Category Theory, Homology Theory and Their Applications I

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zerich

86

m

Category Theory, Homology Theory and their Applications I

1969

Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24 - July 19,1968 Volume One l

Springer-Verlag Berlin. Heidelberg-New York

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. 9 by Springer-Verlag Berlin 9 Heidelberg 1969 Library of Congress Catalog Card Number 75-75 931 Printed in Germany. Title No. 3692

Preface

A conference was held at the Seattle Institute

during

the summer of 1968.

gether research workers

Research Center

The object of this conference

in the fields of c a t e g o r y

w h o applied the results of these theories algebra or topology. ized conference expectation

Thus this was not,

in categorical

of its organizers

theory w i t h i n m a t h e m a t i c s the interplays

of these

algebra being

w o u l d emerge

duration.

Three

space and optimal

invated courses

D. A. Buchsbaum

Regular

F. W. Lawvere

Hyperdoctrines

about 40 seminar

talks were

ceedings

become

I would committee,

[nva!uable

time)

of which however,

after the p u b l i c a t i o n

for a working

w o u l d be highlighted.

colloquium

of four week's

this

received

the pro-

(together w i t h their

the d e s i r a b i l i t y

is the first.

All those

of publishing

Following

the

the table of

appended a list of titles of papers to appear

naturally,

require amendment

if further m a n u s c r i p t s

of this volume.

to express,

to the Battelle

and to the a d m i n i s t r a t i v e in innumerable

participants.

w i t h a view to publishing

suggested

like to take this o p p o r t u n i t y

assistance,

and that

to be turned away owing to consider-

The number of m a n u s c r i p t s

this list may,

our deep gratitude

the conference

theory and h o m o l o g y

at least that number of people,

talks were given b y conference

of this volume we have,

available

the

Local Rings

in several volumes,

in subsequent volumes!

Seminars),

Cohomology

length and somewhat random arrival

contents

size

invited to submit m a n u s c r i p t s

of the conference.

proceedings

to be, a tightly special-

from the conference

in the conference~

within

of lectures were delivered:

Generalized

giving

the m o r e clearly

to-

theory and those

disciplines

(by comparison w i t h the Midwest

had u n f o r t u n a t e l y

J. F. Adams

In addition,

and was not intended

theories w i t h other parts of m a t h e m a t i c s

of invitations,

ations of available

to their own m a t h e m a t i c a l

Memorial

was to bring

theory and h o m o l o g y

that the roles of c a t e g o r y

There were about 80 p a r t i c i p a n t s fully deserving

of the Battelle

Memorial

and clerical

respects,

on b e h a l f of the organizing Institute

for their

support of

staff of the Institute

for their

in the running of the c o n f e r e n c e

and

IV

the preparation of manuscripts.

We were,

indeed,

conference under the auspices of Battelle.

fortunate to be invited to hold our

That the conference participants fully

shared the sense of gratitude of the organizing committee towards the Battelle Memorial Institute is set in evidence b y the following resolution, p a s s e d at the c l o s i n g session of the conference, w h i c h I was charged to make public through the m e d i u m of the conference proceedings.

"This conference at the Battelle Memorial Institute, w i t h its initial object Categories,

Homology and their applications, has n o w reached its termination.

our subjects are far from complete,

the

ference has been intensive, coherent,

Though

exchange of ideas about them at this con-

lucid, and never petty, while the local facilitie~

provided b y the Institute have been infinitely productive.

As p a r t i c i p a n t s

in this

conference, we w o u l d like to thank the Battelle Memorial Institute and its most helpful and efficient staff for the e x t a o r d i n a r i l y fine facilities and services w h i c h have been placed at our disposal.

We are equally indebted to the enthusiasm of the organizing

committee and the w i s d o m of the scientific staff at Battelle for their p r o v i s i o n of this ideal setting for the natural transformation of ideas into theorems."

Cornell University,

Ithaca, November,

1968

Peter Hilton

T a b l e of C o n t e n t s

M.

Barr

D. A.

Coalgebras

Buchsbaum

Lectures

in a c a t e g o r y of a l g e b r a s

on R e g u l a r L o c a l

R. F i t t l e r

Categories

J. F. K e n n i s o n

Coreflectlon maps which

J. L a m b e k

Deductive

S. M a c L a n e

Possible

programs

R. C. P a r ~

Absolute

coequalizers

S. Shatz

Galols

H.

Derived

Stauffer

M. T i e r n e y

and W. V o g e l

Rings

of m o d u l e s w i t h

systems

................ I

. . . . . . . . . . . . . . . . . . . . 13

initial o b j e c t s . . . . . . . . . . 55

resemble

and c a t e g o r i e s for c a t e g o r i s t s

universal

coverings46

(If) . . . . . . . . . . . . . . . 7G .................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

theory ..................................... functors without

Simplicial

derived

125

146

injectives ................ 159

functors

. . . . . . . . . . . . . . . . . . . . . . . 167

F. U l m e r

Acyclic models

M. Z i s m a n

Derived category and Poincar~ duality

and Kan e x t e n s i o n s . . . . . . . . . . . . . . . . . 151 . . . . . . . . . . . . . 205

Papers to appear in future volumes

Generalized c o h o m o l o g y

J. F. Adams J. B e c k

On H-spaces

and infinite l o o p - s p a c e s

H. B. Brinkmann

Relations for groups and for exact categories

S. U. Chase

Galois objects

P. Dedecker

T h r e e - d i m e n s i o n a l non-abelian c o h o m o l o g y for groups

R. R. Douglas,

P. J. Hilton

and F. Sigrist

H-spaces

D. B. Epstein and M. Kneser

Functors between categories of vector spaces

D. B. Epstein

Natural vector bundles

P. J. Freyd

N e w concepts in c a t e g o r y theory

J. W. Gray

Categorical fibrations and 2-categories

K. W. Gruenberg

C a t e g o r y of group extensions

R. Hoobler

N o n - a b e l i a n sheaf c o h o m o l o g y

M. A. Knus

Algebras graded b y a group

M. Karoubi

Foncteurs d~riv~s et K-th6orle

F. W. Lawvere

Hyperdoctrines

F. E. J. Linton

Relative functorial semantics

S. MacLane

Foundations for categories and sets

E. G. Manes

Minimal subalgebras for dynamical triples

P. May

Categories of spectra and infinite loop spaces

P. Olum

Homology of squares and factorization of diagrams

J.

Locally N o e t h e r i a n categories

E.

Roos

F. Ulmer

Kan extensions, cotriples and Andr4 c o h o m o l o g y

COALGEBRAS

IN A C A T E G O R Y OF A L G E B R A S

by Michael Barr

Let A be a c a t e g o r y and ~ = A. m

Then we may form B - A]I, the c a t e g o r y of ~

There

(FU,z,FnU)

where

More e x p l i c i t l y , ~- FU

the a d j o i n t n e s s m o r p h i s m s . u

be a triple on algebras.

is then an a d j o i n t p a i r A ( F ) B and we may ask w h e t h e r or U

n o t F is c o t r i p l e a b l e . G =

(T,n,u)

)B -~ = C

A

)I

and

we m a y form a c o t r i p l e n: 1

)UF = T

Then there is a n a t u r a l

and we are a s k i n g w h e t h e r

are

functor

~ is an e q u i v a l e n c e .

-

For g e n e r a l ~, would,

the p r o b l e m seems v e r y d i f f i c u l t .

of course, be p o s s i b l e

A]I (A ~) m

, 0 o..

m'

this process

to c o n t i n u e

forming categories

Myles T i e r n e y has g i v e n an e x a m p l e

n e e d n ' t e v e r terminate.

that it t e r m i n a t e at a n y finite step.

perhaps,

a c o n c e p t of a d i m e n s i o n of a category.

of sets

to show that

P r e s u m a b l y it is a l s o

possible

Here we give a c o m p l e t e

It

Thus we have,

a n s w e r w h e n A is the c a t e g o r y

(denoted by S), p o i n t e d sets

(denoted by

(I,S)), o r vec-

tor spaces over the field K for w h i c h while

(denoted by V ). I g n o r i n g those --K the functor T is c o n s t a n t , we s h o w that dims = 1

dim(l,S) --

S and

(I,S)

= dimV

= O.

S i n c e the main d i f f e r e n c e b e t w e e n

--K

is that the former c o n t a i n s some m o n o m o r p h i s m s

-2-

which does

do not split

(any #

) X), this concept of dimension

seem to be some kind of h o m o l o g i c a l

For the m e a n i n g and statements orems we refer to

[Be] and

it

According we must consider phisms

to the dual of the t r i p l e a b l e n e s s of w h e t h e r

it preserves

and creates

so UF-split.

phisms.

In any concrete

Lemma

theorem

isomor-

But U reflects

and an F-split p a i r is al-

the q u e s t i o n

these q u e s t i o n s of T r e f l e c t i n g

for T isomor-

c a t e g o r y we may call a triple consis-

tent if it has a model of c a r d i n a l i t y

a t least 2.

that A is one of the three categories m.i

The f o l l o w i n g

the-

of F - s p l i t pairs

are complete.)

Thus we need only c o n s i d e r

First we consider

F reflects

equalizers

all limits

itself.

assume

tripleableness

PRELIMINARIES

(since all of our categories isomorphisms

of the various

of the category.

[Li].

the question

and w h e t h e r

measure

lemma is from

We h e n c e f o r t h

mentioned

above.

[La].

I,

Let ~ =

(T,n,.)

proof.

Any o b j e c t A r A with

be a c o n s i s t e n t

triple

in A. ..m

Then

is I-i.

cogenerator.

Then

A is a ]~-algebra, can be c o m p l e t e d

to

for all A' c A, A' c

cardinality

> 2 is a

A x for some set X.

so is A X , and the given e m b e d d i n g A' c_#A X

If

-3-

A! ( ~A' ~ so that hA'

TA'

;A x /R

is i-i.

From now on we assume ~[ is consistent. seen that on S there are 2 i n c o n s i s t e n t

triples

It is e a s i l y (TX - 1 for

all X is one and TX = 1 for all X ~ ~, T~ = # is the other), while

on

(I,S)

and V

--

Proposition

there

is e x a c t l y

one.

--K

2.

T is faithful.

Proof.

If f ~ g. X---~Y,

then since

n is l-l,

Tf. nX = ,Y.f ~ ,Y.g = Tg. nX so Tf ~ Tg.

Theorem

3,

T

(and hence

Proof. morphisms which

F) reflects

For T, b e i n g

and m o n o m o r p h i s m s ,

is both

isomorphisms.

faithful,

and A has

reflects

the p r o p e r t y

SPLIT

EQUALIZERS

In order to apply the c o t r i p l e a b l e n e s s

qualizers. categories.

that a map

is an isomorphism.

2,

is n e c e s s a r y

both epi-

to have a c o m b i n a t o r i a l

description

It turns out to be the same

(almost)

theorem,

it

of split ein all three

-4-

Theorem

4. dO X ~ Y dr

Two maps

can b e p u t

into

a split

equalizer

di agr a m

t satisfying

td = E,

lowing

conditions

(i)

Equalizer

(ii)

d ~ is

(iii)

If d ~

Condition map

s d ~ = X,

are (d~

sd I

dt,

d~

dld

I) ~ r (or X = Y =

r

1 = d x'

%E

may

t

then

d ~X

=

be d e s c r i b e d

making

the

d 1 X. by

saying

following

d~

= d 1 x'

f

then

commute

d~

(kernel

The n e c e s s i t y only

is a

/Y

P is the p u l l b a c k

(i) r e f e r s

there

~-X

d

Proof.

that

diagram

E

dition

fol-

satisfied.

P

where

if the

i-I.

(iii)

u: P

~ s

pair)

of

(i)

to S anyway.)

a n d E the e q u a l i z e r .

and As

(ii) for

is

clear

(iii),

dl x = d l s d ~ x = d l s d l x ' = dldtx,

(con-

if

= d~

'

=

d o s d l x , = d O s d O x = dOx.

It is s e e n in any

concrete

that

these

category.

heavily

on the e x p l i c i t

cases.

It is o n l y

conditions

Their

sufficiency

categories

necessary

will

at hand.

to f i n d

s: Y

be n e c e s s a r y depends

very

We must

consider

) X

with

sd ~ = X

-5-

and

d~

I = d l s d I,

dt = s d 1 by

the

Case

for

then

nature

S.

t" Y

=

X

dOx

Write

s

+ Y

Y = Y O

)Yo ( a b u s i n g n o t a t i o n ) .

= dlx O

.

Then

can b e

chosen

so t h a t

of equalizers.

u

d~

;E

define

where

1

Choose

s : Y

X

x

by

Y

= ImX.

o

z X

O

Then

with

s IYo =

(d~ -I

and

O

i Y1 i s c o n s t a n t x o. sdlx

then

= x o and

d O x , = d l x ', a n d

sd~

Clearly dlsdlx

then

= X.

= dosdlx.

dlsdlx

= dlsd~

If

If d l x

d I x~

imd ~

= d~

', t h e n

' = dlx ' = d~

' =

dOsdOx , = dosdlx.

Case should

be

(i,~)o

taken

Case write

d ~ is

and

d ~ is

the

s Imd

~ ~

s o Y s Im d ~ Im dlc d,

d",

X o + Y2" d I have

~"

X1

we

Im dl-T'~Y = X o.

can

the

same

except

that

xO

In

the

equalizer

= d~

that

= d l x '.

we

terms

may

may

d I and

assume

Y = X~ + X

of

,

Then

choose

this

Y2

as

d~

8"

X1

y

1

so

= dOx

E = Xo.

+Y2

the

follows.

8

) Y2

are

= dlx

that

decomposition

representations

and

We

)Xo.

assume

of d ~ a n d

Moreover,

Hence

matrix

> Xo

exactly

d: E

inclusion.

d =

where

E be

where

i-i,

is

the b a s e p o i n t .

Let

X = Xo + X1

since

Y

to be

~K"

Now

This

arbitrary.

maps

and

-6-

Le t S

=

,

0

X1

and then the r e q u i r e d e q u a t i o n s

3t

t

ot

=

0

PRESERVATION

are clear.

OF T - S P L I T E Q U A L I Z E R S

In this we show that T - s p l i t e q u a l i z e r s ed in cases

(I,S)

and V K and e x a m i n e

in S.

The methods

that T

(and hence F) is co-VTT.

Proposition

are really quite

the sole vulgar,

are p r e s e r v -

failure of this

since

they prove

5

T (and hence

F) reflects

conditions

(ii)

and

(iii)

of t h e o r e m 4. Proof.

d~

If

X

)Y

and Td ~ is i-i, then

bY.d ~ = T d O . p X and ~X is I-i, so pY.d ~ and hence d ~ is I-i. For the other part we use the diaqram. p|

W m

w

IE---+ X

E'

d

!

)X//~X

l dl

|

!

~T

)TX

d~ ~

Td o

Td I

-7-

It is only n e c e s s a r y property ~Y.d~

of P' allows = Td~

Tdl.v'.~ d ~ .V

=

Theorem

to show that a map

d~

~: P

= Td~

=

= Td I.~Xov = ~y.dl.v

= dlv.

~P'

The u n i v e r s a l

as indicated.

Td~

= Td I 9 d'

and by lemma

,U

Now ' ,W

=

i, bY is i-i, so

d 1 .V.

6.

In cases

Proof.

(i,~)

There

and ~K' F is co-VTT.

is n o t h i n g

4.

There later w h i c h

left to prove.

S

are easy examples w h i c h w i l l become

show that T does not n e c e s s a r i l y

clear

reflect condition

(i) of t h e o r e m 4.

Theorem

7.

m

If T# ~ #, then there

(l,U) : A ~

)(i,~)

such that

A~

commutes,

is a f a c t o r i z a t i o n

(l.U))(I,S)

w h e r e V is the usual u n d e r l y i n g

there is a left a d j o i n t

(I,F) [

(I,U)

and

functor. (I,F)

Moreover,

is co-VTT.

-8-

Proof. to be

I--A-~T#

map in

(F~,A)

on objects

= UF~

=

U*)UA

(%,UA).

by letting

(F(X-{I}),A)

being

to a functor

Note

a functor,

which

~X)

)A

~X,

Define

(I,F)

Then

(I,U)A), a unique

and hence,

(I,U)A

is the unique

= U.

(I,F) has

(I,U)

is the reason

Define

= F(X-{I}).

I---~X'~-'-~X-{I}

first as an object

Theorem

V.(I,U)

Then to

)T~. ,~ F~

-- (I,S) (i

obvious.

that

where

(I,F) (i

left adjoint

is co-VTT.

a: 1

Clearly

= (X-{I},UA)

isomorphism

(I,F)

Pick a point

the last extension

by t h e o r e m

6,

cannot be e x t e n d e d

for the indirect

definition

to of

function.

8,

If T~ ~ %, then F is cotripleable. Proof. WX= since

1

If

Wz

S

1 + X, then Wl (I,F)

tripleable.

is co-VTT,

) (I,S) V.

is defined

But then

F =

it is only n e c e s s a r y

It is, in fact,

co-CTT,

by (I,F).W,

and

to show that W is

as may readily

be checked.

That is, E

is an e q u a l i z e r

Y

if and only if is.

WE

Theorem

9,

Suppose (in S).

e ~ , e 1 are

If ~----~TI

the two distinct

~T2 is an equalizer,

maps

of 1

)2

then F is co-trip-

-9-

leable.

Proof. E'

d')

Td~

TX

It is s u f f i c i e n t ,

"~TY

is

a split

E

d~ d ) X ~ Y

evidently,

equalizer,

to s h o w

then

that

the e q u a l i z e r

Td I is n o n - e m p t y .

d •

At

any

will

be

empty,

rate

condition

a map this

P

(iii)

is

)E so t h a t

means

we

can

find

reflected,

which

if E = %, so is P. a map

2

~Y

means

there

But

if P is

so t h a t

~O

x

i $i e o

i

i

)2

e This

commutes.

means

each

of the d i a g r a m s

d~

x

dI

}Y

x

1-----~e ~ 2 This

commute s 9

;Y

l'"-~ el

provides

us w i t h

a commutative

-------~ T 1 so t h a t

E' = ~.

Now we

This

that

diaqram

) T2

completes

suppose

2

the proof.

T# -

~, b u t

that

Te @

~T2

~ E---~ TI

Te 1

is an e q u a l i z e r .

Since

the u n d e r l y i n g

if

-

functor

creates

ture of a ~

equalizers,

algebra,

this structure.

map.

i0-

it is clear

so that there

Also we

Let T # be defined

let

n#~:

that E has

is a map #--~E

=

to check

(Tl,n # , u # )

Clearly

every

only new fines

free algebra

a functor

faithful model

S~ #

) S]~ which

and almost onto.

for T

but not

for ~#.

cotriples

with

the inclusion

_

_

difference, failure

again,

X =

Moreover,

on S.

already

since

checked

E is the

commute

is that

~ is a

object E, while ~ .

in

that the only

(ST)~.

of the d i a g r a m T

S I

to commute

not quite = ~.

exactly--F##

commute. Thus

= E, F% = ~ - - a l s o

In fact the comparison

(ST)~__ is as in the following.

