a', y
(**I
*) I t Thc
i i y /(Y)
irnnsc (11
ME
=Bna',
p < y and hence 1' E w,
aEysp*
c A . Moreover,
Bn;:'#Jy
happen that a set x is both a s u b w ;uid :in c l c i ~ i c n t01 last formula is to be undersioc? ;I\ i l i c \ . I ~ L I C 01 I c X under,'.
111 the
\. ,it
3,. c :
\t
\.
0
1101
111
;O;
;I\
t11t
$2. Representations of thin categories
91
Really, cc E y implies a E B n y + and y 4 A . If 3 n y + E W, we would have y E B n y + and hence y E w,, so that p I y. Thus, a < y in contradiction with B n y + being in W,. The inclusion c 2 (in fact, the equation A" = B) is an immediate consequence of (*) and (**). Let ~ E C IFor . A E V , put f(2)= B with B = A n B+. If B n y + is in W,, we have y g j3 and hence A n y + = A n p' n y' = B n y + E Wt. Thus, c A. Now, we will prove that (2) if there is a morphism G(a)-+ G(P) in IncI, then a I 1. Let such a morphism exist. We have w, E V, and hence there is a B E V, with Gn=) B. Let y be the least ordinal in B such that y I p. Then, B n y + E B (indeed, if 6 E B n y + , we have 6 5 y and hence 6 < y conso that B n y' = w, n y'. tradicts the minimality of y). Hence, B n y' E Thus, y E w, and hence cc I y I /?I. By (1) and (2), a morphism G(a) G(B) exists iff a 5 p. So far, however, one can have several morphism between G(a) and G@). For this reason, the construction will now be modified. Put
+,,
-+
F(a) =
{A I A" E G(a)
and, for no
B E G(cI),B 2 A")
We will prove first that there are enough inclusion minimal sets in G(a). Namely,
(3) For every B E G(a) there is an 2 E F(a) such that the contrary. Then one has BiE G(a) such that
B =I 2. Suppose
Since there IS no deChoose ordinals fl, such that B, n fl: E B,\B,+ creasing sequence of ordinals, we have a natural k with p k 5 p k + 1 . We have Bk+ n pl+ E Bkf c B k and hence Bk+, n p:+ E w,,, , and Bk + 1 n p:+ = Bk n y E W,. Hence l j k + I = y since, by the former, /?Ik+ ,: and by the latter y is such. Thus, is the maximal element in Bk+ n8+ we have +
Bk+l
But this and n p:
=
pk
Bk n p:
5 Bk+, E
pkffl
=
Bk
n /?Ik++l.
gives the contradiction B,+ I n 8: = B, A B;+ n 4 Bk+ 1 . By (3)we obtain immediately:
Wpkwhile Bk n p:
(4) A morphism F(m) -+ F ( p ) exists iff a morphism G(a) -+ G(p) exists. Thus, the proof of the theorem will be concluded showing that there is at most one morphism F(a) + F(P). Let there be two. Thus, we have an
92
Ch. 111, UNIVERSALITY OF S(P+)
B,,B,EF(B) such that A XI Bl u B2. Since neither B2 c El, there are ordinals yl, y2 with
A E F ( ~and ) distinct
BI c 8,
nor
Thus, we have
B 1 n :y
Further, for some di
E
B1\B2,
Bi n:y
E
Bi n:y
B, n:y
Wyc =
E
B2\Bl
(i = 1,2).
A n 6:
E
Wat.
Comparing the maximal elements we obtain y i = 6,. Let, say, y1 5 y,. We have B , n :y = B2 n:y n :y = A n:y n :y = A n :y = B1 n :y E W,, in contradiction with B, n:y
=
B , n :y $ B,.
H
2.4. Proposition. There are objects ( Y ( x ) ,S(x)) in S(P+)indexed by all sets x such that (1) I f x y there is no morphism (Y(x),S(x)) + (Y(y),S(y)). (2) From ( Y ( x ) ,S(x)) into its@ we have the identity only. Proof. Since there is a one-to-one mapping of the universal class onto the class %'ad of all cardinal numbers, it suffices to find such objects Further, by 1.2, it suffices to find such objects in S ( P + ,P', P'). for a E %'aid. For a ~ % ' a a . ~ put d F(a) = (a, (ri(a))ie3) with ro(a) = { A A c a, cardA = 2 } ,
+
I
h ( a )= (a} > r2(a)= a (every
fi E a is also a subset of a ) .
Now, let for g: a + fi, P+(g)(ri(cx))c ri(fi).Because of y o , g is one-to-one. Because of r1 it is onto. Thus, necessarily a = 0.We have: (1) if y E 6 E a then g(y) E g(6). Indeed, P + ( g ) ( y+ 1) is an ordinal by I-,,it contains g(y) and since g is one-to-one, it cannot contain g(6). Let g =k 1. Take the least y wit! g(y) y. Since g is onto, y = g(6) for a 6 E a. By the minimality of y, y E 6. Thus, by (l), g(y) E g(6) = y. But this contradicts the minimality, since we obtain, H by (I)? S(S(Y))E g(y) E 7 .
+
2.5. Theorem. For every thin category there is a full embedding into S(P+). Proof. Obviously, for an object of S ( P + ) there is a proper class of isomorphic ones. Thus, by 2.3 and 1.2 it suffices to find a full embedding of lncl into S ( P + , P + , P + , P+).
93
32. Representations of thin categories
First, take for every set x the object ( Y ( x ) ,S(x)) from 2.4. For a set X put
F(X)= (X u
u
{ ( M ,x)) x Y ( x ) ,( r i ( X ) ) i E 4 )
XEMEX
(we omit multiplying X by ( 0 ) and the other summand by (1) and take them for d ;joint as they are) with
r*(x)= { A 1 A
r,(X
=
{{ M }I M
For a morphism f : X
E
c
x),
x}u { { ( M ,x)}
+ X’
Ix
x B
EM E
x,
E
s(x)) .
of lncl put
where for M E X , g ( M ) = f ( M ) for x E M E X and y E Y ( x ) ,
g ( M , x, y ) = f’(M)’ f (MI
x, y ) if
xEf(M) ,
if x @f ( M ) .
We have obviously P+(g)( r o ( X ) )c ro(X‘). Since
we see also easily that P+(g)( r i ( X ) )c ri(X’) for i = 1,2,3. Thus, F ( f ’ )E morphS(P’, ..., P’), and a functor lncl + S(P+,. . ., P’) obviously one-to-one, is defined. To simplify the notation, let us write, from now on, for P+(g).Now, let g:
xu
(J { ( M ,x))
x ~ M E X
x Y(x)
--f
x’u (J {(w, x’)} X’E
M’EX‘
x Y(x’)
be a mapping such that, for i E 4. ( r j ( X ) )c ri(X’). Using ro we xee that @ ( X )c X’. Define f : X + X ’ putting j ( M ) = g(M). ( { ( M ,x)) x Y(s)) E r , ( X ) and hence We have ( M } u
u
XEM
{f(M)}u (J $7({(M x)} x y ( x ) ) = { M ’ }u
u ({(K
x’)> x Y ( x ’ ) ) .
X’EM’
X € M
Since , f ( M )E X‘, we obtain M‘ = f ( M ) and the following inclusions: (1)
(2)
u u
{(’(M), x’)> x Y ( x ‘ )=
X’Ef(M)
$7({(M> x)} x Y ( 4 )
XGM
u
xaM
43
@(((M,
= {f(M)S u
y(47
u ({(f’fW
X’€f(M)
x’)} x Y(x’)P
94
Ch. 111. UNIVERSALITY OF S(P+)
Using r2 we see that y ( { ( M ,x)} x Y ( x ) )is equal to either {(M’. x‘): x Y ( x ’ ) or to { M ‘ } . By (2), in both cases M’ = f ( M ) . If g ( { ( M , 1): Y(Y)) = = {(f(M),x’)} x Y(x‘), we have a mapping h : Y ( x ) Y ( s ’ ) \ticn that g ( M 1x, Y ) = ( f ( M ) ,x,h(y)). For a BES(X)we have then L~
--f
((f(M),x’)} x P + ( h )(B) = a{(Mx)} x B ) E I”@’) Thus, P + ( h ) ( S ( x ) )c S(x‘) and hence, by 2.4, x mapping. Thus, we have
=
x’ and h is the identity
either g(M, x, y ) = ( f ( M ) ,x, y ) for all y E Y ( x ) , or
for all y
g ( M , x,Y ) = f ( M )
Finally, if x € f ( M ) it has to be the first case-otherwise in (1) would not hold. Thus, ( F ( X ) ,g, F(X’)) = F ( f ) .
8 3.
E
Y(x). the first inclusion
Categories Y ( F ; (7: 2 ) )and realizations of concrete categories
3.1. Let F : Set + Set be a functor, (T, 5 ) a preordered class. The category 9 ( F ; (T, 5 ) ) is defined as follows: The objects are couples ( X , a) with X a set and CI a mapping of F ( X ) into 7: Morphisms from ( X , a) to (X‘, M’) are triples ( ( X ,CI),~; ( X ’ , a’)) with f : X + X’ such that
for every u E F ( X )
~ ( uI ) a
’ ( ~ ( (u)) f)
The composition is defined by the composition of mappings.
3.2. Lemma. For every mapping ,f: X such thut g: X
+
X’
and
+ X‘
(g x g) ( R ) = (f x
there is an R c X x X
f)( R ) implies g
=f .
Proof. If J i: a constant mapping, take R = X x X. Now, let Y = . f ( X ) have at least two elements. Choose an antireflexive well-ordering < on Y and put R = {(x, Y ) f(x) . f ( Y ) } . = <. Thus, (f x f)(R)
I
95
$3. Realizations of concrete calegories
Let g : X
+ X’
be such that also (g x g) ( R ) =
g(x)
=
<. We have
Y.
Really, since card Y 2 2, J) E Y appears in a couple (yl, y 2 )E < and we have (yl, y 2 ) = (g(xl),g(x2)). On the other hand, x E X appears in a couple (xl, x2)E R, hence (g(x,), g(x2)) E < and hence g(x) E Y. Denote by ,j the mapping Y + X ’ defined by j ( y ) = y and define a mapping 11: Y-+ Y by h(y) = f(x) , where g(x) = y . (If g(x) = g(x’) there cannot be f ( x ) < f(x‘) since this would give (x,X’)E R and hence g(x) i g(x‘). Thus, h is defined correctly.) Hence, we have f = j h 0 g‘ where j g‘ = g. Thus, 0
0
Hence
(h x h)(<)
=
(h x
/I)
((9’ x g’)(R))=
(J’ x f ) ( R )
=
<.
(1) Y < Y’ h(y) < h(y’), (2) y < y’ => there are x,x’with x 4 x’, h(x) = y and h(x’) = y’. Let y be the least element (in <)such that h(y) =+ y. If h(y) < y we have, by (l), h(h(y))i h(y) in contradiction with the choice of y. If y i h(y), there are, by (2),x and x’ with h(x) = y, h(x’) = h(y) and x < x‘. By (I), 1i(x’)= h(y) implies x’= y and hence h(x) = y x although x < y. Thus, h is the identity mapping of Y and hence f = j h g’ = j g’ = g.
+
0
0
0
3.3. Theorem Every concrete category is realizable in Y(P’ c: Q, (7; I)) u,ith a suitable preordered class (7; I).(Q is the functor from 1.3.11. given by
Q(x)= X
x
Proof. Let
X, Q ( f )(x,Y ) = (f(x),f(y)) .)
(a, U ) be a concrete category. On T = {(a,R ) I a E obj ‘UL, R c U(a) x
U(a))
define a preorder I putting
iff there is a cp: a 4 a‘ in such that ((P’ o Q ) ( U c p ) ) ( R )= R‘.
(a, R ) I (u’, R’)
For an object a of ‘21 put
F(a) = (U(a),a,) If cp: a
+ a‘
with
sra(R)= (a, R ) .
is a morphism, we have for every R E P + Q(a)
x a ( R ) = (0, R ) i
0
((1’3
(P’
Q u‘P) ( R ) )= aa,((P+Q ) (U CP)) (R))
96
Ch. 111, UNIVERSALITY OF S(P+)
so that U ( q ) carries a morphism F(a) -+ F(a’). On the other hand, let f: U ( a )-+ U(a’) carry a morphism F(a) + F(a’). Thus, for every R c = UfQ)x u(a), (1) (4R ) 5 (a’, (f x f)(R)). Choose an R satisfying the implication from 3.2 for our f: By (1) there is a q : a -,a’ such that
(f x f)( R ) = ((P+ Q) ( U V ) )( R ) = (u(9)x ~ ( c P()R) ) . 0
Thus, by 3.2, U ( q ) = f:
0 4.
W
S ( P + )is universal
4.1. Lemma. For a set X define a ( X ) and b ( X ) subsets o f P + P f ( X )b y
10 + M
IXEM)
a ( X ) = ( { M }u { { x } b ( X )=
{{@I>
cX},
.
A mapping g : P + ( x ) -, P + ( X ) is equal to P + ( f )for an f: X P + ( g )( a ( X ) )c a(X‘) and P + ( g )( b ( X ) )c b(X‘).
-+
X ’ ijff
Proof. Obviously,if g = P + ( f ) ,the inclusionshold true. Let P+(g)( u ( X ) ) c c a(X’) and P + ( g )( b ( X ) )c b(X‘). First, { { x } }with x E X are exactly all the one-point sets from a ( X ) . Thus, P + ( g ) ( ( { x ) )=) {g({x>)> = { { x ’ } } for an x’ E X‘ and we can define an f: X -+ X‘ by
g ( { x } )= { f ( x ) ). For a non-void set M c X we have so that there is an M c X ’ with (1)
I
{ g ( M ) )u { { f ( x ) )x
EM>=
P‘} u {b‘> I
M’l
If M’ = {xb) is a one-point set, we have
M M ) ) u { { f ( x ) i1 x 4
=
{{xb})
so that f ( x ) = xb for all x E M and g ( M ) = {xb} = P + ( f ) ( M ) . If cardM’ 2 2, M‘ cannot be equal to an { f ( x ) )and hence, by (1), g ( M ) = M’ and { { f ( x ) } x E M } = { {x’} x’ E M’}. By the last equation we can see easily P + ( f )( M ) = M’ = g(M). Finally, using b, we obtain g(0) = 8 = P + ( f ) ( @ ) .
I
I
97
94. S(P+)is universal
4.2. Lemma. For a set X define r i ( X ) c P + P + Q ( X ) ( i e 5 ) putting (a and b from 4.1) r o ( X )= a ( x x X ) , r , ( X ) = b(X x x ) ,
I
% ( X ) = {{{(x, .)>I x E X >> G ( X ) = { {{(4 Y), (4x)>> Y ) E x x x> r4(X) = Y ) , ( Y , Y)>> (x, Y ) E X x X I .
I(.? I
{{k
5
A mapping g : P + Q ( X )+ P+Q(X’) is equal to P’Q(f) for f : X f i r i E 5, P + ( g )( r i ( X ) )c ri(X’).
-+
X‘ iff,
Proof. For g = P’Q(f) the inclusions obviously hold true. Let P + ( g ) ( r , ( X ) )c ri(X’) for all i. Because of ro and r l , by 4.1, g = P+(h) for some h : Q ( X ) + Q(X’). Thus, it suffices to prove that h = Q(f). We have {{h(x,x))) = P+P+h({{(XJ)))) = p+g({{(x,
so that we can define a mapping f : X
+ X’
41))=
W’7X’)H
ET.2
7
by
h(4 x) = (f(x),f(x)) .
Now, we obtain, using r3, {{h(%Y)?(f(x)Lf(x)))} = P + P + ( h ) Y ) , (x7 x)H) = = {{(x’, Y’), (x’, x’)}} , ((“7
using r4
+
({+> Y ) , (f(hf ( Y ) ) ) =
@’7
Y’), (Y‘, Y’)> .
y’), x‘ = f ( x ) and Y’ = f(y); if x’ = Y’, Thus, if x’ y’, h(x,y) = h(x, Y ) = ( f ( X ) ? f(x)) = ( f ( Y ) ,f ( Y ) ) . Anyway7 h(x3 Y ) = (f(4 f(Y)). (XI,
4.3. Theorem. S(P,+)is universal. Proof. By 3.3. and 1.2. it suffices to prove that for every preordered class (7; 5 ) there exists a full faithful functor F from Y ( P +o Q, (IT; 5 ) )into some S(P+,..., P’). Thus, take a (7: I). By 2.5 there are objects y(t) = = ( Y ( f )S(r)) . in S(P+)for t E T such that (1) if r 5 t’. there is exactly one morphism y(t)- + r’(l’);denote its undei !I ing mapping by cp(t, t‘), (2) If t I I ’ there is no morphism y(t) + y(t’).We will construct a functor F : Y(P’
0
Q, (z 5 ) )+ S(P+,..., P’)
98
Ch. Ill, UNlVERSALITY OF S(Pi)
(more precisely, we should make the two summands disjoint by, say, multiplying them by distinct one-point sets) with
r , ( x ,a)
=
r , ( X ) from 4.2 for
1
i E5 ,
Y ~ ( Xa), = ( A A c P ’ Q ( X ) } ,
r,(X,a)
=
u
M c X x X
I
(A A
I
= {M} x
Y(a(M))} 9
r,(X, a) = ( ( M I u A M c X x X
and A c { M ) x Y ( c i ( M ) ) } ,
r 8 ( X ,a) = ( ( M I x A M c X x X
and
I
A
E
S ( a ( M ) ) }.
For a morphism J : ( X , a ) + ( X ‘ , a’) define
F ( f ) = g : F ( X , u) + F(X’, M ’ )
with
g ( M ) = ( P + Q ( f ) )( M ) for M € f “ Q ( X ) , g(M, Y ) = ( ( P + Q ( f )() M ) ,(cp(t,t’))(Y))
where t = a(M), t’ = u ’ ( P + Q ( J ) ) ( M ) .It is easy to see that for g thus defined really P + ( g )(r,(X, E ) ) c r I ( X ’ a’). , Obviously, F is a faithful functor. Now, let g carry a morphism F ( X , a ) + F(X‘, a’). Using r 5 we immediately conclude that S ( P + Q ( X ) )= P + Q ( X ’ ) .
(3)
(Again,
g is short for P + ( g ).)
Using r6 we see that for every M c X x X there is an M‘ c X’ x X‘ such that (4) g ( ( M } x Y(E(M))) ( M ’ } x Y(@’(M))’ Thus, we have mappings
and defined by (5)
h : P + Q ( X )+ P + Q ( X ’ ) , k : P + Q ( X )+ P + Q ( X ’ )
I,:
Y ( a ( M ) )+ Y(E’(M’))
g(M) = h(M)
3
g ( M Y ) = (k(M)>
’
99
$5. Bibliographical remarks
Now, by 4.2, using ri with i E 5, we obtain immediately that
(6)
h
Using r7 we get
=
for an f:X
P’Q(f) a
--f
X’.
!3((M)u ( { M ) x Y(a(M)))= ( h ( M ) )u !3((M>x Y(a(&f)))= ( N ) u A for an A c ( N ) x y(a’(N)).
Thus, by (4) and (5), h ( M ) = N (7)
k
=
k(M), so that also
=
P’Q(f).
Now we have l M : Y ( a ( M ) )+ Y ( a ’ ( P + e f ( M ) )for ) , an A E S ( ~ ( M we ) ) have ( M ) x A E r 8 ( X ,a) and hence g ( ( M ) x A ) = {P’Q(f) ( M ) ) x ~ M ( A r) 8E( X ‘ , a ’ ) ,
so that 1,(A) E S(a’(P+Qj(M))). Thus, by (1) and (2),
MADCLIFF1
(8) a ( M ) 5 a’(f’+Q(J) ( M ) ) (9) IM = p(t, t’) for t = a ( M ) and t’ = a ’ ( P + Q ( f )(M)). By (8), = ( ( X ’ ,a’), f , ( X , a)) is a morphism, by (6), (7), (9) and (5), (F(X’, a’), 97 F(X’, a)) = F ( Q ) . 4.4. By 4.3 and 11.7.7 we immediately obtain
Corollary. Every S(F) with a faithful contravariant F , in particular S(P-), is universal. 4.5. By 4.3, 11.7.7 and 11.6.10 we obtain
Corollary. Under (M), Graph and every Alg(d) with E d 2 2 are universal. Consequently, every concretizable category is algebraic. 4.6. Corollary. The category ‘i? from l 1.3 is universal.
0 5.
Bibliographical remarks
The first example of a universal category was constructed in [T,]. That category even has a stronger property than universality but it is far from being what one would call “naturally defined”. The fact that S(Pf) contains all constructively defined concretizable categories as full subcategories was already known in 1966 (the report about it is in [HPT], where one can find further references). It took, however, three more years before the full embeddability into S(P ’) was proved for general concretizable categories.
100
Ch. 111, UNlVERSALITY OF S ( P +)
Then, in 1969, Z. Hedrlin proved the theorem on embeddability of a general thin category into Incl. This was a basis, together with KuEera's own lemma on selecting a single mapping by binary relations (here 3.2), for the proof of the universality of Graph under (M) which was given by L. KuEera in the same year. Finally, L. KuEera observed that when working with S ( P + )one may omit the assumption (M). The results were reported in [H5] and a full account of the proof was presented in [K1].
Chapter IV COMBINATORICS In Chapter 11, the category of all graphs (= sets with binary relations) and their homomorphisms ( = relation-preseving mappings) was shown to be alg-universal. The present chapter is concerned with the question of alg-universality of categories of graphs (relations) subjected to special conditions. Thus, in $ 3 we prove the alg-universality of the categories of symmetric graphs and of acyclic graphs, and in $4 we show that for symmetric graphs, moreover, a prescribed chromatic number can be assumed. In $ 5 the category of (antireflexive) partially ordered sets is shown to be alg-universal. $ 6 deals with a slightly changed notion of homomorphism, and $ 7 with graphs with loops (here, of course, the problem is the almostalg-universality). $ 8 concerns couples of equivalence relations, and in $ 10 a problem by S. Ulam is discussed.
8 1.
Graphs, symmetric graphs, undirected graphs
1.1. The category Graph was introduced in 11.2.1. Now, we will adopt the terminology of graph theory for its objects and for notions concerning them. Thus, an ( X ,R ) E obj Graph (recall that X is a set and R a subset of X x X) is called graph (more precisely, directed graph), the elements of X
\!3
Fig. 4.1
102
Ch. IV, COMBINATORICS
are called vertices or nodes, and the elements of R its (directed) edges. An edge (x,y)is said to start in x and terminate in y. Small graphs will sometimes be described by figures consisting of labelled vertices and arrows x.+.y indicating that ( x , ~ ) R. E Thus, e.g. Fig. 4.1 describes the graph ((0, 1,2,3}, ((0,o), (0, I), (1,2), (2,3), (3,l)))
'
Sometimes the labels of the vertices will be omitted.
1.2. The concrete category UndGraph
is defined as follows: The objects (called undirected graphs) are couples ( X , R ) with X a set and R a set of non-void at most two-point subsets of X . The morphisms from ( X ,R ) into ( X ' , R') are triples ( ( X ' ,R'),f,( X , R ) ) with f : X + X ' such that for {x, y>E R always (f(x),f(y)) E R'. For the forgetful functor we will take the natural one. 1.3. Again, for ( X , R ) an object of UndGraph, the elements of X are called vertices or nodes, and the elements of R are called edges. We say that an edge {x,y} joins x and y. Hence, the expression that x is joined with y (in ( X , R ) ) indicates that {x,y } is in R. Instead of a formal description of an undirected graph one may use a figure with edges represented by lines (preferably, but not necessarily, straight ones). Thus, e.g. Fig. 4.2 represents the undirected graph
Cjo, 1,2,3,4}, ((0, I}, {0,3}, (0,4}>(1)- (1, 2), (2,3}>(3>4)>14))).
1.4. Obviously, UndGraph is a full concrete subcategory of S(P'). 1.5. We see easily that UndGraph is equally realized with the full concrete subcategory SymGraph
of Graph generated by all the symmetric graphs (i.e. those ( X , R ) where
$ 1 . Graphs, bymmetric graphs
103
( x , y ) E R iff (y,x ) E R ), the isofunctor being induced by the mappings of objects sending ( X , R ) from UndGraph to (X>”{(X, Y)
I { X ’ Y } E R } ).
1.6. Conventions. (1) If it is evident from the context that an undirected graph is meant, one also uses the word graph only for it. (2) The morphisms of Graph and of UndGraph are usually referred to as (graph) homomorphisms. (3) Using the symbol x R y (cf. 1.4.1) in connection with an undirected graph indicates that { x , y>E R . 1.7. Let ( X , R ) be a graph or an undirected graph. A sequence x o ,x l , ..., x, of vertices is said to be a path (a quasi-path, resp.) of length n, joining xo and x , in ( X , R), if for i = 0, 1, ..., n - 1, x i R x i + , (either x i R x i + , or x i + R x i , resp.). A path of length n 2 1 joining x with itself is called cycle of length n (for directed graphs, this notion was already introduced in 1.4.1). Cycles of length 1 are often called loops. 1.8. A graph or an undirected graph is said to be connected if for every two distinct vertices x , y there is a quasi-path joining x and y. It is said to be strongly connected if any two distinct vertices are joined by a path. For undirected graphs, of course, “connected” and “strongly connected” mean the same. 1.9. Obviously, we have: Proposition. A graph ( X ,R ) is connected $for every two distinct x, y there. is a sequence xo, x1, . .., xn such that x = xo, y quasi-path with xi+ 1.
=
x , and, for i
=
E
X
0, 1, ..., n - 1, xi is joined by a
1.10. Obviously again, we have Proposition. An image of‘ a path (qii~isi-path.resp.) under N homomorphism is a path (quasi-path, resp.). Consequently, if ( X ,R ) is (strongly) connected, and .f’ (s a homomorphism of ( X ,R ) onto ( x S), then also (I:S) is (strongly) connected. 1.11. Considering the construction in the proof of Theorem 11.5.3 we see that a stronger statement was actually proved. Because of its usefulness for further application, we will formulate it now in an explicit form:
104
Ch.IV, COMBINATORlCS
Theorem. Denote by Graph, the full concrete subcnteqorl?of Graph generated by the ( X , R ) satisfying the following three conditions: (1) ( X , R ) has no loops, (2) ( X , R) is strongly connected, ( 3 ) cardX 2 2. Then every Re1( A ) is strongly embeddable into Graph,. Consequently, Graph, is ulg-universal Remark. Take notice of the following consequence of the conditions: For every x E X there is an edge starting in x and an edge terminating in x. EXERCISES
1. A graph is connected iff it is not a coproduct of two non-void graphs in the category Graph. 2. Every finite monoid is isomorphic to the endomorphism monoid of a finite strongly connected graph without loops. 3. Let m be a cardinal number. The full subcategory of Graph generated by the graphs whose every component has the cardinality less than m is not alg-universal. 4. Let @: Graph + Graph be defined as follows:
degG
=
supdzx, X€X
--
degG
t
=
supzgx XEX
Denote by G(G) (G(Tt), G(5, it), resp.) the subcategory of Graph generated by the G with deg G I m (GG I n, deg G I m and &g G I n, resp.). For no cardinals m, n, G(iii,Ft) is alg-universal. On the other hand, G ( 3 and G(2) are alg-universal. (Hint: Use E.1.3 and E.1.4).
105
92. The “arrow construction”
6. The one-point graphs and the finite chains without loops are the only rigid objects of G ( 0 .Thus, this category is not alg-universal.
0 2.
c
The “arrow construction” in its simplest form
2.1. In this section we will discuss the method of replacing edges by more complicated objects mentioned already in the Introduction, 1 (de Groot’s construction) and used also in I1 $ 4 for proving that there are enough rigid symmetric graphs. To recall the idea: The point is to find an object-the “arrow”-to be glued into graphs instead of edges. This object has to be sufficiently rigid to behave under morphisms like a directed edge, i.e. apt to be mapped only identically onto itself. The method will be described in detail in its simplest form concerning arrows which are graphs themselves, Afterwards, generalizations will be mentioned only briefly to avoid complications which are unnecessary at the present point. The generalized construction will be used particularly in Chapter VI.
2.2. Take a system d = ((A,r ) ; a, b) where (A, r ) is a graph and a and b are two distinct vertices selected in it. Let (X, R ) be a graph, (x, y) one of its edges. Define a mapping (the summands are taken as disjoint-we distinct one-point sets) putting %,,)(a) = x
9
cp,x,,,(b) = Y
omit the clumsy multiplying by
9
= (X’
‘P(x,,,(.)
Y,
4
otherwise. Now, put
( X ,R ) *
where
=
(xu ( R x (A\(a,
b})),R )
iff there is a (u, v) E r and an (x, y) E R such that ‘ ~ ( ~ , , , ( = u)5 and ‘p,x,,,(~)= rl. For a graph homomorphism f : ( X ,R ) -+ (Y, S ) define
(l,y ~ )E
f
by
* d :x u(R x
@\(a, b)))-+ Y
u (s x
( f * d)(x) = f ( x ) and ( f * d )(X? Y , ).
=
@\{a, bj))
( f ( x ) ,f ( Y h
4.
2.3. Lemma. For every (x, y) E R, (P(~,,): ( A , r) + (X, R ) * d is a horno(X, R ) * d --* morphism. For a homomorphism f : ( X , R ) -+( X ’ , R’), (f* d): -+ (XI, R‘) * d is a homomorphism.
106
Ch. IV, COMBINATORICS
Proof. The first statement follows immediately by the definition of Now, let f’ be.: homomorphism. Obviously, we have for (x, y) E R
Let
(to,t l ) be in R. Thus, we ti.Thus, by (I),
have a (u,,,
U~)E r
R.
and an (x, y ) E R with
q(x,y)(ui)=
( f * 4(ti)= q ( f ( x l , f ( y ) ) ( u i ) so that ( ( f * d ) ( t O ( f) *, d ) ( t l )is) in R‘. 2
2.4. Theorem. Let 8 be a subcategory of Graph (not necessarilyfull). Let there be an d = ((A,r ) ; a, b) such that (1) ( A , r ) E obj 8 and for every ( X , R ) E obj Graph,,, ( X , R ) * d E obj 8, ( 2 ) ‘p&,iX,R) are the only morphisms ( A ,r ) -+ ( X , R ) * d in 8, ( 3 ) fur f ~ m o r p G r a p h , , f e d is in morpB. Then 6 is alg-uniuersal. Remark. Obviously, if 8 is a full subcategory of Graph, the condition ( 3 ) is satisfied automatically. Proof. We will prove that @: Graph, + 8 given by
@ ( X , R )= ( X , R ) * d ,
@(f)=f*d
is a full embedding. We see easily (using also (1) and (3)) that @ is a one-to-one functor into 8. Now, let g: ( X ,R ) * d -+ ( X ,R’) * d be a morphism in 8.By (2), we see that (4) for every (x, y ) E R there is an (x’, y’) E R’ with g ( P ( ~ , ~=) ‘p(x,,y,). Take an x E X. By the Remark in 1.11 there is a y E X such that x = q,x,,,(a). Consequently, ~ I ( . Y ) = yq~(~.,.,(a) is in X ’ and we can define 0
J’:X-+X‘ putting f ( x ) = g(x). Now, let (x, y ) be in R. Take the (x’, y’) from (4). We have x’ = q(xr,Y,)(a) = = g’p(x,y,(a)= g(x) = f ( x ) and Y’ = cp,x,,,*,(b) = qcP,x,y,(b) = g(Y) = f ( Y ) . Thus, f is a homomorphism ( X , R) -+ ( X ’ , R‘). Finally, for (x, y, u) E R x x @\(a, b } ) we have g(x9 Y ,
so that g
=f
4 = S c P ( x , , ) ( 4 = c P U ~ X ) , f ( Y ) ) ( 4 = (f(x), f ( Y L 4 ’
*d
=
@ ( f ) .Thus, @ is full.
$3. Two applications
107
2.5. For simplicity, we have been working so far with an arrow d based on a graph ( A , F ) .Instead of this, objects of some other nature can be taken (in the example inJntroduction 1, the object was a topological space). Further, it happens sometimes that ( X , R ) * d do not allow those transformations we want to have exactly, but that the situation can be amended by extending them by some new points, the role of the added part being, say, to reduce the motion of the original one. Also, the construction can be based on some other alg-universal category instead of Graph,. We shall see soon that e.g. UndGraph is alg-universal. Then, if we possess an object the only non-identical mapping of which is an involution, we can replace edges of undirected graphs by it. EXERCISES
1. Find several examples of rigid G such that no ( G , a , b ) satisfies the condition from 2.4.
3 3.
Two applications of the arrow construction: Symmetric graphs and acyclic graphs
3.1. Theorem. The category SymGraph (and therefore also the category UndGraph) is alg-universal.
Proof. Actually, all we need is to apply 2.4 to the facts already proved in 11. 34. Take the d = (( r, s),a, b)
from 11.4.5. From the definitions in 2.2 we see immediately that for any ( X , R ) , ( X ,R) * a! is in SymGraph. 2.4.(2) follows by the argument of 11.4.9. In that case a rigid ( X , R) was concerned, but what was really used was Just the antireflexivity of R.
3.2. Lemma. Let in a! = ((A,r), a, b) the gruph ( A , r) be connected. Then, for uny connected ( X ,R), ( X , R ) * .d is connected. Proof follows immediately from 2.3, 1.10 and 1.9. 3.3. Thus, the construction used in 3.1 proves a stronger statement, which will also be useful in further applications. Denote by SymGraph, the intersection of SyrnGraph and Graph, (see l.ll), by UndGraph,
the full subcategory of UndGraph corresponding to SymGraph, under the isofunctor from 1.5. Hence, the objects of UndGraph, are the con-
108
Ch. IV, COMBINATORICS
nected undirected graphs without loops with more than one vertex. We have Theorem. UndGraph, is alg-universal.
Proof. In the construction from 3.1 it suffices to check the validity of 2.4.(1) more closely. This condition is satisfied by 3.2. 3.4. Lemma. Let ( A ,r ) contain no cycle, no path joining a and b, and no path joining b and a. Then, for an arbitrary ( X , R),
(x,R ) * ( ( A ,r), a, b)
has no cycles.
Proof. (Notation from 2.2.) Let to,..., 5, be a cycle in ( X , R ) * d.Since ( A , r ) contains no cycle, there is a natural number i and an (x,y ) E R such that 4, 4 ( ~ ( x , y , ( A ) , Ci E ( ~ ( x , y , ( A )and ti+ 1 E ( ~ ( x , y , ( A ) . Let i be the least number with this property and let j be the largest one such that {ti,ti + 1 9
Then, by the definition of
.
'-5
tj}
c ~ ( x , y , ( A .)
E,
{ti>t j > c {(~(x,y)(a)y( ~ ( x , d b l } so that the sequence
(~(i,i)(ti), pG,\,(tj) ---7
contradicts the assumption.
3.5. Theorem. The category of connected graphs without cycles is alguniversal. Proof. Put A = {0, 1,2, a, b} and consider the ( A , Y) indicated in Fig. 4.3. We will show that d = ((A,r),a,b) is a suitable arrow. The validity of the condition 2.4.(1) follows from 3.2 and 3.4.
To check the conditi'on (2), let us first describe explicitly the relation of ( X , R ) * d.We have (for x, y E X ) :
i?
109
$4. More about undirected graphs
(i) < E x (ii)
~
iff i" = (x,Y,2 ) or iff iJ = x , iff ( = y or or or
r E ( x , y,2 ) .
5
=
(y,x , 4 ,
5 = ( x ,y, 0)
r
=
(4I', 2),
Let f : (A,r ) + ( X ,R ) * d be a morphism. Since arO, by (iv), f ( 0 ) =k ( x , y,2). Since Or 1, by (iHiv), f(0) =# (x,y,1). If f(0) = x E X,we have by (i) either f ( a ) = ( x , y,2) or f ( a ) = (y,x,2). This, together with 2 r a , contradicts (iv). Thus, for some (x, y) E R, f ( 0 ) = (x,Y,0) > and we obtain immediately by (ii) also f ( a ) = x. By (iHiv), holds only for
5 = ( x , y , l), so f(2)Rx
(x,Y,0)E 5 that f ( 1 )
=
(x,y,1). Now, we have
and f ( 2 ) E ( x , Y , I ) ,
so that, by (i) and (iii), f ( 2 ) = (x, y,2). Finally, ( x , y,2)Rf(b), and hence f ( b ),= x or f ( b ) = (x, y,1). The second alternative is, however, excluded by f ( b ) R ( x ,Y , 1). Thus, f = V(X,Y). EXERCISES
1. The full subcategory of Graph generated by the graphs such that every vertex lies on a cycle of a given length k 2 2, and no vertex lies on a shorter cycle, is alg-universal. 2. The full subcategory of SymGraph generated by the graphs such that every vertex lies on a cycle of a given odd length k 2 7, and no vertex lies on a shorter odd cycle, is alg-universal. (Hint: Replace the 7-cycles in the construction from 77.4.5by longer ones.) 3. Like E.3.2, but with k = 3. (Hint: In 11.4.5, add central points to the cycles.) 4. Like E.3.2, but with k = 5. (Hint: In 11.4.5, replace the edges of s) by pentagons in a suitable way.)
(x
5 4.
More about undirected graphs
4.1. Since the underlying set of ( X ,R ) * d depends on R. Theorems 3.1 and 3.3 do not answer the question as to whether there is a strong embedding
110
Ch. IV, COMBINATORICS
of Graph into UndGraph (which, as we already know by 11.5.3, would imply strong embeddability of any Re l(d), and consequently, by 11.5.8, of any small concrete category into it. This question will prove even more essential in Chapter VII). In this paragraph this question will be answered positively. Moreover, we will prove that there is a strong embedding of Graph into the category of undirected graphs with prescribed chromatic number 2 3 (see below). 4.2. Throughout this paragraph, the graphs are always undirected graphs without loops. 4.3. A subgraph of graph (X, R ) is any graph ( Y S) with Y c X and S c R. It is said to be a proper subgraph of (X, R ) if Y iX . A subgraph ( Y S) of (X, R ) is said to be full if, for y , y ’ ~Y yRy‘ implies ySy’. 4.4. Denote by 6, the graph
It is sometimes called complete graph (without loops) with n vertices. 4.5. An n-coloring of a graph ( X ,R ) is usually defined as a mapping cp from X into {0,1,..., n - 1} such that, for x R y , q ( x ) cp(y). Thus, it is a homomorphism (X, R ) 6,. A graph (X, R ) is said to be n-colorable if there exists an n-coloring of (X, R). If it is n-colorable but not ( n - 1)-colorable we say that it is n-chromatic or that its chromatic number is n, and write x ( X , R ) = n. The 2-chromatic graphs are often called bipartite graphs.
+
-+
4.6. Remark. The arrow used in 3.1 is 3-chromatic and, moreover, it has a 3-coloring cp with q(a) = q(b). Consequently, the graphs ( X , R ) * d are 3-chromatiq so that one has the following proposition : The category of connected 3-chromatic graphs and their homomorphisms is alg-universal. W 4.7. Lemma.
If there is a homomorphism f: ( X , R )
X(X, R ) I X(X’, R’). Proof. There is a cp: ( X ‘ , R‘) -+ Bx.Xr,Ro. Consider cp of:
-+
( X ’ , R‘) then
W
4.8. Lemma. Let ( X ,R ) be a graph with X finite, f a mapping of X onto itself: I f f is RR-compatible, it is an isomorphism of ( X , R ) onto itself: Proof. Consider the mapping g : R + R given by g((x, y } ) = { f ( x ) , f ( y ) ) . Since f’ is one-to-one, g is also one-to-one. Consequently, since R is finite,
111
$4. More about undirected graphs
g is also onto. Thus, if ( f’(x),f’(~’)}is in R, {x,y } is also in R and hence the inverse mapping to f is RR-compatible. H 4.9, Construction. Let wf2 3, n 2 3 be natural numbers. Denote by
G(m, n) the undirected graph ((0, 1, ..., mn}, R) with iRj iff 0 < Ii - jl < n or {i,j) = (0, nzn) or { i , j > = (0, (m - 1)” - 1). (See Fig. 4.4.) 6
+
4.10. Lemma. G(m,n) is rigid and x(G(m,n)) = n 1. Proof consists in four steps: I. Every proper subgraph (E;S ) of G(m,n) is n-colourable: Let k 4 Y. Denote by r(i) the remainder of i after dividing it by n and put
cp(i)
=
{r(i’ r(i - 1)
for i < k for i > k .
We will prove that cp is an n-coloring. Thus, ~ ( ( m 1) n - 1) is in ( n - 2, n - 1) and hence differ from 0 = cp(0). If {O,rnn) c Y we have 0 < k < mn and hence cp(0) = 0 n - 1 = cp(mn).If i < j and j - i < n, we have neither r(i) = r ( j ) nor r(i - 1) = r(j - 1). Thus, the only citsc left is i < k < j . Then, ~ ( i =) cp(j) means that ~ ( i=) r ( j - 1) so that j - i - 1 is divisible by n and hence equal to zero. This is impossible, since in this case, j - i 2 2. 11. G(m,n) is not n-colorable: Let cp: G(m,n) -+ 6, be an n-coloring. Since cp(O), ..., cp(n - 1) must differ
+
112
Ch. IV. COMBINATORICS
from one another, we can assume q(i)= i for i < n. Since we have n R i for i = 1, .. ., n - 1, necessarily q ( n ) = 0. Let q(i)= r(i) for i < k , let k 2 n. Since k R i for i = k - n + 1, k - n + 2,..., k - 1, we have q ( k ) = r(k) so that, by induction, q(i) = r(i) for any i. In particular, q(mn) = 0 = q(0) in contradiction with 0 R mn.
+
111. By I and I1 we see immediately that x(G(m,n)) = n 1. Moreover, by 4.7, we see that every homomorphism f : G(m,n) -+ G(m,n) is onto, and hence, by 4.8, it has to be an isomorphism.
IV. Thus, it suffices to prove that there is no non-identical isomorphism f : G(m,n) + G(m,n). Denote by p(i) the number of vertices joined with i. Thus, we have = n + l , P(0) p(m4 n, p((m - 1) n - 1) = 2n - 1 , otherwise for n - 1 I i 5 (m - l ) n for O < i < n - l
+1
p(i) = 2n - 2 , p ( i ) = p(mn - i) = n
We have p ( f ( i ) ) = p(i) and hence (since, for n 2 3, n =k 2n necessarily f({l, mn, mn - 1)) c (1, mn, mn - 1 ) .
+i-1 -
1, 2n - 2 )
We have mnRmn - 1 and 1 is joined with none of mn, mn - 1. Thus, f (1)
=
1 and f ((inn, mn - 1)) = (ma, mn - 1)
Now, 0 is the only point joined with 1 and some of mn, mn - 1.Consequently, f(0) = 0 By pf(i) = p(i) again we see that
f({i, mn - 1)) = {i, mn - i) (In the case of i that f(0) = 0.)
=
for
1 < i < n -2.
2 also p(0) = p(i) = p(mn - i), but we already know
Since out of i, mn - i (1 < i < n - 2) only i is joined with 0, we obtain f(i)= i
for
i
Finally, let k 2 n - 1 and let it be already proved that f ( i ) = i for i < k.
$4. More about undirected graphs
Since k is the only vertex joined with all k we have also f ( k ) = k.
-
n
+ 1, k - n + 2, ..., k
113 -
1,
rn
4.11. Lemma. (1) There *exists a connected 3-chromatic rigid undirected graph (B, e) such that - every b E B is contained in a 7-cycle and there is no odd cycle with the length less than 7, - there are b,, b,, ..., b4 in B with mutual distances of at least 6, - there exists a 3-coloring cp o f ( B ,e) with cp(bi)= 0 for i = 0,1, ..., 4. (2) For every n 2 4 there exists a connected n-chromatic rigid undirected graph (B,e) such that - every b E B is contained in a subgraph isomorphic to B,(see 4.4), - there are b,, b,, ..., b, in B with mutual distances of at least four, - there exists an n-coloring cp of (B,e) with cp(bi)= 0 for i = 0,1, ..., 4. Proof. (1) One can take (B, g) = ((0, 1, ..., ni - l), ((i, i + 1) I i = 0, ..., m - 2)) * d with the d from 3.1 (cf. 11.4.5), and with m sufficiently large. ( 2 ) One can take G(m, n - 1) with a sufficiently large m. 4.12. Theorem. For every n 2 3 there is a strong embedding of Graph into the category of connected n-chromatic undirected graphs and their homomorphisms. Proof. Take the (B,e) and b,, b,, ..., b4 from 4.11. Define a functor G : Set -+ Set putting G(X)=X x 3 u X x XUB,
(We assume the summands disjoint; More exactly, we should put, say, G ( X )= ( X x 3) u ( X x X x { 3 ) ) u ( B x (4))), and defining, for f : X + X ' , G ( f ) by formulas
( f ( 4i)
G ( f )(x, Y ) = ( f X() ' f ( Y ) )
for ( x , i) E X x 3, for (X> Y ) E x x x
G(f)(b)
b
for b e B .
G ( f )(x, i)
and
=
=
For an object ( X , R ) of Graph put
114
Ch. IV, COMBINATORICS
where 4
R =QVUR, i=O
Ro
=
R,
=
(((x, j), b,) 1 x E X , j
=
0, 1,2) ,
{ ( r ,( ~ , ( r j) ,- 1)) I r E X x X , j
I
R 2 = {((x,2),(x,j)) X E X ,j R,
=
((rJ3)IrEX x
R,
=
((hb4) r E R>.
I
=
=
1,2} where r
=
( ~ , ( r x) 2, ( r ) ) ,
0, I},
x},
This relation is indicated in Fig. 4.5. For f: ( X , R ) + ( X ‘ , R‘) a morphism of Graph define F(f):F ( X , R ) -+ F(X‘, R’) putting
F(f)= (F(X’>R’),qf), F ( X >R ) ) . Obviously, F is a one-to-one functor from Graph into the category of connected undirected graphs. Moreover, F ( X , R ) is n-chromatlc, since, by the definition of R, we can take the coloring cp of ( B , g ) from 4.11 and extend it over F ( X , R ) putting p(x, 2) = cp(x, y ) = 1, p(x, i ) = 2 for i = 0, 1.
Thus, it sufficesto prove that F is full. Let g: F ( X , R ) + F( S) be a homomorphism. The full subgraph of F ( X , R ) generated by ( X x 3) u ( X x X )
$4. More about undirected graphs
115
is bipartite: it may be colored by *, sending X x ( 2 ) and X x X to 0 and X x 2 to 1. Thus, there is no cycle of an odd length in X x 3 u X x X . Now, we have to do one s t y differently for n = 3 and n > 3. The case of n = 3: Let
1.
51,
’..>5 k
be a cycle meeting both X x 3 u X x X and B with the shortest possible odd length. Let k I 7. Then, because of the distances between b,, ..., b,, there is exactly one of the vertices bi among E p It cannot be the only element of B among Cj, since otherwise the remaining elements of the cycle (c) would form a path
’..)q k - 1 with k - 1 even and q1 E X x 3 u X x X , which would contradict the q17
necessary $(ql) =+ $(ylk- 1) (following from q1Rq2R . . . R q k - ). But bi are the only elements at which a cycle can leave B. Thus, after relabelling (c) so that E X x 3 u X x X , we have Y < s such that 5, = <,= b, and t jE B for r 5 j 5 s. Since there is no cycle of an odd length shorter than seven in B, we have s - r even and obtain a contradiction in a form of an odd-length cycle
51, ... 4,, 5 S + l , “.,ti 9
meeting also both B and X x 3 u X x X . Thus, each cycle of an odd length 1 7 in F ( X , R ) is contained in B, and since B is covered by such cycles, g(B) The case n > 3: There is no cycle
B.
40,51,52
meeting both X x 3 u X x X and B. (Obviously, only one of t j can be in B, but the remaining two cannot be joined.) Thus, since B is covered by triangles, we obtain immediately g(B) = B . Thus, since (B,e) is rigid, we have for all
y(b) = b bEB.
Now, take an x E X . We have b,R(x, 2)R(x,i ) R b , and hence
b2R’(g(x),2)R’g(x,i)Rbi (i
=
0, 1).
116
Ch. IV, COMBINATORICS
Hence, g(x) E X’ x (2) LJ B. But if g(x) E B we have g(x) =!= b, (because of b 2 R ’ g ( x ) ) and hence also g(x, i) E B in contradiction with the distance of b, and b,. Thus, g(x) E X’ x (2). Define f : X -+ X’ by g(x2 2) = (f(42 ) .
Now, we infer from (*) that either g(x, i) = (f(x), i) or g(x, i) = b,, i The second alternative contradicts g(x, i) R’b,. Thus, g(x, i ) = ( f ( x ) ,i),
i
=
=
0, 1.
0, 1 .
Finally, we have (x, O)R(x, y ) R ( y , 1) and (x, y)Rb,. Hence, ( f ( x )O , ) b ( x ,Y ) R ’ ( f ( X ) >1) and 9 ( X > Y)W%
7
which yields g(x, y ) = ( f ( x ) ,f(y)). Thus, g = G(f). For (x, y) E R we have (x, y ) R h, and hence (,f(x),f(y))I?’b,, so that ( f ( x ) ,j ( y ) )E R‘. Thus, .f is R R’-compatible. a 4.13. Remark. The category of bipartite graphs is obviously not alguniversal, because every bipartite graph with more than two elements has a proper endomorphism. On the other hand, every group is isomorphic to the group of automorphisms of a bipartite graph: see E.4.5. 4.14. Let n = {0,1, ..., n - 1). Denote by C(n)the graph
1
by g (n, (n x n)\{(i, i) i < .>) 9
n
the full subcategory of SymGraph generated by the ( X , R) such that for every x E X there is a subgraph of ( X , R ) isomorphic to C(n) and containing x. In Chapter V we will use the following Theorem. %,, is CI kq-i I i i i IW-S(I I
Proof. This is obvious for n = 2. Let n 2 3. Consider the G(m,n) from 4.9. We see that (*) For any two vertices x, y E G(m,n) there are subgraphs A , , . . ., A , isomorphic to C(n) such that x E A , , y E A , and the cardinality of A j n A j + , is n - 1. In the arrow construction from 2.2 put d = G(m,n) and take for a, b arbitrary non-joined elements. We see immediately that no subgraph of ( X , R ) * .disomorphic to C(n)meets two subsets of the form cp,,.y,(G(m,n))
$5. Partially ordered sets
117
in more than one point each. Thus, by (*), for a homomorphism f : G(m,n) + --t ( X , R ) * r;4 there is an (x,y ) E R with cp(x,y,(C(m, n)) I> f(G(m,n)). Since ‘p(x,y,(G(m, n)) is isomorphic to G(m, n), we see by 4.10 that f = Thus, the statement follows from 2.4. EXERCISES
1. Let k be a non-void small category. The category Graphk is alguniversal. More generally, if Ji is alg-universal and has initial and terminal objects, then Rk is alg-universal. 2. For a graph G denote by .Graph the category the objects of which are couples ( X , ‘p) where X is a graph and cp: G + X is an embedding of G into X as a full subgraph; the morphisms from ( X , cp) + (X’, cp’) are those f : X + X‘ for which f o cp = cp’. The category .Graph is alguniversal. (Hint: Use, e.g., the following construction: For G = (X,, R,) choose a rigid ( X , x {l),S ) and a zo E X,. Put
3. For every pair of monoids ( M I ,M , ) there is a graph G , and its full subgraph G, such that the endomorphism monoid of Gi is isomorphic to Mi. 4. There exists an alg-universal subcategory R of UndGraph such that for every G E obj R we can fix a coloring cG satisfying cG = cG,o f for all compatible f : G + G’. 5. Every group is isomorphic to the automorphism group of a bipartite graph. 6 . The category Soc(k) is defined as follows: The objects arecouples ( X ,R ) where X is a set, and R c ( A l A c X , cardA = k), the morphisms ( X , R ) -+ (X’, R’) are the mappings f : X -+ X’ such that, for every A E R, f ( A )E R‘. For k 2 2, Soc(k) is alg-universal. (Cf. CHIN,].) (Hint: Modify the construction of 4.9.)
Q 5. Partially ordered sets 5.1. A partial ordering on a set X is a transitive antireflexive binary relation, i.e. a relation R such that
118
Ch. IV, COMBINATORICS
(i) x R y and y R z imply x R z , (ii) never x R x . A couple ( X , R ) where X is a set and R a partial ordering on X is called partially ordered set in an abbreviated form, poset. The homomorphisms between partially ordered sets are often called monotone mappings.
In this paragraph we will prove that the category Poset
of partially ordered sets and their monotone mappings is alg-universal.
5.2. The length of a partially ordered set ( X , R), denoted by l ( X , R), is the supremum of all natural numbers n such that there is a path X ~ R X I...R R x , in ( X , R). For a partially ordered set ( X , R ) let us further define two functions cp and cc/ (more exactly, cp(x,Rl and $ ( x , R ) ) on X as follows: cp(x) is the supremum of all n such that there is a path x, Rx, R ... R x , - ~ R x in ( X , R ) , $(x) is the supremum of all n such that there is a path x R x , R ... R X , - ~ R X ,
in ( X ,R).
5.3. Obviously. we have Proposition. (1) If cp(y) and $(x) are finite, then x R y implies cp(x)< cp(y) and $(x) $(Y). (2) If f : ( X , R ) + ( X ’ , R’) is a monotone mapping then for every x E X
’
cp(f(x))2 cp(x) and + ( f ( x ) )2
+(XI.
W
5.4. Construction. The partially ordered set ( A , <)is defined as follows: A consists of (1) all integers and (2) all couples (i,j) of integers such that
if i 5 0, if i = 2n 2 0, if i = 211 + 1 > 0,
O S j 5 -i+3, O l j 5 3 , 0 5.j 5 2i 3 .
+
119
$5. Partially ordered sets
The relation
< is defined by (i,j) < (i‘,j‘) i < (it, j ’ ) never t < i.
iff iff
i
‘i i = i’ =
and j < j‘ or i = i’ - 1 ,
The partially ordered set obtained is indicated in Fig. 4.6 (where the arrows depict a subrelation of <, generating < by transitivity).
i i
i
l(3.9)
i
i i
i
Fig. 4.6
For a fixed i define A(i) as the set of all (i,j) with (i,j) E A . 5.5. Lemma. ( A , <) is rigid.
Proof. First, we observe that (for cp from 5.2)
(1)
cp(i) = 0 ,
q(i,j ) = j
+1
Let f : ( A , <) -+ ( A , <) be monotone. By 5.3.2, and (1) we have f(i, 0) = ( k , j ’ ) .
For j > 0 we have (i, 0) < (i,.j) and hence (k,j ’ ) < f ( i , j ) , so that f ( i , j ) = ( k ,j ” ) .
=
120
Ch. IV, COMBINATORICS
Thus, we have a mapping g:
z+z
(Z is the set of integers) such that
f ( 4 )= A(g(i)).
Put
A(i) = max (cp(i, j ) 1 (i, j ) E ~ ( i ).)
By 5.3.(2) we obtain A(g(i)) 2 A(i) .
(2)
We have (3)
A(i) = lil + 4 , A(i) = 4 , A(i) = 2i
for i 2 0 , for i = 2n 2 0 , for i = 2n + 1 2 0 ,
Since i < (i, 0) and i < ( i + 1, 0), f(i,O) and J(i a common lower bound and hence Ig(i) - g(i (4) By induction, we obtain easily
+ 1)1 I
+ 1, 0)
have to have
1.
- s(j)l2 li - jl. (5) Let i be a negative integer. By (2) and (3) we immediately obtain that g(i) + 2n > 0. Suppose that g(i) = 2n + 1 > 0. Since g(i - 1) can be neither 2n nor 211 + 2 we see that for all j < i also g ( j ) = 2n + 1 in contradiction with (2) and (3).Thus, g(i) is negative, and we conclude (again by (2)and (3)) that (6) for i < 0 , g(i) Ii .
Now, take an n 2 0. Suppose that g(2n + 1) = i < 0. Hence, for all m 2 n, g(2m + 1) < 0 (otherwise, by (5), for some m > n either g(2m 1) = 0 or g(2m + 1) = 1 in contradiction with (2)).Thus, we obtain, by (5),
+
g(2m
+ 1) 2 i - 2(m - n)
which contradicts (2)for sufficiently large m. Obviously, G(2n Thus, by (2) and (3), (7)
for n 2 O
g(2n
+ 1) 2 2n + 1 .
Thus, by (6) and (7), g( - 1) < 0 < g(1) and hence, by (4), g(0) = 0 ,
+ 1) + 2m > 0.
121
$5. Partially ordered sets
and by (5) always Thus, by (6) and (7),
5 lil .
c
g(i) = i for the negative integers and for all the odd ones. By (4), finally, also g(i) = i for the even ones. Consequently, we have
f ( i , j ) = ( i d ).
By (1) and 5.4, j ‘ 2 j . By induction we see easily that if j’ > j , then k’ > k for all k > j , which leaves no possible value for the largest j . Thus,
f(i,j) = (i,j). Finally, f ( i ) = i, since i is the only element 5 < ( i 1, 0).
+
5 with both 4 i (i, 0) and
rn
5.6. Theorem. There exists a strong embedding of Graph into Poset. Consequently, Poset is alg-universal.
Proof. Define a functor G: Set + Set putting G ( X ) = X u ( X x 2) u ( X x X u A )
(with A the set from 5.4; again, we assume the summands disjoint without making them explicitly so), G ( f ) ( x ) = f(x) > G ( f ) ( x , i ) = ( f ( x ) ,i ) , G ( f )( x ,Y ) = ( f ( x )f,( Y ) )
and
G(f)(a)
=a.
For ( X , R ) E obj Graph put F ( X , R ) = (G(X),R )
u 3
where R is the smallest transitive relation containing (< u Ri)with i i=O as in 5.4 and Ro = R, = R2 = R3 = R, =
{((x,j),x) I x E X , j = 0, 1) {(r,(zlr),j ) ) I r E X x X , j = 0, 1)
1j
0, 1) ((4, r ) r E X x X > ((6,r ) Y E R>. {(2j,
(XJ))
I I
=
where r
=
(zo(r),zl(r))
122
Ch. IV, COMBINATORICS
Thus, the situation is as in Fig. 4.7 (again, only a subrelation of
R by transitivity is indicated).
w generating
4.1
For a homomorphism f : ( X ,R) -+ ( X ’ , R’) put F((X’,R’),f ,
(x, R))= ( F ( K R‘),G(f), F ( X R ) ) .
It is easy to check that thus a one-to-one functor F : Graph + Poset is obtained. Hence, it suffices to prove that F is full. Let g: F ( X , R ) -+ F(X’, R’) be monotone. No ( E G(X)\A is contained in a path of length five and every E A is contained in such one. Since g sends paths to paths, we see that g ( A ) c A . Hence, by 5.5,
<
for
UEA,
g(a) = a .
Take an X E X .We have O w x and 2 w x , and hence Ow’g(x) and 2w’g(x). Hence, by the definition of R, g(x)E X ’ and we can define f : X -+ X’ putting
f(x) = Q(4.
We have 0 R ( x , 0)R x. Thus, 0 g(x, y )a ’ f ( x ) , and hence necessarily g(x,O) = (f(x),O). Similarly, using 2 instead of 0, we obtain g(x, 1) = = ( f ( x ) 1). , Further, we have 4 R ( x , y ) w ( x ,0) and (x, y ) R ( y , l), so that
123
$5. Partially ordered sets
4R’g(x, Y ) R ’ ( f ( x )0) , and g(x, y ) R ’ ( f ( y ) I), , which yields g(x, Y ) = ( f ( x ) , f ( y ) ) . Thus, g = G(f). Finally, if ( x ,y ) E R, we h$ve 6w(x, y ) and hence 6R’(f ( x ) ,f (y)),so that ( f (x),f ( y ) )E R‘. Thus, f is an RR’-homomorphism. 5.7. A functor F : K
is said to reflect isomorphisms if a is an isomorphism whenever F(a) is. If K, L are thin without distinct isomorphic objects and if we interpret them (see 1.3.6) as reflective partially ordered sets the functors are (see 1.3.11 F) the order-preserving mappings. The condition “to reflect isomorphisms” means now that for x I y and x y we have F(x) $: F(y). Thus, if we define partial ordering < (in the sense of 5.1) on K , L putting +L
+
x
iff x s y
and x + y ,
the isomorphism reflecting functors are exactly the < -monotone mappings. Hence, we obtain, as an immediate consequence of 5.6, Proposition. The category of small categories (infact, already the category of smull thin cutegories) and of their isomorphism rejlecting functors is alguniversal. 5.8. The embedding from 5.6 differsfrom all those constructed previously (to its disadvantage) by sending finite objects to infinite ones. The question, arising naturally as to whether this can be amended, is answered negatively. In the finite set theory, Poret would not be alg-universal. Moreover, we have the following Proposition. A partially ordered set of afinite length is either rigid or it has a proper endomorphism. Consequently, no non-trivial group is the endomorphism semigroup of a partially ordered set of a finite length. Proof. Let 1(X,R ) = n. Thus, there is a path
X O R X I R X... ~x,-~Rx,
in X . For a general x E X put f ( x ) = xg(,.) (‘p from 5.2). By 5.3, f is monotone. Now, either X $: {xo,...,xn} and then this f is a proper endomorphism, or X = { x o ,..., x,). In the latter case, let g be an endomorphism. We have cp(xi) = i, $‘(Xi) = = n - i. By 5.3, ‘p(g(xi))2 i and $(g(xi))2 n - i . Thus, if g(xi)= x j , we have j 2 i and n - j 2 n - i, so that j
=
i.
124
Ch. IV, COMBINATORICS
5.9. Remark. On’the other hand, we will see in the next paragraph that every finite group is isomorphic to the autornorphisrn group of a finite partially ordered set.
0 6.
Graphs with strong homomorphisms
6.1. Recall the well-known composition of subsets of Cartesian products : If R is a subset of Y x X and S a subset of 2 x E: a subset S R of Z x X is defined by 0
(1) S c R = = {(z, x) E 2 x X there is a y E Y with (y, x) E R and (z, y) E S } .
1
If we represent mappings f : X ranges Y -see 1.3.5 A) as
-+
Y (neglecting, for a moment, their
4I
(2) Nf(x), x x>7 the usual composition gof of mappings obviously coincides with the composition given by (1).
I
6.2. Remark. We choose therepresentation {(f(x), x) x E X } off instead of the more common { ( x , f ( x ) ) I x ~ X }The . reason is that we want to preserve for R c Y x X the convention of writing yR x for (y, x) E R (see 1.1.3.). Then, the equality y = f ( x ) is represented by yfx, while the other representation would lead to the more confusing xfy.
6.3. Let ( X , R), (X‘7R’) be (directed) graphs. We see that a mapping f : X -+ X’ is an RR’-homomorphism iff f o R c R‘cf. (Really, if xRy implies f ( x ) Rf(y) and if x’(foR)y, we have an x with x’fx (i.e., x’ = f ( x ) ) and xRy. Consequently, x’ = f(x)R’f(y) and f(y)fy, so that x‘(R’ 0f ) y . On the other hand, if f 0 R c R’ of and if xRy, we have ( f ( x ) , y ) ~ f o Rc R’of and hence there is a z with f(x)R’z and zfy, Thus, z = f(y) and hence f ( x )R’f(y) ). 6.4. We say that a mapping f : X
+ X‘
is a strong RR’-homomorphism if
f c R = R’of. Obviously, the identity is a strong homomorphism and (due to the evident associativity of the composition o), strong homomorphisms compose to
125
$6. Graphs with strong homomorphisms
form strong homomorphisms. Thus, graphs and their strong homomorphisms form a category, which will be denoted by
*
Graph,.
Similarly, we will denote by Poset, its full subcategory generated by the partially ordered sets (see 0 5). 6.5. We have
IsoGraph,
=
IsoGraph, IsoPoset,
=
IsoPoset .
Indeed, if f 0 g and g 0f are identities and if f 0 R c R' f and g 0 R' c R o g , wealso have R ' o f = f o g o R ' o f c f o R o g o f = f o R . 0
6.6. Theorem. Poset, (and, consequently, Graph,) is alg-universal. Moreover, there is a full embedding of Graph, into Poset, sendingfinite graphs to finite graphs. Proof will be based on 2.4. Take A = (0,1, a, b ) and define (A, r ) as indicated in Fig. 4.8. Put d = ( ( A ,r), a, b).
It suffices to check the conditions (l),(2) and ( 3 ) of 2.4 (this is the first time we have to check also (3), since Poset, is not a full subcategory of Graph). For an (X, R) E objGraph, (we assume X n R = 8) we have ( x , R ) * ~= ( X U ( R x 2 ) , R ) with
R described by for
never ( R x , ( i i ( x , y , ~ ) iff ( = ( x ,y , I) or ( = x or ( = y , ( R ( x , y , 1) iff ( = y . X E X
Obviously, is a partial ordering. Thus, the condition (1) of 2.4 is satisfied. Let f : X -+X ' be RR'-compatible. By 2.3 and 6.3 we have ( f * d) R c c R' ( f * d)so that, to check (3), it suffices to prove that 0
0
-
R'og
=
goR
126
Ch. IV, COMBINATORICS
for g = f * a?. We have g(x) = f ( x ) , g(x, y , i) = ( f ( x ) , f ( y ) ,i). Take a ( 5 , q ) ~ w ’ o g Thus, . 5R’g(q) and hence, by q $ X . If q = ( x , y , l ) , we have < R ’ ( f ( x ) , f ( y ) 1, ) and hence 5 = f ( y ) , so that 4 g y and y R q , and hence (5, q) E g R. If q = (x, y , 0), we have
(w),
0
either
or
5
=
( f ( x ) , f ( y ) 1) , and hence
5 = f ( x ) and hence
or finally
4%(x,
Y 3
1)Rr
5gxjb
3
3
5 = f ( y ) and hence SSYjb.
) strong homomorphisms (which To check (2), observe first that ( P ( ~ , ~be follows from a quite analogous procedure to that in checking (3)). Now, let f: A + X u ( R x 2) be such that f o r = R of: In particular, it is rR-compatible so that we see from brlrO and that
-
(w)
(*I
Also, since the equality would imply f ( l ) w ( x ,y , 1) and hence f ( 1 ) E X . Thus, there is an ( x , y ) E R with
f(0) = (x, Y , 0) Hence further (by (*) and
I
(w))f ( 1 ) = (x, y , 1 ) and f(b) = y .
Finally, we have (x,O)ER o f c f o r. Thus, there is an Q E A with f ( a ) = x . Since x y , the only Q left is the a so that f(a) = x .
+
6.7. By 6.5 and 6.6 we obtain immediately (cf. 5.9)
Corollary. For every finite group G there is a finite partially ordered set such that its automorphism group is isomorphic to G.
6.8. Here, the arrow construction gives all one can expect, namely the extendability of Graph (and hence, under (M), that of every concrete category) in Poset,. There is no strong embedding of Graph into Graph, for the same reason for which there was none into the categories of algebras:
$7. Graphs with loops
127
Suppose there is one, say F , and let G be its underlying functor. Take an X 8. Consider the j : ( X , @)-+ ( X , X x X ) given by j ( x ) = x, and its image F(j): (G(X),R ) --* (G(X),S ) .
+
We have G ( j )o R = S 0 G(j),but G ( j ) is the identity so that R = S . Hence, there is a morphism from F ( X , X x X ) into F ( X , @)and hence F is not full. 6.9. In Chapter VI 0 1 we will see an interesting consequence of the fact that Poset, is alg-universal. Namely, after a suitable translation one sees that it is, in fact, a category of topological spaces and their open local homeomorphisms so that it gives the alg-universality of this important topological category. EXERCISES
1. Denote by Graph,, the category of all graphs ( X , R) and all the mappings f : X -+ X’ such that (x,y ) E R iff ( f ( x ) , f ( y ) E) R‘. Then Graph,, is not alg-universal. (Hint: Consider a morphism such that cardf-’(x’) > 1 for an x’. Study the subgraph generated by f - ‘(x’).) 2. Denote by Graph,,, the category of all graphs ( X , R ) and all the mappings f : X + X ‘ such that ( f ( x ) , f ( y ) ) ~ R 3’ ( x , y ) ~ RThen . Graph,,, is alg-universal. (Hint: It is equally realized with Graph.)
0 7.
Graphs with loops
7.1. Obviously, the full subcategory of Graph generated by the ( X , R ) with reflective R cannot be alg-universal because of the constant mappings which are always homomorphisms. But the constants are also the only reason why it is not. In this paragraph, we will prove that it is almost alg-universal in the sense of 1.8.13. Throughout this paragraph denote by
2l the full subcategory of Graph generated by the ( X , R) such that (i) for every x, x R x , (ii) if x R y and y R x then x = y , (iii) ( X , R ) is connected. We will prove that 2l is almost alg-universal.
128
Ch. IV, COMBINATORICS
7.2, Lemma. Let f : ( X , R ) + (X’, R’) be a homomorphism, let (X’, R’) be in a.I f x R y and y R z, and if f ( x ) = f ( z ) , then also f ( x ) = f f y ) . Proof follows immediately from (ii). 7.3. Let ( X , R ) be a graph. Define an equivalence
E
1
on {(x, Y ) E R x =k y } putting (x,y ) E ( u , 0 ) iff either (x,y ) = (u, u) or there is a sequence ( ~ 0 Y3O ,
zo)? ( X I ,
...) ( X n , Y,, zn)
~ 1 zl)? ,
of triples of elements of X such that xi R yi R z R~ x ,~ xi
and for i
=
* yi * * x i , zi
{x, Y }
= { X O > Yo, Z O }
(u’ u}
c {xn, Y,, Z n }
> >
0, ..., n - 1, l{xi,yi>zi}n {xi+l>Yi+i>zi+l}/ 2 2
Denote by E(x,y) the equivalence class containing (x,y ) and put
I
E(x, y) = ( z E x q z ,u) E E(x,Y ) v
(u, z ) E E(x,Y))} .
7.4. Lemma. Let f:( X , R ) + (X’,R’) be a homomorphism, (X’, R’)Eobj 2I. Let, for an (x, y ) E R, x $; y and f ( x ) = f ( y ) . Then, for all z E E(x,y),
f(4 = f (x).
Proof. If x R y R z R x and f ( x ) = f(y) we have f ( z ) = f(x) by 7.2. Now, the statement follows immediately by induction using the definition of E . I 7.5. Denote by A
the object of ‘2I with the underlying set (a, b,c,d,e} and the relation indicated in Fig. 4.9 (where the loops are omitted).
129
$7. G r a p h s with loops
7.6. Lemma. A homomorphism f:A + A is either the identity or a constant. Proof. We have E(a, b), = ((a,b), (a, c), (e,a), (d, b), (b,e), (c,e), (e,d ) } . Thus, by 7.4, if f ( x ) = J ( y ) for any ( x ,y ) E E(a, b), f is a constant mapping. The only remaining possibilities of common values for two distinct X, y are f ( a ) = f ( d ) , f ( b ) = f ( c ) and f ( c ) = f ( d ) . By 7.2 we have; in the first case, f ( a ) = f ( c ) ,in the second, f(d) = f ( b ) ,and in the third, f ( c ) = f(e). Thus, in any case, f is a constant again. Hence, iff is not a constant, it is an isomorphism of A onto itself. Since e is the only elment in which two arrows start as well as terminate, we have
f(e)= e . Consequently, f((a, d } ) = {a, d } and f ( { b ,c } ) = (b, c}. Since from a and c two arrows start and only one does from b and d , we have f l u ) = a, f(d) = d, f(c) = c and f ( b ) = b. rn 7.7. By the definitions in 7.3 we obtain immediately
Lemma. Let f: ( X , R) + (X’, R’) be a homomorphism such that for every 3-cycle x , y, z in (x,R), x += y z x implies f ( x ) f ( y ) f(z) f ( x ) . Then, for every ( x ,y ) E R,
+ +
f(&
Y))
+
+
+
rn
E ( f ( x ) f, ( Y ) ) .