Also

(on the nose!)

and almost onto,

# is a coalgebra

This de-

to be full,

From this it follows

faithful that

T #~ ~ # .

an algebra.

is easily

category

It is

~#X = ~X for X ~ ~,

a single p r e d e c e s s o r

S~#---~S ~.

being

if

S Y# has an initial

on each

is full,

,

The only difference

in ST , E is not initial but has the induced

the unique

**

is also a ~ - a l g e b r a ,

and it was

defining

.

a triple

"l~#-algebra

denote

n#X = nX and

that with becomes

TE u # ~ E

T#X =

by

E trivial

the struc-

makes

~ is such

that

The

-

Theorem

10.

Suppose Tr = r is non-empty. (r

=

Then

this

but the e q u a l i z e r

(S~) G may be d e s c r i b e d

(Y,Y) = i and

comparison S over,

11-

as S U {y] w h e r e

(X,y) = r for any other X E S.

~)(s1r)~ is just the i n c l u s i o n

induced

Note

E----@TI--'~T2

triple on S U

that if ~

{y} has

SY

functor.

for which

(in w h i c h

Tr = r

is a triple w i t h T#r ~ r

E ~ r and these

case it is easily

models--

that Tr = E).

T = ~),

this

constants.

constant natural set of natural

we may

between

for which Tr ~ r

owing to the lack of empty

On the other hand,

can't happen.

This

seen--

More-

as its algebras.

define ~ by Tr = r and get a i-I c o r r e s p o n d e n c e triples

The

E is called

if E = r (as w h e n

the set of pseudo-

is justified by the fact that it is the set of transformations

transformations

of U

of U r

of 0 ~ rears its ugly head again).

)U

(whereas

Tr is the

) U -- the i n d e t e r m i n a n c y

Then we may restate

our

results.

Theorem

Ii.

If the set of p s e u d o - c o n s t a n t s set of constants, as above.

then

(ST)~__ = S.

of]~is

Otherwise

equal

it is S

to the

U{y}

-

12-

REFERENCES [Be]

Jo Beck, "The Tripleableness

[La]

Fo W~ Lawvere, Theories",

[Li ]

Theorems,"

(to appear}~

"Functorial Semantics of Algebraic thesis, Columbia University,

F~ E~ Jo Linton,

(1963) ~

"Applied Functorial Semantics,

(to appear in Zur~oh Leotu~e Notes).

II, N

- 1S -

LECTURES ON REGULAR LOCAL RINGS by D.A. Buchsbaum w

Koszul complexes The aim of these lectures is to show how homological methods may be applied to local ring theory.

That homology is tied up naturally with algebra, can be seen easily by looking at the following example:

Let

R

be an element of

be a commutative ring (all rings are assumed to have an identity element), and let R.

Then we may consider the rather trivial complex

>0

where

R

x

> R

R

is by

x,

>0

>0

means multiplication by the element

Ho(K(x)) = R/(x),

and

while

Hl(K(x)) = O:x = (rs

H1

X

>R

x.

>R

x

H0

is a zero divisor.

K(x),

we see that

measures how divisible For an arbitrary R-module

M,

K(x~RM:

X

>0

H 0 ( K ( x ~ M ) = M/xM

and

>0

>M

>0

>M

H I ( K ( x ~ M ) = O:x= [maMlxm=O].

homology groups to tell us how divisible the module is a zero divisor for

>0

If we call this complex

Thus in a sense

tells us whether or not

we may also consider the complex

and observe that

x

M

is by the element

Again we may use these x,

and whether or not

x

M.

Now let us suppose that we have two elements induced by multiplication by

x,y

in

R.

Then we have the map

K(x)

~

> K(x)

y:

>0

>0

>R

>0

>0

>R

X

>R

>0

>R

>0

and it is the most natural thing in the world for homologists to consider the mapping cylinder of this map of complexes:

> 0

where

g(rl, r2) = rlx + r2y

and

> 0

f(r) = (-ry, rx).

H0(K(x,y)) = R/(x,y); H2(K(x,y)) = 0:(x,y) =

> R

f

> F@R

6

> R

Calling this complex K(x,y),

[rER/rx=0=ry].

> 0

we have

Hl(K(x,y)) = [(rl, r2)/rlx+r2y=0]/[ (-ry, rx)].

-

If we suppose that H2(K(x,y)) = 0

and thus

is an integral domain, and that neither

and that

the map sending (x)

R

14

[(rl, r2)/rlx+r2y = 0]

(rl,r2)

> r 2.

x

and

nor

If y

R

y

is zero, we see that

is mapped isomorphically onto

Under this map, the submodule

Hl(K(x,y)) = (x:y)/(x).

would be zero if and only if

x

x:y = [r~R/ryE(x)}

((-ry, rx)}

is sent onto the ideal

were a unique factorization domain, then

were relatively prime.

by

Hl(K(x,y))

Therefore we again see that a certain

amount of homological information is intimately connected with arithmetical questions. We could proceed step by step in this way, building up complexes efficient to consider the following general construction. R-module p df:AA

A

to >

R.

p-1 A A

Then we may define a complex by

df(al~...Aap)= E(-1)

It is well-known, and easy to check, that

df(c~6) = df(=)^~+(-1)Po~df(6).

i+l

K(f)

Let

direct sum of

R

R,

Also, if

then we can define a morphism

with itself s times, and K(f)

f

GEAA

and

^ a. ql

means we omit

8eAA,

a.. l

then

by

Namely, if

f:R s

> R

Xl,...,x s where

Rs

is a sequence denotes the

is defined by f(l, 0,...,0) = Xl,...,f(O,O,...,l ) = x s. K(Xl,...,Xs).

Returning to the general case of a morphism

f:A

> R,

we state two fundamental properties

K(f):

Proposition i.i.

If

a0

Proposition 1.2.

If

x

complexes

where

Before indicating some general properties of this complex, we should

In this case, we denote the complex

of the complex

be a morphism from the P (K(f))p = A A and defining

^ f(ai)al^...^a. ^...^a l pp

df2 = O.

but it is more

> R

by setting

see how it subsumes the sort of thing we have been discussing. of elements of our ring

f:A

K(x,y,z,...),

x:K(f)

is an element of

A,

then

is any element of

R,

and

f(ao)Hp(K(f)) = 0

9~ R

f:A

for all

p ~ O.

a morphism, we have the map of

> K(f):

3 > AA

> AA ~ >

The mapping cylinder of this map of complexes is

K(~)

2 > AA

> A

AA

> A

where

f

> R

> R

> R

~:~R

is defined by

~((a,r)) = f(a)+9x. The proof of Proposition 1 is obtained by writing a homotopy s(al~...^ap) = a0^al^...^ap,

and showing that multiplication by

P s:AA f(a0)

p+l > A A

defined by

is chain homotopic to the

-

zero morphism.

15

-

Proposition 2 is proved by observing that for any module

P P p-i A, A ( ~ R ) ~ A~9 A A.

From Proposition 2, we obtain

Corollary 1.3.

If

f:A

> R

is a morphism,

x

an element of

R,

and

> R

~:~BR

defined

as in Proposition 2, we have the exact sequence:

+X

--> Hp_l(K(f))

> Hp(K(f))

In particular, if

Xl,...,Xs+ 1

> Ho(K(~)) --> 0 .

> Hp_l(K(f)) --> ... --> H0(K(f))

is a sequence of elements of

R,

we have the exact sequence:

(E): ... -> Hp(K(Xl,...,Xs)) --> Hp(K(Xl,...,Xs+l)) -> Hp_l(K(Xl,...,Xs)) ~Xs+~Hp_l(K(Xl,-..,Xs)) --> --"

Using the fact that arbitrary R-module

K(Xl,...,Xs) M,

and

K(Xl,...,Xs+l)

are free chain complexes, we also get, for an

the exact sequence:

(ECM)):----->Hp(M(Xl,..,Xs)) -->Hp(MCXl,.-.,Xs+l)) --> Hp_l(MCXl,...,x s) ~Xs+~Hp_iCMCxl,...,x s) --> ...

where

M(Xl,...,Xs)

is the complex

K(Xl,...,Xs~ ~ .

From the remarks made about the meaning of

we are led to make the following

Hl(K(x,y)),

, definition.

Definition:

Let

M

be an R-module, and

said to be an M-sequence if is not a zero divisor for

Proposition 1.4.

If

a)

Xl,...,x s

M/(Xl,...,Xs)M ~ 0

M/(Xl,...,Xi_l)M.

Xl,...,x s

a sequence of elements of

For

and

s,

for each

i=l, this means that

is s/aM-sequence, then

The proof proceeds by induction on

b)

the case

Then

i, 1 < i < s, xi

Xl,...,x s

the element

being self-evident.

for all

Remark:

It is not true in general that

to show that

xi

p > O.

To go from

one merely uses the exact sequence E(M) to show that Hp(M(Xl,...,Xs) ) = 0 for p > l, x fact that Ho(M(Xl,...,Xs_l) ) s > Ho(M(Xl,...,Xs_l)) is a monomorphism (since Ho(M(Xl,...,Xs_l) ) = M/(Xl,...,Xs_l)M )

is

is not a zero divisor for M.

Hp(M(Xl,...,Xs) ) = 0 s=l

R.

s

to

s+l,

and the

Hl(M(Xl,...,Xs)) = O.

Hp(M(Xl,...,Xs) ) = 0

for all

p > 0

implies that

Xl,...,x s

- 16-

is a M-sequence.

For example, let

for the residue classes of Hp(K(~,~)) = 0 R,

for

x

p > O.

and

in

R,

Xl,...,x s

M(x 1 .... ,Xs) ~ M(x (1),...,x (s)) ~

z

with

where

is a zero divisor in

R,

~

k

we see easily that

Therefore, we obviously have that

and any sequence of elements

clear that

R = k[x,y,x]/y(x-l,z)#

in

R,

a field. x,z

is an R-sequence,

Hp(K(~,~)) = 0

and any R-module

M,

z,x

~

and

so that

(since for any ring

we have

is a permutation of the set

so that

If we write

[1,...,s}).

However,

it is

fails to be an R-sequence.

As a result of the above remark, it is natural to ask when we can prove the converse of Proposition 4. and

To this end, let us suppose that xi, 9 ..,x s

R

Let

R

R.

Then

M

if

x.

is the intersection of

and ~

an ideal in the radical of

M = O.

is in the radical of

R

is noetherian and

M

finitely generated, we conclude that

)

t,

if

in

R.

Let

R

is an epimorphism if and only if

Using these facts, we can prove

be a noetherian ring,

a sequence of elements in the radical of i)

s

we can then conclude that the map

1 >HD(M(Xl,...,~i,...,Xs))

PK (M(Xl,...,~i, .... Xs) ) = O.

Proposition 1.6.

R,

Xl,. 9

X.

HD(M(xI,9

Hp(M(Xl, .... Xs) ) = 0

R.

M

a finitely generated R-module, and

Xl,...,x s

Then

for some

p,

we have

Hp+k(M(Xl,...,xt))

= 0

for all

k ~ 0

1 < t < s:

ii)

if

Hl(M(Xl,...,Xs) ) = O,

hypotheses, we have (or for

R

is a finitely generated R-module for any sequence of elements i

and all

a finitely generated R-module,

Recall that the radical of

a finitely generated R-module,

If we now use our assumptions that Hp(M(Xl,9

R.

M

We now have Nakayama's lemma at our disposal, which says:

be a ring,

such that O?'M = M.

Moreover,

is a noetherian ring,

elements in the radical of

all the maximal ideals of

Lemma 1.5.

R

Xl,...,x s

then

Xl,...,x s

is an M-sequence.

is an M-sequence if and only if

Thus, under these

Hp(M(Xl,9

= O

for all

p > O

E(M),

together

p = 1).

The proofs of these statements are obtained by repeated use of the exact sequence with the observations immediately preceding the Proposition9

Corollar~ i~ only if

The hypotheses being as in Proposition 6, we have

x (1) ..... x (s)

Corollar~ 1.8.

If

R

is an M-sequence for every permutation

is a noetherian ring,

M

Xl,...,x s ~

of

is an M-sequence if and

[13...,s ).

a finitely generated R-modnle,

~

a prime ideal of

R

-

such that

M

= M~RR~# 0

(i.e. y

which is an M-sequence, then

Remarks:

In general, if

we denote by R,

and

RS

noetherian.

R

is noetherian and ~

ring with one maximal ideal. is flat over

w

R

R

S

a sequence of elements in

with respect to Rs

by

R~,

R~,

is an

M -sequence.

a multiplicatively stable subset of S.

R~.

However, i f ~

If

a prime ideal, then

In the case of

R~

R

S

of

R,

is a prime ideal of

is noetherian, then

is a local ring

i.e.

the unique maximal ideal is ~ R ~

for any multiplicatively stable subset

.

RS

is

a noetherian Finally,

RS

R.

Local rin~s We shall now assume that

will be denoted by of

E

is

ideal of

that

~

we have

Let

E

el,...,e t

Proof.

k;

R

is a local ring, with maximal ideal

~.

The residue field

all R-modules will be assumed finitely generated.

and all proper ideals of

R#

Lemma 2.1.

E~E

= 0

R

are contained in

if and only if

E = 0

~.

If

E

generate

E~E

R/~

In this case, the radical is an R-module and ~

an

because of Nakayama's Lemma.

be an R-module (always finitely generated), and as a k-vector space.

Then

el,...,e t

el,...,e t

elements of

generate

E

E

such

as an R-module.

Although the proof of this is well-known# we will reproduce it here since it serves as a

prototype for so many others. E

Xl,...,x s

is any commutative ring~ and

, we denote the ring of quotients

If

and

considered as elements in

the ring of quotients of

S=R-~

-

is in Supp (M)),

Xl,...,Xs,

R

17

generated by

el,...,e t.

is the cokernel of F/~F

F

> E/~

> E.

L = 0

Tensoring over > O.

so that

F

i.e.

E

Lemma 2.3.

If

E

such that

Ker g

~E

with

> E

of rank

and map it onto the submodule of

F

> E

is exact and

where

L/~ E.

contains a minimal generating set of

E;

E/~E

over

k.

F

el,...,e t

so

generate

is an R-module, there exists a free module ~F.

> 0

is now an epimorphism,

L

= O.

any

This number is equal to

Moreover, any subset of

may be extended to a minimal generating set of

is contained in

> L

we have the exact sequence

> E/~E

> 0

E

k,

t

have the same number of elements.

the dimension of the vector-space

independent modulo

R

However, F/~F

Any generating set of an R-module

two minimal generating sets of [E~:k]

F

We then have the exact sequence

> L~L

This tells us that

Corollary 2.2.

We take a free module

E

linearly

E.

and an epimorphism

g:F

>E

-

Lemma 2.4.

If

E

is an R-module such that

Lemma 2.5.

For every R-module

Proposition 2.6. If

R

E,

18

-

Tor~(k,E) = 0,

then

E

is free.

hdRE ~ h % k .

is a local ring,

gl.dim R = h ~ k .

The proofs of all the above statements are completely straightforward.

For instance, 2.3 depends

on nothing deeper than the fact that an epimorphism of a vector space onto another of the same dimension is an isomorphism. generating set of

One then chooses

F

to be free on the number of generators in a minimal

E.

The main result we are heading for now is that dimension of

R

~:k]

<_ gl.dim R,

i.e.

that the global

is never less than the minimal number of generators of the maximal ideal of

R.

The idea of the proof is the following We take a minimal generating set K(Xl,...,Xn).

di(Xi) c Z~Xi_1

this resolution with K(Xl,...,Xn) we see that

~.~.

k

R = h~k

and

Thus

f.

morphism

~:X k

K(Xl,...,Xn)

X1

> Xk_ 1

Xp+ 1

k:

>0

> k

is a subcomplex of this resolution. TorR(k,k)= X p ~ .

K(Xl, ...,Xn)

Tensoring

Since

is non-zero in dimension

n,

and we have the desired result using the facts that

> XI

is of the form:

n > AR n

8 n> ...

2 8^ >AR n ~

of our resolution to be

Assume now that

> ...

and form the complex

n = ~/~:k].

We may therefore choose morphism

> R

K(Xl,...,Xn)

hdRk >_ n

Z~,

%-

xI

makes the boundary maps zero, so that

TornR(k,k) ~ O.

a free module

% .> > x2 --

and such that

0

Xp

% x3

is a subcomplex of the resolution, and

The complex

iv)

of the maximal ideal

We then show that we can find a free resolution of

>

such that

Xl,..., x n

Xl,...,Xp

restricted to > R

such that

> k i) - iv)

Rn

Rn

with

f-~ R

dl:X 1

have been found such that k AR n > 0

is

> R

defined to be the k i) Xk = X~ ~ ARn: ii) the

8k; i i i ) K e r ~ C ~ k

is exact.

are true with

If k

p ( n,

and iv)

we will show that we can find

replaced by

p + 1.

Our conditions i)

- 19 -

P+IRn A

and ii) tell us that the composition is contained in {Sp+l(s

% = Ker % .

^...^ a. )} ~p+l

as showing that if

~Zp C ~ X p

Now

8p+l(~)

Suppose, then, that

~Pn

then each

5p+l: A R n

> PR n

R n.

is in

8p+l( )

Rn

is to say that

p + l . p+l p+l A R~A R n = R / ~ @ A En,

u

~p+l

is zero. Thus p+l are a basis of A R n,

then

But this is the same

ip+l is in ~. is such that Since

8p+l( ) is

%C~,

in

~Z,

we have

n

we must have

8--p+l: A R * ~ A

R~

8p+l( ) a ~2~-ARn.

>

Rn

Rn

is a monomorphism, we are done.

i(~) = 0 and thus that ~

p p and ~I6Rn/~2~R n = ~

5p+l(l~AiRn)

> Xp_ I

That is, we want to show that if

is in ~ % .

is in AR ,

induces a map

If we can show that

~Zp.

p+l in A R n

8p+l( ) P

= Z~-AR- ~ ~ZfCxp. Since

8p+l( A R n) c

P {Eil~...^ s i } p+l

are linearly independent modulo

+I) s ~Zp,

d P

> X

Rn

-- ~ 7i I ... IP + I " ail~...^a.lp+I

p+l is in ~ A R n.

then

P

We will show that if

^...^ s

Z 7i I .. . ip+ . 1 8p+i(s

_•

@ PR n

= 0

or ~ E ~ A

and the map

5p+ 1

since

For to say that

P+lRn"

But

sends the basis element

^

9 Clearly this map is a monomorphism, to r.(-l)J+ixi | s ^...^s ^...^ s lI lj ip+ 1 j

i| s Ip+l and we are done.

Since we have shown that

{ 8p+l(ail ~ ...^ s

)]

are linearly independent modulo

that these elements may be chosen as part of a minimal generating set for generating set

[Sp+l(ail~... ^ s

)} U [Zl,...,Zq}.

Letting

X~l

%,

~%,

we know

i.e. we have a minimal

be the free module on

q

~p+l generators, we have Xp+ 1

> Zp

is exact.

p+l A R n ~ Xp+ I > ~

Xp+ I

mapping onto

is clearly in ~/~(p+l and

Zp.

The kernel of the map

Xp+ 1

> Xp

> ...

> X1 ~>R~>k'

Thus we have completed the inductive step, and we have proved

Theorem 2.6.

If

R

is a local ring, then ~ : k ]

<_ gl.dim R.

Our next objective is to define the dimension (or Krull dimension) of a local ring show that

dim R < ~/~2 :kS.

R,

and

To do this, we shall use the method of Hilbert-Samuel polynomials which

>0

-

20-

we introduce in a slightly more general setting now.

w

Hilbert-Samuel We let

Z+

Pol~nomials.

denote the set of positive

integers,

and let

G

will be the group of integers or the Grothendieck

group of some category of modules.

Definitions.

f

elements

have

Let

f:Z +

ao,...,a d

f(n) =

> G

in

d n Z (i)ai i=O

G

be a function,

such that for all

is called a polynomial

n s Z+,

n

Generally,

be an abelian group.

function if there are

sufficiently large (i.e.

n >> 0), we

n! where

(~)

+

denotes the binomial coefficient

If

f:Z

>G

i](n-i)]

is any function, we define

Zif:Z+

difference

(functio~ of

f.

Lemma 3.1.

A function

f:Z +

> G

For any integer

> G

by

Zif(n) = f(n+l)-f(n). s > i,

we define

is a polynomial

Zif

is ca/led the first

~sf = ~(~s-lf).

function if and only if

~f

is a polynomial

function.

Proof. all

If

f

is a polynomial

n >> O.

n+l

z[( i

Consequently,

n >> 0

we have

is a polynomial

Thus

if

L~f(n)=

n >> O, Ag(n) = 0

f(n) =

for all

d n ~ (i)ai for some i=O Zkf(n) = f(n+l)-f(n) =

f(n) =

ao,...,a d

G,

and

n

z(n+l)a -i

in

- i

- z - () i-ai

d

n

Conversely,

have

fUnction, we have

so that

d+l n r (i)ai i=O

d • ( )b i i=O g(n)

for

for

n >> O,

is a constant

a O.

function.

define

g(n)=

f(n)-

Thus, letting

d n ~. (i+l)bi. i=O

a i = bi_ I

for

Then for

i > O,

we

n >> O.

We have proved Lemma 3.1 in detail, general than the usual one.

since the definition of polynomial function is slightly more

However, with 3.1 established,

we shall only state the facts we need,

and omit the proofs.

Lemma 3.2.

If

d d' f(n) = iZ= (i)ai = iE= (i)ai n n . 0 0

Thus, the degree,

d,

of

f

is a polynomial

is a well defined integer,

function,

then

d = d'

and the coefficients

ai

and

of

a. = a~. : l f

are

uniquely determined. Now consider a commutative noetherian ring

R,

and let ~

be a full abelian subcategory of the

- 21

category of R-modules.

For each integer

-

s = 0,1,2,...,

let

be the category of finitely s

generated graded

R[Xl,...,Xs]-modules

E =

E E v_>O

i)

if

s = O,

E

such that ~

is in ~

for all

u;

is in ~

for all

u

v

s > O,

if

E~

and t h e g r a d e d

X

modules Finally, let

fo

.

>~E

)

and Coker (rE

K(o~),

of

i.e.

fo

s >~E

)

are in ~ s

to an abelian group

is additive

G

with respect

i"

which factors

to exact sequences

Then

Theorem 3.3. fE

X

s

be a function from the objects of ~

through the Grothendieck group, in ~

Ker (?E

R [ X 1 , . . . , X s - 1]

Let

E = ZEv

be an object in ~ s

and define

is a polynomial function of degree less than or equal to An example of such a set-up occurs when

R-modules of finite length.

R

In that case, an

finitely generate~ and if each

E

fE:Z+

> G

by

fE(~) = fo(Ev).

Then

s-l.

is a local ring, and ~

R[Xl,...,Xs]-module

is the full subcategory of

E = E~

is an R-module of finite length.

is i n ~ s

if it is

We may then choose

G

to be

v

the group, Z, of integers and

fo(E) = length of

E.

This is of course the most usual example.

see that it is not the only example, we may c h o o s e ~ R-modules, and

G

Corollar[ 3.4.

If

to be

R

K(~)

is a local ring,

containing some power ~n and the function

itself with

E

fo

~ (E):Z+

> Z

to be the category of all finitely generated

the usual map.

a finitely generated R-module and ~

of the maximal ideal defined by

To

then

E/[E

an ideal of

R

is an R-module of finite length,

x~(E;v)= length of

E~VE

is a polynomial function

whose degree is less than or equal to To see why this is a corollary of 3.3 we observe that we have exact sequences

and therefore R[Xl,...,XsS ~(E)

Remark:

~ ( E ; ~ ) - - l e n g t h of ~ E $ ~ + l E . where

s-- ~ / ~

The graded module

:kS, and each ~ V E ~ V + l E

andS2

is a module over

is an R-module of finite length.

is a polynomial function of degree <_ s-l, and hence

If ~ i

~/E~+IE

x~E)

Thus

is what we claimed it to be.

are two ideals containing some powers of the maximal ideal ~,

i.e.

if

-

nI ~i

n2

D ~

and ~ 2 D

~

,

X n(E )

and

x~(E)

~(E).

~(R).

If

have the same degree.

R

Thus, if

~

n

<4

< ~'

n,

we must have

and hence equality.

is a local ring, the dimension of

The dimension of an R-module

In particular, they

equal.

This is seen easily by observing that for any integer

deg(~n(E)) ~ deg(X~E)) ~ deg(%(E))

Definition.

-

)(~j.(E) and )<,~,2(E) are

then the degrees of

are all equal to the degree of

29

E

R

is the degree of the polynomial function

is the degree of the polynomial function

a noetherian ring and ~

is a prime ideal of

R,

Proposition 3.5.

be a local ring, and let

then the height of T

X~(E).

If

is the dimension of

R

is

R~.

I

Let

R

~

of

s

be the smallest number of elements required to l

generate an ideal In particular,

dim R ~

R

which contains some power of the maximal ideal

~.

Then

dim R ~ s.

~:k].

Although we will not be using this fact immediately, it is important to note that for a local ring

R,

the following integers are equal: a)

the dimension of

R;

b)

the smallest number of elements required to generate an ideal containing a power of the maximal ideal;

c)

the length of the longest chain of prime ideals in ~0

3~1D'''D'~h

is defined to be the integer

R

where the length of the chain

h.

The proof that these three integers are equal is not completely trivial. to Zariski-Samuel,

The reader is referred

Commutative Algebra, for those details about noetherian rings which are mentioned

here but not proved. For various reasons, it is useful to be able to compare the dimension of a module local ring

R

with that of

E/(x)E

if

x

extremely helpful to know what happens to

is an element of dim E/(x)E

when

R. x

E

over a

From our point of view, it is is not a zero divisor for

E.

To

handle this problem, we quote the Artin-Rees Theorem:

Theorem 3.6. M,

and

I

Let

R

be a noetherian ring,

an ideal of

R.

M

a finitely generated R-module,

Then there exists an integer

h > 0

M'

a submodule of

such that for all

n ~ h

we have

(InM) N M' = I n - h ( I ~ N M').

Corollary 3.7.

If

R

is a local ring and

x

is an element of the maximal ideal

~

which is a non-

-

zero divisor for an R-module

E,

23

-

there is an integer

h > 0

such that for all

u > h,

MVE:x c ~ - ~

where ~ E : x = [e s E/xe s The proof depends upon the Artin-Rees Theorem and the easy observation that x(~E;x) = ~ E

D (x)E. For then we have

from which we conclude:

Proposition 3.8. Let

If we let = E/(~V,x)E,

R

be a local ring, If

x

ThUS length

an R-module, and

and

and ~ = ~ ( x ) ,

X~v-h

x

an element of ~.

E, we have

Then

dim E/(x)E ~ dim E -1

and

> (~,x)E~E

EI(~,x)E = ~(E/~E)

for

x a~,

Since

length

so that

d i m E ~ dim E -i.

h

and all

length

- length

> E/(~,x)E

E~E.

But

Since

> 0

(~V,x)E/~VE ~ (x)E/~VE A (x)E ~ xE/xC~E:x)

with equality holding if so that length

(E/~E:x)

v ~ h. (E~-hE)

x

x

E~E:x

E / ~ E - length (Z~,x)E/~E ~ length

If, however,

= length

suitable fixed (EKE)

E~E:x,

we know that ~ - I E c ~ E : x

E/(~V,x)E = length

(~,x)E~E)

> E/~VE

- ~(~,x)E~V~E).

( ~ , x ) E / ~ E ~ length

E.

we are interested in the length of

we have an exact sequence:

we have length

length

E

is not a zero divisor for

E = E/(x)E,

0

Thus

O (x)E) = ~ ( - h ( ~ : x ~

dim E/(x)E = dim E -1.

Proof. E~E

N (x)E = ~ - h ( ~

~E:x c~-~.

dim E/(x)E ~ d i m E -1. hence

x(~E:x) = ~ E

is not a zero divisor ~ length E / ~ - ~ .

E/~VE - length

is not a zero divisor for

E,

we have

and by 3.7, length (E/~UE:x) ~ length

Thus length

E/(~U,x)E =

length

which immediately implies that

E~V+IE

(E~V-hE)

for

(E/Z~E) - length (~,x)E/~VE dim E < dim E -i. m

In the above proof we said we were interested in the length of E~E

to emphasize the fact that

= R/(x).

dim E

as an R-module is the same as

This is mainly to ensure that when

dimension of the local ring

R.

E~E

Of course,

E = R, dim R

E~E~

instead of the length of dim E

as an R-module where

as an R-module is seen to be the

E~E

As an immediate consequence of 3.8 we have

Corollar~r 3-9. Let

R

be a local ring,

E

an R-module, and

Xl,...,x s

dim E/(Xl,...,Xs)E = dim E-s.

an E-sequence.

Then

-

In particular,

w

-

s <_ dim E.

Codimension and finitistic global dimension We will now let

such that E/~E | if

24

E/Z~'E ~ O. R~

~ O,

XlJ...,x s

6KR~

R

be a noetherian ring, Since

a finitely generated R-module,

there is some prime ideal

and this prime ideal ~

obviously contains

is an E-sequence contained in ~ ,

and thus, by 3.9,

in ~

E/~'E # O,

E

s ~ dim E @

~

.

is bounded, and any E-sequence in ~

then

~in

R

anG

~

an ideal in

such that

grC. From Corollary 1.8, we see that

Xl,...,x s

is also an

E @R R ~ -

sequence in

Hence the number of elements in an E-sequence contained may be extended to a maximal E-sequence in g~'.

prove that any two maximal E-sequences in ~

We shall

have the same number of elements but first we must

review the notion of associated prime ideals.

Definition.

If

monomorphism

E

is an R-module, a prime ideal ~

R~

associated to

> E.

Proof.

E,

denoted by

Ass(E),

E

if there is some

is the set of all primes

E.

Using the fact that

Lemma 4.1.

The associator of

is said to be associated to

If

E

R

is noetherian, we have

is an R-module, then

Clearly if

E = 0

we have

Ass(E) = ~

Ass(E) = ~.

if and only if

To show that

the set of all ideals (~' such that there is a monomorphism empty, and thus there is a maximal such ideal &~O"

E # 0 R/6~"

E = O.

implies > E.

Ass(E) ~ ~,

one considera

This set of ideals is not

One then proves easily that such an ideal

O~"0

is necessarily prime. Another useful lemma is the following:

Lemma 4.2.

Let

0

> E'

> E

> E"

be an exact sequence of R-modules.

Then

Ass(E) : Ass(E') U Ass(E"). The proof of this lemma is trivial, and one can now prove

Proposition 4.3. is a finite set.

If

R

is a noetherian ring and

E

is a finitely generated R-module, then

Ass(E)

-

Proof.

25

One first proves this for cyclic modules,

ideal of

R.

i . e . modules of the form

R/g~

where

is an

One then assumes that the theorem is true for modules generated by less than

elements, and assuming that 0

-

> E'

> E

E

is generated by

> E"

0

where

el,...,en+ 1 E'

n + i

one considers the exact sequence

is generated by

el,...,e n.

Applying 4.2 one

gets the desired result.

Definition.

If

hei6ht of R,

E

R

is a noetherian ring and

to be min height ~ w h e r e

~

E

a finitely generated R-module, we define the

runs through all primes in

we generally abuse notation and call the height of

R/~

Ass(E).

If ~

is an ideal of

the height of the i d e a l S .

An important result relating the height of an ideal with the number of generators of the ideal, is the Krull Principal Ideal Theorem. ideal generated by elements or equal to Ass(R/~),

r.

This states that if

Xl,...Xr,

In particular,

R

is a noetherian ring, and ~ i s

then any minimal prime of

height ~

~ r.

Ass(R/~)

has height less than

This follows from 3.5 since i f ~

then ~R~pdg contains some power of ~ R ~

and thus height

7=

an

is minimal in

dim R~o ~ r.

# Remarks:

l)

For any R-module

E,

the set of zero divisors of

E is the union of all primes in

Ass(E). (2)

Thus, if

E

is a finitely generated R-module (with R noetherian) and if ~

that every element of ~ that

~e

c ~ R/~ R

= 0.

For if ~ i s

for some ~ R/~

7

a Ass(E)

> R/~

If the ideal ~

E,

as above, then ~ (since

and a monomorphism

> R/~ (3)

is a zero divisor of

> E

R/y

then there is a non-zero element

c U7

Ass(E)

is an ideal such

where ~

is finite).

> E.

runs through

e

Ass(E)

of

E

such

and hence

We therefore have the eplmorphism

The image of 1 under the composite map

is the desired element

e E E.

in Remark 2 above is a maximal ideal then ~

a Ass(E).

We are now ready to prove the main theorem of this section.

Theorem 4.4.

Let

R

elements

Xl,...,x n

s+q = n,

where

q

be a noetherian ring, such that

E/g~E ~ O.

E Let

an R-module, and ~ yl,...,y s

an ideal of

R

generated by

be a maximal E-sequence in 0/'. Then

is the dimension of the highest non-vanishing homology of the complex

Furthermore,

HqCECxl'''"Xn)) ~ (Yl'''"Ys)E:~/Cyl''''~s )E

E(Xl,...,Xn)

-

Proof.

The proof proceeds by induction on

a zero divisor for ~e

= O.

Since

E

s.

26

When

-

s=O,

it means that every element of ~

so that by Remark 2 above, there is a non-zero element

Hn(E(Xl,...,Xn) ) = 0 : ~

O,

we see that

q = n

e

in

E

is

such that

and

Hn(E(Xl'''"Xn)) = ( YI'''" Ys )E:LT//( YI'''" Ys )E = 0: ~ . When

s > O,

~e consider the exact sequence

0

> E

Yl

> E

> E

Yl H~+l(g(Xl,...,Xn)) where

~

> H~(E(Xl,...,Xnl)

is zero since

Yl

and using induction, we have s+q = n.

~( YI'''" Ys )E:

. Thus

Noting that

(s-l) + g = n.

Finally, since

and multiplication by

is in ~

H~_l(E(Xl,...,Xn) ) ~ H~(E(Xl,...,Xn) ).

have

>

is the dimension of the highest non-vanishing homology group of

H~_l(E(Xl,...,Xn)

Yl

E(Xl,...,Xn). on

. . .

Thus,

H~(E(Xl,...,Xn) ) and on

H~(E(Xl,...,Xn)) = 0

Y2'''" Ys

and get the

>

> Hq(E(Xl,...,Xn))

H~+l(E(Xl,...,Xn) ) = O, H~(E(Xl,...,Xn) ) ~ 0,

> 0

while

is a maximal E-sequence in 6~ ,

Since, however, we have just shown that

q = T-l,

we

H~(E(Xl,...,Xn) ) ~ ( Y,'''" Ys )~: ~ / ( Y,'''" Ys )~

/( YI'''" Ys )E'

and since

Hq(E(Xl,...,Xn) ) =

H~_l(E(Xl,...,Xn)) ~ H~(E(Xl,...,xn

we have

Hq(E(Xl'''"Xn))~ ( YI'''" Ys )E: ~ / (

Corollary 4.5.

If

R, E, and ~

YI'''" Ys )E"

are as in 5.4, then any two maximal E-sequences in ~

have the

same length.

Definition. depth (~:E).

The length of a maximal E-sequence in ~ If

R

is called the L"-depth of

is a local ring, then codim E = depth (~:E)

where

~

E,

denoted by

is the maximal ideal of

R.

An important consequence of 4.4 is

Theorem 4.6. Let

R

be a local ring, and

E

an R-module such that

h~E <

~.

Then

codim R = h ~ E + codim E

Proof.

Let ~ = (Xl,...,Xn) , where

~

is the maximal ideal of

of the highest non-vanishing homology group of

K(Xl,...,Xn),

R.

and by

Denote by qE

q

the dimension

the corresponding integer

-

for the complex

E(Xl,...,Xn).

9.7

-

and codim E = n-qE,

Since codim R = n-q

we want to show that

qE-q = h ~ E . Now when

h~E

exact sequence

= O,

0

E

is free and clearly

> L

> F

> E

qE = q" > 0

Suppose that

with

F

h~E

~ i.

Then we have an

a free module, and

h~E = 1 + h~L.

This gives us an exact sequence:

> HqL(L(Xl,...,Xn))

Hl+qL(E(Xl,''',Xn))

> HqL(E(Xl,...,Xn))

> HqL(F(Xl,...,Xn))

> HqL_l(L(Xl,...,Xn)).

By induction on

hdRE ,

qL- q > O,

HqL(F(Xl,...,Xn) ) = 0

then

we know that

Ht+qL(E(Xl,...,Xn) ) = 0 done.

for all

qL- q = hdRL and thus

t > i.

so what must be shown is that Hl+qL(E(Xl,...,Xn) ) ~ O,

In this case, we may assume that If

YI'''"

Ys

L

and

F

qL = q'

i.e.

when

have been chosen so that

If

while clearly

Thus in this case, we would have

Our problem, then, is to resolve the case when

i + qL = qE"

qE = i + qL hdRL = 0

and we'd be

or

L

is free.