7.8. Construction. For a graph without loops ( X , R) put
F ( X , R ) = (xu ( R x 4) u 2, W )
W
defined by:
=
(x,y,O) or
(the summands are taken as disjoint), with for an X E X , xR<
iff
(=x
or x R 5
or
5
5
=
(z,x,l),
and, for an ( x ,y ) E R, ( x ,y , 0)Rt
iff
5
=
( x ,y , 1 ) R t
iff 5
=
or
( x ,Y , 0) or 5 = ( x ,Y , 1) 5 = ( x , y , 2 ) or 5 = 0 ,
5 = x or 5 = ( x , ~2,) , or 5 = y or 5: = ( x , y , 3 ) , or 5 = ( x , y , I) or t; = 1 .
( x ,y , 1) or
(.Y.,..~)R< iff 5 = ( x , y , 2 ) ( . Y , . v , ~ ) R ~iff = ( x , y , 3 ) OR< iff 5 = O or ( = 1 or < = ( x , y , 3 ) , Id< iff 5 = 1 or 5 = ( x ,y , 2 ) .
130
Ch. IV, COMBINATORICS
Furl
The
4.10
7.9. Theorem. The category ’% of connected antisymmetric graphs with loops and their homomorphisms is almost alg-universal. Proof. Let 23 be the category of graphs without cycles (which is already known to be alg-universal, see 3.5). Construct a functor F : 23-a
taking the F ( X , R) from 7.8 for the values in the objects, and defining F ( f ) for a homomorphism f : ( X , R ) -P (XI,R’) by F ( f ) (x)
=f(x) 7
F ( f ) (x, Y , i) = (f(x), f ( Y ) , i)
2
F ( f )( i )
=i‘
97. Graphs with loops
131
We see easily that every F(f) is a (non-constant) homomorphism and that F is a one-to-one functor. Now, let g !F ( X , R ) + F(X’, R’)
be a non-constant homomorphism. Let us summarize some obvious properties of F ( X , R). First, we check easily that
(1) any cycle to, tl, t2 with cIJ{x, Y)‘ Further, we see that
5, =k 5, for
i =k j is contained in some
(2) for ( x , Y ) =k (U>u) > JC,,(X,Y ) n C,,(U, 0)i I 1 and that (3) JCOI(X?Y ) f-7 CIj(X9 Y)l 5 1 . Also obviously, (4) the full subgraph of F ( X , R ) generated by QI(x,y ) is isomorphic to the A from 7.5. Now, by (l), (2), (3) and (4) we see immediately that ( 5 ) E(o, 1) = {0,1} and for (5, y) (0, l), E ( < q ) = Q, ( x . y) for some 1, x , y. (And, of course, Q o ( x . j ) IS cqual to, say, E ( x , y ) and Q l ( x , y ) to E((x,y , o), 0) .) Hence, by 7.7,
+
(6) g(Ql(x,Y ) ) = Qj{x’,Y’) for ~ o m je, x’, Y’ . We have
*
(7) do) 9 0 ) since otherwise, by (6), (4) and 7.6, for any ( x , y ) the whole Q , ( x , y ) is sent to g(0). In particular, g(x, y , 0) = g(x, y , 2), which again by (6), (4) and 7.6 implies that also Q o ( x , y ) is sent to g(O), so that g would be a constant. Moreover, (8) g is one-to-one on every Q,(x,y ) , since otherwise we see easily by 7.6 (used once for i = 1, twice for i = 0) that g(0) = g(1). We cannot have g(Qo(x,y ) ) = Q1(x’,y’) since according to the isomorphism from (4) and 7.6 it would mean that g(x, y , 0) = 0 in contradiction with the necessity to map also Q l ( x ,y ) isomorphically ( (x,y , 0) is its bottom point and 0 is a bottom point of no copy of A in F(X’, R ’ ) ) .Thus, we have, by (6) and 7.6 again,
132
Ch. IV, COMBINATORICS
(9) for every (x, y ) an (x’, y’) such that g(x) = x’, g(y) = y’ and g(x, y, i) = = (x‘, y’, i) for i = 0,1,2. Thus ( ( X , R ) is connected), we can define f : X + X’ by f ( x ) = g(x) and we see that it is an RR‘-homomorphism. Finally, consider Q,(x, y). We know already that g(x, y, 0) = (f(x),f(y), 0) and that Ql(x, y) is isomorphically mapped onto a Qi(x’, y’). This leaves only i = 1, x’ = f ( x ) and y‘ = f(y), and
(10) g(x, J’, 3 ) = ( f ( X ) ? f(Y), 3 ) g(0) = 0 Thus, by (9) and (lo), g = F(f). 2
7
g(1) = 1 .
R
EXERCISES
1. The category of all closure spaces and their continuous mappings is almost alg-universal. (Cf. [H2]). (Hint: The category of graphs without cycles is realizable in it.) 2. The category of all symmetric graphs with loops is not almost alguniversal. 3. The category of all transitive graphs with loops is not almost alguniversal.
0 8.
Sets with two equivalences
8.1. Among binary relations, the equivalences (is. the reflexive, symmetric and transitive ones) play a fairly important role. From the point of view of choosing compatible mappings among all mappings, however, a single equivalence is uninteresting. Very few groups can be represented as automorphism groups of an equivalence (due to the obvious fact that an equivalence permits all permutations inside the equivalence classes and all permutations of equally large equivalence classes, e.g. only two non-trivial abelian groups are representable this way), not to speak about representations of monoids as endomorphism monoids. With two equivalence relations, the situation changes radically. In this paragraph we will prove that every group can be represented as the group of all invertible mappings preserving two equivalences on a set simultaneously. Moreover, for concrete groups, extensions of this kind exist. The proof will be done by the method described in 1.8.12, constructing an alguniversal category with the isomorphisms coinciding with the two-equivalences-preserving permutations.
$8. Sets with two equivalences
133
8.2. A quasiequivalence on a set X is a relation 4 c X x X such that there is an equivalence e with q = e\A where A is the diagonal. In other words, a quasiequivalence is,a relation q such that never x q x ,
(1)
(2) (3)
XqY xqy
and y q z
* Yqx, and x + z
* xqz.
8.3. Define concrete categories EE, Q E and QQ as follows: In all the cases the objects are triples ( X , el, e2) where X is a set and ei (i = 1,2) equivalences on X . The morphisms from 2 = (X, e l , e2) into 2’= (X’, e;, e;) are triples (3,f, s’),where f : X -+ X’ is a mapping which is in E E: e,ej-compatible for i = 1,2 in B E : (e,\A)(ei\A‘)- and e,e;-compatible in QQ: (ei\d)(ei\A’)-compatible for i = 1,2
(where A , A‘ are the diagonals in X x X and X’ x X’). The composition is defined in the obvious way, and for the forgetful functor the natural one is taken. 8.4. Obviously,
ISOE E
=
ISOQ E
=
ISO QQ
8.5. Obviously, B E (QQ, resp.) is equally realized (see 1.4.8) with the __full subcategory Q E (QQ, resp.) of Re1 (2,2) generated by the ( X , rl, r2) such that r1 is a quasiequivalence and r2 an equivalence (both ri are quasiequivalences, resp.). E E is itself a full subcategory of Re1 (2,2). EE is itself a full subcategory of Re1(2,2).
8.6. Theorem. B E is alg-universal. Proof will be done by constructing a full embedding __
F : SyrnGraph,, --t B E . Put F ( X , R ) = (R,q, e), where (x, Y )4 (XI?Y’) iff x = y’ and x’ = y , (x, y ) e (x’, y‘) iff x = x’ . (Obviously, e is an equivalence. Since ( X , R ) has no loops, q is a quasiequivalence.)
134
Ch. IV, COMBINATORICS
For a compatible f:(X, R ) -+ ( X ' , R') define F(f) by F(f)(x,y ) = = ( f (f(Y)). 4 Since every point of X is joined in R with some other one, we see easily that F is a one-to-one functor. Let ( X , R),( X ' , R') be objects of SymGraph, and let g: R -+ R' preserve q and e. For x E X choose a y e X with (x, y) E R and define f (x) E X ' by ( f ( x ) 2) ,
=
g(x,Y ) .
Since g preserves the equivalence e, f ( x )does not depend on y and we obtain X'. Now, take an (x,y ) E R. We have (x, y)q(y,x ) and a mapping f : X hence, for q(x,Y ) = ( f ( x ) z, ) and q(y,x ) = ( f ( y ) ,u), -+
(f
( 4 3 . )
4 ( f (Y),
4.
Thus, f ( y ) = z and hence
9(X?Y ) = ( f (f4 ( Y ) ).
Finally, for an (x,y ) E R we have ( f ( x ) f, ( y ) )E R' and hence f is RR'compatible. Thus, F is full. rn
8.7. By 8.6, 8.5 and 1.8.12 we obtain immediately Corollary. Every group is isomorphic to the automorphism group of an oh1c.c.t of EE. Mormrer, for wery concrete group G c X x there is an exre/l.sw I I I Is0 EE. S.,,. Since every constant mapping preserves any equivalence relation, monoids cannot be, in general, represented as endomorphism monoids of sets with a collection of equivalences, whatever their number may be, particularly, by objects of EE. In view of 7.6, it is somewhat surprising that also with quasiequivaiences, in particular in QQ, the monoid representation problem has a negative solution. In the rest of this paragraph we will prove that e.g. the monoid
M = ((1, u, v},
0)
with the multiplication defined by 1 x = x 0 1 = x ( x = 1, u, v), x y = u otherwise, is isomorphic to no endomorphism monoid of a set with a collection of quasiequivalences. 0
0
0
8.9. Lemma. Let q be a quasiequivalence on X , let f: X
-+
X be such that
$8. Sets with two equivalences
135
Then f o r x q y we have f ( x ) = x $ f ( y ) = y. Proof. By the symmetry, it suffices to prove that f ( x ) = x implies e f ( y ) = y. We have Y q x and x = f ( x ) q f ( y ) .
*
Thus, if f ( Y ) Y , we have Y 4 f ( Y ) and consequently f (Y)4 ( f ( f(Y))) = f (Y) in a contradiction with the antireflexivity.
8.10. Proposition. Let (qi)i.J be a family of quasiequivalences on a set X . Then the monoid of all f : X -,X such that for euery i x q i y implies f ( x ) q f ( y ) ,is not isomorphic to M from 8.8. Proof. Let M be represented by the endomorphisms of (X,(qi)ieJ), denote by 1 the identity mapping of X , by u and u the mappings corresponding, in this representation, to the homonymous elements of M . We have u =t= 1 and u o u = u, so that both
I
A = (x u(x) =t= x } and
B
=
I
( x u(x) = x }
are non-void. For X E B we have u(x) = u(u(x))= u ( x ) = x .
Thus, since u =t= u, there is an a E A such that .(a)
For X E A we have
U(.)
*
U(.).
=I= x
(otherwise u(x) = v(u(x))= x ) . Thus, a
Consequently, u(u(a))= .(a)
* * u(a) * .(a)
=+
a.
u(a) and hence
u(a)E A .
Put C
= (X
I 3x0,. ..,
X, E X
3i0, ..., in- 1 E J
(U
Suppose that ~ ( aE )C . We have x i , . .., x,and consequently
= XO
& xjqi,xj+ 1 & X = &)) .
such that
aqioxlqil ... x,-lqi,-lu(a)
u(a)qio"(xl)qil ... 4 X n - l)qin-t ~ ( u ( a )=)
4.)
136
Ch. IV, COMBINATORICS
in contradiction with 8.9, since .(a)
4). Now, define f : X
+X
=/= u(u) and uu(u) = .(a), Thus,
$c.
putting for X E C , for x $ C .
f ( x ) = .x f ( x ) = u(x)
Since x q i y implies that either both x and y are in C or both are in its complement,fpreserves all the qi. Since f(a) = a, fdiffers from u and u. Since f(u(a))= uu(u) = .(a) $. u(a), f i s not the identity. Thus, there exists an endomorphism of ( X , ( q & ) which i s not in M , in contradiction with the assumption.
0 9.
A technical lemma
9.1. The aim of this paragraph is to prove a proposition which is not very interesting in itself, but which proves helpful in some constructions in the forthcoming chapter. Roughly speaking, it states that there is suitable category of sets with couples of symmetric relations and their homomorphisms, such that (m) it is alg-universal, (p) the homomorphisms with respect to the couples of relations coincide with those with respect to their unions. 9.2. Denote by
6 the following concrete category: The objects are triples ( X , R,, R,) such that (1) for i = 1,2, ( X , Ri) are objects of UndGraph without loops, (2) R i n R z = @ , (3) for every X E X there are r i € Ri such that X E ~ n , r,. The morphisms from 8 = (X, R,, R,) to = (X’, R;, R;) are triples (g’,f,2) such that f : X -+ X’ and for ri E Ri (f x f )(ri)€ RI (in other words, such that both ((X’,R!),f,(X, Ri)) are in UndGraph). The composition and the forgetful functor are the obvious ones.
x‘
9.3. The functor Qi:
6 + UndGraph
is given by @ ( X ,R,, R , ) = ( X , R , u R,) and @ ( f ) ( x ) = f ( x ) .
137
$9. A technical lemma
9.4. Theorem, There is a strong embedding
F : Graph -+ Q such that @ restricted to its &age is a realization.
Proof. Consider the embedding constructed in 4.12 for n = 4. Denote by % the image of Graph under this embedding. It suffices to prove that there is a strong embedding F of % into Q with Qi F(N) full faithful. First, observe that we have a rigid (B,e) such that
I
(1) Every object of % is of the form
.
(xu B, R )
where the union is disjoint, and the full subgraph generated by B is (B,e), (2) for every two b, b’ E B there is a sequence b
in B such that for every i
=
=
bo, b,, ..., bk
=
b’
0,1, ..., k - 2
bi, bi+l,bi+z is a cycle, (3) every 3-cycle in ( X u B, R ) is contained in B, (4) every element of X is joined with an element of B, (5) there are c, d, e in B such that no two of them are joined and such that there is no b joined with both c and e. (6) The c, d, e from ( 5 ) may be chosen in such a way that for b E {c, e> there cannot be P R b R x or BRx with b E B and X E X . Now, construct putting where
F: %-+Q
F ( X u B, R ) = ((xu 8)x 2, R,,It2)
(t,i ) R l ( q , j ) (t,i ) R 2 ( q , j )
iff ( i = j and CRq) or EX and q ~ { c , e } ) or ( i ;(~ c , e } and VEX),
iff either (t = q and i += j ) or ((t,i) = (c, 0) and ( q , j ) = (4 1)) or ((t,i ) = (4 1) and (q,j) = (c, 0)).
138
Ch. IV, COMBINATORICS
For a homomorphism f put F ( f ) (c, i) = (f(t), i). Obviously, F is a one-to-one functor. Since @ is faithful, it suffices to prove that Qj F is‘full. Put i? = 8 , u and consider a homomorphism 0
g: ((X u B) x 2, R ) -+ ((X’ u B) x 2, R’).
First, we see that (7) no 3-cycle in ((X u B) x 2, meets X x 2 . (In a cycle (x, i), (5, il), (q, i2) there cannot be i = i, = i2 by (3) and (6). Thus, let i , =+ i. Then = x, while (x, 0) and (x, 1)have no common vertex to join.) Further, we immediately see that (8) no 3-cycle meets both B x ( 0 ) and B x (11, so that, since 3-cycles go into 3-cycles, we obtain by (2) that
w)
<
= x ( j }. Write q(i)for this j . By (l), since (B, e) is rigid, we have (9)
g(B x (i})
g(b, i) = (b, cp(i)).
Obviously, q(0) =+ q(1). There cannot be q(0) = 1, since (c,O)R(d, 1) and ((c, I), (40 ) )B R. Thus, (10)
g(b, i) = (b, i )
for b E B , i = 0, 1 .
Now, for ( x , i ) ~ Xx (i) we have (c,i)R(x,i)R(e,i),hence (c,i)R’g(x,i)R(e,i), so that g(x, i) E x x ( i } . (11) Define f : X u B + X’ u B by
0)= g(x, 0 ) . For (x, EX x (1) wehave g(x, l)R(f(x),O) andby (11) g(x, 1 ) ~ xx(1). Thus, g(x, 1) = (f(x), 1). Finally, if CRq, we have (C, O)R(q,0) and consequently (f(C),O)R’(f(q),0). Thus, either f ( [ ) R ’ f ( q ) or (after possible interchange) f ( 5 ) E X‘ and f ( q )E (c, e). In the second case, by (11) and (lo), 5 E X and q E (c, e} so that by (6) we do not have ( R q . Thus, f is RR‘compatible and we have g = @ F ( f ) . rn ( f ( 0 9
$10. On a problem by S. Ulam
10.1. The problem we have in mind is the following: Is there a system Y of 2“” countable (directed) graphs such that there is no homomorphism between any two distinct A, B E Y ?
$10. On a problem by S. Ulam
139
More generally, one can formulate the problem as follows: Let A = (R, U ) be a concrete category, let a be an infinite cardinal. Is there a system Y of object2 of R such that (1) c a r d 9 = 2", (2) for every A E Y , cardU(A) = a, and (3) A(A, B ) = 8 for distinct A, B E Y ? We will refer to this problem briefly as q(R, a); thus the special case above constitutes the %(Graph, oo). 10.2. Obviously, if %(R,a) holds and if there is a full embedding @:
R
-+
2 such that
card A = a
card @(A)= a ,
then we have %(2,a) as well.
10.3. We obtain immediately %(Re I(2,2), a)
for any infinite OL.(Indeed, for a set X take a rigid relation Ro. Now, put Y = ( ( X ,(KO, R))I R c X x X arbitrary}. 10.4. According to the form of the embedding from 11.5.3, 4.12, 5.6,
11.2.5, following the observations 10.2 and 10.3 we have' %(%,a) with an arbitrary infinite a answered positively, e.g., for R any of the following categories: Graph, SymGraph, Poset,
k-chromatic graphs with k 2 3 , Alg(A) with E d 2 2 .
(Let us note here that for Graph, M. Karpinski found an independent solution.) 10.5. The embeddings in the next chapter will yield a number of further positive solutions of the %(R, a), e.g. for semigroups, rings, bounded lattices, etc.
10.6. If the category R in question is a concrete subcategory of Rel(d) with a finite A (which was the case with all the examples mentioned so far), the positive solution of the %(R, a) says that the system Y is as large as it possibly can be: there are no more than 2" non-isomorphic objects of the cardinality a. In a category with more than 2" non-isomorphic objects of the cardinality a, the problem should be more properly put with the
140
'
Ch. IV, COMBINATORICS
number of the isomorphism types instead of 2". Thus, e.g. in Y P ' ) the problem concerns a system of cardinality 2'". (This is also positively solved: first, take a rigid ( X , ro) E S ( P + ) ; second, use the same trick as in 10.3 to obtain a system of objects of Y P ' , P ' ) with no non-trivial morphisms and the cardinality 22". Finally, use the embedding from 111.1.2.) 10.7. In case of directed graphs and finite c1, the cardinality of a set Y of graphs such that there are no homomorphisms between any two of its members can be considerably larger than 2". Hell and NeSetfil ([HlNe,]) proved the following: If n 2 18, there is a rigid symmetric graph with n vertices and k = g n z - 3n - 2) edges. Consider all the directed graphs obtained by orienting the edges. The system has the cardinality 2k.
0 11.
Bibliographical remarks
As already mentioned in the Introduction, the idea of the arrow construction was used by de Groot in [dG] to obtain a topological result. In connection with graph problems it appears in [HP,] and in several later papers. The strong embeddability of Graph into SymGraph was first proved in [P5](here, we present a construction based on the same idea, but modified to obtain stronger results). The construction of G(m,rz) that we are using to obtain suitable color properties appeared, essentially, in [HlNe,], where it was used for other purposes (see E4.6). The alg-universality of Poset was proved in [HMD]. We adopt its construction of a basic rigid partially ordered set; to obtain a strong embedding, the proof is radically altered. The category of strong homomorphisms was investigated sooner than many of G o others; the results appear as early as in [HP,] and [PI]. The fact that the category of graphs with loops is almost alguniversal was proved in [H,] (our proof is based on the idea of the proof given there). The results on the categories of couples of equivalences were proved in [NP]. There are many results concerning representations of categories, semigroups and groups by combinatorial means. Because of limited space, in choosing the material we were forced to leave out some very interesting results (particularly, we had to neglect the rich literature concerning automorphism groups). Let us mention some of them at least. The category of graphs with a given common subgraph is alg-universal (cf. E 9 4 where a special case is proved), similarly, the category of graphs with a common given factor-graph is ([HM], [H3]). In [Ko5] the alg-
$1 1. Bibliographical remarks
141
universality of the category of graphs covered by copies of a fixed graph is proved. Of the results of automorphism groups let us mention the characteristics of groups of planar graphs ([Ba,,,]), many further results on groups of special graphs (e.g. [Ba,,,;,], [BaIml,2], [SaLo], [Iz], [Mn]), the fact that every finite group is isomorphic to the automorphism group of a projective plane ([M3]), results concerning the representations and the realizations of particular groups ( e g [Iml,3], [Now]), and the solutions of extremal problems concerning representation (e.g., [Ba6], [GeHiQ], [Ha], [MCQ], [Qi,,]); by [N] given a group G and a subgroup H , one can find a symmetric graph with the automorphism group isomorphic to G and such that one obtains a graph with the automorphism group isomorphic to H by orienting the edges. Also, it is worth mentioning that there are several interesting papers in which the automorphism groups and the subobject structure are simultaneously investigated.
Chapter V ALGEBRA We already know that the category of all algebras of a given type A is alg-universal iff C A 2 2. So far, however, with the exception of the commutativity of groupoids (see 11.2.3), we have not paid any attention to the special requirements concerning the operations. In this chapter we are going to discuss the categories which arise from algebraic structures commonly used : groups, R-modules, monoids, semigroups, rings, lattices, and varieties of unary algebras. Further, some categories of structures closely related to the algebraic ones are investigated, namely the categories of small categories and the categories SetA with A a small category. The first section is devoted to elementary results; we present negative results on the categories of groups, R-modules, and commutative semigroups. The second section deals with the category of semigroups (which is shown to be alg-universal) and the category of monoids (which is shown to be almost alg-universal). In 5 3, the category of commutative rings with 1 is shown to be alg-universal, in 3 4 the categories of lattices are discussed. Section 5 is concerned with some varieties of unary algebras, $ 6 with categories of small categories with several choices of morphisms, and, finally, in $ 7 some problems concerning the categories SetAare discussed. In the greatest part of this chapter (practically, in the whole sections 1-5) the categories investigated consist of some kind of algebras and of all their homomorphisms in the usual sense. If this is the case, we refer to the category by simply specifying the nature of its objects. Thus, when we speak of the category of groups, we mean the category of groups and their homomorphisms, etc. With a few exceptions, the categories investigated are, for some obvious reason, algebraic in the sense of 1.8.4. Of course, in such a case the (almost)alg-universality is all one can have (cf. Appendix B).
143
$1. Some easy results
0 1.
Some easy results
1.1. Recall the definition of the categories Alg(d) from 1.5.7. In 11.5.5 + it was proved that Alg(d) is alg-universal if E d 2 2 .
1.2. Proposition. The category of groups is not almost alg-universal. Proof. We will prove that every non-trivial group with more than two elements G has non-trivial automorphisms. (Consequently, e.g., the discrete category with three objects cannot be represented.) If G is not commutative, there are a, x E G such that a x xa. Thus, the mapping sending x to uxais a non-trivial automorphism. If G is commutative, the mapping sending x to X - is an automorphism. Suppose it is trivial, i.e., xx = 1 for every x. By Zorn's lemma, one sees that there is a maximal subset M of G such that, for every finite K c M, n { x I x E K } 1. It is easy to check that every x E G can be uniquely written as a product of elements of M and hence every permutation of the elements of M can be extended to an automorphism. Since cardG 2 3, cardM 2 2.
+
+
1.3. Proposition. Let R be a commitfritice ring. Then the category of R-modules is not almost alg-universal.
Proof. Consider an R-module M with card M 2 3. For every r E R the mapping F sending x to rx is an endomorphism. Let every ? be a constant or the identity. If ? is a constant, we have F(x) = ?(o) = r . o = o (the neutral element of M). Thus, there is a subset R, c R such that
for ~ E R andforevery , XEM,r . x = o , for ref K O and for every X E M, r . x = x. Take an arbitrary non-trivial automorphism cp of the underlying abelian group of M (there is such by 1.2). We have, for r E Ro, cp(rx) = cp(o) = o = = rcp(x),and for r I# R,, cp(rx) = cp(x) = rcp(x) so that cp is an automorphism of the module. Thus, every R-module M with card M 2 3 has a non-trivial non-constant endomorphism. 1.4. Lemma. Let x be an element of a semigroup S such that there are positive integers k, I with Xk+'
Then there is an n such that X"
=
Xk
.
. XI1 = x n .
144
Ch. V, ALGEBRA
Proof. We see immediately that for any natural p , q Xk+pl+q
Choose p , q such that p l
=
k
=
Xk+q.
+ q and put
n
=
k
+ q.
1.5. Corollary. N o non-tricial group is isomorphic to the endomorphism monoirl of a finite semigroup. Indeed, in a finite semigroup we have, for every x, positive k , 1 with Xk+l
= Xk.
H
1.6. By 11.2.3, the category of commutative groupoids is alg-universal. With the category of commutative semigroups this is not the case. We have Proposition. No non-trivialfinite group is isomorphic to the endomorphism monoid of a commutative semigroup. Proof. Let the endomorphism monoid of S be a non-trivial group. By 1.4, for every x E S and any m, n, m =k n
implies x m $. x " .
Thus, the mappings qn:S -+ S sending x to commutative, they are endomorphisms.
X"
are all distinct. Since S is
H
1.7. Remark. (1) We have not excluded the possibility of the category of commutative semigroup being almost alg-universal. The potential candidates for representing objects of other categories are the commutative semigroups S such that (*) there is an element 0 E S such that 0.x =0 =x.x
forevery X E S .
This is indicated by the fact that every group is isomorphic to the automorphism semigroup of a commutative semigroup satisfying (*) (see Ex l), and, on the other hand, by the fact that every commutative semigroup S with cards 2 3 which does not satisfy (*) has a non-identical non-constant endomorphism. Indeed, suppose this is not true. Then the mapping sending x to x 2 is either a constant or identity. In the former case, we have x 2 = a for all x, in particular also aa = a and hence the mapping sending x to ax is a homomorphism. If it is a constant, we see that ax = a for all x (since, in particular, aa = a), which is the excluded case, if ux = x for all x, S is a group and the statement follows by 1.2.
$2. Semigroups and monoids
145
In the latter case we have x . x = x for all x. Thus, cpx defined by cp,(y) = xy is always a homomorphism. Hence, there is a J c S such that for x E J cpx = id, i.e. xy = yx = y for all y , and for x E SJ\ cpx is a constant and hence x . y = y . x = x . x = x for all y . Thus, if x $; y and x E J , necessarily y E SJ\ and vice versa. We have, however, three distinct elements in S . (2) Also, there is still a possibility of an almost alg-universal category of R-modules with non-commutative R. Here, the evidence hints to the modules where in the additive structure always
(**)
x+x=o.
O n the other hand, if (**) does not hold, the correspondence sending x to - x is a non-trivial automorphism. But if R is the free ring with two generators, every group is isomorphic to the automorphism group of an R-module satisfying (**). (See Ex. 2). EXERCISES
1. Every group is isomorphic to the automorphism group of a commutative semigroup (Hint: Consider an undirected graph ( X , R') with the prescribed automorphism group. O n X u R u (0) define an operation 0 putting x y = {x, y > if {x, y > E R , 5 0 q = 0 otherwise.) 2. Let R be the free ring generated by u l , u2. Every group is isomorphic to the automniphism group of an R-module. (Hint: Consider an object (X, a l , a 2 ) of A Ig( 1. 1) having the prescribed automorphism group. On 3 the set of all finite subsets of X define A + B = ( A u B)\(A r B). The ring R operates on 2 by the rule a i A = (x card(a; '(x) n A ) is odd).) 0
I
9 2.
Embeddings into the categories of semigroups and monoids
2.1. In this paragraph we will prove that the category Smg
of all semigroups (in contrast with the category of commutative semigroups, see 1.6) is alg-universal. Also, we will see that the category Mon
of monoids is almost alg-universal.
146
Ch. V, ALGEBRA
According to 1.5, a representation of a finite monoid as an endomorphism monoid of a semigroup has to be done, in general, with an infinite semigroup. Thus, the constructions below cannot be replaced by finitistic ones (cf. the similar situation concerning posets, IV.5). For the reasons which will become apparent in Chapter VII, we prefer proofs of universality by means of strong embeddings. A proof of this kind will be given in sections 2.11-2.25 (most of the space will be devoted to properties of a certain rigid semigroup, on which the proof is based). We decided, however, to present a much simple proof concerning a mere full embedding first. ,
2.2. Recall that the category Graph,
of strongly connected graphs ( X , R ) without loops such that cardX > 2, and all their compatible mappings is alg-universal (see IV.l.ll). Let us notice that, according to the strong connectedness, for an x E X in an object ( X , R ) of Graph, there is always a y such that x R y .
2.3. For a set X denote by the free monoid (semigroup, resp.) generated by X , i.e. the set of all words (all non-empty words, resp.) in the alphabet X with the concatenation for multiplication. The empty word will be denoted by o (tacitly assuming on F o ( X ) we will denote by Fo(X)/ that o $ X ) . Given a congruence the factorsemigroup (a similar notation is used for the monoids); the congruence class containing w will be usually denoted by
-
-
.
LWJ
A word w E F ( X ) can be uniquely expressed in the form w = x;llx.;lz... xy with rn, 2 1 and xi
+ xi+,
The sequence xl,..., xk will be called basic sequence of w, the number k will be denoted by and called length of w, the number
w
k
2(w) = C m i i= 1
will be called full length of w.
147
82. Semigroups and monoids
2.4. Let r, r’ be relations on Fo(X), F,(X’) respectively, -, -‘ the congruences generated by r, r’. Evidently, if a homomorphism f : F o ( X )-+ Fo(X’) preserves the relations, it preserves the congruences and hence we have a homomorphism f : F0(X)/ -+ F,(X’)/ ’
-
defined by f([w])= [f(w)].
-
2.5. The functor S: Graph,
is defined as follows: where the congruence
-
yx’y
-+
Smg
s(x,R ) = F,(X)/
-
is generated by
-
yxy
for all (x,y ) R~.
I f f : ( X ,R ) -+ (X’, R’) is compatible, the homomorphism g: F o ( X )-+ Fo(X‘) defined by f(x,, ..., x.) = f(xi)..,f(x.) preserves the congruences (see 2.4) and hence we can define S ( f ) : S ( X , R ) -+ S(X’,R’) by S ( f ) = $7. Similarly, using F instead of F,, we define a functor
M : Graph,
-+
Obviously, M ( X , R ) = S ( X , R ) u { [ o ] ) .
Mon.
-
2.6. Observations. (- from 2.5) (1) If yx’y yxy then xRy. (2) If w w‘ then w and w’ have the same basic sequence. (3) If w w’ and w =+ w‘, then E(w) 2 3. (4) If w = x;ll ... x ; k , W’ = x;’ ... x;. (xi =# w w‘ and max (mi,ni)> 2, then mi = ni. (Recall that there are no loops!)
--
-
-
2.7. Lemma. I f u, u are in F,(X) and uuuu uuu then u E X . Proof. We have l(u,u) = l(u) + l(u) - c(u, u ) where c(u, u) = 1 if the last member of the basic sequence of u coincides with the first member of the basic sequence of u. If uuuu uuu, by 2.6.2 we obtain Z(u) - c(u, u ) = 0, so that u = x”.By 2.6.4, 2n I2. Thus, u = x.
-
- -
2.8. Lemma. S ( X , R ) contains no idempotent element. Proof. If u u u, we have (applying 2.7 with an arbitrary Then x x2 in contradiction with 2.6.3.
1 % ) ZI
= x E X.
1
148
Ch. V, ALGEBRA
2.9. Theorem. S is a full embedding. Consequently, Smg is alg-universal. Proof. Obviously, S is one-to-one. Let g: S ( X , R ) -+ S(X‘, R’) be a homomorphism. Take an x E X choose a y such that x R y. Thus, y x 2y y x y. If g([x])= [u] and g ( [ y ] ) = [u], we obtain uuuu uuu and hence u E X ’ by 2.7. Put u = f ( x ) . Since g is a homomorphism,
-
-
g([x, ..
.%I)
=
[f(x1)...f(xn)l .
m
By 2.6.1, f is RR’-compatible. Thus, g = S(f).
2.10. Theorem. M is an almost full embedding. Consequently, Mon is almost-universal. Proof. Obviously, M is one-to-one. Recall that M(X,R) = S(X,R)u ([o]). Let g : M(X, R) + M(X’, R ) be a homomorphism. By 2.8, g([o])= [o]. Let g([u])= [o] for some u E Fo(X). Then obviously g ( [ x ] )= [o] for an x E X (actually, for all the members of the basic sequence of u). Choose an arbitrary y € X and a sequence xo = x, X , ,..., X , = y in X such that x i R x i + , . We have g([xi+lx?xi+l])= g([xi+lxixi+1])Since g([x])= [o],we have g ( [ x l ] ) g. ( [ x J ) = g ( [ x l ] )and hence g ( [ x J = = [o] by 2.8. Proceeding by induction we obtain g([y])= [o]. Thus, g maps M(X, R ) constantly onto [o]. Hence, if g is not a constant, g ( S ( X , R ) )c S(X’,R’) and we see that g = M(f) by the previous theorem. 2.11. The semigroup D: Let a, b be two distinct elements. Put D = Fo( (a, b))/ where
-
- is the congruence generated by ab2
Put
-
baba.
M = ( [ a ] ,[ab],[ba],[aba], [bab], [baba] = [ab’]), N = D\M (thus, [b],[b2]E N ) . Convention. If there is no danger of confusion, we omit the square brackets. (Thus, we will often write e.g. baba = ab’.)
-
+ w’ then either
2.12. Observation. If w w’ and w a suitable order, w = ab and w’= baba.
l(w) 2 4 or, in
52. Semigroups and monoids
149
2.13. Proposition. (1) N is a subsemigroup of D, (2) ij 11, u, w are in M , then uuw is in N , ( 3 ) if 11, u, M U are in M , then uu E (aba, baba) (4) if uu and exactly one of u, u are in M , then the other one of u, u is b or b2, and uu =I= aba. Proof. If uu E M , we have by 2.12 I(u) + P(u) I 4 and @) + I(u) = 4 only if uu = baba. In the last case, (u, u) is some of (b,aba), (ba,ba) and (bub, a). If I(u) + t(u) I 3, (u, u ) is some of (a, b), (b,a), (a, ba),(ab,a), (b, ab), (ba,b), (a, b2),(ab, b) .
Checking all these candidates for (u,u) one obtains immediately (l), (3) and (4). To prove ( 2 ) take a uuw E M . Then l(u) 1(u) l(w) I 4 by 2.12. Checking such words one sees immediately that always at least one of the u , u , w consists of the single b and hence is not in M .
+ +
2.14. Consider the mapping (not a homomorphism)
$: q a , b ) )
+
F((a, b})
defined as follows: If w = w'ab2w" and w" =I= uab'u (i.e., our ab2.is the last occurence of this letter group in w),put $(w) = w'babaw", otherwise put $(w) = w . Put w = (w $(w) = w } .
I
Lemma. For every w there is an n such that
V(w)E w. Proof. First, we compute easily that (*) If k 2 2 and if M E W starts with a,
t ) 2 k - 1 ( ~ b 2 k ~ )= b 2 ( ~ b ) 2 k - 3 ~ 2 b a ~ , *2k-'(ab2k+lu)
=
b2(ab)2k-3a2babu.
Now, let us proceed by induction over (w). Let w = w'ab2w", W"E W Consider the following two cases: (a) w' = 0 : We have w = abr+2asu.Then either r I 1 and then $(w) = = bababraSuE or r + 2 2 4 and by (*) for 2k = r 2 or 2k 1 = =r 2, $ 2 k - 1 ( w ) W ~ Moreover, we see that in this case always the P ( w ) which is in W is of the form b'au with t 2 1 (and, of course, au E W).
+
+
+
150
Ch. V. ALGEBRA
(P) w’ = qasbr with s 2 0, r 2 0 and r + s > 0: let n be such that I,”U~’W”)E W By (a) we have then $“(ab2w”) = b‘au with t 2 1 and
au E W Thus, if t+hn(ab2w”)E W we have $“(w) = v’ab’v” such that v” E W and l(v’) < f(w’). Now, use the induction hypothesis.
2.15. Define by Gqw) =
*
$: F({a, b))
-, F({G b})
(w) where n is such that t,!P+I(w)
=
$“(w).
Observations. (1) If w, W ’ E (ab2, baba} then $(w) (2) For every w, w
-
$(w).
=
$(w’).
4
2.16. Lemma. For every u, v E F((a, b)) there are non-negatiue integers r, s such that p+‘(uv) = *r(u$(u)), lp+‘(UU) = *s(*(u) u) . Proof. By 2.14 we have an r such that ~ ) ~ + ‘ ( u= v ) ~ ( U U ) If. v $ W then ut+b(v)= $(uu), if U E ut+b(v)= uu. In both cases $ r + l ( ~ v )= tj‘(ut,h(v)). Looking for the s, in the case of u E W we can put s = r with the r such that t,br+‘(uu) = $‘(uu) again. If u $ W write u = u’ab’u” where u” E W Thus, $(u) = u‘babau”. Let s be the smallest nonnegative integer such that $s+l(u”v) = ~+P(u”u)= w. Thus, w E W and lp-l(u”v) $ W Now, we have $‘($(u)
2;)
=
lp(u’babau’’v) = u’babaw ,
lp+I(~ =)lp+1(U’ab2u’’v) = $(u’ab’lp(u’’v)) = $(u’ab2w) = u‘babaw .
=
2.17. Lemma. For any two w, w’ E F((a, b}), $(ww’) = $($(w) $(w’)). Proof. By 2.16, tj?(wIw;) = $(wlt+b(w;)) = $($(wl) wi). Consequently, $(ww’) = $($(w)$(w’)) and we obtain the statement by iteration.
-
, there is exactly one G E W 2.18. Proposition. For every w ~ F ( { ab}) such that w W. Consequently, for every w there is exactly one congruent b‘aS1bas2b... bas.
with t 2 0, n 2 1, s, 2 0 and si > 0 for 1 I i < n. Convention. This btas1bas2... bas“ will be called normal form of w. Proof. The existence follows immediately from 2.15. Now, let w1 w2 for wl, W ~ W E Thus, there exist words ql, ...,qn such that w1 = 41,
-
151
$2. Semigroups and monoids
w2 = q,, q i + = uixvi and qi 2.15.1 we obtain g(4i- 1) =
$($(Mi)
=
uiyvi with x,y
@($(x)@(ui))) 2
$($(Mi)
E
(ub', baba). By 2.17 and
$($(Y) G(ui))) = q ( 4 i ) .
Consequently, w I = $ ( w l ) = $(w2) = w2. We see easily that $(w) = w iff w is of the form indicated in the second part of the statement. rn 2.19. Proposition. D is right cancellative, i.e., if vu = wu, then u = w. Proof. We compute easily that the normal form of (ab)"b is
'
ba(ba2)"- ba . Consequently, if b'us1bas2... busn is the normal form of w, the normal form of wb is
bi+ 1
biasib ...
1
if n = 1, s, = 0, if n 2 1, s, 0, if 1 I i < n, s, = 0, si > 1
+
b'aslb ... asnb ba(bu2)"-'-'ba
and s i + l = ... = s n P 1= 1, if n 2 2, s, = 0, s1 = ... = s,-'
b'+'a(ba2)n-2ba
=
1.
Thus, the normal form of wb determines the normal form of w.uniqsely. Now, let, for v, w E D, v =i= w. By the fact just proved, wb ==! vb, and consequently ub' wb'. Let u = b'aslb ... bask be given. Consider the (distinct) normal forms . b'laP1b. .. baPn, bi2aq1b... ba4- of ub', wb'. Thus, we have again distinct normal forms for vu and wu, namely
+
b'IaPlb ... bapn+slbas2 ... bask, bt2uq1b.. . bah+slbas2... bask, so that vu
+
wu by 2.18.
2-20. Since ab2 and baba contain the same number of b's, we obtain immediately m n Lemma. I f aP1bql... ahbh urlbsl... ar"bs",then 4l = s,.
-
1
1=1
i=l
Proposition. D is rigid, i.e., i;f f: D 3 D is a homomorphism then f is the identity mapping. Proof. Put u =f(a), u = f ( b ) . Then uu2 uuuu. By 2.20 we see immediately that u = up with a p 2 1 . Now, let b'ab'baS2... basn be the normal form of v.
-
152
Ch. V, ALGEBRA
First, let us suppose that n L 2. Then for any w, the normal form of wu terminates in ... bay*... ba". This particularly holds for zmz and hence it should also hold for ouuu, which, however, terminates ... bdn'". Thus, u = b'd' Suppose that s1 > 0. Then obviously t =+ 0. If t > 0 is even (and, hence, 1 2 ) , the normal form of uu2 terminates in while that of uuuu terminates in a S 1 + l + P . If t is odd one gets analogously a discrepancy between ... as' and ... as'+*. Hence u = b'. The normal _form of uuuu terminates in ... u p for t odd and in ... a p f l if t is even. Since the end of the normal form of uv2 is ... a, we see that t is odd and p = 1. Let us suppose that t 2 3. Then, by (*)in 2.14, uu2 = ab2' = b2(ab)2*-3 a2ba, while the normal form of uuuu starts with b', r 2 3. Thus, finally, f ( a ) = a, f ( b ) = b, so that f is identity. 2.22. The functor @. We preserve the notation from 2.11. First, let us define a functor G: Set + Set
putting G ( X ) = ( X x M ) u N (the union is assumed disjoint), G ( f )(x, u) = = (f(x), u) for (x, u)E X x M , G ( f )(u) = u for u E N. Further, define mappings p x : G ( X ) -+ D by P x b , ). = u, P X ( 0 ) = 0. If ( X , 0) is an object of Alg (2), i.e., if 0 is a binary operation on X , put
a)= ( G ( X ) ,.)
@(X,
9
where (the operation in D is indicated by juxtaposition) if 5, v E G ( X ) , P(<) P(V) E N , put i" .11 = P(C) P(v), (p) if u, u, uu E M , (x, u ) . ( y , u) = (x 0y , uu), ( y ) if u, uv E M and u E N , (x, u) . u = (x, uu), if u, uv E M and u E N , u . (x, u) = (x, uu). Thus, px is a homomorphism @ ( X ,0) + D. Using 2.13.2 we check easily that the operation . is associative. Finally, let f : ( X , 0)-+ ( X ' , 0)be a homomorphism. Define (R)
q f ) qx, : 0) @(X', 0) -P
153
$2. Sctnigroups and monoids
= G ( f ) ( t ) .Obviously, this is a homomorphism, so that putting @(f)(t) we obtain a functor @: Alg (2) -+ Smg .
2.23. Lemma. Zf b . 5 = b . ( x , aba) in @(X,0 ) then 5 = ( x , aba). Proof. Since p i s a homomorphism, we have b p ( 0 = p(b) p ( i ) = = P(b. t) = p(b. ( x , aba)) = baba. Since b E N and baba 4 N, by 2.13.1 p ( 5 )M ~ and hence t = (y, u ) with U E M . The only U E M such that bu = baba is aba. Thus, ( y , baba) = b . 4 = b . (x, uba) = ( x , baba) and hence also v = x. M
2.24. Lemma. If g : @ ( X ,0) -+ @(XI,0) is a homomorphism, then for any x, y E X pg(x, baba) = pg(y, baba). Proof. We have (x, baba) . b = babab = (y, baba) . b. Thus, p g ( x . haba) . . p(6) = p g ( y , buba) . p(b) and the statement follows by 2.19. 2.25. Theorem. @ is a strong embedding of Alg ( 2 ) into Smg. Proof. Obviously @ is one-to-one and it is carried by G. Thus, it suffices to prove that it is full. -P 0)be a homomoprhism. For every x E X Let g: @ ( X ,0) define a mapping 9,: D + G ( X ) putting cpx(u) = ( x , u ) for x ~ M \ ( a b a ) cp,(aba) = (X 0x , aba) , cpx(u) = u for X E N . @(XI,
This cpx is not a homomorphism, but
944.cpx(u) = ( P X b ? 4 holds with the exception of u, u E M and uu = baba. (Indeed, it obviously holds for U U E N . If U U E M , at least one of u, u has to be in M by 2.13.1; if one of u,u is in N , uu 4 aba by 2.13.4 and the equation holds, if u, u, uu E M, by 2.13.3 uu is aba or baba and in the former case the equation holds again.) In this exception, cpx(u).cpl(u) = ( x
Hence, by 2.24,
0x, baba) ,
q,(uu) = (x, baba) .
p g q , is a homomorphism D
-+
D.
154
Ch. V, ALGEBRA
Thus, by 2.21, pgqX is the identity and we obtain immediately Define
g(u) = u
for
UE
N,
g(x,u)= (x’,u) for
UE
M.
f:X + X ’
by the equation g(x,u ) = (f(x), u). We will show now that g = G(f).To this end we have to prove that
(*I
g(x,).
=
(f(x),4
for all u E Ad. We have it already for u = a. Further, when multiplying the equation g(x,u ) = (f(x),u ) by b from the right and from the left, we obtain (*) for u = ub,bu, and when repeating the procedure, we obtain it also for u = ub2 and bub. The remaining u = ubu is dealt with using 2.23: we have b . g(x,aba) = g(b) . g(x,aha) = g(b . (x,u ~ u )= ) g(x,baba) = = (f(~), baba) = b . (f(x),U ~ U .)
Finally, we see that f is a homomorphism ( X , 0) -+ (X’, 0): We have
4.MY),
4
(f(x)0f(Y)>aha) = (f(x), ba) = g(x, .9(Y?ba) = = s(x 0y, aha) = (f(x 0y), a q .
6 3.
Categories of rings
3.1. In this paragraph we will show that the category Rng
of all commutative rings with unit and all the ring (not necessarily unit preserving) homomorphisms is almost alg-universal and that its subcategory Rng1
formed by the unit preserving ring homomorphisms is alg-universal. One can prove more, namely, the alg-universality of the category of integral domains. The detailed proof of this statement is very long and technical. Of this, we will present just an outline at the end of the paragraph.
3.2. Notation. Throughout thic parayraph,
G ’ will designate the category of all undirected graphs without loops, and of their compatible mappings, (By IV.4.12, G is alg-universal.) As usual, the
155
$3. Categories of rings
ring of integers will be denoted by Z , the ring of polynomials in x E X with coefficients in Z by Z ( X ) . 3.3. The functor Y . Let J be the ideal in Z ( X ) generated by {x3 - 5 . 7 1 X E X > , T ( X ) = Z ( X ) / J and q x :Z ( X ) + T ( X ) the corresponding epimorphism. For an object ( X ,R ) of G define Y ( X ,R ) as the subring of T ( X )generated by the set M(X,R)= {5xJxEX}u{xyJ{x,y}ER}uZ.
A mapping f : X --* X ’ extends uniquely to a ring homomorphism Z ( f ) : Z ( X ) -+ Z ( X ‘ ) . Obviously, Z ( f ) maps the ideal J into the corresponding ideal in Z ( X ’ ) . Thus, there is a unique homomorphism T ( f ) :T ( X )+ T ( X ’ ) such that cpx. Z ( f )= T(f)qx. If f: ( X ,R ) --* ( X ’ ,R’) is a morphism of G , T ( f ) maps M ( X , R ) into M(X‘,R‘) and hence also Y ( X ,R ) to 9’(X’,R’). Denoting Y (f ) : Y ( X ,R ) + sP(X’, R’) 0
0
the domain-range restriction of T ( f ) ,we obtain a functor
9: G
+
Rng.
Obviously, Y is one-to-one. The proof that Y is almost full will be given in 3.10. Before that, we have to prove several auxiliary statements.
3.4. Proposition. For any ( X , R ) a i d any x , y E X
iff { X , Y } E R .
X.YEY(X,R)
Proof. If { x , y > E R, we obviously have x . y E Y ( X ,R). Now, let x y be in Y ( X ,R). Denote by P the set of all w E T ( X ) which may be expressed as w
=
x y . ... . xmn
+
with m k {0,1,2), x i € X and xi x j for i + j . Further, put = { x i mi > 0), l(w) = card u(w). Let S be the set of all CTET ( X ) which may be expressed as
1
0
=
k
U(W)
=
+ Ckvv + zm,w
where u, w are in P, U(V)E R, k, k, E {0,1, ..., 4) and for every w either l(w) > 2 or m, is divisible by five. T ( X ) is a free abelian group over P and
156
Ch. V, ALGEBRA
hence the expression is uniquely determined by 0.One checks easily that S is a subring of T ( X ) .Since all the generators of Y ( X ,R ) have the described form, we have S 3 Y ( X ,R ) and hence, in particular, x . y E S. The statement follows. 3.5. Denote by J , the ideal in Z ( x , , ...,x,, e) generated by
+ +
i = 1 , 2 ,..., n ,
x?-35,
and e2 e 1. Put L, = Z ( x l , ..., x,,e)/J,. More generally, denote by J ( X ) the ideal in Z ( X u {el) generated by x 3 - 35 for X G X and e2 Q 1 (we assume that e $ X ) , and put L ( X ) = Z ( X u { e } ) / J ( X ) .If X c Y we have the natural embedding Z ( X u {e])c Z ( Y u {el), and we also see that for 4, v] E z ( x u (e}), 5 - v] E J ( X ) iff 5 - r E J ( y). Thus, the embedding further induces monomorphisms
+ +
L ( X ) -+ L( Y ). To simplify the notation, we will work with this as with an inclusion, thus, L ( X )c L ( Y ) . This convention is in a handy agreement with the usual simplification of notation in which one writes the elements of L ( X ) as their representatives in Z ( X u {el) and indicates the congruence modulo J ( X ) by the equality sing. We will write L = L(@). Thus, in our convention, L c L ( X ) for all X . Observation. No integer is a divisor of zero i n I.,,. 3.6. Lemma. If q3 - j4.7 = 0 in L,, then v] = 5eJxifor u j E (0, 1,2} and i~ {1,2,..., n } . Proof. We proceed by induction: (a) L , is an integral domain since x: - 35 and e2 e 1 are irreducible. Hence, q3 - 54. 7 has three roots in L , at most, and we see that 5x1, 5x1@,5x1e2are its distinct roots. (b) For j E (0, 1,2) define mappings ' p j : L,+ -+ L, putting
+ +
,
Cp,(P(Xl,
.. x , + 1)) .1
,
= P(" 1,
..., X"?@ ' X I ) ,
where the p ( x , , ..., x,+ ,) E L,,, are assumed expressed as polynomials with coefficientsin L,. Obviously, 'pj are ring homomorphisms. Let q 3 - j4.7 cx;, for some a, b, c E L,. We have be 0 in L,,+ Then v] = u bx,, to prove that c = 0 and either a = 0 and b = 5 ~ " with m E {0,1,2), or b = 0. Put q j = 'pxv]). Then - 5 4 . 7 = 0 in L,, so that, by the in, some mjE {O, 1,2} and y j E { x l , ..., x,]. duction hypothesis, y l j = 5 ~ " ~ yfor
+
,+ ~7
157
53. Categories of rings
+
+
By a simple computation we obtain qb eql e2q2 = 3cx: and consequently 3CXf = 5@""y0 + 5@"' ' y l + 5@"'+' y 2 . +
Since the left hand side of this equation is divisible by three, yo = y , = xi for some i. Thus, 3cx: = 5xi
2
1
ernJ+j. Multiplying
i= 0
by x 1 and dividing by five we obtain 7 .3 . c
=
xlxi
y,
=
the equation
2
1
=
e"'j+j. i =I n-
The right
'&"J" .
hand side, however, is divisible by seven only if = 0. Thus, c = 0 and we have either (1) rn, = rn, = rn2 mod 3 or (2) rn, = rno + 1 mod 3 and rn2 = rn, 2mod 3. In the first case, cqj= 3a, and hence 3u = 5xi xemJ= 3 . 5 . Q m o . xi. Thus, a = 5@"'Oxi and hence q j = 5emoxi= a and consequently b = 0. In the second case, 3a = Cqj = 5 x i 1 ~ " = J 0 and hence a = 0. Thus, 5xie"' = q j = bejx,. Consequently, xi = x 1 and q = 5 ~ " ~ x , + ~ .
+
3.7. Lemma. If q2 = q in L, then either q = 0 or q = 1. Proof by induction: (a) Since L , is an integral domain, the polynomial q2 - q has two roots at most. (b) Let q 2 = q in L,+,, q = a + b ~ , ++ ~cxIil for some a, b,c in L,. We have to prove that c = b = 0. Let us use again the homomorphisms ' p j from the proof of 3.6 and put q j = cpj(q).By the induction hypothesis, q, E We have 3cx: = q o + Qql + e2q2 and hence the right-hand side is divisible by 3. Since 1 + e = -e2, 1 + Q 2 = -e, e + Q2 = - 1, if one or two of the qi were 1, either 1 or Q or e2 would be divisible by 3 which it is not. Thus, either qo = q1 = q2 = 0 or qo = q , = y12 = 1, so that c = 0. Since 3a = qo + q l + q2, we have a = yo and consequently bxl = 0. Multiplying by xf we obtain 35b = 0 and hence finally b = 0.
p,q.
3.8. Lemma. If q3 = 0 in L, then q = 0. Proof. This is true in L , since it is an integral domain. Put, again, q j = qj(q).Let q = a + bx,+ + cx;+ with a, b, c in L,. By the induction hypothesis, q j = 0. Hence, 3a = qo q l q2 = 0, 3c = qo Qq1 + e2qz = o so that a = c = 0. Thus, finally, b3x,3+, = 35b3 = 0 and hence also b = 0.
,
, + +
+
158
Ch. V, ALGEBRA
3.9. Proposition. For every set X , in the ring T ( X ) ( 1 ) 0 and 1 are the only idempotents, (2) t3 = 0 iff 5 = 0, and (3) t3 = 54.7 iff 5 = 5x for some X E X .
Proof. Since every element of L ( X ) (see 3.5) can be considered .as an element of a suitable L({x,,...>xn}) = Ln
c
L(X)
3
we see by 3.7 that 0 and 1 are the only idempotents in L ( X ) , by 3.8 we see that t3 = 0 iff t = 0 in L ( X ) , and by 3.6 that 5ejx ( j = 0,1,2) are the only solutions of the equation t3 - 5 4 . 7 in L ( X ) . Now consider the subring of L ( X ) consisting of the elements which may be expressed as polynomials in the elements of X . Obviously, this is isomorphic to T ( X ) . Since 5 ~ j is x an element of this subring only if j = 0, the statements follow. 3.10. Theorem. The functor Y (see 3.3) is an almost full embedding and every Y (f ) is a unit preserving homomorphism. Consequently, every cutegory Si such that Rng, c W c Rng is almost ulg-universal. I n particular, Rng, is alg-universal. Proof. Y is obviously one-to-one. Let g : Y ( X ,R ) 4 Y ( X ’ ,R’) be a homomorphism. First, we will prove that if it does not preserve the unit, it is constant to zero. Indeed, by 3.9.1 we have then g ( l ) = 0 and hence g(n) = 0 for every n E Z; further, ( g ( 5 ~ )=) ~g ( ( 5 ~ )=~ g) ( 5 4 . 7 ) = 0, so that, by 3.9.2, g(5x) = 0 ; finally, for { x ,y } E R we have ( ~ ( X Y ) )=~ g(x3y3)= = g ( 5 2 . 72) = 0 and hence again g(xy) = 0. Thus, all the generators of Y ( X ,R ) are being sent to zero so that the whole Y ( X ,R ) is. Now, let g be a unit preserving homomorphism. Then g(n) = n for n E Z and hence ( g ( 5 ~ )= ) ~g ( ( 5 ~ )=~ g) ( 5 4 . 7 ) = 5 4 . 7 so that, by 3.9.3, there is an x’ E X such that g(5x)= 5x‘. Define f: X + X’ putting f ( x )= x’. If { x ,y } E R, we have 25g(xy) = g(5x. 5y) = g(5x). g(5y) = 25f ( x ). f ( y ) . Since 25 does not divide zero in Y ( X ’ ,R’), we have g(xy) = f ( x ). f ( y ) . Finally, by 3.4, f ( x ). f ( y ) E Y ( X ’ ,R’) only if { f ( x ) ,f ( y ) }E R‘ so that f is a morphism ( X ,R ) + (X‘, R‘).Thus, g = Y (f).
3.11. The result of 3.10 can be strengthened considerably; namely, one has the following Theorem. Denote by Id,, the category of integral domains of characteristic zero possessing units and their ring homomorphisms, by Id; its subcategory
159
$3. Categories of rings
formed by the unit preserving homomorphisms. There exists a strong embedding
9: Alg ( 1 , l ) -+ Id; Consequently (since a non-zero homomorphism into an integral domain obviously preserves the unit), every category R such that Id: c R c Id, is almost alg-universal. Outline of proof. The proof contains a rather complicated part concerning a statement playing a role analogous to that of 3.9 in the proof of 3.10. In this outline it (the statement (S) below) will be formulated only. The functor F is defined as follows: For every set X choose a disjoint copy X and fix a bijection X H X of X onto X (one could work with X x ( 0 ) and X x {l}, and the bijection ( x , O ) ~ ( xl), , but this notation would be too clumsy). Put P ( X ) = Z ( X u X)and denote by J(X)the ideal in P ( X ) generated by
+x3
51XEX). Put T(X)= P(X)/J(X) and denote by c p x : P(X) T ( X )the corresponding (x3
-
--f
homomorphism. If ( X ,a, b) is an object of Alg ( 1 , l )put M ( X , a,
a) = z u {5P I P
T(X))u (cpx(P) I P E P ( X ) >C ( P ) > 3) u U{cpX(X*.CI(X))[XEX) u {cPx(X.B(x))IXEX), f
where c(p) is the smallest degree of a non-zero monomial term of p . Finally, F ( X , a, is the subring of T(X)generated by M(X,a, B). Since all the polynomials x 3 + X3 - 5 are irreducible, T ( X ) is an integral domain, with characteristic zero obviously. Consequently, this also holds for 9 ( X , a, For a homomorphism f : ( X , a, p) -+ (X’, a’, p’) one defines F(f) in the obvious way and obtains a one-to-one functor
a)
a).
9: Alg(1,l) -+ Id;. Finally, denote by Po(X)the set of all polynomials ~ E P ( Xwith ) coefficients in (0,1,2,3,4}, with the total degree s 3, and the degree of 5 I 2 for any x E X.Put
I
’
D ( X ) = 5T(4 + {cpx(P) P E p(x),C(P) 3) . Now, the fact we are asking the reader to believe is that (S) (1) every t E T ( X ) may be written uniquely as t = cpX(p) d with peP0(X)and d E D ( X ) , (2) for any t , s E T ( x ) , t 3 + s3 = 54 iff { t , s } = {5x,5%) for some
+
XEX.
160
Ch. V, ALGEBRA
We will show that 9 is carried by an F : Set + Set. Put X u = .(.) x E X}, X , = {x .B(x) x E X}.Thus, F ( X , a, is generated by { 1) u D ( X ) u X , u X,. We see that a product of any two 5, q of X , u X , is in D(X). Consequently, every W E P ( X a, , can be written as c &y, d with d E D ( X ) , yi E Xuu X b , c, ciE Z, (the integers modulo 5 ) and with only finitely many non-zero ci. By (S 1) such a representation is unique. Thus, denoting by W,(Y) the group freely generated by Y in the variety of abelian groups with the additional identity 5z = 0, we see that the underlying set of F ( X ,a, p) coincides with that of W,(X, u X , u { 1>)0 D ( X ) (the biproduct of abelian groups; from D(X)only the additive group structure is taken). Furthermore, X u and X , are in the natural one-to-one correspondence with the X and X. Thus, we can replace the X , u X , by X u X.It is easily show that we can define the desired F by taking for F ( f ) the underlying mapping of W , ( f u f u id) @ D ( f ) (where f(Z) = f(x) and D ( f ) is the restriction of T(f)). Finally, let g: F ( X , a, p) -+ F ( X ’ , a’, p’) preserve the unit. Take an x E X. We have, by (S 2), ( 5 ~ +) (ST)’ ~ = 54, hence (g(5x))j (g(5Y)))3 = 54, and hence {g(5x),g(SZ)} = { 5 y , 57) for a ~ E X ’ .Put y = f(x), so that {g(5x),g(5T)} = (5f(x), 5 ~ 7 ) By . (S I) we see that = {x”
I
a)
1
a)
+
+
+
(*)
u2uEF(X’,a‘,p’)
iff
EX'
and
u =
al(u)
We have 5’g(x2 . .() = g((5x)’). g(5a(x)) = ( 5 ~ ) ’.5v for a u E {f(x),f(x)} and u E {f(a(x)),f(a(x))}.Thus, u2v = g(x2 .(.)) E F ( X ’ ,_ a‘, P), _ _ and-hence, by (*), u E X’ and u = Consequently, u = f(x) and a’(f(x)) = a ’ ( ~= ) =u = Now we see that g(5x) = 5f(x) for every x E X and, by (S 2) again, g(5F) = 5 f 7 . Moreover, f(a(x)) = a’(f(x)). Similarly, by (S l),
fo).
(**I
a.
UVE
F ( X ’ , a’, p‘)
iff u = p’(u)
Consequently, f(B(x)) = /l’(f(x)).Obviously, g is a restriction of T ( f )and hence g = F ( f ) . rn Remark. The full proof is given in [FS,].
5 4.
Categories of. lattices
4.1. In this paragraph we investigate some categories of lattices. We show that the category of all lattices is almost alg-universal and that the category of the lattices with 0 and 1 and their 0,l-preserving homomorphisms is alg-universal. Further we present some negative and some positive results concerning distributive and complete lattices.
161
54. Categories of lattices
4.2. As usual, a lattice is a set with two binary idempotent commutative and associative operations called join (denoted by v ) 2nd meet (denoted by A ) and satisfying further the identities X V (X A J’) = X == X A ( X V J’)
.
Often, an equivalent description of a lattice as a partially ordered set in which non-empty finite sets have suprema and infima is used. Speaking on homomorphisms, however, we always mean the homomorphisms with respect to the operations. 4.3. The free lattice generated by a set X will be denoted by F X . Recall that for every mapping f: X -+ L where L is a lattice there is exactly one homomorphism g: F X --t L such that g X = f (as is always the case with free algebras), in particular for every f : X -+ X ’ we have a unique
I
extension
Ff: F X
+ FX’
.
In the following we will often work with the so called set of lattice polynomials over X , denoted by 9 X , which is the smallest set such that (i) X c Y X , (ii) if p1,p2 are in Y X , then the expressions (pl v p 2 ) and (pl A p 2 ) are in Y X . Although the symbols v and A (or, rather, (... v ...) and’ (... A ...)) behave like operations in 9 X ,Y X is not a lattice. In fact, it is a free algebra with two binary operations unrestricted by any equations. We define recursively by
e: 9 X +F X
4.)
-x
e(P1 v P z )
=
4 7 1 A P2) =
for X E X ,
4 P I ) v e(Pz), .(PI)
A
e(Pz).
(On the left-hand sides, v and A are just symbols, letters in the expressions, on the right-hand side, however, they indicate lattice operations.) 4.4. It is well known (see e.g. [Bff],) that if p, q are in 9 X and a = e(p),
b
=
e(q), then a I b iff some of the following assertions holds:
(0) (1) (2) (3) (4)
p=q=xEX, P p q q
Pz e(p1) 5 b and 4 P 2 ) 5 b , p1 A p 2 and either e ( p , ) 5 b or e(pz) I b , = q 1 v q2 and either a 5 e ( q l ) or a 5 e ( q z ) , = 41 A 4 2 , a e(4J and a 5 4 4 2 ) . = P1 v
=
9
162
Ch. V, ALGEBRA
4.5. The length (also called rank) l(p) of a polynomial is defined recursively as follows: 6) l(x) = 1 for all X E X , (4 l(P1 v P 2 ) = l(P1 A P 2 ) = l(P1) + @ 2 ) . The capacity c(p) is defined by C(X) = {x} for all X E X , (ii) C ( P I v P 2 ) = (+I v P 2 ) ) u C ( P 1 ) u C ( P 2 ) , C(P1 A P 2 ) = {+I A P2)} u C(P1) u 472). Thus I maps PX into the set of positive integers, c maps it into exp F X .
(4
4.6. Let us denote by N the smallest equivalence on P X such that (i) if p i N pi, i = 1,2, then PI v P 2 N P; v Pi, P1 A P 2 = P i 4Pi, (ii) if p i N pi, i = 1,2,3, then (P1 v P 2 ) v P 3 = P; v (Pi v P i ) and (P1 A P 2 ) A P 3 ‘v P; A (Pi A Pi). Obviously, if p = p’, then l(p) = @’), e(p) = e(p’) and c(p) = c(p’). ,,,Define
V
Pi. i= 1
m
Api by
i= 1
1
Apl
Lemma. Let p , q E P is an r E 9 X such that either (a) l(r) < l(p) or (b) l(r) < l(q)
pi
= pl,
i= 1
i= 1
=
(“
)
/ \ p i ~ p r n i l , and similarly
i= 1
X be such that e(p) = e(q) while p+ q. Then there e(r) = e(p) and and c(r) c c(p), and c(r) c c(q).
Proof will be done by induction over k = /(p) + l(q). For k = 2 the statement is evident. rn n (A) First, suppose that p z q N Aq, and that p i and qj are not
Api,
meets, i.e., either pi
i= 1
= VPik k
j= 1
or p i E X and similarly for qj.
Since e(p) 2 e(q), we have e(pi)2 e(q) for all i = 1,..., m. Thus, by 4.4 either (a) e(pik)2 e(q) for some k, or (B) e(pi) 2 e(qj)for somej. Analogously, since also e(q) 2 e(p), we have either (7) e(qj,)2 e(p) for some I, or (6) e(qj)2 e(pi) for some i. If (a) holds for an io E (1, ..., m), we have e(piOk)2 e(q) 2 e(p). We have e(pi) 2 e(p) for all i. Thus, e(pi,k A A p i ) 2 e(p). Since e(p,,) 2
163
$4. Categories of lattices
2 e(Pi,k), we have e A
APi
(ir1
)
i*io
2
e(Pi,k A
APi) 2 e(p). Put r
i+io
= Piok A
/\pi. We have e(r) = e(p) and check easily the other required
i+io
properties as well. The case (y) is quite analogous. In the remaining case, (D) holds for every i and (6) for every j . Denote by f(i) the set of the j E (1, ..., n } for which e(pi) 2 e(qj), by g(j) the set of the i E (1, ..., m } such that e(qj) 2 .(pi). If there is a k E g(f(io)) such that k $. i, we can put r = A p i , analogously i+k
if there is a k Ef(g(j,)), k =!= j,. Thus, let us suppose that g(f(i)) = = (i} and f(g(j)) = ( j ) for all i,j. Then, m = n, f a n d g are singlevalued and g = f - '. Since p q, pi, qi, for some io. By the induction hypothesis we have an ro such that e(ro) = e(pio)= e(qfcio,) and either l(ro) < l(pi) and c(ro) c c(pio), or l(ro) < l(qf(io))and c(ro) c c(qfCio,).In the first case put r = r , A A p i , in the second i+io = ro A Aqi.
+
i i io
The case of p Let p
111
N
N
vpiand q
Vpi and
i= I
N
+
vqi is dual to (A).
A w j . Since e(p) 5 e(q), we have e(pi)I e(q) I,
q
N
i= 1
and e(p) I e(qj) for all i,j. Since e(q) I e(p), we have either (a) e(q) I e(pi,) for an i, or (D) e(p) 2 e(qjo) for ajo. If (a) holds put r = pio, if (8)holds put I' = q,(,.
rn
4.7. Lemma. For every a E F X there is a pa E Y X such that (4 e(Pa) = a, if e ( ~=) a, then @ a ) I @) and +a) c c(P), (7) if e(p) = a and l(p) = /(pa),then p 2: pa.
(a)
Proof. Choose for p a the polynomial with the minimal length from among those with e(p) = a. Then, (7) follows immediately from 4.6. To prove (b), we have to show that c(p,) c c(p) for any p with e(p) = a. If p N pa, we have c(p) = c(p,). If not, there is, by 4.6, an r such that e(r) = a and I(r) < l(p) and c(r) c c(p). If r > pa, we obtain c(p,) = c(r) c c(p). If not, find an r1 with e ( r l ) = a, l(rl) < l(r), c(rl) c c(r). Since the length decreases, this procedure has to stop after a finite number of steps. rn Remark. By (y), pa is uniquely determined up to the equivalence It is called the canonical polynomial of a.
=.
164
Ch. V, ALGEBRA
4.8. Lemma. Let
Pd
be the canonical polynomial of d, let
Api n
P d ir
i= 1
with no p i a meet. If d = A d j then for any i E ( 1, ..., n} there is a j such j= 1 that e(pi)2 dj. m Proof. We have e(pi)2 A di. If pi = x E X , then by 4.4, e(pi)2 d j for m
some j . Let p i
j= 1
2:
Vpik.By 4.4 either k
e(pi)L d j for some j or e(pik)2 d
for some k. If the latter holds, however, then for r
2:
pik A
Apj we have
j+i
e(r) = d and l(r) < l(pa), which is a contradiction. Thus, .(pi) 2 d,.
H
4.9. Lemma. Let,for a, b, c E F X , aAb=aAc=d
Then d = a A (b v c). Proof. Evidently d I a A (b v c). Let p a be the canonical polynomial of d. (a) If pd
2:
Api and pi are not meets, we have, by 4.8, for every i, either
i= 1
e(pi)2 a or e(pi)2 b and either e(pi) 2 a or e(pi)2 c. Let M be the set of the i with .(pi) 2 a, N = (1, ..., n)\M. Put
Then La
A
= q1 (b v c).
Pd
(b) If pd
q 2 , e(q,) 2 a and e(q,) 2 b v c. Thus, d
A
vpi,we have e(Vpi) 2 n
2:
i= 1
a
A
i
=
e(q, A q 2 ) 2
b and hence, by 4.4, either
d 2 a or d 2 b or e(pio)2 a A b for some io. In the last case, however, e(pio) = d and (pi,) < @ d ) in contradiction with the choice of p d . Similarly, from e ( V p i )2 a A c we obtain d 2 a or d 2 c. Thus, either d 2 a or d2bvc. 4.10. For an a E F X , the capacity c(a) is defined by C(.)
= C(PJ
(=
C(P) e ( p )= a
by 4.7).
Lemma. We have c(a v b) c ( a v b ) u C(.)
u c(b)
c(a A b) c { a A b} u .(a)
LJ
Proof is straightforward.
c(b).
165
$4. Categories of lattices
4.11. A bounded lattice is a lattice L with a least and a largest elements. We will denote them by 0, and 1, respectively (if there is no danger of confusion, simply by 0 and 1). A (0, 1)-homomorphism h : L + L' between bounded lattices is a homomorphism such that h(0) = 0 and h(1) = 1. The category of bounded lattices and (0, 1)-homomorphisms will be denoted by Lb
(thus. L, is a subcatesory. not a full one, of L, the cateyory of all lattices and all homomorphisms). The free bounded lattice over a set X will be denoted by
-
FX, the (0, 1)-homomorphic extension of a mapping f: X X' by Ff,. Obviously, F X , = F X u (0, l} with the union disjoint. 9
4.12. The lattice L(X,R). For an object ( X , R ) of G (see 3.2) consider, first, FX,. An element a E F X is said to be join-singular (meet-singular, resp.) if there is an (x,y} E R such that a 2 x v y (a I x A y , respectively). We say that a is singular if it is join- or meet-singular. Put
I
L ( X , R ) = ( 0 , l ) u ( a a E F X and no b E .(a) is singular} L ( X , R ) is a lattice. Indeed, if a, b and a v b (a A b, resp.) are in L ( X , R), then obviously a v b = sup (a, b ) (a A b = inf {a, b}, resp.) in L ( X , R). We will show that otherwise sup (a, b} in L ( X , R ) is 1 and similarly inf (a, b} = 0. Thus, let a, b E L ( X ,R), a v b $ L ( X , R). By 4.10, c(a v b) c ( a v b} u c(a) u c(b), SO that a v b has to be singular, and since a is not, a v b has to be join-singular. Hence, if d is in F X and d 2 a v b, then d $ L ( X , R). Thus, 1 = sup {a, b} in L ( X , R). Similarly for the infimum. Convention. The join and meet in L ( X , R ) will be denoted by v, ii respectively. Denote by I = z(X, R ) : L ( X , R ) + F X , the inclusion mapping. Though z is not a homomorphism, we have z(a y b) = (a) v z(b) whenever a y b =I= 1, and z(a i ib) = z p ) A z(b) whenever a K b =k 0.