L c~F.

is a maximal R-sequence, it is also a maximal F- and L-sequence, and 4.4 tells us

that

and the map

Hq(L(Xl'''"Xn))

= ( YI'''"

Ys )L:~/( YI'''"

Ys )L

Hq(F(Xl'''"Xn))

= ( YI'''"

Ys )F:~/( YI'''"

Ys )F

Hq(F(Xl,...,Xn) )

Hq(L(Xl,...,Xn) )

( YI'"" Ys

( YI'''"

YI'"" Ys)L

is the natural map of

Ys )F:~/( YI'''"

Ys )F"

If we show that this map is not

a monomorphism, we are done. Since Since

L

( YI'''" is free,

w s zL - ( YI'''" ~ 0

in

Ys ) : ~ D

( YI' " " "' ys) ,

zL~

( YI'''"

Ys )L'

we have

( YI'''"

Ys )L"

Ys )F

zLc

w a ( YI'''"

Ys )L:~/( YI'''"

( YI''" ~' Ys )F:~/( YI'''"

But

there is a

Ys )L"

monomorphism, and the proof is complete.

z~F c ( YI'''"

Ys )L:~

However,

so that our map

z #( YI'''"

w

but

Ys ) with

Ys )F"

w # ( YI'''"

z~c

( YI'''"

Ys )"

Choosing an element Ys )L

and thus

is mapped to zero in

Hq(L(Xl,...,Xn) )

> Hq(F(Xl,...,Xn) )

is not a

-

A particular consequence of 4.6 is that if h~E

( %

then

h~E

Definition:

If

R

f.gl.dim R,

to be

28

R

-

is a local ring~ and

E

an R-module such that

~ codim R.

is a commutative ring, we define the finitistic ~lobal dimension of sup h ~ E

where

E

R# written

ranges over all finitely generated R-modules of finite

homological dimension. Before winding up this section, we state the following technical lemma:

Lemma 4.7.

If

R

is a local ring,

is not a zero divisor for

Proposition 4.8.

Proof. with

If

R

E,

then

R,

a finitely generated R-module, and

x s ~

an element which

hdRE/XE = 1 + hdRE.

is a local ring, then

We have already seen that s = co~m

E

f.gl.dim R = co~4m R.

f. gl.dim R ~ codim R.

we have (by 4.7)

h~2R/( Y I ' ' ' "

However, if

Ys ) = s = codim

YI'''"

Ys

is an R-sequence,

and hence the equality.

Putting 2.6, 3.5, 3.9, and 4.8 together, we obtain

Theorem 4. 9 .

If

R

is a local ring, we have

f.gl.dimR=

w

codlin R ~_ dim R (_ ~ : k ]

~_ gl.dim R.

Re6ular local rin~s. We are now ready to apply what we have done to the study of s me important properties of regular

local rings.

Definition.

A local ring

R

is regular if

dim R = ~/Y~-:kJ.

The following is a well-known property of regular local rings.

Proposition 5.1. maximal ideal Xl,...,x n

~.

Let

R

be a regular local ring, and

Then for each integer

L,

the ideal

is a (maximal) R-sequence.

Our first main result about regular local rings is:

Xl,...,x n

a minimal generating set of the

(Xl,...,xi)

is a prime ideal.

In particular,

-

Theorem 5.2.

A local ring

R

29

-

is regular if and only if

gl.dim R < ~.

If

R

is regular,

gl.dim R = dim R.

Proof.

If

R

is regular, 5.1 tells us that

gl. dim R = n < ~. dim R

Conversely, if

h ~ k = h~R/(Xl,...,Xn) = n = dim R,

gl.dim R < m,

then

gl.dim R = f~

R

and hence

and by 4.9, we have

=

Using this characterization of regular local rings, we obtain

Theorem 5.B.

If

R

is a regular local ring and ~

is a prime ideal of

R,

then

R~,

is a regular

local ring.

Proof. gl.dim R

L e n a ~.4. If

= h % / ~ R

R

~ hdRR~

<

is a regular local ring,

~.

E

Thus

R

is regular.

a finitely generated R-module, and ~ s

Ass(E),

then

hdRE ~_ height ~ .

Proof.

We have

hdRE >_ hdR. E ~

maximal ideal of

R~

,

Since ~

w/ha~e

codim E

R~ = 0

is obviously in as an

Ass(E ),

Ry-module.

Thus

and ~ R hdR

E~

is the

= codim R ~

dim Rnf = h e i g h t ~ . A straightforward application of the Krull Principal Ideal Theorem yields the following

Lemma 5.5.

If

R

is a noetherian ring, ~

an ideal of

is not a zero divisor for

R/Q( and

particular, if

is an R-sequence, then

Lemma 5.6.

Proof.

Let

zero-divisor for

R/(Xl,...,xt) ,

Ass(R/(Xl,...,xt) ).

height~ and thus

But if

hdRR/(Xl,...,xt) = t. s = t.

R.

~

Then

.

t = depth (6~;R).

we must have ~

x

an element of

R

such that In

height (Xl,...,Xs) = s.

depth (~;R) ~height ~

be a maximal R-sequence in 6~ i.e.

and

then height (~,x) ~ 1 + height 6~.

be an ideal in a regular local ring

By localizing we know that

Xl,...,x t

~s

Xl,...,x s

R/(~',x) ~ 0,

R,

depth (

Suppose

;R) = height

height 6E= s

and let

Since every element of ~

is a

contained in some prime ideal

E Ass(R/(Xl,...,xt) ) we have, by 5.4,

Thus, since 6~ c 7

implies height ~" ~ height ~ ,

we have

s
-

Le-~a 5:7. for

E.

If

Let ~

R

be a noetherian ring,

is in

Ass(E)

ideal ~ " ~ Ass(E/xE)

Definition.

and ~ '

E

30

-

a finitely generated R-module, and

is a prime ideal containing

(~,x),

x

a non-zero divisor

then there is a prime

such that ~' D ~ " D ~ .

We say a chain of prime ideals ~ 0 c ~ l c ... C ~ n

is no prime lying strictly between ~ i

and ~ i + l "

is saturated if for each

i

there

We say a ring satisfies the saturated chain

condition for prime ideals (s.c.c.) if any two saturated chains of primes between any two given primes ~c~'

have the same length.

Theorem 9.8.

Proof.

Every regular local ring satisfies the saturated chain condition for prime ideals.

It obviously suffices to show that if ~

Suppose, then, that

s = height ~ .

Clearly there is an element there is a prime ~ "

Remarks:l)

is saturated, then height ~ '

Then there is an R-sequence

Xs+ 1

in ~ ' - ~

E Ass(R/(Xl, ...,Xs+l) )

we cannot have ~ " = ~ .

c ~'

Thus y "

= ~'

such that

Xl,...,x s

Xl,...,Xs+ 1

such that ~ ' ~ " D ~ .

and height ~ '

= 1 + height

which is maximal in ~ .

is an R-sequence. Since height

Using 9-7,

" = s+l > height

= s+l = 1 + h e i g h t ~ .

By constructing a local ring that does not satisfy the s.c.c., Nagata was able to show that

not every local ring is a factor ring of a regular local ring.

For clearly, a factor ring of a ring

satisfying the s.c.c, must also satisfy the s.c.c. 2)

We see that if

3)

It can be shown that if

element of

R,

R

is a regular local ring, then

then height ~ , x )

R

dim R/4s

satisfies the s.c.c, and if ~

+

h e i g h t ~ = n = dim R. is an ideal of

R

and

x

and

~ 1 + height ~ .

we now come to the Cohen-Macaulay Theorem:

Theorem 5. 9 . Let

R

is unmixed

every prime ~ ~ Ass (R/6~)

Proof.

(i.e.

has height

The main point of the proof is to show that

follows easily. s=l

be a regular local ring, and 6~'= (Xl,...,Xs)

To show that

Xl,...,x s

s),

Xl,...,x s

and

an ideal of height s. Xl,...,x s

is an R-sequence.

is an R-sequence, for then the rest

is an R-sequence# we proceed by induction.

we are done (since a regular local ring is an integral domain).

s = height (Xl,...,Xs) ~ 1 + height (Xl,...,Xs_l).

But then

Then

Assuming

s > l,

Certainly when we have

s-1 ~ height (Xl,...,Xs_l) 5 s-1

so by

- 31-

induction We have divisor for

Xl,...,Xs_ 1

R/(Xl,...,Xs.1)

is an R-sequence.

It is then trivial to show that

xs

is not a zero

(by using a height argument), and so we have the result.

We end these lectures with an outline of a proof of the fact that every regular local ring is a unique factorization domain.

The proof given here is due to Kaplansky and is extremely elegant.

As

the reader ~-ill see, the tone of the proof is very different from most of what has gone before.

It

would perhaps be nice if a proof could be found using the notion of R-sequences.

Definitions.

An element

x

of an integral domain

prime ideal.

An integral domain is a unique factorization domain (ufd), if every non-unit of

a finite product of prime elements of

Theorem 9.10.

is a prime element if the ideal

(x)

is a R

is

R.

Every regular local ring

Outline of proof.

R

R

is a unique factorization domain.

We have already noted that a regular local ring is an integral domain.

Also,

R

contains prime elements since every element which is part of a minimal generating set for the maximal ideal of

R

is prime.

Let

x

be such a prime element and let

be the multiplicative set consisting

of all non-negative powers of

x.

Then it can be shown that

UFD.

RS

is a UFD.

Since

~

every prime ideal of height l) is principal.

Hence we must show that

that every minimal prime ideal of

(i.e.

RS

R

S

is a UFD if and only if

RS

is a

is a noetherian domain, it suffices to show If we could

show that every such ideal were free, it would follow that it is principal.

As a first attempt at

showing freeness, we show that minimal prime ideals are projective.

be such an ideal, and

any maximal ideal of that

R'

or

of

R',

Then since

is a regular local ring.

induction on dim R, R~

R' = ~ .

R~

we see that

R~

Since

hdR,~= 0

is a UFD.

h~,~

and ~

is an R'-module, then

E'

E

E

R'

over

R, E

we can find free R'-modules

and all maximal ideals ~Z

of

dim R,

so that by

being either a minimal prime ideal of

when }~

ranges over all maximal ideals

are principal.

R'

To see this, we first observe that if

For

we obtain one for

E' = E | R' E'

for some

by tensoring with

is a projective R'-module, then

is a projective ideal it is easy to see that

p > 1

dim R ~ <

we have

is projective.

particular, it is easy to see now that if i.e.

R~

has a finite free resolution.

and choosing a free resolution for

of length one

Hence y R ~ ,

= sup h

Next we show that projective ideals of E'

5 gl.dim. R' < gl. dim. R < %

Furthermore, it is easy to see that

we know that

itself, is free.

gl.dim. R ~

Let ~

E

R-module R'.

E,

In

has a free resolution

F 0 and F 1 such that E ~ F 1 = F O. Finally, when P P P AE = 0 for p > 1 since ( A E ) ~ A E ~ = 0 for

(because

E~

is principal for all such ~ ) .

Applying the

-

fact that

n A(A~

have

n 1 AF O ~ A E

R'~

p B) ~ E A A @ @

n-1 A Fl~

all projective ideals of have shown that

R'

n-p A B

R'

is a UFD.

32

to the situation

1 AE@

R'

(since

are principal, Thus

R,

F1

-

E~

F I = F O,

and letting n

must have rank n-l),

= rank of

and thus

and all minimal prime ideals of

R'

1 R' ~ A E

FO,

we

= E.

Since

are projective, we

too, is a UFD, and the outline of the proof is complete.

Since all the results mentioned in these notes are known, I have not given a detailed bibliography. Most of what has been said has its source in papers by Serre and Auslander-Buchsbaumwith material and some amplification in the already mentioned volume by Zariski-Samuel. on polynomial functions is new and not yet in print. concerned with multiplicities,

background

Perhaps the material

It will appear in a paper by Marshall Fraser

but the elements and purely formal properties of generalized polynomial

fUnctions have also been discussed by P. Wagreich.

-

CATEGORIES

33

-

OF MODELS W I T H

INITIAL

OBJECTS

by Dr. R. Fittler

INTRODUCTION The c o n s t r u c t i o n erations viewed

in the sense

as a prototype

universal

structures.

of free algebras w i t h

of Birkhoff

(cf.

for the c o n s t r u c t i o n In order

map

i: N

) I~I

freely g e n e r a t e d

(underlying

is a uniquely

determined

theory

.

Usually

such that

algebra

homomorphism

con-

of free an

by a set N if there

set of ~)

f: N " .~IMI, where M is an arbitrary

can be

the s u b s e q u e n t

of the definition

Let T be a fixed a l g e b r a i c

algebra ~ is called

Cohn)

op-

of more general

to motivate

cepts we need a little m o d i f i c a t i o n algebras.

[1], P.M.

finitary

is a

for each map

of type T there

g: ~----)M

which makes

the d i a g r a m

N

g

f

commutative. be viewed

The model M of T together

as a model M' of the

with

(algbraic)

f: N

(see definition

together with

) I~I)

i. N

14).

can

theory T' consisting

of the theory T together with new i n d i v i d u a l for each element of N

)IMI

constants, The model ~'

has the p r o p e r t y

that

one (

for each

-

model M' of T' there (in the sense define

g: ~

model ~

which

these maps

are called

theory T').

certain

I~l---~IMI

between

F-maps.

~

9) states F-initial

of this paper

model ~

theory T by the is exactly

the u n d e r l y i n g

sets

of T

They

formulas ~(see 4) each h a v i n g in T the sentence

"~x

/%

~(y)---~x

T.

(~ ( x ) •

one e l e m e n t which

u(x)

are identified

Furthermore,

an n-ary

if

formula

(~ (x I) t% ~2(x2) 1

is provable

~x

n

u-

u(x)).

set

the

(i.e., there Two such

in

is

formulas

is provable

in

..., x ) of F holds n Un if and only if the formula

Vx I

...

= y)

~x(~(x),M---~,~(x))

~i'

2

"'''

fulfills

~0 for the n-tuple ~x

(essentially

set of ~ is a quotient

that one can prove

u(x),

(see 2).

almost e x p l i c i t l y

property

exactly

one map

More p r e c i s e l y

the

, if it exists,

The u n d e r l y i n g

~xVy

to

(characterization

that one can construct

of the set of all unary

u(x)

>M'

will be called F-initial.

theorem

the following way.

g-~'

This suggests

conditions.

a given set F of formulas

The main

nique)

one h o m o m o r p h i s m

of a first order

fulfills

are maps

preserve

theorem

-

that for each model M of T there

)M,

which

is exactly

of the algebraic

an initial

condition

34

a(Xl,

A ... /% p(x ) - - ~ S ( X l , n

in

..., x n) )

in T.

A model ~ which a F-generated

model

tensor product in the category

of T.

of modules,

is constructed

in this way

(see definition localization

of all models

7).

is called

Examples

of rings,

direct

are limits

of a first order theory T where

35

the morphisms

are F-maps.

Another at left adjoints all models

class of examples of appropriate

result

i0 and remark

t- T

12).

ories in the sense of Birkhoff, in the sense of Lawvere completeness

theorem

[3].

(cf.

)T'

categories

the theory T in the of the theories

(see

If T and T' are algebraic

the-

one gets algebraic

functors

[2]) without mentioning

the individual

OF MODE~S

by x, y individual Definition

their predicate

constants.

Let

and a, b, c, .o. formulas

and

variables.

i

A set F of formulas

Formulas

i

of a theory T is called a generalized

formula if it contains

(i)

constants

By a,8,~,u we denote

of Generalized

x, y of individual

GSdel's

it.

Let S, T be first order theories with equality.

p, q, r, ... denote

of

The functors

We will use subsequently

CATEGORIES

2.

T, T'.

from interpreting

theory T' by means of a map

i.

functors between

of two first order theories

in consideration

definition

is obtained by evaluating

the formulas

variables

Under substitution ual variables

and if it is closedz

of individual

and individual

(ii)

Under conjunction

(iii)

Under disjunction

x = y for each pair

variables

constants.

by individ-

-

(iv) 3~

Remark formula is a special case of what H.J. Keisler

calls a"generalized

atomic set of formulas"

Let us introduce

models MT, NT,

preserve a(ml,

the c a t e g o r y ~

FpT" ... of T and its morphisms

the maps between

the underlying

o.., f(mn))

holds in NT~

the set of F-maps

M

Definition 9

a z F

the formula

then

By

The isomorphisms

F[~,

NT]

of

we denote

T~

of F-initial Models ,m

i

Let F be a generalized T is called F-initial unique F-map

are

)N T

6.

f, ~-----)N T

are

In [3] Keisler calls these

We call them F-maps.

are called F-isomorphisms.

[3]).

IMTI ----->INTI which

all formulas of F, i.e., if for

maps F-expansions.

.

sets

(cf.

Its objects

...,m n) holds in M T where m. ~ IM I i T

a(f(ml,

~F,T

-

Under existential quantification.

A generalized

4.

36

gz ~

|

formula of the theory T.

A model ~ of

if for each model M T of T there is one )M . T

Remark

All F-initial models of T are F-isomorphic.

-37 -

7.

Definition_ of F - q e n e r a t e d

Models

Let T be a theory and F a g e n e r a l i z e d to c o n s t r u c t

a model of T in the f o l l o w i n g way.

set of u~ary formulas ~x

p e F with

{~(x) A ~ y ( ~ ( y ) - - - ~ y

them d e t e r m i n a n t

in T9

equivalence

classes

Let

~ =

holds

~x n

in T.

Let us call relation

VxVy(~(x) J% ~ ( y ) - - ~ x

a(Xl,

ooo}

be the set of

. 9149 x n)

z F

9 9 /% Wn(X n ) - - ~ a ( x

this d e f i n i t i o n

and if there

ing p r e d i c a t e

constants

model ~ of T then ~

we define

I, .o.,

is not consistent 9

x n))

is an i n t e r p r e t a t i o n

of the remain-

of T in ~ such that one obtains

is called

an F - ~ e n e r a t e d

If it

a

model of T9

Remark

All F - g e n e r a t e d

models

Characterization i

= y)

in ~" if the sentence

(UI(Xl) A

is c o n s i s t e n t

99

that

in T.

In general

89

For

We try

Let ~ be the

an e q u i v a l e n c e

{ (w),(~),

of ~ .

(~n))

~x 2 .o9

is provable

We define

~ -- 7 if the sentence

is provable

"a((~l) , ..0,

the p r o p e r t y

= x)} is provable

formulas.

on ~ by s e t t i n g

Vx I

formula of T.

ml

i

of T are F-isomorphic 9

Theorem

9

A model ~ of T is an F - g e n e r a t e d is F-initial 9

model of T if and only if

-

Proof.

First

~ I being equivalence we define

g((~))

holds in M T. of~

38

let ~

-

be F-generated.

classes

The elements

(u) of determinant

formulas

of

,,

as the only element m ~ M T such that ~(m)

Thus g is well defined.

Because

of the structure

it is clear that g is a F-map and it is the only one. In order to show that a F-initial

generated

it is necessary

each element theory.

a c I~ I.

predicate

constants

of

~I

contains

formula

the language

of T and two unary a E ~ I.

those of T and for each n-tuple and each

a(Xl,

~a for

let S be the following

Pa and qa for each element

axioms of S include of elements

to find a determinant

For this purpose

Its language

model of T is F-

..., Xn)

r F

The

al, . .., a n

the two sen-

tences :

Vx I Vx 2 9

%IXn(Pal(x I) ;% ... ;% Pan(X n) ---~ a(Xl,

... Xn))

Vx I

~ X n ( q a l ( x I) A

"'" xn))

9

9

....

if a(al,

..., an) holds x Pa(X) ^

in~

is a model, Pa and qa both a, for each a r ~ I -

that the sentence

in S for each a ~ I~I.

the sentences

for each a r ~ I.

because ~

by the singleton

It follows is provable

, and furthermore

-~x qa(X)

S is consistent being interpreted

"'" ^ qan (xn) ---~ a(Xl'

(~x)

Because

(Pa(X),/--~,qa(X)) if it were not

-

39

-

there would exist a model N of S with different interpretations for some Pa and qa" fl and f 2 ~

)N

Hence there would exist two different maps

assigning to each a c k01 an element fl(a)

of Pa in N and an f2(a) of qa in N respectively. a model of T it follows that fl and f2 are F-maps, diction to the assumption that ~

Viewing N as in contra-

is F-initial.

For a fixed element a I r I~I the proof of the sentence ~X(Pal(x),~--~,qal(x))

in S uses only finitely many axioms.

Those which are not already in T are contained in a finite collection C of sentences

like:

V x I ... Vxk(Pal(Xl)A...APak(x k) ~ 8 ( X l ,

..., xk))

Vx I .-. ~xk(qal(xl)a...^qak(x k) ---~8(Xl,

..., x k))

x pa l(x), Sx qal(x),

..., SXPa k(x) and ..., SXqa(X;

where 8 is an appropriate

conjunction of formulas ~ from F.

Let the theory ~ consist of T together with the collection C of axioms.

We know already that

VX(Pal(x)~=~qal(x))

is provable in ~.

We claim that the unary formula ~ (x l) , defined by

- 40-

Zx 2

~x 3 ...

is a determinant is contained ~gx ~ (x)

8(x I, x 2, ..., x k)

k

formula in T.

in T because

It remains

.

The formula

it is true in ~

, hence

in

is provable

in T.

Assume

it

then there would exist a model N of T and two dif-

ferent elements in N, i.e., c2,

in ~

to show that

~x Vy(~ (x) /% , (y) ---~x = y) were not;

It is clear that the formula

in F and that ~(a I) holds

is provable

every model 9

~x

..., Ck

8(ci, c2,

such that ~(b I) and ~(c I)

bl, c I r INI

there are elements

b 2, . .. , b k

and

in N such that 8 (b I, b 2, ... , bk)

..., Ck)

and

Thus N could be extended

hold in N.

hold

to a

model ~ of ~ by interpreting

pa l(x)

by x = bl, pa 2(x)

by x = b2,

Pak(x)

by x = b k

qa l(x)

by x = Cl, qa 2(x)

by x - c2, 9 .o, qa k(x)

by x = c k

which violates

~X(Pa I (x) ~ q a

99

l(x) ) 9

We get all determinant

formulas

in this way because

is a model of T. Because ~

is F-initial

holds in ~

a formula

(for ~(x , 9149149 x ) r F 1 n if and only if the formula:

~Xl,

9149149 VXn(~al(Xl) A

and al,

a(al,

9149 a

..9 ;% ~an(Xn) ---~S(Xl,

holds in each model of T, i.e.,

..., a n ) n

r I~ I)

..9

Xn ))

if and only if it is provable

-

41-

in T.

APPLICATIONS

10.

Definition

of a m a p

t: T

)T'

of a t h e o r y T i n t o

a

t h e o r y T'

t assigns formula

to e a c h n - a r y

t(a)

formula

of T' s u b j e c t

a of T a n - a r y

to the

following

con-

ditions : (i0.i)

a and t(a)

(10.2)

t preserves

x = y a n d x = a.

(lO.3)

t preserves

the s t r u c t u r e

(e.g.,

have

t(--Ta)

the s a m e

=

free v a r i a b l e s .

of the

-'~t(a),

formulas

t[~xa

(x)] = V x t ( a ( x ) ) ,

etc.) .

(10.4)

If the s e n t e n c e

a is p r o v a b l e

in T t h e n t(a)

is p r o v -

a ble in T'.

F o r the sake of s i m p l i c i t y w e dividual

constants

assume

and variables

t h a t T' c o n t a i n s

the in-

of T'.

11.

A map

t: T

-~T'

an i n t e r p r e t a t i o n

between

theories

of the m o d e l s

M

induces

in an o b v i o u s w a y

, o f T' as m o d e l s T

of T.

-

12 ~

49

-

Remark

If F and F' are g e n e r a l i z e d ively,

and if, moreover,

induces

a functor

underlying

t*.~

formulas

of T and T' respect-

t(F) cI

F' then the map

F' ,T'

) 11~F, T

)T'

t- T

c o m m u t i n g with the

set functors.

13. For a language T c o n t a i n i n g e q u a l i t y but w i t h o u t axioms

and a g e n e r a l i z e d

that the F-maps correspondence SM(F,T).

IMTI.

with

There

formula F, K e i s l e r observes

) NT

constants

by some elements

let

t*'~F,,T,)~F,

14.

Definition

constant

of S M(FfT

[3]

holding

are

in M

which T of formulas

In order to generalize

be a functor d e f i n e d l

say

for each e l e m e n t

the free variables

of M T.

T

theory,

of the theory SM(F,T)

are the sentences

are c o n s t r u c t e d by r e p l a c i n g a c F

of a certain

is one i n d i v i d u a l

The axioms

in

with M T fixed are in one-one

the models

The p r e d i c a t e

those of T.

m

MT

additional

that,

as in 12.

)

Let M

be a model of T. The theory S (F,T') has the same T M predicate constants as T'. The i n d i v i d u a l constants of S (F,T')

are those of T' and a d d i t i o n a l l y

there is a new one

M

for each e l e m e n t m c IMTI o these elements

We do not d i s t i n g u i s h

and the c o r r e s p o n d i n g

individual

between

constants.

-

the axioms of S

43

-

(F,T') include those of T' and the sentences M

of the form

(tin)(ml, ..., m n)

the m. are elements l

15.

of

M

where

and

T

a(ml,

m(Xl,

..., x m)

..., m ) n

r F,

holds in M . T

Lemma

The pairs

(f, NT,) , where

in one-one

correspondence Proof.

f: M T

;t*(NT,)

with the models of SM(F,T').

Let us just establish

both directions.

A pair

(f: M T

scription of NT,

as a model of

of the new individual constants Conversely,

is an F-map are

the correspondence

in

)t*(N T ), N T ) is a de' ' SM(F,T'), the interpretation m c JMTI

given by the map f.

S (F,T'), we get a M model LT, of T' by forgetting the interpretation of the individual constants f:

MT

)t*LT,

ual constants

if L is a model of

m r IMTI

(T'-reduct).

The map

is given by the interpretation m c IM I T

of the individ-

in L. Q.E.D.

Let

t" T

F' be generalized t(F) c F'.

)T'

be a map of theories and let F and

formulas in T and T' respectively

such that

By Remark 12, this gives rise to a functor.t*:

~I~

%~ F' ,T'

Fe T

.

-

16.

Definition --

A model ~

c

if there T dence

-

of t*-Free Models .

.

Generated

i

%'~

i

is called

F' P T' M

44

is an F-map

a t*-free model g e n e r a t e d

i- M

by

) t*~ such that the correspon-

T

j- F' [ ~,NT,]

by M T

9

)F[ M T ,t*(N T' ) ], j (f) = t*(f)

"i

is a bijection0 17 o

Remark

This

is the usual universal

well known

that ~

definition

is d e t e r m i n e d

by M

of free objects.

It is

up to an F'-isomorphism. T

If for every

M

generated by MT.

(M T) functor

of t*

(cf.

Let

M

formula

the theory

then ~

r ~

there

FeT

is a t*-free

can be viewed

model

as the left adjoint

be a fixed model of To FiT

F' of T' generates SM(F,T' )

a generalized

by "closing"

ition 2 of a generalized 18 o

c ~

[5], S. MacLane).

T ized

T

The general-

formula

F' according

F''

of

to the defin-

formula.

Lemma t

A model ~

of

T'

is

a t*-free

if and only if the pair (according

to Len~ma 15)

model g e n e r a t e d

by

MT c ~ F , T

(i: M

) t ~ , ~) corresponds T to an F ' ' - i n i t i a l model of S (F,T'). M

-

This follows directly 19.

45

from the definitions

5 and 16.

Corollary

A model ~ of T' is only if ~

t*-free generated by M ~ ~T~ T FwT

is an F' '-generated model of the theory

S (F,T'). M

is t*-free generated by M T if it is F''-

Proof. initial

if and

(by Lemma 17).

orem 14, we know that ~

Because of the characterization is an F''-initial model of

if and only if it is an F''-generated

the-

S (F,T') M

model. Q.E.D.

REFERENCES

Ill

Cohn, P.M.

Uniuzrsal Algzbra, Harper and Row, New York,

1965. [2]

Godel, K., Die Vollst~ndigkeit

Monarch. Math. Phg6o

Funktionen-Kalk~is. 349-360,

[3]

der Axiome des Logischen 37;

(1930).

Keisler, H.J., Formulas",

"Theory of Models With Generalized

J.

Atomic

of Symbolic Logic, 25, (I)~ 1-26,

(1960).

[4]

Lawvere,

F. W.,

ries",

"Functorial Semantics

Pr0c. Nat.

of Algebraic Theo-

Acad. S~/., 50, (5); 869-872,

(1963).

[5]

MacLane, 71,

S., "Categorical (i); 40-106,

Algebra",

(1965) 0

Bull. Amzr. Math. S o t . ,

-

46

C O R E F L E C T I O N MAPS W H I C H R E S E M B L E

UNIVERSAL COVERINGS

by

J. F. K e n n i s o n

1.

INTRODUCTION

If X is c o n n e c t e d and s e m i l o c a l l y [9, p. of

78])

then one can r e g a r d the set

c: X

> X, w h e r e

c o v e r i n g of X.

1-connected

~I(X)

c is the u n i v e r s a l

as any fibre

simply connected

This o b s e r v a t i o n g e n e r a l i z e s

locally 1 - c o n n e c t e d spaces such as

(see

to some n o n - s e m i -

X = R x R \ {(2-n,0) In e Z+},

in the sense t h a t there e x i s t s a H u r e w i c z

fibration

(which is a c a t e g o r y t h e o r y g e n e r a l i z a t i o n

c: X

> X

of the u n i v e r s a l

s i m p l y c o n n e c t e d covering)

such t h a t any fibre of c can be re-

g a r d e d as

last example,

~I(X).

In this

h a v e an i n t e r e s t i n g n o n - t r i v i a l

topology

moreover,

the fibres

(see 5.8 the final

e x a m p l e of this paper).

Categorically,

e a c h of the above m a p s

can be r e g a r d e d as a c o r e f l e c t i o n

unique

c o n n e c t e d spaces.

is s i m p l y c o n n e c t e d and locally p a t h w i s e

if S is any o t h e r such space, lift

S

> X

> X,

of X into the c a t e g o r y of

s i m p l y c o n n e c t e d and locally p a t h w i s e is, X

c: X

S

then e v e r y m a p

which commutes with

c.

(That

connected, > X

and

has a

It is n e c e s -

sary to add that all spaces h a v e base p o i n t s w h i c h are p r e s e r v ed by maps.)

-

47

-

W e shall i n v e s t i g a t e a particular

coreflective

p o i n t e d spaces

this p h e n o m e n o n by c o n s i d e r i n g , subcategory Cn of the c a t e g o r y of

(i.e. spaces w i t h b a s e points).

s m a l l e s t full r e p l e t e c o r e f l e c t i v e the c o n t r a c t i b l e

spaces.

subcategory

(A s u b c a t e g o r y

Cn

is the

c o n t a i n i n g all

is r e p l e t e if it is

c l o s e d u n d e r the f o r m a t i o n of e q u i v a l e n t objects.) let

c: X

> X

We shall

be d e f i n e d to be the c o r e f l e c t i o n

of X in-

w

to

Cn .

If X is c o n n e c t e d and s e m i l o c a l l y

1-connected,

c is the u n i v e r s a l covering, b u t in general, ration w i t h t o t a l l y p a t h w i s e d i s c o n n e c t e d G(X),

c is a Serre

fibres.

fib-

We show that

the fibre of c, has n a t u r a l g r o u p and t o p o l o g i c a l struc-

tures d e f i n e d on it. ~I(X).

As a group,

G(X)

One of the t o p o l o g i c a l p r o p e r t i e s

d i s c r e t e iff c is a c o v e r i n g and TI).

is a q u o t i e n t of is t h a t

G(X)

(for X l o c a l l y p a t h w i s e

is

connected

G is f u n c t o r i a l b o t h as a g r o u p and as a space.

ing b o t h of the s t r u c t u r e s

on

and 5 . 7 ( a ) ) - - i f

Us-

G(X), we can d e t e r m i n e w h e t h e r

or n o t X is s i m p l y c o n n e c t e d in the sense of C h e v a l l e y 5.6(b)

then

X is locally p a t h w i s e

(see 4.5,

c o n n e c t e d and T I.

For such spaces, we can also o b t a i n some i n f o r m a t i o n a b o u t the u n d e r l y i n g t o p o l o g y of the u n i v e r s a l p r o - c o v e r i n g of Lubkin, [7].

The u n i v e r s a l p r o - c o v e r i n g is c o m p a r e d and c o n t r a s t e d w i t h

the c o r e f l e c t i o n X

in the examples.

We do n o t k n o w if group, b u t by 5.2, the f o r m

G(X)

G(X)

is always a t o p o l o g i c a l

a large class of t o p o l o g i c a l groups

for a r e a s o n a b l y w e l l - b e h a v e d

X.

are of

As a corol-

-

48

-

lary to 5.2, we show that if ~ is any group and N is any n o r m a l subgroup

of ~, there is a connected,

T 2 space X w i t h universal

~I(X)

connected

= ~, G(X)

covering

locally p a t h w i s e

= ~/N

and

relations

coreflective

between

subcategories.

see

[5] and

al and could be applied proper

class)

ed spaces nected tions

The methods

to any one of the large n u m b e r

connected,

and the s u b c a t e g o r y

Thus this paper liminary

report.

ed c o r e f l e c t i o n 3.2(b),

of spaces

locally p a t h w i s e

connect-

are simply

and is to some degree of

considered,

3.6 and its consequences,

con-

There may also be a p p l i c a -

In the last section was b r i e f l y

(i.e.

in the sense of Lubkin

is not e x h a u s t i v e

gener-

that lie b e t w e e n

of spaces w h i c h

in the sense of Chevalley.

[6]

For addition-

used are quite

subcategories

on

[4], and

can be found in

in the next section.

[6].

of simply

in the category

of Isbell,

These results

of c o r e f l e c t i v e

the s u b c a t e g o r y

the

is b a s e d p r i m a r i l y

the b i c a t e g o r i e s

and are stated w i t h o u t p r o o f al details,

m X

of X.

Our c a t e g o r y t h e o r y m a c h i n e r y certain

c: X .

connected,

3.7,

[6], a closely and analogs

[7]. a prerelat-

of 3.1(a),

3.8 and 3.9 were ob-

tained.

Our t e r m i n o l o g y ular,

a complete

category

is b a s e d is both

remark that the term "bicategory" meanings,

on Freyd,

[2].

left and right has acquired

In p a r t i c complete.

We

several diverse

but we are u s i n g it only as d e f i n e d below.

- 49 -

Note

F or c o n v e n i e n c e , throughout

2.

this paper,

BICATEGORY

a coreflective

be a s s u m e d

STRUCTURES

subcategory

shall,

to b e f u l l and r e p l e t e .

AND COREFLECTIVE

SUBCATEGORIES

Definition

Let B b e phisms

on B.

provided

with

a category.

Then

(I,P)

Let I and P be classes

is a l e f t b i c a t e g o r y

structure

on

that: LB0:

Every

LBl:

I a n d P are c l o s e d u n d e r

LB2:

Every member

LB3:

Every morphism

fl e I

and

equivalence

f0 e P.

is in I n P.

f can b e If

factorization,

alence

e for which

ef 0 = h

(I,P)

in a d d i t i o n

and

factored

f = gh

is any o t h e r

If,

with

then

in B.

as

f = flf0

g e I

there exists

and an e q u i v -

ge = fl"

to the a b o v e ,

is a b i c a t e g o r y

compositions.

of I is a m o n o m o r p h i s m

h e P

then

of m o r -

every

f ~ P

is epi,

structure.

Definition

Let

(I,P)

b e a left b i c a t e g o r y

X is an I - s u b o b ~ e c t

of Y if t h e r e e x i s t s

f e I.

Similarly,

Y is a P - q u o t i e n t

g: X

) Y

g r P.

with

structure f: X

on B.

Then

> Y with

of X if t h e r e e x i s t s

-50-

2L: 1

Lemma

(Diagonal

Let g E P

and

fm = ng.

(I,P)

be

f e I Then

Property)

a left bicategory

and morphisms

there

exists

structure

m and n be

a morphism

given

r with

on B.

Let

with

rg = m

and

fr = n.

2.2j~ T h e o r e m

(Isbell)

Let B be

complete

powered.

Let M b e

the

of all e x t r e m a l

iff

class

e = mg

(M, E #)

and

the

class

m e M

well-powered

of all m o n o m o r p h i s m s

epimorphisms imply

is a b i c a t e g o r y

is a l e f t b i c a t e g o r y

and e i t h e r

(that

of B and E # is,

e e E#

m is an e q u i v a l e n c e ) .

structure

structure

of B

or c o - w e l l -

on B.

on B,

Then

Moreover,

if

(I,P)

t h e n E # c_ p.

Remark

It f = flf0 smallest

follows with

fl

subobject

from

the d i a g o n a l

e M

and

f0

through

which

an o b j e c t

of B.

property

e E#

then

(2.1),

that

fl r e p r e s e n t s

if the

f factors.

Definition

L e t A be has

the u n i q u e

morphism fh=

g.

g: A

lifting > Y

property there

Then with

a morphism

respect

is a u n i q u e

f: X

to A if

h: A

~ X

> Y

for e v e r y with

-51-

Definition

Let

A

be a subcategory

the set of all m o r p h i s m s perry with respect We define for which

which

and

2.3 and 2.4,

B.

We define

have the unique

to every object P(A)

p = ig

of

of

I(A)

lifting

to be pro-

A.

to be the class of all m o r p h i s m s i c I(A)

imply

stated below,

i

p

is an equivalence.

are the main

results

of

[6].

2.3 T h e o r e m

Let category. full of

C

be a complete,

Let

A

subcategory A.

Working

be defined tients

C

in the category Let

A*

(I,P)

(c)

A*

for which

of

A.

If

B

I = I(A)

and

P = P(A)

of all P-quo-

Then:

of

B

structure

coreflective

B

is

has a represen-

is I-wellpowered

is a left b i c a t e g o r y

is the smallest

be the

of members

in C iff A = A* and

if each object

set of I-subobjects). (b)

let

B

be the subcategory

of members

(i.e.,

Let

of c o p r o d u c t s B,

is c o r e f l e c t i v e

"I-wellpowered" tative

of all quotients

of coproducts A

and c o - w e l l - p o w e r e d

be any full subcategory.

as above.