4.13. Lemma. For a, b E L ( X , R ) a
vb
=
1 and a K b
=
0 iff (a, b} E R u {(0,1}}.
Proof. If (a, b} E R, then a v b and a and a iib = 0.
A
b are singular, so that a v b = 1
166
Ch. V, ALGEBRA
+
Now, let a v b = 1 and a 7 b = 0. Obviously, if (a, b) n (0, 1) 0, (a, b) = (0, l}. Thus, we will assume a, b E L ( X , R)\{O, 1). The a v b and a A b are singular, and since a is not, a v b has to be join-singular and a A b meet-singular. Hence, there are (x,y), {u, v > E R such that x v y I S a v b and u A v 2 a A b. Consequently, x I a v b and u 2 a A 6. By 4.4, either x I a or x < b and either u >a or u 2 b . (LX) If x < a, a I u, we have by 4.4 x = a = u so that y i x v b and v 2 x A b. Since ( X , R ) has no loops, y < x and v 2 x cannot hold and we obtain y I b i v ; hence y = b = v again by 4.4. (fl) The case x < b, b I u is quite analogous. (y) If x I a, u 2 b, we have y I a v b and hence, by 4.4 either y I a or y 5 b. The first of the inequalities would imply x v y < a in contradiction with a E L ( X , R). Thus, y < b. Similarly, a A b < v implies a < v. Consequently, x < a I v and y < b I u so that by 4.4 x = a = v and
y=b=u. 4.14. Denote by TC = n ( X , R ) : F X , --t L ( X , R ) the unique (0, 1)-homomorphic extension of the inclusion mapping X -+ L ( X , R). Define a congruence on F X , putting
-
Lemma.
-
a
-b
iff .(a)
=
n(b).
is thefinest congruence on F X , such that
x v y - 1 and
x
A
y-0
forall
(x,y)~R.
Proof. Let k be the finest congruence such that x v y ri 1 and x A y 0 for all (x, y) E R . Since x v y = 1 and x K y = 0 for (x,y} E R, we have .(a) = n(b) whenever a r i b. Thus, it suffices to prove that for every a E F X , there is a b E L(X, R) such that a ri b. This is obvious for a E (0, l}. Thus, let a E F X . We will prove our statement by induction over the length of the canonical polynomial pa. For a E X the statement holds. If a = a , v a2 (or a = a , A a,) and a , b, E L ( X , R),a2 r i b, E L ( X ,R), then a A, b, v b, (or a ri b, A b2). If b , v b, 4 L ( X , R),necessarily b,, b, E L ( X , R)\(O, l} and b, v b, 4 {0,1>.By 4.10, b, v b, is join-singular and hence there is an ( x , y ) E R such that x v y I b, v b,. Since x v y ri 1, we have b, v b, ri 1, and hence a ri 1. Analogously for b, A b, $ L ( X , R). 4.15. For convenience, let us denote by 2(X,R ) the factor-lattice k X,/ -. Of course, it is isomorphic to L ( X , R).
64. Categories of lattices
167
From now on, we will use the symbol n = n ( X , R ) for the factorization epimorphism F X , + 9 ( X , R). Thus, to summarize the previous results, - 7t is the factorization by the finest congruence such that x v y 1 and x A y 0 for all {x, y} E R, - we have mappings z(X, R ) : 2 ( X , R ) + F X , such that z(n(x))= x for all x E X n o z = id (7) if a v b 1 (a A b 0, resp.) then z(a v b) = z((a) v z(b) ((a A b) = ~ ( aA) ~ ( b )resp.) , - a v b = 1 and a A b = 0 in 9 ( X , R ) iff {a, b} = {n(x), ~ ( y ) }for some {X’ Y } E R.
[;I
-
-
+
+
4.16. Theorem. Every category R such that LbC52CL
is almost alg-universal. In particular, Lb is alg-universal. Proof. A triangle in an undirected graph ( X , R ) is any {x, y , z } such that {x, y } , { y , z } and { z ,x} are in R. By IV.4.14 the category Graph,,
the full subcategory of UndGraph generated by the connected ( X ,R ) without loops such that every X E X is contained in a triangle, is alguniversal. Let us define a functor
9: Graph,
+
L
as follows: 9 ( X , R) is the lattice from 4.15; for a morphism f : ( X , R ) -+ (X’, R’), 9 ( f ) is the uniquely determined 2 ( f ) : 2 ( X , R ) -+ 9 ( X ’ ,R’) such that 9 ( f ) n = n Ff, -+
0
-
-
0
(this is correct since Ff, obviously preserves the congruence given by the x v y 1 and x A y 0 for {x, y } E R ). Obviously, 9 is a one-to-one functor. We will prove that it is almost full in two steps: (A) First, we will show that a non-constant homomorphism h : 2 ( X , R)+ 9 ( X ’ , R‘) is a (0, I}-homomorphism. For a w E F X , write W for n(w). If h(0) = 1 or h(1) = 0 then obviously h is constant. Thus, we can assume h(0) 1. Suppose that h(0) = d + 0. Consider a triangle {x, y, z } in ( X , R) and put T = {X,j7,Z,O, 1). If h(u) = h(u) for some U , U E 7; u $: v, then (consider u v v and u A v ) necessarily h(0) = h(1) so that h is a constant. -+
+
168
Ch. V, ALGEBRA
Thus, h is one-to-one on T. Hence, in particular, h(u)4 (0,l) for u E (X,y, ,TI, so that (h(X),h@),h(3) E T(X’,R’)\(O, 1). Since d = h(0) = h(X) A h(7) = h(X) A h ( q
we obtain, by 4.15 (y) and 4.9
44 = @(X))
A
(NL)) v @(a)
in FX’, so that by 4.15 (B), d = h(Z) A (h@) v h ( q ) = h(X A (7v 3) = = h(X A 1) = h(X) which is a contradiction since d = h(0).Thus, h(0) = 0. Analogously one proves that h(1) = 1. (B) Now, let h : T ( X , R ) -,2’(X’, R’) be a (0, 1)-homomorphism. For an x E X choose a triangle ( x , y, z> in ( X , R). As we have seen in (A), h is one-to-one on T. Hence, h(2) v h(j) = 1 and h(Z) A h(J) = 0 so that there is (see 4.15) an f ( x ) E X’ such that h(X) = f f . Thus, we obtain a mapping f : X -+ X’. We see immediatelythat it is RR‘-compatibleand that h = 9 ( f ) . 4.17. Denote by
D (Db, resp.)
the full subcategory of L ( L b , resp.) generated by the distributive lattices, i.e. by such ones in which always (u
A
b) v c
= (u v c) A
( b v c) ,
(a v b) A c
=
(a A c) v (b v c) .
Proposition. (1) D is not almost alg-universal, (2) Db is not alg-universal. Proof. (1) Let L be a distributive lattice with at least three elements, let a E L be neither the least nor the greatest element. Put h(x) = x A a. Since L is distributive, h : L -+ L is a homomorphism and we have h2 = h. Since a is not the greatest element, h id, since a is not the least one, h is not constant. Thus, e.g. no group with more than two elements is an endomorphism monoid of a distributive lattice. (2) Denote by T the trivial bounded lattice consisting of 0 and 1. For a bounded distributive lattice L with at least three elements take the maximal congruence 0 such that (0,l) $ 0 (it exists by Zorn’s lemma). 0,i. Suppose that there is an Z E L/O, ii =I= Define a congruence on L by
+
-
X - y o X A i i = y A a
-
(X is the @-congruence class of x). Then 0 in contradiction with the maximality of 0.Thus, L / 0 N T and we obtain a homomorphism
169
$4. Categories of lattices
h : L + L by composing the embedding T c L with the factorization mapping L + L/O. This h is not invertible. Thus, no non-trivial group can be represented as the endomorphism monoid of a bounded distributive lattice. rn Remark. We will see below, however, that every group can be represented as the automorphism group of a bounded distributive lattice. 4.18. A complete lattice is an ordered set L in which every subset has a supremum and an infimum. (We will indicate them by I\ respectively; in the case of sets of two elements we will write again x v y, x A y.) If L, L' are complete lattices, a mapping f: L + L' is said to be a complete homomorphism if f ( l \ A ) = A f ( A ) and f ( v A )= V f ( A ) for all A c L. Evidently, every complete lattice is bounded with 0 = = AL and 1 = AS = V L , and every complete homomorphism is a (0,l)-homomorphism. A (complete) distributive lattice is called (complete) Boolean algebra if for every x E L there is a unique X'E L such that x v xc = 1 and x A xc = 0. A complete lattice is said to be completely distributive if for any collection { A , m E M } of subsets of L,
v,
v@
I
V = A (V(a(m) aeA A (VAm)= V ( A ( 4 m ) meM msM
( A A m )
aEA
where A
=
)(A,. msM
A set of generators of a complete lattice (Boolean algebra) L is a set K c L such that the smallest complete sublattice (Boolean algebra) of L containing K is L itself. Denote by DL (DB, resp.) the category the objects of which are couples (L,K ) where L is a completely distributive lattice (Boolean algebra, resp.) and K a set of generators, and the morphisms from (L,K ) to (L', K') are the complete homomorphisms h: L + L' satisfying h(K) c K'.
4.19. Recall the functor P - : Set + Set from 1.3.14. (F). For any set S, the P - S ordered by inclusion, as it is well known and very easy to verify, is a completely distributive Boolean algebra, and for every f: S + ?; P-f:P - T + P - S is a complete homomorphism. Moreover, conversely, every complete homomorphism g: P - T + P - S is P-f for some f: S + T
170
Ch. V, ALGEBRA
Define cp = cps:
s
P-P-S
--f
putting cp(s) = ( Z c s 1 s E z}. We see easily that thus a transformation (&: Id + P - P - is defined. Lemma. = {cp(s) 1 s E S } is a set of generators of the complete Boolean ulgebra P - P - S , while the complete lattice generated by is the set L of all T EP - P - S such that X E T , Y x X => Y E T . 0
s
s
Proof. The second statement follows immediately from the fact that L is a complete lattice and every T satisfying the condition is equal to ( A cp(s)). Now, for an arbitrary X c S,
v
X E T SEX
{X} = (Acp(4 A SEX
so that an arbitrary T EP - P - S , being
(Y Vx seY Acp(s)Y7
v { X } , can be expressed in cp(s).
X ET
4.20. Construction. For an object ( X , R ) of G (see 3.2) let B ( X , R ) be the complete Boolean algebra P - P - ( X u R ) (we will suppose the X , R disjoint). Let L ( X ,R ) be the complete lattice generated by cpXUR(X u R) in B( X , R ) . Let 2 be the congruence on B ( X , R ) (or L(X, R))generated by
Put
cp(x) v q ( y ) v cp({x, Yj) = 1 S{X,R)=
=
{z
c
A
for all {X' Y } E
R .
(4x1 v CP(Y)v 4 b y I ) )
(x,Y)ER
x u R 1 (Vr E R ) ( Z n (r u ( r } ) )=+ 0) .
We see immediately that T 1: T' iff T n S(X,R)= T' n S(X,R).The factorization epimorphisms will be denoted by &X,R): A(X,R):
B ( X , R ) + B(x, R ) = B ( X , R ) / E , L(x,R ) L(x,R) = L(x, R)/= .
4.21. For Z E X U R put Z = j c p ( z ) , ~ = ( X I X E X )R, = { F l r E R } . We will use the fact mentioned above that the classes Z are in a one-to-one correspondence with the elements of the form q ( z ) n S(X,R).
Lemma. If zl, z2 E X u R are distinct, then TI =k Z2. Proof. For z, E X , z2 E R, cp(z2)
S(X,R)\(P(Z1)
S(X,R)
.
$4. Categories of lattices
171
m}
4.22. Lemma. I f Z, v Zz v Z3 = 1, then {Z1, Zz, Z3} = {Z,7, for an { x ,y } E R. Proof. We have S ( X , R ) = S(X,R)n (cp(zl)u cp(z2)u cp(z3)).If zi E X for i = 1,2, 3, R E S ( x , R ) but R 4 (cp(z1)u cp(z2) u cp(z3))= C, if zi E R for i = 1,2, 3, { x l ,x 2 ,x 3 } u (R\{zl, z2, 2 3 ) ) E S,,,,)\C for arbitrary xi E zi. If z1 E X and z 2 , z 3 eR choose x i ~ z i \ ( z 1 ) ( i = 2,3); we have { x 2 , x 3 }u u (R\(z2, z 3 } )E S(x,R)\C. Thus, in a suitable order, z l , z2 E X and z 3 E R. If z3 { z l ,z 2 } , there is an x E z~\{z,, z 2 } and we have { x } u (R\{z3}) E E S(X,R)\C. Thus, z3 = { Z I , Z Z } .
+
4.23. Theorem. DB and DL are alg-universal. Proof. By IV.4.14 the full subcategory Graph,
of G generated by the ( X ,R ) in which every vertex belongs to a subgraph with four vertices such that any two of them are joined, is alg-universal. Define a functor B:Graph, -, DB
x,
putting B(X, R ) = ( B ( X ,R), X u R ) (B from 4.20, R from 4.21). For a morphism f: ( X , R ) + ( X ’ , R’) define B(f) from B(f) o j = jo P-P-g where g(x) = f ( x ) , g({x,y } ) = { f ( x ) , f ( y ) }(the checking of the correctness of this definition is straightforward). Obviously, 6@ is one-to-one. Now, let h: B ( X , R ) + B(X’, R’) be a morphism, i.e., a complete homomorphism h: B ( X , R ) -+ B(X’, R’) such that h(X u R ) c X‘ u R’. First, we will prove that h ( X ) c X’.Otherwise, h(X) = for an X E X , {x‘,y‘} ER’. Choose distinct x ~ E X ,i = 1,2,3 such that {x,xi} and { x i , x j }for i j are in R. We have
+
x v xi v
=
1
for i
=
1,2,3.
Hence, v h(Yi) v h ( ( x , J ) = 1. By 4.22, {h(xi),h ( ( x , } = {T’,7’} and hence h cannot be one-to-one on !.i?,..F2,F3j.This is ;I contradiction, since, for i + j , X, v Xj v {xi,xj) = 1 and hence /I&) v h(Yj) v h({xi,x j } )= 1
172
Ch. V, ALGEBRA
so that h(Ti)
+ h(Tj) by 4.22. Thus, h ( X ) c 8’.Define
fm
f : X+X’ by = h(T). Take an ( x ,y } E R. Since we already know that f ( x ) ,f ( y )E X’, we immediately obtain by 4.22 that h ( ( x , ) = ( f ( x ) , f ( y ) } .Consequently we also see that f is RR’-compatible. Now, using the fact that q (see 4.19) is a transformation, it is easy to verify that h = B ( f ) . For the case of DL, one defines a functor 2’: Graph, + DL putting 9 ( X , R) = ( L ( X ,R), X u 8) and taking for 9(f) the domain-range restriction of a(f).Obviously, 2’ is one-to-one. To see that it is full, notice that a complete homomorphism E(X, R) -+L(X’, R’) can be extended to a complete homomorphism B(X, R) -+ B(X’, R’). rn 4.24. Let L be a complete lattice. An element a E L\(O} is called meetirreducible (or simply, irreducible)if for any A c L a = M implies a E A.
A minimal irreducible element is an irreducible a such that no b < a is irreducible. Lemma. a is a minimal irreducible element in L(X, R) iff a E 8 u 8. Proof. (A) Let Z = M, z E X u R. Then q(.)
S(X,R)
=
{ZES(X,R)
I zEZ}
S(X,R)
for every TEA. If 24 A , we may choose ZT E T\(cp(z) n S(X,R))for each T EA. Put W = Z p By 4.19, W ET for all T E A so that W EAA.Since
u
TEA
z $ W we have W $ 5.Hence Z 4 AA. (B) Take a b with b < Z, Z E X u R. Thus, b s (ZES ( , R ) Z E Z } . Denote by 8 the system of all o: b + X u R such that o(2)E Z for every Z E b. Since L ( X , R) is completely distributive, we have
I
b=V
A ( S ( x , R ) n ~ ( t )=) UES A ZVe b ( S ( x , R ) n c~(o(Z))).
Z s b teZ
To prove that b is not irreducible it suffices to prove that b i V(S,,,,) n ZEb n q(o(Z)))for all o E 8 : Let there be a Z 0 e b such that o(Z,,) 4 z. Choose a VES(X,R) such that z B I/: Then {C(zO)} u V E (S(X,R) cp(‘(Z)))\b.
v
Zeb
Otherwise, o(Z) = z for all Z
E b.
Then
v (S(x,Rln cp(o(Z)))= Z 4 b.
Zeb
(C) Now let a be a minimal irreducible element. We have a
=
A (VA,,,)
msM
45. Unary algebras
where A , c X u a is irreducible, a
=
173
(since any element may be expressed this way). Since V A , for some m. Since a is minimal, card& = 1.
4.25. By 4.24 and 4.23 we immediately obtain Corollary. The category L* of a11 completely distributive lattices and all complete homomorphisms preserving minimal irreducible elements is alg-
W 4.26. Since every isomorphism preserves minimal irreducible elements, we obtain further (see 1.8.12). Corollary. Every group is isomorphic to the automorphism group of a completely distributive lattice. W 4.27. Remark. It was proved in [MM] that Z, the cyclic group of order 3, is an automorphism group of no Boolean algebra. universal.
EXERCISES
1. The category of completely distributive lattices and their complete homomorphisms is not alg-universal.
8 5. Unary algebras 5.1. In this section we are going to investigate the alg-universality of some varieties of unary algebras. The basic constructions which we will use, as they concern embeddings of categories like Graph into categories of algebras, yield embeddings, which necessarily fail to be strong (cf. IV.6.8). Concluding the section, however, we will prove that all of them imply strong embeddings of Alg(1, 1) into the varieties in question. 5.2. To simplify the notation, we will write
A k I(J) for the category Aig((Ai)ieJ)where A i = 1 for all i . Thus, the objects of Alg ( J ) are systems ( X , where cpi: X -+X are mappings and the morphisms ( X , (cpi)) + (X’, (cpf)) are the mappings f satisfying cp: o f = f o qi for all i E J . We will often write, as we did before, ( X , cpo, ..., cp,) instead of ( X , (qi)i=o, l,..,,). The equations defining a variety of unary algebras have particularly simple form: They are either of the type Wl(X) = WZ(X)
where wi are words of symbols of the operations to be interpreted as com-
174
Ch. V. ALGEBRA
positions of the concrete operations, requiring that the compositions indicated by w1 and w2 coincide, or of the type w1(x) = W 2 ( Y )
requiring that the indicated compositions have a constant common value. If 8 is a set of equations concerning unary operations indexed by J , we will denote by v(8; J ) the corresponding variety, i.e. the full subcategory of Alg, ( J ) generated by the algebras satisfying all the indicated equations. 5.3. Recall the realization of Alg(d) in Alg(4’) with A i A‘ from 11.1.3. In our case, it gives a realization of Alg, ( J ) in Alg, (J’)for J c J’ in which with an algebra ( X ,(pi)J) we associate the ( X , (pi)J,) where pi = idx for i E J’\J. Consequently, if we already know that V(E; J ) is alg-universal, we also know that every V(& u 8,; J u J1) where the equations from El
are of the first type from 5.2 and do not contain symbols of operations indexed by J , is alg-universal. (Moreover, when modifying the case of A , A’ with unary and nullary operations, we see that one can allow equations of the second type in &, stating exactly that some of the operations are constant, if there is such in E, as well). This stresses the importance of the varieties in Alg (1,l) (although, sometimes the problem of alg-universality of a variety with a larger J cannot thus be reduced, as one will see e.g. in 7.12). 5.4. We see easily that the varieties of unary algebras with the equations of the first type from 5.2 coincide with particular categories SetK,namely with those where the small categories K are monoids. Some facts about the varieties of unary algebras not mentioned in this paragraph will be presented in this broader context in 0 7. We will now present a few elementary negative results.
5.5. Proposition. Let 6 be a full subcategory of Alg ( J ) such that for every non-void ( X , there is an xo E X with qi(x0)= xo for all i E J . Then 6 has no non-trivial rigid object and therefore it is not alg-universal. proof. Really, the f : X -,X defined by f ( x ) = xo is always a homomorphism. rn 5.6. Proposition. N o V(8, J ) such that E contains all the equations ofthe form pi ‘pi= ‘pi pi (i, j E J ) is alg-universal. 0
0
175
s5. Unary algcbras
Proof. Let ( X , ( V ~ )be ~ ) in V(&,J). Suppose it is rigid. Then all the -+ X , being homomorphisms according to the equations, are identities. Hence every mapping X X is a homomorphism so that cardX = 1. Thus, there is no non-trivial rigid object in V(d,J ) . cpi: X
--f
5.7. A subset Y of an algebra A = ( X , ( V , ) ~is) said to be connected if for any two x, x' E Y there are x1 = x, x2, ..., x, = x' such that for every j = 1, ..., n - 1 there is an i € J with either xj = or x j + l = cpi(xj). Every maximal connected subset of A (it is referred to as a component of A ) is evidently a subalgebra. Proposition. Let m be a cardinal number, let 6 be a full subcategory of A l g l ( J ) such that each component of any object of 6 has the cardinality less than m. Then G is not alg-universal. Proof. Really, G has only a set of non-isomorphic rigid objects: An algebra with a large enough underlying set has necessarily isomorphic distinct components. Interchanging them gives a non-trivial automorphism. rn
5.8. Proposition. Let G be a full subcategory 01 A l g , ( J ) such that for every object ( X , ( V ~ )the ~ ) cps; are one-to-one mappirigs. Then G is not alguniversal. Proof. Put n = card J . Since cpi's are one-to-one, card ( { cpi( y ) I i E J } u u cp- ' ( y ) )I 2n for every y. Consequently, the cardinality of no com-
u
if
J
ponent exceeds Apply 5.7.
1
+ 2n + (242 + (243 + ... 5
K~ . n .
rn
5.9. Convention. ( 1 ) Write 1, (or. if there is no danger of contusion, simply 1 ) For the identity mapping of X onto X . For a cp: X + A' define as usual cp" by cpo = 1, cp"+' = cp 0 cp".
(2) We will write simply
V(4 instead of V ( € ; (0, 1)). If € = { e l ,..., ek},we will write simply V(el,...,ek) (thus, e.g. V(q2 = 1, @cp = i,h) is an abbreviation for
5.10. As a corollary of 5.8 we immediately obtain Proposition. Let ( r ~ , )be ~ ~a ,system of nutural numbers. V( { 9"'= 1 ~ E 1;J J )
is not alg-universal.
I
rn
176
Ch. V, ALGEBRA
5.11. Proposition. I = V(p2 = 'p,'!+I = +) is alg-uniuersal. Proof. In this proof, the sing indicates the addition modulo 8. For a graph ( X , R ) define @ ( X ,R ) = (r,'p, +)
+
where Y = X x (0,1, ..., 81 u R x (91, and
+ 1) = (x, i + l ) , +(x, i ) = +(x, i + 1) = (x, i + l ) ,
for i I 6 even for i I 7 odd
q(x, i ) = cp(x, i
'p(x, 8) = (x, 3) 'p((x9
9
Y), 9) = (x, 5 ) 7
8) = (x,0) +((% Y ) , 9) = (Y, 0 ) . +(%
The situation is visualized in Fig. 5.1 ('p is indicated by full arrows, $ by dashed ones):
,--\
5.1
If f: ( X , R ) + ( X ' , R') is compatible, define
@(f): @(X,R ) + @(XI,R') putting @(f)(x, i ) = ( f ( x ) ,i), this way a one-to-one functor is defined.
@(f)(x, y, 9) = ( f ( x ) ,f ( Y ) , 9).
@: Graph + I
Obviously, in
177
55. Unary algebras
Now, let g: @(X,R) -+ @(X’,R’) be a homomorphism. Since only the with i odd I 7 (i even I 6, resp.) are fixed under cp ($, resp.), we see that g ( x x (0, 1, ...)7)) c X’ x (0, 1, ..., 7)). 0
(x, i )
Further, considering cp, $, cp$, $cp, &cp, $cp$, etc. we see that for i I 7, if g(x, i) = (x’, j ) then g(x, i + k) = (x’, j k). Suppose that g(x, 8) = (x’, i) with i I 7. Then g(x, 0) = g +(x, 8) = +(XI, i) and g(x, 3) = g cp(x, 8) = cp(x’, i) in contradiction with (*), and a similar contradiction is obtained supposing g(x, 8) = (x’, y’, 9). Thus,
+
g ( x x (8)) = X’ x (8) and we can define f: X -,X‘ by g(x, 8) = (f(x), 8). Using alternatively and cp, we obtain g(x, 0) = (f(x), 0), g(x, 1) = ( f ( x ) , 1) etc. Thus, g(x, i) = = (f(x), i) for i 5 8. Now we see immediately, since $ g(x, y, 9) = (f(x), 0) and cp g(x, y, 9) = = (f(x), 5) that g(x,y,9) = (f(x),f(y),9) and hence f is compatible and 9 = @(f).
+
5.12. Let R be an n-ary relation on X , i.e., R c X x ... x X (n times). We say that R has no loops if for every (xi, ..., X,)E R there are i,j with xi =i= xj. R is said to be cyclic, if (x,, xl, ..., x,- 1) E R whenever (xl,’..., X,)E R. Denote by C, the category of sets with cyclic relations without loops and their compatible mappings.
Lemma. For every n 2 2, C, is alg-universal. Proof. C2 is alg-universal by IV.3.3, and it can be realized in every C, n 2 2 as follows: For ( X ,R ) from C2 consider the (X, defined by
a)
iff {xl, ..., x,)
(xl, ...,
=
(x,y)
for an (X,Y)ER .
5.13. Proposition. W, = V(cp” = 1, $’ = $) is alg-universal for ewry n 2 2. Proof. Define a functor as follows:
@:
c, --* w,
= ( R x ( 0 ) )u ( X x ( 4 2,..., n @(X,R ) ...)% - 1, 0) @(XI, .)x n 9 0) = (xn7 *(Xl? ..., x,, 0) = (Xl,a) 9
3.
7
+ l)),
178
Ch. V, ALGEBRA
cp(x1 i)
=(x.i+l)
cp(% n)
=
for i = l , ..., n - 1 ,
(s,I ) ,
+ 1)
=
*(X? i )
(x,n + l ) , = (x,i ) for i
tj(x, n + 1)
=
cp(x, n
= 1, ..., n ,
( x , 1).
The situation is visualized in Fig. 5.2 (the full arrows indicate cp, the dashed ones indicate tj): ,--,
-
.
-.
,--\
- - - _- _ _ _ _ _ _ _ - - - - - -
/
Fig. 5.2
If f: ( X , R ) (X', R') is compatible, define @(f):@ ( X ,R ) -+ @'(X',R') by @(f)(x?i, = (f(x),i, for 2 l 7 @(f) O) = (f(.1)7 ...7f(Xn), O). One checks easily that @ is really a one-to-one functor into W,. Now, let g : @ ( X ,R ) @(X',R') be a homomorphism. First, we see that the (x, n + 1) are the only elements fixed under cp (there are no loops in R !). Consequently, g(X x ( n + 1 ) ) c X' x { n + l} and we can define
-
...?
xfl,
f: x - X '
by (f(x), n
+ 1) = g(x, n + 1). Consequently,
g(x, i )
=
g(cpi-l$(x, n + 1)) = cp'-'$(f(x), n + 1)) = (f(x),i ) for 1 < i < n .
For an r = (xl, ..., x,) E R we have tj g(r, 0) = g(x,, 0)E X x ( n } so that g(R x ( 0 ) )c R' x ( 0 ) u X x (n}. Since also t j c p g ( r , O ) ~ Xx {n}, necessarily g ( x l , ..., x,, 0) = (xi,..., x;, 0). We have (f(xi),n) = g t j c p n - i + ! ( ~ l , . . . , ~ n= ,O) = tj cp,-'+ yx;, .. ,x;, 0) = (xi, n) .
Thus, f is compatible and we have g
=
@(f).
179
$5. Unary algebras
5.14. If all the equations from & are consequences of equations from &, then obviously V(bl; J ) is a full concrete subcategory of V(&; J ) . Thus (recall 5.3), if V(&,; J ) is alg-universal, if J' 3 J and if every equation of 8'either follows from 8,or is of the first type from 5.2 and does not concern the operations indexed from J , then V(6,; J ) is realizable in V(&; J).
5.15. Theorem. Let (mi)isJ , (ni)ie be collections of natural numbers such that,,for every i, mi n, 2 1. The variety
+
W = V((cpli
=
(PT~"'~ I i E J ) ; J )
is alg-universal iff either mi, m j > 0 or mi > 0 and nj > 1 for two distinct indices i, j E J . Proof. If mi,mj>O, we have, by 5.14, V((cpi= cp', cpj = cpf); {i,j)), which is alg-universal by 5.11, realizable in W . If mi > 0, m j = 0 and nj 2 2, we have V ( [ q i = ipi, 2 9;' = l}; {i,j}),alg-universal by 5.13 realizable in W . On the other hand, if the condition is not satisfied we have either an io E J such that mi, > 0 , mi = 0 for i $. io and ni = 1 for i i o ,
+
and then, obviously, W can be full embedded into Alg(1) and hence it is not alg-universal, or, mi = 0 for all i E J and then we use 5.10. Remark. 5.15 covers all the varieties such that no equation concerns more than one operation and all are of the first type mentioned in 5.2. The condition mi + ni 2 1 is no restriction: The tautology cpf = cpf can be replaced by equally tautological cp! = cp!.
5.16. Observations and remarks. In 5.11 there is actually more proved than stated. It is easy to verify that every @ ( X ,R) satisfies, besides cp2 = cp and $' = $ also (cp$)" cp = cp and (t,hcp)" $ = $. In [PSI a full description of the alg-universal subvarieties of I is given. It is proved there that W c I is aIg-universal iff it contains v(q2= cp, $ 2 = $, (cp$)" cp = cp, (Il/cp)'( II/ = (I/) for some k 2 3. Similarly, the algebras @ ( X ,R) in 5.13 satisfied more than the equations stated, namely also
... cp"*
cp"*
for any ki such that
1
ki
i= 1
=
n.
cp"*
=
*
180
Ch. V, ALGEBRA
5.17. So far all the equations we have dealt with have been of the first kind mentioned in 5.2. We are now going to consider also the second. Proposition.
T = V(cp'(x) = cp"Y), $'(x) = $"Y)? cp'(x) = cp+(Y)? V
( X ) =
$cp(Y))
is alg-universal. Proof. Take the category G' of all connected symmetric graphs with at least two points and all their compatible mappings. By IV.3.3, G' is alg-universal. Obviously, if ( X , R ) is in G' then for every x E X there is a y =k x such that ( x ,y ) E R (and, of course, also (y,x ) E R ). Construct a functor @: G'-+ T as follows: @ ( X ,R) = (xu R u (a, b}, cp, $) (the union is assumed to be disjoint, a, b distinct) where q(a) = q(b) = cp(x) = a ,
(for all x E X ) ,
$(a) = $(b) = $(x) = b
V(X,Y ) = x
9
$(x, Y ) = Y .
If f : ( X ,R) -+ ( X ' , R') is compatible, define @(f): @ ( X ,R ) -+ @(X',R') by @(f)( x ) = f(x) for x E X,@(f)( x ,Y ) = ( f ( x ) f, ( y ) ) , @(f)(c) = c for c = a, b. One checks easily that really a one-to-one functor G' -+ T is obtained. Let g : @(X,R ) -+ @(X', R') be a homomorphism. Since a (b, resp.) is the only element fixed under cp ($, resp.), we have g(a) = a and g(b) = b. If x E X , we have a y with ( x ,y ) E R, hence x = cp(x,y ) = $(y, x ) and g(x) = = cpg(x,y ) = $g(y, x ) SQ that necessarily g(x)E X' (no other point is both q(<)and $(q)). Thus, we have a mapping
f: x -+ X' defined by f ( x ) = g(x). Finally, g(x, y ) = (x',y') E R', since otherwise cpg(x,y) gcp(x,y) = g(x), and we have x' = w ( x , y ) = g(x) = f ( x ) and similarly y' = f ( y ) . Thus, f i s a compatible mapping and @(f)= g.
+
5.18. Theorem. Let numbers. The variety
( M , ) ~ (ni)ieJ, ~ ~ , (pi)icJ,(qi)ieJbe
collections of natural
w = v((cpy(x) = cpY'+"(X),cpp(x) = cppt+q'(y)I id}; J)
is alg-universal iff there exist distinct i, j Pk
22
and either
EJ
such that
nk = 0 or mk 2 2
for
k
=
i,j.
55. Unary algebras
181
Proof. If the condition holds, we have by 5.3,
I
V ( { d ( x ) = cpZ(y) k
= i,j);
{i,j}),
which is alg-universal by 5.17, realizable in W . Now, let for some k either pk I1 or nk > 0 and mk 5 1. In the first case obviously (Pk is a constant. In the second case, choose an 1 such that mk In, = pk + r with r 2 0. Then cpp(x) = qP+'"'(x)= cp;cpkpL(x) = = cp;cppk(y) = cp?(y) so that again cpk is a constant. Thus, if the condition does not hold, all the (Pk with the exception of at most one are constant so that W is realizable in Alg (1,0,0, ..., 0), which is not alg-universal, see 11.5.5. R
+
5.19. Remarks. The variety Srng of all semigroups and the variety Cornrn of commutative grupoids are alg-universal (see 2.9, 11.2.3), but Srng n Cornrn, the category of all commutative semigroups is not. Simi-
larly here we have the case of alg-universal I and W,, (see 5.11, 5.13) and I n W,, which fails to be so (since cp" = 1 and cp2 = cp imply cp = l), or the case of W,, n W,, with m. I I relatively prime. Naturally. the question concerning the minirnality of a variety with respect to the alg-universality arises. Let us say that a varietyM is a minimal alg-universal variety if it is alg-universal and none of its proper subvarieties is so. It is not known whether e.g. Smg is a minimal alg-universal variety. I and W, are not minimal (see 5.16). It is not known whether every alg-universal variety in Alg(1, 1) contains a minimal one. In the following paragraphs 5.20 - 5.22 we will show that the minimality with respect to the equations of the first type mentioned in 5.2 only does not coincide with the minimality with respect to all equations (5.20), and give two examples of minimal alg-universal varieties (5.21, 5.22).
5.20. Example. For the variety V = V(cp2 = cp3 = cp$, $2 = t,h3 = $cp) V for T from 5.17. Thus, V is alg-universal and we obviously have J not minimal. Let us, however, add to the defining equations of V a new one of the type ~ ( x = ) w'(x). According to the equations already present, we can assume w, W ' E {l, cp, i,b, q2, Obviously, if some of w, w' is 1, we obtain cp = 1 or $ = 1 (also if q2= 1 we have cp = qcp2= cp3 = cp2 = 1, and similarly for $' = 1) and the category is not alg-universal. Obviously it is not alg-universal if we add cp = $. The remaining possibilities all imply cp2 = $' (e.g., cp = i,b2 gives cp2 = $4 = t,h3 = i,b2) and the latter gives cp$ = $cp so that the category is not alg-universal by 5.6.
$'I.
182
Ch. V, ALGEBRA
5.21. Example. We will show now that the T from 5.17 is a minimal alg-universal variety. According to 5.20, we cannot add an equation of the type s(x) = s’(x); also, we cannot add W(X) = w’(y) with w =k w’, because then even the larger variety determined by W(X) = w’(x)is not alg-universal. Thus the new equation just states that some of 1, cp, $ is constant, which leads obviously to a category which is not alg-universal. 5.22. Example. In the following example we will show that a variety can be minimal alg-universal even if all the defining equations are of the type W(X) = w’(x).Consider
w =V(cpZ
=
1, *z = *, cp*cp*
=
$).
Since in the proof of 5.13 every @ ( X ,R ) is in fact in W, we see that W is alg-universal. Now, let us add a new equation. Let it be of the form W(X) = w’(y).Let w contain $. We can assume that it terminates with otherwise we consider the equivalent equation wcp(x) = w’cp(y) (recall that cp2 = 1). Then, our equation implies ((P$)~’(X)= w”w’(.v). But (cp$)” = and hence $ is constant. If none of w,w‘contain $, then w(x)= w’(y)states that a one-to-one mapping is constant, so that the underlying set of the algebra contains only one point. Thus, we are left with the case of an additional equation of the form W(X) = w’(x).
+,
First, let us consider the following three cases: (a) w’ = 1: If w contains $, then $ has to be one-to-one and we can use 5.8. If it does not, we have w = qk.Since it should not follow from the original equations, k is odd and consequently cp = 1 so that the variety is again not alg-universal. (b) w = cp$, w’ = $cp: use 5.6. (c) w = $, w‘ = cp$: Then $ (I)(.)) = $(x) = cp($(x)) and our variety is not alg-universal by 5.5. Now, we will show that any equation W(X) = w’(x)which is not derivable from the original equations implies some of the cases (a), (b), (c) just discussed. Because of (a) we may suppose that w,w’ =k 1 so that we have w7
W’
E (cp,
*7
cp*, *cp,
*cph cp*%
*cp*cp)
Let, first, both w and w’terminate with $. If w’ = $, we have either w = cp$ or w = $cp$ (the other cases would be derivable from the original equations). In the former case, we may apply (c) immediately, in the latter after multiplying the equation by cp from the left. If w’ = cp$, w is either $
s5. Unary algebras
183
or $cp$. When multiplying by cp we obtain cp$ = $ and use (c). If w‘ = $q$, the only possible w is $. Again we use (c) after multiplying by cp. If both w and w’ terminate with cp, multiply by cp from the right. The obtained equation is one of those already discussed. Finally, let w terminate with cp and w‘ with $. If w = cp, we obtain 1 = wcp = w’cp and can use (a). If w = $9, either w’ = $ and we obtain $ = cpw$ = cp$2 = p$ and use (c), or w‘ = cptj and we use (b), or w’ = $cp$, so that ( ~ $ = 9 cp$cp$ = $; consequently cp$ = $cp and we can use (b) again. If w = cpJIcp, either w‘ = tj and then immediately cp$ = $q, or w’ = cpJ/ and we obtain $ = $cp, which has already been discussed, or w’ = $q$ and we obtain $ = $40 again. Finally, if w = $cp$cp either w’ = $ and we obtain cpw = $cp = cp$ and use (b), or w’ = cp$ or $cp$ and we obtain $cp = cpw = $ which has already been discussed.
rn
5.23. We will conclude this section by a proposition on strong embeddings of Alg (1,l). For technical reasons, we will need a generalization of the
notion of strong embedding. Let I/: 6 -+ 2, V’: 6’-+ 2’ be functors. A full embedding @: G -+ 6’ is said to be a VV’-embedding if there exists a functor F : 2 + 2’ such that V’ o @ = F V; the F is called underlying functor of @. Until now, the case of 2 = 2’ = Set has been the only one considered . (cf. 1.6.11). Now, we will also be concerned with the functor 0
B,: ReI(n) -+ Set x Set (ReI(n) = Rel(A) with A = (n), see 1.5.6; the index n will be omitted
from now on) defined as follows: B ( X , R ) = ( X , R); if f : ( X , R ) (X‘, R’) is a morphism, then B ( f ) = (1;9 ) where g: R -+ R’ is the domain-range restriction of f x ... x 1: The symbol B is also used for the restrictions of B to subcategories of Re I (n). Observation: The arrow-construction yields a BB-embedding whenever it yields a full embedding. Thus, there exists a BB-embedding of Graph into SymGraph, (see IV.3.3). Also, the full embedding of SymGraph, (considered as a full subcategory of C,) into C,, described in 5.12, is a BB-embedding. Its underlying functor is equivalent to the F defined by F ( X , Y ) = ( X , Y x (2,-’ - l)), F(1; 9) = (1;g x id) (the verification is left to the reader as an easy exercise). -+
5.24. Denote by U : Alg(1, 1) -+ Set the natural forgetful functor. While Graph cannot be strongly embedded into Alg ( 1 , 1) (see IV.6.8), it can be BU-embedded there (the full embedding of Graph into the smaller cate-
184
Ch. V, ALGEBRA
gory I described in 5.11 is a BU-embedding). On the other hand, Alg(1, 1) can be UB-embedded into Graph. Indeed, let F : Set --t Set x Set be defined by F ( X ) = ( X x 3, X x 4), F(f) = (f x id, f x id). Then, SP: Alg(1,l) -+ Graph defined by
( + I
@ ( X ,cp, I))= X x 3, 1)) x
u R i) 3
i=o
R~
=
(((x,i), ( x , i
Rz
=
R3
=
(((x,2)?(cp(x),1)) x EX} 7 (((x1 o), ( I ) ( X ) ? 2)) x E
I 1
is a full embedding such that B @ 0
E
X)
where
for
i
=
0,1,
x)
=
F’ U where F’ is equivalent to F. 0
5.25. Proposition. Let there exist a BVembedding of Graph into 6, let (3, W) be a concrete category. Ifthere exists a VW-embedding o f 6 into 3 Alg (1,l) can be strongly embedded into (3, W). Proof. We have a BW-embedding of Graph into J. By 5.24 there is a UB-embedding of Alg (1, 1) into Graph. The composition is a strong embedding of Alg (1,l) into (3, W).
Corollaries. In all the constructions in this section, the categories embedded (Graph in 5.11, C, in 5.13, and SymGraph, in 5.17) are such that there exist BB-embeddings of Graph into them. Further, the full embeddings con-
structed are obviously BVembeddings so that we always obtain the strong embeddability of Alg (1, 1) into the variety in question. Let us mention here that J. Sichler proved, in [S,], a more general statement: If Graph can be fully embedded into a variety of unary algebras, then Alg (1,l) can be strongly embedded into it. EXERCISES
1. Prove that
v = v(cp2= 1,
=
I), (I)4yI) = (I)cpI3)
is alg-universal (Hint: Consider @: SymGraph, --r V given by @ ( X ,R ) = = (E: cp, I)) where Y = R u X x (0, 1, ..., 6) and cp(x, y ) = ( y , x), @(x,y ) = (x, 1) for all (x, y ) E R, q ( x , 0) = ( x , O), cp(x, i) = ( x , j ) for ( i J } E {(1,2), (3,4), ( 5 , 6 ) } , @(x,0) = ( x , 3) = I)(x, 3 ) = I)(x, 2), I)(x91) = = (x, l), $(x, 4) = (x,5) = @(x7 5) = @(x,6 ) for all x E X . )
185
g6. Categories of small categories
2. Prove that for any t 2 2,
v, = v(cp2= 1, $’
=
$>($cpp)’*
=
$)
is alg-universal. (Hint: Consider Qi: SymGraph, -+ V, given by @ ( X ,R ) = = (Y, cp, $), where Y = R u X x (0, ..., 6}, cp(x,O) = (x,O), cp(x, i) = ( x , j ) for ( i , j } E ((1, 2), (3, 4}, (5, 6}), $(x, 0) = (x, 1) = $(& 1) = $(x, 6 ) , $(x, 2) = (x, 3 ) = $(x, 3), $(x, 4) = (x, 5) = $(x, 5) for all x E X , cp(x, y ) = = ( Y , x), $(X,Y) = (x,5) for all (X,Y)ER .) 3. Construct a strong embedding of Alg (1, 1) into V(VZ = 1, $’ = $7 (cp$y = $).
(Hint: Consider @: Alg(1, 1) -+ V given by @ ( X ,a, b) = (I:cp, $) where Y = X x (1, ..., 9,1, ..., P,1,2,?) and cp(x, i ) = ( x , j ) if { i , j } E ( { l , ~ ) L { ~ , ~ } , (3,3}),(4,5>,(3,31, (6,7), (5,s>,(8,9>, (%g)>>,cp(x,i ) = (x, i) if i E (1,2,3); $(x,T) = (x, i ) = $(x, i) for i = 1,2,3, $(x, 4) = (x, I), $(x, 5) = (x,2), $(x, 6) = (x, 2), $(x, 7 ) = ( ~ ( x )3),, $(x, 8) = (x, 3), $(x, 9) = (B(x), 1) and $ ( x , q = ( x , n such that $(x, i ) = (z,j).
0 6.
Categories of small categories
6.1. Unlike in most parts of this chapter, where the morphisms are always common homomorphisms, in this section the main concern is the choice of morphisms. We investigate the categories of small categories with various types of functors (full, faithful, product preserving etc.). 6.2. We have already two results on categories of small categories. First, the basic category Cat
of small categories and all functors is not alg-universal (because of the constant functors), but it is almost alg-universal, since its full subcategory of monoids is (see 2.10). Second, the category Cat,
of small categories and all isomorphism-reflecting functors is alg-universal (see IV.5.7). It is worth noting that the negative result on finiteness from IV.5.8 can be extended from the thin categories to the general ones (see E 6.4).
186
Ch. V. ALGEBRA
6.3. Theorem. Let N designate the category of small categories and the fullfunctors, Catisothe category of all isofunctors onto. Then no category G such that
Catisoc 6 c N is alg-universal. Proof. Denote by M the monoid with three elements 1, p, v and with the composition defined by p p = p = pv, vv = v = vp. Suppose that is an object k of G such that its endomorphism monoid is isomorphic to M . Denote by rn, n the functors representing p, v respectively. Consider an arbitrary a E obj k. We have m(a) = m(rn(a))= rn(n(a)) and since rn is full, there are a : m(a) + .(a) and p: .(a) -+ m(a) such that m(a) = rn@) = lm(a). Then .(a) = n(rn(a))= = n(rn(B))= n(p). We have pa: m(a) -+ .(a) and hence pa = m(y) for a y : a + a. Thus,
Im(o)= rn(ga) = rnrn(y) = m(y) = pa. Analogously we prove that ap = so thzt a = p-' and rn(a) and .(a) are isomorphic. We will show now that m(a) .(a) for some aeobj k. Indeed, if rn(a)= .(a) for all a E obj k, we must have an a : c -+ d such that m(a) .(a). We have
+
+
rn(a): n(c) -+ n(d) so that, since n is full, there is a p: c + d such that m(a) = n@). Then, however, .(a) = nrn(a) = nn(/3) = n(p) = m(a) which is a contradiction. Due to the fact just proved, there are a, b, objects of k , which are distinct but isomorphic. Choose an isomorphism I , : a b, put Eb = zfl- ' and for x E obj k\(a, b ) put I , = 1,. Now, define a functor (a morphism in G) --f
f: k - + k putting f(a) = b, f ( b ) = a, f ( x ) = x otherwise, and f(9)= for any cp: x --+ y . We have f 1 and f = 1 so that f =i= rn, n. Thus, G(k,k ) is not isomorphic to M .
+
I
6.4. A system {O,,: x + y x, y E obj k } is said to be a system of zeromorphisms of k if, for every p: y + z and v: x + y in k,
p
0
o,,
=
o,,
=
o,,
0
v.
Obviously, if a system of zero-morphisms exists, it is determined uniquely (really, if also {O;,. is such, O,, = O,O:, = O;,). Let {O;,} be the system of zero-morphisms of ki,i = 1,2. A zero-morphismpreserving functor f : k, -+ k, is a functor such that f(Oi,) = O;(y)f(x) for any x, y E objk,.
187
56. Categories of small categories
6.5. Theorem. Let A (B, respectively) be the ca/rgory d u l l smoll Lore
yories (small categories admitting systems of zero-ii~oi.pliisnis.respectively) and all their faithful functors (faithjul zero-morphism preserving functors, respectively). Every category 6such that 5 c G c A is alg-universal. Proof. We will construct a functor @: Graph, -+G
(for Graph, see IV.l.ll) as follows: First, denote by M the monoid (1,0, a, a’} with a3 = 0, 0 . 5 for any 5. For an object ( X , R ) of Graphoput
=
5 .O
=0
@ ( X ,R ) = k
where objk = X and k(x,x ) = ( ( x ,1, x), ( x ,0, x),(x, a’, x ) } for x E X , k(x, Y ) = ((Y, 0, x), (Y, 4 X)’ (Y, a’.)} for ( x ,Y ) E R , k(x, Y ) = ((Y, 0, x), (Y, a2,x)} for x =k Y , (x, Y ) $fR, with the composition defined by (z,t i , j ) (y,5, x ) = (z,q5, x). Obviously, k has a system of zero-morphisms, namely ((y,0, x ) 1 x, y E obj k}.. For a morphism f : ( X , R ) -+ (X’, R’) define @(f): @ ( X ,R ) -+ @(X’,R’) putting
@(f) ( x ) = f(x)
@(f) (Y?5, x) = (f(Y), t , f ( x ) ) .
9
Obviously,every @(f)is faithfulzero-preservingfunctor, and @isone-to-one. Let g: @(X,R) -+ @(X’,R’) be a faithful functor. Since obj @(X,R ) = X, we can define f: X -+ X’ by f ( x ) = g(x). Since g is a functor, we have g(x, 1, x ) = ( f ( x ) ,l,f(x)). Since g is faithful, we have g(((x,1, x),(x, 074, (X? a’, x)}) = = ( ( f ( X ) , Lf(X))?
(f(x),O , f ( X ) ) , ( f ( x ) a2,f(x))) ,
Consequently, g(x, a’, x ) = ( f ( x ) ,a’,f(x)), since otherwise g(x, a’, x ) = = ( f ( x ) O,f(x)) , = ( f ( x ) ,O,f(X)) ( f ( x ) ,O , f ( X ) ) = g(x, a’ . a’, x) = g(x, o,.). Thus, g(x, 5, x ) = ( f ( x ) ,5,ffx)) * We have always
(*) g(y, 0, x ) = g(Y, 0, x ) . g(x, 0, x ) = (f(Y)? = M Y ) , O,f(x)).
t,f(X)).
(f(4O,f(x))=
188
Ch. V, ALGEBRA
If (x, y ) E R we have, since g is faithful, either (f(x),f(y)) E R‘ or f ( x ) = = f ( y ) = x’, in which case, by (*), {(y, a, x), (y, a’, x ) ) has to be mapped onto {(x‘, 1, x’), (x’, a’, x‘)}. Thus, we obtain the contradiction (i is such that g(y, mi,x ) = (x‘, 1, x ’ ) ) g(y, 0, x ) = g(y, ai,x ) g(x, 2,x ) = (x’, 1, x’) (x’, a2, x’) = (x’, a’, x’) .
Thus, f is RR’-compatible. If (x, y ) E R, by (*) g maps {(y, a, x), (y, a’, x ) ) onto {(y’, c1, x’), (y’, a’, x’)}, and if (x, y ) $ R, g sends (y, a’, x ) into {(y’, a, x’), (y’, a’, x’)) (here, (y‘, a, x’) does not have to exist). Thus, the proof will be concluded by showing that never g(y, a’, x ) = (y’, a, x’). Indeed, there is a z such that (y, z) E R and a 5 such that g(z, 5, y ) = (z’,a, y’). Thus, if g(y, a’, x ) = (y’, a, x’), we have (z’, a’, x’) = g(z, 5, y ) g(y, a’, x ) = = g(z,
6.7. Let us briefly recall the definitions concerning limits in categories, mainly in order to refresh the notation (cf. E 1.3). A diagram in a category k is a functor 9:D - k
-
where D, its scheme, is a small category. A (lower)bound of 9 is any system b 9(rn))mEobjDof morphisms of k such that g ( p ) 0 p,,, = for all p : rn -+ n in D . A limit of 9 consists of an object a (called the top of the limit) and a bound (A,,,:a 9(m)),,, such that for every bound (pm: b -+ D(m)), there is exactly one 8: b -+ a such that A,, = pm for every rn E obj D . If the scheme of the diagram 2 is discrete (see 1.3.5 I), we speak of the product of the collection (93(rn))mEobjD. We say that f : k -+ k preserves the limit of 9 if, for a limit (Am: a -+ 9(rn)), of 9,(f(A,): f(a) -+f9(rn)), is a limit of f 9. We say that f preserves limits (products, products of pairs,
(am:
-
-+
0
an
0
$6. Categories of small categories
189
resp.) if it preserves all limits of diagrams in k (all limits of diagrams with discrete schemes, all limits of diagrams with discrete schemes with two objects, resp.). 6.8. Now we will describe a category k which will play an important role in further constructions. The objects of k are the sets ui = { U J , aj2,ai3, Ui4} (i = 1,2), bi = {bil, bi2, b;3,- bi4, _ bi5, - bi6) (i = 1,2), c = {cl, c2, c3, cl, c2, c3) (if there is no danger of confusion, the symbols a i , b i , cin the indication of the points of aj, bj, c are omitted). The morphisms are all the mappings generated by c1,pl,02,p2, i l , el, i2, e2 situated as seen in Fig. 5.3. and
described by the diagram in Fig. 5.4 (where the arrows indicate where the points are sent to):
Fig. 5.4
We verify easily that
(E)
Cipi = lo,= ejzi
(i
=
1,2)
P101P202P101 = Pzo2Pl~lP2~2.
190
Ch. V, ALGEBRA
6.9. Lemma. (1) cardk(c, x) = 3 = cardk(x, c) for any x =/= c, (2) cXdk(c, C) = 6, (3) cardk(a,, ui) = 2. Proof. In this proof always {i, j > = { 1,2}. The lists of possible morphisms follow from (E). (1) k(c, ai) = {a:,oipjoj, oipjajpiai),> k(c, bi) = { l p : , 1ia,pjaj,liaipjojpiaif. In both cases, the morphisms indicated are distinct (look at the values in 1 and i). k(ai, C) = {pi,pjajpi, piaipjajpi}, k(bi7 C) = {piei, pjojpiei,pioipjajpiei}. In both cases the morphisms indicated differ in the value in 1). ( 2 ) k(c, c ) = {pl'17 pZa2>p2a2p1019 pl'1p2'2? p1a1p202p101? '}. Any two of the morphisms indicated differ in 1, 7, 2 or 2. (3) k(ai,ai) = {l, aipjojpi)and ~ , p ~ ~ ~ = p2. ~ ( l ) rn 6.10. Lemma. Put ti
=
riei,
v.i = 1.o.p.a. i j j M. i
i .
W e have cardk(b, bi) = 3, k(b, bi) = (1, ti,vi> and for every 1: bi x =/= bi7 ITi = A. Proof. We have q(1) = 2, vi(l) = 3. Since every A: bi x with x bi can be written as A'@:, we have A t i = A'eiziQi = l e i = A. rn --+
--+
+
6.11. Checking easily the remaining cases we obtain Lemma. For any x, y E obj k, card k(x, y ) 2 2. 6.12. Lemma. N o collection of objects with more than one member has a product in k. Proof. Suppose that y is product of (xi)i=l , , . , n , n 2 2. Since k(c, x i ) 2 3, there are at least 3"-morphisms c --+ y. This is a contradiction, since k(c, y ) I 6. rn 6.13. Lemma. Let for a functor f : k -+ k hold f(bi) = bi, i = 1,2. Then f is the identiy,functor lk. Proof. Since f(ei)f(ii) = 1, we have either f(ui) = ai, f(ei) = ei and f(zi) = t i or f(ui) = bi, f(ei) = f(ii) = 1. Let the latter hold with, say, i = 1. We have la, = a l p 1 ,hence l,, = f(l.,) = f(ol)f(pl). But then necessarily f ( o l )=f(,ul) = 1 so that f ( c ) = b , . Since a2p2= la,, we obtain a contradiction f ( a 2 )E {al, a 2 ) . Thus, f(ai)= ai, f ( g i ) = e: and f(li) = Zi.
191
$6. Categories of small categories
Further, we obtain f(vi) f ( p i ) = lai for both i f(c) = C, f(pi) == pi and ,f(ai) == oi.
=
1,2. Consequently,
rn
6.14. Construction. Here again (i,j} = (1,2}. Let k, k' be two distinct and disjoint copies of the category k. Let us denote by x', 5' the object resp. morphism corresponding to the x, 5 of k . The point of the subsequent construction of categories hl, h , is adding freely a product bi x bf to the sum of k and k'. Define hi as follows:
objhi = o b j k u o b j k ' u ( p } (p+objkuobjk'), k and k' are full subcategories of hi, hi(x, y ) = 8 for x E obj k and y E obj k' u (p), and for x E objk' and y E obj k u ( p ) , hi(p,bi) (hi(p, bf), resp.) consists of expressions 5 0 R (5 0 R', resp.) where R, R' are newly added symbols, 5 : bi bi in k ( 5 : bf + bf in k', resp.)of course we write R instead of 1 0 R, all the morphisms p -+ x are of the form 5 0 R with 5 : bi --, x, similarly the morphisms p -+ x' are 5' n' with 5': bf -+ x: The composition l1 o (t20 TC) is, of course, (tl 6;) 0 R. (Thus, we see that cardhi(p, x) = card ki(bi,x) and similarly for x'.) hi(p,p ) = ((a,8) a: bi -+ bi, p: b: bf), and we define the composition further by (a,p). (y,d) = (a y, fi 0 d), x 0 (a,p) = a 0 n, R' (a,p) = p 0 R'. -+
0
0
I
-+
0
0
The category hi is visualized on Fig. 5.5.
6.15. Observation. p with
( 7 1 , ~ ' ) is
a product of bi and bi in hi.
rn
6.16. Lemma. Let (q,: y x . ~=,....." )~ be a product in hi, let n 2 2. Then n = 2, y = p und either ~1~ = n, f1, = n' or q 1 = n', q2 = n. Proof. Let all the'x, be in one of k, k', say in k. Then again, for the same reason as in 6.12, cardhi(c, y ) 2 9 which is impossible. Further, none of the x, is p , since then we would have to have, for an analogous reason again, cardhi(p, y ) 2 2 .9. Thus, some of the x i are in k and some are in k'. -+
192
Ch. V, ALGEBRA
Then, necessarily, y
= p.
Since, by the definition of product,
cardhi(p, P) =
n cardhi(p,xs) m
>
s= 1
we see that n = 2 and cardhi(p,xs)= 3. This leaves us only with the possibilities of (xl, x,), in a suitable order, equal to (c, c’), (c, b;), ( b , c’) and (b,, b;). In the first two cases, q , = t n for a t: bi -+ c and we have, by 6.10, q l (z,,1) = 5 n(q, 1) = tzin = Sn = = q , (1,l) and q , (zi, 1) = t12n = 11, (1, 1) in contradiction with the unicity in the definition of limit. The third case is analogous to the second. Thus, (x,, x2) = ( b ,bj). Necessarily (see 6.15), (ql,q,) = (n,n’),since no non-identical morphism p -+ p is isomorphism. 0
0
3
0
0
m
6.17. Lemma. Let (Am: p + 9 ( n i ~ ) , , , ~ be ~ ~ a limit of u diagram 4 hi. Put A = {9(m)lmEobjD}. Zf p $ A then A n o b j k 8 =t= A n o b j k ’ .
9:D
+
Proof. If, say, A c objk, consider the tm:bi 9 ( m ) such that A, = = t , o n. Obviously, (t,: bi + 9(m)), is a bound of 9 and there is no --f
4: bi + p .
6.18. Recall the category 6 proved to be alg-universal in IV.9.2. Its objects were constituted by some of the triples (X, R,, R,) such that (i) X is a set, R,, R 2 c expX and R , n R 2 = 8, (ii) for every I E R1 v R,, cardr = 2, (iii) for every x E X there are ri E Ri such that {x} = r , n r2. The morphisms from (X, R,, R 2 ) into (X’, R;, R;) are the f : X + X’ such that for any i and any r E R,, (f x f )( r )E R:. In the following section we will describe a functor Qi: &-,Cat.
This is the idea of’its construction: To obtain @(X,R,, R2), replace every x E X by a copy k, of our category k ; now in the places where ( x , ~ )RE, add a product of b,, and b,, to k, u k, (so that the replacement of x and y as a whole looks like a copy of h l ) . Similarly, represent the pairs {x,y> from R 2 by extending the k, v k, to a copy of h,. 6.19. Now, we will describe the @(X,R,, R,) more formally. For every
x E X choose a copy k, of k (the objects and morphisms corresponding to u and t will be denoted by u, and t,), so that k, n k, = 8 for x y.
+
For every r E R , u R 2 choose an element pr, so that pr =Ipr. = for r =I= r’
193
a6. Categories of small categories
and pr $ U k , . Now, put obj @ ( X ,R,, R 2 ) =
u
X€X
I
obj k , u { p , r E R , u R 2 }.
The morphism structure of @ ( X ,R,, R,) is (up to isomorphism) uniquely determined by the following assumptions : (a)for every i and r E Ri there is a full embedding 40,: ht @(X> R1, R2) such that, if r = { x ,y } , then cp,(u) = u,, cpr(u‘)= uy for u E k and cpr(p) = p,. +
(a)
morph@(X,R,, R,)
=
u
cp,(morphhi).
reRl U R Z
For a morphism f : ( X ,R,, R2)+ (X’, R;, R;) the functor @(f):
is defined by @(f)(ux)
= #I(,)
@ ( f ) (pi)
= Pr,
@ ( f ) (qr(t)) =
=
@(XR,, R2) -.+ @(X’,R;, R;) for U E k for r = ( x , y } E R , u R , , 7
r‘ = { f ( x ) , f ( y ) ) for r = {x,Y} E R , u R2 r’ = {fix), f ( Y ) } .
c~r,(t)
Convention. We will write (, cp,(n’) for r = { x ,y } .
=
cp,(t), 5,
=
7
qr(t’),TC,,= cp,(.n),
nYr=
6.20. Lemma. Eet ( X ,R,, R 2 ) be an object of 6,put 1 = @ ( X ,R , , R,). Then (1) If 1(u, u) $; 8 =l= l(u, u ) then u, u E k, for exactly one x E X . (2) There is no object u E I such that I(u,ui)ji 8 for some ui E ks, with x,, x 2 ,x3 distinct. ( 3 ) I f l(u,p,) 8 then u = p,. (4) Let u be such that l(u, u) $; 0 $; 1(u, w) for some u E obj k,, w E obj k,, x j i y . Then r = { x , y } E R l u R 2 and u = p , . I f Z E X , z $ r and u E k , then l(pr,z) = 8. ( 5 ) If r = { x ,y } E Ri, then ( T C ~ , :p , + b,,, TC,,,: p , ---* by,) is a product in 1. (6) Let ( q s : u + us),, ,,,,,” be a product in 1, n 2 2. Then it is one ofthe products from (5). Proof. (1) - ( 5 ) follows immediately from the construction. Now, we see by (2), (3), (4) that there is an r E R , u R , such that u, usE cp,(hi). Then, (6) follows easily from 6.16 and from the condition (B) in 6.19.
+
,
194
Ch. V. ALGEBRA
6.21. Proposition. For every f: ( X , R,, R,) + (X’, R ; , R;), @( f ) is a limit-preserving functor. Proof. Put 1 = @(X, R,, R,), 1’ = @(X’, R;, R;), F = @(f). Let (A,: a + 9(m)), be a limit of a diagram 9: D -+ I. Put A = ( 9 ( m )I meobj D]. Since, obviously, 1 has no terminal object, A 8. First, suppose that a E obj k , for an x E X . Then, obviously, A c obj k,, F(A) c objk,,,, and (F(L,): F(a) -+ F9(m)), is a limit of F 0 9: Really, if (fl,: b + F g ( m ) ) is a bound of F 9, we have either b E kfc,, and we can use the obvious fact that (F(A,)), is a limit of the domain-restriction of F 0 9 to k,,,), or b = p r for an r E R : such that f ( x ) ~ r In . this last case, fl, = t,o n,,x,r for some 5,: bi,(,, F9(m). Then, (t,), is a bound of F 0 9 in k,,,,, we have a unique 5 such that F(il,) 0 5 = 5 , and we obtain fl, = F(1,) 0 (t 0 nf(x)r). If also fl, 5 F(A,) 0 q, necessarily q = q’ 0 z,)~ and we see immediately that q’ = 5. Now, let a = pr for some r E Ri. Put r = (x, y } , F = (f(x), f(y)}. We have A c objcpr(hi) so that F(A) c objcpi(hi). Let (fl,: b -+ F9(m)), be a bound of F 0 9. If pr E A then F(p,) = pi E F(A) so that, by 6.20.(3), b = pi. If pr 4 A , then by 6.17, F(A) n obikfc,, =+ 8 =+ F(A) n objk,,,,. Then again b = p F since only for b = pi we have l’(b, u) =+ 8 fm all u E F(A). Thus, we have no other bounds of F 0 9 than those in cpi(hi) and we see that (F(L,&, is a limit of F 0 9. rn
+
0
-
6.22. Theorem. Let A be the category of all small categories and the limit preserving functors, B the category of all small categories and the functors which preserve the limits of pairs ofobjects. Every category 53 such that
ACRCB is alg-universal. Proof. By 6.21, @ maps Q into A. Denote by Y:Q-R
the range restriction of @. It is obviously a one-to-one functor. Let F : 1 = @ ( X ,R,, R 2 ) + I‘ = @ ( X I ,R;, R;) be a morphism in R. Thus, it preserves products of pairs of objects. Take an x E X . For any u, v E k,, l(u, v) =+ 8 =+ 1(u, u). Thus, by 6.20.(1),f ( k , ) c k,, for a uniquely determined x’E X’. Put x’ = f(x). Thus, we have a mapping
f:x
-
X’
Let F : 1 = @ ( X , R , , R 2 ) + I‘ = @(X’,R ; , R;) be a morphism in R. Thus; such that f ( k , ) c kfc,, for every x.
96. Categories of small categories
I
195
1
Now, put Bi = {biz x E X},B: = (bi, u E X ' > (i = 1,2). Since the only pairs with products in t' are {biu,b,) with {u, u } E R: (see 6.20.6)) and since {bix,biy} has a product in 1 whenever {x, y] E Ri (see 6.20.(5)), F maps B , v B2 into B; u B2 and f is R, u R,, R; v R;-compatible. By IV.9.4, f is then R,R;-compatible for i = 1,2 so that by 6.18(iii), F(bi,) = birc,,. By 6.13 F maps k, onto k,,,, sending cp,(<) to ( ~ ~ ( 5Finally, ). since F preserves products, p , is sent to pi and we see easily that F = 'u(f).
6.23. Remark. The @(f) in our construction do not preserve colimits. One can prove, using a considerably more complicated construction, that even every R between the category of all limit and colimit preserving functors, and the category B from 6.22 is alg-universal. EXERCISES
The category (A)
of partial algebras of the type A = (&)beB is defined as follows: The objects are (X, (q,)heR) where c o b : D(co,,) -+ X are mappings such that D ( 0 b ) c X A b . The morphisms from d = ( X , ( q , ) )to d' = @',(mi)) are triples (A'. 1.2) where f : X X ' is a mapping such that (1) if wb(a)is defined, co;l(fc a) is defined, (2) wb(f0 u) = f(q,(a)) for all b and all c1 E D ( u b ) . 1. Define U : Cat + Set putting U ( k ) = morphk. Prove that (Cat, U ) can be realized in Palg (2, 1,l). 2. Prove that Palg ( A ) is alg-universal iff Z A 2 2. 3. (a) Find a strong embedding of Graph into Palg (2). (b) Define a functor G: Set -+ Set putting
G ( X ) = (X x X x (0,l)) u X (the union is assumed disjoint), G ( f ) (x, y, i) = (f(x), f(y), i), G(f) (x) = f ( x ) . For a graph (X, R) put @(X,R) = (G(X),a, fl) where u(x, y, 1) = (x, Y, 0) for all x, Y , a(x, y, 0) = (x, y, 0) for (x,y) E R, otherwise undefined, D(x, y , 1) = (y, x, 0), fl(x, y, 0) = x, otherwise undefined. @(f)= G ( f ) . Prove that @ is a strong embedding of Graph into Palg(1, 1). 2 2, then there is a strong embedding of Graph (c) Prove that if into Palg ( A ) . 4. A non-trivial group cannot be represented as the monoid of all isomorphism reflecting functors of a finite category into itself (cf. IV.5.8).
196
6 7.
Ch. V. ALGEBRA
Remarks on categories of functors
7.1. The aim of this section is to inform about a few results concerning the categories SetA
of functors f : A -,Set and their transformations ( A is a given small category). Not much systematic study has been done in this area (with the exception of the case of thin A). All we can do is to present a few illustrating examples, scattered results and open problems. It is perhaps not necessary to recall that SetAis always an algebraic category (see E 11.5.1). Thus, the question to be asked is merely the alguniversality.
7.2. A small category A is said to be rich (poor, resp.) if SetAis (is not, resp.) alg-universal. 7.3. If g: A
-+
B is a functor, we have a functor
5: SetE+ SetA G ( T ) ~= T ~ ( ~ If) . g is a
defined by i j ( f ) = f o g , factorization, i.e. if it is one-to-one on the objects and onto on the morphisms, one checks easily that i j is a full embedding. Less obvious is the fact (observed first by Freyd) that if A is a full subcategory of B then SetAis fully embeddable into SetB (the full embedding of SetAinto SetBis not, of course, induced by the embedding of A into B in the way mentioned; actually, it is its adjoint). This leads to introducing the following definition: We say that A precedes B,and write A S B,
if A is a full subcategory of a factorcategory of B (i.e. of a C such that there is a factorization B --* C). One can prove easily that is transitive. Due to the above fact if A I B and A is rich, also B is rich. We say that A is a minimal rich category if it is rich and every rich B IA is isomorphic to A . 7.4. There are two cases (opposite in a way) of small categories of ininterest: On the one hand, the monoids (= categories with one object), on the other hand the thin categories. We see that for monoids M , N we have M I N iff M is a factormonoid of N , if M is a monoid and A I M then A is a monoid.
97. Categories of functors
197
Similarly, for thin categories S , T we have S 5 T iff S is fully embeddable into 7; if T is thin and A 5 7; then A is thin. Consequently, a monoid is minimal rich iff it is rich and none of its non-isomorphic factormonoid is rich, a thin category is minimal rich iff it is rich and none of its non-isomorphic full subcategory is rich. 7.5. In the coming sections 7.5-7.12 we are going to discuss the case of monoids. Obviously, one has to do here with varieties of unary algebras. More exactly, SetMis isomorphic to any
V(6;J ) (see 5.2) where J is a set of generators of M and 6 a system of equations equivalent to { U O B= y ICI,BEM ) . (On the other hand, every V ( € ; J ) such that all the equations in 6 are of the first type mentioned in 5.2 is isomorphic to a SetM.Really, take the free monoid generated by J and factorize it by the congruence generated by €.) Since, if the systems of equations 6 and 6' are equivalent, the varieties V ( € ; J) and V(€'; J ) coincide, the representation of SetMas a variety of unary algebras just described depends only on the set of generators J of M . We will denote it by V ( M ;J ) .
7.6. By 5.8 and 5.6 we see immediately that every group and every commutative monoid is poor. 7.7. A monoid is said to be strongly minimal rich, if V(M; M ) is a minimal alg-universal variety in Alg,(M). We will see soon that a property SO defined is really stronger than being minimal rich. We have Proposition. M is strongly minimal rich iff; for every its set of generators J , V ( M ; J) is a minimal alg-uniuersal variety in Alg, (J). Proof. Suppose M is strongly minimal while there is a proper subvariety W of V ( M ; J ) , which is still alg-universal. But if we take the additional equations defining W in V ( M ; J) and add them to the original equations of V(M; M), we obtain its proper subvariety isomorphic to .W and hence alg-universal. rn
198
Ch. V, ALGEBRA
7.8. Let M‘ be a (proper) factormonoid of M . We see immediately that
V(M’; M ’ ) is isomorphic to a (proper) subvariety of V ( M ; M ) , namely that obtained by adding the equations representing the congruence by which M was factorized. Consequently, we obtain Proposition. A strongly minimal rich monoid is a minimal rich monoid. 7.9. The converse of proposition 7.8 does not hold. This can be demonstrated by the following example: Consider the monoid M with two generators a, b and defining equations a2 = a3 = a b ,
b2 = b3 = b a .