(a)

well-powered

on

then: B.

subcategory

of

A c A*.

2.4 Proposition Let

~, A*, ~, ~, and

(I,P)

be as above.

Assume

that

- 52 -

B is i-well-powered.

Then the following

statements

are equiva-

lent~ (a)

A

e A

(b)

A r B

and whenever

f: X

> A

is in

I then

N

f is an equivalence. (c)

A z B

and every B morphism

(d)

A z B

and every member of I has the u n i q u e

ing p r o p e r t y

with

3.

CONSTRUCTION

study of the smallest

valent

spaces

Working

spaces.

images)

the category

and

It is shown in

powered,

hence I-well-powered

map

c z X

of all "quotients" of members

is a left bicategory

all equi-

topo-

of all con-

PC, of all pathwise

P = P(Cn)

section.

connect-

(i.e.

of Cn.

be defined as in the

[5, p. 409]

that PC is well-

and so 2.3 and 2.4 apply.

That

system on P C which c a n be used

, the ooreflective

For each

containing

let Cn be the subcategory

previous

to compute Cn

subcategory

section to the

in the category of pointed

of coproducts

Let I = I(Cn)

(I,P)

of the previous

The subcategory

ed spaces is clearly

is,

lift-

OF THE COREFLECTION

coreflective

logical spaces, we shall

continuous

is in P.

(i.e. spaces which are homotopicaily

to a point).

tractible

> A

respect to A.

We now apply the results

contractible

f: X

subcategory

generated by C n.

X r PC, we shall construct

the corefiection

> X

associated with C n .

As explained

in the

-

introduction,

-

this map is a g e n e r a l i z e d

nected covering properties

53

of X.

of c, Cn

We shall

universal

also discuss

simply

con-

some e l e m e n t a r y

and I.

Note

(i) assumed

From here on every space

considered

shall be

to be in PC. (2)

F r o m here on the term "quotient"

in its t o p o l o g i c a l

rather

than

shall be used

its c a t e g o r i c a l

sense.

Notation

In this paper as in

the term ~

[3], that is to denote

shrink to a point.

simply c o n n e c t e d

of PC for w h i c h

[9], w i t h o u t

space shall be a connected,

assumption,

connected in

usual meanings.

that the u n m o d i f i e d

sense in

etc.,

A Chevalle~ connected

(We drop Cheval-

[i], that a space be T2.)

and H u r e w i c z

"fibration",

(or just

locally

coverings.

terms Serre F i b r a t i o n Note

all loops

any r e s t r i c t i o n s

such as local connectedness.

space w i t h no n o n t r i v i a l ley's b l a n k e t

a member

shall be used

The term c o v e r i n @ p r o j e c t i o n

coverin ~) shall be used as in on the base space,

connected

have the former

Fibration

shall have their

terms

sense in

The

"fibering",

[3] and the latter

[9].

321 P r o p o s i t i o n

(a)

Every member

of I is a Serre

fibration w i t h

to-

-

tally pathwise

disconnected

(b) totally

(Y,y0)

pathwise g:

>

section.

(X,x 0)

homotopy

)

identity map on there

exists

follows nl:

map)

that

(A,a 0)

Let

Let

(the constant

By the covering

and

nl(a0) >

the proof

(Y,y0)

of

map)

> A

and

be a

hI

the

property, no = Y0

Then the path loop.

= Y0

totally

and

such that

fn t = gh t. a shrinkable

with

ht: A

homotopy

nt: A----> Y

= n0(a0)

(a) until

(b), let

be given.

g(ht(a0)) ,

I.

(A,a 0) 6 Cn

a homotopy

(the constant lies over

A.

is in

fibres.

h0 = a0

with

PC,

fibration

(X,x 0)

with

As for

in

be a Hurewicz

disconnected

(A,a 0)

fibres,

We shall postpone

the end of this f:

fibration,

disconnected

Proof.

-

fibres.

Every Hurewicz

pathwise

54

(nt(a0))

It readily

and so

is the desired

lifting

of

g.

space,

which

3.2 Corollary (a) quotient simply in

simply

of a c o n t r a c t i b l e

connected

connected space,

and locally

is in

Cn*.

is the

Thus,

every

pathwise

connected

space

pathwise

connected

member

is

Cn*. (b)

Cn*

Every

Every

is Chevalley

locally

simply connected.

of

-

Proof. q: A

> X

(a)

implies

Let X be simply

map.

[9, p.

hence,

space,

and let Let

C:

h with

But it is e a s i l y

connected

namely

X

X

ch = q

c is a h o m e o m o r p h i s m .

locally p a t h w i s e

tient of a c o n t r a c t i b l e

(b)

A ~ Cn.

that c is a q u o t i e n t map.

every connected,

see

connected

Then there exists

shown that c is one-one,

below,

-

be a q u o t i e n t map w i t h

be the c o r e f l e c t i o n which

55

Finally

space is the quo-

the path space d e f i n e d

75].

As shown in

[9, p.

67], every

covering

is a Hure-

wicz

fibration. Thus, if X is locally p a t h w i s e c o n n e c t e d and , in Cn , then every c o n n e c t e d c o v e r i n g of X is p a t h w i s e con-

nected,

hence

in I, hence

trivial by 2.4(b).

Definition

Let path space

(X, X 0)

over X is the set of maps

([0, i], 0) logy.

r P_~C be given.

>

(X, X 0)

The base p o i n t of

P(X,

x0),

the

the c o m p a c t - o p e n

x 0)

is the constant map.

> X

be the map for w h i c h

topo-

= f(1).

When there is no danger P(X)

P(X,

from

together w i t h

We let ~ : P(X, x 0) ~(f)

Then

instead

of

P(X,

x 0).

of confusion,

we shall use

- 56

-

3.3 T h e o r e m

L e t #: P(X) where

c is m o n o

(We a r e

> X b e as above.

in P C and ~0 is an e x t r e m a l

applying

2.2

to

PC). -

c: X

Let

~

"

# as

~ = c~ 0

epimorphism

: P (X)

> X

in PC

and

0

) X.

Then ed w i t h

r

X

Cn

X* r C n *

as

We f i r s t n o t e t h a t P(X)

r Cn

under P-quotients. given

"lifts"

and c is t h e

coreflection

map associat-

X.

Proof:

ment

Factor

(see

[9, p.

It r e m a i n s

in 3.1 c a n be u s e d

v i a c.

Moreover,

~0 r P 75])

to s h o w

as

E # ~ P.

Then

a n d Cn * is c l o s e d

t h a t c E I.

The a r g u -

to s h o w t h a t e v e r y m e m b e r

the

lifting

is u n i q u e

of C n

as c is m o n o

in PC.

3.4 CorollarY

Let If X is T i' of

quotient

~: P(X) then -i

c

(x)

)

-i

X

(x)

and

c:

X

) X

b e as above.

is a t o t a l l y p a t h w i s e

for all

disconnected

x r X.

3_..5 C o r o l l a r y

L e t X be T 1 and let coreflection. and

h(1)

duced by h

= x I.

Let

h:

C:

[0, i]

T h e n the m a p

(which s e n d s Y0

(X , X 0) > X

from

;

(X, X 0)

be a p a t h w i t h c-l(x0 )

to

to Y l iff t h e r e e x i s t s

b e the h(0)

c -I (x I)

= x0 in-

-

~:

[0, l]

> x

with

a homeomorphism. topy class

This

(x 0)

to

-i

~

the u s u a l

Taking

quotients,

mains

c ~ = h, ~(0) homeomorphism

This

which

= Y0

and

depends

h(1)

only

= yl )

is

on the h o m o -

the w e l l k n o w n

continuous

sends

~ to the p a t h

sum for paths we obtain

m a p has

invariant

the loop

having

an e n d p o i n t

the d e s i r e d

an o b v i o u s

map

inverse

from

map

from h +

in common). c-l(x0)

to

and also c l e a r l y

re-

under homotopy.

3.6 P r o p o s i t i o n

(Base P o i n t

Let be a r b i t r a r y

Consider (x I)

(using

c-l(xl).

-

of h.

Prqof: -1

57

f:

and

Chan~e)

(Y, y0 )

let

;

be in I~

(X, x 0)

x I = f(yl ) ~

Then

f:

Let

Yl E Y

>

(Y, Yl )

(X, x l)

is also in I.

Proof: and ~(i)

= YI"

> (X, x I) the c o p r o d u c t > ([0,

i)

mapping there

Let

in PC

(Z, 0)

lows t h a t desired

~:

[0, l]

h = fh.

(Z, 0)

Z =

defined

of g.

i],

i)

product).

can also

(X, x 0) .

Clearly

(Y, y0 )

+

~(i)

= Yl

Z e Cn

h(0) g:

= Y0

(A, a l)

(A, a I)

Then

regard

with

that and

if h is r e g a r d e d

~I [0, i] = ~, h e n c e

lifting

([0,

We

>

be such

(A, a I) E Cn

the w e d g e

(X, x I). into

> Y

Let

Let

(i.e.

is w e l l into

exists

~:

be given.

(X, x I) i],

Let

be

h + g: Z

as m a p p i n g h + g

as

and since

f~ = h + go and thus,

~IA

f ~ I

It folis the

-

58

-

3.7 C o r o l l a r y

Let

(x, x 0)

X r

and

iff

Cp

e

PC

Proof:

b e given.

x0, x I e X

Then

(X, x I) e Cn.

Follows

in v i e w of 2.4(b)

a n d the above.

3,.,8 C o r o l l a r y Let of

(X,

Then

x O) .

c:

c:

(X*, X 0)

Let

>

X 1 ~- X

(X , xl)

>

(X, x 0)

be

be the coreflection

arbitrary

(X, x I)

and

let

xi

is the c o r e f l e c t i o n

= c(x 1).

of

(X, x 1).

Remark

In v i e w our p r a c t i c e

of the a b o v e

results,

we shall

of f r e q u e n t l y

suppressing

c: X

be the c o r e f l e c t i o n

freely

any mention

continue

of the b a s e

point.

Definition

Let

X

the deck translation phisms phisms

g: X

@rou~

> X

Then

for X is the g r o u p of all a u t o m o r -

such that

are n o t r e q u i r e d

of X.

cg = c.

to p r e s e r v e

(These a u t o m o r -

any b a s e p o i n t s . )

3.9 T h e o r e m

Let

c: X

deck translation that whenever

) X group

be the c o r e f l e c t i o n

acts t r a n s i t i v e l y

Xl, x 2 e X

of X.

on e a c h

are g i v e n w i t h

Then the

f i b r e of c so

c ( x I) = c(x 2) ,

-

there

is a u n i q u e

Proof:

both

;

coreflections

-

deck translation

Assume

(X , x I)

C:

59

that

g ( x l) = x 2.

c(x l) = c(x 2) = x0.

(X, x 0) an d of

g for w h i c h

c:

(X, x 0)

)

(X , x 2)

in v i e w of 3.7.

Then (X, x ) are 0 Thus t h e r e

w

are m a p s

g:

(X

with

,

x 1)

(X , x I) cg

=

c

v e r s e of g and so g i s g follows

since

Proof S E Cn

Let

that

fm = ng~

m: S

g r a m of p o i n t e d

a

Let

and

spaces

TOPOLOGICAL

n: S •

This

property

AND

h is t h e

in-

[0, i]

of

as a d i a -

in S, etc.)

there exists implies

respect

any

be such

this

any b a s e p o i n t

Let

be

> X

can r e g a r d

clearly with

be in I.

> S x [0, i]

a n d so b y 2.1,

an d so f is a S e r r e

4.

>

The uniqueness

) X

g: S

spaces by choosing

homotopy

Clearly

f: Y

(In v i e w of 3.6, w e

rg = m and fr = n.

the c o v e r i n g

(X*, x 2)

in PC.

N o t e t h a t g E P by 2.4(c) r with

h:

translation.

and let

> Y

and

ch = c~

deck

(a):

be a r b i t r a r y

(X, x 2)

and

c is m o n o

of 3.1

map.

)

a map

that f has

to all c o n t r a c t i b l e

fibration.

GROUP

STRUCTURES

ON THE F I B R E

Definition

Let (X, x0).

~l(X,

c:

(X*, x 0)

We d e f i n e

G(X,

)

be the c o r e f l e c t i o n

x 0) = c - l ( x 0)

G(X,

x 0)

be the n a t u r a l

x 0)

and

G(X,

x 0)

(X, x 0)

function.

are b o t h

and

let n:

~I(X,

of x 0)

[Notice t h a t

equivalence

classes

of

loops

60

in

~-l(x0),

connected,

see 3.3.

homotopic

Since

-

G(X, x 0)

is totally pathwise dis-

loops must be equivalent in

G(X, x ). 0

Hence we can readily obtain the above function n.] 4.1 Proposition G(X, x 0) n: ~I(X, x 0) G(X, x 0)

can be given a group structure so that

9 G(X, x 0)

is canonically

is a group homomorphism.

isomorphic

Then

to the deck translation

group. Proof: translation G(X, x 0) group.

For each

g for which

x ~ G(X, x 0) g(x~) = x.

in one-one correspondence If we give

G(X, x 0)

associate the deck

By 3.9, this places with the deck translation

the induced group structure,

it

is then easy to show that n is a group homomorphism. Definition Let

n: ~ (X, x 0) > G(X, x 0) be the above homo1 We define N(X, x 0) to be the ke;nel of n.

morphism.

4.2 Proposition: ~l' G

and N are functors from PC into groups.

action of these functors is, to within an equivalence, dent of the base point. marion from

~

1

to G.

The indepen-

We can regard n as a natural transforFinally N(X)

is isomorphic to

=.(x ). r

- 61-

Proof:

Straightforward.

Remarks

G(X, x 0) cal space group,

is n o t only a group, b u t is also a t o p o l o g i -

(as a s u b s p a c e of X ).

it is a t o p o l o g i c a l

a l t h o u g h so far we can only s h o w that the "algebraic"

f u n c t i o n s of one v a r i a b l e xax-lb

for a, b fixed)

fact follows since -i

Perhaps

(such as the f u n c t i o n s e n d i n g x into

are c o n t i n u o u s

G(X, x 0)

for X a T 1 space.

This

is a q u o t i e n t of the loop s p a c e

(Xo). It can e a s i l y be s h o w n that G is a functor from PC to

the c a t e g o r y of groups w i t h t o p o l o g i e s .

If X is s e m i l o c a l l y c:

X

n:

~

> X

1-connected

(see

[9, p. 78])

then

is the u n i v e r s a l s i m p l y c o n n e c t e d c o v e r i n g ,

(X) > G(X) i d i s c r e t e topology.

is a n a t u r a l

i s o m o r p h i s m and G(X)

has the

Definition

X is s e m i l o c a l l y G - c o n n e c t e d if e a c h x e X has a n e i g h b o r h o o d U such that the n a t u r a l map

G(U)

> G(X)

is

trivial.

4.3

Theorem

Let X be c: X

> X

locally p a t h w i s e

be its c o r e f l e c t i o n .

c o n n e c t e d and T

and let 1 T h e n the f o l l o w i n g s t a t e m e n t s

- 62 -

are equivalent:

(a)

c-l(x) X

E

is discrete

(b)

X is s e m i l o c a l l y

(c)

o is a covering

Proof: x0 r X

(a)

set of X* such that

G-connected. of X (in w h i c h

connected "- (b) :

such that

pact-open

topology.

there

Thus,

KI'''''Kn

contained

such that

k E AM(Ki,

starting ~0(A)

M(K,

[0, 1]

G(U)

> G(X)

(b)-----> (c): neighborhood trivial. easily

Let

shown

of a given

~0(~)

=

c_ U}.

of x 0.

compact

sets

UI,.~

n

of X,

Let

U c_ AU i

Then any loop A, AM(Ki,

U i) .

Thus

It is now easy to

x~.

is trivial. Let U be an open, x c X

such that

{Vs} be the path

components

that each r e s t r i c t i o n

the deck translations,

>

where k is the constant

range in U, is in

and so

x 0)

is open in the com-

and open sets

neighborhood

and

~0: P(X,

are n o n - e m p t y

U) = {f E P(X)If(K)

at x 0 and w i t h

be given

Let

~0 -i (W)

Ui) c_ ~0-1(W),

connected

c W A c-l(x 0)

show that

in

of X).

Let W be an open sub-

W n p-l(x 0) = {x~}. 3, then

case c is the

x0 E X

c(x 0) = x 0.

be as in section

be a pathwise

covering

Let

(x * , x 0)

path on x 0 and

for all)

(and hence

Xo

universal

choose

for some

the Vs's

ciV

pathwise G(U)

of

> G(X)

c-l(u).

is one-one.

are canonically

connected is

It is In view of

homemor-

-

phic.

63

-

It can further be shown that each

map using

the fact that

cIUv

) u

(c is a q u o t i e n t map as % is, see Thus

clc-l(u)

clV ~

a universal

"-

(c)

I.

(a) :

connected

Then f is regular g:

valently,

(Y, y0 )

f:

(x , x 0)

G(X,

and

c% 0 = %o

)

x0).

f:

if for each >

Clearly,

Thus c is a covering.

covering

in view of 2.4 (d)

>

(Y, y0 )

(Y, y0 )

(X, x 0)

(X, x 0)

fg = f.

in I is regular

g

be in

there exists

such that

for w h i c h

is such that

>

y E f-i (x0)

(Y, y)

(Thus it follows

that on p. 74 of

It fol-

is in I.

Let

(Y, y0 )

f ~ I and by 2.4(d)) of

75],

Trivial.

Let X be T 1.

automorphism

g:

[9, p.

is a h o m e o m o r p h i s m .

as every other such covering

Definition:

is a q u o t i e n t map.

is also a q u o t i e n t map as U is open.)

lows that each It is clearly

clV~ is a q u o t i e n t

Equi-

iff the map

fg = c (which exists -i

(y0)

is a n o r m a l

that our d e f i n i t i o n

an

as

subgroup

agrees w i t h

[9].)

regularity

is i n d e p e n d e n t

of the base point.

4,4. Corollary

Let X be locally p a t h w i s e is a canonical

one-one

of X and open n o r m a l

Proof: Choose

correspondence

subgroups

group.

and T 1.

between

regular

Then there coverings

of G(X).

Let H be an open n o r m a l

any base point in X .

deck t r a n s l a t i o n

connected

Then G(X)

Consider

s u b g r o u p of G(X). can be r e g a r d e d

the q u o t i e n t

as the

space X / H o b -

-

r a i n e d from X

translation

64

-

by i d e n t i f y i n g x w i t h h(x) w h e n e v e r

h is

(regarded

as)

a m e m b e r o f H.

the d e c k

(The space

X /H

does n o t d e p e n d on our choice as H is normal.)

The map

f: X*/H

> X

i n d u c e d by c can be s h o w n

to be a c o v e r i n g by m e a n s of the a r g u m e n t s

g i v e n in 4.3.

4.5 C o r o l l a r y

Let X be l o c a l l y p a t h w i s e

c o n n e c t e d and T I.

is C h e v a l l e y s i m p l y c o n n e c t e d iff G(X)

Then X

has no p r o p e r open n o r -

mal subgroups.

Proof:

For

X, the c o v e r i n g s

locally pathwise

of X are in e f f e c t i v e o n e - o n e

w i t h the c o v e r i n g s m a r k below).

(connected and)

in the sense of Lubkin,

F r o m T h e o r e m 1 on p. 211 of

any n o n - t r i v i a l

[7],

connected

correspondence (see the re-

[7] it follows that

c o n n e c t e d c o v e r i n g of X gives rise to a n o n -

t r i v i a l r e g u l a r c o n n e c t e d covering.

Remark

In w h a t

follows, we shall r e f e r to the g e n e r a l i z e d

u n i f o r m spaces of L u b k i n s~ages ,.

The m a i n

(called spaces in

facts that w e shall use,

[7]) as L u b k i n from

e a c h t o p o l o g i c a l space is, in a n a t u r a l way, e v e r y L u b k i n space has an u n d e r l y i n g is

(connected and)

f: Y

> X

locally connected,

[7], are t h a t

a L u b k i n space and

topology.

Moreover,

if X

then e a c h c o v e r i n g

is such t h a t Y can be r e g a r d e d as a c o m p a t i b l e

-

65

-

Lubkin space and f as a Lubkin covering. kin covering

every Lub-

of such an X is of this type.

The universal

~ro-coverin~,

limit of all connected denote the underlying locally pathwise regular

Conversely

Lubkin coverings topology

connected,

coverings

of X.

of X.

Xf X is

1.4 on p. 215 of

simply

connected

We shall

[7].)

It follows

let

(connected

then X is the inverse

of X (in view of Theorem

ollary

and)

limit of all

1 on po 211 and Corthat X is Chevalley

iff X = X.

The relationship locally pathwise

X, of X, is the inverse

between

connected,

X

and X

(for connected,

X) is given by:

4.6 Theorem

Let X be and T 1.

Let

continuous

(connected

f: X

> X

function

and)

locally pathwise

be the underlying

connected

topology

associated with the universal

and

pro-c~vering

of X. Then there exists

3: X

The map ~ is the coreflection of all open normal subgroups tient space

such that

of X and G(X) of G(X).

f~ = c.

is the intersection

Moreover X is the quo-

X /G(X).

Proof: is clearly

> X

The existence

in I.

tion and that G(~)

of ~ follows

A direct proof shows

from 2o4(d)

as f

that ~ is the coreflec-

is the above intersection.

It can readily

- 66 -

be shown

that X is locally p a t h w i s e

tient map

connected

and so ~ is a quo-

(see 3.2).

Remark

Examples

5.6

(b), 5.7,

and 5.8 p e r t a i n

~o the r e l a ~ i o n -

ship b e t w e e n X and X*.

5.

EXAMPLES

5 9 1 Lemma

Let X be T I. =mq m: m

Q

-i (x)

of

where > X

q: P ( X )

> Q

is m o n o in PC.

is the largest

,-l(x).

is a q u o t i e n t map and Suppose

that for each x r X,

totally p a t h w i s e

Q = X

and

disconnected

quotien~

> Q

pothesis

and 3~

m = c.

By 3.3, we have that X

of X t h r o u g h w h i c h

f: X

such that

# factors.

mf = c.

we o b t a i n

a map

is the smallest Thus

"sub-

there exists

In view of the q u o t i e n t h: Q

It is easy to show t h a t h is the inverse a/%d

be factored as

.

Then

P rogf: object"

Let ~: P(X)------> X

> X* w i t h

by-

hq =

of f and hence,

0"

Q = X*

m = Co

5.2 Theorem:

Let G be a T O t o p o l o g i c a l for the n e i g h b o r h o o d s

group.

of the identity

Let {H } be a base

element

such that each H

-

is an o p e n s u b g r o u p . locally

pathwise

67

-

Then there exists

connected

T

space

a

(connected

X such that

and)

G(X)

is t o p o -

2 logically

and a l g e b r a i c a l l y

Proof: simply

Let

connected

e a c h ~ let Wl(Sm)

S

(S, s O )

pointed

c S

equivalent

T 2 space

be such t h a t

> ~I(S)

such that

S can be c o n s t r u c t e d ing some 2 - c e l l s ,

as in

[9, p.

a c h a n g e of the t o p o l o g y contain

connected,

such that

~l(S,

S

induces

> S )

a wedge

For

an i n j e c t i o n (Such an

of 1 - s p h e r e s

and a t t a c h -

143-148].)

attempt

Thi s

locally

s O ) = G.

m a p s o n t o Hm.

to c o n s t r u c t

on S - - r e q u i r i n g

some S .

the f a c t t h a t S, w h e n

a pathwise

~l(S

by taking

[A s t r a i g h t f o r w a r d

s o must

be

to G.

X would

involve

that a neighborhood

approach would work except

retopologized,

might

acquire

of

for

some new

loops.]

Let R be the r e a l of X as S • R w i t h

{s

line.

} • R

We define

identified

the u n d e r l y i n g

to a s i n g l e p o i n t

set cal-

0 led x0~

Let

e: S • R

logize

X so t h a t

product

topology

exist c

e

-i

> X

U c X and,

is o p e n

for

~ and a c o u n t a b l e

be the c a n o n i c a l iff

e-l(u)

map.

is o p e n

x 0 e U, w e a l s o r e q u i r e

subset

C c R

such

We topo-

that

in t h e

that there S

x

(R\C)

(u).

We

let

r: X

r is n o t c o n t i n u o u s of s e q u e n c e s .

Thus,

> S

be t h e o b v i o u s

at x 0 b u t r d o e s p r e s e r v e if

f:

[0, I]

> X

projection

function.

the convergence

is c o n t i n u o u s

, so

-

is rf:

[0, i]

> S.

and let

v: ~

equivalence ~I(X,

x O)

on G.

~ -i

-i

Hence

-~ ~l(X,

(x O)

x O)

following

k ~ DM(Ki,

-i

3)

topology

on

to the given topology

an S

implies

compact

UI,...,U n

(U), where

there exists

c N U., w h i c h

= G.

loop into its

of the identity

of the identity

Then there exist

U i) ~ v

paths

4.2).

and open subsets

Clearly

take each

to look at n e i g h b o r h o o d s

Let U be a n e i g h b o r h o o d

[0, i]

= -l(S)

We claim that the q u o t i e n t

It suffices

topology.

are h o m o t o p i c

~l(X)

induced by v, is h o m e o m o r p h i c

(see remark

tient

if f, g

be the loop space of X (as in section

(x) 0

class.

-

Similarly,

on X, then so are rf and rg. Let

68

H

sets

c U.

KI,...,K n

of X such that

k is the constant

and an

in the quo-

t E R

Thus

with

loop at x 0. e(Ss

• {t})

U is a n e i g h b o r h o o d

in the

given topology. It remains tient t o p o l o g y . find compact that

Let H

sets

h ~ NN(Ki,

to show that each H

and

KI'''''Kn U i)

union,

UIi,

nected neighborhood

be given.

and open sets

[0, i] \ h-l(x 0)

of open intervals. W of s

for all but finitely such that

)

We m u s t

UI,,..,U n

such

~ v-I(H~).

Let the open set disjoint

h r v-I(H

is open in the quo-

rh(xn)~ W,

many

be w r i t t e n

Choose

as the

any simply

con-

(in the space S). Then rh(li) c W 0 I i. [If not, let {Xn } be a sequence

for all n, and each I i contains

at most

- 69-

o n e x n.

Then h separates

Let obvious W.

I l, I 2 , . . . , I m

increasing

Let a

Let

9 .. H s a n H 8 ~ H e.

al

xalxa2x...xa ~2 "" " -an

identity hood cW.

order)

be the

n

= v(h)

intervals

for w h i c h

ai = v ( a i ) "

rh(i

sets

r He ,

hence

sends

of He.] of

identifies m q = ~. PC.

... U I m

mf = mg need

connected,

N M ( K i, U i)

In v i e w of 5.1,

and

follows

f, g:

for

(in a d d i t i o n

We shall show or f ~ g

Im c K 1 U

Let

for e a c h

in a n e i g h b o r h o o d

of t.

... K n,

neighborhood

> x

To prove on an o p e n

be such that

Let

in

f = g whenever this

claim, w e

and c l o s e d

at the b a s e

t e [0, I]

of h.

map which

to s h o w t h a t m is m o n o

f r o m the c l a i m t h a t

to a g r e e i n g

and open

2 ~ i ~ n, and t h a t

m': Q

~ Q.

r h ( K I)

r ( U I) ~ W

K2,...,K n

b e the q u o t i e n t

o n l y s h o w t h a t f and g a g r e e

[0, i]

... U

it s u f f i c e s

[0, i]

such that

is the d e s i r e d

> Q

paths.

into of the

of x 0 s u c h t h a t

intervals

I 1 U 12 U

q: P(X)

the i d e n t i t y

x in-

L e t K 1 be a c o m p a c t n e i g h b o r -

compact

so t h a t

homotopic

This e a s i l y

~8~IHs~2H 8

some neighborhood

I 1 U 12 U

Choose

Then

let

so t h a t

of

for

sends

is c o n t i n u o u s ,

t h a t e a c h U i is s i m p l y

We

hli. 1

x

= H 8.

h ( K ) c U.. i -- i

as

Choose H 8

in the

is n o t a s u b s e t

since the map sending

of the c o m p l e m e n t

U2,...,U n

)

(arranged

[This is p o s s i b l e

into a subset

~l(Ul)

1

defined

L e t U 1 b e an o p e n n e i g h b o r h o o d

and

of

f r o m its c l u s t e r p ~ / % t s . ]

b e the l o o p e f f e c t i v e l y

1

i = 1,...,m,

to

~x n}

subset

point).

that either

x = mr(t)

f = g

= mg(t).

We

- 70-

first

assume

that

neighborhood of

m-l(u)

over,

the

by means m-l(u) f ~ g

of x.

regarded

subsets of the

are of

U x H

above

is,

no

U ~

the

form

set

can be

be

shown

and

If

connected outside

shown Thus,

U • {g}

(mf)-l(u).

sequence

X

any

that

in a n a t u r a l

argument.

if U is a s i m p l y

(that

Let

It is r e a d i l y

can b e

in the

works

x ~ x 0.

as

U • G.

to b e o p e n the p a t h

hence

connected

the u n d e r l y i n g

way

in

x = x 0, the

set

Morem

-i

(U)

components

either

sequential

of

simply

f = g

same

or

argument

neighborhood

U can c o n v e r g e

of

of x 0

to x0).

5.3 Corollar~

Let Let

{H a } be

~ be

a topological

a base

for

t h a t N is n o r m a l . )

logy.

Then

there

nected

T 2 space

Proof: place

5~4

of G.

The

exists

X with

subgroup. Let

Let

~l(X)

have

X as in t h e

arguments

then

.

the

locally

= w, N(X)

T0).

of the i d e n t i t y N = NH

G = ~/N

a connected,

Construct same

(not n e c e s s a r i l y

the n e i g h b o r h o o d s

t h a t e a c h H a is an o p e n lows

group

= N,

above

such

(It t h e n quotient

pathwise and

G(X)

proof

fol-

topo-

con= G.

using

~ in

subgroup,

then

apply.

Corollary

If

~ is a n y

there

exists

Wl(X)

= w, N(X)

logy.

group

a semilocally = N

and

and N is any n o r m a l G-connected G(X)

= ~/N,

T 2 space with

X such

the d i s c r e t e

that topo-

- 71-

Thus there exist u n i v e r s a l c: X

> X

equivalent

with

=I(X)

connected

any p r e a s s i g n e d

to any p r e a s s i g n e d n o r m a l

covering

group and

spaces ~l(X )

subgroup.

5.5 C o r o l l a r y

Let ~ be any group and let {Hs} be any c o l l e c t i o n subgroups

that

of

such that: (1)

For all ~, 8, there

is a I w i t h

H A c_ H

(2)

For each u and each g z w there exists

N H 8. 8 such

g-lHsg c_ H .

Then there exists space

X such that

with

{H /N}

a connected, ~l(X)

locally p a t h w i s e

= w, N(X)

= N = NH

a base the n e i g h b o r h o o d s

connected

and

G(X)

T2

= ~/N

of the identity.

5.6 E x a m p le

(a) connected

There exists

(in fact (b)

(hence

connected

a Chevalley

simply

connected

T 2 space X for w h i c h G(X)

and

lo-

is n o n - t r i -

X ~ Cn*).

Proof:

(a)

Follows

from 5.4 w i t h

(b)

Let Z+ be the p o s i t i v e

be the group of all p e r m u t a t i o n s H n be the s u b g r o u p i ~ n.

such that X is n o t simply

can be p r e a s s i g n e d ) .

There exists

cally p a t h w i s e vial

~l(X)

X r Cn

of all

We claim that

f r G

(i) and

of Z+.

~ = N. integers.

For each

such that

f(i)

Let G

n ~ Z +, = i

let

for all

(2) of 5.5 are s a t i s f i e d by

{~n }.

- 72 -

Clearly

C h o o s e m so t h a t Thus there that

H

let

Let

c H n --

s r G

{0~}

minimal many

X with

(b).

for s o m e n. then

Note

be arbitrary.

then

s -- fg

of t h e o r b i t s

"other half" 9

where

implying

Finally Then

ite o r b i t

Let

b e given 9

for

i @ j9

v = sl(u),

sn+l(u) Clearly elements

5.7

U.

and

so

Define

s 2 ( u ) , . .9,

g(z)

g fixes

u r U

- s(z)

if

g: Z + s n (u).

open subgroups subgroup

fixes

to show

any

f e H.

s r H.)

(an o r b i t b e i n g

If s h a s

a

infinitely

to the i d e n t i t y

to the i d e n t i t y

on

on t h e

many elements

and so

a s s u m e t h a t s has o n l y a at l e a s t o n e i n f i n -

It f o l l o w s > Z+ Further

t h a t si(u)

so t h a t define

for all other values.

Thus

and such

= G

f r G

of s

s must have

at l e a s t n e l e m e n t s

f, g r H.

normal

fix infinitely

s r H9

g - i H m g c Hn.

of the i d e n t i t y .

f restricts

and g r e s t r i c t s

f and g thus

b e given.

for s o m e g a n d so

= 0~) 9

of o r b i t s 9

for

= G(X)

that

of o r b i t s

finite number

s j (u)

Then

(It s u f f i c e s

s(0~)

n > 0

t h a t H is a n o r m a l

f r g-lHng

s e t 0a s u c h t h a t

f, g c H

and

i < n.

~l(X)

Assume

b e the c o l l e c t i o n

orbits,

"half"

g r G

for a l l

g(i)

implies

set of n e l e m e n t s , Now

let

claim that G has no proper

4.5,

and t h a t

>

(2)

is a b a s e of o p e n n e i g h b o r h o o d s

We by

m

for

is a s u i t a b l e

{H n }

which

As

(i) holds.

and f fixes

g(v) g(u) Let

-- v =

f = sg -I.

infinitely

many

s = fg r H.

Example

(a)

There exists

a locally pathwise

connected

T 2 space

-

73

-

X s u c h t h a t ~ is n o t C h e v a l l e y s i m p l y connected.

x

(Equivalently,

x.) (b)

The above space X has a u n i v e r s a l

c o n n e c t e d cover-

ing w h i c h is n o t in Cn.

Proof: permutations i n t e g e r s n. la(n) I = h:

Z

Inl > Z

For this e x a m p l e ,

g: Z

s u c h that

for all n. such t h a t

the identity.

Let H h(i)

{A N H } n

n

= -g(n) a: Z

for all > Z

for

is a b a s e

i = i, 2,...,n.

Topo-

for the n e i g h b o r h o o d s

It is r e a d i l y v e r i f i e d t h a t there e x i s t s G(X)

with

be the s u b g r o u p of all

= i

of

a suit-

= G.

It is also e a s i l y s h o w n t h o s e in 5.6)

g(-n)

Let A be the s u b g r o u p of all

logize G so that

able X w i t h

> Z

let G be the g r o u p of all

(using a r g u m e n t s s i m i l a r to

t h a t A is the s m a l l e s t o p e n n o r m a l s u b g r o u p of G.

Hence, by 4.6, w e have

G(X)

o p e n n o r m a l subgroups.

Thus ~ is n o t C h e v a l l e y s i m p l y c o n n e c t e d

(in fact X = X ). in v i e w of 4.4.

= A.

B u t A is a b e l i a n and has m a n y

F i n a l l y the m a p

f: X

- X

is a c o v e r i n g

By c o n s t r u c t i o n of ~ this is the u n i v e r s a l

con-

n e c t e d covering.

Remarks

(i)

In the above e x a m p l e ,

Lubkin universal pro-covering sal L u b k i n c o v e r i n g of X.

X

it is e a s i l y s h o w n that the ~ X

is a c t u a l l y the u n i v e r -

Since Lubkin coverings

L u b k i n space X has no n o n - t r i v i a l

compose,

c o n n e c t e d coverings.

the

On the

-

74

-

other hand, we have seen that ~ has many n o n - t r i v i a l coverings.

Thus

~ ~ X

w h i c h means

connected

X cannot be a t o p o l o g i c a l

space. (2)

It should be m e n t i o n e d

covering

is not always

pathwise

connected.

but X

> X not one-one

5.8 Exam

le

Let in

lows from

and in

(see e x a m p l e s

[6].

~, is simply

[7, p. 232],

0)}.

X with

~ (X) = 0, 1 4, 7, and 22 of [7]).

This

space is d i s c u s s e d

It can be shown that the u n i v e r s a l connected

and hence,

that the group G(X)

~ = X .

It fol-

is the c o m p l e t i o n

the free group w i t h

c o u n t a b l y many g e n e r a t o r s

containing

finitely many generators.

all, but

pro-

of X* if X is not locally

There exist spaces

X = R • R\ { (2 -n,

[7, p. 232]

procovering,

a quotient

that the u n i v e r s a l

of

and open subgroups As a group G(X)

= ~l(X), but there may be o c c a s i o n s w h e n the above d e s c r i p t i o n of G(X) of

is e a s i e r

~l(X)

to w o r k w i t h than the a l g e b r a i c

as a free group w i t h u n c o u n t a b l y

description

many generators.

-

75

-

REFERENCES

[i]

Chevalley, C., Theory of Lie groups, I, Princeton University Press, Princeton,

[21

(1946).

Freyd, Po, Abelian categories, Harper and Row, New York, (1964).

[3]

Hu, S. T., Homotopy theory, Academic Press, New York, (1959).

[4]

Isbell, J. R., "Some remarks concerning categories and subspaces", Canad. J. Math., 9; pp. 563-577,

[5]

(1957).

Kennison, J. F., "A note on reflection maps", Illinois J.

Math., ii; pp. 404-409, (1967). [6]

, "Full reflective subcategories and generalized covering spaces", Illinois J. Math.,

[7]

Lubkin, S., "Theory of covering spaces", Trans. Amer. Math.