By 5.17 (and 7.5), M is rich. Now, let h: M + N be a proper factorization. If h(a) = h(b); V ( N ; N ) is a subvariety of Alg, (1) and hence it is not alg-universal. If h(a) =k h(b) we can represeiit V ( N ; N ) as a subvariety of V ( M ; {a,b } ) obtained by adding equations of the first type from 5.2. Thus, by 5.20, V ( N ; N ) is again not alg-universal so that N is not rich. Hence, M is a minimal rich monoid. Since, however, there is a proper alg-universal subvariety of V ( M ; {a,b } ) (see 5.20), M is not strongly minimal. An example of a strongly minimal rich monoid can be obtained reinterpreting 5.22: Namely, consider the monoid with two generators a, b and defining equations
u2 = 1,
b2 = b ,
abab = b .
7.10. At the first sight, one can be a little surprised at the fact that the property “to be rich” is not preserved by antiisomorphism. The monoid M o p dual to the M from 7.9, hence, with the defining equations a2 =
a3 = b a ,
b2 = b3 = a b ,
is not rich. Indeed, consider an algebra (X,a,/?) satisfying a’ #I2 = p3 = a/?.Put z = a/?(x).We have
=
a3 = /?a,
/?(z)= /?a/?(x)= p3(4= a/?(x)= z , a(.) = a2B(x)= a/?’(x) = a/?/?(x)= p3(x)= a/?(x)= z . Thus, V(MoP;{a,b } ) is not alg-universal by 5.5. 7.11. While the property “to be poor” is preserved by factorization, a submonoid of a poor monoid can be rich. Really, consider the monoid M with two generators a, b and the defining equation a2ba = baba .
199
$7. Categories of functors
If we take a non-void algebra ( X , a, p) from V ( M ; (a, b ) ) and put z = aba(+x), we obtain a(.) = p(z) = z. Thus, M is poor by 5.5. Take, however, the submonoid N of M generated by a and b2. Since our defining equation can be applied to no part of a word in a and b2, we see that N is a free monoid with two generators, and hence rich. Remark. There are, however, monoids which are hereditarily poor in the sense that every submonoid is poor. E.g., by 5.6, every commutative monoid is such. 7.12. We remarked in Q 5 that the question of alg-universality of varieties of unary algebras cannot be reduced to the study of subvarieties of Alg (1,l). Using the terminology of the present paragraph we will support this statement by an example. Namely, we will present a rich monoid with three generators such that each of its submonoids with two generators and each of its factormonoids with two generators is poor. Consider the monoid M with generators a, b, c and defining equations
a = a2 = aba = aca = abca = acba, b = b2 = bab = bcb = bacb = bcab, c = c2 = cac = cbc = cabc = cbac . Let m, n be any elements of M = {l,a, b, c, ab, ac, ba, be, ca, cb, abc, acb, bac, bca, cab, cba),. One verifies easily that always
m2 = m and mnm= m .
~ADCLWB
Consequently, whenever one takes a submonoid or a factormonoid N of M with two generators m, n, the equations
m = m2 = mnm, n = n2 = nmn are satisfied. Consider an algebra (X, p, v) from V ( N ; (m, n}). Put P = = ( x p(x) = x } , Q = { x V ( X ) = x } . If P n Q 8, take a y E P n Q and define cp(x) = y for all x.Then cp is a homomorphism. If P n Q = 8, choose z E P and put y = v(z).
I
+
1
Define f:X
-P
X putting
f(x)= z
for
X E
P,
f(x)
=
y
otherwise.
Again, f is a homomorphism. Thus, V ( N ; {m, n}) contains no non-trivial rigid algebra, hence it is not alg-universal and hence, finally, N is poor.
200
Ch. V, ALGEBRA
Thus, every submonoid and every factormonoid of M generated by two elements is poor. On the other hand, M itself is rich. Really, define @: Graph -+ V(M;{a, b, c > ) as follows: Put @ ( X ,R ) = ( ( R x { 1,2,3)) u ( X x {4,5)),4 P, Y) where
a(x,Y , 1) = (x, 4), P(x, Y , 1) = (x,Y , 2) ~ ( xY,, 1) = (Y, 4) a(x,y, 2) = B(x, Y , 2) = (X? Y , 2) Y ( X , Y , 2) = (X’ Y , 3) a(x, Y , 3) = P(x, Y , 3) = (x,Y , 2) Y(X, Y , 3) = (x,Y , 3) a(x,4) = y(x,4) = (x,4), P(x,4) = (x,5 ) , E(X, 5 ) = y(x, 5) = (x,4) P(x, 5) = (x,5 ) . 9
9
2
The situation is visualized in Fig. 5.6. One verifies easily that @ ( X , R )is in V(M;{a, b, c ) ) and that @ is a one-to-one functor.
Now, let g: @(X,R ) -+ @(XI,R’) be a homomorphism. Since the elements of R x {2) are exactly those fixed under a and /Iwe , obtain g(R x (2)) c R‘ x ( 2 ) .
201
$7. Categories of functors
Similarly, using a and y we see that g(X x (4))
Define a mapping f : X
+ X'
= X'
x (41
by (f(x),4) = g(x, 4). We have g(x, 5) =
= S B ( X ? 4) = B(f(x),4) = (f(X)?5). R x (1) consists of exactly those elements
a ( 5 )X~ x (4)
and
5 for which
B ( ~ ) ER
x (2).
Consequently, f ( R x (1)) c R' x { 1). Using y, one sees that if f(x, y, 2) = = (x',y', 2), f ( x , y, 3) = (x',y', 3). Proving that g = @(f) is now straightforward. 7.13. Apart from a few concrete examples, not much is known about the general question of minimal rich categories. However, it was found that there are precisely 35 minimal rich thin categories. They are shown in Fig. 5.7 (where only the morphisms generating the categories are indicated by arrows). Moreover, it is known that every rich thin category contains a minimal one. Thus, a thin category is rich iff it contains some of the 35 categories above as a full subcategory [TR,].
7.14. Checking easily the few categories with less than four morphisms, and taking in account that if A has two objects, a, b and two non-trivial morphisms a, B: a -+ b then SetAis the alg-universal category of multigraphs, one sees that the minimal number of morphisms of a rich category is four. By the result from 7.13 we see that the minimal number of morphisms of a rich thin category is eight. Rich monoids with few elements were also investigated. It was proved that the minimal number of elements of a rich monoid is five. (The proof of this statement is not difficult but very long due to the large number of cases to be considered. It is given in [S,].) 7.15. Let us note that, similarly as there are infinite minimal rich thin categories, there exist infinite minimal rich monoids. An example is given in [Gg,]. It is the monoid M with two generators a, b and the defining equations aZb2= bZ = b 3 , a2b = b a .
202 kl
Ch. V, ALGEBRA
I
k2
kx 0/
kl6
I
kl7
I
7 A 00
A
k9
/
O
O
o/o\/o
I k20
k25
O
a
Fig. 5.1
0
0
0
P
203
$7. Categories of functors
EXERCISES
1. Prove explicitly that every category with less than four morphisms is poor. 2. The monoid M with two generators a, b and the defining equations a = a3 = b a ,
b
=
b2 = a2b
has five elements and is rich. (Hint: Consider the @: Graph,, -+ -+ V ( M ; {a, b}) defined by @ ( X ,R) = ( X a, p), where Y = X x {0,1,2} u u R x {3}, a(x,0) = p(x, 0) = p(x, 1) = (x,1) = a(x, 2), a(x, 1) = P(x, 2) = (X’ 2), a(x, Y , 3) = (x, 2), P(x, Y , 3) = (Y, 2) .) 3. Prove that M from E 2 is a strongly minimal rich monoid (do not use the unproved fact from 7.14). 4. The monoid M with two generators a, b and the defining equation a2b = ba
is an infinite rich monoid. (Hint: Consider the @: Graph, -+ V ( M ; {a, b } ) defined by @ ( X ,R) = (Y, a, p), where Y = X u R u (0,1,2,3, ...} a(x, Y ) = x, D(x, Y ) = y for (x, Y ) E R, a(.) = 0, = 1 for X E X , a(i)= i + 1, p(i) = 3 + 2i for i~ (0,1,2,...}. To prove that @ is full, observe that X = { Z E Y = az(z)}. Note: Every finite factormonoid of M from E 4 is poor. 5. Let k, be a small thin category such that obj k, = (0,1, ..., n> and k,(i,j) =l= 8 iff either i = j or i is even and j E { i - 1, i + 11. Prove that k, is rich iff n 2 6. Prove that k z P is rich iff n 2 7. (Do not use the unproved fact from 7.13).
a(.)
1 a(.)
(Hint: Consider @: Graph -+ Setk6defined by @(X,R) = f where f ( 0 ) = f(6) = X f(1) = f(5) = x x yo, I} f(2) = x {2,3) u R x {4}, 2
x f(3) = x x f(4) = x x (f(0 (f(2 (f(2
+
+
+
7
(5,6} 3
{2,3,4>> 1))(x) = (x,0) = (f(6 5)) l))(X, 3) = (X’1) = (f(4 -+ 5))(x, 3), 1))(X’2) = (x,0) = (f(4 -+ 5)) (x,2) +
(4
3
7
204
Ch. V, ALGEBRA
( f ( 2 1))(X’Y , 4) = (x,1) (f(4 5)) (x,4) = (X’ 1) > (f(2 3)) (X’2) = (x, 5) > (f(2 3)) (x,3) = (x, 6 ) , ( f ( 2 -+ 3)) (x,Y?4) = (Y, 5) (f(4 --* 3)) (x, 3) = (x, 5) (f(4 3)) (x, 2) = (x, 6) (f(4 3)) (x,4) = (x,6 ) ; 9
+
+
-+
+
2
2
I
+
+
i -+ j indicates the morphism from i into j.)
6. The monoids A , B, 7;are defined as follows: They have two generators, say a, b and they are satisfying the following equations:
A : a ’ = 1 , b2 = b , abab = b , R: = 1 , b2 = b , (ba)’b = ( b ~ ) ~ , T : (I’ = 1 , b2 = b , (ba)’b = b .
(I’
Prove that A, B and 7;with 1 odd prime are exactly the minimal rich monoids which precede the monoid M with two generators, a, b and the defining equations a’ = 1, b2 = b. (Hint: For the proof of their richness use 5.22, E 5.1 and E 5.2. To prove that there is no other minimal rich monoid preceding M , consider a rigid algebra from V(M; {a, b } ) with at least six elements.)
9 8.
Bibliographical remarks
Of the results presented in this chapter, Birkhoff‘s representation of groups as automorphism groups of complete distributive lattices ([Bff]) is the oldest. All the others are much more recent. They appeared after the proof of the alg-universality of Graph, Alg (1,l) and Alg (2), and are based on representing some of these categories. The alg-universality of the category of semigroups was first proved in [HL]. Almost simultaneously, and independently, the construction of the rigid semigroup D of 2.11 appeared ([DN]). The strong embedding of Alg (2) into the category of semigroups (the one we presented in the second part of $2) based on the properties of D appeared in [T3]. Another strong embedding into the category of semigroups appeared simultaneously in [S,]. The alg-universality of the category of commutative rings with unit (see 3.10) was proved in [FS,], and the alg-universality of the category of integral domains (outlined
@. Bibliographical remarks
205
roughl) i n 5 3. too) in [FS,] (recently, J. Kollar found a new elegant proof of t l i i c fact [ KI]). The results concerning the category of lattices summarized in 4.16 appeared in [GS], [S,] (one uses there the results from [ChG], [J], [W], here 4.5-4.15). Theorem 4.23 is from [Ko4]. Of the papers concerning varieties of unary algebras let us mention [PSI containing a complete discussion of the varieties of algebras with two idempotent operations. Theorems 5.15 and 5.18 have not been published before and similarly the results on categories of small categories appear here first. The general idea of investigating varieties of algebras as functor categories is due to Lawvere. The notion of rich small category was defined first in [HL]. The results concerning rich thin categories mentioned in 7.13 (and some others) appeared in [TR,], [TR,], [TR,]. The examples of monoids of 7.11, 7.12 and 7.15 appeared in [Gg,] and [Gg,]. Finally, we would like to mention a very interesting area concerning the so-called testing categories. The basic problem is to find a small category K (a finite one, if possible) such that the alg-universality of a category satisfying some properties (e.g. such as completeness). or belonging to a class of similarly constructed ones. is implied by the embeddability of K into it. Hither belong e.g. the papers [S,] (mentioned in § 5 ) , [R4] and [R,]. Also [TR,] contains a result of this kind, namely that Z, is a testing category for Set' with k thin.
Chapter VI TOPOLOGY Unlike in algebra, where one seldom meets other choices of morphisms than the usual homomorphisms, in general topology we are encountered with a variety of special kinds of mappings. Thus, already in the basic parts of the theory of metric spaces one works, besides continuous mappings, with Lipschitz mappings, contractions and uniformly continuous mappings. Working with topological spaces, one currently uses continuous, open and closed mappings, local homeomorphisms etc. Considering the representation problems in the categories thus obtained we see two obvious main lines: On the one hand, there are choices of morphisms which admit the constants : continuous mappings, uniformly continuous mappings, contractions, etc. In the categories obtained this way, the question to ask is whether they are almost (a1g)-universal. On the other hand, the choices of local homeomorphisms, open and quasiopen mappings etc. are able to exclude the constants. Here, we may ask about the (a1g)-universality. The chapter is divided into seventeen sections. Of those, some are closely interconnected. According to the topics, the sections group as follows: I. Section 1, concerning the alg-universality of To-spaces with local homeomorphisms (this is just a reinterpretation of a previous combinatorial result). 11. Sections 2,3,5 and 6: This is the technical basis of the whole chapter. 111. Sections 4,7' and 8: Here, results on categories of metric spaces are presented (e.g., the almost-alg-universality of the category of metric spaces with all continuous mappings, the universality of the category of metric spaces with open uniformly continuous mappings, etc.).
207
91. An elementary result
IV. Sections 9 - 13: After a negative result of T,-spaces (Q9), the categories of TI-spaces are studied. It is shown, e.g., that the category of T,-spaces and local homeomorphisms is alg-universal, and that the category of TI-spaces and open continuous mappings is universal. V. Sections 14 and 15: Without a set-theoretical assumption, the category of paracompact spaces and all their continuous mappings is shown to be almost universal. VI. Section 16: The dual of the category of compact spaces and all their continuous mappings is shown to be almost alg-universal. VII. Section 17 contains some negative results concerning categories related to those of m-compact spaces with bounded character.
Q 1. An elementary result about To-spaces 1.1. Let ( X , R) be a poset (see IVS.l), i.e. let R be a transitive antisymmetric relation on X . Define a topology T ( R )on X by
iff ( X E U & y R x
U c X isopen
*
~ E U ) .
Obviously, z(R) is a To-topology. For x E X put O#(x) = u R x (recall that R x = { y y R x ) ) . Evidently, Oh(.) is open and it is the smallest open set c0ntaining.x. We see immediately that a mapping f is continuous (open, resp.) iff f(O#z(x))c = O#;(f(x)) (f(O#(x)) = O#(f(X))? resp.1 for all x.
{XI
1
1.2. Lemma. Let ( X ,R), (XI, R') be posets, f: X -+ X ' a mapping. If f o R = R' f then f is open continuous with respect to T(R),T(R'). Proof. We have evidently (f o R) x = f(Rx) and (R' o f ) x = R'f(x). Thus, if f R = R' o f , we have 0
0
f(O#(x)) = (f(x)) u f ( R x ) = {f(x)} u Rlf(x) = O#f(x)
'
rn
1.3. In the remaining part of this paragraph we will adopt the notation from IV.6. Lemma. Let y: X u ( R x 2) -+ X' u (R' x 2) be continuous and locally Then g = f * d for an RR'-compatible one-to-one with respect to f : X + X'. Consequently, f o R = Rr o f . Proof. Since g(O#z(~))c O#g(t) for every 41, and since g is one-to-one on every O#(t), we have cardO#(g(t)) 2 cardOji(i;). Thus, necessarily
~(w),
@I).
g(x, Y , 0)
=
(Xl, Y , , 0)
208
and
Ch. VI. TOPOLOGY
g(x, Y , 1) = (XZ? Y z , 1) or g(x, Y , 1) = ( x 2 , Y,, 0).
Since (x,Y , 1) E Lo&, (XI,Y,, 0). Thus,
*
Y , 0) we have g(x, Y , 1) E O#(xl, Y,, 0) and g(x, Y , 1)
g(x, Y , 1) = ( X l ? Y l , 1).
+
*
Further, Y E oh(.> Y , 1) so that g(y)E o#(x,, y 1 , q and g(y) ( X I , Y l , 0). Thus, 9(Y) = Y,. Finally, x E o#(x, Y , 0) so that g(x)E o#(xl, Y,, 0) and g(x) g(y),g(x, Y , i). Hence, g(x) = x1 so that g = f * d.
+
1.4. Theorem. Denote by Top&,, (Top&-l), resp.) the category of T,-spaces with open local homeomorphisms (continuous locally one-to-one mappings). Let A be a category such that TOP:,, = 3 = TOP&, - 1 ) . Then A is alg-universal. Proof. Consider the embedding F of Graph, into Poset from IV.6.6. We will prove our statement showing that the correspondence sending ( X , R ) to (X. T ( R ) )generates a realization of F(Graph,) in A. Let q carry ~ ( k ’in) )3:. It is continuous locally one-to-one a morphism (X,z(Ri))-+ (A’, and hence, by 1.3, g carries a morphism (rf, R ) + (rf’, R’) in F(Graph,). On the other hand, if g carries a morphism in F(Graph,), it is equal to f * d for an f : ( X , R )+ (X’,R’) and hence, by the formula for f * d , it is locally one-to-one. By 1.2 it is, moreover, open continuous so that it is an open local homeomorphism and hence a morphism in R. rn
5 2.
Some special mappings. Quotients and sums of metric spaces
2.1. Let us summarize a few definitions of special mappings between topological spaces and those between pseudometric ones. If ( X , z), ( Y 9) are topological spaces, a mapping f : X -+ Y is said to be open if, for every open U in z, f(u)is open in 9; quasi-open if, for every non-void open U , there is a non-vgid open V c U such that f ( V ) is open; a local homeomorphism if, for every x EX,’there is a neighborhood U such that f’ maps U homeomorphically onto f (U ) ; a quasi-local homeomorphism if it is continuous and for every non-void open U there is a non-void open V c U such that f maps I/ homeomorphically onto f (V ) .
209
$2. Some special mappings
Let ( X , e), (I:o) be pseudometric spaces. A mapping f : X -+ Y is said to be a contraction if a ( f ( x ) j'(y)) , I Q(x,y ) for all x , y E X ; a local isometry if for every X E X there is an E > 0 such that ( o ( f ( x )f,( y ) )= = e ( x , y ) whenever e ( x , y ) < E ; a quasi-local isometry if it is continuous and for every non-void open U there is a non-void open I/ c U such that o ( f ( x ) , f ( y ) = ) @(x,y ) whenever x,yE v
2.2. Proposition. Let X , Y be topological or pseudometric spaces, + Y a continuous mapping. Let A c X be nowhere dense. Zf f X\A i s open (a local homeomorphism, a local isometry, resp.) then f i s quasi-open (a quasi-local homeomorphism, a quasi-local isometry, resp.). Proof is trivial.
I
f: X
2.3. Let P = ( X , Q ) be a pseudometric space, q mapping of X onto a set I: A finite sequence uo, vo, u1, 01, ...,u,,
0,
of elements of X is said to connect x and y if x
y = q(un) and
= q(uo),
q(vi) = q(ui+
for i < n .
For x , y Y~ put i= 1
e(ui, vi) I uo, uo, ..., u,, v, connects x and y
It is easy to see that a is a pseudometric on Y. It is said to be induced by q, the space ( X a) is called the quotient of P by q, a.nd q is called the quotient mapping. If R is a relation on X denote by R the equivalence generated by R and take a mapping q : X + Y such that
(1)
xRy
iff q(x) = q(y).
The quotient of P by q is then also said to be induced by R,and denoted by
PIR . (Obviously, P/R is determined up to isometry; moreover, if qi: X + Xi are two mappings satisfying (1) there is an isometry z between the induced spaces such that I q1 = q2. 0
2.4. Observation. The quotient mapping is a contraction.
210
Ch. V1, TOPOLOGY
2.5. Let R be a relation on X , q the mapping satisfying (1) from 2.3.
The set
I
C(R)= { x E x 3Y
* x , 4(Y)
=
(denoted also by C(q))is called critical set of R (or of q).
(x,
2.6. Proposition. Lct q i : ( x , . ~4 , ) ai) (i = 1,2) be quotient mappings. Let f:( X l , e l )-,(/Y2.ez) be n contraction such that if q l ( x ) = ql(y) then q 2 f ( x )= q2f(y). Then there is exactly one contraction 9 : (Yl, al) .+ (Y2,a2) such that g 41 = q 2 o f . Proof. By the condition on f and qi there is obviously exactly one mapping g such that g q1 = q 2 o f : To show that it is a contraction it suftices to take an arbitrary ((ui,u ~ ) ) ~ = ~ , connecting .,..~ x and y in Y, and observe that ((f(ui), j ( u i ) ) ) i =1, . . . , n connects
0
2.7. Let ( X , Q )be a pseudometric space. A relation R on X (or the associated mapping q : X .+ Y ) is said to be ldiscrete if x , Y E C(R) and
x
+y
imply that
e(x,y ) 2 1 .
2.8. Lemma. Let ( X , e) be a pseudometric space, q : ( X ,e) .+ (Y, a 1-discrete quotient mapping. Then if a(q(u),q(0)) < min (e(4u),
CJ)
i)>
there are uo, u1 E C(q) such that uo =k u l , q(uo) = q(ul) and o(q(u),4(u)) = e(u,U O )
+ e(u1,u).
Proof. For every (u;. u $ = ~ , , , , , connecting ~ q(u) and q(u) there is a sequence (ui, uJi= 1. ,,,,malso connecting q(u) and q(u) such that (a) {ui,ui i = o,..., m } c {ui, ui i = 0, ..., n } , (b) ui, u j E C(q) for i > 0, j < m, and u j =k uj+l,
1
I
(Indeed, consider a sequence satisfying (a) and (c) such that m is the least possible. If, say, u j = u j + 1, we could have taken the sequence uO,uo, ..., u j - 1, u j - uj, u j + 1, u j + 2 ,..., which would have contradicted the choice of m.) Thus, if a(q(u), q(u)) < e(u, u), we have n
a(q(u),q(0)) = inf(
C e(ui, I (ui, ui) connects
i=O
ui)
q(u),q(u) and satisfies (b), n 2 I}
21 1
42. Some special mappings
If moreover, o(q(u),q(u)) < i,we have, hence, a(q(u),&)) =
= inf (e(u, 0 0 )
+ e(u1, u) Iuo, u1 E C(Y),
00
*
u1
and q(v0) = +I)}.
Since obviously uo and u1 from C(q) such that
e(u7u0) + e(u1, ). < 4 are uniquely determined, the statement follows.
H
2.9. Proposition. quotient mapping. Then (i) o is a metric, (ii) q : ( X , Q )+ (Y, o) is a local isometry (in tact, e(u, u ) < 3 implies o(q(u),4(u)) = e(4 u)). (iii) for x , y ~ q ( C ( q )distinct, ) o(x,y) 2 1, (iv) q X\C(q) is open. (Thus, ifmoreouer C(q) contains no isolated point o f ( X ,e), q(C(q))is nowhere dense in o)and by 2.2 q is quasi-open.) Proof. (i) Let o(q(u),q(u)) = 0. If e(u, u ) = 0, we have u = u and hence q(u) = q(u). If e(u,u) > 0 there are, by 2.8, vo,ul such that q(u,) = q(ul) and e(u, uo) = e(ul, u) = 0. Thus, u = uo and u1 = u so that q(u) = q(u). (ii) Let a = a(q(u),q(u)) < e(u, u) < 3.By 2.8 a = e(u, uo) e(ul, u) with uo, u1 E C(q), uo =+ ul. We obtain e(uo,ul) i e(uo, u) e(u, u) e(u, ~ 1 <) < f + f = 1, in contradiction with the assumption on C(q). (iii) Let o(q(u), q(u)) < 1, u, u E C(q), u v. Then, o(q(u),4(0)) I I e(u, uo) e(ul, u) < 1 for some uo, u1 E C(q). But then e(u, uo) = = e(u,, u) = 0, and hence q(u) = q(u) by (i). (iv) Take an x $ C(q) and a y E Y such that
I
(x
+
+
+
+
+
a(q(x),Y ) < min (i,e(x, c(q))). We have y = q(xl). Suppose that e(x, x l ) > o(q(x),q(xl)). By 2.8, o(q(x),& I ) ) = e(x, uO) g(ul, x , ) with uo E C(q). This is a contradiction, H since e(x,0 0 ) 2 e(x, C(q)).
+
(x,
2.10. Proposition. Let q l : ( X t ,el)-, a,) ( i = 1,2) be l-discrete quotient mappings. Let f : (Xl, el) + ( X 2 ,ez) be a contraction and a local isometrv (open local isometry, resp.) such that if q l ( x ) = ql(y) then q 2 f ( ~=) = q 2 f ( y ) ,and that f(Xl\C(ql)) c Xz\C(q2). Let g be as in 2.6. Then g is a quasi-local isometry and local isometry on Yl\qlC(ql) (or an open local isometry on Yl\qlC(ql), resp.).
212
Ch. VI, TOPOLOGY
Proof. Take a y~ Yl\qlC(ql), y = ql(x). By 2.9 (ii) and (iv) and by the assumption on f there is a neighborhood U of Y such that 4 , and f are isometries on U , q2 is an isometry on f ( U ) , and that q l ( U ) is open. For y, = ql(xl) with x1 E U we have I oz(g(Y),g(y1)) = az(gq,(x), g q h 1 ) ) = az(qzf(x), qzf(x1)) = = e2(f(x),f(x1)) = el(%x1) = Ol(Y, Yl).
If, moreover, f i s open then g is open on Yl\qlC(ql), by 2.9 (iv). 2.11. Let (Xi, Q,), i E J , be a collection of metric spaces such that the diameter of each Xi is 1 at most. On the set X = u X i x (i} define a icJ metric e putting
Further, define mappings E ~ (Xi, : ei)---* ( X ,e) by E,(x)= (x,i). Obviously, E, is an isometry. The collection ( E & ~ , or sometimes simply the space (X, e), is called sum of the collection ( ( X ,ei))iaJ. 2.12. Proposition. Let ,fi: (xIi, eli) (Xzi,ezi) ( i E J ) be contractions (local isometries, resp.). Let ( E ~ , : (X,,, eji)7( X j ,ej))i,J ( j = 1,2) be sums. Then there exists exactly one contraction (local isometry, resp.) f : (Xl, el) -+ -+ (xz, e2) such that for every i . f o E ~ = , E~~ o f i -+
Proof is obvious. EXERCISES
1. Let f : (X, z) + ( x 9) be a quasi-open map such that for every non-void open U there is a non-void open V c U such that f maps I/ homeomorphically onto f(V ) . Then f is not necessarily continuous. 2. Topological spaces and all their quasi-open mappings form a category while topological spaces and all their quasi-local homeomorphisms do not. Topological spaces and all quasi-open quasi-local homeomorphisms form a category. 3. Show that 2.2 may be strengthened as follows: f : X + Y is a quasilocal homeomorphism (or a quasi-local isometry) iff it is continuous and there exists a nowhere dense A c X such that f X\A is a local homeomorphism (or a local isometry). On the other hand, an analogous statement on quasi-open mappings is false.
I
$3. The functors A,etc.
213
4. Denote by P, the category of all pseudometric spaces and all their contractions, by P,‘ its full subcategory generated by all the spaces with diameter 51. Prove that P, has coequalizers but not coproducts. P,‘ has both coequalizers and coproducts and hence (see E 1.3) it is cocomplete. (Hint: In the construction of coequalizers use 2.3 and 2.4.) 5. We recall that a map f: (P, e) -,(Q, 0)is called Lipschitz mapping if there exists a number L such that o ( f ( x )f,( y ) ) i Le(x,y ) for all x, y E P. Denote by P9 the category of all pseudometric spaces and all Lipschitz mappings, by P& its full subcategory generated by the spaces with diam I 1. P9 has coequalizers but not coproducts, P& is finitely cocomplete.
6. Denote by P (or Puor Pi) the category of all pseudometric spaces and all continuous mappings (or uniformly continuous mappings or all isometries). Prove that Pi s Pc s PS s Pu S P and an analogous statement on the corresponding categories Mi,..., M of metric spaces.
8 3. The functors A,&lo,2,do, A/,and M u In this and the following sections we are going to describe some modifications of the arrow construction (see IV.2) useful for embeddings into various categories of metric spaces. So’far, it will be done on the basis of general system of “arrows” (the basic and fundamental classes, see 3.2 below). A suitable system of arrows will be constructed later in 0 6.
3.1. Let 49 be a non-empty class. A B-labeled graph is a triple
(x,R, 9) where ( X , R) is graph and 9 :R + a mapping. If ( X , R, ‘p) and (X’, R’, ‘p‘) are B-labeled graphs, a mapping f:X + X’ is said to be compatible (more exactly, R9R’cp’-compatible)if it is Rd’-compatible and for every (x,y) E R, cP’(f(x),f ( Y ) ) = 9(x,v). If K is a full subcategory of Graph, we denote by K,
the category of @-labeled graphs ( X , R, 9)with ( X ,R) in K and their compatible mappings. Obviously, for every b E B we have a full embedding of K into Ka sending ( X , R) to ( X , R, const,). Thus, if K is alg-universal, SO is K a
214
Ch. VI, TOPOLOGY
3.2. A basic (fundamental, resp.) class is a non-empty set of quadruples (triples, resp.) such that H = ( H , 0)is a metric continuum (i.e. compact metric connected space) and ai are points chosen in H such that diam H = ~ ( a ,u,j ) = 1
j. for i =l=
3.3. From now on, we will denote by G
the full subcategory of Graph generated by all the ( X , R ) such that for every X E X either an (x,y)or a (y,x)is in R. Further, G'
will designate the full subcategory of G generated by all the connected graphs. By IV.I.11, G' is alg-universal. 3.4. The spaces A(X,R, cp), d ( X , R, cp) etc. In this paragraph we will give a formal description of some constructions. The intuitive meaning will be described in 3.5. Let a basic (fundamental, resp.) class 9be given, let ( X , R, cp) be an object of G@ Put W0= H'\(u; i = 0, I, 2 (0,1, resp.)) , cp(r) = (H', (a:)), let (&': H' --* H)rER
I
1
be the sum (see 2.11) of the collection (Hr)rER. Put EL = E' H*,. Now, consider the quotients (see 2.3) H 4H /
-
=
where the relation
-
&*(a;)
A ( X , R, cp)
- is given by
&'(a;)
( H -!$ H /
e'= g o & ' ,
w : X + A ( X , R, 'p)
by
W(X> =
e'(ai) for r
= J ( X , R, cp), resp.)
for r = (xo,xl), s = (yo,yl) and xi= y j
and in the first case, in addition, &'(a;) and define a mapping
-
-
&'(a;) for all r, s E R. Put
eL = g o & (w: X
= (x,,x2)and
x
-+
J ( X , R, cp), resp.)
= xi-
215
53. The functors W , etc.
Now, put
R>'P) d O ( X R, 'P) A s ( X ,R, 'p) A , , ( X , R, 9) A0(x7
= =
" (er(a;)>)
A(x>R, 'p)\(w(x)
d ( x ,R , 'P)\w(x)
9
3
I
Jf'(X, R , v ) \ { ~ ( x ) xR = 8) , = d o ( X , R , 'p) u {W(X) s R = 8).
=
1
3.5. An intuitive description of the spaces from 3.4 The space d ( X , R, 'p) is obtained by taking the graphs ( X , R ) and replacing the arrows by their labels glueing the ao-points in the initial points and the a,-points in the terminal ones. If we have a one-element B7this is the "arow construction" already mentioned in the 'Introduction (and analogous to that of IV.2). Thus, the difference is only in reckoning with possibly various arrow replacements. Constructing. / / ( X . R. q).we have further distinguished points in the spaces used to replace the arrows, one in each; those are then glued together. The constructions with the subscripts 0 are obtained by omitting the distinguished points. Take notice of the fact that although this omission makes the replacement spaces fall apart topologically, it does not do so metrically: at the location of the points (now removed) where two or more of the replacement spaces met, we still have open sets of arbitrary small diameter intersecting each of these same spaces. we omit (starting with .A' again) only those Unlike in d o ,in .,d,? distinguished points in which no arrow starts. On the other hand. .Muis obtained from 2 by omitting all the distinguished points but those in which no arrow starts.
3.6. Observations. (1) d ? ( X , R, 'p) is a complete metric space and it is a completion of do@, R, 'p) and A , ( X , R, 9). (2) . K ( X , R, 'p) is a complete metric semicontinuurn*) and it is a completion of A o ( X , R, q) and A s ( X ,R, q). (3) A 0 ( X ,R, 'p) are dense-in-itself locally compact spaces. (4) If every H', is a semicontinuum then also A s ( X ,R, 'p) is a semicontinuum.
3.7. The mappings A ( f ) , d(f), etc. Let f:( X , R, 'p) (x', R', CP') be a compatible mapping. By 2.12 and 2.6 we see immediately that there is exactly one contraction +
A ( f )A:( X , R, 'P) + A ( X ' , R', cp') *) i.e., any two of its points may be joined by a continuum
216
Ch. VI, TOPOLOGY
such that A ( f ) e' 0
=
e''
for all r E R , r'
=
(fx f)(r).
Quite analogously we define a contraction d ( f ) : d ( X ,R , Q ) -+ d ( X ' , R ' ,v'). Since, obviously, if u 4 w ( X ) also A ( f ) (u) 4 w(X') (and similarly with d ( f )), we obtain, by restriction, mappings d o ( f ) : k o ( X , R, 'P)
Jo(f):
Further, since if x R define
Jo(x,R, 'p)
+
-+
A o ( X ' , R', P') JO(X', R', cp') . 7
+ 8 we have also f ( x ) R ' 4 8, we can, by restriction,
As({): As@,R, 'p) A s ( X ' , R', 40') . Finally, if, moreover, f has the property that x R = 8 * f ( x ) R ' = 8, +
f
we can also define, in the obvious way,
dU(f): A , ( X , R, 'p)
-+
A@',R', cp') .
3.8. By 2.10 and 2.2 we immediately vbtaiit Proposition. ~ ( f )J,( f )d, S ( f )and ~ , ( fare ) quasi-open contractions and quasi-local isometries. Ao( f) and Jo(f) are open contractions and local isometries. Proof. Obviously, A ( f ) 0 w = w J: 0
3.9. Convention. Working with A s ( X ,R, cp) it will be convenient to put, for r E R , H4 = (e')-' ( A s ( X ,R, 40)) and denote by ei: H', 3 d S ( X ,R, 'p) the domain-range restriction of e'. Obviously, H', c H : c H' . Similarly for A,,.
3.10. So far, the members of a basic or fundamental system have been too general to yield suitable material for a variant of the arrow construction. In 4 6, however, we will construct a countable basic system
I
{(H",a& a;, a;) n = 1,2, ...}
$3. The functors A, etc.
217
such that , A,, if F is any of the constructions A, .A'o, 2,J o As, if (X, R, 'p) is a labeled graph, and if f : H; + F ( X , R, 'p) is continuous, then either f is a constant, or f = er H t where Y(Y) = (H",(a:)). In the next section we will show that under these conditions, the constructions will provide us with full embeddings into many categories of metric spaces.
I
3.11. Under the conditions from 3.10, one sees easily that if a mapping f : F ( X , R , 'p) + F ( X ' , R', q') is continuous on every er(H"), then it is continuous. Indeed, the only suspicious points are those of w(X). If f ( o ( x ) )4 w(X'), all the adjacent er(H")are mapped constantly into f ( o ( x ) ) ; if f(w(x)) E o ( X ' ) , those adjacent er(H")which are not mapped constantly into f(w(x))are mapped isometrically onto their copies so that, for every E > 0, the Eneighborhoods are mapped into +neighborhoods. EXERCISES
The following exercises are not directly connected with the contents of 5 3. Their role is to provide the reader with material necessary for some important exercises in 5 4. For a set 6 of pseudometrics on a set X define a set c 6 of pseudometrics on X as follows. u E c 6 iff for every E > 0 there exists a 6 > 0 and el, ..., enE 6 such that ei(x,y) < 6, i = 1, ..., n implies u(x, y) < E. 1. Prove that c has the usual properties of algebraic closure operator
6 c C G , 6,c 6
2
*
CG, c c62, c(c6) = c 6 .
Let us recall that a 6 with 6 = c 6 is called uniformity on X , (X, 6)is called uniform space. A map f : ( X , 6)+ ( X , 6') is called uniformly continuous iff for every u ' 6' ~ and E > 0 there are Q E 6 and 6 > 0 such that o ' ( f ( x ) , f ( y ) < ) E for u(x, y ) < 6. 2. Prove that the uniform spaces and uniformly continuous maps form a category. We denote it by Un. Prove that the functor @: P, -+ Un (see E 2.6) sending ( X , e) to (X, c { e ) ) is a realization.
218
Ch. VI, TOPOLOGY
A proximity 6 on a set X is a symmetric binary relation on exp X such that (i) never 0 6 ~ (ii) A 6 B whenever A n B 8 (iii) ( A , u A 2 ) 6 B iff A , ~ B or A , ~ B (iv) if A , non6 A,, then there exist disjoint U1, U , such that Ainon6(X\Ui), i = 1,2. ( X , 6 ) is called proximity space. A map f : (X,6) -+ (X’,6’) is called proximally continuous iff A 6 B implies that f(A)G’f(B). 3. Prove that all proximity spaces and all their proximally continuous mappings form a category (we will denote it by Prox). 4. Prove that 6: U n -+ Prox sending every uniform space (X, 6 ) to ( X , [ ( A ,B) a(A, B ) = 0 for all (r E (5)) is an embedding which commutes with the natural forgetful functors; prove that 6 is not full. 5. Prove that F: Prox -+ Top sending every proximity space (X, 6) to
+
1
(X,{%C~~(XE% & ) ( { x ) 6 A ) =- % n A + 0 } ) is an embedding, which commutes with the natural forgetful functors; prove that F is not full. 6. Let P, be the category of all pseudometric spaces and the mappings f : ( X , e) + ( X ’ , e’) such that e’(f(A),f(B))= 0 whenever @(A,B ) = 0. Prove that V : P, -+ Prox which sends (X, e) to ( X , {(A,B ) @(A,B ) = 0)) is a realization. 7. Prove that P, is equally realized (see 1.4.8) with P,.
I
0 4.
Some full embeddings into categories of metric spaces
In this section we will assume a basic class with the properties from 3.10 given. The reader who prefers to have its existence proved may read sections 5 and 6 first.
4.1. Notation. The category of metric spaces and their continuous (uniformly continuous, resp.) mappings will be denoted by M
(Mu,resp.) .
4.2. Theorem. Let N be the category of all complete semicontinua and their quasi-open contractions which are quasi-local isometries. Let R be a category such that N c R c M Then R is almost alg-universal.
219
$4. Some categories of metric spaces
Proof. By 3.6 and 3.8, the construction A defines a functor G&+ 52. Since it is obviously one-to-one and since G', is alg-universal, it suffices to prove that in the case of the with the properties from 3.10 a continuous mapping g : A ( X , R, 'p) + A ( X ' , R', q') is either a constant or g = A ( f ) for an f:( X ,R, 'p) -+ (X', R', q'). By 3.11 it suffices to prove that if g o e' is a constant for some r E R, the whole g is a constant. Since the graph ( X , R ) is assumed to be connected, it suffices to prove that if g 0 e' is constant and if es((H')n e'(H')\{e'(a>)) @,also g es is constant. By 3.11, of course, it suffices to prove that g es is not an e'. But this is obvious, since es((H)is assumed to meet e'(H') in two points.
+
0
0
4.3. Theorem. Let N be the category of' all infinite dense-in-itself loccilly compact metric spaces and their open contractions which are local isometries. Let 52 be a category such that NcRcM,.
Then 52 is almost alg-universal. I f 52 does not contain any constant mapping, it is alg-universal. Proof. By 3.6 and 3.8, the construction A. defines a functor G& --+ 52. Thus, it suffices to prove that a uniformly continuous g : JZo(X,.R, q) -+ + A o ( X ' ,R',cp') is a constant or g = A , ( f ) for an f: ( X ,R, q)--+ (X',R', cp'). Since A ( X , R, 9)is a completion of A o ( X , R, q), g can be extended to a (uniformly) continuous 3: A ( X , R, 'p) -+ A'(X', R', q'). By the proof of 4.2, if i j (and, hence, g ) is not a constant, we have i j and we see immediately that g = A o ( f ) .
=
&(f)
4.4. These two statements have many direct or almost direct consequences. The reader is strongly recommended to read the text of the following exercises even if he does not intend to do them.
EXERCISES
1. The categories Pi, Pc,Py, Pu,P are introduced in E 2.5 - 6. Prove that all of them but Pi are almost alg-universal. Analogously for the corresponding categories of metric spaces. 2. If P, is some of the categories Pi, P,, P2, P,, P, denote by P,,,the category of all pseudometric spaces and all open morphisms of P,. Prove that
220
Ch. VI, TOPOLOGY
Pi,, and Po are not alg-universal, while the others are (Hint for- Po: look at 0 9). 3. The categories U n (see E 3.2) and Prox (see E 3.3) are almost alguniversal. By topological properties of an object or a morphism of Prox (or Un) we mean the properties of its image under 9(or 3 L ) (see E 3.4 and E 3.5). 4. The previous exercise may be strengthened as follows. Every category A such that Prox,,l,h c R c Prox (or Uno,l,hc A c u n ) is almost alguniversal, where Prox,,l,h (or Un,,l,h) is the category of all proximity (uniform) spaces and all proximally (or uniformly) continuous open local homeomorphisms. 5. Let Prox,,l,h (or Uno,l,h)be as above, Prox,, (or Un,,) be the category of all proximity (or uniform) spaces and all proximally (or uniformly) continuous quasi-open mappings. Every category A such that Prox,,l,h c c A c Prox,, (or Uno,l,hc A c Un,,) is alg-universal. 6. Denote by Prox,, - (or Unll - the category of all proximity (or uniform) spaces and all proximally (or uniformly) continuous locally oneto-one mappings. Prove that every category R such that Prox,,l,h c A c c Proxll-l (or Uno,l,hc A c Unll-l) is alg-universal. 7. Let G;, be the full subcategory of G (see 3.3) generated by all finite graphs. Then there exists an almost full embedding of G;i, into any category A such that Mcont c A c P, where P is as above, Mcont is the category of all metric continua (see 3.2) and all quasi-open contractions which are quasi-local isometries; in particular, any finite monoid has a representation by all non-constant continuous mappings of a metric continuum.
0 5. .Labeled topologized graphs. The functor P 5.1. A W-labeled topologized graph is a quadruple
(x,t, R, q) where ( X , t ) is a topological space and (X, R, q) is a 2d-labeled graph. We will denote by TGCa
the category the objects of which are the ( X , t, R, q) with ( X , R, q) in GO, and the morphisms are the continuous compatible mappings.
221
$5. Labelled topologized graphs
5.2. Given a labeled topologized graph ( X , t, R , 'p), define a topological space Y(Xt, R, 'p) as follows: (a) the underlying set of Y(X, t, R, 'p) coincides with that of &(X, R, 'p); (b) let 23 be an open base of &(X, R , 'p), let e be its metric and 8,= = (x E A ( X , R, 'p) e(x, a("))< E ) for 0 c X ; then
I
(o~ 1 E > 0, o open in (x,t ) )u (B\o(X)
I B E23)
forms an open base of Y ( X , t, R, 'p). ( Aand o from 9 3.)
5.3. Observation and convention. Obviously., the mappings e' and o from 3.4 are continuous as mappings H' -+9 ( X ,r, R, cp), ( X ,t ) + P(X, t,R, 'p). Speaking of the construction 9, we will use the symbols er, o in this sense. 5.4. Further observations. (1) If ( X , t ) is a T,-space, e'(H') is closed in Y(X,t, R, 'p) and the induced topology coincides with that induced by A@', R, cp). (2) A f o ( X , R, 9)is an open subspace of 9(X,t , R, 9). (3) The topologies of Af,(X, R, 'p) induced by A ( X , R, 'p) and by Y(X,t, R, 'p) coincide. (4) o maps ( X , t ) homeomorphically onto o ( X ) . (5) o ( X ) is a closed subspace of Y(X, t , R, 'p). (6) If t is the discrete topology, Y(X,t, R, 'p) = A ( X , R, 'p).
5.5. Let f:( X , t , R, 'p) -+ (X', t', R', cp') be a morphism in TGB. If U is open in Y(X', t', R', 9') then w - l .."e(f)-' ( U ) = f - l o - ' ( U ) (see 3.8) is open in ( X , t), so that &(f)-' ( U ) is open in Y ( X 7t, R, 'p). Thus, we have a continuous mapping Y ( f ) Y: ( X , t, R, 'p)
+
P(X', t', R', 40')
defined by 9(f) (x) = &(f) (x). Obviously, we are obtaining a functor
9:TGB + TOP. 5.6. Later in this chapter we will use the following Proposition. 9(X, t, R, 'p) is paracompact whenever ( X , t ) is. Proof. Let 4!l be an open covering of 9(X,t , R, 'p). Since ( X , t ) is paracompact, o ( X ) is paracompact (see 5.4.4) and hence there is a system of open sets such that ( W n o ( X ) 1 W EW ' ) is a locally finite covering of
222
Ch. VI, TOPOLOGY
1
o ( X ) refining {U n o ( X ) U E $21, and such that
wc { X E ~ ( X , t , R , c p ) I @ ( X ,~ n ~ ( X ) ) < + } ,
where e is the metric of A ( X , R, 9).Thus, W is locally finite in P ( X ,t , R, 9). Let be an open locally finite covering of { x E A ( X ,R, cp) 1 e(x, o(X)) > l/n), which refines %. Then W u Wnis an open o-locally finite covering of P(X,t , R, (p) refining Q.
u
n= 1 , 2 ,
...
EXERCISES
1. Let a basic class 9?have at least three elements,say (Hk,{ u f ] ) ,k = 1,2,3. For any ordinal number CI, denote by G , = (X,, t,, R,, cp,) the following topologized graph.
X , = CI + 2 (the set of all ordinals fl I SI + l), t , is the order topology, R, = R: v RZ v R: where Ri = ((0, l)}, RZ = {(B, B + 1) 11 5 P I a}, R,3 = {(%a+ I)), qa(r)= (Hk, ( u f ] ) for r E RE, k = 1,2,3. Examine the morphisms in the full subcategory of TGB generated by the class of all G,. 2. Let 92 have three elements at least. Prove that the full subcategory of TGB generated by all ( X , t, R, q)such that ( X , t ) is a compact Hausdorff space is alg-universal.
0 6.
Construction of sufficiently rigid basec and fundamental classes
6.1. Throughout this section, the word continuum always means a nondegenerate continuum (i.e., consisting of more than one point). We will use the following well-known statements (the reader can find their proof in Appendix A): (1) For every continuum there exists a countable pairwise disjoint system of its subcontinua. (2) If P is a compact Hausdorff space, C some of its components, G open in P and C c G, then there is a closed and open U such that K c U c G .
223
96. Sufficiently rigid classes
6.2. In 1967, H. Cook constructed a metric continuum C such that for arbitrary subcontinuum K of C, a continuous mapping f : K C is either a constant or the identical embedding of K into C (see [C,]). (In this paragraph we will only use the existence of such a C. A construction of C is given in Appendix A.) Hence by 6.1.1, we can assert the existence of a countable system -+
%
of metric continua such that if C, D are in %?, if K is a subcontinuum of C and if f : K -+ D is a nonconstant continuous mapping, then C = D and f(x) = x for all X E K . Moreover, we see immediately that we can assume that
u W
=
n= 1
%,,
where every Wn is countable and, for CE'$?,,,,diam C = 2-". For every C E % we will assume fixedly chosen points co(C),cl(C) such that diam C = = distance (co(C),cl(C)). 6.3. The graph (17; W ) .Consider the set T = (0, 1,2} u ( ( n , i , j ) l n= 1,2,...; i
=
0, 1; j
=
0, 1,2)
Let us introduce the following notation for some ordered couples of the elements of T: for 17 > 0 , ( ( n , O , j ) , ( ~Lj)) Zj(n) = ((n,O,j), ( n 1,O,j)) for I I > 0 , Ij(0) = ( j ,(1,O,j)) rj(0) = ( j + 1, (1,l,j)), rj(n) = ((n, 1,j), ( n + 1,l,j)) for n > 0 Sj(0) =
( j ,j
+ I),
Sj(n) =
+
9
(the addition is understood modulo 3; thus, 2
n. = j S j ( I 1 ) ,
lj(lI), Vj(/l)
Ij
=
+ 1 = 0). Put
0, 1, 2,; n = 0, 1, 2, ...>.
The obtained graph (T, W ) is visualized in Fig. 6.1. 6.4. For W E W and positive integers k choose once for ever distinct C(w, k ) E %? in such a way that if w = sj(n), C(w, k ) E Wn+ 1, and if w = lj(n) or r,{n), C(w, k ) E V,,+,. For a fixed k consider first the sum (E,,,: C(w, k ) -, -+ Pk)w.w (see 2.1 1) and then the quotient
by the relation
q: P + Q
-
E ~ ( C ~ ) cWr(cj)
iff the i-th coordinate of w coincides with the j-th coordinate of w'.
224
Ch. VI, TOPOLOGY
Finally, define
Hk
as the completion of Q. I
I
-0
I
,A\
(2.1.22 , ,
6.1
In the sequel, will designate the metric of the H k in question. The space H kis visualized in Fig. 6.2.
225
$6. Sufficiently rigid classes
6.5. Observations and notation. Obviously, Hk\Q three points ak,,at,
4
consists of exactly
I
where as is in the closure of U { w j ( n ) w = s, 1, r ; n = 1,2, a continuum, Hk, = Hk\(ak,, at, a:} is a semicontinuum. We have diam H k = CT(U;, a:) = 1 ( i =+ j ) , We denote by d , : C -+ H k
,..I.
H k is
the mapping q o E, where C(w,k ) = C (if there are such w, k). 6.6. Further in this section, the constructions of &. 2. 9 are assumed to be based on the basic class @ = ( ( H k .okg, a:. u i ) k = I . 2 . . ..I 0 1 - on the fundamental class B = { ( I f k ,a!, at), k = I . 2. ...I. Also in the following sections throughoui ihia chapter, the basic and fundamental classes used will be subsets of the @ just constructed.
I
6.7. Let us recall a well-known proposition on components of closed and open subsets of continua: Let S be a non-empty closed (resp. open) subset of a continuum C, let $3. Then any component of S (resp. the closure of any component of S ) CS \ intersects the boundary of S. (For the proof see e.g. Appendix A, 1.7 and 1.8.)
+
6.8. Notation. If a labeled (topologized) graph with labeling cp: R + B (@ from 6.6) is given, we write cp(r) = (Hr,(a!)). In this sense we also use the symbol H;. Further, we write ci(C,r ) instead of erdc(ci(C))for er from 3.4. We write A(C, r ) for e'dc(C). 6.9. Lemma. Let C be in V, let f : C -+ P ( X , t, R , cp) or f : C A ( X , R, 'p) be a continuous mapping. Then either f(C) c o ( X ) or f is a constant or there is an r such that f = e' dc. Proof will be done for the slightly more complicated case of 9. Take a BE%? and an r E R . Put -+
0
S(B, I) = f -I(@,
4\{c*(&
r), c,(B, r ) ) ) .
If C = S(B,r) for some B , r , the f must be constant by 6.2. Thus, let US suppose that C\S(B, r) =k 8 for all B and r. (a) Let S(J3, r ) = 8 for all B and r. Then, since f(C) is connected, either f is constant or f(c)c ~ ( x ) .
226
Ch. V1, TOPOLOGY
(b) Let S(B, r ) $1 8 for some B and r . By 6.7, f maps a non-degenerate subcontinuum of C into A(B,r). By 6.2, B = C. The set A ( B , s ) intersects six A(D, s)’s at most (see 6.3 and 6.4), none of which, moreover, is homeomorphic to C. Hence, S(D, s) = 0 for each of them. Consequently, f ( C ) c c A(B, r). Now, use 6.2 again. 6.10. Proposition. Let F be some ofthe constructions 4,A’,, 2,d o , At’”, A!, (see 5 3), 9’.Let 2 = ( X , R, 9 ) ( ( X ,t, R, 9)in the last case) be such that F ( 2 ) is defined. Let H be some of the H k jrom 6.6. Zf f : H,
is continuous, then
-+
F(2)
either f is a constant mapping, or there is an r E R such that q ( r ) = H and f = e‘ H,, or F = 9’and f(H0) G I O ( X ) . Proof. Since H o is connected, iff is neither a constant nor f(Ho) c w(X), there is a C E V with d,: C H o and with neither f o d,(C) c o ( X ) nor f 0 d , constant. Hence, by 6.9, f c dc = e‘ dc (and, of course, H = Hr). Since for any two C , D E %? such that dc, d D terminate in H,, there are Ci i = 0, ...,n such that dc, terminate in H,, Co = C, C, = D and d(Ci)n n d(Ci+ $; 8, we see easily (using 6.9 again) that
1
--+
0
f(H0) = e‘(H0).
1
Now suppose that f is neither e‘ H , nor a constant. Since either f 0 dc = = e‘ dc or f o dc is constant, there have to exist C, D such that (1) f o dc = e’ dc and f o d D is a constant. Since we have the Ciwith the properties above, there are C, D such that (1) and that 0
0
d(C) n d(D) =k 8. (2) Furthermore, there have to exist C, D satisfying (1) and (2) such that (3) there is an E C,D such that d ( E ) meets d(C) and d(D) in distinct points or there are E, F such that C, D, E, F are distinct, d(D) n d(E) $1 8, d(E) n d ( F ) $1 0 and d(F) n d(C) $1 8. (Indeed, if the C, D found originally do not already possess the property, they are placed either in lj(n)(see 6.3) and ldn + l), or in rj(n) and r,{n l), or in rj(0)and Ej+,(0),or in sj(0) and lj+,(0),or finally in s,{O) and rj-,(0). In the first two cases, consider s,{n), in the third sj(0) and sj+ 1(0), in the fourth s j + 1(0)and in the fifth sj- l(0).) Now, since f dD is a constant, f o d E $; e‘ d E (the common point of d ( E ) and d(D) does not remain fixed). Thus, f c d, is constant. Similarly, in
+
+
0
227
$7. Some strong embeddings
the second alternative we see that f dF is constant as well. But this implies that the common point of d(D) and d(C) (resp. d ( F ) and d ( C ) )is being sent to the common point of d(D) and d(E), which is a contradiction. 0
EXERCISES
1. Show that for no fundamental class F, the construction 2 considered as a functor from G> into M (see 4.1) is an almost full embedding. 2. Denote by M,, the category of all metric spaces and all continuous quasiopen mappings. Let F = {(C,u,,, u l ) ) be the fundamental class such that C fulfils 6.2. Then 2:G', -+ M,, is a full embedding. Hence, the alguniversality of M,, can be proved by the simple arrow-construction (see IV.2). 3. Prove that M is almost alg-universal by the simple arrow construction. (Hint: start with the category of all connected graphs with all the loops. Replace each edge which is not a loop by a copy of the Cook continuum and use IV.7.)
3 7.
Some strong embeddings into categories of metric spaces
The full embeddings from 5 4 are not strong (in the sense of 1.6.11). Now we are going to show that some of them may be replaced by strong ones. This will be shown to have important consequences in Chapter VII. 7.1. Throughout this section we will use the basic system consisting of the first three elements of the system 33 from 6.6. We will denote it by d .
7.2. For a graph G
=
( X , R) put O(G)
where
=
(2,k, 'P)
= X U (X x X ) u ( X x X\R)
x (0)
(More exactly, X x (21 u ( X x X) x (1) u ( X x X\R) want to have the union disjoint),
x (0}, since we
R=R,uR~vR~,
with
Rl R2 R,
I x,Y E x} = (((x, Y), Y ) I x, Y E x> = {(r, r ) 1 r E R } u ( ( I , (r, 0)) I r E X = ((x, (x, Y ) )
>
3
x X\R}
q(w) = (Hi, (uf)) for w E R , .
,
228
Ch. VI, TOPOLOGY
For a compatible f : (X, R ) + (X’, R’) define O(f):
by
(4
x
-+
Z’
= f ( x )2
0(f) O ( f ) (x,Y )
= (f(X),f(Y))
0(f)(x,y, 0) = ( f ( x ) , f ( y ) ) if it is in R’
3
Off) (x,Y, 0) = ( f ( X ) , f ( Y ) , 0) otherwise .
It is easy to check that thus a one-to-one functor 0: G+Gd
is defined.
7.3. Proposition. 0 is a full embedding. Proof. Let g : cp) --+ (Z’, E‘, cp’) be a morphism. For x , y E X we ~ hence (g(x),g(x, y)) E R; and hence g(x)E X‘, g(x, Y ) = have ( x ,(x,y ) )R,, = (x’,y‘) E X’ x X’, and x’ = g(x). Thus, we can define f : X + X‘ by , f ( x )= g(x) and immediately obtain (using also R 2 ) g(x, y ) = ( f ( x ) , f ( y ) ) . ~ f ( xY,) E X x X\R,wehave ( ( x , ~(x,Y, ) , 0 ) )&,hence ~ ( ( f ( x )f,( y ) ) ,g(x,y , 0 ) ) ~ E R j so that either ( f (x),f ( y ) )E R‘ and then g(x, y , 0) = ( f ( x ) ,f ( y ) ) or ( f (x),f (y))$R‘ and then g(x, Y,0) = ( f ( x )f, ( y ) ,0). Finally, if r = ( x ,Y)E R, (g(r),g(r)) E R; so that g(r) = ( f ( x ) , f ( y ) E) R‘. Thus, f is RR’-compatible. W
(x,w,
7.4. Theorem. Let N be the category of all dense-in-it-self locally compact metric spaces and their open contractions which are local isometries. Let R be a category such that NcAcM,
(for Musee 4.1). Then there exists an almost strong embedding of G into R. Proof. Put %Yo = ,,do0 0 (,,dofrom $3, 0 from 7.2). By 7.3 and by the proof of 4.3, %Yo is an almost full embedding of G into R. Denote by U the natural forgetful functor of Mu.Let A be the disjoint union of the underlying sets of HA, H i and Hi.Define X x X x A
c ( ~ , ~ ) :
by
-+
U T 0 ( X ,R)
229
$7. Some strong embeddings
It is easy to check that we have thus defined a natural equivalence E:
FoVg Uob,
where V is the natural forgetful functor of G and F sends f to f x f x idA. 7.5. Theorem. Let N be the category of all metric semicontinua and their quasi-open contractions which are quasi-local isometries. Let R be a category such that N c R c M
(for M see 4.1). Then there exists ~ i i ulmost i strong embedding of G into R. Remark. Unlike 7.4, which is a stronger counterfact of 4.3, 7.5 does not cover all the cases of 4.2: The non-complete objects of N are essential). Proof. Put %” = As 0 (for A,see 5 3). Asis a full embedding for the reason quite analogous to those for which Af is (see the proof of 4.2). Thus, by 7.3, S is a full embedding. Now, by the definition of Aswe have 0
w, ‘p) = Ao(2,w, ‘p)
Z ( X , R ) = Asp,
LJ
{ *> u o(x)LJ o(x x
x)
(where * designates the point obtained by glueing the a,-points together), and the union is disjoint. Furthermore, if f: ( X , R ) + (X’, R’) is a compatible mapping, f ) follows, at the points of Afo(2, R, ‘p) = S d X , R), the rule for S o ( f ) , sends w ( X ) to w(X’) and o ( X x X ) to w(X’ x X’) following f and f x f respectively, and finally, sends * to *. Thus, the statement follows from the statement on proved in 7.4.
a(
EXERCISES
1. Let Sk be the subspace of the H k from 6.4, formed by the points a:, a:, a! and all the d,(C(w, k)) where either w is lj(n)or rj(n)(see 6.4). The space Sk is visualized in Fig. 6.3. Consider the basic class 9 = {(Sk,(a:)) k = 1, ...}.
1
230
Ch. VI, TOPOLOGY
Prove that ~ 4 l 0 G; : + M (see 4.1) is almost full but cAs: G', M is not almost full. Hence, the basic class 9 cannot be used in the proof of 7.5. instead of d,although in can be used for the proof of 7.4. 2. Consider a basic class formed by the Cook continua joined into triangles (see Fig. 6.4). Prove analogous statements to those of Exercise 1. -+
9
8. Some universal categories of metric spaces
Assuming the condition (M) (see 11. $6) we obtain from the previous results on alg-universality obvious corollaries on the universality of the categories of metric spaces in question. In this paragraph, we will prove the universality of some categories of metric spaces without using (M) (which is essential-see Appendix B).
8.1. In the construction of this paragraph we will use the fundamental class . . F = { ( H Jah, , a:) I j = 1,2,3$ ( H j , u&,ui from cj 6). 8.2. Further notation. The subcategory of G,-(see 3.2, 3.3) with the same objects, but only those morphisms f : ( X , R, 'p) + ( X ' , R', 'p') for which xR = t!l implies f ( x ) R ' = 8 will be denoted by
wGF. (Thus, wG,-has those morphisms f of G,- for which AU(,f) -see makes sense). We define a functor
I
3.7-
n: Set -,Set
putting n(x) = ((x, A ) x E A = x}, U ( f )(x,A ) = ( f ( x ) ,P + ( f )( A ) ) (P' and, further down, P: from 1.3.11.C and 111.1.1).
8.3. Lemma. Let g : A u ( X ,R, 'p) -+ Mu(X',R', 40') be an open uniformly continuous mapping. Then there is an f : ( X , R, 'p) + ( X ' , R', V ' ) in WG, such that g = M u ( f ) .
$8. Some universal categories
23 1
Proof. Since g is open, no g 0 e; is a constant and hence. by 6.10, always
g 0 e; = e;. Thus
s(Jo(x,R, 40)) = JO(X’> R‘, 9’).
Since g is uniformly continuous, we have, consequently, an extension
S: J ( X , R, ‘p) --+
J ( X ’ , R‘, cp’) ,
and here again for every r E R there is an r’ E R‘ such that S 0 e; = e;. Now, we see easily that S = J(f) for a suitable f, that this f is in wG, and that g = A , , ( f ) . 8.4. The functor
is defined as follows where =
Vy.:S(P:)
-+ wG,-
.y(x,R) = (Kw, ‘p)
1
X u P,i(X) u n ( X ) u ((x, 2 , O ) (x, 2 )E n ( X ) and 2 $ R }
(again, the union is assumed to be disjoint and a multiplying by different elements, to make it really so, is ommited),
8.5. Lemma. Let g : Y ( X ,R ) --r .Y-(X’, R’) be a morphism of wGF.Then g = V ( f )for a morphism f:( X , R ) (X’, R’) & i for every 2 E P:(X), --f
g({z) u Rz)= { g ( Z ) ) u &@).
232
Ch. VI, TOPOLOGY
Proof. If g
=
Y ( f ) we have
z } = ( ( f ( x ) J ( Z )I )x E z } = = W’S(Z). = { ( Y , f ( Z 1)Y Ef(Z)l = ((Y?g(z)) 1 YE
I
g(RZ) = (g(x9z ) x
E
On the other hand, let g be a morphism and let g ( ( Z }u R Z ) = (g(Z))u u Rrg(Z).Considering R, and R , we immediately see that
g(x)= X‘ and that if we define
7
g(PO+(X))= Po’(X’) 3 s(n(x)) = and g(RZ) = R’g(Z),
f: by f(x) = g(x), Now, we have
qx’)
x + X’
g(x, z) = ( f ( x ) g(z)). ,
1
( ( x ,g(z))Y E g(z)}= W’g(Z))= S ( R Z )= { ( f ( x ) ,
Ix Ez)
so that we see that g(Z) = P ; ( f ) ( Z ) (and hence also g(x,Z ) = n ( x , Z)). Further, we have g(x,z,0 ) R (x, z)
qf)
so that either P;(f) (Z)ER’and then g(x,Z,O) = n(f) (x,Z),or P:(f) (Z)$R’ and then g(x,Z , 0) = ( n ( f(x, ) Z), 0). Finally, if Z E R, we have (x, Z)R,(x, Z), hence g(x, Z)R;g(x,Z ) and hence P ; ( ~ ) ( Z ) R‘. E Thus, g = V(f). be the category o j a l l metric spuces and their 8.6. Theorem. Let Mo,u open uniformly continuous mappings, N the category of all dense-in-itself metric spaces and all open contractions whicb are quasi-local isometries. Let R be a category such that N c R c Mo,u. Then R is universal. Moreover, there is a strong embedding of S(P:) into R. Proof. Take an f : ( X ,R ) + (X’, R’) in S(PO+).Since the w(Z)E o(P;(X)) are the only points of A u ( V ( X ,R)) at which AuV(f) does not necessarily send a neighborhood isometrically onto a neighborhood, we see by 8.5 that A u V ( fis)open. Thus, we can define a functor
a : S(P,+)+R
by @ ( f ) = A U V ( fObviously, ). 42 is one-to-one. If g : Q ( X ,R ) + %(X’,R’) is open uniformly continuous, then by 8.3 g = A , ( h ) for an h : (3,R, ‘p) + R’, cp’). Since g is open, considering
(x’,
$9. Negative results
233
the neighborhoods of the points o(Z)E w(P:(X)) we see that h ( ( Z }u wZ)= = {h(Z)}u R’h(Z) and hence, by 8.5, h = V ( f ) so that g = %(f). To prove that V is a strong embedding may be left to the reader as an easy exercise (cf. the proofs in $ 7!). EXERCISES
1. There exists a proper class YJi of metric spaces such that if M , M E YJi and if f : M -+ M ’ is open unlformly continuous, then M = M’ and f = 1,. 2. Let Pi,,, P,,,, P2,,, P,,,, P, be the same as in E $4. Denote by Mi,o, M,,,, M2,,, M,,,, M, the corresponding categories of metric spaces. E.4.2 can be strengthened as follows. Let R be some of the categories Pi,,,,..., Po, Mi,o,..., M,. The following statements are equivalent: (i) Every finite monoid can be represented as the endomorphism monoid of an object of R. (ii) R is alg-universal. (iii) si is universal. (iv) 52 is none of the categories Pi,,, Po,Mi,,, M,. 3. The category Un, of all uniform spaces and their open uniformly continuous mappings and the category Prox, of all proximity spaces and their open proximally continuous mappings are universal.
0 9.
Negative results on open and locally one-to-one mappings
In the connection with the result of $ 1, two questions naturally arise. First, can one prove similar results for more satisfactory separation axioms instead of To? (Note that the results on metric spaces from $0 4,7,8 concern other choices of morphisms than those of 0 1.) Second, can one construct strong embeddings there instead of mere full embedding? In this paragraph we are going to prove two negative results giving, in a way, bounds for further investigations in the directions mentioned. Then, the following four sections will be filling the gap. 9.1. Proposition. Let Top, designate the category of topological spaces and their open continuous mappings. The category Graph is strongly embedduble into no subcategory Ji O J Top, containing all the homeoniorphisms between its objects.
234
Ch. VI, TOPOLOGY
Proof follows immediately from the fact that an identity carried morphism of R is necessarily isomorphism. Under a strong embedding, the nonisomorphic identity carried morphism ( X , 8) -+ ( X , X x X ) would have to be sent into an identity carried one. 9.2. Lemma. Let H be a Hausdorff space, f : H + H a continuous mapping such that f f = J: If either f is open or f is locally one-to-one, . f ( H ) is open and closed. Proof. Since f 0 f =f,we have f(H) = { x f ( x ) = x ) and hence, since H is Hausdorff, f ( H ) is closed. Iff is open, f (H) is open. Thus, let f be locally one-to-one. Take x E f ( H ) and a neighborhood U of x such that f is one-to-one on U . Since x = f ( x ) , V = U nf - I ( U ) is a neighborhood of x . For y E V we have y E U and f ( y ) E U . Since f ( y ) = f f ( y ) , necessarily y =f ( y ) . Thus, V c f ( H ) . Since x was arbitrary, we see that f ( H ) is open. 0
1
9.3. Theorem. Let res~.) designate the category of Hausdorff spaces and their open continuous mappings (locally one-to-one continuous mappings, homeomorphisms onto, resp.). A category R such that Hiso c 53 c H, or H o
(Hl(1-
Hiso
c
Hiso,
c HgI - 1)
is never alg-universal. I n fact, there exists a commutative monoid with six elements, which cannot be fully embedded into R. Proof. Consider the mappings h, s of the set { 1,2,3,4,5,6} into itself given by 123456 123456 = (2 1 4 5 6 3)’ = (1 2 1 2 12) where the lower row indicates where the elements of the upper one are being sent to. One checks easily that :
(1)
h4 = 1 , s2
= s,
h2s = s and hs
=
sh
and that consequently the monoid M of the mappings of {l,2, ..., 6) into itself generated by h and s consists exactly of the following distinct elements:
1, h, h2, h’, s, hs .
235
$9. Negative results
Suppose that there is a full embedding of M into R. Denote by
6,F:H
+H
the morphisms of R representing h, s. Obviously, 5 is a homeomorphism onto. By 9.2,qH) is open and closed in H . Consequently also A = S(H) u hS(H) is open and closed. Since, for U E A , E2(a)= a (see (1)) and since fi2 non-void and we have a y E B such that
+ I,,
B
=
H\A
is
y , y y ) , h2(y) and h3(y) are distinct.
(2)
If z E A , we have h(z)E ES(H) LJ h2 F(H) = gF(H) u S(H) = A , if E(z)E A , we have z = E3(h(z)) E h3 F(H) LJ E4 S(H) = K q H ) u F(H) = A . Thus, E maps homeomorphically A onto A and B onto B. Define 1: H + H putting
h(x)
I(x)
=
l(X)
=x
for X E A for X E B .
Thus, 1 is a homeomorphism and hence a morphism in Si. By (2), 1 Since it is a homeomorphism,
+
1 *F,
+ h, E2, h3.
hos.
+
Since hs s, we have 1 1,. Thus, 1 corresponds to no element of M , which contradicts to the fullness of the embedding. EXERCISES
1. The result on open uniformly continuous mapping in 8.6 does not contradict to 9.1. Why? 2. In the categories described below, objects are precisely all Hausdorff spaces, morphisms are the following types of continuous mappings.
H1,Hlh H, Ho,l,h
locally one-to-one mappings local homeomorphisms open mappings open local homeomorphisms.
Denote by M the monoid (1, cr, cr2} with o2 = 03.Prove that M cannot be represented in any of the above categories. 3. Let R be a category defined as follows. The objects are all Hausdorff
236
Ch. VI, TOPOLOGY
spaces; f : H + H’ is a morphism of A iff f is an open continuous mapping and either f is a local homeomorphism, or, for any y E ~ ( H f) , ‘ ( y )is a dense-in-itself subset of H . Prove that ~
(a) R is really a category, (b) Ho,lhc A c H, (H, and Ho,lhas above), (c) the M from the previous exercise can be represented in R (Hint for (c): consider the space H = (x) u C1u(C1 x C2), where the summands are disjoint, open in H and C,, C 2 are two disjoint subcontinua of a Cook continuum).
5 10. Techniques for TI-spaces 10.1. In this section we start investigating the universality of categories of local homeomorphisms, locally one-to-one continuous mappings, and similar choices of morphisms. The reason why in $0 6-8 we obtained results on quasilocal homeomorphisms etc. only were the joints between the “arrow”-spaces. Using the techniques based on variants of the arrow construction we cannot prevent folding two or more joined arrow-spaces together. The typical situation is indicated in Fig. 6.5.
If the joint connecting the arrows to be folded is constituted by their common point, there is no way to save the property of local homeomorphism at the point but to make the point an open set. This is possible, as we saw in 5 1, but it leaves the point hopelessly unseparable from some others. There is, however, no necessity of joining the arrows by a common point. What we really need is just to make the ends stick together, which may also be done as follows: We add to every vertex one more auxiliary arrowspace and define the neighborhoods of the end-points attached to a given vertex in such a way that they always meet the corresponding auxiliary space. This is illustrated by Fig. 6.6.
$10. Techniques for T,-spaces
237
The typical neighborhood of the point a is indicated by the area A u c, the typical neighborhood of b by B u C. If the arrow-spaces are TI, the one obtained is such, too.
No stronger separation property is achieved no matter what arrowspaces are used, but no harm done: we already know by 9.3 that we cannot expect anything better anyway. In this section we are going to give a more proper description of a procedure based on the idea just sketched. This will be used in the following three paragraphs to prove several positive results on the categories of TI-spaces. 10.2. Let P be a Hausdorff space. A decomposing system D on P consists of the following data: (a) a closed discrete subset T of P, (b) a covering 9 of P by metric continua, and (c) a mapping A : T -+ 2 satisfying the following requirements: for D E 9,6(D\T) = B(D) = D n T =k 0 (6is the boundary), (B)ifD,D'E2, D + D ' , t h e n D n D ' c T , ( y ) forevery dET, d E I n t U ( D ] d E D E 9 ) , (6) for every d E T, d E A(d), and ( 8 ) the set g d = {D E 9 tl E D, D $; d(d)) is never void. (ct)
I
(Intuitively, T consists of the points which are going to be reduplicated to obtain satisfactory joints; A(d) is the auxiliary arrow mentioned above.)
10.3. Let D, D' be decomposing systems on P, P' respectively. A continuous mapping f : P + P' is said to be DD'-compatible if f ( T )= T', f(f'\T) = P'\T', for every d E T, f(d(d)) = d(f(d)), and for every d E T and D E g d , f(D) E 9f(d,.
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Ch. VI, TOPOLOGY
10.4. Let D be a decomposing system on P. The space
is defined as follows: The underlying space of PD is
(P\T) u
PD
u (%
dcT
x {d))
9
the neighborhoods of a point x E P\T are the sets containing a neighborhood of x in P, the neighborhoods of a point (D,d) are the sets containing (A\T) u (B\T) u {(D,d ) ) for a suitable neighborhood A of d in d(d) and a suitable neighborhood B of d in D. Define a mapping
6: P D + P
putting 6(x) = x
for x E P\T,
6(D,d ) = d .
10.5. Observations. (1) Since P is a TI-space, PD is a TI-space. (2) 6 is a continuous mapping. ( 3 ) If P is connected, then P D is also connected. (4) If d E 7; D E gd,{x,} converges LO d in P and x, E D\T for all n, then (D,d ) is the only limit of {x,} in PD. 10.6. Let D, D' be decomposing systems on P, P' respectively, f : P a DD'-compatible mapping. The induced mapping is defined by f(x) =f(x)
+
P'
f = fDD': P D + P'D' for X E P \ T ,
10.7. Proposition. (1) f 0 6 = 6'of, ( 2 ) f is con[inuotrs. Proof is straightforward.
f ( W= ) (f(D)>f(d)).
rn
10.8. Lemma. Let f : P + P' be DD'-compatible. Let for every d E T and D E 9 with d E D there be a U open in D, d E U , such that f maps U homeomorphically onto f ( U ) open in f (D). Then for every (D, d ) there is an open neighborhood in PD, which is mapped by f homeomorphically onto an open neighborhood of (f(D),f (d)) in P'D'. Proof. Take a neighborhood A of d in d ( d ) and B of d in D which are mapped homeomorphically onto f (A), f ( B ) open in f (d(d))= d(f (d)), f ( D ) , respectively. Then, (A\T) u (B\T) u {(D,d ) ) is desired neighrn borhood of ( D ,d ) in P D .
$1 1. Alg-universal categories of TI-spaces
239
10.9. Corollary. Let f satisfy the assumptions of 10.8, let, moreover, f be locally one-to-one (locally homeomorphic, open, respectively) on P\T. Then f is locally one-to-one (locally homeomorphic, open, resp.).
10.10. Proposition. Let g : PD -, P'D' be continuous mapping. Then there exists a mapping h: P --+ P' such that h is continuous on each D E 9 and 6' g = h 0 6. I f this h is continuous and DD'-compatible, g = E. Proof. For x E PT \ put h(x) = 6'g(x). Before finding the values in T, let us introduce some auxiliary notation: For D E 9 and a mapping a : A - ' ( ( D } )-+ 9 such that always a(d)E 9 d define a mapping 0
pa: D
+ PD
putting p,(x) = x for x E D\T, pa(d) = (a(d),d) for d E A - ' ( { D ] ) , pL,(d)= = (D,d ) otherwise. We see easily that pa is a homeomorphism onto pL,(D) and that 6 pa is the inclusion D -, P. Now, let d be in T Take an arbitrary a : A-l({A(d)}) 9 with a(c)E and put 0
h(d) = 6'9 P a @ ) . This is correct: if another fl is taken, 6'gpa and 6'gpp coincide on a dense subset of d(d), and hence, since P' is Hausdorff, they are equal. Moreover, we see that for thus defined h, h D = 6-'gp, for any a : A-'((P]) -, 9 with a ( t ) E 9 , and hence it is continuous on D. Let the h be DD'-compatible. Take a d E T and D E gd.Choose x , E DT \ converging to d in P. Then (x,] converges to ( D , d ) in PD so that {g(x,)} converges to g(D, d ) in P'D'. Obviously, (h(x,)} converges to h(d) in P'. Since h is supposed to be DD'-compatible, h(D)E g h ( d ) and h(x,) E h(D)\T'. Consequently {h(x,)} converges to (h(D),h(d))in P'D'. Since g(xn)= h(x,), we have, by 10.5.(4), g(D, d ) = (h(D),h(d)).
1
EXERCISES
1. Let 9 : P D + P'D' be a continuous mapping, let h: P + P' be a mapping such that 6' g = h 6 (see 10.4). Then h is not necessarily continuous. Construct an example. 0
0 11.
0
Alg-universal categories of TI-spaces
11.1. Conventions and notation: Denote by
G the full subcategory of Graph generated by the connected graphs without loops. By IV.1.11, G is alg-universal. In the construction in this paragraph,
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Ch. VI, TOPOLOGY
we will use the basic class JZ! consisting of the first three elements (i.e. ( H i ,a&a;, a;), i = 1,2,3) of the basic class from 6.6. I n the constructions with decomposing systems we preserve the notalioii of ? 10. 11.2. The functor
@:
is defined as follows
G-G,
@(X,R ) =
(x,a,
-
'p)
where 3 = X u {c,, d,} with c, =l= dx and {ex,d,} n X = 8, R = = R , u R , u R , with R , = {(dx,c,)}, R , = {(c, x) X E X } and R , = R , V(Y) = (Hi,(a:)) for r E Ri. For a morphism f : (X, R ) (X', R') put @(f) = g where g(cx) = cx., g(dx) = dx# and g(x) = f ( x ) otherwise.
I
-
11.3. Lemma. Qi i s f i l l embedding.
(z',a',
Proof. Obviously, Qi is one-to-one. Let g: (2, R, 'p) 9') be a morphism. Because of R,, g((cx, d,}) = {c,, d x t } ; because of R , , g(cx) = cx, and g ( X ) c X ' . Since g preserves R Z , its domain-range restriction f : X X ' preserves R. Thus, g = @ ( f ) . R -+
-
1 I .1.Tlie functor S:If ( X , R ) is an object of G, consider first the space
P
=
A(@(X,R))
Let us write ex instead of e(cx,x)and e* instead of e(dx,cx). Thus, we have mappings (actually, homeomorphisms into) e*: Ho-+ P , ex: H ' - P for X E X , er: H 2 + P for r E R . Put c = c(X, R) = e*(ay), d = d(X, R) = e*(a:) and define a decomposing system D = D ( X , R ) on A ( @ ( X ,R)) putting
T = {c, d } u {o(x) I x E X } , 9 = {e*(Ho)}u {e*(H') x E X } u (er(H2) Y ER } , d(o(x)) = e x ( H ' ) . d(c) = d(d) = e*(Ho),
I
I
Put F ( X , R ) = A ( @ ( X ,R))D. Let f : ( X , R ) -+ ( X ' , R') be a morphism. We see easily that then A(Qi(.f)) is D ( X , R)D(X',R')-compatible so that we can define S ( f ) : F ( X ,R )
by S ( f = ) Jq@(f)).
-
Y ( X ' , R')
24 1
512. Strong embeddings
11.5. Theorem. Let N be the category of the uncountable connected TI-spacesand their open local homeomorphisms. Let R be a category such that
N
c R c Top.
Then 53 is almost alg-universal. Proof. Obviously, Y ( X ,R ) is an object of $3. Further, we see easily that A? @ ( jis) an open local homeomorphism on A’@ ( X ,R)\T and that the assumption of 10.8 is satisfied. Thus, by 10.9, Y(f ) is an open local homeomorphism. Now, let g : F ( X , R ) -+ Y ( X ’ ,R’) be a non-constant mapping. By 10.10 we have an h: A @ ( X ,R ) -+ A @(X’, R’) such that 6’ g = h 6, continuous on the elements of 9.By 3.11, h is continuous on the whole A @ ( X ,R). It is non-constant, since otherwise also g would be a constant ( 4 @ ( X ,R)\T is dense in F ( X , R ) and Y ( X ’ ,R’) is T,). Thus, h = 4@( f ) for a suitable f : ( X , R )-+ (X’,R’), and by the second part of 10.10, 9= ). 0
0
$0-
EXERCISES
1. Let B = { ( H I ,ah, a t ) . (11’. a& a:)) be a fundamental class such that { H I , H’) satisfies 6.2. Let N be as in 11.5, Top,, be the category of all topological spaces and quasi-open continuous mappings. Prove that any category R such that N c R c Topqo is alg-universal, using only the fundamental class 9. (Hint: Consider the Y : G -+ G, given by
y ( X , R) = ( X u ( ( 0 ) x X ) , R i u Rz, ‘P) where R ,
9 12.
=
R, R , = {(x,(O,x))
I
EX),
(P(Y) =
(Hk,akg,a:)for rER,.)
Strong embeddings into categories of T,-spaces
By 9.1 we see that we cannot expect a strong embedding counterpart of 11.5. Still, there is a question as to whether there is a strong embedding of the category Graph into the category of T,-spaces and local homeomorphisms (not necessarily open). We will show that the answer is positive. The proof will be based on a combination of the methods from the paragraphs 7 and 10.
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Ch. VI, TOPOLOGY
12.1. We will work with the basic class %? consisting of the first five elements i = 1, 2, ...)5 (Hi, a:, a;, a;), of the 9 l from 6.6. Recall the definition of the functor 8 and the notation of 7.2. We define a functor 0,: G + G , putting o,(x,R ) = (21, R’, cp’) where 8’ = s u ( c x , d x ) (c,, d, are assumed distinct, the union disjoint),
R’ where u(X x
=R,
x)},
if f:( X , R )
=
R u R4 v R ,
u R2 u R 3 from 7.2, R4 = {(d,, e x ) > ,R , cp’(w) = (Hi, (af))
= ((ex,
x) I x E X u
for w E R i;
(X’, R’) is a morphism of G,
G(i)(5)
=
S(i)(t)
for
5€ 2
>
S,(f)(cx) = ex‘, O,(f) (dx) = dx,. 12.2. Lemma. 0,is afill embedding. Proof. Because of R4, a morphism g: O , ( X , R ) + O1(X’,R’) maps {ex,d,) onto (ex.,dx,),because of R , it sends ex to ,c, and maps O ( X , R) into 8(X’, R’). Now, it suffices to repeat the reasoning of 7.3. 12.3. Working with the space A!’,O,(X, R ) we will write more concisely, for x E X u ( X x X ) , ef instead of efxSx)(see 3.9), and e t instead of e:dx2cx). Again, we will use the convention of writing Hf instead of H: if q ( r ) = = (Hi, (uf)). Put d = d ( X , R ) = e,*(a;), c = c ( X ,R) = e,*(a:). On AscO,(X, R ) define a decomposing system as follows:
D
=
D ( X , R ) = ( T ( X ,R), qx,R), 4 , , R J
T = { c ( X ,R), d ( X , R)) u w(X u X x X ) , 9 = (e,*(H:)) u (e,*(H;) 1 x E X u ( X x X ) ) u (e:(H:) r E R , u R2 u R 3 ) , d(c) = d(d) = e*(H:), A(w(x))= ef(H;) for x E X u ( X x X ) .
I
243
$12. Strong embeddings
12.4. Lemma. Let U , V be the natural forgetful functors of G, Top, respectively. Then there is a functor F,: Set .+ Set such that F, 0 U g
VoA00O,. Proof. Recall the functor 9 from 7.5. There was an F such that F z V O2.Our statement follows easily from the observation that E
A,O,(X,
R ) = T ( X , R ) u e:(H,4) u
u
X€XU(X x
o
U
g
e:(H,5).
X)
12.5. Theorem. Let N be the category of all connected uncountable
TI-spaces and all quasi-open local homeomorp~isms.Let A be such that N c 53 c Top.
Then there is an almost strong embedding of Graph into A. Proof. Put B ( X , R) = A , O , ( X , R ) D ( X , R), B ( f ) = A s O I ( f ) (notation from 0 10). We see easily that the assumptions of 10.8 are satisfied and that A s O l ( f ) are locally homeomorphic (though not necessarily open) on the elements of 9 ( X , R). Thus, by 10.9, g ( f )is a local homeomorphism. It is quasi-open since it is obviously open on A , O ( Y, R)\T(X, R). The fact that 9 is almost full follows from 10.10 quite analogously as the corresponding fact in 11.5.Thus, it suffices to find afunctor G: Set .+ Set such that G 0 U g Vo 9. 4 We have vqx,R ) = V A , O , ( X , R ) u (JDi(X, R )
where Di: G
--f
Set are defined as follows:
&(X, R ) = 9
c
i= 1
x ( c )I
D,(X, R ) = g a d x {dj ,
(9
D l ( f ) (D? w(x)) = (-@%Ol(f) wf(x)) similarly the other Di(f). One verifies easily that D i Fi o U where F , ( X ) = X x X x 2, F 2 ( X ) = x x x x 3, F 3 ( X ) = x x ( 0 ) u ( x x x ) x {l), F4(X) = = X x (0) u ( X x X x 3) x { l}, F , ( f ) (x, y, i) = ( f ( x ) , f ( y )i), , similarly for the other Fi(f). (Let us show in detail e.g. that D, z F, 0 U : For z = (x, y ) E X x X put Z = ( z , z ) if z E R, Z = (z, z, 0) if Z E X x X\R. We have 9U(z) = = {e?,')(H:), e:('.Y)(Hf),z ( H 5 ) ); send ( e ~ 3 z ) ( H z ) ~to) , (x, y , 0), (eF.Y)(H:),Z ) to (x,Y , I), ( e W ) ,z ) to (x, Y , 2) .)
r
7
244
Ch. VI. TOPOLOGY
Consequently, by 12.4, where G ( X ) =
u A
i=O
VOBZGOU
F i ( X ) x {i}, G ( f )(<, i) = ( F i ( f )(t),i). (F, from 12.4.) W EXERCISES
1. Let N be as in 12.5, let Topqobe as in E 11.1. By means of a fundamental class with at least four elements, prove that for any category R such that N c A c Top,, there exists a strong embedding of Graph into R.
Q 13. The category of T,-spaces and their open continuous mappings is universal
Similarly as in 0 12 we will combine two methods used before, this time those of paragraphs 8 and 10. Unlike in section 8, we are going to prove only full, not strong, embeddability of S(P;) into our category (we cannot have more, see 9.1-f. also E 9.1). Thus, the counterpart of the functor from 9: 8 will be actually simpler. 13.1. We will work with the fundamental class 9 consisting of the i = 1, ..., 4
(Hi,a;, a ; ) , from 6.6. Otherwise, the notation from 13.2. The functor is defined as follows: where
r?'
=X
u( X
Vl: S ( P , f )-+ W G , ~ 'fl(X, R) = (gl, R', cp')
I
x { 0 ) )u P;(X) u n ( X )u { (x7Z , 0) (x, 2)E n ( X ) and 2 E R)
(the union is assumed disjoint), A
where
6 8 is preserved.
513. T,-spaces and open mappings
245
If f : ( X , R ) -+ (X’, R’) is a morphism of S(P,f), put
( f )(4 ^I. l(f) (x, 0) Vl
V l ( f(2) )
= f ( x )> =
( f ( x ) 0) ,
=
P , f ( f )(2)
2
3
q f (X’ ) z) and % ( f ) ( x , 270) = ( q f( x),Z),O). %(f) ( x ,z)
=
13.3. Lemma. Let g : V l ( X ,R ) -+ V l ( X ’ ,R’) be a morphism of wG,. Then g = V l ( f for ) a morphism f : ( X , R ) -+ (X’, R’) V f o r every Z E P;(X), g((z)u R’z)= { g ( Z ) } u R’Ig(z). Proof is an obvious modification of the proof of 8.5. a 13.4. Recall the construction 2 and the symbols 2,6 from $ 3 . We will work with the functor dV1.The mappings 2 V 1 ( f behave ) like open mappings at all the points except the 6 ( y ) with y E X u n ( X ) . To amend this we will use the method of $ 10 with the decomposing systems defined as follows: First, put for x E X , r = ((x,0),x), x ( ~ ,=~Z?(af) ) z ( X , R ) = 2 ( a i ) for z = ( X , Z ) E I I ( X ) , r = (x,z), L’ = F x , z ) , z ) ( ~ for2 z ) E P,+(x).
u
Put
.
X E Z
(Top,, resp.) be the category of all uncountable 13.5. Theorem. Let connected TI-spaces and at/ open continuous quasi-local homeomorphisms (of all topological spaces and their open continuous mappings). Then every category R such that
is universal,
c R c Top,
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Ch. VI, TOPOLOGY
Proof. Define a functor u2il : S(P;)
-+
A putting
@i(X,R ) = J v i ( X , R ) D ( X , R )
3
@ i ( f )=
Jvi(f).
One sees easily that a1is defined correctly, i.e. that every @l(X,R) is an uncountable connected TI-space and that every “ i l ( f ) is an open continuous quasi-local homeomorphism. Now, let 9 : u2il(X, R ) -+ u2il(X’,R’) be an open continuous mapping. By 10.10 there is an h : A?V,(X, R ) dt%’”,(X’, R’) such that 6’ o g = h 6 and such that every h D for D E 9 is continuous. Thus h is continuous on every 2 ( H i ) with i = 1,3,4, and since it is continuous on the L,, it is continuous on every 7 ( H 2 )as well. Thus h is continuous, obviously non-constant. Thus there is a k : .Irl(X, R ) -+ V,(X’, R’) such that h = A?(k). Since g is open and h coincides with g on neighborhoods of w ( 2 ) for Z E P;(X), we see easily that the assumption of 13.3 is satisfied for k so that k = V l ( f )h, = A?Yl(f). Thus, h is D ( X , R) D(X’,R’)-compatible and by the second part of 10.10, W g = T; = @,(f). -+
0
0 14.
I
-+
Rigid spaces and stiff classes of spaces
14.1. Given a class V of morphisms of a category A, an object a of R is said to be rigid with respect to V if there is no cp: a -+ a in V\{ la}.Thus, the expression “rigid from the introduction coincides with “rigid” with respect to all the non-constant continuous mappings. In this and the following sections we will use the expression rigid as an abbreviation for “rigid with respect to the non-constant continuous mappings” (which is the strongest rigidity one can have in a topological category). A class YJl of topological spaces is said to be stiff if for any P,, P2 E YJl and any continuous f : P, -+ P2 either f is constant or PI = Pz and f is the identity. Thus, P is rigid iff { P } is stiff. 14.2. Taking the
Hkfrom 6.6 we see that:
there exists a countable stiff set of metric continua. 14.3. Proposition. Let A be an almost alg-universal category of some
topological spaces and all their continuous mappings. Then for every m there exists a stiff set A c objA such that & !, .drac = m. I f R is almost universal, there is a stiff proper class A c objA. Proof. Consider a full embedding of a discrete category (see 1.3.5.1) A of the cardinality m (of a large discrete category A ) into A. Put A = = {F(a)I a E obj A } . W
$14. Rigid spaces and stiff classes
241
14.4. Corollary. For every cardinal m there exists a stiffset .A? of cornplete metric semicontinua such that card&?= m. Under the assumption (M)*) there exists a stiff proper class of complete metric semicontinua. Proof follows by 4.2 and 11.6.
14.5. Remarks. (1) In the next section we will see that there is a stiff proper class of paracompact spaces. A quite simple construction of a stiff proper class of paracompact spaces is also sketched in E 14.1. (2) In 0 16 we will see that there exist arbitrarily large stiff sets of compact Hausdorff spaces, and that, under the assumption (M), there is a stiff proper class of such spaces. 14.6. In the next section we will need a statement on powers of rigid spaces (Theorem 14.8 below). If A is a set, the A-th power of H is denoted by H A (its underlying set is the set of all mappings A + H , the product topology coincides with the topology of pointwise convergence-see, e.g., [Ke],). We denote by pa: H A -+ H the projections sending y to y(a). Further, for y E H A and b E A we introduce continuous qyb:
H +HA
defined by Pb q y b = id, pa 0 (Pyb = consty(,) otherwise. In the following paragraphs 14.7- 14.9, 0
H is a rigid Hausdorff space with more than one point. 14.7. Lemma. Let g : H A -+ H be continuous. Let, for a y E H A and b E A, g cpyb be the identity. Then g = P b . For z € H A put F, = { a € A .(a) =+ y(a)). Denote by Y,, the set 0
1
of all z € H A such that card(F,\{b)) 5 n. Since Y =
I
I
u Y, 4
n= 1
is dense in H A ,
it suffices to prove that g Y = pb I: This will be done by induction. Let g Y, = p b Y,,. Take a z E Y,,,, and choose a c €F,\{b). We will distinguish two cases: (a) z(b) $. y(c): Then, define x E H A by X(C) = y(c), and .(a) = .(a) for a =I= c. We have x E Y,, so that g(x) = x(b) = z(b). Now g cpxc is either the identity or a constant. Since, by the definition of x, cpx.(y(c)) = x, we have g(cp,,(y(c)) = g(x) = z(b) y(c). Thus, g o cpxc is a constant to z(b), so that dZ)= ~ ( ' P X C ( ~=( ~z(b) ) ) = Pb(Z).
I
1
0
+
*) see 11. $ 6
248
Ch. VI, TOPOLOGY
(B) z(b) = y(c): Since H is rigid, obviously none of its points is isolated. Thus, there is a net (hJ1converging to z(b) in H such that h, $I z(b) for all z. Define z, E H A putting z,(b) = h,, z,(a) = .(a) otherwise. The net (zJ converges to z and, by (a), g(z,) = pb(zI).Thus, g(z) = &,(Z). 1 14.8. Proposition. Let g : H A -+ H be continuous. Then either g = Pb for some b E A or g is constant. Proof. If, for some b and y, g 0 (Pby = id, g = Pb by 14.7. Thus, suppose that all the g o (Pby are constant, while g itself is not. We have some z, z' E H A such that g(z) =+ g(z'). Choose a neighborhood U of z' in H A such that g(z)$g(U). There is a Y E U such that y(a) i .(a) only in finitely many a E A , say a,, ..., a,. Define yi (i = 0, ..., n) putting yo = z, yi(aj) = y(aj) for j I i, yi(a) = .(a) otherwise. Thus, Y i = ~ p y , o+, ,(z(ai + 1))
9
Yi+ 1 =
~ y , a +,
Since g 0 qyIa, + , are constant, we obtain S(Z) = 9(YJ which is a contradiction.
=
,(y(ai+1)) .
= dYn) = 9(Y)
14.9. Proposition. For an h E H denote by c f the element of H A sending all the a E A to h. Let h, k be distinct, let g : H A + H B be a continuous mapping such that g(c;f)= c i and g(c:) = c f . Then there is a mapping f : B + A such that g(y) = y 0 f for d l y E H A . Proof. Since p:(g(cf)) = h i k = p:(g(ct)), p:g is not a constant. Thus, by 14.8 there is an a = f ( b ) such that p:g = p:. Hence, for f : B + A thus defined, g(y)(b)= y(a) = y(f(b)). EXERCISES
I
1. Let O d be the class of all ordinals, { G , a E Otd) be the class of topologized graphs from E 5.1. Prove that (for a suitable basic class) { P ( G g ) c1 E O d } is a stiff proper class of paracompact spaces. 2. Let C be a subcontinuum of the plane defined as follows (cf. [dG]): Let C, be a circle in the plane and let {ai,j)i,jbe a double sequence of distinct natural numbers >2. Let ( P ~ , be ~ } a countable dense subset of C1. We affix to each pltja closed chain C , , j of aIJ of links which are contained in the interior of C , ( P , , ~excepted) and mutally disjoint. Next, we take a countable dense subset on the union of all Cl,j such that each p 2 , j has a neighborhood in C, u u C , , j homeomorphic to an arc. Affix to
I
j
249
$1 5. Paracompact spaces
each p 2 , j a chain C 2 , jof uz,j of links contained in the interior of that link to which p 2 , j belongs and such that all the new chains are mutually disjoint. Proceed by induction. The diameter of the Ci,j converges to zero. Now, C is the closure (in the plane) of the union of the countable chains obtained this way. (See Fig. 6.7).
(a) Prove that C is rigid with respect to all homeomorphisms af C onto itself. (b) Prove that C is rigid even with respect to all quasi-local homeomorphisms. (c) Prove that C with suitable ao, u1 E C can be used for the arrow construction in the category of metric spaces and quasi-open quasi-local homeomorphisms (see E 2.1). , (d) Take a triple sequence { u i , j , k ) i , j , k of distinct numbers >2. Construct a countable fundamental class = { ( H k ,a!, u:) k = 1,2, ...) in accordance with the construction above. Prove that this fundamental class is suitable for an embedding of labeled graphs into the category of metric spaces and quasi-open quasi-local h0,meomorphisms. Modify all the proofs of results about embedding into this category by means of 9. 3. Prove that, in general, the continuum C is not rigid with respect to all continuous quasi-open mappings. (Hint: Choose suitable {ui,j}.)
I
0 15.
The category of paracompact spaces is almost universal
Thc basic choice of rnorphisms in topology is certainly the continuity. If we look at the results available so far, we see that in the cases with
250
Ch. VI, TOPOLOGY
all the continuous mappings for morphisms we have had only the alguniversality. The universality independent of a set-theoretical assumption was proved for categories with other choices of morphisms (e.g. open continuous mappings in 13.4). In this section we will fill this gap by proving that the category Par
of all paracompact spaces and all their continuous mappings is almost universal. 15.1. In the construction we will use the basic class consisting of the first four elements of the one from 6.6, i.e. of
i = 1,2,3,4.
(H',ab,a;,a;),
Let us denote by H the full subcategory of Top generated by spaces homeomorphic to (H4)Z v D where D is an arbitrary discrete space, v designates the usual topological sum, and (H4)Z has the meaning from 14.6 (the superscript 4 is an index, not a power, of course). 15.2, Lemma. For i = 1,2, 3, a mapping of Hh = H'\(u&
a;, ui} into
any object of H is a constant. Proof. Let f : H', -+ (H4)Z v D be continuous non-constant. Since Hb is connected, f(Hfo) c (H4)Z.Denote by g : H', -+ (H4)Z the range restriction off: Let p z : (H4)Z-+ H4 be the projections. By 5.10, we have p z g = const. Thus we can define a mapping x E (H4)Z by x(z) = p,(g(h)) (h arbitrary). Then, p z o g = p z const, for all z E 2 so that g = const,. 0
0
15.3. Define
a:, a: putting
U?(Z)
= a4 (i = 0,l) for all
E
(H4)2
z E Z , and denote by
A: 2 = (0,1> -+ H4 the mapping defined by A(i) = ai (i = 0, 1). For a set s c 2' put
s=
(AO$I$ES}.
15.4. The functor @: Define a functor @: S(P,)"P
---f
TGa
25 1
$15. Paracompact spaces
(for S(P2) see 1.5.8, 1.3.14.F; TGB from 5.1) putting @(Z,s) = (K7,R , 9)
where
x = (H4)Z u s(z, s), s(z, s) = ((H4)Z\S) x
(0) u { a t , a?) x (1) ,
z coincides with the product topology on (H4)Z, it is discrete on S(Z,s),
and (H4)Z and S(Z,s) are open and closed in z, R
with
R,
dr)
=
=
((Y2
Ro u R1 u R2
I
(Y, 0)) Y E (H4)Z\S)
I
R l = ((Y7 Y ) Y E 5; R2 = {((if, (‘I:. I ) ) I =
>
/ = 0.
?, ( ( I : -
11 ,
for
( H I . (cI:))
y)((aJZ,(a,Z,1))) = ( H ’
7
2))
for
r ER, uR , , =
0, 1 .
For a morphism f : ( Z ,j) + (Z’.s’) define @ ( f )= g: @(Z’,s’) -,@(Z,s), where 9($) =*of for $€(H4)2’,
($oh 0)
for $E(H4)z’\S and $ o f € (H4)z\S,
g($,O)
=
9(W)
=$of
otherwise,
g(a,”‘, 1)
= (a?, 1)
for
= 0,
I.
Checking that @((f)thus defined is a inoiphisiii of TGa is straightforward. 15.5. Proposition. di is a full embedding. Proof. Obviously @ is one-to-one. Let g: @ ( Z s‘) , + @(Z,s) be a morphism. Since it preserves the labels. we have g((H4)Z’c (H4)Z, g(a;‘) = a;, g(a,Z’, 1) = (a:, 1) for j = 0, 1 Since it is continuous, there is, by 14.9, a mapping f : z -+ z’such th,it y($) = II/ o f for all $ E ( H 4 r .One verifies easily that this f is ss’-compatible and that g = @ ( f ) . ,
15.6. Recall the construction of 9 from 5.2. Since every @(Z,s) is paracompact, by 5.6 9 @(Z,s) is paracompact too. Define
Q’ : S(P2)0p + Par putting Q’(Z, s) = 9@ ( Z ,s), Q + ( f ) = 9 d i ( f ) . Theorem. Q’ is a full embedding. Consequently, Par is almost universal.
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Ch. VI, TOPOLOGY
Proof. Q’ is obviously one-to-one. If g : 9@(Z’,s’) + B @(Z,s) is nonconstant, by 6.10 (applied quite analogously as in 4.2) g = P ( h ) for an h: @(2, s‘) + @(Z,s). By 15.5,h = @ ( f ) for an f : ( Z ,s) -,(Z’,s’).Obviously, the dual of a universal category is universal. Thus, the statement follows from 111.4.4. 15.7. In the remaining part of this section we will show that there exists, moreover, a full embedding of S(P2)opinto Par behaving like a strong one (except for the contravariance). This will be shown to have further consequences in Chapter VII. We use the notation of 3.4 (in particular, o and er). Further, we put
A(2, s) = o((H4)Z\S) x ( 0 ) ) . Observation. For a morphism f : ( Z ,s) -+ (Z’,s’),
Q’(f)
(Q’(z’, s’)\A(z’,
s’)) c Q + ( Z ,)\A(& ,
s).
15.8. Lemma. Let X be a paracompact space, let A, B be closed, B c Int A. If A\B is paracompact, X\B is paracompact. Proof. Let @ be an open covering of X\B. Choose an open G c X such that B c G c G c IntA. The 42 covers the paracompact X\G. Choose a locally finite open (in X\G) covering V of X\G, refining %. Put Vl = ( V n (X\G) V E V } . Since A\B is paracompact, there is an open locally finite covering V2+of A\B refining @. Put V2= = { V n IntA VE V2+). Then Vl u V2is a locally finite refinement of 42.
I
1
15.9. Proposition. Q ‘ ( Z , s)\A(Z, s) is paracompact. Proof. Apply 15.8 for B = A ( Z ,s), A = {e‘(y) r E Ro, o(y, a:) It f where (T is the metric of H’. Since A is metrizable, A\B is paracompact. g
1
15.10. Define a functor
Q: S(P2)op-+ Par
s) = Q’(z,s)\n(z, s), Q ( f )( Y ) = Q’(f) (Y). We have by Q(z, Theorem. Q is an almostfull embedding, and there exists an F : SetoP-+ Set where U , V are the natural forgetful functors. such that U Q = F Proof. Let g : Q ( 2 , s) + Q ( 2 , s’) be a non-constant continuous mapping. For an r E R, g o el,: H: -+ Q(Z’,s’) is either a constant or e.d for an r’ E R‘. Consequently, it may be continuously extended to H’.Thus, the whole g can be continuously extended to g t : Q’(Z, s) .+ Q+(Z’,s’). 0
Thus, g
=
0
Q(f ) by 15.6.
253
$16. Compact Hausdorff spaces
The description of the F may be left to the reader as an easy exercise.
8 16.
Compact Hausdorff spaces
16.1. In this paragraph we will show that the category of compact Hausdorff spaces and their continuous quasi-open mappings is alg-universal, and that the category of compact Hausdorff spaces and all their continuous mappings is dual to an almost alg-universal category. While the former needs just an easy reasoning on compactifications, the latter is more complicated.
16.2. The Cech-Stone compactification of a topological space P will be denoted, as usual, by P P , the extension of a continuous f: P -+ P' by Pf: In this section, the spaces to be compactified will always be metric ones and hence it is convenient to regard /3 as a functor
p:
M
--f
Cornp,
where M (Comp, resp.) is the category of metric (compact Hausdorff, resp.) spaces and all their continuous mappings. 16.3. First, we recall two well-known facts on compactifications of metric spaces : Lemma. Let P be a metric space. Then (1) no point of PP\P is a limit of a sequence ofpoints of P ; (2) ifG is open in P and ifits closure in P is compact, then G is open in fiP. Proof. (1) Let lim xn = x E pP\P, x, E P . Then, the set (x, II = 1,2, ...} n-m
I
+
is a discrete closed subset of P. We can assume that x, =+ x, for n m. By Tietze-Urysohn theorem we have a continuous mapping f : P Z ( I designates the unit interval) sending x2, to 0 and x ~ , +to~ 1. This f; however, admits no extension over PP, because we would have to have f ( x ) = lini f ( x n ) . -+
n-
L
(2) Take an x E G and a continuous f:P -+ I such that f ( x ) = 1 and f(P\G) = (0). Let g : p P -+ I be the extension off. Since the closure G is compact, it is closed in P P . Consequently, g(PP\P) = 0, so that H = = ( y E B P g(y) > 0) is a neighborhood of x in B P and we have H c G. Thus, G is open, being a neighborhood of all its points.
1
16.4. A metric space P is said to be quasi-locally compact if there is a nowhere dense T c P such that P\T is locally compact. Proposition. If P is quasi-locally compact, PP\P is nowhere dense in P P .
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Ch. VI, TOPOLOGY
___ Proof. By 16.3.(2), PP‘ P c (/;P\P) u T. 16.5. Theorem. Let P, P‘ be quasi-locally compact metric spaces. Then g : PP + PP‘ is a quasi-open continuous mapping $ g = Bf for a quasi-open continuous f : P + P‘. Proof. Denote by 7; T‘ the nowhere dense subsets of P, P‘, respectively, from the definition in 16.4. Let f : P -+ P’ be quasi-open, let U be a non-void open subset BP. By 16.4, U n P is non-void, hence f ( U n P ) contains a non-void V open in P’. If V = V’ n P’ for a V’ open in PP’, we have V 3 W = V’\(PP’\P‘); W is non-void by 16.4, and g ( U ) 3 W Now let g: BP + P P be a quasi-open mapping. Take an x E P\T and suppose that g(x)$P’. By 16.3.(2) we have U , c P such that X E U,, U , are open in PP, and diam U , < l/n. Since P‘ is dense in B P and since g is quasi-open, g( U , n P‘) is non-void. Choose y, E g( U,) n P‘ and x, E U , such that y, = g(x,). We have x = limx, and hence g(x) = limy, in a contradiction with 16.3.(1). Thus,
g(P\T)
c p‘ .
Since the points of T can be expressed as limits of sequences from P\T, we obtain, using 16.3.(1)again, g(P) c P’ . Thus, g = b f where f:P + P’ is the domain-range restriction of g. Finally, let U c P be non-void open in P. Take a U’ open in P P such that U = U ’ n P . We have a non-void open V‘ c g(U’). By 16.4, V = V’\(BP\P) is non-void. Obviously, V c f ( U ) . 16.6. Theorem. Let Comp,, be the category of all compact Hausdorff spaces and their quasi-open continuous mappings, Compi,,,,, the category of all connected compact Hausdorff spaces and their quasi-open quasi-local homeomorphisms. Let R be such that ComP;,,,,,
= 53 = ComPq,.
Then R is alg-universal, Proof. Consider the full embedding
A?: G&+ M,, from 0 6. Every A ( X , R , ‘p) is obviously quasi-locally compact. Thus, by 14.5, observing that iff is a quasi-local homeomorphism Pf is one as well, we see that A? gives a full embedding of G%into 9. 0
$16. Compact Hausdorff spaces
255
16.7. The crucial fact in the previous theorem was that a quasi-open continuous mapping g: fl A ( X , R, 'p) + j? A ( X ' , R', 'p') is always equal to Pf for a suitable f : A ( X , R, 'p) + A ( X ' , R', 9'). A substantially stronger fact that every non-constant g is a Pf would imply that even every R with Comp;,,,,,
c A c Comp
is almost alg-universal. In [T,], the desired proof is given only for the case of ( X ' , R', cp') with card X' smaller than the first measurable cardinal. Thus, the mentioned result follows only under the assumption of non-existence of measurable cardinals. The remaining part of this section is devoted to proving a more satisfactory fact, namely that, regardless of set-theoretical assumptions (cf. [T, 1]), Comp is dual to an almost alg-universal category. 16.8. Let R be a full subcategory of S ( P - ) (for S ( P - ) see 1.5.8,1.3.14.(F)) generated by all the ( X , r ) such that (*) a subset Y of X is in r iff all the finite subsets of Yare in r. Lemma. A is alg-universal. Proof. Recall that UndGraph, is the category of all connected undirected graphs without loops and all their compatible mappings. By IV.3.3, UndGraph, is alg-universal.Define a functor @: UndGraph, + 52 by putting @ ( X , R ) = ( X , r ) , where r = { Y c X I { x , y } c Y * { x , y } $ R } , @(f) = f for morphisms. Since @ is a full embedding, R is alg-universal. 16.9. Let H be a TI-space, let h,hl E H be distinct. If X is a set, denote by H X the space of all mappings from X into H with the usual product topology. If Y c X , define h y E H X . putting h,(y) = hl whenever y E I: h,(y) = ho otherwise. If Y c expX, put t = {h, Y Er } . Lemma. I f ( X , r) is an object of R, t is a closed subset of H X . Let f Proof. The closure of P in H X is necessarily a scbset of {h,, be a point of {h,,, h , } X which is not in P. Put Y = {y E X I f ( y ) = h , ) so that Y $ r. By (*) there exists a finite subset F of Y such that no subset of^ containing F is in r. Consequently, = {g E {h,, h l } x I g(y) = hl for all y E F } is a neighborhood off in {h,,, h,}' which does not intersect F.
I
256
Ch. VI, TOPOLOGY
16.10. Construction. Let H , A, B, C be four distinct continua belonging to the system %? from 6.2. Let h,, h l , a, b be four distinct points of H . If X is a set, denote by a, and bx the points of H X such that a,(.) = a, b,(x) = b for all x E X . Choose two distinct points in each of the other continua, say, ao,al in A, bo,b, in B and co,cl in C . Construct a functor Y : Sop 3 Comp
as follows: If ( X , r) is an object of 53, then Y ( X ,r) is the space obtained from H X v (7 x C) v A v B (where f is the set from 16.9 and v designates the topological sum of spaces) by the identifications we will list. We identify a, with a,, bo with b,, (hy, c), with h,, and a , with b , and all the (hy, cl) where Y runs through r. By 16.9, i. is a closed subset of the compact H X . Hence, H X v (7 x C ) v A v B is a compact Hausdorff space. Consequently, its quotient space Y ( X ,r) is also compact. Obviously, it is a Hausdorff space. To simplify the notation, let us suppose that A , B , 7 x C , H X are subsets of Y ( X ,r), and a , = b , = (hy, cl), (h,, c,) = hy etc. If f : ( X , r ) -+ (X’, r’) is a morphism of S , define Y ( f ) = g : Y(X’,r’) -+
qx,
I)
by putting g(a) = a of for all UEH,’, g(z) = z for all Z E Au B, g(hy, z) = (h, of, z ) for all (hy, z) E 7 x C (note that if h, is in 7 and f is a morphism of 53, h, of is in i).On sees easily that Y is a faithful functor. We have to prove that it is almost full. 16.11. Lemma. Let ( X ,r) be an object of R, let L be one of H,A, B, C. Let g : L -+ Y ( X ,r ) be a non-constant continuous mapping. Then either L E ( A , B } and g(y) = y for all y E L, or L = C and there is exactly one Y E r such that g(y) = (h,, y ) for all y E L, or L = H and g ( L ) c HX. Proof. We restrict ourselves to the case of L = C, the others being either analogous or simpler. (a) Put G = g-’(A\(a,, all).If G = C , g is constant by 6.2. Thus, G =# C. (b) Suppose that G =+ 0 and denote by B(G) the boundary of G. Choose a z E G and, for each closed subset F of G containing z, denote by K F the component of F containing z. By 6.7, KF contains a point of the boundary of F. Hence, K = U K F intersects any open neighborhood of B(G) so that F
the closure of K intersects B(G). Let y be a point of this intersection. Since every K , is a subcontinuum of G, g is constant on G by 6.2. Since
$16. Compact Hausdorff spaces
257
K F 1u KF, c K,, u F 1 , g is constant on the whole of K . Consequently, g is constant on the closure of K , and hence g(y)E A\(ao, a,). We conclude that G = 8, i.e. g(C) does not intersect A\{ao, al}. (c) We can prove analogously that g(C) intersects neither B\(b,, b,} nor HX\((ao, bo) u (i x ( ~ 0 ) ) ) . (d) Consequently, g(C) c {ao,b,} u i x C. Since g(C) is connected and has more than one point, g(C) c 7 x C. Define 71: 7 x C -+ C by n(hy,y)= y, and put h = 71 09. Since r is totally disconnected and g is non-constant, h is also non-constant. By 6.2, h(z) = z for all z E C. Thus, g - l ( ( h y , c l ) )= cl. Let g(C) intersect ( h y ) x (C\{cl}) for at least two distinct Y’s. Then C\(cl} is not connected. (e) Now, we show that C\(cl} is connected. Let us suppose the contrary, is. that C\{cl) = P u Q with P and Q non-empty, open and disjoint. Put I(z) = c1 for all Z E P , l(z) = z otherwise. Thus, a non-constant nonidentical continuous mapping is defined in contradiction with 6.2. Hence, there is exactly one Y such that g(C) c (hy} x C. Now, use 6.2 again. W 16.12. Lemma. Let ( X , r ) be an object of 52, X’ a set, g : H X ’ -+ Y ( X ,r) a non-constant continuous mapping. Then g ( H X ’ )c H X . Proof. Suppose that g(z)EY(X,r)\Hx for a Z E H ~ ’For . an X E X ’ define cpx: H -+ HX’ by putting (cpx(y))(x‘) = z(x’), whenever x’E‘X’\(x}, and (cpx(y))(x) = y. By 16.11, g cpx: H -+ Y ( X ,r) has to be constant. Proceeding by induction (as in 14.7) we obtain that g(v) = g(z) whenever u E HX’, and v(x) = z ( x ) with the exception of a finite number of x’s only. Since these v’s form a dense subset of HX’, g is constant on HX‘. This is W a contradiction. Q
16.13. Proposition. Y is almost firll. Proof. Let ( X , r ) , ( X ’ , r ’ ) be objects of R, let g: Y(X’,r’) -+ Y(X,r)be a non-constant continuous mapping. Suppose that g is constant on either A or B or HX’or ( h y ) x C for some YE^'. Then, by 16.11 and 16.12, g has to be constant in contradiction with the assumption. Thus, by 16.11 and 16.12 again g(HX’)c H X and g(y) = y for all y E A u B. Particularly, g(ux ) = ax and g(bx.) = bx. Consequently, by 14.9, there exists a mapping f: X X‘ such that g(z) = z o f for all z E HX’.Then g(hy, c l ) = (hy of, cl), 1.e. f : ( X , r ) (X’, r’) is a morphism of R. We see that g = Y(f). W -+
-+
16.14. Theorem. CQmp is dual to an almost alg-universal category. Proof follows immediately from 16.8 - 16.13.
W
258
Ch. VI, TOPOLOGY EXERCISES
1. There is no strong embedding of Graph or GraphoPinto Comp. 2. Iff is a morphism of R, then Y ( f ) (notation from 16.8- 16.13) need not be quasi-open. Characterize the f with quasi-open Y ( f ) .
3. Prove that a large discrete category (see 1.3.3 and 1.3.6.1)can be fully embedded into any category R such that ComP;,,,,,
Ji
= COmPqo
(for the notation, see 16.6). (Hint: For any ordinal number a, put G, = (X,, R,, rp,), where X , = a + 1 (the set of all ordinals b I a), R, = ((b, B 1) /?< a } , rp,((O, I)) = = (H', (a:}), rp,(b, p 1) = ( H I , {a!)) for > a, where {(ITk, (a!}) k = 0, I} is a fundamental class. Take a compact Hausdorff topology qa on the underlying set of 2 ( G m )such that qa coincides with the order-topology of o(X,) and with the topology of d ( G , ) on each er(H'), r E RE.)
+
0
+
I
I
17. Some negative results
17.1. Although we have alg-universal categories of metric spaces and of compact spaces, no category of compact metric spaces including all their homeomorphisms is alg-universal. The reason is simple : the cardinality of a compact metric space does not exceed 2'O so that they are not sufficiently many, e.g. for representing small discrete categories of large enough cardinalities. More generally, we have the obvious Observation. If there is a set A c objR such that every object of R is isomorphic to an element of A , then 52 is not alg-universal. This observation is trivial, but proving its premises in a concrete case is not always so. In this paragraph we will present a generalization of a result by Archangelskii according to which the Observation can be applied to the categories of m-compact spaces with bounded characters.
17.2. Later in this paragraph we will also use a proposition slightly stronger than Observation 17.1, namely Proposition. Let R be a subcategory of 2 such that (1) every isomorphism rp: a + b of 2 with a, b E objR is in R, (2) there is a set A c obj2 such that every object of R is a coproduct (in 2)of some elements of A . Then 53 is not alg-universal (not even almost 2cardA such alg-universal). I n fact, there is a set B c objR with cardB I ,
259
$17. Some negative results
that if b E objsi is not isomorphic to an object of B, the endomorphism 1 such that B' = 1. monoid of b contains a Proof. Let B be the set of the coproducts a = ai such that a E obj R,
+
v
+
i6J
ai E A and ai aj for i =+ j . If b E objsi is not isomorphic to an element of B, we have a coproduct ( v i : ai --* b)iEJwith j , k E J such that aj = ak and j =I= k. Define B: b -+ b by Bvj = vk, B v k = v j , 6vi = vi otherwise. 17.3. A category is said to be topological if its objects are some topological spaces and if every homeomorphism between them is morphism. Corollary. Let si be a topological category such that each of its objects is a disjoint union of open closed subsets each of which has the cardinality less than m. Then 53 is not alg-universal (and, again, not even almost alg-universal). 17.4. The weight (= total character) w(p) of a topological space P is the least cardinality of a base of P. For x E P, the local character x(x) = X&) is defined as the least cardinality of a base for the neighborhood system of x. We put, as usual,
x(p) = SUP {X(X) I x E p } .
17.5. As usual, a cardinal number is understood as the least ordinal of a given cardinality, hence it is the set of all smaller ordinals. In this sense (recall 14.6) the symbol P" is used. Lemma. For an infinite cardinal number m, ,(I")
I m.
Proof. The interval I itself has a countable base B=
a. Put
{ )( UaI U, E 42 for finitely many a, U , = I otherwise} . a<m
Obviously, 9 l is a base for I" and cardB
=
m.
17.6. Lemma. Let P,be a Hausdorff space Ivith x(P) 5 m. I f cardA I 2 " for A c P, then card A 5 2" ( A is the closure of A).
260
Ch. VI, TOPOLOGY
Proof. For an x E A find a net N , = ( x ~ of) elements ~ ~ ~ of A such that cardB Im and N , converges to x. Since P is Hausdorff, N , $: N , for x y.
*
17.7. Let m be an infinite cardinal. A topological space is said to be m-compact if each of its open coverings %! has a subcovering V with cardV < m. Using De Morgan's formulas one sees immediately that a space is m-compact iff for every system 9 of closed sets with the m-intersectionproperty (i.e., such that for every 9' c 9 with c a r d 9 ' < m, 0) one has 4= 0. Remark. The notion of KO-compactcoincides with that of compact, the notion of Kl-compact with that of Lindelof. The KO-intersection property is usually called finite intersection property.
n9
09' +
17.8. Lemma. Let P be a Hausdorff m-compact space with x(P) I m, let Q be a Hausdorff space with w(Q) 5 2". For every continuous mapping f of P onto Q there is a closed F c P such that cardF 2" and f F is still onto. Proof. Choose a base B of Q with cardB I 2". Put
I
Y =[ n B j 1 B j E B and cardJ I m>\(0}. We have
j EJ
card Y I (2")" = 2".
For every S E Y choose an ts E J - '(S), put A By 17.5, cardF 5 2". Take an arbitrary y E Q. Put Y(y) = (SE
qY
ES}
,
=
I
{ t, S E Y ) and F
=
2.
I
d ( y ) = {F n f -yS) S E Y ( Y ) ) .
Obviously, the intersection of a subsystem 9' c Y(y) with cardY' I m is in Y ( y ) . Take such an Y' and put T = We have
09'.
so that Thus, d ( y ) has the m-intersection property and since P is m-compact, n d ( y ) 0. Since Q is a Hausdorff space, ( y } = s ~ Y ( y ) }Thus, .
n(q
+
Fnf-'(y)= Fnn(f-'(S)ISEY(y)} =
Fnn(f-'(S)ISE.Y(y)}
nd(y)+ 0 . 2
=
617. Some negative results
26 1
17.9. Convention. The least cardinal greater than m will be denoted by
m+. Lemma. Let P be an m-compact space with x(P) I m. Let there be given, for every ordinal A < m', a closed FA c P, a T,-space Qn and a continuous fn:P -+ Qn such that (a) for x < I , F, c Fn and fn-'(fn(F,)) = F,,
(D) f n ( F ~=) Qn, ( Y ) iff&) = fn(Y) and x < A
Then P =
u
A<m+
FA.
Proof. Put F
=
u
d<m+
then f ( x ) = f ( Y ) .
FA. Since x(P)
m, F is closed by (a). For X E P ,
G,(x) = F nfn-'(fn(x)). Evidently Gn(x) is closed, by (p) it is non-void and by (y) we have G,(x) c G,(x) for x < I .
I
Thus, {G,(x) A < m + } has the m-intersection property and, hence, since P is m-compact, there is a Y E Gn(x). Thus, we have y ~ f ~ - ' ( f ~ ( x ) )
n
l<m+
for all A < m and hence also x ~ f ;' ( f n ( y ) )for all I < m +.Since y E F , we have y E Fn for some A < m'. Hence, x ~ f f i ' ~ (l(Fn)) f ~ += FA c F. Since x was arbitrary, P = F. +
17.10. Lemma. Let P be a completely regular m-compact T,-space with x(P) < m. Then there are FA and f n : P -+ Qn ( A < m + ) satisfying (a), (fi) and (7)from 17.9 and such that Qn c 12mand cardF, 2". Proof. For a collection (gb: P -+ Qb)b.B of continuous mappings denote by [ g b ] b ~the ~ mapping 9: p X Q b defined by g(x) = ( g b ( X ) ) b s B (for +
B
finite B we will write [ g l , ..., gn] instead of [giIi=' ,..., ,). For every x E P choose a base for the neighborhood system %x such that card@, 5 m. For every x E P and U E @x take a continuous h': P -+I such that hU(x)= 1 and hU(P\U) = (0). Now, we will construct the F,,,f, and Qn by transfinite induction. Let g 1 be an arbitrary continuous mapping P -+ 12m.Put Q1 = g,(P), denote by f i : P -+ Q1 the range restriction of g l , and choose an F , by 17.8 (see also 17.5). Let us already have the F,, Q,,f, for x < A. Put
262
Ch. VI, TOPOLOGY
and let f n : P Qn be the range restriction of gn. Again, find an FAby means of 17.8 (if cardF, I 2" for x < A, card"Y-I 2". m = 2" so that Qnc 1'"). TakeanxEF, ( x < A ) a n d a y E P w i t h x - + y . W e h a v e a V ~ Y s u c h that h"(x) = 1 and h"(y) = 0. Then g,(x) -+ gn(y)and hence y $ f '(fn(x)). Thus, fn-'(fn(F,))= F, (and, moreover, fn is- one-to-one on Since f,(F,) = Q A = f,(P), obviously F, c Fk Thus, the condition (a)is satisfied. The (p) is obvious from the construction. If gn(x)= gl(y), we have in particular f,(x) =f,(y) for every x < ?, (recall the definition of g n ) so that also (y) is satisfied.
h,).
17.11. By 17.9 and 17.10 we immediately obtain Proposition. Let m be an infinite cardinal number. I f P is a completely regular m-compact T,-space with x(P) 5 m, then cardP 2". 17.12. By 17.11 and 17.1 we obtain Theorem. Let m, n be cardinal numbers. Let K be the category of all completely regular m-compact T, -spaces with local characters less than n, and all their continuous mappings. Let K C K be an arbitrary subcategory. Then K is not alg-universal (not even almost H alg-universal). 17.13. We recall that an x E P is called accumulation point of a set M if for every neighborhood U of x, (U\{x}) n M is non-void. Lemma. In an m-compact space P, every set M with cardM 2 m has an accumulation point. Proof. Suppose the contrary. For every x E P choose an open neighborhood U(x) such that U(x)n M c {x}. Let {U(x) x E N } be a subcovering of {U(x) x E P } such that cardN < m. We have
I
I
cardM
=
card((
u
xeN
U(x)) n M)
=
card
u
EN
(U(x) n M ) I cardN < m .
17.14. Lemma. Let m be an infinite regular cardinal number. Let there exist an open covering %! of a space P such that card@ < m and that for is an m-compact space. Then P is m-compact. each U E @, the closure Proof. For an open covering "Y- of P choose subsystems VU(U E a) such that UVu 3 0 and cardVU <: m. Put 9 = U("Y-" U E @>. Then 9 c "Y-, c a r d 9 < m and U Y = P. H
I
263
$17. Some negative results
17.15. Let %?! be an open covering of a space P. A subset H c P is said
to be a %?!-componentif H
u %
=
n=O
H , where H ,
E%
and H , ,
is the union
of all the U E % intersecting H,. Obviously, the %-components form a disjoint open covering of P . 17.16. Let %?! be a covering of a space P , U E 42. The star of U , denoted
Y t ,u is the set of all VE%?!intersecting U . Lemma. Let m be an infinite regular cardinal number. Let % be an open covering of a TI-space such that, for every U E 92,card Y / , U < m and that, for every U E % , the closure 0 is an m-compact space. Then every %-component H is m-compact. Proof. H = for a %’ c % with card%?!‘ < m. Thus, the statement follows by 17.14.
u%‘
17.17. A covering % of a space P is said to be point-finite if every x E P is contained only in a finite number of members of 42, locally finite if every x E X has a neighborhood intersecting only finitely many members of @. It is said to be irreducible if for every U E %, U\u{ V E% V 9 U > 4= 8. Lemma. A point-finite covering contains an irreducible subcovering. Proof. Using the axiom of choice we may assume that % = { U, CI < fi} where B is an ordinal. Put Vo = Uo if U\, Ua 9 8, otherwise Vo = 8.
1
u
If
V, are defined for all
a> 0
y < a, put V, = Ua if Ua\(
I
u V, u u
y
a
U,) 9 8,
otherwise V , = $3. Let V be the set of all the non-void V,. Take an x E P. Since 02 is point-finite, there exists the largest a such that x E U,. If x # V,, necessarily X E V, for some y < a. Thus ..Y- is a covering. Obviously it is irreducible. 17.18. Lemma. Let %?! be an irreducihle open covering of a T,-space P such that for every U E 62 the set U9’Pf,U is contained in an m-compact subspace. Then card .(It,U < in for ecrry U E %?!. Proof. Suppose that there is a Vo E % and a V c % such that card Y -= nr and Von V + 8 for all V E V .Choose X“E V\u{ U E % U 9 V } and put M = { x V V EV ) .We have card M = m and M c 9’tq,V0 and hence M is contained in an m-compact space. By 17.13, M has an accumulation point z. This is a contradiction, since then a V, E % containing z has to contain more than one (actually, infinitely many) of the x p
I
I
264
Ch. VI, TOPOLOGY
17.19. A space is said to be locally m-compact if each of its points has an m-compact neighborhood. Proposition. Let nt be a regular cardinal number. Let P be a paracompact locally m-compact space. Then P = u9 where 9 is a disjoint system of open m-compact subsets of P. Proof. For an x E P choose an open neighborhood U ( x ) such that U ( x ) is m-compact. Since P is paracompact, there exists a locally finite open covering % of P which is a star-refinement of ( U ( x )I x E P } , i.e. such that for every U E % there is an X E P such that uYC,U c a(.). By 17.17 there is an irreducible subcovering V of a. By 17.18, cardYt,V< m for every VE Y . By 17.16, the Y-components of P are m-compact. Thus, we can take for 9 the set of all Y-components. 17.20. Theorem. Let m, n be infinite cardinal numbers. Let L be the category of all paracompact locally m-compact spaces with local characters less than n, and all their continuous mappings. Let K C L
be a topological category. Then K is not alg-universal (not even almost alg-universal). Proof. If U c P is open and x E U then obviously xu(.) 5 xp(x). Thus, the statement follows by 17.19, 17.11 and 17.3. 17.21. Remark. The assumption that K is topological (in the sense of 17.3) is essential. Recall that by 4.3 the category of metric locally compact spaces with uniformly continuous open mappings is alg-universal.
0 18.
Bibliographical remarks
The first result on representations by means of topologically defined mappings we know about is de Groot’s representation of groups ([dG]) mentioned in the Introduction. A topological representation of semigroups appears in [HP,], where, essentially, the construction we present in 9 1 is used. (A method of topological interpretation of a combinatorial fact was also used later in [H,], where the graphs with loops are shown to form an almost universal category, and interpreted as closure spaces-see IV 4 7, and 9 11 in Chapter IV.) The first results concerning representations of semigroups and categories by means of topological spaces of a more satisfactory separation properties seem to be those of [BHP] and [dM] (in essence, a.o., a certain category of metric spaces with quasi-local homeo-
518. Bibliographical remarks
265
morphisms is proved to be alg-universal there, see also E VI.7.1.). The existence of Cook’s continuum [C,] led to a rapid development in the field. Thus, one gets, in [T7], the almost alg-universal embedding into the category of metric spaces and all continuous mappings, and the almost universality of the compact Hausdorff spaces and continuous mapping (under a set-theoretic assumption) in [T7], and the almost-universality of the category of paracompact spaces in [Ko2]. As for the results presented in this ~hapter~4.2 appeared, essentially, in [T7], 4.3 was announced in [TR,] and proved in [TR,]. The results of $ 7 were announced in [T,] but the full proof is given here first. Also the results of 4 8 have not appeared before. The negative results of 0 9 are partly contained in [dM] (for the method used see E 9.2), but 9.3 is stronger. The results of 9 11 were announced in [T,], but the proofs are first presented here. The universality of the category of TI-spaces and open continuous mappings has not appeared before. The result on the powers of rigid spaces from 14.8 was announced in [Hr,] quoting [Hr,]. The essence of 15.6 appeared in [KO,], 15.10 has not been published before. The results of 4 16 appeared in [T7,11]. The negative results in $ 17 are a suitable interpretation of a known topological fact. The result on the bounded cardinality of compact Hausdorff spaces with bounded characters from [Ar] is generalized here for higher cardinals (using a simplified Ponomarev’s version of the proof presented in [Pv]). There are several related results which could not be, owing to lack of space mainly, presented here. Let us mention e.g. that for some topological categories K currently used (M, M,, comp etc.) the full characterization of the schemes k for which the category of presheaves with values in K is alg-universal is known ([TR,], [TR,]). In [K4], the category of locally compact semigroups and continuous homomorphisms, and the categories of locally compact algebras of non-trivial types, are shown to be universal. For any T,-space the category of the regular spaces without non-constant continuous mappings into Y and all. their continuous mappings is almost universal ([TI,]). Also, the results of [Ko3] and [T,] are closely related with the topic.
Chapter VII STRONG EMBEDDINGS AND STRONGLY ALGEBRAIC CATEGORIES Roughly speaking, if a concrete category is universal with respect to full embeddings, then it is so with respect to extensions, as well (see 1.8.3). Let us now consider strong embeddings. Most everyday-life concrete categories have the following property: each set carries only a set of objects. This property is obviously inherited by strongly embedded categories. Thus, with reasonable categories we cannot expect a strong embedding universality covering really all concrete categories. Many of the universal categories we spoke about, however, have at least the property that all concrete categories satisfying some, not very restrictive, condition are strongly embeddable into them. This is discussed in the first three sections of this chapter. The others are concerned with the strong embeddability into categories of algebras which, on the contrary, is a very restrictive condition.
5 1.
Strong embeddings of categories S ( F ) into S(P+)and Graph
In this section we will prove that every S ( F ) (see 1.5.8) is strongly embeddable into S ( P + ) . In the next section this will be shown to imply strong embeddability of a fairly wide class of concrete categories. 1.1. Lemma. S ( F ) i s realizable in S(P- 0 F). Proof. For ( X , Y) E obj S ( F ) define @ ( X ,r) = ( X , i )E obj S ( P - 0 F ) by putting i: = ( M c F ( X ) M n r = 0}
I
Obviously, for a g: F ( X ) + F ( X ) we have g(r') c r
iff P - ( g ) ( f ) c i' .
267
$1. S ( F ) into S ( P + )and Graph
Thus F(f)(r’) c r iff (P- F ) ( f )( i )c P’ induces a realization. 0
SO
that the correspondence @
rn
1.2. Lemma. Every S(F) is realizable in an S(G) with afaithful G. Proof. Define G by G ( X ) = F ( X ) x (0) u X x {l}, G(f)(u, 0) = = (F(f)(u),0), G ( f )(x,1) = (f(x),1). G is faithful and we have the obvious monotransformation F -+ G sending u to (u, 0). Thus, the statement folrn lows by 11.7.4.
1.3. Theorem. Every S(F) can be strongly embedded into S(P+).
Proof. By 1.1 and 1.2 we can assume F to be covariant faithful, by 111.1.2 it sufficesto find a strong embedding into S ( P + ,P”). By 111.4.3 and 1.8.3 there exists an extension @ of the concrete category
(Set,F ) in S ( P + ) .Thus, we have a monotransformation p: F - + U o @
where U is the natural forgetful functor of S ( P + ) .Put G = U @. G is obviously faithful and we have 0
@ ( X )= (G(X),R,(X))
with R o ( X ) c P+(G(X)).
Because of the monotransformation p, by 11.7.4 it suffices to find a strong embedding Y : S(G) + S(P+, P’). Put Y ( X ,I ) = ( G ( X ) ,Ri(X,.r))ie*)
1
with R o ( X , r ) = R o ( X ) and R , ( X , r ) = { A A c r } . For a morphism f = ((X’,r’),J ( X , Y)) put
Y(f) = ( Y ( X ’ ,r’), G ( f ) , Y ( X , I)) . ’
Since G = U o @, we have G ( f ) ( R , ( X ) )c R,(X’). If A € R , ( X , r ) , we have A c r and hence G ( f ) ( A )c G ( f ) ( r )c r‘, and hence G(f)(A)E E R,(X’, r‘).Thus, Y ( f ) is in S ( P + , P’). We see easily that Y is a one-to-one functor (since @ was). Obviously, V OY = G o U for the natural forgetful functors U , I!and hence it remains to prove that Y is full. Take a morphism
0 = ( Y ( X ’ ,r’)’9, qx,I)) of S ( P + , P’). Thus, g: G ( X )-+ G(X’) is such that g = G ( f ) . Further, we have I E R I ( X ,I ) , hence P + G ( f )(r)E R1(X’,r’) and hence G ( f )(r) c r‘. Thus, f = ((X’,r’),x ( X , r)) is in S(G) and we have 0 = Y(f).
268
Ch. VII, STRONG EMBEDDINGS etc.
1.4. By 111.4.4 we obtain immediately Corollary. Let F : Set + Set be an arbitrary functor, let G: Set + Set be contravariant faithful. Then there is a strong embedding of S(F) into S(G). 1.5. By 11.6.10 we obtain Corollary. Under (M), every S(F) can be strongly embedded into Graph. EXERCISES
1. The category Un of uniform spaces (see E VI 9 3 ) is realizable in S(P- Q). 2. The category Prox (see E VI 9 3 again) is realizable in S(Q P-). 3. The category Mi,M, of metric spaces and isometries into (contractions, resp.) is realizable in S(Q x const El). Find a realization of PI9, the category of metric spaces and the Lipschitz mappings, in an S(F) with an uncomplicated F . 0
0
9 2.
Which concrete categories are strongly embeddable into S ( P + )
Obviously, not every concrete category can be strongly embedded into S(P+).By Proposition 2.2 which we are going to prove below, it is evident that such a category is necessarily regular and that it cannot have a proper class of mutually non-isomorphic objects with equally large underlying sets. (These two necessary conditions are very easy to prove directly from the definition of strong embedding. The reader may do it as a simple exercise.) These two conditions are not yet sufficient. A category strongly embeddable into an S(F) has to have the property that one can, after adding some new objects at places where necessary, achieve the decomposability of morphisms into one-to-one and onto underlied ones, and preserve the property of having not too many non-isomorphic objects with underlying sets of a given cardinality (see the condition (D) below). This last condition is already sufficient, which will be shown in Theorem 2.5. It can be replaced by a condition formulated in terms of the category itself, without speaking of a possible expansion (see [KP,]). But we choose to stay here at the condition (D) because of its easy controlability in everyday-life categories.
2.1. Proposition. The following two statements on a concrete category ('8,U ) are equivalent:
$2. Categories strongly ernbeddable into S(P+)
269
(i) There exists an F such that (a, U ) is strongly embeddable into S(F). (ii) There exists an F such that (W,U ) is realizable in S(F). Proof. The implication (ii) (i) is trivial. (i) 3 (ii): Let us have a faithful G: Set -+ Set and a full embedding @: ‘2I -, S(F) such that for a E obj a,@(a)= (CU(a),r(a)), for cp: a -, b, @(cp) = (@(a’),GU(cp), @(b)). We have r(a) c FGU(a) so that Y(a) = (U(a),r(a))is an object of S(F o G). We will prove that Y induces a realization of (W, U ) in S(F 0 G) showing that g : U(a)-+ U(b) carries a morphism cp: a b iff FG(g)(r(a))c r(b). Let g = U(cp). Then FG(g)(r(a))c r(b) because @(q) is in S(F). Let FG(g)(r(a))c r(b). Then, since @ is a full embedding, there is a 9:a -+ b such that GU(cp) = G(g). Since G is faithful, we obtain g = U(cp). --f
2.2, Corollary I f (W, U ) is strongly embeddable into an S(F), then it is regular (see 11.6.5) and the following condition holds: ( S ) There exists a class d c objW such that every a E obj W is isomorphic to a member of d and that for every cardinal a
I
{ a a ~d and card U(a) 5 a ) is a set. Proof. The regularity is obvious. The condition ( S ) is evidently inherited by concrete subcategories so that it suffices to show it for S(F). Therefore, it suffices to take d = {(a,r ) r c F(a), c1 a cardinal number) .
I
2.3. A concrete category is said to have the property (D) if (D) for every morphism a of W there is a decomposition with U(a,) one-to-one and U(a2)onto.
CI
= a1 0 a2
2.4. Lemma. A regular category (a,U ) satisfying ( S ) and (D) is realizable in a regular concrete category (23, V )satisfying ( S ) and such that (1) for every morphism 8: b -+ b and every decomposition fl o f i of U(8) into two mappings there are morphisms Pi with = U(8,) and 8 = =
81 82. O
Proof. Construct (S,V’) as follows: The objects of 23’ are triples ( X , a, p ) where X is a set, a an object of W and p a mapping of a subset D(p) of X onto U(a). Denote by j ( p , X ) the mapping of D(p) into X sending x to x. The morphisms from ( X , a, p ) into (X’, a’, p’) are triples ((X’, a’, p ‘ ) , f ;( X , a, p ) ) where f: X -+ X is a mapping
270
Ch. VII, STRONG EMBEDDINGS etc.
- 1. - 1. -
suGh that there exists a mapping g: D(p) --+ D(p‘) and a morphism a: a such that the diagram
x
(2)
D p)
X‘
--+
a‘
U(a)
I
p a )
U(a’)
D(p’)
commutes, i.e. f o j ( p , X ) = j ( p ’ , X ’ ) o g and U(a)o p = p’ o g. The composition is defined in the obvious way, the forgetful functor V’ sends ( X ,a, p ) to X and (x’,A2)t o 5 Obviously, (B’,V’)has the property (S). Take the class d from the formulation of ( S ) for 2l and put
93 = ((a, p , a) 1 a a cardinal, a E d ].
Also obviously, there is a realization of (‘$ U ) in I, (B’,V ’ ) : It suffices to associate (U(a),a, l,,,,) with a. (B’, V’) has the property (1): Let f = ((X’,a’, p ’ ) , f ,( X , a, p ) ) be a morphism of B’, let f be equal to f i o f 2 with f 2 : X -, Y and fl : Y -+ X’. We have a g: D(p) --+ D(p‘) and an a : a a‘ such that the diagram (2) commutes. By the property (D) of (%, U ) there exist morphisms x2: a --+ b and a l : b -+a‘ such that a = a1 o a2, U ( a l )is one-to-one and U(a,) is onto. Put D(q) = f2(D(p))c Y and construct mappings g1: D(q) D(p’), g 2 : D(p) -+ D(q) and q: D(q) 4 U(b) by putting 9l(Y) = f(x) (= g(x)) where Y = f&), --+
--+
g2(x) = f2(x)7 4(Y) = U(a2)
(4.)) where Y = fZ(X). (The definitions of g1 and q are correct: if y =f2(x) =f2(x’), we have f(x) = f l f Z ( X ) = f l f Z ( X ’ ) = f ( x ’ ) and U(a1) (P(X)) = U(a) ( P ( 4 = = p’(g(x)) = p’(f(x)) = p’(f(x’)) = U ( q ) U(a,)(p(x’)) and U(al) is one-toone.) We have f2 Oj(P3 = j(4, y) 92 and
w%)
x)
U(a2)
O
O
P
= 4 92
directly from the definitions,
fl
y)
0
s2= fl o f 2
0
j(p,
x)= f o j ( p , x)=
= j(p’, X’) O 9 = j(p’, ~ ( a , ) o q o g 2=
u(+p
X ’ ) 91 9 2 O
O
2
= P‘”91”92
and since g 2 is onto, f o j(q, Y ) = J(p’, X‘) g1 and U ( a l )0 q = p‘ 0 gl. 0
52. Categories strongly embeddable into S ( P + )
~ h u s ,f1 = ((x’, P’, a’),fl, (z: 4, b)) and .7z morphisms in S’ and we have
=
27 1
(( E: 4, b),f2, (x,P, a)) are
f=f,
-
of2.
Now, (23’,V‘) is not necessarily regular. Consider the equivalence defined by b , b , iff there is an isomorphism cp: bl + b , with V’(cp)= lVr(bl,. Since (a, U ) is regular, we see that
-
( u ( a , ) , a , ,1 U ( a l ) )
- (U(az),az>
1,
[(,.I)
if1
~ 1 1=
a2.
Thus, we can take representatives of the equivalence classes of objects of 23’ so that (U(a),a, I,,,,) represent their classes, and we can take for (23,V ) the full concrete subcategory of (S’,V )generated by these representatives.
2.5. Theorem. The following two statements are equivalent: (i) (a, U ) is a concrete subcategory of a regular concrete category satisfying ( S ) and (D). (ii) (a, V ) is strongly embeddable into S(P+). Proof. (ii) * (i). By 2.1 and 2.2 it suffices to prove that every S(F) has the property (D). Take a morphism ((X’,r’),J ( X , r)) of S(F). Put ’Y = f ( X ) and define fi: Y + X’ and f,: X -+ Y by f , ( y ) = y , f 2 ( x ) = f ( x ) . Put s = F(f1)-l (r’).Thus, we have F(f1) = r‘ and since F(f,)( F ( f 2 )(r)) = F(f)( I ) c r‘ also
(4
F ( f 2 ) (r)
=w-1)-
(r’) = s .
Thus, we have the decomposition
(xs))
((X’,r’),f,(x, r)) = ((X’,r’),f1,
O
(( Y 4, f2,
(x, r)) .
(i) * (ii): By 2.4 it suffices to prove that every regular concrete category
(a, U ) satisfying (S) and (1) from 2.4 is realizable in an S(F). First, note that, under the circumstances, for a set X ( a I U(a) = X > is
a set. Indeed, suppose it is a proper class. Since, by (S) and by the regularity, there is only a set of non-isomorphic a with U(a) = X,there has to be a B c (u U(a) = X } with cardB > cardXX and such that any two b, c E B are isomorphic. Choose a b o E B and, for every b E B an isomorphism cpb: bo + b. According to the cardinality of B there are b,c, b =+ c, with U(cp,,) = U(cpc).Thus, U(cp,0 cp; ’) = 1, in contradiction with the regularity.
I
272
Ch. VII, STRONG EMBEDDINGS etc.
I
For a set X define first a preorder ion G ( X ) = {a U(a) = X } by a
Put
< a’
iff
U(a) = 1 for some a : a
-+
a‘.
F ( X ) = { M I M ~ G ( X ) , ( ~ E M , U
For a mapping f: X
Y define F ( f ) : F ( X ) 3 F ( Y ) putting
+
F(f)( M ) = { b I U(b) = Y &
3a E M 3cp: a -+ b, U(cp)= f )
( F(f)( M ) is in F( Y ) : Let b be in F(f)( M ) and B: b -+ b‘ be a morphism with U(6) = ly. We have a cp: a a + b‘ with U(Bcp) = f.) F is a functor: We have
pep:
-+
b with
UE
M and U ( q ) =f, and hence
F ( l , ) ( M ) = ( b ] U ( b ) = X , 3 a ~ M 3 c p a: - + b with U ( c p ) = l ) = M . Further, we have for f : X and
-+
Y and g: Y
--f
2
I
F(g of) ( M ) = { c U(c) = 2, 3a E M , 3cp: a 3 c, U(cp)= g of}
F(9) ( F ( f ) ( M ) ) =
= { c ( U ( c ) = Z ,3 b E F ( f ) ( M ) 3 c p 1 :b - t c , U ( c p l ) = g ) = = (C U(C)= 2, % E M 3 ~ 2 u: -+ b 3 ~ 1 b: -+ ~ ( U ( c p 2=) f & U ( ~ p , ) = g ) } .
1
Thus, always F(g) ( F ( f )( M ) )c F(g of) ( M ) and under (1) obviously also F(9 o f ) ( M ) = F ( g ) (F(f)(MI). Now, for an a E obj % define
@(a)= (U(a),r(a))E objS(F)
putting
r(a) =
{MEF(U(U))/UEM}.
If cp: a + b is a morphism, we have obviously b E FU(cp)( M ) for M E r(a), and hence FU(cp) ( M )E r(b). Thus, U(cp) carries a morphism @(a)-+ @(b). On the other hand, let f : U(a) U(b) be such that F(f)(r(a))c r(b). We have, in particular, so that
1
6 = {a’ IN: a -+ a’, ~ ( a=)
1
E r(a),
1
b E {b’ U(b’) = U(b)3 ~E’6 3cp’: a’
-+
b‘, U(cp‘) = , f ).
Thus, we have a 9’: a’ --* b with U(q0’)= f and a’ E 6, and hence also an a: a -+ a’ with U(a)= 1. Hence, for cp = cp‘ 0 a: a -+ b, U(cp)= f. Thus, @ induces a realization.
273
53. Strong universality
2.6. By 2.5 and 1.5 we immediately obtain Corollary. Under (M), the following two statements are equivalent: (i) (a,U ) is a concrete subcategory of a regular concrete category satisfying ( S ) and (D). (ii) (a,U ) is strongly embeddable into Graph. rn
0 3.
Strong universality
3.1. In the previous section we saw that although a fairly reasonable category cannot have the property that every concrete category whatsoever is strongly embeddable into it, the category S ( P + ) (and, under (M), the the category Graph) is universal in this sense with respect to reasonable concrete categories. 3.2. A concrete category is said to be (almost) strongly universal if every regular concrete category satisfying (S) and (D) (see 2.2 and 2.3) can be (almost) strongly embedded into it. By 2.5 and 2.6, S ( P + )and Graph are strongly universal. Thus : A concrete category (a,U ) is strongly universal iffthere is a strong embedding of S ( P + )or of Graph into ('? U).I, . m
3.3. Recall the definition of the strong co-embedding (11.6.4) of a concrete category (%, U ) into (23,V ) : it is a full embedding F : aoP + 23 such that there is a G: Set'P + Set with Vo F = G o U . We are going to show that a strongly universal category is universal also with respect to strong co-embeddings. 3.4. Proposition. (1) Zf (R, U ) is (almost) strongly universal then for every (2,V ) satisfying ( S ) and ( D ) there is an (almost) strong co-embedding into (R, U). (2) Let (2,V ) be strongly universal and let there exist an (almost) strong co-embedding of (2,V ) into (R, U). Then (R,U ) is (almost) strongly universal. Proof. (1) By the proof of 11.7.3, and by 11.7.2 there is a strong COembedding F of S ( P - ) into S(P-). We have strong embeddings F, : (2,V ) + -+ S ( P - ) and F,: S(P-)--t (R, U). Consider F, - F F,. (2) follows immeddiately by (1). 3.5. Corollary. Zf (R., U ) is strongly universal, universal.
(SOP,
P-
0
U ) is strongly W
274
Ch. VIT, STRONG EMBEDDINGS etc.
5 4.
Examples of strongly universal categories
4.1. In this paragraph we will indicate the use of the assumption (M)
(i.e., interpreting alg-universality as universality) by simply writing the symbol (M) before the statement.
4.2. Among the categories of (directed) graphs with graph homomorphisms, let us point out the following strongly universal ones (see IV.5.6):
(M) the category of connected graphs without cycles, (M) the category of partially ordered sets.
-
-
4.3. By IV.4.12 we see the strong universality of (M) the category of connected undirected graphs with a prescribed chromatic number 2 3 (and their homomorphisms). 4.4. By VI.8.6, among others,
the category of metric spaces and their open uniformly continuous mappings is strongly universal, - (M) the category of locally compact metric spaces and their uniformly continuous mappings is almost strongly universal. -
4.5. By VI.12.5, (M) the category of connected TI-spaces and their continuous mappings is almost strongly universal, - (M) the category of connected TI-spaces and their local homeomorphisms is strongly universal. -
4.6. By VI.16.10 and 3.4.(2),
the category of paracompact spaces and their continuous mappings is almost strongly universal. -
4.7. Further examples of strongly universal categories are the S(F) such that either there is a monotransformation from Q into F or F is contravariant faithful.
0 5. Strong embeddings of
Alg(d) into Alg(d’)
In 11.5.5 we saw that there was a full embedding of Alg(d) into Alg(d’) whenever Cd‘ 2 2. By the constructions used one does not see whether
275
55. Alg(d) into Alg(d')
the same holds for the strong embedding. It actually does, and this is what we are going to show in this section. 5.1. Lemma. For every A = ( A b ) b , B there is a set c such that there is a strong embedding of Alg(d) into Alg((l)csc). Proof. Choose a set A and one-to-one mappings P b : A b -+ A such that A n B = 8 and put C = A u B. Define Qi:
-+
Ak((1)CEC)
by U' Qi = Q A o U ( U , U' are the natural forgetful functors, for 1.3.14. E) and @ ( X ,( 0 b ) B ) = (X", (cp&) where 0
for c E A for c E B
see
QA
(qc(a))(a) = ~ ( c ) , (%(a))(a) = oc(u Pc) O
(Thus, the cpc are constant mappings.) If f : X + X' is an om'-homomorphism, we have for c E A ((QA(~0 ) CPC)
=(
f
0
(a))(a) = ( f o ~ c ( a (a) ) ) = S(a(c)f=
a) (c) = (
d ( f 0
a))(a) = ((d 0 Q A ( ~ ) (a)) ) (a)
3
for C E B ( ( Q A ( ~ )0 CP,) (a))(a) =
=
d ( f 0
a
0 PC)
(f
0
~ c ( a )(a) > = f ( ( ~ c ( a(a)) ) ) = f(wc(ao KC))
= ( q c ( f 0 a)) (a) = ( ( ~ 0c Q
=
A ( ~ ) (a)) ) (a).
Thus, the formulas really define a functor. Obviously, it is one-to-one.
Now, let g: X A + X'A be a pp'-homomorphism. If .(a) = x for every a E A , we have cpc(a)= a for every c E A and hence cp: g(a) = g cpc(a)= g(a) for every c E A. Thus, for a constant a, g(a) is a constant, and we can define f: X+X' by f(x)
=
g(const,) (c) (c E A arbitrary). For a general (g(4 =
(4= ((d 9 )(4(4 = ((9 O
O
cpc
c1:
A
-+
X we obtain
(4(4 =
(g(Cpc(4)) (a) = g(constC4cJ (4 = f ( a ( 4 .
Thus, g = Q A ( f ) .Now, take an a: A b such that a = B o P b . We have ob(foa) = d ( f B o
Pb)
=
-+
X and choose a /?: A
('?b(fo B)) (a) == ( Q A ( ~ )( ( P b ( P ) ) ) (')
= ( f o ( P b ( B ) ) (a) = f ( O b ( B
Pb))
=f(ob(ff)).
Thus, f i s an om'-homomorphism. Hence, Qi is full.
=
+
x
276
Ch. VII, STRONG EMBEDDINGS etc.
5.2. Lemma. For every C there is a strong embedding of Alg((l)cEc) into Alg(2) and Alg(2,O).
Proof. Define a functor F : Set -+ Set putting F ( X ) = (X x
c)u x u c u (0,1,2}
(more exactly, (X x C x (0)) u ( X x { 1)) u . .. to have the union disjoint),
F(f)(x, c) = (f(x),c) F(f)(x) = f ( x ) for X E X , F(f)(u) = u otherwise. 3
By 11.5.5 and I1 0 3 there are (i = 0 , l ) only for f = idc.
C -+ C such that
i,b0,
fo
+i= rc/i o f
Define a one-to-one functor @:
putting
U ’ o
@ = F o U,
Ak((l)C€C)
-+
Alg(2)
@(X7(q&)= ( F ( X ) ; ) where
(x, c). 0 = (qc(x),c), (x, c). 1 = x , c . 0 = $o(c) o . o = 1,
(x, c ) . 2 = c ,
c . 1 = l+b1(c), 1.1 = 2 ,
7
CI
.p
=0
c.2
=
2.2
= 0,
1,
otherwise
(the checking that a homomorphism is sent to a homomorphism is straightforward). Now, let ( X , (qC),)and (X’, (q$) be given and let g: ( F ( X ) ,.) (F(X’),.) be a homomorphism. First, CI . a is always in (071,2}. Thus, we have g((0, 1,2}) c (0, 1,2), since i = ( i - 1).(i - 1) (when counting with 0. 1,2, the subtraction and addition will be understood modulo 3). Since g(i + 1 ) = = g(i). g(i), we see immediately that if g(0) = j , we have g(i) = I i. We have 0 = 0 . 1. Thus, g(0) = g(0). g(1) = j . ( j 1) = 0. Hcnce. also g(1) = 1 and g(2) = 2. For c E C we have 1 = g(1) = g(c .2) = g(c) . 2 so that necessarily g(c) E C. Since we have further -+
+
+i(c) =
g(c . i)
=
g(c) . i = l+bi g(c)
+
9
we see that g(c) = c for all c E C. For an (x, c) E X x C we have g(x, c) . 2 = g((x, c) .2) = g(c) = c E C so that g(x, c) E X’ x C.
211
95. Alg(d) into Alg(d‘)
For an
XE
X choose an arbitrary c. We have g(x) = g((x,c). 1) =
= g(x, c) . 1 E X ’ so that we may define
f:X+X‘ by f ( x ) = g(x). Now, if g(x, c) = (x’, c’), we have x’ = g(x, c). 1 = = g((x, C). 1) = g(x) = f ( x ) , c’ = g(x, C ) . 2 = g((x, C ) ‘2) = g(c) = c, so that g(x, c) = ( f ( x ) ,c). Thus, g = F ( f ) . Finally we have (cp:f(x),c) = (f(x), c ) . 0 = g((x, 4 . 0 ) = S(%b)>c) = (f cp,(x), c) , so that f is a homomorphism. Hence, @ is full. If we choose, say, 0 for the nullary operation, we obtain the statement on Alg(2,O).
5.3. Lemma. There is a strong embedding of Alg(2) into Alg(1, 1) and A k ( L AO). Proof. We will present a construction suitable for the strong embedding into Alg(l,l, 0). A construction of a strong embedding just into Alg(1, 1) may be done in a substantially simpler way. Define a functor F : Set + Set putting F ( X ) = ((x2u x)x (0, l})u {0,1> (again, the unions are understood disjoint), F ( f )(X’ Y , i) = ( f ( x ) ,f ( hi) F ( f ) (x, i)
F(f)(i)
= ( f ( x ) ,i) =i
,
9
for i = 0,1.
Define a one-to-one functor @: Alg(2) + Alg(1, 1)
putting U’ @ 0
=
F
0
U and @ ( X ,.)
cp(x, y, 0) = (Y, x, 1)
cp(x, y, 1) = (X’ Y , 0) cpfx, i)
= (x, 0)
cp(i)
=
1- i
=
( F ( X ) ,cp, $) where
$(x, !’. 0) = (L0) $(& Y ? 1) = (x . Y? 1) $(x, i) $(i) = O
1 - i) (i = O,I).
= (x,
(Again, it is easy to verify that homomorphisms are being sent to homomorphisms.) Let ,a carry a homomorphism @(X;) to @(X,.). We have $ g ( o ) = = g $(O) = 0 and hence g(0) = 0, since nothing else is fixed under *. Consequently also g(1) = g cp(0) = cp g(O) = q(O) = 1.
278
Ch. VII, STRONG EMBEDDINGS etc.
Similarly, g ( X x (0)) c X’ x (01, since cp g(x, 0) = g cp(x,0) = g(x, 0). Thus, we can define a mapping f: X + X’ by g(x, 0) = ( f ( x ) 0) , .
We obtain immediately Now, we have
g(x, 1) = 9 $(&0) = $ ( f ( x ) ,0) = ( f ( x ) ,1).
$ g(x, Y , 0) = g(x, 0) = ( f ( x ) ,0) so that g(x, y , 0) is in (X” x (0)) u ( X ’ x { 1)). If, however, g(x, y, 0) = (u, l), we have g(y, x , 0) = g (p2(x,y, 0) = cp2(u,1) = (u, 0) # Xf2 x (0) u X’ x (1). Thus, g(x, y, 0) = (u, u, 0) and we obtain further $(u, u, 0) = 9 $(x, Y , 0) = (f(x),0) > (u, 0) = $ cp2(% u, 0) = 9 $ cp2(x,Y , 0) = ( f ( Y ) , 0)
(U’O) =
so that g(x, y, 0) = ( f ( x ) f, ( y ) ,0) and consequently also g(x, y, 1) = = 9 9 ( Y , x,0) = (p(f(y),f(x),0) = ( f ( X ) . f ( Y ) , 1). Thus, 9 = qf).Finally, we have (f(X.Y),
1) = 9(X.Y, 1) = 9$(X,Y, 1) = $ ( f ( X ) , f ( Y ) , 1) = ( f ( X ) . f ( Y ) ,
1 ) 7
so that f is a homomorphism. Hence, @ is full. To obtain the statement on Alg(1, 1,O) choose 0 for the nullary operation. W 5.4. By the preceding lemmas and by 11.1.3 we obtain immediately Theorem. If Ed’ 2 2, there is a strong embedding of Alg(d) into Alg(A’) for an arbitrary A. W EXERCISE
1. Find a strong embedding of Alg(2) into Alg(1,l) simpler than that of 5.3.
8 6.
Some negative results and the notion of strong algebraicity
6.1. In contrast with the results mentioned in 64 we observed in some cases that a concrete category is universal without being strongly universal. This was the case e.g. with - Alg(d) with I d 2 2 (11.5.5) (under (M)), - the category of graphs and strong homomorphisms (IV 8 6)(under(M)),
$6. Strong algebraicity
279
- the category of topological spaces and open local homeomorphisms -
(V1.11.5)(under (M)), the category of topological spaces and open continuous mappings. (V1.9.1, VI.11.5)
In all of these cases the reason why they were not stronglv universal was in the following property, which is obviously inherited by strongly embedded categories: If an invertible mapping carries a morphism, it is an isomorphism.
6.2. We already know (see 111.4.5)that under the set theoretic assumption (M)every concretizable category is algebraic.Thus, the notion of algebraicity
does not describe a particular type of a category. Among concrete categories, however, the fact that Alg(A)is not strongly universal enables us to point out those which behave “more algebraically” than the others.
6.3. A strong ff-embedding of (a,U )into (23,V )is a full faithful F: % -+23 such that there is a faithful G with V OF = G OU .
(a,
6.4. We say that a concrete category U ) is strongly algebraic if there is a strong ff-embedding of (%, U ) into an Alg(A).
6.5. Remark. We choose this slightly more general definition instead of requiring the strong embeddability into an Alg(d), mainly for the reason that a strongly embedded category inherits the regularity, which, we feel, has not much to do with the algebraic behavior. We do not wish a strongly algebraic category to fail to remain so after merely reduplicating an object. It is very easy to see that for regular categories the strong algebraicity coincides with the strong embeddability into an Alg (A). 6.6. Theorem. The following statements are equivalent: (1) (%, U ) is strongly algebraic, (2) there is a strong ff-embedding of U ) into Alg(A) with afixed A with C A 2 2, (3) there is a strong ff-emhetltlinq of (41. U ) irito the cate9ory of commutative groupoids, (4) there is a strong fflembedding of (ill U,) into the category of semigroups. Proof. Follows immediately from 6.4, the proof of 11.2.3, and V.2.25. H
(a,
6.7. The following proposition is sometimes handy in proving that a concrete category is not strongly algebraic:
280
Ch. VII, STRONG EMBEDDINGS etc
Proposition. In a strongly algebraic category (a,U ) : (1) I f a : a 3 c and p: b 3 c are morphisms in a, U(B) one-to-one and U ( a ) = U(P)of; there is a y : a -, b in 2I such that U(y) = f : (2) If.: c 3 a and 3!,: c -+ b are morphisms in 'u, U ( p )onto and U(a) = = U(/?),there is a y : b -+ a in 2I such that U(y) = $ $0
Proof. Obviously Alg(2) has the properties (1) and (2). Denote by V the natural forgetful functor of Alg(2). We have a full faithful F : 2I -+ Alg(2) and a faithful G with V OF = G o U. Let U(a) 5 U(p)of for cc: a -+ c, p: b + c, let U(p)be one-to-one. We have F(a): F(a) -+ F(c), F(p): F(b) 3 .+ F(c) and VF(a) = GU(a) = GU(0)o G(f) = VF(P) G ( f ) . Since (1) holds in Alg(2) and VF(fl) = GU(P) is one-to-one (there is a q such that cp o U(fl) = 1 so that G ( q )o GU(P) = 1) we have a y': F(a) -+ F(b) with G(f ) = V(y'). Since F is full, there is a y : a -, b with y' = F(y). We have, hence, G ( f ) = VF(Y)= GU(1/). 0
Since G is a faithful, f
=
U(y).
rn
6.8. By any of 6.7(1) and 6.7(2) we see that a strongly algebraic category has to have the invertibility property mentioned at the end of 6.1. We will now show a simple example of a category which is not strongly algebraic although it has the invertibility property. Take the category of compact Hausdorff spaces and their open (not necessarily continuous) mappings. If we have a one-to-one open mapping f of a space X onto I:the mapping f - ' : Y -+ X is continuous, hence a homeomorphism and hence open. On the other hand, take the subspace C of the real line consisting of 0 and all the l / n with n positive natural. Further, let A, B be two non-homeomorphic countable compact spaces and let a (fl resp.) be a one-to-one mapping of A ( B resp.) into C with a(A) = C\(O} @(B)= C\(O}, resp.). Obviously, CI and P are open and determine a unique f: A -+ B such that p o f = a. The f is one-to-one onto so that if it were open, it would be a homeomorphism. Thus, the implication in (1) does not hold.
6 7.
Categories A((Fi,di)iEJ)
The categories introduced in this paragraph play an important role in a criterion of strong algebraicity which will be proved in Q 8 and used in 0 9.
28 1
$7. Categories A((F,,A,),,,)
7.1. Let J be a set, Fi for i e J (covariant or contravariant) knctors Set -+ Set, A i for i E J types. The concrete category
A((‘,.
1t)iEJ)
(for a finite J, one writes, e.g., A((F,, Ao), ..., (Fn7 A,)) ) is defined as follows: The objects are systems ( X ,(ai)iGJ)where X is a set and, for every i E J , cti is an algebraic structure of the type A i on Fi(X).The morphisms from 8 = ( X ,(ai)iEJ)into 2’ = (X’, (a$EJ) are triples (3,J2)where f: X -+ X’ is a mapping such that for every i E J , F i ( f ) is an a,a:-homomorphism for covariant Fi7an ct;ct,-homomorphismfor contravariant Fi. The composition given by the rule
(P’J’, P) (S’,f,2)= (P’,f’ o f , S), the forgetful functor sends 2? to X and (8‘,f, 2)to$ 0
7.2. We write EJ
)
instead of A((Fi,(l)iEJ)(i.e., in the case when for every i, A i = with Aio = 1 ).Thus, the objects of A((Fi)i,J)are the systemg 2? = (X, (qi)iEJ) where qi are mappings Fi(X) -+ Fi(X) and the morphisms from 3 into ( X , (q$isJ) are carried by the f: X + X such that Fi(f) qi = Fi(f). If, moreover, the J is a one-point set, we simply write 0
0
7.3. Lemma. Let A be a non-void set. For an A-nary operation a on Y (ie., a: YA -+ Y-see 1.1) deJine a: Y A + Y A by putting (z(5))(a) = ct(5) for every a E A. Then g: Y Y is an act‘-homomorphism f l QA(g)o ii = a’0 Q,(g) (for Q, see I.3.14.E).
Proof. Let g be an aa’-homomorphism. We have
’
((QA(g)0 @) (5))(a) = (9 a(5))(a) = g(or(5))= = a’(g 5) = a(QA(g)(5)) = ((E Q A ( d (O)(a). 0
(Q)
0
On the other hand, if Q,(g) a’(g
0
0
0
ti = i’o QA(g),we have
5) = ((z’ QA(g))(5))(a) = ((QA(g) 4(5))(a) = g(a(5)) 0
0
*
Ch. VII, STRONG EMBEDDINGS etc.
282
7.4. Lemma. For a mapping cp: Y-+ Y define
p: P-( Y ) -+ P-( Y ) (for P - see 1.3.14.F) putting QI = P-(cp). If cp: Y + E: cp’: Y‘ + Y’ and g : Y + Y’ are mappings, then
iff P - ( g ) o q ’
gocp = cp’og
=
qoP-(g).
Proof. This is an immediate consequence of P - being a one-to-one H contravariant functor.
7.5. Lemma. Every A((Fi,L I ~ ) ~can , ~ ) be realized in an A((Gk)keK) with covariant functors G,. Proof. Let d i = (Aib)bsBi. We can assume that no A , is void (possible nullary operations may be replaced by constant unary ones). Put K = UB, x (i). is J
For i such that Fi is covariant put G(b,i)= QAzb Fi, for i such that Fi is contravariant put G(b,i,= P- QA,, Fi. Now, for an object 0
’ (x, 0
0
=
Of A((Fi;.,
di)ieJ)
with
cli
= (‘ib)beB,
@)’(
=
(‘i)i,J)
put
(x,
@(b,i))(b,i)eK)
-
where, for Fi covariant, & i ) = Eib, for Fi contravariant, fl,,i, = I?,, (notation is an object of A((Gk)kEK) and by 7.3 and 7.4 we see of 7.3 and 7.4). @)’( immediately that @ induces a realization.
7.6. A covariant functor F : Set + Set is said to be pointed if there are elements such that, for every f : X
P(x)
F(X)
+ X‘,
F(f)(P(X))= d X ’ ) (in other words, if there exists a transformation from the constant functor C sending every mapping to the identity of a one-point set, into F).
7.7. Theorem. For every system (Fi,Ai)i,J there is a pointed covariant faithfulfunctor F such that A((Fi,di)ieJ)is realizable in A(F). Proof. By 7.5 it suffices to prove that for every system (Fi)i, of covariant functors, A((J‘JisJ) is realizable in an A ( F )with a pointed covariant faithful F .
$8. A criterion of strong algebraicity
283
Take elements a and b, which are not in J , and put
F ( X ) = U ( F i ( X ) x {i]) u ( X x { a ) ) u (1 x { b ) ); ieJ
for a mapping f : X
-+
X' define F(f):F ( X ) + F ( X ' ) by for i E J ,
F(f)(u,i ) = (Fi(f)(u),i) F(f)( x , a) = ( f ( x ) ,a) F(f)(0, b) = (0, b) . 9
Obviously, F is a covariant, pointed and faithful functor. ('pi)ieJ)be an object of A((Fi)isJ). Put @ ( X ,(tpi)isJ) = ( X , 'p) NOW,let (X, where 'p: F ( X ) -+ F ( X ) is defined by
q(u, i ) = ('pi(u), i )
for i E J ,
'p(x,a) = (&a)
and
'p(0, b) = (0, b) .
Thus, ( X , 'p) is an object of A(F). Let f: X -+ X' carry a morphism We have
F(f)('p(u, i) = (Fi(f)tpi(U)? i) F(f)('p(x, 4)= (f(x), a) =
=z
(X,
-+
2' = ( X , ('pi)isJ).
= ('p:Fi(f),i) = 'p'(F(f)(U' i))
Cow)(x, a))
9
7
F(f)('p(0,b)) = (0, b) = 'p'(F(f)(0, b))
@(z)
so that fcarries a morphism -+ @(r?'). On the other hand, iff carries a morphism @(r?)-,@(g'), we have, for i E J , qi(u),i) = ~
( V(U, f ) i)
( ~ o Fi(f) i i)) F i ( f ) o 'pi = 'pi o Fi(f). Thus, f carries a morphism 8 p.Hence, (~i(f)
= CP' F(f)9'(
i)
=
so that @ induces a realization.
6 8.
9'(
-+
A criterion of strong algebraicity
8.1. Lemma. (Under (M)) For every concrete category (a, U)there is an extension @: 2I -+ Graph with the monotransformation p: U -+ Yo@ such thatfor every c t : a 4 a' from morpha the following holds:
"(4)
u E PU(
iff ( v @(4) (4E pa'("(a')) .
284
Ch. VII. STRONG EMBEDDINGS etc.
Proof. Let @’: 2l
-+
Graph be an extension, @’(a)= ( V @’(a),R(a)),
let p’: U
-+
V@’ be the monotransformation. Construct a functor @”: 2l
-+
Re1(2,2,1)
putting @”(a)= (U(a) x (0) u V@(a)x (l}, (Ri(a))i=o,l,2)with (u, i) &(a)
(u, i) R,(a) ( u , j )
iff i iff i
(u, i) E R,(4
iff i = l ,
(UJ)
=j =
1 and (u, U ) E R(a),
= 0, j =
1 and u = p‘“(u),
@”(u)(u, 0) = (U(u)(u), O), @”(a)(u, 1) = (V@’(a)(u), 1).
Obviously, @” is a one-to-one functor. Let g : U(a) x ( 0 ) w V@’(a)x (1) -+ U(a’) x ( 0 ) u V@’(a’)x (1) be Ri(a)&(a’)-compatible for i = 0,1,2. Because of R,, g( V @ ( a )x (1)) c c (V@(a’)x (l)), and hence we have an h: V@’(a)-, V@’(a’) with (h(u),1) = g(u, 1). Because of R,, h is R(a)R(a’)-compatible and hence h = V@’(u)for an a: a -+ a’. Take a u E U(a). We have (u, 0) R,(p’*(u),l), hence (u,i) = g(u, 0)Rl(h(p’“(u)), 1). Thus, i = 0 and h p’“(u) = p’“(v). On the other hand, h p’”(u)= V @(u)p’“(u) = p”’( U(u)(u)) so that g(u, 0) = = (U(0c)(u), 0). Thus, @’ is a full embedding. Now, denote by V’ the natural forgetful functor of Re1 (2,2,1) and take the full embedding Y :Re1(2,2,1)-+ --f Graph from 11.5.3. Define monotransformations p”: U -+ V’ v: V’ -+ V OY by $’“(ti) = (u,O), vb(u)= u. We see that the embedding @ = Y O@” with the monotransformation p = VW p” (where v@i = v @ , , ( ~ ) ) have the required properties. 0 @‘I,
0
8.2. A covariant functor F: Set + Set is said to be algebraically selective if there is a type A and a full faithful @: Set -+ Alg(A) such that Uo@=F
(where U is the natural forgetful functor).
8.3. Proposition. I f F is algebraically selective, then A(F) is strongly algebraic. Proof. Let d = (A,),,, be a type, @: Set -+ Alg(d) a full embedding with U @ = F. Put 0
=
(F(X)7
(ab(X))bcB).
$8. A criterion of strong algebraicity
Take a c 6B, put C = B u {c} and A‘
= (Ab)b,c
285
with A,
=
1. Define
Y : A ( F ) -+ A l g ( d ’ )
putting Y ( X ,a)= (F(X),(ab(X2 ~ .,(X,a) = a,
uq
l ) ) ~ , with ~ )
a x a )).(
ab(X,a) = C l b ( x ) for b E B and
F(f)).( . We see easily that Y is a strong embedding. =
rn
8.4. Lemma. Let there exist transformations
such that ~p is the identity. Then A ( F ) is realizable in A ( G ) . Proof. For ( X , a) E obj A ( F ) define @ ( X ,a) = ( X , fl) E obj A ( G ) putting fl = px 0 a 0 EX. Let for f : X X’ hold a’c F ( f ) = F(f’)0 M. Then --$
B’oG(f)=p~a‘o~oG(f)=poo‘oF(f)o~= =
p o F ( f ) o a o s = G ( f ) o p o a o ~ =G ( f ) o f l .
On the other hand, let
0
0
a’ F ( f ) = E 0 p o 0
=
p. Then po F ( f ) = 0’ G ( f ) o p
G ( f )= G ( f ) ~
’ E O 0
E O
0
=
& o G ( f ) o p o p =F ( f ) o & o p o p= F(f)oa.
8.5. Theorem. (Under (M)) Every A((Fi,di)i,J)is strongly algebraic. Proof. By 7.7, A((Fi,di)i,J) is realizable in an A ( F ) with a pointed covariant faithful F. Thus, by 8.4 and 8.3, it suffices to prove that for every pointed covariant faithful F there is an algebraically selective G and transformations
FAGAF
such that EV is the identity. Take functor F , let p be the symbol from 7.6. Consider tbe extension @: (Set, F ) -+ Graph
and the monotransformation
p: F + H = V o @ from 8.1. Further, define a functor R : Set + Set by
@(XI=
(fw, R(X))
286
Ch. VII, STRONG EMBEDDINGS etc
( R ( X )is, hence, a subset of H ( X ) x H ( X ) ) and
qf)(u7
Finally, define G : Set -+
(4H ( f )(4).
(H(f), Set by 0)
=
G ( X ) = H ( X ) u R ( X ) u {0,1> (the summands are assumed disjoint, the multiplication by indices making them really so is omitted again),
G ( f ) ( u ) = H(f)(u)
G ( f )(U' u) = qf)(u, u) G(f)(i) = i
Define transformations
for u E H ( X ) , for (U' u) E q x ), for i = 0 , l .
FAGAF
by ~ ' ( u = ) p'(u), E'(~'(u)) = u and ~'(5) = p ( X ) if 5 4 vX(F(X)). Obviously, v is a transformation. If f: X -+ X ' and u E F ( X ) , we have EX'
G ( f )(P"(U)) = &"(H(f) PX(U)) =
= EX' P X ' ( F ( f )(u))= F ( f )
(4= F ( f ) EX( P X( U ) )
3
if 5 6 v X ( F ( X ) )we have G ( f )(5) 4 v''(F(X')) by 7.1 and the definition of G so that again EX' G ( f )(5) = P(X') = F ( f )(P(X))= F ( f ) E X ( ( ) . Thus, also E is a transformation and we have EP =.id. Now, define a functor Y : Set -+ Alg(1,l) by putting Y ( X ) = ( G ( X ) ,(a~),=,,,)where for U E H ( X ) , for (uo,.,) E R ( X ) > for j = 0, 1 ,
a,(.)
= a,(.)
a,(u0,.,) = ui
=
0,
7
ai(j) = i .
For f : X -+ X put Y ( f )(u) = G(f) (u). Obviously, Y is a faithful functor. Thus, if we prove that it is also full, G will be proved to be algebraically selective. Let g: G ( X )-, G ( X ' ) be such that for i = 0, 1, We have, in particular,
ai.g = g o q .
g(i) = g a,@)= at g(i)
99. Applications
287
so that, since i is the only element fixed under ai, For UEH(X),
for i
g(i) = i
=
0,1
.
g(u) = 9 ao(u) = g(0) = 0 so that g(u)E H(X'). Define h: H ( X ) 3 H ( X ' ) by h(u) = g(u). Finally, for (uo, u,) E R ( X ) we have a0
aig(uo,u,)
g(ui) = h(ui) so that g(u,, u,) = (h(u,), h(u,)) E R(X'). But this, by definition of H and R, implies the existence of an f : X -,X' such that h = H ( f ) and consequently g = G(f).
6 9.
=
Applications
9.1. Proposition. (Under (M)) The category of complete lattices and all their complete homomorphisms, and the category of complete Boolean algebras and their complete homomorphisms are strongly algebraic. Proof. It suffices to represent the general union and intersection as unary operations on P + ( X ) .
9.2. Proposition. (Under (M)) The categories of
- all topological spaces and their open continuous mappings, - all topological spaces and their closed continuous mappings, - compact Hausdorff spaces and their continuous mappings are strongly algebraic. Proof. Denote by WL'A ($&A, resp.) the closure (interior, resp.) of a set A in a given topology. A mapping f: X -+ Y is continuous iff any of the following two statements holds : (1) for every B c I: f - ' ( $ % t B ) c $ ~ t f - ' ( B ) , (2) for every A c X , f ( v t A ) c W t f ( A ) . f i s open iff (3) for every B c I: f - ' ( $ n t B ) =3 $m.tf-'(B). (Really, i f f is open, f ( $ ~ t f - ' ( B ) ) is open and contained in B so that it is contained in f ~ t t BOn . the other hand, if f - ' ( $ ~ t B ) =I Y~zt f -l(B), we have, for an open A, $.Ztf(A)
=f
f - $Htf(A)
3
f ( Y % t f - ' f ( A ) )3 f ( 9 9 2 t A ) = f ( A )
so that f ( A ) is open.) Analogously for a closed f:
288
Ch. VII, STRONG EMBEDDINGS etc
Thus, by (1) and (3), f is open continuous iff
F(f) 9nL = 9 n t P - ( f ) , 0
0
by (2) and (4), f i s closed continuous iff P + ( f )0 vt
=
v& P + ( f ) . 0
Thus, the first category is a full concrete subcategory of A(P-), the second is a full concrete subcategory of A(Pf). The third is a full concrete subcategory of the second one. 9.3. Let F : Set + Set be a functor. Define a concrete category
S(F) (S(F), resp.) as follows: The objects are couples (X, r) with r c F(X), the morphisms from (X, r ) into (X’, r’) are triples ((X’,r’),f; (X, r)) with f : X -+ X’ such that F ( f )(r) c r‘ and F(f)(F(X)\r) c F(X’)\r’ ( ~ ( f (r) ) = r‘, resp. ) . The forgetful functor sends ( X , r) to X and ((X’,r’),fi ( X , r)) tof: 9.4. Proposition. S(F’ and g ( F ) are strongly algebraic. Proof. Obviously, S(F) is realizable in A(P+F,(0)). Now, define G : Set -+ Set putting G ( X ) = F ( X ) x (0} u 2 x (11, G ( f )(u,O) = = (qf)(u), O), G ( f )(i, 1) = ( 4 1). Define unary operations o(x,r)on G ( X ) by putting o(u, 0) = (0,1)
if u E r ,
o(u,o)=(1,1)
if u # r ,
o(i, 1) = (i, 1) .
Obviously, F(f)( r ) c r‘ and F(f)(F(X)\r) c F(X’)\r’ = o G ( f ) , so that S(F) is realizable in A(G).
iff G ( f ) o =
0
0
N
9.5. (Wyler’s choices of morphisms) Let F , G be covariant set functors. The category W,(F, G ) (W*(F,G)? resp.) is defined as follows: The objects are couples (X, r) where r : F ( X ) -+ G ( X ) is a mapping
(r c G ( X ) x F ( X ) , resp.),
289
$9. Applications
the morphisms from ( X ,r ) into (X’, r’) are triples ((X’,r’),f; (X, r)) with f: X + X’ such that G ( f )o r = r’ o F(f) (in the latter case, the o is the composition of relations). The forgetful functors send ( X , r) to X and ((X’,r’),J (X, r)) to$ 9.6. Lemma. For r c M x N define f : N -+ P’(M) by f ( n ) = ( m (m,n) E r } . Let g : M --f M , h: N -+ N be mappings, r c M x N , r’ c M x N’ subsets. Then the following two statements are equivalent: (i) g o r = r’ o h, (ii) P’(g) o f = 7 o h. Proof. Let (i) hold. Take an n E N . We have =
I
(7 h) (n) = (m’ I (m’,h(n))E r } = (m’ 1 m’(r’ h) n} = = (rn’ 1 m’(g 0 r) n> = g((m I rnrn)) = 0
0
= (P’g
0
F) ( n ) .
On the other hand, let (ii) hold. Let m’(r’ o h) n. Thus, m’r’h(n) and hence m‘ E 7(h(n))= P’(g) (F(n)). Hence, there is an m with m’ = g(m) and rn E f(n), i.e. mrn so that m’(g r) n. If m’(g r) n, we have m‘ E g((m mrn}) = f’(h(n)) so that m’(r’h)n. 0
I
0
9.7. Proposition. W,(F, G ) and W,(F, G) are strongly algebraic. Proof. By 9.6 (put M = G(X), N = F ( X ) , g = G(f), h = F(f)), W*(F, G) is realizable in W,(F, P’G). Thus, it suffices to show that W,(F, G) is realizable in an A(H). Put H ( X ) = F ( X ) x G ( X ) , H(f)(u, v) = (F(f)(u),G ( f ) ( v ) ) .For a mapping ‘p: F ( X ) + G ( X ) define cp: H ( X ) + H ( X ) by @(U> 0) = (u, cp(u)). If G ( f ) o c p = c~‘oF(f), we have (H(f)n@)(u, u) = (IS(!) (u), G(.f)(cp(u)))= = (F(f) (u), ‘ p ’ ( F ( f )(u))= (cp’ qf)) (u,u), if H(f)cp = cp’ H(f), we have (F(f) (‘p’ F(f)) = (cp’ H(f))(u, 4 = (H(f1 cp) (u, 0) = = (F(f) (u),( G ( f ) ‘p) (u)) (in the case of a void G ( X ) , the equations are satisfied triviallv). Thus, sending(X, ‘p) to ( X , cp), we induce a realization of W,(F,G) in A(H).
(4
(4)
O
O
O
O
O
0
9.8. Corollary. The category Graph, (see IV.6.5) is strongly algebraic. (It coincides with W*(I,I ) where I is the identity functor.)
290
Ch. VII, STRONG EMBEDDINGS etc.
EXERCISES
1. Prove that die category OF all sets and their one-to-one mappings is strongly algebraic. 2. Prove t h a t [he category of all sets and their onto mappings is strongly algebraic. 3. Prove that the category of topological spaces and their quotient mappings is strongly algebraic.
6 10. Bibliographical remarks The notion of strong embedding appeared explicitly in [P5]; some of the full embeddings constructed before had been actually strong, though. The critical Theorem 1.3 is due to L. KuEera. It is quite recent and has not been published yet. The results of Q 2 are an application of those published in [KP,]. The strong embeddings of Q 5 appeared in [PSI. The essence of the remaining $5 6-9 was contained in [PT,]; some of the results presented here are, however, considerably stronger.
Appendix A COOK CONTINUA In Chapter VI we made an extensive use of the existence of a continuum such that there is no nonidentical non-constant continuous mapping of a subcontinuum into itself. Continua with this property were first constructed by H. Cook in [C,]. The construction there is only sketched and the proofs require, to a certain extent, the familiarity with the theory of continua. Therefore, we decided to present here a detailed construction starting with elementary notions of the theory.
9 1. Continua and their basic properties 1.1. First, let us recall some well known facts about connected sets and spaces : The closure of a connected set is connected. Consequently every component of a space is closed. The product of any collection of connected spaces is connected. Let 4. B be closed (resp. open) subsets of a space, let A u B and A nB be connected. Then A and B are connected.
1.2. As an obvious consequence of the definition of a compact space we have the following Lemma. Let d be a system of closed subsets of a compact space P such that A, B E d implies A n B E d. Then, for every open set U containing the intersection of d there is an A E d with A c U . 1.3. Proposition. (a) A component C of a compactspace is the intersection of all closed-and-open subsets of P containing C .
292
App. A : COOK CONTINUA
(b) Let C be a component of a compact space P, let U be open and C c U . Then there is a closed-and-open G with C c G c U . Proof. (a) Denote by 9 the system of all closed-and-open subspaces of P containing C. If C' = G 2 C, then C' is not connected and hence GEQ
C' = A u B with non-void closed disjoint A and B and C c A . Take U I, U,;open and disjoint with UA 3 A and U B =I B and put U = UAu U,. By 1.2 there is a G E such ~ that C' c G c U . We see easily that then G n L AE d,contradicting (C; n UA)n U , = 0. The statement (b) follows
immediately by (a) and 1.2. w Remark. 1.3.(b) has already been used in Chapter VI. In the sequel it will be used without further mentioning.
1.4. Proposition. Let (xn),and (y"), be convergent sequences in a compact metric space P, let x and y be their limits, let, for every n, the components of x, and y , coincide. Then the components of x and y coincide. Proof. Let C , be the common component of x, and v,. Suppose P = X u Y with X , Y disjoint open, and x E X , y E Y. Then for every n either C, c X or C, c Y contradicting the convergence of the sequences.
1.5. A continuum is a compact Hausdorff connected non-empty space. If it consists of one point, it is said to be degenerate, otherwise it is said to be nondegenerate. If a subspace H of P is a continuum, it is referred to as a subcontinuum of P, and if H =k P, as a proper subcontinuum. 1.6. A system X of sets is said to be monotone if, for any K , K' EX, either K c K' or K' c K . Proposition. The intersection of a monotone system of subcontinua of a space P is n continuum. Proof. Let X be the system. We may assume P E X . Put N = K. h E X
Obviously, H is compact Hausdorff and non-void. If H = A u B with A , B non-void, disjoint and closed. there are open disjoint U,, U , with U , I> A , UB 2 B, and we have. by 1.2. a K E X such that K c U , u U,; hence K is either not connected or contained in either UAor U,, which is a contradiction. rn
+
1.7. Proposition. Let H be a continuum, F closed, and 0 F s H . Then each component of F intersects the boundary of F . Proof. Let C be a component of F and C n B = 0, where B is the boundary of F . By 1.3.(b)there is a G, closed-and-open in F such that C c G c F\B.
52. Indecomposable continua
293
The F\B is open in H , and hence G is open in H . Since F is closed in H , G is closed in H . Since H is connected and C non-void, necessarily G = H , in contradiction with F H.
+
1.8. Proposition. Let H be a continuum, G open, and 8 =l= G H . Then the closure of any component of G intersects the boundary of G. Proof. Let C be a component of G, g L C its closure. Let us suppose that W t C n B = 8, where B is the boundary of G. Then VLC = C c G. Take an open L with C c L c W t L c G and apply 1.7 for %?LL to obtain a contradiction. 1.9. Corollary. I n every non-degenerate continuum there exist infinitely many disjoint non-degenerate subcontinua. (Indeed, find a sequence (G,), of open subsets with disjoint closures and denote by C, the closure of a component of G,. Every C , intersects the boundary of G,, and hence it is non-degenerate.)
5 2.
Indecomposable continua
2.1. Convention. From now on, the continua are assumed to be metrizable. Many of the statements, however, hold without this assumption as well. 2.2. A continuum H is said to be decomposable if H = H , u H , with H I and H , proper subcontinua. Otherwise, we speak about an indecomposable continuum. Proposition. A continuum is indecomposable @ each its proper subcontinuum is nowhere dense in it. Proof. If H = H I LJ H , for proper subcontinua H,, then the H iare not nowhere dense in H (we have Vt(H\H,) c H , ). On the other hand, let there be a proper subcontinuum L which is not nowhere dense. Then 8 V/(H\L) = H lso that we have either a decomposition H = = L u Vt(H\L), or %‘L‘(H\L) = El u E , with disjoint non-void closed Ei. In the latter case, put A , = L u El. Then A , u A , and A , n A 2 are connected, hence by 1.1 A , and A , are connected and we have a decomposition H =AIuA,. .
+
2.3. Let H be a non-degenerate continuum. Write x-Y if there is a proper subcontinuum H’ of H containing both x and y. The is always reflexive and symmetric, and if H is indecomposable, relation
-
294
App. A : COOK CONTINUA
it is obviously also transitive. Thus it is an equivalence. The equivalence classes are called composant of H . 2.4. Proposition. A composant of a non-degenerate indecomposable continuum is meager (= of thefirst category) in it. Proof. Let L be a composant of a continuum H , and W a countable basis of open sets in H . Choose an U E Land put 9?'= { B e g a $ B += 0). For B E 98' let M E be the component of H\B containing a. Denote by L' the union of all M E with B E 9. We have obviously L' c L. On the other hand, for X E L we have a proper subcontinuum S of H with a , x E S . Take a B,EB' such that B, c H\S. Obviously, S c M E , so that X E L ' . Thus L = L', which is meager since every ME is nowhere dense by 2.2. H By Baire category theorem, a non-degenerate continuum is not meager in itself. Consequently, an indecomposable continuum has uncountably many composants. In fact, it was proved in [Ma] that it is a union of 2'O distinct composants.
I
2.5. Lemma. Let L be a proper subcontinuum of' a non-degenerate continuum H . Then there exists a subcontinuum L' such that L s L' 5 H . Proof. Choose a non-void open U with VL!U n L = 0.For L' we can take the component of H\U containing L. Since, by 1.7, L' intersects %'t U, we have L =i= L'. Proposition. A composant of a non-degenerate indecomposable continuum is dense. Proof. Let H be the continuum, and S the composant. Obviously, g t S is a continuum. If VL!S $. H , we have a proper subcontinuum L 2 g / S by the lemma. But then L c S which is a contradiction. H
5 3.
Limits of diagrams in Top
3.1. The notion of a limit was introduced in E.I.3. In E.I.3 we also discussed the concrete form of products and equalizers. Now, we will present a concrete form of a limit of a general diagram in Top (see 1.3.5.E)). Given D : K + Top, construct first the product P = )( D(k) and denote by P k the natural projections P we have pk(x)= xk ). Put
Q
= {X
EP
k6objK
-+
Ivp: k
D(k) (thus, if x E P, where x +
k',
pk,(X) =
and consider it as a subspace of P. Denote by restrictions of P k .
=
(Xk)k,&jK,
D ( p ) pk(x)} 0
(Pk:
Q -+ D(k) the domain
$3. Limits of diagrams in Top
295
It is easy to see that (Q, ( ( P k ) k ) is a limit of D. (The reader is advised to verify this directly even if he has already done E.I.3.22.)
3.2. We see easily that if all the D(k) are Hausdorff spaces, then Q is closed in P . If objlv is at most countable and the spaces D(k) are metrizable, then P is metrizable. Thus, we have in particular: If all the spaces D(k) are compact Hausdorff (compact metrizable and objK at most countable, resp.) then Q is compact Hausdorff (compact metrizable, resp.). rn 3.3. Let D, D': K -+ Top be diagrams, ( X , ( A k ) ) , ( X ' , (4)) their limits and z = ( T k ) : D D' a transformation. The (uniquely determined) f :X X' such that & o f = z k o & is denoted by limz. In the concrete form of the limits from 3.1 we have (limz) ( ( X k ) k ) = ( z k ( X k ) ) k . -+
-+
3.4. Denote by N the inversely ordered set of natural numbers considered as a thin category (i.e., objN is the set of natural numbers and there is exactly one morphism m -+ n, say A:, iff m 2 n, otherwise there is none). A diagram D : N + Top will be called inverse spectrum. In most cases, it will be described in the form
((xfl), (fnrn)) ' where X , = D(n), f," = D(A:). (Obviously, it suffices to determine the mappings fi' '.) Its limit will then be denoted by (X7(fa)) . It will often be suitable to assume that it is in the concrete form from 3.1. 3.5. Let N' be an infinite full subcategory of N and let D': N' + Top be the domain restriction of D : N -+ Top. We see immediately that:
( X ,(f,),,Go,,,N) If then ( X , (fn)ntob,N)
is a limit of D , is a limit of D' .
3.6. Proposition. Let ( ( X f l )(fn)ll)) , be an inverse spectrum, ( X , ( f , ) ) its limit, and C a closed subset of X . I f fn(C) = f n ( X ) for infinitely many n, then C = X . Proof. If there is an XEX\C, which is open, then there are natural numbers n l , ..., n, and neighborhoods U , of fn,(x) in X f l , such that
nf n ; '( U,) J
I =
I
c X\C.
Choose an m 2 max ( n l ,. .., n,) such that fm(C)= f m ( X )
fc(
and a neighborhood I/ of f m ( x ) such that V ) c U , for i We have V n f m ( X ) 8, contradicting V n f m ( C )= 8.
+
=
1, ...,j .
rn
296
App. A : COOK CONTINUA
3.7. Convention. Let (X, (fn)) be a limit of an inverse spectrum ((X,), (Lm)). If the X , are metric spaces with bounded metrics on,we metrize X (see 3.2) usually by
where d,
=
diamX, if diamX, > 0, d,
=
1 otherwise.
3.8. An inverse spectrum ((Xn),(f;lm)) such that all X , are continua is called a continuum-valued inverse spectrum (briefly, CV-spectrum). If, moreover, all the f; are onto, it is called approach. Proposition. Let (X, (f,))be the limit of a CV-spectrum ((X,,),(f;)). Then X is a continuum. If ((X,), (fnm)) is an approach and if some of the X, are non-degenerate, then X is non-degenerate. Proof.. Put P = )( X,, and denote by p , the projections P -+ X,. Put n= 1
Q,
=
{ x ~ P I p ~ (= xf i)n o p n ( x ) for i
Being homeomorphic to X X , ,
X
m
m=n
= n= 1
=
0, ..., n } .
Q, is a continuum. We see that
Q,, and hence it is a continuum by 1.6. If the f; are onto, all the f,,
are onto so that if X, is non-degenerate, X cannot be degenerate.
3.9. Proposition. Let (X, (fn)) he the limit of a CV-spectrum ((X,), (fnm)). Let us have subcontinua H , of X, such that f ~ i l ( H , + l c ) H,. Put H = { x E X I ~ , ( X ) E H ,f o r a l l n } . Then (H,(h,)) is a limit of ((H,),(hT)), where h;, h, are the domain-range restrictions off;, f,. Consequently, H is a continuum. Proof. Straightforward.
8 4.
Snake-like and circle-lie continua. Solenoids
4.1. Notation. R designates the set of all real numbers, I the closed unit interval, K the complex plane, and C the unit circle ( z E K (z(= l}. R, I, K and C are considered metric spaces with the usual metric.
1
4.2. Let (H, (h,)) be a limit of an approach (see 3.8) ((H,), (hy)).If all H , are homeomorphic to I (resp. C), then H is called snake-like continuumabbreviated SC (resp. circle-like continuum-abbreviated CC). If H , = C for all n and if hi+'(z) = zPn (where z is a complex number and p , an integer), we say that H is a P-adic solenoid, where P = (p,,),. Since the
$4. Snake-like continua etc.
297
h:+' have to be onto, (pn(2 1. If (pnl > 1 for all n, H is called polyadic solenoid. 4.3. Proposition. A polyadic solenoid is a non-degenerate indecomposable continuum. Proof (notation as in 4.2): H is a non-degenerate continuum by 3.8. Suppose that H = A u B with proper subcontinua A, B. Put A , = h,(A) and B, = h,(B). By 3.6 there is an n such that A,, B, are proper subcontinua of C. Choose an a E A,\& and a b E &\A,. Put L(x) = (h:+')-' (x). We have L(a) c A,+l\B,+l,L(b) c Bn+l\A,+l. Since (p,( > 1, both L(a) and L(b) contain at least two points and the points of L(b) are in distinct components of C\L(a). This is a contradiction, since L(b) c B,+, c c C\L(a). rn 4.4. Proposition. A non-degenerate proper subcontinuum of a snake-like
or circle-like continuum is a snake-like continuum. Proof (notation as in 4.2, and H, = C for all n or H , = I for all n): Let L be a non-degenerate proper subcontinuum of H . Put L, = h,(L) so that hF(L,) = L,. For n 2 no, where no is sufficiently large, L, has to be a proper subcontinuum of H,, and hence homeomorphic to 1. Thus, the statement follows from 3.5 and 3.9. rn 4.5. A finite sequence 9 = (Dl, D,,..., D,) of open subsets of a topo-
logical space H is called linear (resp. circular) chain, abbreviated L-chain (resp. C-chain), iff (i) H
=
u D i , Di\uDj
i= 1
j+ i
+ 8 for all i,
(ii) Di n D j f 8 for i < j iff j = i -t- 1 (resp. j = i + 1 or i = 1 and j = n). (The sets Di need not be connected.) 4.6. A continuum H is said to be L-chainable (C-chainable, resp.) if for each open covering %? there exists an L-chain (C-chain, resp.) 9 = = (Dl, ..., D,)refining %? (i.e. such that for every D,there is an X E V such
that Di c X ) . An L-chainable and C-chainable continuum is said to be LC-chainable. If e is a metric on H , and 9 = (Dl,..., D,)a chain, put e(9)= maxdiamDi. Lemma. Let ( H , Q )be a metric continuum. H is L- (C-, resp.) chainable ifffor any E > o there exists an L-chain (a C-chain, resp.) 9 with e(9)< E. Proof. If H is chainable, consider the covering with open ~/2-balls. On the other hand, given an open covering W,consider a chain with e(9) < E, where e is the Lebesgue number of W (i.e., an E > 0 such that
298
App. A : COOK CONTINUA
for an L c H with diamL < E there is always an X E ~ with ? L cX ; it exists since H is compact). 4.1. Lemma. A snake-like continuum is L-chainable, a circle-like one is
C-chainable. Proof. Obviously, I is L-chainable and C is C-chainable. Let (H,(h,)) be the limit of an approach ((H,), (hr)) with H , = I ( H , = C, resp.). Take the metric a from 3.7 (a, is the usual metric in I resp. C). For an E > 0
find an no such that
m fl=no+1
2-" < +E, and a 6 > 0 such that a,(hF(x), hF(y))<
< $E whenever a,,(x, y ) < 6 and n I no. Let 9 = (Dl,..., D,) be a chain < E for D'= (h,il(Dl),..., h,il(D,)). in H,, with a,,(9) < 6. Then we have ~(9') 4.8. A zero-adic circle-like continuum is an H such that ( H , (h,)) is a limit of an approach ((H,), (hr)),where H , = C and h:+' = p,, a, with arbitrary mappings onto a,: C -+ I and p,: I -+ C. Lemma. A zero-adic circle-like continuum is snake-like. Proof. Put F, = I, f;+' = a, bn+ f, = a, h,, 1. We see easily (cf. 3.5) that (H,( f n ) ) is the limit of ((F,), (fnm)). rn Corollary. A zero-adic CC is LC-chainable. rn 0
0
0
4.9. Proposition. Let H be a non-degenerate continuum. Then: (u) H is L-chainable iff it is snake-like, (p) H is C-chainable iff it is circle-like, ( y ) H is LC-chainable iff it is a zero-adic circle-like continuum. We already know that an SC, a CC and a zero-adic CC are L-chainable, C-chainable and LC-chainable, respectively. Thus it suffices to prove the converse. The proof can be based on the same principle for all three statements. We will, therefore, carry out the argument for the most complicated ( Y ) only. Lemma. Let H be a non-degenerate LC-chainable continuum, Q its metric. Then there exists a sequence {9,} such that (i) each 92n+l is an L-chain and each g2,,is a C-chain in H , (ii) for every D E 9,+there is an E E 9,such that for every D'E 9,+ D n D 8 we have D' c E, (iii) e(9,) I 2-". Proof. Let 9,= {D;, ..., D;(,)} be defined. Take an E > 0 such that E 5 2-"-' and such that every X c H with diamX < 3~ is contained in some 07.Then take for 9,+an arbitrary L- or C-chain with a(9,+ < E.
+
rn
$4. Snake-like continua
299
etc.
Proof of the proposition: I. Let (9,) be the sequence from the lemma, 9,= (D'i, ..., Di(,,J. We may assume that k(n) > 1. Denote by dl the point of the k(n)-dimensional Euclidean space with i-th coordinate 1 and all the other coordinates 0. If DY n D; 8, denote by s,(i,j) the segment joining dY and d;. Put L, = Us,(i,j). Thus, L 2 n - 1is always homomorphic to I , and Lz, to C. 11. We will show that for each n there is an m > n and a mapping A of L, onto L, such that (ct) m - n is odd, A maps each sm(i,j)either linearly onto a segment joining d; with gd; d:), or onto a point d; or 3d2, d:) such that DT u DT c D; and 0% n Dt =I8. = Really, for every p , q such that 02,n0:: 8 choose a point xp,qE D;nD; and an E > 0 with E < min e(xp,q,H\(Di n 0:)).Choose an m such that
+
(a)
+
+
+
P.4
m - n is odd and 2 - m + 3< E . For any choose i = i(p, q) with x ~ , DT. ~ E By the choice of m, Dy&j are subsets of D; n Di for j = 1, ..., 4, if there is a C-chain, + and - are considered modulo are so many sets in grn(if 9,,, k(m)). Since k(n) > 1, at least DT+j or Dy-j are in 9,. Now, put i(df+ j ) = Hd;
i(dy+
=
+ d;) d;,
for j
=
0,2,4,
i(dy+ 3) = d;
If there is no Dy+j in grnfor some j = 1, ..., 4, replace i + j by i - j . For the other i put
and Df is a subset of no intersection of two members of 9,
.
We see that the linear extension of this iover the segments maps L , onto L, and satisfies (ct) and 111. Now, define an approach ((H,), (hr))by induction putting H , = L1, H , + = L,, h;+,l = 2, where rn and A are chosen by 11. IV. For X E H denote by $,,(x) the union of the segments s,(i,j) with x E Dl (there are at most three of these, namely, if x E D; n D;+1, sn(I - 1, I), s,(1, 1 1) and sn(I 1, I + 2)). Obviously, h~+l($,+l(x)) c
(a).
c $,(x),
and
+
n hr($,(x)) m
m=n
+
contains exactly one point, which will be
denoted by h,(x). We have obviously hy(h,(x)) = h,(x). On the other hand, assume we have a sequence (y,),, with Y,EH, and h:(ym) = y,. Let y, be
300
App. A: COOK CONTINUA
the union of all Dl with yn E sn(i, j ) (there are at most three of these). One sees easily that
n T,, contains exactly one point, say x, and that h,,(x) m
)I=
= y,.
I
Now consider thc mappings h,: H -+ H,,. For every n there is an 111 > n such that / I ~ ~ ( I ~ ~ , ~ c ( . Y.Id )) $,(x) (the interior of $,(x) in H,) and hence h,(x) E 9,t $ n ( . ~ ) . Let U be open in H,, and h,(x) E U c Y ~ t $ , ( x ) There . is a p > n such that h:($,(x)) c U . The intersection V of the DP with x E Df is a neighborhood of x and we have h,( V )c U . Thus, h, is continuous and we see that ( H , (h,)) is a limit of ((H,), (hr)). V. Since each H,, is homeomorphic to C, H is a CC, and since factorizes through a mapping onto I, H is a zero-adic CC. 4.10. We have evidently
Lemma. Let (L,g) be an L-chainable continuum. For every E > 0 there is an L-chain € = { E l , ..., En} such that @(a) < E and Ei\ U V t E , =I=8 for all i. j+i
I
Proposition. An indecomposable snake-like continuum is a zero-adic circlelike continuum. Proof. Let H be an indecomposable SC. Thus, it is non-degenerate and L-chainable. To prove that it is also C-chainable, for e a metric on H and an arbitrary E > 0 we must find a C-chain 9 with @(a) < E. Let € = = ( E l , ..., En) be the L-chain from the lemma. Suppose that n 2 3. Put El = E1\VtE,, 8, = E,\%'tE,-, and choose disjoint proper subcontinua L , and L , of H such that Eln Li $: 8 and Enn Li 8 for both i = 1,2 (see 2.3 -2.5). Choose x 1 EL,n El and x, E L1 n En.Let 6 > 0 be such that: (i) 6 < + e ( LL,), 6 < +e(xt, H\EJ (t = 1, n), (ii) if diamX < 6, then X c Ei for some of i = 1, ...,n. Let 9 = (Gl, ..., G,)be an L-chain in H with e(9) 6. There are natural numbers ii, 5 such that L , n G, 8 iff ii s j I 6. Find a minimal sub-chain G,, Go+,, ..., G, of G,,G,+l, ..., G6 such that either x1 E G, and x, E G, or x1 E G, and x, E G,. For k = 2,. .., n - 1 denote by Ak (resp. Bk) the union of all Gjc Ek with a I j S b (resp. Gjc Ek with j < a or j > b). Put 9 = ( E l , A,, ..., A,- 1, En,B,- ..., B,). 9 is a C-chain. Now, the statement follows from 4.9.
+
-=
+
,,
6 5.
Crooked mappings
5.1. Recall that I designates the unit interval of real numbers. A path in a space P is a continuous mapping g: I -+ P. If g is one-to-one, g(I) is called arc. The following statement is evident:
301
$5 Crooked mappings
Lemma. Let g: I P be a path, g(0) =+ g(1). Then there exist a, b E I such that a < b, g(0) == g(a), g(1) = g(b) and g(x) $ {g(O),g(1)) for a<x
5.2. Let P, Q = ( Q , e) be metric spaces and E > 0. A continuous f : P --t Q is said to be &-crookedif for each path g: I -+ P there are t , , f z E I such that t , I t,, e(fg(o), fg(t2)) I E and e(fg(tl), fgfl)) I E. Observations. (a) &-crookednessis not a topological notion. (b) Iff is &-crookedand g continuous, then f o g is &-crooked. (c) Iff, g : P Q are continuous, iff is &-crookedand if max e ( f ( x ) ,g(x)) 5 6, -+
xeP
then g is (E + 26)-crooked. (d) Let [a, b], [c, d] be closed intervals, and g a linear homeomorphism of [a, b] onto [c, d]. Iff: P + [a, b] is &-crooked,then g o f is (as)-crooked, where a = (d - c)/(b - a).
5.3. Let p be an arc. A crook on p , or, more precisely, an [a, b, c ; d]crook on p , is a continuous mapping f: p -+ p such that there is a homeomorphism h of I onto p and real numbers 0 I a’ < b‘ < c’ < d’ I 1 such that a = h(a’), b = h(b’), c = h(c’), d = h(d‘) and f ( b ) = d , f maps h([O, a’] u [c’, 11) identically onto itself and homeomorphically h([a’,b’]) onto h([a’, d’]) and h([b’,c‘]) onto h([d’, c’]). 5.4. Proposition. For any E > 0 there is an &-crooked mapping of I onto itselfwhich is (I composition of n finite nirniher of crooks 017 1. Proof. This will be done by induction. Assume we have, for E = (tz - 1)an &-crooked h : I I such that h(i) = i for i = 0 , l composed of a finite number of crooks. Put I, = [0,1/2n], I , = [1/2n, l/n], I, = [l/n, 13 and denote by ,Ii the increasing linear homeomorphism of Ii onto 1. Define 1i: 1 4 1 --f
by putting li(t)= 2,; h ,Ii(t) for t E I i , li(t) = t otherwise, and put 1= 1 ~ 0 1 ~ 0 / 3 .
’
(Thus l(t) = 2; h &(t) for t E Ii.)Since obviously every li is a composition of finitely many crooks, 1 is one. Now, let y: 1+1
be the mapping which linearly sends I, onto [0, (n - l)/n], I , onto [l/n, (n - l)/n] (decreasingly) and I , onto itself. Thus y is a crook, and it suffices to show that f = 7.1 is l/n-crooked.
302
App. A : COOK CONTINUA
+
Let g : I -P I be a path. Put g‘ = f o g. We may suppose that g’(0) g’(1) (otherwise, we may choose t l = t 2 = 0) and, by 5.1, that g’(x)4 (g’(O), g’(l)} for 0 < x < 1. Thus we have either for an i
g(/) c Zi
or
0I g’(0) <
1
=
1,2,3
n-1 < g’(1) I 1 n
and
~
or
In the first case the statement follows from 5.2(d) since the restriction : Zi -+ f ( I i ) off is (l/n)-crooked. In the second case, put
I
t l = inf { t g’(t) = ( n - I)/.} ,
I
t 2 = sup ( t g’(t) = l/n} .
Analogously in the third one. 5.5. Recall that C designates the unit circle in the complex plane K . Let p be the closed interval [G, ii 2x1, q an [Z,6,T; 4-crook on p , 5 < 6 < C < d < G 2x. The mapping f : C -P C defined by f(eif)= ei4’@) is called [a, b, c; d]-crook on C, where a = ei’ etc. Let a 0 be a real and B a complex number, and let 1: K -+ K be defined by l(z) = c1z fl. Iff is an [a, b, c; d]-crook on C, then 1 o f 0 1is called an [l(a),l(b),l(c); l(d)]-crook on l(C).
+
+
+
+
5.6. Let n 2 3 be a natural number. Denote by Z j ( j = 0, ..., 2n - 1) the closed interval [j/2n, ( j 1)/2n], by 6, the mapping of I,, u 12,+ into [0, 2 - 3/2n] sending Z21 linearly onto [21/2n, (21 2n - 1)/2n] and 121+r onto [(21 + 2n - l)/Zn, (21 2)/2n)]. Put A j = {elnit t € Z j } and define .+ C by*) q,(exp(2xit)) = exp(2nidl(t)).Finally, define y~,:A,, u
+ +
+
yI:
I
C-PC
by q(z) = q,(z) for z E A,, u A l l + 1 . We see easily that q is a composition 0 ... ijl Go of a finite number of crooks on C. (Indeed, we have q = where ijfl-l(z)= V ~ - ~ ( Z for ) Z E A , , , - ~u A 2 f l - l , Y],-~(Z) = z otherwise; . if qn- 1, ..., GI + are defined, we define GI by ijl(z)= z for z E C\(A2, u A,, + 1), ijl(z)= y~,(z) for z, q , ( z ) ~ A , u , A , [ + otherwise G&)E A Z l f 2 such that
-
-m- ... 1
Yll(Z)
=
0
m
%(Z).)
*) In the cases of complicated argument
0
x we write exp(x) for ex.
303
95. Crooked mappings
Proposition. For any E > 0 there exists an &-crookedmapping of C onto itself which is a composition of a finite number of crooks on C. Proof. Choose a natural n 2 3 such that lexp(27~it)- exp(2ni(t
+ (3/2n)))lI
E.
Put 3, = ~ ( 2 7-~(1/2n))-’ and choose a A-crooked composition h of finitely many crooks on 1. Put ‘ p j = g j h 0 9,:’ where g j is the linear mapping of Z j onto 1. Define sj:A j + C by 9,{exp(2dt)) = exp(2niq,(t)), 9: C -+ C by 9(z) = Sj(z) for z E A j . Obviously 9 is a composition of finitely many crooks. Put f = 9. Thus, f is again a composition of finitely many crooks. We will show that f is &-crooked. Let g: I -+ C be a path. Put g’ = f o g . We have to find t,, t , ~ l ,tl I t 2 such that Ig’(0) - g’(t2)15 E and Ig’(tl) - q’(1)l I E. If Iq’(0) - q’(l)l I E put tl = t 2 = 0. Suppose Ig’(0) - g’(1))l > E. By 5.1 we may assume that g’(x)$ {g’(O), g’(l)} for 0 < x < 1. Then, by the definition of q and the choice of n, g(l) c A j for a j = 1, ..., 2n - 1. By 5.2(d) and the choice of 1 the restriction off to any A j is &-crooked. 0
0
5.7. Lemma. Let H i = ( H i ,oi), i = 0, ..., n be compact metric spaces, and 9,: H i - + H i continuous mappings. Put f = g, 0 ... 0 9,. Then for every E > 0 there exist di > 0 such that $ i j i : HiP1-+ Hi are continuous and sup ai(gi(x),iji(x))< ai, then sup o,,(f(x),f(x))< E for f = g,,0 ... 04,.
,
X
X
-Proof by induction: Since g,, is uniformly continuous, there is a J > 0 such that o,,(gn(x),gn(y))< 1:/2 whenever on-,(x, y ) < 6. Put 6,, = r:/2 and choose the dl, ..., 6,- for 2 it1 the place of E. 5.8. Proposition. Let Y be a countable subset of C. Then for any E > 0 there exist [a,, bi,c,; di]-crooks yi ( i = 1, ..., n) such that f = y1 0 ... 0 y, is &-crooked and, for i = 1, ..., n, {ci,d i } n = 8 where Y, = 1 = = (c, d i ) u y; ’( YJ.
x+
Proof. First, choose crooks 6,, ..., 6, on C such that 6, 0‘d2 0 ... 0 6, is ~/2-crooked.Recall that if y is a crook, y-’(x) has three points at most, so that will be countable whatever crooks y i we choose. By 5.7 we can find a crook y1 (and y2, y 3 , etc.) sufficiently close to 6, (a2, d3, etc.) such that {c,, d , ) n Y = 8 (and, further, { c 2 ,d 2 } n Y2 = 8 etc.). Note. The statement is, of course, true for any circle in K .
304
6 6.
App. A : COOK CONTINUA
Hereditarily indecomposable continua
6.1. A continuum H is called a Peano continuum if for any x E H and any open U 3 x there is an open U‘, such that ~ E U c’ U and every Y E U‘ can be joined with x by a path in U . Observations. (a) Evidently, I and C are Peano continua. (b) Every component of an open set in a Peano continuum is open.
6.2. Lemma. Let H be a Peano continuum, U c H open connected. Then any two x, y E U can be joined by a path in U . Proof. For z E U find an open U , with z E U , c U such that for any point u of U , there is a path g with g(I) c U , g(0) = z and g(1) = u. Let x, y E U be given. Since { U , z E U } is an open covering of U and since U is connected, there are U,,, ..., Uzn such that x E U,,, y E U,,, and U z z n U , , + , =k 0. Choose xi E U z , n U,,,, and join x with y via z1, X l , Z 2 , x2, ..., z,. Corollary. If H , are Peano continua, X H , is a Peano continuum.
I
n
6.3. A non-degenerate continuum is said to be hereditarily indecomposable if each of its subcontinua is indecomposable. The expression “hereditarily indecomposable continuum” will be abbreviated by HIC. 6.4. A metric space P is said to be &-crookedif the identity mapping 1,: P + P is &-crooked. Lemma. Let P = (P, e) be a metric space, H , 2 H , 3 ... a sequence of Peano subcontinua of P. Let H , be &,-crooked and let limE, = 0. Then n’m
m
fl H , 1
n=
1
is a one-point set or an HIC.
Proof. The n H , is a continuum by 1.6. Let it be non-degenerate. Let L be a subcontinuum of O H n and L = A u B with proper subcontinua A , B of L. Choose an aEA\B, a bEB\A and an n such that E, < < min (@(a,B), e(b,A)). Let U,, U , c H , be open connected and such that A c
UA,
B
c UB,
@(a,gl UB) > 8, , e(b,ge U A ) > 8,
(they exist by 6.1(b)). Take an X E U , n Us.By 6.2 there are paths CI,p: I + H , with ~ ( 0= ) a, a(1) = p(0) = x, p(1) = b; a(/)c UA and p(I) c U,. Define g: I + H , by g(t) = c((2t) for 0 I t < 7, g(t) = p(2t - 1) for f I tI 1. Since H , is &,-crooked, we have t,, t , ~ l with t , I t2, e(a,g(t2))S E, and e(b,g(t,)) I E,. But then necessarily t 2 < and t l > f which is a contradiction.
4
305
56. Hereditarily indecomposable continua
6.5. Let f : (P, e) -+ (Q, Q) be a continuous mapping between metric spaces, let E be a positive real number. We define L(G f
1
as the supremum of all 6 such that e ( x , y ) < 6 implies o ( f ( x ) , f ( y ) < ) e. If (P, e) is compact, we have L ( & , f )> 0. Proposition. Let F = ((F,,),(f/)) be an approach (see 3.8) let euery = (F,,o,) be Peano and let every fl-l be &,-crooked, where E, < < min L(2-"di,fi"-') for di = diamF, > 0. Let (F,(f,)) be a limit o f F .
F,,
i = 1,..., n - 2
Then F is an HIC. Proof. Put P Put H,
=
m
(P, e) = )( F, (see 3.7), let n,: P n= 1
=
I
{ x E P fij(nj(x))= ni(x)
-+
F,, be the projections.
for i < j In) m
aJ
Obviously, H , is homeomorphic to )( Fm and F to H m=n
=
(IH,. We show
n= 1
that H , is 23-n-crooked, which will prove the statement by 6.4. and 3.8. Let g: I -+ H , be a path. Since f,"-l is &,-crooked, there are t, I t 2 with Q n - l(fn"- l.,S(O), fl-Pf19(t2))I En, Q n - l(fn"- Inn9(tl),fn"-1nn,s(1))I EnThus, we have 0n-1(nfl-1(x)7
nfl-l(Y))
'fl
for ( x ,y ) = (g(0),g(t2)) and for ( x 7 y )= (g(tl),g(l)). By the choice of obtain oi(ni(x),ni(y))< 2-"di for i In - 2. Thus,
E,
we
6.6. Remarks. By 6.5, one can construct easily a snake-like or circle-like HIC. The first HIC was presented in [Kn], others followed in [B,] and [Mo,]. In [B2], R. H. Bing proved that all snake-like HIC are homeomorphic (they are sometimes called pseudoarcs),and showed that all the HIC constructed in the papers quoted are such. Numerous interesting properties of pseudoarcs are known, e.g. : A pseudoarc is a plane continuum (i.e. homeomorphic to a subcontinuum of the Eucliedean plane; every snake-like continuum is a plane one)-
306
App. A : COOK CONTINUA
see [B,]. It is homeomorphic to any of its nondegenerate subcontinua ([Mo,]). It is homogeneous (i.e., for any two x , y there is an autohomeomorphism sending x to y --see [B,], [Mo2]). It is the only homogeneous non-degenerate SC ([B3]). Every SC is a continuous image of a pseudoarc ([Mi]). A pseudoarc can be expressed as a limit of an ((H,),(hr)) with H , = I and all the h:" coinciding [Hn]. It was proved in [B2] that "almost all" plane continua are pseudoarcs.
6.7. We recall that a continuous mapping r : Q -+ L is called a retraction if L c Q and r(x) = x for x E L. Lemma. Let Q be a HIC, L its proper non-degenerate subcontinuum, r: Q + L a retraction. Then there exist non-degenerate disjoint subcontinua A, B such that r(A) = B. Proof. Let B(L) be the boundary of L. Choose distinct x , y E L, x E B(L). By 2.4 there is a composant D of Q which does not contain L. Since D is dense in Q (see 2.5) and G = Q\L is open, D n G is dense in G. Thus, there are sequences (xn),(y,) in D with limx, = x and limy, = y and such that all the x , are in G. Let Qn be a proper subcontinuum of Q containing both x, and y,. Since r(x) = x y = r(y),we have x, y , and r(xm)+= r(ym) for an m. Thus, Qm and r(Qm)are non-degenerate. Since x , 4 L, we have Q\,L 8. Since Q, c D and L\D 8, obviously L\Q, =k 8. Since Q is an HIC, Q, u L is not a continuum so that Qmn L = 8. Since, on the other hand, r(Q,) c L, we may put A = Q, and B = r(Qm).
+
+
+
+
6.8. Proposition. Let H be an HIC, Q a subcontinuum o f H and f: Q -+ H a continuous mapping. Then either f is a constant, or f( x ) = x for all x E Q, or there exist non-degenerate disjoint subcontinua A, B of H such that A c Q and f ( A ) = B. Proof. Let f be non-constant so that L = f(Q) is non-degenerate. Put R = { x E Q f ( x ) = x } . Obviously R is closed. I. Let there be a y E Q such that f ( y )4 R. (a) First, we prove that there is a U open in Q containing y , and a G open in H containing f(y), such that V / U n %'/G = 8 and f-'(G) = U . This is evident if f ( y ) 4 Q. Thus, suppose that f(y) E Q. Since f(y) 4 R we have f(f(y)) f(y) so that there is an open neighborhood G' of f(y) such that %'/ G' does not contain f(f(y)). Then M = f - ' ( V t G ' ) is compact, it is a neighborhood of y in Q, and f(y) 4 M . Let G c G' be an open neighborhood of f ( y ) such that its closure does not intersect M , put U = f - ' ( G ) . Then %'/U c M and hence V e U n V / G = 8.
I
+
97. Monotone, atomic and confluent mappings
307
(p) Let T be the component of U containing y , put A = WL'T Since A intersects the boundary (in Q ) of U (see l.S), it is non-degenerate. Hence, B = f ( A ) containing f ( y ) and intersecting the boundary of G is also nondegenerate. 11. Now, let f ( y ) E R for all y E Q. Then R = L c Q andfis a retraction. If L = Q, f(x) = x for all x E Q, if L s Q, use 6.7.
9 7.
Monotone, atomic and confluent mappings
7.1. A continuous mapping f : F + G where F , G are compact metric spaces is said to be monotone if f - ' ( z ) is non-void connected for all z E G. Lemma. Let f : F G be monotone. I f Q is a subcontinuum of G, then f - ' ( Q ) is a subcontinuum of F . Proof. Suppose that f - ' ( Q ) = A u B with disjoint non-empty closed A , B. Since f - ' ( z ) are connected, we have f(A)n f ( B ) = 8. Since A and B are compact, f ( A ) , f ( B ) are closed and we obtain a decomposition Q = = f ( A ) u f ( B ) in contradiction with the connectedness of Q. m Corollary. A composition of two monotone mappings is monotone. -+
7.2. Lemma. Let B = ((F,), (Lm)), 93 = ((G,,), (gr)) be approaches, B 9 a transformation with all cp, monotone. Then lim cp, '(see 3.3.) is monotone. Proof. Let ( F ,(f,)), (G, (9,)) be limits of B, 29 respectively. The mapping 'p = lim cp, is onto since all f,,, 'p,, and g, are onto and G is compact. For an x E G put x, = g,,(x).We have cp- '(x) = ( z E F fn(z)E cp, '(x,,) for all n]. By 3.9 this is connected since cp,-'(x,) are subcontinua of F,,. (cp,,):
-+
I
7.3. A continuous mappingfof a compact metric space F onto a compact metric space G is said to be preatomic if for any subcontinuum H of F with non-degenerate f ( H ) , H = f - ' ( f ( H ) ) . An f is said to be atomic if it is preatomic and monotone. Observation. A composition of two (pre)atomic mappings is (pre)atomic. Remark. By [EH], a preatomic mapping is atomic (we will not use this fact, though). 7.4. Lemma. Let F, 9 be approaches, (cp,,): F -+ 9 a transformation such that all cpn are atomic. Then limcp, is atomic. Proof. (The notation from 7.2.): By 7.2 it suffices to prove that cp is preatomic. Let Q be a subcontinuum of F with cp(Q)non-degenerate. Suppose that there is an x E q-'(cp(Q))\Q. Then there is an m such that fm(x) $fm(Q).
308
App. A : COOK CONTINUA
Since q(Q) is non-degenerate, gi(q(Q)) is non-degenerate for an i 2 m. Then, q; '((pi ofi(Q))= A(Q) does not contain h(x) so that q,(f;-(x))is not in 'piof;(Q) so that finally q ( x ) is not in q(Q), which is a contradiction. 7.5. Lemma. Let 9 = ((Fn),(Lm)) be an approach such that all f," are atomic, ( F , (f,)) its limit. Then eachf, is atomic. Proof. Take an no and put 9 = ((G"), (gr)) with G, = Fn0 and g r identical. Consider a transformation (q,):9 -+ 9 with q, = f: for n > no. Since obviously f,,= limq,, the statement follows from 7.4. 7.6. Lemma. Let G be an HIC, F a continuum, and let f:F + G be atomic. Let, for every y E G, f ' ( y ) be either a one-point set or an HIC. Then F is an HIC. Proof. If F is not an HIC, there exist two intersecting subcontinua Q1 and Q 2 of F such that neither contains the other. We have f(Q1 u Q2) non-degenerate (otherwise, f-'J(Q1 u Q2) would contradict the assumption). Since f(Q1) u f(Q2) = f(Q1 u Q 2 ) is a subcontinuum of G, we have either f(Q1) c f(Q2) or f(Q2) c f(Q1). Let the former be the case. Then we obtain Qz = f-'(f(Qz)) = f-'(f(Q1 u Q2)) = Q I u Qz, which is a contradiction. 7.7. Lemma. Let 9 = ((F,),(fnm)) be an approach such that (i) F, is an HIC, and (ii) every fn"' is atomic and, for y E F,, ( f:+ ')- ' ( y ) is either a one-point set or an HIC. Let ( F , (f,))be a limit of 9. Then F and each F, is an HIC. Proof. Each F, is an HIC by 7.6. Now, let Q1 and Q2 be two intersecting subcontinua of F such that neither contains the other. Let n be the least integer such that L(Q1u Q 2 ) is non-degenerate. Then fn(Q1 u Q2) is an HIC, since it is a subcontinuum of the HIC F,. Thus, we have (after possible relabeling), $(Q1) cfn(Q2). Since f, is atomic (see 7.5) we have Q2 = = fn-'(fn(Q2)) = f; '(f,(Ql u Q2))= Q1 u Q2 which is a contradiction.
'
7.8. A continuous f: F -,G is said to be confluent if for any subcontinuum Q of G each component of f-'(Q) is mapped by f onto Q. 7.9. Lemma. A continuous mapping of a continuum onto an HIC is confluent. Proof. Let f : F + G be the mapping, Q a subcontinuum of G and L a component of f-'(Q). Suppose that L =k F. I. Put U , = (x E F @(x, L ) < l / n ) (e is the metric of F), let L, be the
I
309
$8. Upper semicontinuous mappings
closure of the component of U , containing L. Since L, intersects the boundary of U,, L ,
=+ L
and we have L
n L , and L, W
=
n=
1
+
t L,
for all n.
11. Since L c L, we have Q n f(L,)=k 8. Since L, is a continuum and L a component off - ‘(Q), we have f(L,)\Q =k 8. Since G is an HIC, Q u f(L,) is indecomposable and we obtain Q c f(L,).
111. Thus, Q c
nf(L,). Since L,, m
c L,, we have n f ( L , ) = f(L).
fl=l
Thus, Q c f ( L ) .Since, on the other hand, f ( L ) c Q, we obtain f(L)
6 8.
= Q.
Upper semicontinuous mappings
8.1. In this and the following paragraphs, the symbol f: P + Q is used also for multivalued mappings (strictly speaking, such a mapping is actually a mapping P + expQ). I f f is multivalued and A c P, we put f(A)= = f(x) and f is said to be onto if f(P)= Q in this sense. It is called
u
xeA
upper semicontinuous mapping (abbreviated : USC mapping) if for every x E P and every open U 2 f(x) there is an open V 3 x such that f (V )c U . The convention on f(A) allows us to define the composition of multivalued mappings by the formula ( g o f ) (x) = g(f(x)). We see easily that if f : P + Q is USC, A c P and U 3 f(A) is open, then there is an open I/‘ I> A such that f(V)c U . Thus, a composition of USC mappings is USC.
8.2. For a sequence (A,) of subsets of a topological space P we put lim sup A, = n+m
n %‘f( u A,). W
W
n=l
m=n
Proposition. Let f : P + Q be USC and let Q be a regular TI-space.Let (x,) be a sequence in P converging to x, let f (x) be closed. Then
Proof. Suppose that there is a y E lim supf(x,)\f(x). Choose disjoint open U 3 f(x) and V 3 y, and find an open G 3 x with f(G) c U . Then, for a sufficiently large n, f ( x , ) c U while f(x,) n I/ =+ 8 for infinitely many n. 8.3. Proposition. Let P, Q be compact metric spaces and f : P a (single-uahed)continuous mapping onto. Then f - is USC.
+
Q
310
App. A : COOK CONTINUA
Proof. Suppose the contrary. Then we have a ~ E and Q an open U 2 f -'(y) such that f -'(G)\U = 8 for l any open G 3 y . Hence there is a sequence (y,) converging to y and points x , E f - ' ( y , ) such that x , 4 U . Let x be the limit of a subsequence of (x,). We have y = f ( x ) , and hence x ~ f - ' ( y )c U , which contradicts x U . 8.4. A multivalued f: P + Q is said to be continuum valued (abbreviated CV) if each f ( x ) is a subcontinuum of Q. Obviously, a single-valued continuous mapping is an USC and CV mapping.
8.5. Proposition. Let P , Q be metric spaces, f : P -,Q an USC and CV mapping, H a subcontinuum of P. Then f ( H ) is a continuum. Proof. I. First, we prove that f ( H ) is compact. Let (y,) be a sequence of points of f ( H ) . Choose x , E H such that y, E ~ ( x , ) . Let x be the limit of a subsequence of (x,). Let us suppose that no subsequence of (y,) converges to a point of f ( x ) . Since f ( x ) is compact, there exists an open U 2 f ( x ) such that no y , is in U . The mapping f is USC so that there exists an open V 3 x with f ( V ) c f ( U ) . This is a contradiction because V has to contain some x,. 11. Suppose that f ( H ) = A u B with closed disjoint A, B. Then H = = (x E H f (x) c A } u (x E H f ( x ) c B } with disjoint summands. It is easy to see that the summands are closed. Thus, some of them, say the first, has to be void. But then obviously A is void. Hence, f ( H ) is connected. Corollary. The.composition of USC and CV mappings is USC and CV.
1
1
8.6. Lemma. Let P, Q be continua, f : P -+ Q an USC and CV mapping, T c Q. Then ( x E P T c f ( x ) ) is closed. Proof. Let (x,) be a sequence such that for each n T c f(x,,),let x be its limit. Then, by 8.2, f ( x ) 2 lim supf (x,) 2 T.
1
8.7. Proposition. Let P , Q be continua, f a n USC and CV mapping of P onto Q . Then there exists a subcontinuum H of P such that (i) f ( H ) = Q, (ii) for any proper subcontinuum H' of H,f(H') Q. Proof. Denote bv 3 the inclusion ordered set of the subcontinua H such that f ( H ) = Q . Let X ' be a monotone subset of 3,put X = (I H .
+
H€X'
By 1.6 X is a continuum. Take a y E Q . For H E Y?' put S ( H ) = = {x E H y ~ f ( x ) ) Obviously, . each S ( H ) is closed and the intersection of
1
311
$9. Mappings of a circle into itself
any finite number of sets S ( H ) is non-void. Since P is compact, so that y Ef ( X ) .Thus, X
E 8.Since
n S(H) =+ 8
HEX'
P E 8 we can now use Zorn's lemma.
8.8. Proposition. Let ( P , e),(Q,o)be compact metric spaces, f: P + Q an USC and CV mapping. For every E greater than the diameter of any f ( x ) there is a 6 > 0 such that e(x, y ) < 6 implies o(f(x), f ( y ) )= inf a(z, z') < E. z e f ( x ) ,Z ' E f ( Y )
Proof. Suppose the contrary. Hence, there are sequences (x,,) and (y,) converging to x such that o(f(x,), f(y,,)) 2 E. Choose an q > 0 such that diam U < E for U = { z E Q o(z, f ( x ) ) < 111. There is an open G 3 x such that f ( G ) c U . Thus, f(x,,), f ( y , ) c U for n sufficiently large, which is impossible. R
I
8 9.
Mappings of a circle into itself
9.1. Besides the conventions and abbreviations from use some others. First, denote by
$3 4 and 8, we will
@
the set of all USC and CV mappings cp of C onto C such that every p(z) is a point or an arc of the length at most 1. Let a cp from @ be given. For any t E R choose a half-line pt starting in 0 such that pt does not intersect q(eit), and choose a continuous branch log of logarithm on K\p,. Let at be an open interval on R containing t and such that for s E at,cp(ei")does not intersect p t either. Denote by Z the set of all integers. For any k E Z and any S E % , put 1 $t,k(S) = i log Z -k 2 k ~ Z E q(e'")
{
1
I
Obviously, $ t , k is an USC and CV mapping of at into R and (See 8.5, is a union of an increasing sequence of continua)
(4
gr $t,k = ((&Y )
is a closed connected subset of
1
at?
at x
Define T,: R -+ R by T,(t) =
u
ke Z
Y
$t,k(S)}
R $t,k(t).Obviously, T, is independent
on the choice of the branches of logarighms and on the choices of'p,'s and
(ii)
if s E 42t,
then
T,(s) =
u
keZ
$t,k(S).
312
App. A : COOK CONTINUA
I
Put gr T, = { ( t ,y ) y E T,(t)). One can verify easily that (a) for any t E R, all the components of gr T, intersect ( t } x R, (b) for any distinct t , t' E R with Uzi, n a,, =k 8 there is a unique pair (k, k') of integers such that gr $t,k n gr $t',k' 4 8. Hence, for any t, the components of gr T, are in one-to-one correspondence with the sets { t } x $ t , k ( t ) . Every component of grT, intersects ( t } x R exactly in this set. By (i) and (ii), every component of grT, determines a USC and CV function $1
R+R.
This has the following properties. (0) $(R) contains an interval of the length 27t. (1) $(x) is always a closed interval (possibly degenerate) of a length I 1, 1 (2) %(sup $(27t) - sup $(o)) is an integer, ( 3 ) for any X E R sup $(x
+ 27t) - sup $(x) = inf $(x + 271) - inf $(x) = =
sup *(27c) - sup $(O) .
(The properties (0)-(2) are obvious, ( 3 ) may be proved starting from the values on [0,27t] of all such Ic/ and extending them by suitable shifts; then these extensions must coincide with the values of the $'s outside [0,27t] .)
9.2. Denote by Y the set of all USC and CV mappings $: R that (0)-(3) from 9.1 are fulfilled. For $ E Y put
-+ R
such
1 W($)= -( SUP $(27t) - SUP W)) 27c
9.3. Clearly, every $ E Y determines a cp E @ by
I
q(e") = (eiZ z E $(t)>. (This definition is correct by (2) and (3) from 9.1.) Conversely, by 9.1, every cp E Q, determines countably many $'s from Y , but they differ only by the additive constant 2k7t; denote by cp* some of these $'s.
313
59. Mappings of a circle into itself
Observation. If cpl, cp2 and cpl (4%
O
q2 are in
0
cp2)* =
cp:
O
40;
Q,,
then
+ 2zn
for an integer n. If cp E Q, is single-valued, cp* is also single-valued. 9.4. For cp E Q, the values W ( q * )and M(cp*) obviously do not depend on the choice of cp*. Put
W(cp)= W(cp*) M(cp) = M(cp*). Observations. (1) If cp is single-valued and if a natural number n divides W(cp),then cp has a continuous single-valued n-th root. (2) W ( q )= 0 iff cp factorizes through a mapping into 1. Remark. For cp E Q, single-valued, W ( q )is the usual degree of q. Convention. If C1, C 2 are circles in the plane and q : C1 + C , a singlevalued continuous mapping, put W(cp)= W(LX;'0 cp 0 a l ) where cci: C + C i are linear. 9
9.5. Lemma. Let cpl(z)c cp2(z)for all z E C. Then W(cpl) = W ( q 2 )and M(cp1) 5 M(cp2). Proof. We can choose q: and cp; such that cpf(z) c q?(z). Thus, W(cp:) = W ( @ ) by (1) and (2) in the definition of Y . The inequality M(cp7) I M ( q z ) is evident. 9.6. Lemma. For any $ E Y , any integer m and x E R we have sup $(x
+ 2m.n) - sup $(x) = inf $(x + 2m.n) - inf $(x)'=
2.nm W($).
Proof. Suppose m 2 0. We have
+ 2m.n) - sup $(x) = = (sup$(x + 2mn) - sup$(x + 2(m- 1 ) ~ )+) (sup$(x + 2(msup $(x
1)n)- sup$(x)).
Since the first summand is equal to 27t W($),the statement for the suprema follows by induction. An analogous procedure applies for m < 0 and for the infima.
rn
9.7. Lemma. If $2 E Y , and if t+h2 is single-aalued, then $1 $2 E Y and W(*l $ 2 ) = W($l). W(ICI2). Proof. Obviously (see Corollary in 8.5) h,t1 . t+b2 is a USC and CV mapping satisfying (0) and (1) from 9.1. For an x E R we obtain (using (3) for $2) 0
3 14
App. A: COOK CONTINUA
+
supt,h1(t,b2(x 271.)) = S U ~ $ ~ ( $ ~ ( X+) 271. W(t,h2))which is equal, by 9.6, to SUP $1(*2(x)) + 271. W($l). W($2). Thus, 1 -(SUP *1($2(x + 2n)) - SUP $1(*2(x))) = W($l). W($2) 2x (analogously for the infima). Thus, have W(91 *2) = W(*l). W(*2).
0
$2
also satisfies (2) and (3) and we
O
9.8. Lemma. If $2, t+h1 0 4b2 E Y and W($l O $2) = W($l). W($Z). Proof. I. Let X E t,h2(O) be such that @l(X)
if
$l
is single-valued, then
= SUP $1($2(0)).
Since inf $2(0) Ix 5 sup t,b2(0),we have
+ +
inf t,h2(271.)= (inf $,(2x) - inf $2(0)) inf t,h2(0)= = 2x W($J inf $2(0) s 271. W(t,h2) x s
+
+
2x W($z)+ SUP $Z(O) = SUP $2(2Z)
and hence x 271. W($J E $2(2x). Thus, by 9.6, $1(x + 2n: W($2))- $1(x) = 2x W($l). W($Z). 11. Analogously, choosing y E $,(2n) such that $,(y) = sup 14~($~(271.)), we obtain $l(Y) - $l(Y - 2x W($2))= 271. W($l). W($2). 111. Thus, *l(X + 271. W($2))- $1(4= $l(Y) - $l(Y - 271. W($2)). Put A = h ( Y ) - $I(X + 271. W($2)) B = $&) - $1(Y - 2n W($2)) * 9
The x and y are such that A 2 0 and B 2 0. Since by the equation above A + B = 0, we obtain $l(Y) = $1(x + 271. W($z)). Consequently, 271. W(*l O $2) = $l(Y) - *2(x) = = $1(x + 2x W($2))- $l(X) = 2n W($Z).W($1).
9.9. Lemma. If +hl, $2 E Y , $2 is single-valued and W($l) = 0, then M($1) 2 M($l $2). Proof. Choose xo, x1 E [0,2x] with inf $l($z([O, 271.1))= inf $~($z(xo)) and sup $1($2([0,2x1)) = sup $1($2(x1)). Further, choose to,t l E [O, 271.1 O
315
$10. Non-homeomorphic solenoids
and integers no, nl such that $,(xi) = ti 2n M($l
O
$2)
= SUP *1(*2(Xl)) =
sup rC/,(t,
+ 2nni for i = 0 , l . Then, - inf*1($2(xo)) =
+ 2 7 4 - inf $,(to + 2xn0).
Since W ( I / I= ~ )0, we obtain by 9.6 2l.c M($l
O
$2)
=
SUP $&l) - inf$,(to)
2n M(*l).
9.10. Lemma. If I/I~, $2, ri/2 E Y and if $ J ~is single-valued, M($l $ 2 ) 2 IW($l)lProof. By (0),(2) and ( 3 ) from the definition of Y , $2([0,2n]) contains an interval of the length 2n. Thus, there are xo, x1 E [0,2n], yo E $2(xo) and y, E ~ + b ~ (such x ~ ) that y1 - yo = 2n and we have 0
O
271 M($l
O
*2)
2
l*l(Yl)
=
/$l(YO
-
*l(YO)J
=
J*l(YO
+ 2n) - $l(YO)J
+ Yl
- Yo) -
=
*l(YO)I
= 2+qih)/.
9.11. Using 9.7 and 9.9 we obtain the following proposition, which will play an important role in the next paragraph. Proposition. Zf ql, (p2 E @ and if 'p2 is single-valued, then 'p, 'pzE @ and W(p1 0 q 2 )= W ( q l ) W((p2). . ZJ; moreover, W(cpl) = 0, then M(cpi) 2 2 M(v1 v2). Similarly, by 9.8 and 9.10 we obtain Proposition. If q l ,cp2, ql 0 q2E @ and if 'pl is single-valued, then W(q1 O v2) = W(v1). W(v2) and M(v1O v2) 2 pV(v1)I. 0
O
0 10.
a
Upper semicontinuous mappings between circle-like continua. Non-homeomorphic solenoids
10.1. The conventions and abbreviations used here were introduced in 4.1 and 9.1 -9.4. Furthermore, an approach ((H,), (hr)) with H , = C is to be called circle approach. If, moreover, W(h;+')= q,, let us call it a Q-adic circle approach with Q = (q,),= l , 2 , , . . ; if moreover h;+'(z) = zqn, it will be called solenoidal Q-adic approach (thus a Q-adic solenoid is a limit of a solenoidal Q-adic approach). A limit H of a Q-adic circle approach is called Q-adic circle-like continuum. If q, = q for all n, then H is also called q-adic circle-like continuum (observe that "0-adic CC" coincides with "zero-adic C C defined in 4.8; cf. 9.4). If lqnl > 1 for all n, H is called polyadic circle-like continuum (abbreviated PACC).
316
App. A : COOK CONTINUA
Let P , Q be non-degenerate continua, f : P -+ Q a multi-valued mapping. It is said to be proper-subcontinuum-valued if every f ( x ) is a proper subcontinuum of Q. The expression “proper-subcontinuum-valued upper semicontinuous mapping” will be abbreviated to “PSV and USC mapping”. Evidently, every PSV and USC mapping is CV and USC, so that the results of 9 8 may be applied. 10.2. Lemma. Let ( H ,(h,)) be a limit o f a circle approach ((H,), (hr)),and (S, (s,)) a limit ofa circle approach ((S,), (sr)). Let s;+’(z) = zqnfor all Z E S,+ where q, is an integer, (qn(> 1. Let there exist a PSV and USC mapping f of H onto S. Then there exist increasing sequences (ni) and (mi) of natirral numbers, and PSV and USC mappings cpi: Hmi+ Sni such that for i < j always
,,
(1)
and
s20cpj
cpio
hz
are in @ ,
(2)
W(s2 cpj) = W(cpi h z ) ,
(3)
M ( s 2 cpj) 5 M(cpi h z ) .
0
0
0
0
Proof. (a) First,. we will show that there is an no such that, for all x E H and all n 2 no, s,(f(x)) S,. Suppose the contrary. Then, there is a sequence (x,) in H such that s,(f(x,,)) = S,. Consequently, for an m 2 n, sn(f(xm))= s ~ ( s m ( f ( x m= ) ) )s,. Let x be a limit of a subsequence ( X k n ) of (x,). Then, we have s,(f(x)) =, lim sup s,(f(xk,)) = S , for each m, so that by 3.6,
+
n-tm
f ( x ) = S, which is impossible. (b) Now, we will show that there is an n, such that for all n 2 n , and all X E H , s,(f(x)) is an arc of a length less than i. For n > no put
r, =
n
n-1
i=no
qi. We have s,n0(z)= z’” and hence the preimage of an arc in S,,
of length d < 2n: consists of disjoint arcs of the length d/r,. Thus, since s,(f(x)) is connected, it suffices to take for n, any k 2 no such that 2x/rk < 3. (c) Put ni = n1 + i - 1. Let e be the metric of H from 3.7. Choose real numbers ci > 0 such that for x , x’ E H with e(x,x’) < ei there is always an arc of a length less than 3 intersecting both sni o f ( x ) and sni of(x’) (see 8.8). Then, choose mi such that the diameter of h,’(z) is less than ei for all z (see 3.7), and that mi+ > mi. Define a multivalued d i : Hmi -+ Sni by di(z) = s,; o f 0 hLi1(z).By 8.3, each di is a USC mapping. Further, each di(z)is contained in an arc of Sniof a length less than 1, and for i < j we have di 0 h z ( z ) 7 s2 dj(z). 0
317
$10. Non-homeomorphic solenoids
(d) For a positive integer i and a z E Nmi,let cpi(z)be the shortest arc in S,,, containing di(z). Thus, 'pi: H,,,, ---* Sni is an arc-valued mapping. It is USC and each cpi(z) is less than 1 in length, so that epic@. For i < j we have sz(cp,{z)) c cp,(hz(z)). Hence, s z cpj and cpi h z are in @ and, by 9.5, W(s2 q j )= W(cpi h?) and M ( s z 0 q j )I M ( q i h z ) . rn (1
0
0
0
0
10.3. Lemma. Let L be an SC, and S a polyadic solenoid. Then there is no PSV and USC mapping of L onto S . Proof. Suppose there is such a mapping6 Let H c L be a subcontinuum such that f ( H ) = S and f(H') =I= S for each proper subcontinuum H' c H (see 8.7). Since S is indecomposable (see 4.3), H is so, too. By 4.10, H is a zero-adic CC. Let (H,(hn)), (S,(sn)) be limits of circle approaches ((H,),(h:)), ((&),(ST)) with W(h:) = 0 and si"(z) = zqn, )qnl > 1 for all n. Obviously, f H is a PSV and USC mapping of H onto S. Take the sequences (mi), (ni)and the PSV and USC mappings 'pifrom 10.2. We will show that W(cpj)= 0 for all j 2 2. Indeed, since W ( h 2 )= 0, we have
I
0 = W(cpl) W(h2)= W(cp,0 h?) =
Since W(s2)=
n
n,
-1
k = ni
Now, choose an
I
w(sz
3
cpj) = W(s2)W(cpj).
qk =k 0, we have W(cpj)= 0.
2 2.
By 9.11 we have for j 2 i
in contradiction with the finiteness of M(cpi).
rn
10.4. Lemma. Let H be a CC, S a polyadic solenoid, P a continuum. Let h: P -+ H be a PSV and USC mapping onto, f : H -+ S a continuous single valued mapping onto. Then f 0 h is a PSV and USC mapping of P onto S . Proof. Obviously, f h is USC, and f ( h ( x ) ) is always a continuum. Since h(x) is a proper subcontinuum of CC, it is a snake-like continuum rn (see 4.4) and f(h(x))f S by 10.3. 0
10.5. Let Q = (q,), R = (r,) be sequences of non-zero integers. Q is said to be a factor of R if there exists a t such that for every n 2 t there is n
m
i=f
i= 1
an m such that nqi divides n r i . 10.6. Proposition. Let Q = (qnb R = (r,) be sequences of non-zero integers, S a Q-adic solenoid and H an R-adic CC. I f Q is a factor of R, there is a continuous single-valued mapping of H onto S .
318
App. A : COOK CONTINUA
Proof. Let (H,(h,)) be a limit of a circle approach (H,,(hr)) with W(hi+') = r,, (S, (s,)) a limit of a solenoidal Q-adic circle approach (S,, (sr)),
t be the integer from the definition of factor. Put p o
j- 1
=
1, pj = nqt+i. i=O
There is an increasing sequence mo, m,,... of natural numbers such that
mo = 1 and for each j > 0 p j divides
n
m,-1
i= 1
Ti.
Since
fi ri
m -1 i= 1
=
W(hy),
(l/p,) (&)* is in Y with (h';.l)* and Y from 9.3 and 9.2. Hen& there exists a p,-th root of h y for any j > 0. Denote it by 5, = Pd(hy). Put to = id. We prove that the diagram below commutes. We have
t,-10h2~= , P~-$/(h;?l-')oh,":.,= "-$/(h?), ,$:o 5, = [PC/(hy)]q'+J-I = PI-'J ( K J ) .
,
Thus, 5
=
limtj is a continuous mapping of H onto S .
10.7. Proposition. Let P be an SC, H a PACC. Then there is no PSV and USC mapping of P onto H . Proof. Let H be a Q-adic CC, S a Q-adic solenoid, Q = (4,) such that (qnl > 1 for all n. By 10.6 there is a continuous single valued mapping f of H onto S. Thus, the existence of a PSV and USC mapping of P onto H contradicts 10.4 and 10.3.
10.8. Proposition. Let H be a Q-adic CC, G an R-adic CC, Q = (q,), R = (r,) with non-zero q,, r,. I f there exists a PSV and USC mapping of G onto H , then Q is a factor of R. Proof. (a) If Q is not a factor of R, there exists an increasing sequence l,, lz, 13, ... of positive integers and a sequence A,,1 ,A3, ... of primes such that, for some ki,
n
l,+1-1
=
J=l,
divides
IT qj but it divides no
1,+1-1
j=i,
t
n r j . Put 4: =
j= 1
q,, Q' = (4:). Obviously, 1q:I > 1 for a1 i. Let S be a Q'-adic
solenoid. Q' is a factor of Q and hence there is a continuous mapping h of H onto S . Let f be a PSV and USC mapping of G onto H . By 10.4, h o f is a PSV and USC mapping of G onto S .
3 19
$11. Cook continua
(b) Let (G, (g,,))be a limit of a circle approach ((G,,), (gy))with W(g;+')= r,,, and ( S , (s,)) a limit of a solenoidal Q'-adic circle approach ((S,,),(sr)). Take the sequences (mi) and (nJ,and the mappings qi:Gmi--t Snifrom 10.2. (c) Suppose that W ( q l ) 0. Then we have by 9.11
+
W ( X : )= W(q1 0 g::)
~ ( ~ p 1 )
z=
W(S;; qi) = W(S::)W(qi) 0
we see that 1=n,
l j divides W ( q l )so that W ( q l )exceeds any number, which is
a contradiction. Thus, W ( q l )= 0. (d) Thus, by 9.11, we obtain for any i
in contradiction with the finiteness of M ( q l ) . 10.9. Corollary. (1) Let Q, R be two sequences of non-zero integers, H a Q-adic solenoid and G an R-adic CC. Then the following statements are equivalent: (i) There exists a continuous single-valued mapping of G onto H . . (ii) There exists a PSV and USC mapping of G onto H . (iii) Q is a factor of R. (2) Let q, r be positive primes, H a q-adic and G an r-adic solenoids. Then H is homeomorphic to G q = r.
5 11.
Cook continua
11.1. A Cook continuum is a continuum H such that for any subcontinuum Q of H and any continuous f : Q + H either f is a constant or f ( x ) = x for all x E Q.
11.2. The abbreviation HIPACC stands for "hereditarily indecomposable polyadic circle-like continuum". 11.3. Theorem. Let fl = ((Hn),(h:)) be an approach such that ( 1 ) H, is a HIPACC, (2) every h:" is an atomic mapping such that,for any Y E H , (h:'l)-' is either a one-point set or a HIPACC,
(y)
320
App. A : COOK CONTINUA
( 3 ) for every n and every non-degenerate subcontinuum P of H , there is ( P ) contuins a HIPACC, an m 2 n such that (A:)-' (4) ifCi is a HIPACC in Hni, i = 1,2, and ifthere exists a PSV and USC mapping of C , onto C,, then n , = n, and C, = C2. Let (H,(h,)) be a limit of X'. Then H is a Cook continuum.
Observation. If 2 satisfies (1)-(4) then H and H , are HIC (see 7.7), the mappings hr, h, are atomic (see 7.3 and 7.5) and confluent (see 7.9).
Proof. Let Q be a non-degenerate subcontinuum of H , f : Q + H a nonconstant continuous mapping. Suppose that there is an x with f ( x ) = x. l By 6.8 there are non-degenerate disjoint subcontinua A , B such that A c Q, B = f(A).Put g = f A.
1
(a) Since B is non-degenerate, there exists an n such that hn(B)is ' nondegenerate. By (3), for some. m, (/I:)-' (h,(B)) contains a HIPACC. Since hr is atomic, (hr)-' (h,(B)) = h,(B). (b) Thus for a suitable m, h,(B) contains a HIPACC. Let i be the smallest such m, C a HIPACC in hi(B).Let L be a component of g-'(h;'(C)). Since hi g is confluent, (see 7.9), hi(g(L))= C. Let L' be a minimal subcontinuum of L such that hig(L') = C (see 8.7), j the smallest m with h,(L') nondegenerate. Put C' = h,(L'). We will show that c' is a subcontinuum of a HIPACC C". If j = 1 we can put C" = HI. If j > 1, /I-~(C') = {z> and we can put C" = (hj-')-' ( z ) by (2). Thus, we have the following alternative: either C' = C" is a HIPACC, or C' C" and C' is an SC by 4.4. (c) Define 7 : c' + C putting ~ ( x =) hi g(hj)-' (x). We will show that it is a PSV and USC mapping of C' onto C. Since hj is monotone, every ~ ( x is) a continuum. z is USC by 8.3. Since C' = hj(L') and hig(L') = C, z is onto. Since C' is non-degenerate, (hj)- (x) is a proper subcontinuum of L' for any x . By the minimality of L', hi g hJ: '(x) C. (d) By 10.7, since C is a HIPACC and z is PSV and USC, C' is not an SC. Thus, C' = C" is a HIPACC so that, by (4), i = j and C = c'. (e) Since hi is monotone, h,(A) and hi(B)are disjoint in contradiction with the inclusion C c hi@) n hi(B). 0
'
11.4. Hence. to prove the existence of a Cook continuum it suffices to
construct an approach satisfying (1)- (4). This will be the subject of the following paragraph.
$12. The double diagram D
5 12.
321
The double diagram D
12.1. Conventions. A plane continuum (a subcontinuum of the Euclidean plane K ) will always be considered metrized by the metric of K. N denotes the set of all positive integers. The symbol
*
is preserved for a one-to-one mapping of N x N x N onto the set of all primes p 2 2. The symbols inv f(x) ,
inv f(P)
are synonymous with f-'(x), f - ' ( P ) , respectively, and are used in cases of complicated symbols standing for fi The symbol N is also used for the thin category (partially ordered set) ( N . 1,(see 3.4). 12.2. Let D: N x N + Top be a functor. Let us have, in a notation similar to the one used above, say D = ((H,,,), (h::kn')), such that H,,, are plane continua, h;:;"' are onto, and the following conditions are satisfied: (a) For every n E N and every non-degenerate subcontinuum P of Hl,, there exists an n' > n such that invh:;,"'(P) contains a circle. (b) H,,, = C for all m E N and every Hm,ncontains only a finite. number of simple closed curves, all of them circles, say C,!,,, ..., Cx,, ( I depending on m, n), C;,, pairwise disjoint. (c) h;$" are atomic mappings. For a ZEH,,,, invh;:;+l(z) is either a one-point set or a circle. The preimage invh;;;+l(Ck,n) is never a circle. (d) For every zE H,,,, invh,",; ""(z) is non-empty finite. invh,",: ',"(C;,,) is a union of finitely many circles in HmCl,,. h;,:'," maps any c;+'," onto some CL,,,, and the domain-range restriction tp; + ,: Ck + ,n -+ Ci,, has the following properties: W(q&+,,,)= $(m 1, n, i) and if m > n, i (Pm+ 1 ,n is E, + ,-crooked, where E,, < min (L(2-" diamHi,j, hyj')).
+
,
i,j= 1, ..., m - 2
(for the definition of L see 6.5). 12.3. For 'an EN denote by D, the approach obtained from D by fixing the n. Obviously, (h,":;+'): Dn+l+ D, is a transformation. Let (H,,(h,,,)) be a limit of D,, = lim h;:;+', put ,%? = ((H,), (h;)). In this section we will show that
m
,%?
satisfies (1)-(4) from 11.3.
322
App. A : COOK CONTINUA
12.4. Lemma. Let P be a non-degenerate subcontinuum of H,. Then hl,,(P) is a non-degenerate subcontinuum of Hl,fl. Proof. By (d), if hl,,(P) is a one-point set, every h,,,(P) is finite, and hence, being connected, it is a one-point set. But then P is degenerate. 12.5. Lemma. For m e N choose a Ck, so lI7at :.7:/ '."(C:;; Put C = { Z E H ,I ,h,,,(z)E C k f lfor all m e N > .
Then C is a HIPACC. It is P-adic with P
=
-
1,n)
=
Ckn.
{t,b(m,n, im)>:= 1.
Proof. It' is P-adic by (d), hereditarily indecomposable by 6.5 and by the crookedness condition in (d). 12.6. Lemma. Q = hR::+('C;, is always a one-point set. Proof. Suppose that Q is non-degenerate. If z E Q, inv h;;;+ '(z) is not a circle because it would yield a proper subcircle CL,fl+l.Thus, by (c), it is always a one-point set. Thus, the restriction of hE::+' to I Z ~ , , +is~ a homeomorphism, and hence Q is a circle. Hence, since h;$+' is atomic, invh,";;+'(Q) = C;,fl+l is a circle in contradiction with (c).
12.7. Lemma. Zf n' 2 n, inv hR;;'(C;,J contains a circle. Proof. Put j = n' - n. If j = 0, the statement is evident. Let inv hm.n+J m,n (CLJ contain the circle Ck,n+j.Then, by (c) and (b), inv hi$Zj+'(z) is a circle for a z E Ck,n+ j . Thus, the statement follows by induction. Ip 12.8. Lemma. Let P be a non-degenerate subcontinuum of H,,, which does not meet any circle of H,,,n. Then P is an arc. Proof. Let t be the smallest number such that Q = hE:y(P) is nondegenerate. Thus Q is a non-degenerate subcontinuum of a circle (see (c)). By 12.7, it is not a whole circle, and hence it is an arc. By 12.7 we also see that, for j with t I j I n, invh,":i(z) is a one-point set for each z E Q. Thus, h,";; maps P homeomorphically onto Q.
+
12.9. Lemma. Let P be a subcontinuum of Hm,flsuch that P\Ck3, 8 and P n Ck,n 8. Then P 3 Cggn. Proof. Since P\Ck,, 8, we have n > 1. By 12.6, hE;;-l(Ck,n)is a onepoint set, say {.>. Since P n CL,n 8, z E h,":,"-l(P).By (c),since P\Ck,, 8 and inv h,":: - '(z) = C;,,, h;:: - I(P) is non-degenerate. Since is atomic, the statement follows. Ip
+
+
+
+
323
$12. The double diagram D
12.10. Lemma. Let C be a HIPACC in H,. Then,for every m, II,,~(C)is a circle. Proof. Put C, = h,,,(C). All C, are non-degenerate, since Cl is (see 12.4). Suppose that C, meets no circle in Hl,. Then, by (d), C, does not meet any circle in H,,,. By 12.8, hence, C, is an arc so that C is an SC in contradiction with 10.7. Thus, C,meets a circle, say D,. Using (d) we can find, by induction, circles D, in H,,, such that D, n 2, 8 and h:,: ',"(Dm+ = = D,. If D,\C, 8 for a t E N , we have by (d) D,\C, += 8 for all m 2 t so that C, D, and hence, by 12.9, C, D, for all m 2 t, and C is an SC, which is a contradiction. Thus, C, I> D, for all m. Put
p
+
+
Z'
=
I
(x E H , h,,,(x) ED, for all m E N } .
By 12.5, it is a HIPACC and a subcontinuum of Z.By 4.4, a proper subcontinuum of C is an SC. Thus, Z' = C and consequently C, = D,.
s
12.11. Proposition. The approach A? satisfies (1)-(4) from 11.3.
Proof. (1) follows by 12.5. (2) By (c) and 7.4, hi+' is atomic. Take a z E H , and put y = hl,,(z). If invh::;+'(y) is a one-point set, h;+'(z) is a one-point set, since invh::;+l(h,,,(z)) is such. If invh::;+l(y) is a circle, then, by (c) and (d), invh~;~+l(h,,,(z))is a circle for all n so that invh",'(z) is a HIPACC by 12.5. (3) Let P be a non-degenerate subcontinuum of H,. By 12.4, h1,,(P)is non-degenerate. Hence, by (a), there is an n' > n such that inv hi::'(& ,,,(P)) contains a circle. Thus, by (c), (d) and 12.5, inv h;'(P) contains a HIPACC. (4) Let Ci c H,,be a HIPACC, i = 1,2. By 12.10, there are indices t(m, i ) such that hm,,&) = C:>,i). Put Pi= (+(m, ni, t(m, i))),= 2, .... By 12.4, Ci is a Pi-adic CC. By 10.8, if there is a PSV and USC mapping of C, onto Cz, Pz is a factor of PI, which, according to the choice of $, implies s PI = Pz and hence n, = n2, t(m, 1) = t(m, 2) and C, = CZ-
,,
The remaining three sections contain an explicit construction of a diagram D with the properties (a)-(d). At first sight, the existence of such a diagram does not seem to be surprising and the reader may simply believe it. We add a concrete construction because, in fact, there are still several technical dificulties to be overcome.
324
6 13.
App. A: COOK CONTINUA
Construction lemmas
13.1. First, let us introduce a few conventions and recall some wellknown facts from the topology of plane. A simple closed curve f will always be assumed to be a subspace of the complex plane K (homeomorphic to the circle). The circle with the centre in z and radius E will be denoted by C(Z,E)(and by C , when z = 0). By Jordan theorem (see e.g. [KIb), K\f consists of two open connected sets with f as a common boundary. The bounded one will be denoted by I n t y . Put D($)
=
$ u Intf .
A set of the form D($) is called a (closed) disc. The boundary of a disc D in the plane will be denoted by B(D). (Thus, B ( D ( 2 ) )= 8.)The following statement holds (see e.g. [KIb): Every homeomorphism of a simple closed curve f onto a simple closed curve f ’ can be extended to a homeomorphism of K onto itself. Evidently, under this homeomorphism h, h ( D ( f ) )= D ( f ’ ) , h(Int$) = = Int 9’. The above theorems will be used in the sequel without particular quoting. If p is an arc, then Intp designates the p without the end points. A simple closed curve f and an arc p are said to form a @-curve if Intp c Intf, and the end-points of p , say a and b, are in $. Then we have arcs q l , q2 such that $\{a, b } = Intq, u Intq,, and p u q 1 and p u q2 are simple closed curves. D ( p u 4 , ) and D(p u q,) are said to be the halves of the @-curve. 13.2. Lemma. Let $ be a simple closed curve and p an arc in K. Then for any E > 0 there is only afinite number of components of p n D($) and of p n Int$ with more than E in diameter. Proof. Suppose there is a sequence L,, L,, ._.of distinct components of p n D($), L, with the end-points a,, b,, and that there are x, EL, with dist(x,, {a,, b,}) > +E. A limit x of a subsequence of (x,), is obviously in p . Then U n p is disconnected for any neighborhood U of x with diam U < *E, in a contradiction with the local connectedness of p . The statement on p n Intf follows from analogous reasoning. H 13.3. Let f be a simple closed curve, p an arc with the end-points a, b, A c I n t y a nonempty closed connected set. Denote by 9 the system
325
$13. Construction lemmas
of the closures of all the components of p n I n t j which do not intersect A u (a, b}. An L E 9 and f form a @-curve.Denote by f L the subarc of 9 with the same end-points as L and with A n Int($, u L) = 8. Obviously, f , n f L .is either f , or f L .or 8 or a common end point of Y Land jr. Lemma. For an L E 9, there are only finitely many L , , ..., L, in 9 such that L , = L and j L , YL,+for i = 1, ..., n - 1. Proof. Suppose there is an infinite sequence L , , ..., L,, ... such that j , , s f L , +1. Put T
u m
=
n= 1
j,,. T is an open or a half-open arc on $ with
end-points a, b. We have a = lima,, b = limb,, where a,, b, are end-points of f L nin a suitable order. If a $. b, then diamL, 2 $la - b] for infinitely many n contradicting 13.2. If a = b, consider f , = (f\fLn) u {a,, b,]. We have
u 00
n= 1
f,,= { a } so that limdiamj, = 0. By 13.2also limdiamL, = 0.
This contradicts the inclusion A c Int(f, u L,), since diam(A u (a}) > 0.
13.4, By 13.3, every f , is contained in a maximal system of all L with maximal 9,. Put f ( p 4 = (f\
u
fL)U
LET’
u
LET’
rn
YL..Let 9’ be the
L.
Lemma. 2 ( p , A ) is a simple closed curve. Proof. Choose homeomorphisms h, of f L onto L such that hL(x)= x for the end-points x of L. Let a mapping h of 9 onto $ ( p , A ) be defined by h(x) = hL(x) for x E $, and h(x) = x otherwise (since f L n YL,contains at most a common end-point, this definition is correct). Since (see 13.2) only a finite number of L’s has more than a given E > 0 in diameter, h is continuous. Thus, since it is one-to-one and since 3 is compact, h is a homeomorphism. rn 13.5. A crossing of two subsets p , q c K is a point x E p n q such that there is a disc D c K and a homeomorphism h of D onto D(C)(C is the unit circle) such that x E D, h(x) = 0, h(p n D) = (z E D(C) Imz = 0} and either h(q n D) = (z E D(C) Rez = 0} or h(q n D) = (z E D(C) Rez = 0 and Imz 2 0). Let p be an arc in K, let r] > 0 be given. An arc q c K is called an r]-modification of p , if the end-points of p and q coincide and
1
I
e(p, q) = max(sup dist(x, q), XEP
SUP YE4
dist(y, P)) < r] .
I
326
App. A : COOK CONTINUA
Proposition. Let p , q be arcs in K, and Y a countable subset of q, let the end-points of p not be in Y v {a, b}, where a, b are the end-points of q. Let y > 0, S > 0 be given. Then there is an q-modijkation p' of p such that (ct)
p\T
= p'\T
(/?) every z E p'
( Y ) P' n y = 0.
where T
=
I
{z E K dist(z, q ) < S},
n q is a crossing of p' and q,
Proof. Let h be a homeomorphism of K onto itself such that h(q) = { z E K I I m z = O and R e z ~ [ - l , l ] } . Put $ = {zEKIdist(z,h(q))=$ where ij > 0 is such that h-'(D($)) c T Only a finite number of components of h(p)n D($), say L:, ..., Lt either contain an end-point of h(p) , or intersect h(q). Put $* = f(E, A ) (see 13.4), where 5 = h(p), A = h(q). Choose a homeomorphism g of K onto K such that g(z) = z for z ~ h ( q ) , and such that g($*) = C, (= (z IzI = 2)). For LT c D ( f * ) , i = 1, ..., s, put Li = g(L*). Further, put Si= {z E Int C, I dist(z, Li) < 9) where 9 > 0 is such that Sin S j = 0 for i j , and dist((g 0 h)-'(z), p ) < y if dist(z, g h(p))< 9. For each Li take an arc Li in D(C,)such that Lihas the same end-points as Li,and, for a sufficiently small 5 > 0, Lin D(CZTt) is a polygonal arc in Siwhich does not intersect g h(Y u {a, b ) ) and where every common point of Liand g h(q) is their crossing (see Fig. A.1). =
I
+
gh(4
A. 1
Replace the Li by Li in g h(p) and denote by jT the obtained arc. Then, p' = (gh)- '(F) has the required properties.
I
13.6. Lemma. Let p be an arc in K, and {b, I E J } a disjoint system of its subarcs, {b:1 i E J } a disjoint system of arcs in K such that for all
327
# I 3. Construction lemmas
z E J , b, and b: have the same end-points and b: n (p\
u
XE
b,)
=
J
8. Put
4 = (P\Ub,) u U b : . ZEJ
reJ
I f q is compact and ij’there is a continuous mapping g of q onto p such that g(b:) = b, for all z E J and g(x) = x for x E p\UInt b,, then q is an arc. J
Proof. Choose homeomorphisms h, of b: onto b, preserving the common end-points. Define h : q -+ p by h(x) = h,(x) for x
otherwise.
h(x) = x
E b:,
The mapping h is one-to-one onto, and hence, since q is compact, it suffices to prove that it is continuous. Let a sequence (x,) in q converge to x and let (h(x,)) converge to y. It suffices to prove that y = h(x). The only nontrivial case is that of X , E b:* where z, 1 ., for n n’. Put 6 = p\UIntb,.
+
+
~ E J
Since g(x,,)E bln and (g(x,,))converges to g(x), we have iim diamb,, = 0 and g(x)E so that g(x) = x = h(x). Since h(x,) E b,,, (h(x,)) converges to g(x) = y and hence h(x) = y.
13.7. Put M
=
D(C,)\IntC. Let p : M + D(C,)
be a continuous mapping such that
(*)
p(z)
=z
p-yo) =
c
for
ZE
C2
p- l(z) is a one-point set for z
E D(C,)\(O)
.
An arc p in D(C2)is said to be a p-arc if 0 E p and if there exist continuous real functions r and cp defined on p - (0) such that
(i.e., “ p - ’ ( p - (0)) spirals positively around C ) . Thus, the mapping h(z) = into M inverse to p. = r(z)eirP(’)is a homeomorphism of p\(O) Example. Define A: D(C,)\(O) --* M by A(z) = (ie + 1)ei[@-(1’2)i(1’e)1 for z = @ei*,let p : M -+ D(Cz) send z 4 C to A-’(z) and Z E C to 0.
328
App. A: COOK CONTINUA
Then every segment meeting 0 is a p-arc. See Fig. A.2.
P
t -
Fig. A.2
13.8. A set U c K is said to be one-connected if it is homeomorphic to an open bounded connected U' c K with connected K\U' (it should be noted that a homeomorphism preserves all the properties of U' required except the boundedness). The following statement will be of use: If a one-connected U does not contain 0, then there is a continuous branch of logarithm on U . Lemma. Let p be a p-arc, let p and C, form a @-curve, let @(p, C,) be its ha& and S = pL-l(@(p,C2)\(O}). Then there is a continuous branch of logarithm on S, which will be denoted by log, suck that l i m I m l o g z = +a.
I++l
zcs
Proof. Let pl, p , be subarcs of p with p = p , u p , and p1 n p z = (0). Since S is contained in a one-connected L' $I 0, there is a branch of logarithm on S. We can choose one, log, so that Imlogz = cp(p(z)) for p(z) E p,\{O). Then there is an integer k such that Imlogz = cp(p(z)) 2kn for ~ ( z ) E E p,\(O). For an q with 1 < q < 2 put f , = p(C,). Since 0 E Inty, and f , c Int C,, f , intersects p\{O). For i = 1,2 put Zi,= f , n pi. Put $(q) = min(infcp(z), inf(cp(z) 2k7c)). We have obviously lim $(q) = f c o
+
ZI,
z2rl
+
,-1+
(since Zi,converge to (0)). Since 9, @(p, C,), we have C, S, and the end-points of each component of S A C, are in p-'(p\{O]). Hence; if y E S n C,, then either IImlogy - cp(z)l < 2n for some z E Z,, or IImlogy - cp(z) - 2kxl < 2x for some z E Z,,. Thus, lim Im logy = =
IYI+
+co.
1+
13.9. Proposition. Let p be a p-arc suck that p and C, form a @-curve, let T be compact, p c T c @(p, C,). Put S = p - ' ( T ) . Then the domain-range restriction ,ii: S T of p is an atomic mapping (see 7.3). --f
329
$13. Construction lemmas
Proof. Evidently, p is monotone, so that we have to prove that if Q is a subcontinuum of S with non-degenerate ,G(Q), then ji-' P(Q) = Q. This obviously holds if 0 $ p(Q). Thus, suppose that 0 E P(Q). It sufficesto prove that C c Q. Suppose ~ , E C \ Q and choose a tb, with yo = ei*. There is a 6 > 0 such that U n Q = 8 for U = ((1 + q ) ei($+')10 I q I 6, 111 I 6). Put E = )rnin(h,diamQ) and choose a component L of Q n D(C,+,) intersecting c . By 1.7, L intersects C,,, and hence it is non-degenerate and we have L n C $: 8, L\C =# 8 and L c D(C, +e). Let B be a component of L\C. By 1.8, %?lBn C 8 so that dist(B, C) = 0. Take the function log from 13.8. The connected set Imlog(B) has to contain the interval (a, +a)for a real number a. On the other hand, B n U = 8
+
and hence, for G, = (2nn
+ I) - 6, 2nn + I) + 6),
Imlog(B) n (
u
+m
G,) = 8,
-00
which is impossible.
H
13.10. Let p be a p-arc, put L = p- ' ( p ) . Let 2 be a simple closed curve such that 2 c Int C,, 0 4 D($), C\D($) =k 8 and C n D ( 2 ) is connected. Let (a, t E J} be the system of all components of L n D($). Thus, an a, is either a one-point set or an arc with the end-points in $. If C n D($) += 8, denote by a, the component contained in C. Let L' c D(C,) be compact and f : L' -+ L a continuous mapping such that f is identical on L\Intf = L'\Int$ ,
I
1
there is a disjoint system {a: z E J > such that a: is a one-point set iff a, ,is so, otherwise it is an arc whose end-points coincide with those of a,, a; = a, and L' n D ( 2 ) = u a : , teJ f maps a: onto a,. Such an f will be called almost identical mapping. 13.11. Proposition. Let p be a p-arc, and f : L' 3 L = p - ' ( p ) an almost identical mapping. Then p(t') i s a p-arc. Proof. (a) Put q = p(L'), b, = p(a,), b; = p(a$ g = p 0 f p-'. Since g is single-valued and USC (see 8.3), it is a continuous mapping of q onto p . Thus, q is an arc by 13.6. (b) We have 0 E q. Since D ( 2 ) is contained in a one-connected U # 0, there is a continuous branch of logarithm I on D(2).Put E = supIm I ( y ) 0
- infImA(y). Since p is a p-arc, there are r and D(b)
D(Y)
4p
satisfying (**) from 13.7.
330
App. A: COOK CONTINUA
Put $(y) = Iml(y) + 2k1n for y E a:, where k, is such that $(y) = cp(p(y)) in the end-points. We have Iq(p(y)) - $(y')( I E whenever y e a , and y' E a:. For z E q\{O} put @(z)= lp-'(z)l, put $(z) = $(p- '(2)) for p - ' ( z )E a: and $(z) = cp(z) otherwise. Then &(z) e'"'"') = z, lim$(z) = 00 and, 2-0
+
rn
since p- : q -+ L' is USC, lim @(z)= 1. Thus, q is a p-arc. z+o
5 14.
The d-process
14.1. In this section, p: M satisfying (*) from 13.7.
-+
D(C,) is always a continuous mapping
14.2. The data of an &'-construction: Let H be a continuum in K. Let the following system of data be given: (a) a finite non-empty subset F c H , (b) a disjoint system D = {Dz z E F } of discs such that z E IntD,, the component q2 of D, n H containing z is an arc, q, and B(D,)form a @-curve and the whole of D, n H is contained in one of its halves (see 13.1),which will be denoted by 0,. We will call q, support of z. (c) A system X = (h, z E F ) of homeomorphisms h, of D, into D(C,) such that h,(z) = 0 and p , = h2(qz) is a p-arc. Put P, = h,(@,). (d) A positive number E such that C(z, E) c IntD, for all z E F . (e) A system 9 = {g, z E F } of homeomorphisms g, of D,\Int C(z, E ) into M = D(C,)\Int C such that gz(C(z,E)) = C and p g,(y) = h,(y) for all z E B(D,). a = (F, D,2,E, 9) is said to be an d - d a t a system.
I
I
I
14.3. If a = (F, D,2,E, 9) is an d - d a t a system, we put
H(a) = (H\ and define
u
ZEF
Int Dz) u (U(9; I(PZEF
l(hz(Dz
n HI))))
r(a): H(a) --t H
u
by r(a)( y ) = h;' p gz(y) if y E D, n H(a) for a z E F , r(a) (y) = y otherwise. We say that H(a) and r(a) are created by a. Further, put B = H u D,.
a(.)
ZEF
= (H\UIntD,) u U q ; ' ( p - ' ( h _ ( D ; ) ) ) = H \ u I n t C ( z , ~ )and define zcF
zeF
zrF
R(a)--+ B if y E D, n B(a)for a z E F, i(a):
by F(a)(y) = h,- p g,(y) By 13.9, r(a) is an atomic mapping.
F(a)( y ) = y otherwise.
331
$14. The d-proces
14.4. An iterated d-process. An inductive description will be given of a sequence of continua H i and of d-data systems ai on them. In the course of the induction we will also point out some particular neighborhoods of some points in H i , which we will call admissible neighborhoods. (1) H , = C. The admissible neighborhoods are defined for all y as the discs D with y E IntD and B(D)n C consisting of two points. We put formally F, = 0 and choose an arbitrary E , > 0. (2) Let there be, for i = 1, ..., n - 1, H i + l = Hi(ai), ri = r(cLi) and ii = i(ai) where ai= (Fi, Di,%i,ei,9i) are d-data such that Fi c X i =
u u
i- 1
= Hi\ =
j=l
DEB,
B(D), Fi n C(z, ci-
=k 0 for z
E
Fi- 1 , and that Di
=
(Dz 1 z E Fi} consists only of admissible neighborhoods.
The admishihle neighborhoods in H i + 1are defined for the Y E X ~=+ ~ =
Hi+l\U
j=1
Uo(n\: B,
provided ~ E X ~ + ~ \ U D D, , is admissible iff D c Xi+,and it is an admissible neighborhood of y in H i ; provided y E X i + n (IntD,\C(z, ci)) for a z E Fi, then D is an admissible neighborhood of y iff it is a disc such that D n C(z,ei) = 0, D c IntD, and Fi(D) is an admissible (in Hi) neighborhood of Fi(y); provided y E C(z, ci) for a z E Fi, then D is an admissible neighborhood of y iff it is a disc such that y E Int D, D c Int D, and B(D) intersects C(z, ci) in only two points. We say that H,, r,, A,, 7,, X, etc. are created by the iterated d-process.
A system D
=
1
(0, Z E F } such that F and
14.2 for H = H , such that F c X , = H,\
u
n-1
j=1
D satisfy (a) and (b) from
u B ( D ) and F n C(z, &.-I) =+
0
3),
for all z E F,- 1, and such that every D, is an admissible neighborhood of z in H,, is called a refinement of ID,- 1.
14.5. Lemma. Let D be an admissible neighborhood of an XEX,. Then a component of D n H , is either a one-point subset of B(D) or an arc with the end points in B(D).
Proof. In fact, one proves easily by induction that every disc contained in an admissible neighborhood has this property. (If x E C(z,en- 1) for a z E F,use the fact that r,- is atomic.) H
332
App. A: C O O K CONTINUA
14.6. Lemma. Let D be an admissible neighborhood of a ZEX,, q, be the component of D n H , containing z. Then q, and B(D)form a @-curve and D n H , is contained in one of its halves.
Proof follows by induction from the definition of admissibility.
14.7. Observation. If H , is created by an iterated d-process, then, by 14.6, F, and D,,= { D , z E F,) can be formed so that D, is a refinement ,. Then, let us choose for z E F, a p-arc p , such that p , and C, form of 3,a @-curve,and a homeomorphism h, of D, onto D(C,)such that h,(z) = 0 and h,(q,) = p,. Put %,, = (h, z E F,}. Obviously we can then choose E, and 9, to obtain an &-data system CI, = (F,, D,,%, E,, 9,).Thus, a next step in an iterated &-process is always possible.
I
I
14.8. Proposition. The circles C(z, E , - , ) with Z E F , - , are exactly all the simple closed curves contained in H,. Proof. Let 2 be a simple closed curve. Put f ' = r(a,)($). If there is a z E F,- n f ' , then f ' = { z ) ,otherwise, since r(a,) is atomic, C(z, E,- ,) f , which is impossible. Thus, f c C(z, E,- ,) and hence f = C(z,8,- ,). Supso pose that F,- n f ' = @.Then f ' is a simple closed curve in H,that either f ' = H , = C (if n = 2) or f equals to some of the circles C(z, E,-,) with z E Fn-, (if n > 2). This contradicts the fact that F,, intersects all of these circles (see 14.4 (2) i = n - 1).
,
14.9. Let H , be created by an iterated d-process, H , = H,-I(c~,-I), CI,-,=
(F,-,,
a,-,,
E,-,,
9,-l).Let there be given a countable
Y, c X,. Let D be an admissible neighborhood of y E X,(admissible in H,).
Let qy be the component of D n H , containing y, and p an arc such that p c IntD and p n qy = (a, b ) where a, b are the end-points of p . Let a, b $ y,. Lemma. Let v] > 0, 6 > 0 be given. There is an v]-modification p' of p (see 13.5) such that (i) p' n qv = {a, b } , p' c IntD, p' n Y, = 8, (ii) p'\T = p\T for T = ( Z E K dist (z,H,) < S), (iii) all the common points of p' and H , are their crossings. Proof. (a) If n = 1, H , = C and we can put p' = p . (b) Let n = 2. Thus, we have H , = C and H 2 = H,(cII), a, = = (F,, a,,Xl,el, Y1). The non-trivial case is that of y E C(z, E , ) for some zE F,. Let f be a simple closed curve, p c Int f , 2 c Int D. Let q be a subarc of qz, the support of z in H , , such that z 4 q. Then ij = r(cI1)-'( 4 ) is an arc in H,. Since dist (2, B(D)) > 0, by 13.2, only finitely many corn-
1
333
$14. The d-proces
ponents of ij n D intersect 2.Since q,\{z} is a union of a countable number of such q c q,\(z), only countably many components of r(al)-l(q,\{z}) intersect 3 and each of them has a positive distance from the union of the others. Denote by Lo the component of D n H , contained in C(Z,E J . We see that only countably many components of H , n D intersect 3,say Lo,L,, L,, .... For k 2 1, diSt(Lk, L j ) = 9, > 0. Choose 6, < min(6, +gk) j9k and put & = {X E K dist(x, Lk) < 6,) .
u I
Modify p in each & by 13.5 with q k < rni11(2-~q,dist(p, j ) )Since . Tk are disjoint and limqk = 0, the resulting set is an arc having all the properties required. (c) The only non-trivial case in an inductive step is that of Y E C(z, E,- ,) for a z E F,- 1 . If p n Int C(z,E,8, i.e. if p n H , = { a , b), put p' = p. Suppose that p n IntC(z, E,- l ) = 8. Let q2 be the support of z (in H,- 1), S = r(x,,-,)-' (q,\{z}). By (b), there is an (q/2)-modificationp + of p such that S n n p + = 8, p + n IntC(z, = 8, the common points of p+ and C ( z ,8,- ,) are exactly the end-points of p that p\T = p+\T where T = {x E K dist (x, S) < td), and such that every common point of p + and S is their crossing. Put Y,,- = r ( E , - 1) ( Y,,), L = r(c1,- 1) (p+). L is a simple closed curve and we have z E L c Int D, and (I,\(.}) n Y,- n q, = 8. Since ?(a,,- 1) maps D,\D(C(z, 8,- 1)) homeomorphically onto D,\(z), every common point of L\{z) and q2\{z) is their crossing. Thus L.consists of z and countably many arcs Lo,L1, L,, ... the end-points of which are in q, and the interiors of which are in D,\q,. Let ( M i )be the subsequence of (Li) consisting of all those Li which are in 0,(see 14.2 (b)). Since every common point of L\(z} and q,\{z} is their crossing, every M , has a positive distance from U M i . By the inductive hypothesis, each M j can
+
I
i*J
be modified in such a way that it does not meet K - l , it has exactly the end-points in common with q, and each of its common points with H,-I is their crossing. If we use q j and 6, sufficiently small, we can construct the p' using the preimages of modified M i under r(c(,- l). 14.10. Proposition. Let H , be created by an iterated process d , k t F,, c X,, befinite, F,, n C(z, E, - 1) 4= 8 for all z E F,, - 1. Thenfor any countable Y, c H , there is a refinement 9,= (0, z E F,} of such that n B(D,) = 8 for z E F,. Proof. For a z E F,, choose an admissible neighborhood D: of z in H , so that 9' = {D: z E F,- 1) is a refinement of a,,1 . Let q, be the c ~ m ponent of H , n 0: containing z. Choose a,, b, E (4, n IntD:)\Y, so that z
I
I
334
App. A : COOK CONTINUA
lies on q, between u, and b,. Let L, be a simple closed curve intersecting q, precisely in u, and b,, and such that L, c Int 0:. Then qz cuts L, into two arcs one of them meeting H , exactly in the end-points. Modifying the other A B(D,) = 8. one following 14.9 we obtain
,
14.11. Let (H,,):= be a sequence such that every H , is created from the = H,(a,) with a, = previous H i by an iterated d process, Hnil = (F,, a,, Z,,, E,, 9"). Let us be given countable subsets % c X , such that Fn c
%,
~ ( a n ) ( % += l) K
.
Then, the sequence (HI, Y,), ( H 2 , Y2), ... is called d-sequence.
14.12. Now, we will start to construct a diagram D with the properties from 12.2 constructing an d-sequence such that
will be its fir'st row.,
Let Y,' = ( y , . ' I . m ) ~ , be a dense sequencc i n H I = C. Put F , = {yI,l,l 1 and choose an admissible neighborhood D oly such that B(D) P, Y,' = 6. Put Dl = { D ) . Choose Zl, E ~ Y1 , arbitrarily so that a, = (F,, a,,X',, E ~ g1) , is an d-data system. The induction step: Let H , be given and let countable subsets Y,i = = {yi,,,,, m E N } , , 1 , .,., 2 n - I , such that % = Y,i is dense in H,, be defined.
,., , ,
1
I
u
I
I
Put F, = { Y ~ , , , ~i = 1,. .., Y - ' } and choose a refinement a, = ( D z z E F,) of a,- such that B(D,) n = 8 for all z E F, (see 14.10). Then, choose Z,,E,, 9,such that a, = (F,, a,, Z,,,E,, 9,)is an d-data system. Put H , + l = H,(a,,). Now define Y,i+l, i = 1, ..., Y - l , putting
yn2+"iL+i is a
countable dense subset of C ( Y ~ , ,E,). , ~ , Put %+, Then (H,, Y,) form an d-sequence.
0 15.
=
u c+,. i
Processes 3?,%?, 9,& and construction of D
15.1. The 93-process. Let ( ( I f l ,I;)), be an d-sequence. Let n be a natural number and let H ; be created by an iterated d-process, H, = Hh- ,(a;ah-1 = (Fi-,, a;-,, Zk-,,& ; - l , 9;-l).
$15. Construction of D
335
Let f : H ; -,H , be a continuous mapping such that (n) for every y E H,,, f-'(y) is non-void finite; every x E f - I ( Y,) has a neighborhood D, admissible in HI, over which f can be continuously extended to a mapping which maps D homeomorphically onto a neighborhood of f(x) admissible in H , ; 2 is a simple closed curve iff f ( f ) 1s' so. Construct H L + 1 , r' and g as follows: Put =f-'(X), Fi = f - ' ( ~ , ) . By (n), F,,' intersects any circle in HI, (see 14.8)and F,' c Hk\
u
n-1
j=1
lJ B(D'). /)ET;
Let DL = ( D k l y ~ F i )be a refinement of such that f can he continuously extended over Dl (we will preserve the symbol f for the extension) so thatf maps Dl homeomorphically onto a neighborhood off(y) admissible in H , and f(Dl) c Df(y). B(Db) n = 0. Put 2;= (h; y E F') where h;(x) = (hf(,,o f ) for all x E D;, and choose EI, and 9; = {g; y E FL} such that a; = (F;, DI,, X'k, &I,, Y,) are d-process data. Now, put H,+ = Hk(or;), r' = r(aI,). For x E HI,+l\ D; put g(x) = r(a,)-'(f(x)), for x E
1
1
u
Y€F:,
~ D ; \ I n t c ( z , ~ & )put ) g ( x ) = g&(g;(x)). are created by a a-process.
We say that the Hh+l, r' and g
15.2. Remarks. r' and g are obviously continuous. We have f o r ' = H,, was created by an iterated d-process so that r' is atomic. It is easy to check (see 14.8) that g satisfies the conditioli (n + 1) from 15.1 and, hence, the a-process can be applied for g again. = ~(a,)o g. The space
x))l
15.3, The '#-process. Let ((Ill, be an d-sequence. Consider H,+ = H,(%), a, = pfl> a,,x',,E,, g,,) for an IZ.Let a system m = (m, z E F,) natural numbers be given. We will construct a continuum HI,+l and 1\ D, = a local homeomorphism c,: HI,+, + H,, such that H,+ = H,+
,\ u D, Z€F,
I
u
zsF,
and on this set c, is the identity, and every C(z,E,) is
mapped onto itself by the formula c,(y) = z + (y - 2)"' (i.e., it is wound around itself m, times). Take Z E F, and its support q, in H, (EGe 14.2), put p z = h,(q,) and P, = h,(@,) (with the 0,from 14.2). Choose an arc t, such that t, n p z = {0}, t, and h,(B(D,)) form a @-curve and P, is contained in one of its halves, denoted by K . Put R = p-'(K\(O)). Let log be a continuous branch of logarithm on R, let $ be its imaginary part. Denote by S the set of all = /x/limZ ei*L(x)imz with x E R.Let CJ: S u C-, -+ R u C be given by the formula a(y) = y"'. Then, (T maps S homeomorphically onto R and it is a local homeomorphism of S u C onto ~ ' ' ~ 2
336
App. A: COOK CONTINUA
R u C. Let I : p(S u C) -+ T, be given by A(0) = 0, I ( y ) = ( P O p - ' ) ( y ) otherwise. This 1 maps p(S u C) homeomorphically onto T,. Put p i = = A-l(pz). Obviously, p i is a p-arc (see 13.7). 0 0
Choose a simple closed curve $ c IntCz such that I-'(t,) and 9 form a @-curveone half of which is constituted by I-'(%). Find a homeomorphism T~ of D ( 3 ) onto h,(D,) extending A. Define a; = (FL, T I ; , ~ ; , E ; , ~ ; ) as follows: FA = F,, 3;= a,, E; = E,, Pn = (h: z E F,) where hz = = 'z; h,, 9; = (9: z E F,) with gh a homeomorphism mapping D,\Int C(z, En) onto D(p-'(Y))\Int C, &(y) = g,(y) for Y E C(z, cn), and p g:(y) = h:(y) for y E B(D2).Put HA, = H,(a;) and define c, by putting c,(y) = g;'(a(g:(y))) if y E H,,, n (D:\IntC(z, E ) ) for a z E FA = F,, c,(y) = y otherwise. We see that HA+ and c, have the properties mentioned above. We say that they are created by a %'-process. 0
1
I
,
15.4. Remarks. (a) One sees that HA,, and H , , , are created by the iterated &-processes with coinciding Hi, ai (i = 1, ..., IZ - 1) and H,.
337
$15. Construction of D
(b) c, satisfies (n + 1) from 15.1. Really, if y E H,, ,, c; ' ( y ) is non-void finite, f is a simple closed curve in H,,,iff cm(,$) is (see 14.8). Thus, it suffices to show that every y E c, '( x , ') has a neighborhood D admissible in HA+, such that c, can be extended over D so that D is mapped homeomorphically onto a neighborhood of c,(y) admissible in H,+'. If y E H,, D:, this is'evident. For y E C(z, &A) with a z E F.' the statement
u
ZE
FA
,
follows from the definition of 0. If y E HL+ n (D:\C(z, the statement follows from the choice of the arc t,.
&A)
for a z E F ,
x)),
15.5. The VW-process. Let ((H,, be an d-sequence, let us have collections m, = ( n ~ ,1,z~E F,) of natural numbers. Let c,,: C -+ C be given by c,,(y) = y4, m, = q. We construct a new d-sequence ((If;, alternating the use of W- and W-process as indicated in the following diagram :
r)),
(m).
Put Y,' = (c,,,, of,)-' By 15.2 and 15.4 we can really alternate the processes and we see that ((HA, mi)) is really an &-sequence. We say that it is created by a VW-process.
,
15.6. Let ((HI,I;)), be an d-sequence, HI + = H,(a,), a, = (Fl,a,, gI). Let, for an n, a'be a refinement (see 14.4) of a,.Let H' be a continuum in K, g: H' + H,, continuous onto. We say that g is almost identical, with changes in 3'(cf. 13.10) if there are two disjoint systems (a,),(a:)of arcs resp.), the end-points such that: a, (a:, resp.) is always an arc in H,,, (H, of a, and a: coincide, g maps a: onto a,, g maps H'\U Int a: = H,+ Inta,
,
,\u I
1
identically onto itself, for every z both a, and a: are contained in a common D E 9' with the end points in B(D), and a, = a: whenever a, c C(Z,E,) for a z E F,. 15.7. The %process. Let ( ( H I ,F)),be an &-sequence, n a natural number, 9' a refinement (see 14.4) of 9,. Let g: H + H,, be almost identical, with changes in 9'.We are going to construct a continuum L and mappings I : H' L and f: L + H , such that f 1 = r(a,) g and H' and 1 are created
,
--+
0
0
338
App. A : COOK CONTINUA
from L by the &-process. First, observe that g-'(C(z, 8,)) = C(z, E,,) for every z E F,, but g need not be identical in C(z,E,,). Define L by the condition L\
u
Int D,
=
u
H'\
Int D, ,
Z€F,
Z€F,
and using the 1 identical and f coinciding with g in L\UIntD, and for a z E F,, define L n D,, f on L n D, and g on H' n D, as follows: Let q, be the support of z in H , (see 14.2). Put q, = (r(a,))-'(qz), t, = g-'(q,). By 15.6, t, c D,\Int C(z, E,). Choose a homeomorphism g: of D,\Int C(z, E,,) onto g,(D,\Int C(z, E,)) coinciding with gz on B(D,) (one can put g: = g, but this is not necessary). Define a mapping ij: g:(t,) -+ g,(qz) by gg:(y) = g z g ( y ) for any Y E t,. Then g is almost identical in the sense of 13.10 so that, by 13.11, pg:(t,) is a p-arc; denote it by p i . Put p , = h,(q,) and choose a homeomorphism s, of h,(D,) onto itself, identical on h,(B(D,)) and such that s,(O) = 0, s,(p:) = p,. Define l(Y) = h; sz c1 9 z ( Y ) for y E H n D,, L n D, = 1(H' n Dz).One can see that H and 1 are created from L by the &-process with data =
1
{Fm Dm
En,
1
y} 9
where G' = (9: z E F,} and H' = {s; h, z E F,}. For every z E F,, define f i n L n D, by f ( z ) = z, f ( y ) = r(a,,)g 1-'(y) otherwise. 0
15.8. Note. It is easy to see that f is almost identical, with changes in
a, u {D E D'ID n D,
=
8
for all z G F , , } .
(observe that l(t,) = q, in 15.7 so that q, is the support of z in L.) be an d-sequence, n a natural 15.9. The iterated %process. Let ((Hl, number, B' a refinement of a,. Let fn + : Ha+ --t H,+ be almost identical. By 15.8,a %process can be applied again and again. By an iterated %process we obtain a commutative diagram
1
HI
H;
+--
-
[o
1 1
Hz
+--- H 3 t
...
+
Hn
H;
t-
Hj
...
t
H,,
C-
t
-
Hn+l
-
Hn+z
C-
Hn+l
where the & are almost identical mappings with changes in refinements of Di, and HI+, is created from Hi by the &-process with data a;=
339
$15. Construction of D
I
= (Fl,a, Zf,E ~ 9;) , where 8;= {s;' h, z E Fi).Since H , = C andf, is almost identical, H i = C. One can prove by induction that for all Z, Fi+,n Y ( ~ : ) - ~ (=k z ) 8. Thus, H,+,is created by an iterated &-process (see 14.4). We see that if B(D) n Y,+ = 8 for all D E a',then ((Hi, are the first n + 1 members of an &-sequence with = L-'(x). 0
,
x'
x))l
x'));:;
15.10. The &-process. Let ((HI, be an d-sequence, n a natural X,,,E,, 9,). Take a z E F, and number, H,+ = H,,(a,) with a, = (F,,, a,, an (a,b, c ; d)-crook (see 5.5)
,
e,: C(Z, E,) C(Z, E n ) in C(z, E,,) with c, d # We are going to construct a continuum Hh+ and an f: W,,+1 + H,,+ 1, almost identical, with changes in a refinement of a,.Further HL+I and H,+ will contain the same circles C ( y ,E,) (y E F,) and f will map C(z, E.) onto itself as e, (we say that HL+l and f : H ; + , + H n f l are obtained by an &-process from the given &-sequence and the crook e,). Let Yz be a simple closed curve, such that 2,c Int D,,intersecting C(z, E,,) in exactly two points, and such that the arc abcd of C(z, E,,) is in Inty, - ( s e e Fig. A.4).
x+,.
+
,
\
/
Fig. A.4
Put D: = D(f,)\Int C(z, E,,). Suppose for a moment that a,fi, y, 6 are disjoint arcs in 0:with one of the end-points a, b, c, d respectively, the other one (say a', b', c', d') on 9,with the interior points in IntD:\D(C(z,&,,)) (see Fig. A.4), and such that (i) we have q n 0, = 8 for the arc q = a'b'c'd' of 9,and the 0, from 14.2.(b), 0 = (F(an))-'(0), 7 from 14.3, (ii) (Y 6) n Y,+, = 8, (iii) each common point of y (resp. 6) and HL+ is their crossing (see 13.5). We will denote discs given by some of a, fi, y, 6 and by subarcs of 2,and C(z, E,,) by the sequence of their end-points. Further, put D = abcddc'b'a' and L, = D n H,+,. Let E , : D: --f D: be a continuous mapping sending a, y and 6 identically onto themselves,
340
App. A: COOK CONTINUA
abb'a' homeomorphically onto D, bcc'b' homeomorphically onto dcc'd' c D, M , = D: - abcc'b'a' identically onto itself, and y to e,(y) whenever Y E D, n C(Z,%).
By 14.5, a component of L, is either one point or an arc. Since q n L, = 8, the end points of each such arc component are in a u 6. Let T be the union of those components of L, which intersect either a or both y and 6, R (resp. a) the union of those components of L, n cddc' which intersect 6 (resp. both y and 6). Since H,+l is compact, ?: R and R are closed subsets of D (see 1.4). Now, put H:
=
(abb'a' n Ez- ' ( T ) )u (bcc'b' n E;
'(a))u R u R ,
K+l= ( H , , + , \ I n t D , ) u ~ : ,
f(y) = y otherwise. and define f : HL + -,H,,, by f ( y ) = E,(y) for y E Hz, Now, we must prove that, for a given (a,b,c; d)-crook e, with c,d $ X + , there are Yz and arcs a, B, y, 6 with the properties required. First, choose a simple closed curve d such that C(z, 8.) c Intf, f c IntD,. Choose an arc q c f such that q n @, = 8, and take arcs Z, p, 7,swith one of the end points a, b, c, d, respectively, and with the second, say a', b', c', d', in Intq and such that they are placed on f in the same order as a, b, c, d on C(z,en). Then, join the end-points of q by an arc q' intersecting C(z, E,,) in exactly two points between d and a, and a simple closed curve $, = = q u q' with a, b,c,dEInt$,, f z c D, is obtained. Thus, cL,p,y,8
\
Fig. A S
341
$15. Construction of D
satisfy the requirements with a possible exception of (ii) or (iii). Now, modify the arcs by 14.9 with 6 sufficiently small that the new one remain in D:,and q sufficiently small to prevent intersection, and so that the new arcs satisfy (ii) and (iii).
15.11. Remarks. (a) If (iii) were not required, a new simple closed curve might appear, as indicated in Fig. AS (one can also see there why we have defined H: by means of ?I R, R in 15.10). (b) One can see that f is almost identical, with changes in ID' = (D(Xz)). a' is a refinement of ID,, so that the %process may be applied. (c) Put Y;+ = f - ' ( Yn+l). By (ii) an8 (iii) we can see that j satisfies 15.1 (n + 1). Thus, the B-process can be applied. 15.12. The 9&98'-proeess. Let ((Hl, I;))l be an d-sequence, n a natural number. Take a z E F,, and a crook e,: C(z, E,,) -,C(z, E,). As indicated in the following diagram, let us apply the &-process to H,,, and proceed to the left with the %process. Then proceed to the right with the 98-process.
H1 .<-.
Hn .........Hn-1 .4-.-.-.-.-
H2
Hn+1
Hn+2
Hn+3
Hk+1
K + 2
K + 3
-.......... .<-.-.-.-.-
.<
Hi
H;
Hh-1
H:,
Put I;' = h-'(X).Then ((Hi, X'))l is an d-sequence and we say that it is created by the 9&@-process.
15.13. Now, a diagram D satisfying (a), (b), (c) and (d) from 12.2 may be constructed as follows: The first row, say (HI,,,),has been constructed in 14.12. Now, the (rn 1)th row (H,+ l,n) provided (Hm,,) is already given, can be constructed as follows: Apply the a&B-process on (Hm,,,)sufficiently many times to obtain an d-sequence (K,) and mappings k,: K , + H,,,, such that, for n = I,. .., rn, k, are &,-crooked on each circle in K , (see 5.6 and 5.8). Then, apply the WB-process with the prescribed primes +(rn + 1, n, i ) (see 12.1).
+
Appendix B MEASURABLE CARDINALS AND NON-ALGEBRAIC CATEGORIES
By 111.4.5, under (M), i.e. if there are not too many measurable cardinals in the set theory used, every concretizable category is algebraic. In this Appendix, we will show that this is not the case under non(M). This result was proved in 1971 and published in [KP,].
0 1.
a-additive measure
1.1, Definition. Let a be an infinite cardinal number. An a-additive measure on a set X is a mapping p: e x p x
such that
+2 =
{0,1>
(1) P ( X ) = 1, (2) for every disjoint collection (A,),,a of subsets of X, p ( U A l ) = isa
It is said to be non-trivial if p ( { x ) ) = 0 for every X E X .
1
1.2. Remark. The symbol in the equation in (2) designates the usual arithmetic sum. Thus, the condition (2) says that for a disjoint system ( A L (2a): p ( A J = 1 for at most one z, and (2b): p ( U A , ) = 1 iff there is a z with p(A,) = 1. tEa
$1. a-additive measure
343
1.3. Proposition. Let p: expX + 2 be an a-additive measure. Then (a) 4 8 ) = 0, (b) for every disjoint system ( A b ) b s B with cardB I a, p(
(4 /@\A)
= 1 - P(A), (d) for A = B, 4'4) I I@)? (e) for every system (A&B
bcB
Ab)
=
1
p(Ab) 9
boB
with cardB I a, p(
u
beB
Ab)
=
y","
p(Ab)
Y
(f) for every system (AbjbeBwith cardB I a, p(
A,
u
Ab)
beB
=
2%
p(Ab)
.
Proof. (a) follows by (2a) in 1.2 if we consider the system (A,),€,with = 8 for all z. (b) We can assume B c M. Define A: = A, for z E B, A: = 8 otherwise, and apply the equation (2) and (a). (c) We have p(X\A) p ( A ) = p ( X ) = 1 by (b). (d) We have p(B) = p ( A ) @\A) by (b). (e) We can assume B = to be an ordinal. Put
+
+ a
A;
=
Ay\UAa. d
Obviously, (A;) is a disjoint system and UAI, = U A Y .Thus, by (d) we have, for every 6, p(Aa) I p(
u
YEB
Ay), and hence
YSB
YES
on the other hand, by (2) and (d),
(f): By (c) and (e) we obtain p(nAb)
= p(X\U(X\Ab)) =
=
- p(u(x\Ab))
1 - rnax(1 - p(A,,)) = minp(A,).
344
App. B: NON-ALGEBRAIC CATEGORIES
1.4. Proposition. Let p: expX + 2 be a mapping, a a cardinal number. Then the following statements are equivalent:
(1) p is an a-additive measure, (2) for every A, p(X\A) = 1 - p(A), and for every system (A,),,,, (3) {A I p ( A ) = I} is an ultrafilter (see 11.6.1) closed with respect to the intersections of a elements. Remark. By the equivalence (1)=(3) we also see that every ultrafilter f closed with respect to the intersection of a elements gives rise to an aadditive measure p defined by p ( A ) = 1 for A E $, p ( A ) = 0 otherwise. Obviously, ,u is non-trivial iff f is non-trivial.
Proof. (1) +. (3): Let (1) hold, put f = { A Ip(A) = l}. We have X E $ by 1.1, 0 $ $ by 1.3(a). If A c B and A E ~we, have B E $ by 1.3(d). If A , E for ~ z~a, we have n A , E f by 1.3(f). Finally, let A u B E $. Then either A E f or B E $ by 1.3(e). (3) * (2): Let (3) hold. Exactly one of A and X\A has measure 1 (one at most, since A n (X\A) = 0 # f , one at least, since A u (X\A) = X E 2). Thus, ,u(X\A) = 1 - p(A). If, for some A,, p(A,) = 1, we have p ( U A , ) = 1 since U A x =I A, and A, E f . If p(A,) = 0 for all z E a, we have always X\A, E f and hence X\UA, = n(X\A,)Ef, so that p ( U A , ) = 0. Thus, p(UAI) = = maxp(A,). (2) * (1): Let (2) hold. We have
P(0) = 0 ,
since otherwise p(A) = 1 for every A (Really, put A , = A, A, = 0 for z 0: p ( A ) = p ( U A , ) = max(p(A), .), in contradiction with p(X\A) = = 1 - p ( A ) .Thus, p(X) = 1 - p(0) = 1.
+
40))
Let (A,),,, be a disjoint system. We have p(A,) = 1 for one A, at most (if p(A,) = p(A,) = 1 and z $: x we obtain a contradiction 0 = p(X\A,) = = P((X\A,)" A,) = max(p(X\A,), p(A,)) = I), and p ( U A , )= maxP(A,) = 1 iff there is a z with p(A,) = 1. Thus, p is an a-additive measure by 1.2.
52. If an a-measure exists
345
Q 2. If an a-additive measure exists, it is on a rather large set
In this section we will show that the condition (M) is quite reasonable. We will do it by reproducing the well-known fact that a measurable cardinal is necessarily strongly inaccessible (see 2.6 below). (Actually, no countably additive measure exists on the first inaccessible cardinal either, similarly on the second next etc. The proof of these statements is, however, much more complex than the simple inference we are going to present here. The reader may find it e.g. in [KM],.) If the Godel-Bernays set theory is free of contradiction, it remains so after adding the axiom of non-existence of strongly inaccessible cardinals, simply because no construction makes an inaccessible cardinality necessary: A union of accessibly large sets over an accessibly large one is still accessibly large (see 2.6(i)), a product X x Y can be represented as U X x ( y ) , and, finally, YGY cardXYi ~ a r d ( 2 ~= ) ' card2XxY so that (see 2.6(ii)) X y is accessibly large if X and Y are. 2.1. Proposition. Let ,u be a (non-trivial) a-additive measure on a set X, let ,u(Y)= 1. Then v: exp Y + 2 defined by v(A)= p ( A ) is a (non-trivial) a-additive measure. Proof is trivial. H 2.2. Proposition. Let ,u be an a-additive measure on X , let f:X + Y be a mapping such that p ( f - ' ( { y ) ) ) = 0 for every y E Z: Then v: exp Y + 2 defined by v(A) = p ( f - l ( A ) ) is a non-trivial a-additive measure. Proof. v(Y)= p ( X ) = 1. If (A,),,, is disjoint, (f-l(A,))z,, is disjoint and U f - ' ( A , ) = f - ' ( U A , ) . Thus, v ( U A , ) = v(A,).
2.3. Proposition. Let X = U X , , let there exist a non-trivial a-additive YCY
measure on X . Let ,u(X,) = 0 for all y. Then there is a non-triuial a-additive measure on the set I: Proof. For an X E X choose an ~ ( x )YE such that X E X ~ (For ~ ) .the mapping f: X + Y thus obtained obviously always f - l ( { Y ) ) = x,. Thus, p ( f - ' ( { y ] ) ) = 0 and we have the measure on Y from 2.2.
2.4. Proposition. Let p: expX -+2 be a non-trivial a-additive measure which is not B-additive. Then there exists a non-trivial a-additive measure on B.
346
App. B: NON-ALGEBRAIC CATEGORIES
Proof. Since p is not fl-additive, there is a system (A,),,, such that p ( A , ) = 0 for all z and p( U A , ) = 1. Restrict p on U A , by 2.1 and apply 2.3. I,,
I,,
2.5. Proposition. Let p be a non-trivial a-additive measure on X. Then cardX > 2". Proof. Suppose the contrary. Then (by 2.2 used for an invertible f ) there is a non-trivial a-additive measure p on X c expa. For an X E put ~
P(x)= {AIXEAEX},
N ( x ) = (AIX$AEX}.
Since X is a disjoint union of P(x) and N(x), we have p(P(x)) (by 1.3(b)).Put B = {x E a p(P(x))= l}.
+ p ( N ( x ) )= 1
1
Then for x $ B p(N(x)) = 1. We have
Thus, by 1.3(f),either p(@)
=
1 or p ( { B > )= 1 which is a contradiction.
2.6. An uncountable cardinal number x is said to be strongly inaccessibleif
(i) whenever x = cardUA, then either cardB bsB for some b, (ii) x > I implies x > 2'.
=x
or card&
=
x
2.7. Theorem. Let a be an infinite cardinal, let x be the smallest cardinal such that there is a non-trivial a-additive measure on x. Then x is strongly inaccessible. Proof. Let p be a non-trivial a-additive measure on x. Let x = U A b . beB
By 2.3 and 2.1 there is either a non-trivial cc-additive measure on B or on some Ab so that either cardB or some of the card& has to be x. If x > A, p is A-additive by 2.4. Thus, by 2.5, 1c > 2'.
9 3.
F-measures
3.1. Let cc be an infinite cardinal, F : Set"P-+ Set a functor. A mapping p: F(X)+F(l)
(1 is the one-point set (0)) is called a-additive F-measure on X if for every
347
93. F-measures
mapping q : Y
-+
F ( X ) with card Y I a there is a $: 1 -+ X such that F(+)Ocp
It is said to be non-trivial if p 3.2, Convention.
<:
+
P"(P. F(<) for every =
5: 1 -+ X .
is the mapping 1 -+ X with <:(O)
= x.
3.3. Lemma. A mapping p: p - ( x ) -+ 2
=
P-(l)
is a (non-trivial) a-additive P--measure iff it is a (non-trivial) a-additive measure. Proof. Let p be an a-additive measure, q : Y -+ P - ( X ) a mapping, = ( y E Y p(cp(y))= i } (i = 0, l), and card Y I a. Put
1
A
=
n 4 Y ) n n (x\q~(Y)).
YSYI
YEYO
By (1.3(c) and (f), p ( A ) = 1, and hence A =k 8. Take an a E A. Since (p-(
On the other hand, let p be an a-additive P--measure. Put $
=
= { A c X l p ( A ) = l}. We see that for A E and~ B =I A we have B E $ (using q : 2 -+ P - ( X ) defined by q(0) = A, q(1) = B :
P(cp(1)) = *-l(v(U
2
+-l(do)) = P(Cp(0))= 1 )
9
(+from 3.1), that for A u B E f either A E f or B E f (using 40: 3 -+ P - ( X ) defined by cp(0) = A, q(1) = B, q(2) = A u B), that 8 f and x E (using q,cp': 1 3 P - ( X ) defined by q(0) = 8, q'(0) = X ) , and that for d c f with c a r d d s a we have nd E f (using cp: d u (d} +P-(X) defined by cp(A) = A for A ~ dcp ,(d)= ). Thus, by 1.4, p is an a-additive measure.
+
nd
Now, p is trivial as measure iff p ( ( x } ) = 1 for an x E X , which holds iff p = p-(t,X). 3.4. Theorem. Let F : SetoP-+ Set befaithful, let cardF(1) I a. Zfthere exists a non-trivial a-additive measure on X , there also exists a non-trivial a-additive F-measure on X . Proof. For U E F ( X )and v ~ F ( 1put )
1
N(u, 0 ) = (x E x F ( g ) (u) = 11) .
348
App. B: NON-ALGEBRAIC CATEGORIES
Let p be a non-trivial a-additive measure. Since cardF(1) I a, and since for every u E F(X) x = N(u, u)
u
UCF(1)
is a disjoint union, we have exactly one V(U)E F(1) such that p(N(u, u(u)))= 1. Thus, a mapping v : F(X) -+ F(1) is defined. Take a mapping cp: Y 3 F(X) with card Y I a. Define cp': Y x F(1) -+ P-(X) by cp'(y, u) = N(cp(y), u). By 3.3 there is an x E X such that p c cp' = P - ( ( I ) 0 cp'. Thus, P(N(cp(Y)70))
hence
u = .(cp(Y))
=
1
iff x E "cp(Y)7 u) > iff F(5:) (cp(Y))= 0
9
i.e. v 0 cp = F(5:) 0 cp. Thus, v is an a-additive F-measure. For every x E X there is a u E F ( X ) with N(u, I?(<:) (u)) = (x}. It suffices to take the u = ( F ( f ) )(u), where F(
9 4.
The existence of non-algebraic categories under non(M)
4.1. Theorem. Under non(M), SetoPis not algebraic.
Proof. Should it be so, there would be a full embedding F : SetoP+ -+ Graph (see 11.5.6). Put G = U F where U is the natural forgetful functor of Graph. For convenience, let us represent binary relations on X as subsets of X2, i.e. as sets of mappings 2 + X . 0
By non(M) and Theorem 3.4, there is an a-additive G-measure v: G(X) + We have F ( X ) = (G(X),R ) and F(1)= (G(l), S) where R, S are binary relations. Since v is non-trivial, it is not of the form G(5) and since F is full, there is a cp E R such that v 0 cp $ S . But there is a II/: 1 + X such that v cp = G(II/) o cp. This is a contradiction since G(II/) preserves the relations. 4 -+ G(1).
0
$4. Non-algebraic categories
349
4.2. Corollary. Under non(M), no universal category is algebraic. In particular, S ( P + )and any S(F) with contravariantfaithful F are non-algebraic. Proof. If R is universal, there is a full embedding of SetoPinto R and therefore there cannot be one of R into Graph. For the second statement see 11.7.7. 4.3. Lemma. If (Set, F ) has an extension into an algebraic category, then S(F) is algebraic. Proof. Let @ be an extension of (Set, F ) into Graph. Put G = U o @ where U is the natural forgetful functor of Graph. We have a monotransformation p : F -+ G . Define R ( X ) c G ( X ) x G ( X )by @ ( X )= (G(X),R(X)). Obviously, we have a full embedding Y : S ( F ) -+ Re1(2,l)
defined by Y ( X ,r) = (G(X),R(X),p"(r)). 4.4. Corollary. Under non(M) there are concrete categories (R,U ) with a full embedding, but no extension into an algebraic category (e.g., (Set, P')).
RESEARCH PROBLEMS
1. There are a number of results on particular questions of representation and realization of groups which do not have a monoid (or category) counterpart (see the references mentioned in the second half of the Bibliographical remarks in Chapter IV). It would be interesting to analyze the possibilities of generalizations there. 2. The following interesting problem was communicated by L. Babai: By [Fr2], every finite group is isomorphic to the automorphism group of a graph with the total degree 3; on the other hand one can see easily that infinite groups force an unlimited increase of the degree. Do such groups exist that the graphs used for their representation necessarily contain a topologically complete graph with a basic vertices (i.e., that there are a vertices which are mutually joined by non-intersecting paths)? It is worth mentioning that L. Babai proved that when representing finite monoids, in contrast with Frucht’s result above, there appear inevitable topologically n-complete subgraphs. 3. There is one very fundamental open problem, the solution of which .will probably require a lot of combinatorics: Under non (M), is the large discrete category algebraic (i.e., embeddable into Graph)? We are, in fact, unable to exclude the possibility that the statement would not require (M), but still would be independent on the Godel-Bernays axiomatic system. 4. A rigid graph is said to be critical (co-critical, resp.) if when omitting (adding, resp.) any arrow one obtains a non-rigid graph. The two-point graph with one arrow joining the two points is both critical and co-critical. Are there any other rigid graphs both critical and co-critical? This
Research problems
351
problem has an obvious undirected variant, and both of these problems have, further, automorphism variants. There does not seem to be any evident interdependence between the four problems thus obtained. 5. We know that the category of commutative semigroups is not alguniversal. Still, is it not almost alg-universal? More important, however, would be the answer to the question : is the category of commutative monoids (with the unit preserving homomorphisms) almost alg-universal? 6. Let R be the free ring with two generators. Is the category of R-modules almost alg-universal? 7. There are many open problems concerning minimal alg-universal varieties (actually, with the exception of the unary case, almost all these problems are open). Does every alg-universal variety' contain a minimal one? There are many interesting particular problems, e.g. : What is the minimal alg-universal subvariety-if it exists-of the variety of semigroups, or of commutative groupoids? 8. Similarly, there are open basic problems concerning the minimal rich categories. Is every rich category preceded (in the ordering from V, 4 7) by a minimal rich one? In particular, does every rich monoid factorize to a minimal one? 9. Obviously, if St has a zero object then for no category A , RA is alguniversal. Characterize the categories for which there is always an A with RA alg-universal. 10. The category of small categories and the limit preserving functors is, under non(M), not algebraic. Thus, there is still a possibility that it may not be only alg-universal, but also universal independently on (M). 11. In many cases of the categories of topological spaces one has obtained only the alg-universality in the cases where the category itself is not algebraic (under non(M) ). In all of these cases the question on the universality is open. 12. The category of compact Hausdorff spaces, in particular, is not algebraic under non(M) (see [KP,]). Does one need the assumption (Yd) is proving the almost universality there? 13. There are some characteristics of topological spaces which have not yet been considered in connection with our problems. In particular, there is the dimension (or, more precisely, there are dimensions). 14. Is the necessary condition for being strongly algebraic from VII.6.7 also sufficient? If not (which seems probable), find a better one.
352
Research problems
Is the category of Banach spaces strongly algebraic.7 (It satisfies the condition from VII.6.7 for a reason considerably more sophisticated then it is the case with the categories proved to be strongly algebraic in Chapter VII.) 15. In [KP,] there was proved that under non(M) there are many nonalgebraic categories. Actually, roughly speaking, the everyday-life categories which are not algebraic for some obvious reason (e.g. for being directly a category of algebras or relations) tend generally to be non-algebraic. An important question is: (Under non(M)) Does there exist an algebraic category with no small left adequate? (For the notion of left adequate see [I1].) A positive answer to the problem 3 above would provide us with interesting examples. The large discrete category itself does not possess nice categorial properties, but its algebraicity implies the algebraicity of the complete cocomplete thin category Ofid of ordinals (with the usual ordering from smaller to larger ones; it is worth noting that it is not known whether the algebraicity of OfidoPis also implied by the algebraicity of the large discrete category).
BIBLIOGRAPHY Books: ARCHANGELSKU A,, PONOMAREV V. : Osnovy obSEej topologii v zadaeach i uprainenijach, Nauka, Moskva (1974). BERGEC. : Theorie des Graphes et ses Applications, Dunod, Paris (1958). BERGEC. : Graphs and Hypergraphs, North-Holland, Amsterdam (1973). BIRKHOFF G. : Lattice Theory (3rd ed.), Publ. Amer. Math. SOC.,New York (1967).
CECHE. : Topological spaces, NCSAV, Praha (1966).
COHNP. M. : Universal Algebra, Harper and Row, New York (1965).
FREYD P. : Abelian Categories. An Introduction to the Theory of Functors, Harper’s Series in Modern Mathematics. Harper & Row, Publishers, New York (1964). GILLMAN L., JERISONM . : Rings of Continuous Functions, Princeton, 1960. GRATZER G. : Universal Algebra. Princeton, 1968. (2nd ed.) D. van Nostrand, Springer-Verlag (1979). HARARY F. (ed.): Proof Techniques in Graph Theory, Academic Press (1969). HERRLICH H., STRECKER G. E.: Category Theory. An Introduction, Allyn and Bacon Inc., Boston (1973). KELLEY J. L. : General Topology, D. van Nosti-and Co., Inc., Princeton, New Jersey, Toronto-New York-London (1955). C . : Topologie I, 11. KURATOWSKI Monografie Matematyczne, Warsaw (1950).
354
Bibliography
[M]b
KURATOWSKI C., MOSTOWSKI A. : Teoria mnogosci, Monografie Matematyczne, Warsaw-Wroclaw (1952). MACLANE S. : Categories. For the Working Mathematician, Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, Berlin-HeidelbergNew York (1971).
[MB],,
S., BIRKHOFF G.: Algebra (2nd ed.), MACLANE The Macmillan Comp., New York (1968).
[Mi],
MITCHELL B.: Theory of Categories, Pure and Applied Mathematics 17. Academic Press, New York-London (1965). ZYKOVA. A.: Teorija konetnych grafov I., Nauka, Novosibirsk ( I 969).
[KM],
[z]b
Papers:
ADAMEKJ., KOUBEK V.: Products of graphs as a representation of semigroups, Recent advances in graph theory (Proc. Symp. Prague 1974), 13- 16, Academia, Prague 1975 J . , KOUBEKV., TRNKOVA V.: Sums of Boolean spaces represent every AVAMEK group, Pacific J. Math. 61 (1975), 1-6
J.: Bounded endomorphisms of lattices of finite height, ADAMSM. E., SICHLER Canad. J. Math. 29 (1977), 1254-1263 AVAMSM. E., SICHLERJ.: Homomorphisms of bounded lattice with a given sublattice, Archiv der Math. 30 (1978), 122- 128
R. D.: On collection of pseudo-arcs, Abstract 337 t, Bull. Amer. ANDERSON Math. Soc. 56 (1950), 350 ANDERSON R. D.: Atomic decomposition of continua, Duke Math. J. 24 (1956), 507 - 514 ARCHANGELSKU A. V. : The power of bicompacts with the first axiom of countability, DAN SSSR 187 (1969), 967-970
J.: Zum Cayleyschen Darstellungssatz, Math. Ann. 154 ARMBRUST M., SCHMIDT
(1 964), 70 - 72.
BABAIL.: Automorphism groups of planar graphs, I, D i m . Math. 2 (1972), 295 - 301 BABAIL.: Automorphism groups of planar graphs, 11, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdos on his 60th birthday), 29-84, Colloq. Math. SOC.Janos Bolyai, Vol. 10, North-Holland, Amsterdam BABAIL.: Groups of graphs on given surfaces, Acta Math. Acad. Sci. Hungar., 24 (19731, 215-221 BABAIL.: A remark on contraction of graphs with given group, Acta Math. Acad. Sci. Hungar., 25 (1974), 89-91 BABAIL. : Automorphism groups of graphs and edge contraction, Discrete Math. 8 (1974), 13-20
Bibliography
355
BABAIL.: On the minimum order of graphs with given group, Canad. Math. Bull. Vol. 17 (4), 1974 BABAIL.: Asyniniciric trees with two prescribed degrees, Acta Math. Acad. Sci,
Hungar., 29 (1977). 193-200 BABAIL., IMKNi i W . On groups of polyhedral graphs, Discrete Math. 5 (1973) 101 - 103 BABAIL., IMRICHW . : Sense preserving groups of polyhedral graphs, Monatsh. Math. (197.3, 1-2 BABAIL., LovAsz L.: Permutation groups and almost regular graphs, Studia Sci. Math. Hungarica 8 (1973), 141- 150 BINGR. H. : A homogeneous indecomposable plane continuum, Duke Math. J. 15 (I 948), 729 - 742 BING R. H. : Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51 BINGR. H.: Each homogeneous non-degenerate chainable continuum is a pseudoarc, Proc. Amer. Math. SOC.10 (1959), 345-346 BIRKHOFF G. : On groups of automorphisms (in Spanish), Revista Union Math. Argentina 1 1, I 5 5 - 157 BOUWER I. Z.: Section graphs for finite permutation groups, J. Comb. Theory
6 (1969), 378-386
BROWNM.: On the inverse limit of Euclidean n-spheres, Trans. Am. Math. SOC. 96 (1960), 129-134 BLJKOVSK~. L., HEDRL~N Z., PULTRA,: On topological representations of semigroups and small categories, Mat.-fyz. Easopis SAV 15 (1965), 195- 198
J. J.: Confluent mappings and unicoherence of continua, Fund. CHARATONIK Math. 56 (1964), 213-220 CHENC. C., GRATZERG . : On the construction of complemented lattices, J. of Algebra 11 (1969), 56-63 CHVATAL V. : On finite and countable rigid graphs and tournaments, Comment. Math. Univ. Carolinae 6 (1965), 429-438. CHVATAL V., SICHLER J.: Chromatic automorphisms of graphs, J. Comb. Theory (B) 14 (1973), 209-215 COOKH. : Upper semicontinuous continuum valued mappings onto circle-like continua, Fund. Math. 60 (1967), 233-239 COOKH.: Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241 -249 DANTZIGD. VAN: Uber topologische horncyenc ( ' n n l i n t i a , Fund. Math. XV (1930), 102- 125 DLABV., NEUMANN B. H.: Semigroups with few endomorphisms, J. of the Australian Math. SOC.10 (1969), 162- 168 W. : Primfaktorzerlegung und Automorphismen des Kardinalproduktes DORFLER von Graphen, Glasnik Mat., Ser. 111 9 (29) (1974), 15-27
Bibliography W. : Set-systems and their automorphisms, Recent Advances in Graph DORFLER Theory (Proc. Second'Czechoslovak Sympos., Prague 1974), 153- 166, Academia, Prague 1975
Z.: On atomic mappings, Colloq. Math. 27 (1973), EMERYK A., HORBANOWICZ 49 - 55 S., MACLANE S. : General theory of natural equivalences, Trans. Am. EILENBERC Math. Soc. 58 (1945), 231 -294
Ful
GeHiQ]
FELSCHER W. : Birkhoffsche und kategorische Algebra, Math. Ann. 180 (1969), 1-22 FREYD P. J.: Concreteness, J of Pure Appl. Alg. 3 (1973), 171- 191 FRIED E., SICHLER J. : Homomorphisms of commutative rings with unit element, Pacific J . Math. 45 (1973), 485-491 FRIED E., SICHLER J. : Homomorphisms of integral domains of characteristic zero, Trans. Amer. Math. SOC.225, (1977), 163- 182 FRUCHTR. : Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compos. Math. 6 (1938), 239-250 FRUCHT R.: Graphs of degree 3 with given abstract group, Canad. J. Math. l(1949) 365-378 FRUCHT R., GEWIRTZ A,, QUINTAS L. V.: The least number of edges for graphs having automorphism group of order three, Recent Trends in Graph Theory (Proc. Conf., New York 1970) Lecture Notes in Math., vol. 186, Springer-Verlag, Berlin and New York, 1971, 95- 104 FUHRKEN G.: On automorphisms of algebras with a single unary operation, Portugaliae Math. 32 (1973), 49 - 52 A. (see also) Frucht R., Gewirtz A., Quintas L. V. GEWIRTZ GEWIRTZ A., HILL A,, QUINTASL. V.: Extremum problems concerning graphs and their groups, Combinatorial Structures and their Application (Proc. Calgary Internat. Conf., Calgary 1969), 103- 109, Gordon and Breach, New York 1970. P. (see also) Hedrlin Z., GoralEik P. GORALC~K GORALE~K P.: On translations of semigroup 111 (Russ.), Mat. Eas. SAV 4 (1968), 273 - 282 Z.: On translations of semigroups I1 (Russ.), Mat. fas. GORALE~K P., HEDRL~N SAV 4 (1968), 263 - 272 Z.: On reconstruction of monoids from their table GORALE~K P., HEDRL~N fragments, Math. Z. 122 (1971), 82-92 GRATZERG. : (see also) Chen C. C., Gratzer G. GRATZERG., SICHLERJ.: On the endomorphism semigroup (and category) of bounded lattices, Pacif. J. Math. 35 (1970), 639-647 GREGORG.: On rich monoids, Comment. Math. Univ. Carolinae 16 (1975), 735 - 744 GREGORG.: The infinite minimal rich monoid, Acta Univ. Carolinae Math. et Physics 18 (l977), 23 - 34
Bibliography
357
GRWT DE J. : Groups represented by homeomorphism groups, Math. Annalen 138 (1959), 80-102 GROOTDE J. WILLER. J. : Rigid continua and topological group-pictures, Archi" der Math. 9 (1958), 441 -446 G . : The least number of edges for graphs having dihedral automorphism HAGGARD group, Discrete Math. 6 (1973), 53 - 78
Z.: see also HEDRL~N Bukovsky L., Hedrlin Z., Pultr A,; Pultr A,, Hedrlin Z. ; VopCnka P., Pultr A,, Hedrlin Z . ; GoralEik P., Hedrlin Z. HEDRL~N Z. : Commutative systems as generalized semigroups (Russ.), Dokl. AN SSSR 164 (1965), 483-486
Z. : Non-constant continuous transformations form any semigroup with HEDRL~N a unity, Nieuw Archief voor Wiskunde XIV (1966), 230-236 Z.: On endomorphisms of graphs and their homomorphic images, HEDRL~N Proof techniques in graph theory, Acad. Press, 1969 Z. : On universal partly ordered sets and classes, J. Algebra 11 (1969), HEDRL~N 503 - 509 Z.: Extension of structures and full embeddings of categories, Actes HEDRL~N du Congres Internat. des Mathematiciens 1970, tome 1, Paris 1971, 319-322 Z., GORALE~K P.: On translations of semigroups I (Russ.), Mat. Eas. HEDRL~N SAV 4 (1968), 161- 176 Z., KASTLJ.: On extensions onto semigroups of semigroups with two HEDRL~N generators, Semigroup Forum 3 (1971), 86-88 Z., LAMBEKJ.: How comprehensive is the category of semigroups? HEDRL~N J. Algebra 11 (1969), 195-212. Z., MCDOWELLR. H.: Partly ordered sets with given monoid of orderHEDRL~N preserving mappings, Preprint Z., MENDELSOHN E.: On category of graphs with a given subgraph HEDRL~N with applications to topologyand algebra, Canad. J. Math. 21 (1969), 1506- 1517
Z., PULTRA,: Remark on topological spaces with given semigroups, HEDRL~N Comment. Math. Univ. Carolinae 4 (1963), 161 - 163 Z., PULTR A,: Relations (graphs) with given finitely generated semiHEDRL~N groups, Monatsh. Math. 68 (1964), 213-217 Z., PULTRA.: Symmetric relations (undirected graphs) with given semiHEDRL~N groups, Monatsh. Math. 69 (1965), 318-322 Z., PULTRA,: 0 predstavlenii malych kategorij, Dokl. AN SSSR 160 HEDRL~N (1965), 284 - 286 HEDRLIN Z., PULTRA.: On rigid undirected graphs, Canad. J. Math. 18 ( 1966), 1231- 1242 HEDRL~N Z., PULTRA,: On full embeddings of categories of algebras, Illinois J. of Math. 10 (1966), 392-406
Bibliography Z . , PULTRA. : On categorial embeddings of topological structures into HEDRL~N algebraic, Comment. Math. Univ. Carolinae 7 (1966), 377-400
Z . , SICHLER J. : Any boundable binding category contains a proper class HEDRL~N of mutually disjoint copies of itself, Alg. Universalis 1 (1971), 97- 103 Z., PULTRA., TRNKOVAV . : Concerning a categorial approach to HEDRL~N topological and algebraic theories, General Topology and its Relations to Modern Analysis and Algebra 11, Proc. 2nd Prague Top. Symposium 1966, 176- 181 HELLP.: Rigid undirected graphs with given number of vertices, Comment. Math. Univ. Carolinae 9 (1968), 51 -69 HELLP.: Full embeddings into some categories of graphs, Alg. Universalis 2 (1972), 129- 141 HELLP.: On some strongly rigid families of graphs and the full embeddings they induce, Alg. Universalis 4 (19741, 108- 126 HELLP., NESETRILJ. : Rigid and inverse rigid graphs, Combinatorial structures and their applications, Gordon Breach 1970, 169- 171 HELLP., NESETRIL J. : k-societies with given semigroup, Combinatorial structures and their applications, Gordon Breach 1970, 301 -302 HELLP., NESETRILJ.: Graphs and k-societies, Canad. Math. Bull. 13 (1970), 375 - 381 HELLP., NESETRILJ.: Groups and monoids of regular graphs (and of graphs with bounded degree), Canad. J . Math. 25 (1973), 239-251 [HChKN] HELLP., CHVATAL V., KUCERAL., NESETRILJ.: Every finite graph is a full subgraph of a rigid graph, J. Comb. Theory 11 (1971), 284-286 HEMMINGER R. L.: On the group of a directed graph, Canad. J. Math. 18 (1966), 211-220 HEMMINGER R. L. : The group of and X-join of graphs, J. of Comb. Theory 5 (1968), 408-418 HENDERSON G. W.: The pseudoarc is an inverse limit with one bounding map, Duke Math. J. 31 (1964), 421-425 HERRLICH H.: On the concept of reflections in general topology, Proc. Symp. on extension theory of topological structures, Berlin 1967 HERRLICH H. : Topologische Reflexionen und Coreflexionen, Lecture Notes in Mathematics 78 (1 968), Springer-Verlag Berlin-Heidelberg-New York HILLA , : see Gewirtz A,, Hill A,, Quintas L. V. Z . : see Emeryk A,, Horbanowicz Z. HORBANOWICZ IMRICH W. (see also) Babai L., Imrich W. IMRICH W. : Graphen mit transitiver Automorphismengruppe, Monatsch. Math. 73 (1969), 341 - 347 IMRICHW . : On graphical regular representation of groups, Infinite and finite sets (Colloq. Kesztehely, 1973; dedicated to P. Erdos on his 60th birthday), Vol. 11, 905-925, Colloq. Math. SOC.Janos Bolyai, Vol. 10, North-Holland, Amsterdam
Bibliography
3 59
IMRICH W . : On graphs with regular groups, J. Comb. Theory Ser. B 19 (1975), 174- 180 IMRICHW., WATKINSM. E.: On graphical regular representations of cyclic extensions of groups, Pacific J. Math. 55 (1974), 461 -477 ISBELLJ. R.: Adequate subcategories, Illinois J. Math. 4 (1960), 541 -552 ISBELL J. R.: Two set-theoretical theorems in categories, Fund. Math. 53 (1963), 43 - 49 ISBELLJ. R. : Subobjects, adequacy, completeneess and categories of algebras, Rozpravy matematyczne XXXVI (1964)
[JI
ISBELL J. R.: Structure of categories, Bull. Amer. Math. SOC.72 (1966). 619-655 ISBELL J. R.: Small adequate subcategories, J. London Math. SOC.43 (1968), 242 - 246 H . : Regulare Graphen beliebigen Grades mit vorgegebenen Eigenschaften, IZBICKI Monatsch. Math. 64 (1960), 15-21 B.: Sublattices of a free lattice, Canad. J. of Math. 13 (1961). 256-264 JONSSON KASTLJ.: see Hedrlin Z . , Kastl J. KAT~TOV M.: Remarks on Boolean algebras, Colloq. Math. I1 (1951), 229-235 KNASTER B. : Un continu dont tout sous-continu est indecomposable, Fund. Math. I11 (1922), 247 - 286 KOLLARJ. : Some subcategories of integral domains, J. Algebra, 54 (1978), 329-331 KOUBEKV. : (see also), Adamek J., Koubek V.; Adamek J., Koubek V., Trnkova V. ; Miinzova M., Koubek V. KOUBEK V.: Set functors I and 11, Comment. Math. Univ. Carolinae 12 (1971), 175- 195 and 14 (1973), 47-59 KOUBEKV. : Each concrete category has a representation by T,-paracompact topological spaces, Comment. Math. Univ. Carolinae 15 (1974), 655 - 663 KOUBEK V. : On categories into which each concrete category can be embedded, Cahiers de Topologie et Geometrie Differentielk, Vol. XVlI (1976), 35 - 57
V.: On the category of modular lattices, to appeaI KOUBEK KOUBEKV. : Graphs with given subgraphs represent all categories, Comment. Math. Univ. Carolinae, 18 (1977), 115-127 KOUBEK V. : On categories into which each concrete category can be embedded I1 contravariant case, Cahiers de Topologie et Geometrie Differentielk XVIII (1977), 249 - 269 KOUBEK V., REITERMAN J. : Set functors I11 - monomorphisms, epimorphisms, isomorphisms, Comment. Math. Univ. Carolinae 14 (1973), 441 -455 KOUBEK V., REITERMAN J.: Factor categories of the category of sets: description and concreteness, J. Pure Appl. Algebra 4 (1974). 71 -77
360 [KoNR]
Bibliography KOUBEKV., NESET~IL J., RODL V.: Representing groups and semigroups by products in categories of relation, Alg. Universalis 3 (1974), 336-341 KUCERAL. : Lectures from the theory of categories (Czech), Charles University 1970 - preprint. KUEERAL. : On bindability of products and joins of categories, Comment. Math. Univ. Carolinae 12 (1971), 373-376 KUCERAL.: Every category is a factorization of a concrete one, J. Pure Appl. Alg. 1 (1971), 373-376 KUEERAL.: On universal concrete categories, Algebra Univ. 5 (1975), 149- 151 KUCERAL. : Representations of monoids and categories by homomorphisms of semigroups with unity, to appear K U ~ E RL., A PULTRA,: On a mechanism of defining morphisms in concrete categories, Cahiers de Topologie et Gtomttrie Diffkrentielle 13 (1972), 397 -410 KUEERAL., PULTRA. : Non-algebraic concrete categories, J. Pure Appl. Alg. 3 (1973), 95- 102
J. : see Hedrlin Z . , Lambek J. LAMBEK LovAsz L. (see) Babai L., Lovasz L. S . (see) Eilenberg S., MacLane S . MACLANE MANIP. : Automorphismen von polyedrischen Graphen, Math. Ann. 192 (1971), 297 - 303 S . : Sur les continus indecomposables, Fund. Math. X (1927), MAZURKIEWICZ 305-310
D. J., QUINTAS L. V. : A stability theorem for minimum edge graphs MCCARTHY with given abstract automorphism group, Trans. Amer. Math. SOC.208 (1975), 27 - 39 R. H.: (see) Hcdi-lin Z. MCDOWFIIR. ‘LICDOWELL MCKENZIER., MONKJ. D.: On automorphism groups of Boolean algebras, 10. Infinite and finite sets, Keszthely (Hungary), 1973 MENDELSOHN E. : see also Hedrlin Z., Mendelsohn E. MENDELSOHN E. : Sip-indicators, products, and graphs with given semigroups, Preprint (1967-8) MENDELSOHN E.: Every group is the colineation group of some projective plane, Preprint (1969)
E.: On the monoid of endomorphisms of automata, Preprint (1970) MENDELSOHN MENDELSOHN E. : On a technique for representing semigroups as endomorphism semigroups of graphs with given properties, Semigroup Forum 4 (1972), 283 - 294
J. : A functional conception of snake-like continua, Fund. Math. MIODUSZEWSKI 51 (1962), 179- 189 A. B. : Topological representation of semigroups, General MIRANDA DE,PAALMAN Topology and its Relation to Modem Analysis and Algebra 11. Proc. of the 2nd Prague Top. Symposium 1966, Academia Prague 1967, 276 -282
Bibliography
361
MOISEE. E.: An indecomposable plane continuum which is homeomorphic to each of its non-degenerate subcontinua, Trans. Amer. Math. SOC. 63 (1948), 581-594 MOISEE. E.: A note on the pseudo-arc, Trans. Amer. Math. SOC.64 (1949), 57 - 58 MULLERV., PELANT J. : On strongly homogeneous tournaments, Czech. Math. Journ. 24 (99) (1974), 378-391 MULLERV., PELANT J., NESETRILJ.: Either tournaments or algebras? Discrete Math. 1 1 (1975), 37-66 MUNZOVIM. : Transformations determining uniquely a monoid, Transformations determining uniquely a monoid 11, Comment. Math. Univ. Carolinae 1 1 (1970), 595-618, 15 (1974), 311-333 MUNZOVA M., KOUBEK V.: Selections of mappings by graphs, Proceedings of the graph symposium held in Prague, June 1974, Academia, Prague (1979, 387 - 392 NESETRILJ. : see also: Hell P., NeSetiil J.; Hell P., Chvatal V., KuEera L., NeSetiil J.; Muller V., Pelant J., NeSetiil J. NESETRILJ. : On symmetric and antisymmetric relations, Monatsh. Math. 76 (l972), 323 - 327 NESETRILJ., PULTR A. : Representing of monoids by mappings preserving equivalences, Algebra Universalis 1 (19711, 155- 162 NOWITZL. A., WATKINS M. E.: Graphical regular representations of non-abelian groups, I, Canad J. Math. 24 (1972), 993- 1008
PELANT J. : see:
Muller V., Pelant J., NeSetiil J. ; Muller V., Pelant J.
PLATTC.: A note on endomorphism semigroups, Bull. A.M.S. 13 (1970), 47-48 PLATTC.: One-to-one and onto in algebraic categories, Alg. Universalis 1 (1971), 117-124 PLATTC.: Iterated limits of universal algebras, Alg. Universalis 1 (1971), 167- 181 POHLOVA V. ( K ~ J R K O V ~ P O H LV.) OV : Factorization A and non-algebraic categories, Comment. Math. Univ. Carolinae 14 (1973), 241 -262 PONOMAREV V.: Lectures held in Prague in 1973. PULTRA. : see also: Bukovsk? L., Hedrlin Z., Pultr A.; Hedrlin Z., Pultr A,; Vop8nka P., Pultr A., Hedrlin Z.; KuEera L., Pultr A,; NeSetiil J., Pultr A. PULTRA. : Concerning universal categories, Comment. Math. Univ. Carolinae 5 (1964), 227-239
362 P 2 1
Bibliography PULTRA. : On selecting of morphisms among all mappings between underlying sets of objects in concrete categories and realization of these, Comment. Math. Univ. Carolinae 8 (1967), 53 - 83 PULTRA.: Limits of functors and realizations of categories, Comment. Math. Univ. Carolinae 8 (1967), 663 - 682 PULTRA.: A remark on selective functors, Comment. Math. Univ. Carolinae Carolinae 9 (1968), 191 - 196 PULTRA.: On full embeddings of concrete categories with respect to forgetful functors, Comment. Math. Univ. Carolinae 9 (1968), 281 - 305 PULTRA. : Eine Bemerkung iiber volle Einbettungen von Kategorien von Algebren, Math. Annalen 178 (1968), 78-82 PULTRA., HEDRL~N Z . : Relations (graphs) with given infinite semigroups, Monatsh. Math. 68 (1964), 421 -425 PULTRA., SICHLER J. : Primitive classes of algebras with two unary idempotent operations, containing all algebraic categories as full subcategories, Comment. Math. Univ. Carolinae 10 (1969), 425-445 PULTRA., TRNKOVA V. : On realization and boundability of concrete categories in which the morphisms are choiced by local conditions, Comment. Math. Univ. Carolinae 8 (1967), 651 -662
[QJ [Q2I
PULTRA,, TRNKOVA V.: Strong embeddings into categories of algebras, Illinois J. Math. 16 (1972), 183- 195 QUINTAS L. V. : see also: Frucht R., Gewirtz A,, Quintas L. V.; Gewirtz A,, Hill A,, Quintas L. V.; McCarthy D. J., Quintas L. V. QUINTAS L. V. : Extrema concerning asymmetric graphs, J. Combinatorial Theory 3 (1967), 57 - 82 QUINTAS L. V.: The least number of edges for graphs having symmetric automorphism groups, J. Combinatorial Theory 5 (1968), 115 - 125 REITERMAN J.: see also: Koubek V., Reiterman J. ; Trnkova V., Reiterman J.
REITERMAN J. : An example concerning set-functors, Comment. Math. Univ. Carolinae 12 (1971), 227-233 RIEGERL. : Some remarks on automorphisms in Boolean algebras, Fund. Math. 38 (1951), 209-216 RODLV.: see Koubek V., NeSetiil J., Rod1 V. ROSENBERG I.: Strongly rigid relations, Rocky Mountains J. of Math. 3 (1973), 631-639 R OS I C KJ.~: Strong embeddings into categories of algebras over a monad, I, 11, Comment. Math. Univ. Carolinae 14 (1973), 699-718, 15 (1974), 131 -147 R OS I C KJ.~: On extensions of full embeddings and binding categories, Comment. Math. Univ. Carolinae 15 (1974), 631 -653
Bibliography
363
ROSICK? J. : Codensity and binding categories, Comment. Math. Univ. Carolinae 16 (1975), 515-529 ROSICKY J . : Concerning binding categories, Czech. Math. J. 25 (1975), 362-368 SABIDUSSI G.: Graph multiplication, Math. Zeitschr. 72 (1960), 446-457 SABIDUSSI G.: Graphs with given infinite groups, Monatsh. Math. 64 (1960), 64 - 67 SABIDUSSI G.: Vertex-transitive graphs, Montsh. Math. 68 (1964). 426-438 SABIDUSSI G. : Graphs with given group and given graph-theoretical properties, Canad. J. Math. 9 (1957), 515-525 SCHMIDT J. : see also Armbrust M., Schmidt J. SCHMIDT J. : Direct sums of partial algebras and final algebraic structures, Canad. J. Math. 20 (1968), 872-887 SICHLER J.: see also: Fried E., Sichler J.; A d a m M. E., Sichler J . ; Hedrlin Z., Sichler J.; Chvatal V., Sichler J.; Gratzer G., Sichler J.; Pultr A., Sichler J. SICHLER J. : Category of commutative groupoids is binding, Comment. Math. Univ. Carolinae 8 (1967), 753-755 '
SICHLER J.: '%(I, 1) can be strongly embedded into category of semigroups, Comment. Math. Univ. Carolinae 9 (1968), 257-262 SICHLER J . : Concerning minimal primitive classes of algebras containing any category of algebras as a full subcategory, Comment. Math. Univ. Carolinae 9 (1968), 627 - 635 SICHLER J.: Thesis (1968) SICHLER J. : Non-constant endomorphisms of lattices, Proc. Amer. Math. SOC. 34 (1972), 67 - 70 SICHLER J.: Testing categories and strong universality, Canad. J. Math, 25 (197% 370 - 385 SICHLER J . : One-to-one and onto homomorphisms of bounded lattices, Alg. Universalis 1 (1971), 267 SICHLER J. : Weak automorphisms of universal algebras, Alg. Universalis 3 (197% 1-7 T K U K O V .~: see also:
Adlimek J.. Koubek V., Trnkovi V.; Hedrlin Z . , Pultr A,, Trnkova V.; Pultr A., Trnkova V.
TRNKOVA V. : Universal categories, Comment. Math. Univ. Carolinae 7 (1966), 143- 206
364
Bibliography TRNKOVA V. : Universal category with limits of finite diagrams, Comment. Math. Univ. Carolinae 7 (1966), 447-456 TRNKOVA V. : Strong embedding of the category of all groupoids into the category of semigroups, Comment. Math. Univ. Carolinae 9 (1968), 251 -256 TRNKOVA V.: Some properties of set functors, Comment. Math. Univ. Carolinae 10 (1969), 323-352 TRNKOVA V. : When the product preserving functors preserve limits, Comment. Math. Univ. Carolinae 11 (1970), 365 - 378 TRNKOVA V.: On descriptive classification of set-functors I and 11, Comment. Math. Univ. Carolinae 12 (1971), 143-174 and 12 (1971), 345-357 TRNKOVA V. : Non-constant continuous mapping of metric or compact Hausdorff spaces, Comment. Math. Univ. Carolinae 13 (1972), 283-295 TKNKOVA V . : Strong embeddings of the category of graphs into topological categories, Recent advances in graph theory (Proc. Symp. Prague 1974), Academia, Prague 1975, 51 1-515
TRNKOVA V.: Representation of semigroups by products in a category, J. of Algebra 34 (1975), 191 -204 TRNKOVA V. : On a representation of commutative semigroups, Semigroup Forum 10 (1975), 203-214 TRNKOVA V. 1 Vse malyje kategorii predstavimy nepreryvnymi nepostojannymi otobraienijami bikompaktov, DAN SSSR 230 (1976), 789 - 791 TRNKOVA V. : Categorial methods are useful for topology, General Topology and its Relation to Modern Analysis and Algebra IV, Proc. 4th Prague Top. Symp., 1?76, Lecturc Uotes in Math. Springer-Verlag (1977), 21 1-225 TRNKOVA V., REITERMAN J. : When categories of presheaves are binding, General Topology and its Relation to Modern Analysis and Algebra, 111, Proc. 3rd Prague Top. Symp., 1971, Academia, Prague 1972,447-450 J . : On full embeddings of categories of algebras into TRNKOVA V., REITERMAN categories of functors with thin domain, Comment. Math. Univ. Carolinae 12 (1971), 419-422
TRNKOVA V., REITERMAN J . : The categories of presheaves containing any category of algebras, Dissertationes Mathematicae 124 (1975), 1-58 TRNKOVA V., REITERMAN J . : Topological categories containing any category of .ilgebras, Czech. Math. J . 27 (1977), 626-643 ~ J L M E RF.: Properties of dense and relative adjoint functors, J. of Algebra 8 (l968), 77 - 95
VI N~ R EJ.K: A new proof of the Freyd’s theorem, J. Pure and Appl. Als. X f 1976), 1-4 Z.: A rigid relation exists on any set, Comment. VOPBNKA P., PULTRA,, HEDRL~N Math. Univ. Carolinae 6 (1965), 149- 155
365 WATKINS M. E. : see : Imrich W., Watkins M. E.; Nowitz L. A,, Watkins M. E. [W]
P. M . : Free lattices, Ann. of Math. 42 (1941), 325-330 WHITMAN WILLER. J . : See Groot de J., Wille R. J.
INDEX A
abstractization 24 adjoint 38 algebra 41 algebraic a. category 16, 53 a. operation 22 strongly a. 17 a. structure 41 algebraically selective functor 284 algebraically universal category 53 alg-universal category 53 almost (alg-) universal 55 almost identical mapping 329 approach 296 circle a. 315 Q-adic circle a. 315 solenoidal a. 3 15 arc 300 atomic mapping 307 automorphism 5,41 automorphism group 5
B big category 28 Boolean algebra 169 bound (lower and upper) 35 bounded lattice 165
C canonical polynomial 163 capacity 162, 164 carrier 24 category 8, 28 algebraic c. 53 algebraically universal c. 53 big. c. 28 cocomplete c. 35 complete c. 35 concrete c. 12,39 concretizable c. 39 discrete c. 30 extensionally universal c. 53 finite c. 28 large c. 28 poorc. 196 regular concrete c. 44 rich c. 196 small c. 28 strongly algebraic c. 279 thin c. 30 topological c. 259 universal c. 53 Cayley-MacLane representation Cayley representation 26 chain circular c. (C-chain) 297 linear c. (L-chain) 297
50
367 chainable 297 chromatic 110 n-c. graph 110 c.number 110 circular chain 297 cocone 35 coequalizer 34 colimit 35 collection 40 coloring 110 compatible mapping 22,41 complete c. category 35 c. graph 110 c. homomorphism 169 c. lattice 169 completely distributive 169 composant 294 concrete c. category 12,39 c. monoid 8,24 c. subcategory 13,39 concretizability 10 cone 35 confluent mapping 308 connected graph 103 strongly c. g. 103 constants 22 continuum 292 decomposable c. 293 degenerate c. 292 circle-like c. 296 hereditarily indecomposable c. 304 indecomposable c. 293 non-degenerate c. 292 Peanoc. 304 Q-adic circle-like c. 315 snake-like c. 296 c. valued mapping 310 zero-adic circle-like c. 298 contraction 209 contravariant functor 31 coproduct 33 cosingleton 34 covariant functor 31
covering irreducible c. 263 locally-finite c. 263 point-finite c. 263 crook 301 crooked c. mapping 301 c.space 304 cycle 67 proper c. 67 D decomposable continuum 293 decomposing system 237 degenerate continuum 292 diagram 35 directed d. edge 102 d. graph 101 discrete category 30 distributive lattice 168 domain 28
E edge 102 embedding full e. 43 strong ff-e. 279 VV-e. 183 endomorphism 5,41 endomorphlsm monoid 6 epimorphism 30 epitransformation 33 equally realized 40 equivalence (natural e.) 32 equivalent categories 33 evaluation functor 37 extension 14,43 e. in the stronger sense 44 extensionally universal category F faithful functor 39 finite category 28
53
368 forgetful functor 31, 39 full f. embedding 43 f. functor 42 f. subcategory 10,29 f. subgraph 110 lunctor 31 algebraically selective f. 284 contravariant f. 31 covariant f. 31 evaluation f. 37 faithful f. 39 forgetful f. 31, 39 full f. 42 natural forgetful f. 39 pointed f. 282 f. reflecting isomorphisms 123 underlying f. 48 zero-morphism-preserving f. 186
G generators 169 graph 101 I-labelled g. 213 ?&labelled topologized g. 220 connected g. 103 directed g. 101 g. homomorphisms 103 n-chromatic g. 110 n-colorable g. 110 symmetric g. 102 undirected g. 102 group (automorphism group) 5
H hereditarily indecomposable continuum 304 homomorphism 22,41 graph h. 103 I identity 28 indecomposable continuum
293
initial object 34 inverse spectrum 295 continuum valued (CV) i s . 296 irreducible 172 i. covering 263 minimal i. element 172 isofunctor 39 isomorphic i. categories 39 i. concrete categories 13 i. concrete monoids 25 concretely i. 40 i. monoids 25 i. objects 30 isomorphism 5,30
J joined vertices 67 join-singular 165
L large category 28 lattice bounded 1. 165 complete I. 169 distributive 1. 168 1. polynomials 161 LC-chainable 297 L-chainable 297 left adjoint 38 length full 1. of a word 146 1. of a cycle 103 1. of a path 103 1. of a polynomial 162 1. of a poset 118 1. of a word 146 limit 35 linear chain 297 Lipschitz mapping 213 local 1. homeomorphism 208 1. isometry 209
369 locally finite covering 263 lower bound 35
0
M (M) (the assumption (M)) 77 mapping almost identical m. 329 atomic m. 307 m. carrying a morphism 40 continuum valued m. 310 &-crooked m. 301 Lipschitz m. 213 monotone m. 117, 307 open m. 208 preatomic m. 307 quasi-open m. 208 quotient m. 209 underlying m. 40 upper semicontinuous m. 309 m-compact 260 measure a-additive m. 342 a-additive F-m. 346 meet-singular 165 monoid 6,24 endomorphism m. 6 concrete m. 8,24 monomorphism 30 mono tone m. mapping 117 m. mapping between metric spaces 307 m. system of sets 292 monotransformation 33 morphism 5,8,28 multigraph 29
N natural n. equivalence 32 n. forgetful functor 39 n. transformation 32 naturally equivalent 32 node 102 non-degenerate continuum
object 5, 8, 28 initialo. 34 terminal 0. 34 one-connected set 328 operation algebraic 0. 22 ordering partial 0. 117
P partial ordering 117 path 103, 300 Peano continuum 304 pointed functor 282 pojntdnite covering 263 poor category 196 poset 117 preatomic mapping 307 preserve (co)limits 36, 188 product 33 proper p. cycle 67 p. subgraph 110 pseudoarc 305 pullback 34 pushout 34
Q quasi-local q.4. homeomorphism q.4. isometry 209 quasi-path 103 quotient 209 quotient mapping 209
208
R
292
range 28 rank of a polynomial 162 reflect (to r. isomorphisms) 123 regular concrete category 44 relation 21 relational structure 41
370 representation Cayley r. 26 Cayley-MacLane r. 50 rich category 196 minimal r. c. 196 strongly minimal r. c. 197 right r. adjoint 38 r. translation 26 rigid automorphism r. 15 endomorphism r. 15 r. object 15 S semigroup 24 s. with unit 6 singleton 34 singular join-s. 165 meet-s. 165 skeleton 33 small categorji 28 solenoid P-adics. 296 polyadic s. 297 strong s. co-embedding 78 s. embedding 15,48 s. ff-embedding 279 strongly s. algebraic 17, 279 s. embeddable 17,48 s. inaccessible 346 structure 4 algebraic s. 41 relational s. 41 structure type 4 subcategory 28 fulls. 10, 29 subcontinuum 292 subgraph 110 propers. 110
sum of a type 62 symmetric graph 102 T terminal object 34 thin category 30 topological category 259 topologized &labelled t. graph 220 total character 259 transfer property 44 transformation (natural t.) 32 translation (left or right) 26 type 40
U underlying u. functor 48, 183 u. mapping 40 u. set 40 undirected graph 102 unit 6, 9, 24, 28 universal algebraically u. category 53 u. category 53 extensionally u. category 53 upper bound 35 upper semicontinuous mapping 309
V vertex
102
W weight (of a space) 259 Y Yoneda functor 37
INDEX OF SYMBOLS GENERAL
91B 36 aoPp 30 [B,'LI] 36 ko'LI 125 morph'LI 8 obj'LI 8 1, 28 1, 31 SPECIAL OBJECTS
110 G(m,n) 111 K 321 n 63 N 321 R 41 z 120 =n 1 Z ( X ) 155 B,
Alg, (.I 173 ). Cat, Cat, 185 Catico 186 Comm 181 Comp 253 ComP,,, COmP;o,qlh 254 D, D, 168
DB 169 DL 169 EE 133 G 214 G' 180 G 239 G, 214,213 G" 214 G(i?t), G(i?), G(5,ii) 104 Graph 59 Graph, 104 Graph, 167 Graph, 171 Graph, 125 Graph,,, Graph,,, 127 .Graph 117 H,,, 234, 235 Id, 159 Id; 158 lncl 90
372 213 L, L, 165 L* 173 M , M,,. 213,218,233 Mcont 220 Mon 145 P, P,,, 213,218 Palg(d) 195 Par 250 Poset 1 18 Poset, 125 Prox 218 Prox,,,,,, Prox, 220 Prox, 233 B E , QQ 133 Rel(d) 41 Rng, Rng, 154 S(F), S(Fm ...>FnL S((%J) 42 S(F), S ( F ) 288 Y ( F ; (7; I) 94 ) Set 29 Smg 145 Soc(k) 117 SymGraph 102 SymGraph, 107
K,
TG, 220 Top 29 Top, 233 Top,, 241
TOP;, 1 t p TOP&," 208 TOP,':,',, 245 U n 217 Un, 233 Unqo 220 UndGraph 102 UndGraph, 107 V(&) 175 V(& J ) 174 W@', G), W,(F, G ) 288 WG, 230 ~
SPECIAL FUNCTORS
A', do, 2, J,,,As,A'u 213-217 0 228 9 221 P+ 31 P - , PA 33 Q 31 QA 33 V 231