Soc., 104; pp. 205-238,

[8]

(1962).

MacLane, S., "Categorical algebra", Bull. Amer. Math. Soc. 71; pp. 40-106,

[9]

(to appear) .

(1965).

Spanier, E. H., Algebraic topology, McGraw-Hill, New York, (1966).

-

DEDUCTIVE

76-

SYSTEMS AND CATEGORIES *

II. STANDARD

CONSTRUCTIONS

AND CLOSED

CATEGORIES

by J o a c h i m Lambek**

0. INTRODUCTION We wish

to explore

(1) p r e - o r d e r e d (2) deductive (3) categories By a p r e - o r d e r e d reflexive,

in mind by

(i)

with

structure.

anti-symmetric,

to give a rigorous

"structure"

illustrate

A semilattice

(A,T,A),

structure,

systems,

but not necessarily

Instead we shall

triple

sets with

between

set we mean a set with a transitive

It w o u l d be premature we have

the c o n n e c t i o n

where

~

and

relation

definition

"deductive

and ~.

of w h a t

system".

with an example.

(with largest

element)

is a p r e - o r d e r e d

set,

is a T

an

Familiarity with Deductive Systems and Categories (I) is not presumed. Thanks are due to the Battelle Institute for inviting me to give these lectures and to S. E i l e n b e r g for his s t i m u l a t i n g comments. Mathematics

Dept.,

McGill University,

Montreal

- 77

element

of

A,

A

and

-

a binary

operation

B

C~A

on

such

A

that

A ~ T, C~AA

For

(2) system

D(X)

Its

as

terms

all

finally

(It is

understood

A

>

B

are

listed

i:

A

4:

C C

A

axioms

5 6

formula

there

elements B

are

X

of

terms,

are

expressions

all are

is

so

else

B

is

a deductive

follows:

(or a x i o m

2:

A

~,

also

is

AAB.

is

T

a term.)

of

the

form

terms.

schemes)

5:

> B B A ----~ C C

'

are

is

and

rules

of

inference

; A A B C -----> B

written

>

C

6:

,

3: C

)

'

in tree-form.

A

C

For

> A >

C A

example,

a proof: 1

> A A B >B A

Any

X

set

follows:

> A A B > A

AAB A A B

& C~B.

nothing

and

,

following

and

that

Proofs the

A

formulas

as

- A

are

if

where

Its

discrete

described

a term;

Its

any

~

appearing

A B

as

1 AAB----->AAB AAB >A

4 >

B

a proof,

A A

in p a r t i c u l a r

the

last

T

, )

A B

B

- 78 -

formula,

is c a l l e d p r o v a b l e

the a b o v e t h a t A A B

> A,

b u t o n l y the

and

9 A A B, A A B

A A B

> B A A

A cartesian

(A,T,A,~,8),

where

A: A • A

natural

A

> A

HomA(A,B).

Thus

is the u s u a l

also

the

1

when

9

[A,T]

are

• [C,B]

-

,

[C,A A B]

T

terminal

object

is "the"

.

is s h o r t A

of

for

and

A

product.

see l a t e r t h a t the d e d u c t i v e

but

8

system

n o t o n l y the free s e m i l a t t i c e , category

generated

interested

by

X.

in the t h e o r e m s

in the s e c o n d

D(X) but

In the of the

case t h e p r o o f s w i l l

play

role.

More D(X)

and

and

[A,B]

f i r s t c a s e w e are o n l y

an i m p o r t a n t

is an o b j e c t of

and

to c o n s t r u c t

system,

T

set,

free c a r t e s i a n

deductive

is a q u i n t u p l e

one-element

categorical

We s h a l l c a n be u s e d

are all theorems,

as f o l l o w s :

8 (A,B,C) : [C,A]

is "the"

category

is a b i f u n c t o r ,

isomorphisms

1

> B,

is a c a t e g o r y ,

~(A):

Here

Thus we see from

f i r s t is an axiom.

(3)

A,

A A B

or a theorem.

X

generally

one c a n f o r m a d e d u c t i v e

is a p r e - o r d e r e d

0: X

set by adjoining

> Y

system the a x i o m

- 79

whenever

X ~ Y

in

adjoining

a f a m i l y of a x i o m s

-

Df: X one

for e a c h m a p

it is i m p o r t a n t when

f: X

f @ g, e v e n t h o u g h

developed have

ignored;

equivalent?

Df

namely,

D

they assert

when

For technical as d i s t i n c t

g

the s a m e

on c o n c e p t s

are two p r o o f s

reasons axioms

formula.

and techniques

one q u e s t i o n

consider

by

,

X.

and

However

For example,

4 A A B A A B

is

lean heavily

by l o g i c i a n s .

is a c a t e g o r y

> Y

> Y

to r e g a r d

We s h a l l

X

X; a l s o w h e n

t h e y s e e m to

of the s a m e

formula

the p r o o f :

1 ) A A B > A

5 A A B -~-~ A A B A A B > B

6 A A B

Nobody would some

sense,

d e n y t h a t this p r o o f m i g h t b e r e g a r d e d equivalent

to the s h o r t e r 1

AAB

On the o t h e r hand, following

> A A B

proofs

there

A

A A

>AAB.

is a g o o d c a s e

AAA

In fact,

they

for c o n s i d e r i n g

as i n e q u i v a l e n t : 1 > A AA ~A

'

admit different

in

proof:

1 4

as,

5 A AA A~A

.> A A A >A

"generalizations".

the

-80-

Our p r o g r a m w i l l be to look at two kinds of structured

categories

constructions to discuss

in some detail:

standard

and c l o s e d categories.

also c a r t e s i a n

but time did not p e r m i t

It had been p l a n n e d

closed categories

inclusion

in this volume.

ment of this topic and also the d e v e l o p m e n t theory of s t r u c t u r e d

categories

and combinators, A treat-

of a general

must be deferred.

i. S T A N D A R D C O N S T R U C T I O N S

By a closure (A,T)

where

satisfying

A

s y s t e m we shall u n d e r s t a n d

is a p r e - o r d e r e d

the f o l l o w i n g

rules:

T(T(A)) A ~ B

write

T2(A)

for

A

and

it is useful

A.

etc.,

set

,

~ T(B)

It will be c o n v e n i e n t also

for c o n s t r u c t i n g

X.

of the form

Its terms T(A)

course,

instead of d e s c r i b i n g

we can,

in this case,

expressions

of

to look at a d e d u c t i v e

by a p r e - o r d e r e d expressions

B

,

~ T(A)

> T(A)

T(T(A)),

As a device

when

T0(A)

system

D(~)

are elements A

for

free closure

A.

systems

generated of

is a term.

X

and

Of

the set of terms recursively,

list all terms e x p l i c i t e l y

of the form

T

set with an o p e r a t i o n

A ~ T (A)

for all elements

a pair

Tn(x),

where

X

as b e i n g

is an e l e m e n t

of

to

- 81 -

X

and

n

is a n o n - n e g a t i v e

are e x p r e s s i o n s terms.

The formulas

A

where

> B

They also may be listed e x p l i c i t e l y

expressions present

of the form

integer.

of the form

the axioms

Tm(X)

A

of

and

D(X)

are

B

as b e i n g It remains

> Tn(y).

to

and rules of inference: 1A

A Q

A

> A

~ B

, P

B

A P~

> C

C

X ~

Y , whenever T2(A)

A h(A)_. T(A)

m(A[-

X ~ Y

T(A)

in

X,

,

P

A T(A)

T(P)_

The last rule of inference, as a rule for g e n e r a t i n g a given proof

P: A

) B T(B)

for example,

a proof

T(P):

m u s t be i n t e r p r e t e d T(A)

> T(B)

> B.

An example

of a proof

in

D(X)

is the following:

1 A

from

A>

T (A) T 2 (A)

A > T (A) - T 2 (A)

T 2(A) m(A)) 9 T (A)

denoted unambiguously

by

m(A)T2(IA ) .

T(A)

,

-

We now c o n s t r u c t

A

> B

as in

D(X),

is p r o v a b l e

D(X) ~

system.

-

the free closure

g e n e r a t e d by the p r e - o r d e r e d the terms of

89

set

A ~ B

in

in

D(X).

T(A)

seen that

means

F(X)

that F(X)

is a closure

F(X)

In w h a t sense is it free?

sets and order p r e s e r v i n g the c a t e g o r y of closure

maps.

mappings.

systems

mappings which preserve

T

Thus a strict map

A

> B

H:

A

of

(~,T)

A.

first g e n e r a l i z i n g forgetful

a left adjoint universal

(B,T)

is a map

It is a sore point of our theory

the f o l l o w i n g

functor

F

of

U

U: Clo

system

> U(F(~))

strict map

"~

and

statements

of "adjoint". > Ord.

There

k",

without is an

As is well known,

is d e t e r m i n e d by a S o l u t i o n to the Given any p r e - o r d e r e d

F(~)

so that for every closure

H': F(X)

set

and an order p r e s e r v i n g

and every order p r e s e r v i n g m a p p i n g a unique

>

= T(H(A))

the concept

m a p p i n g problem:

find a closure 6(~): ~

let us call them strict

in the above cannot be r e p l a c e d by

o t h e r w i s e we could not make

obvious

Clo we shall mean

in Ord such that

for all elements "="

By

of p r e - o r d e r e d

and those order p r e s e r v i n g

exactly,

H(T(A))

that

are

is the same in

By Ord we shall mean the category

H:

F(X)

Its elements

X.

and

It is easily

system

> A

H: ~

> ~

for w h i c h

X, mapping

system

(A,T)

there exists

- 83 -

U(H')

6(X) = H. UF (X)

F (X) k H'

(x)

\4

x

H

It r e m a i n s (X) (X) = X (X) X

order.

X ~ Y

Now, H:

X

H':

> A >

H' (T(A))

in f a c t

the

to

verified

list

formulas or

m = n

closure with or

or

system

the p r e - o r d e r n > i,

and

that

that

and verify

that

in

X,

hence

is any

then

provable

map

and,

by

where F(~)

elements

relation

are

out that

X ~ Y

Clo,

= H.

it is in f a c t

are p r e c i s e l y in

X

and

whenever successor

the

m = 0

to t h e

the n o n - n e g a t i v e

is the

= Tn(H(X)).

in

is i s o m o r p h i c

m ~ n T

map

U(H') ~(X)

They

and

and

H' (Tn(X))

is a s t r i c t

D(~).

system

Define

H' (X) = X

be p o i n t e d

Thus

for w h i c h

by

for w h i c h

of

closure

mapping.

at o n c e H'

> Tn(y)

whose

stipulate

X ~ Y

(~,T)

or all

theorems

n > i.

X

D(X) ,

preserving

strict

Tm(x)

if

in

that

it s h o u l d

all

of

recursively

(A,T)

only

X

We

F(X).

assume

Here easy

DX,Y

in

= T(H' (A)),

It is e a s i l y

~(~).

Indeed,

any order

F(X)

(~,T)

element

is an a x i o m

therefore,

A

to d e f i n e

for e a c h

preserves > Y

-

m = 0

integers, or

function.

m = n

-

We s h a l l see later, constructions, X ~ Y

in

that

F(X)

6(X)

84

-

in the c o n t e x t of s t a n d a r d

is full in the sense t h a t

only when

X ~ Y

in

X.

It w i l l also

f o l l o w f r o m w h a t w i l l be said later that Clo is e q u a t i o n a l (tripleable)

where

A

n: 1 A

o v e r Ord.

A standard construction

is a q u a d r u p l e

is a c a t e g o r y ,

- A

> T

satisfying

and

T: A

~: T 2

u. (nT) = 1

T

> T

is a functor,

say f r o m H: A

(A,T,n,~) > B

to

= ~" (Tn)

and

u. (uT) = u. (Tu).

and w h o s e maps, (B,T,n,~),

which preserve

and

are n a t u r a l t r a n s f o r m a t i o n s

Trip w e s h a l l m e a n the c a t e g o r y w h o s e o b j e c t s standard constructions

(A,T,n,~)

By

are small

c a l l e d s t r i c t maps,

are those f u n c t o r s

the s t r u c t u r e e x a c t l y ,

in the

sense that

HT = TH,

Hn = nH, Hu = uH .

W h a t has b e e n said in a p o l o g y for u s i n g s t r i c t m a p s

in

Clo also applies here.

We w i s h to c o n s t r u c t a left a d j o i n t to the o b v i o u s forgetful

functor

U: T r i p

s l i g h t l y m o d i f y the d e d u c t i v e

~ Cat.

To this p u r p o s e we

system

D(X)

m a y be any small category,

u s e d earlier.

Now

X

set,

and we m u s t s p e a k of the o b j e c t s r a t h e r t h a n of the

e l e m e n t s of

X.

not just a p r e - o r d e r e d

-

A l s o the a x i o m

D

Df:

that

Df ~ D

f: X

together with

objects

of

F (X)

are e q u i v a l e n c e certain

classes

~ B.

n, u, a n d

i

A

=

6 (X) : X

of p r o o f s

relation

~

We w r i t e

[P]

are d e f i n e d

[1A] , [P] [Q] = n(A)

=

reflexive

relation

satisfies

the f o l l o w i n g

(1)

D(X),

between

composition

o n l y by v i r t u e

thus:

=

[m(A) ] ,

~(A)

6 (X) (f) =

[Of].

the e q u i v a l e n c e

transitive,

of t h e s e

to a

of m a p s ,

[h(A)],

R{S ~ QS

F (X)

p r o o f s , to be

of

relation

symmetric, D(X)

rules

' rules

and

which

rules:

PR

of

according

[T(P)],

The s u b s t i t u t i o n

The

for t h e s e t of all p r o o f s

on the s e t of p r o o f s

P~Q

because

The maps

=

to s p e c i f y

We t a k e it to be the s m a l l e s t

> U (F (X)) .

D(X). of

construction

[pQ] , T([P])

(X) (X) = X,

It r e m a i n s

It is u n d e r s t o o d

t h e free s t a n d a r d

Identity maps,

6 (X)

,

X.

are the t e r m s of

defined presently.

T,

in

a functor

equivalence

Q ~ P: A

) Y

of a x i o m s

f ~ g.

We n o w c o n s t r u c t F (X)

X

) Y

whenever

g

-

is r e p l a c e d b y a f a m i l y

X,Y

one for e a c h m a p

85

P~Q T(P) ~ T(Q) are t h e a b o v e

- 86 -

definitions

of

(2)

[P] [Q]

and

Rules w h i c h

T ( [P] )

justified.

a l l o w us to c o n c l u d e

that

UF(X)

is a c a t e g o r y :

IBP The

l a s t rule,

following

_-- P,

P1 A

for e x a m p l e ,

A

(3)

R)

B

.

the e q u i v a l e n c e

(4)

C P'- A B

n

Rules w h i c h

(6)

assure

A

that

, T(PQ)

assure h(B)P

Rules which and

D

of the

B

>

C

T

that

n

, T(P)m(A)

assure

Q

> C P > D

C

.~ D

is a functor:

~ T(P)T(Q) are n a t u r a l :

and

~ m(B)T2(p)

the c o m m u t a t i v i t y

condi-

~:

m(A)h(T(A))

~ IT(A)

m(A)T(m(A))

~ m(A)m(T(A))

Rules w h i c h

assure

1x

DIx

PROPOSITION mapping

> B

A

~ IT(A)

T (P)h (A)

(5)

)

R

A

> D

Rules which T(I A)

universal

asserts

C

A

for

(PQ)R = P(QR)

two proofs:

B Q;

tions

- P,

, Dfg

I.

problem

that

~ DfDg

(F,6) for

~ m(A)T(h(A))

,

9 ~ (X)

is a functor:

9

is a s o l u t i o n

U; that

is,

of the

for e a c h

small

- 87 -

category and each map

X,

each

functor

H': F(X)

small standard H: X

>

Proof.

there exists

> A,

(A,T,n,u)

Given

constructions

such t h a t

H: ~

> A,

(A,T,n,~),

a unique

U(H')

~(X)

strict

= H.

we c o n s t r u c t

H'

as

follows:

H'(X)

Before

defining

P

D(X)

in

H' (T(A))

= H(X),

H'([P]),

recursively

= T(H' (A))

we d e f i n e

for e a c h p r o o f

H' (P)

thus:

H' (1A) = IH, (A) ' H' (PQ) = H' (P) H' (Q), H' (T(P)) H' (h (A)) = n (H' (A)) , H' ( m ( A ) )

It is e a s i l y

verified

sequently,

on the

w e are

length

H' (P) = H,(Q)

of the p r o o f

justified

hence

H'

one q u i c k l y

=

verifies

H' (P)

H(f)

,

of e q u i v a l e n c e .

.

that

H' ( T C A ) )

=

TCH' CA)),

H' ( T ( [ P ] )

H' (n(A))

= n(H' (A)),

H' (u(A))

is i n d e e d

=

in d e f i n i n g

H' ([P])

Moreover,

~ (H' (A)) , H' (Df)

that

P - Q---->

by i n d u c t i o n

=

= T(H' (P)),

a strict map

=

T(H' ([P]),

= u(H' (A)),

in Trip.

Con-

- 88 -

Finally

U(H')

6(X)(X)=

and these equations

REMARK a functor F

becomes

6:1

and

into a natural

in

Cat.

[F(X),

However

categories

of

In the p r e s e n t

the i s o m o r p h i s m

~(X)(f)=

As is w e l l known,

the left a d j o i n t

9 UF.

U(H')

force the d e f i n i t i o n

1.

~

H(X),

F

U

can be made into such that

with adjunction

situation

(A,T,n,u)]

,

H'.

transformation

~

this is not typical

and depends

of

H(f)

one can e a s i l y e x t e n d

[X,A]

in

Ens

to one

for s t r u c t u r e d

on the fact that

T

is a c o v a r i a n t

functor. REMARK that Beck's

Trip

2.

It may easily be shown at this

is e q u a t i o n a l

conditions.

this example structured

(tripleable)

However

to be typical,

Proof. we c l a i m that

the v e r i f i c a t i o n

is too easy in

and we shall defer

(a)

P ~ Df

2.

6(~)

it to another

is full and faithful.

Given a proof for some map

A c t u a l l y we shall show more generally > Y

length of

over Cat by v e r i f y i n g

category.

PROPOSITION

P: A

stage

has this form. P.

P. X f: X

> Y 9 Y

in in

D(X), X. I

that any proof

We p r o c e e d by i n d u c t i o n

on the

- 89

If

P = 1y,

then surely

there is n o t h i n g to prove. all ruled out. R: A

> B

we have

Q ~ Df

Therefore

P = T(A),

Q: B

> Y.

for some

assumption, P ~ DfDg

R ~ D

(b)

0

Suppose

) Y.

a proof

all o c c u r r e n c e s

of

and

i,

> Z.

later.

2

that

is the

and one n o n - i d e n t i t y

The @ e n e r a l i t y

P'

H ( 1 ) , and

in

H(

Let

H: ~

H(0)

and 9

of a proof

H: 2

D(2)

> X

such that P 9

results in

proofs have

for the moment, 9 ~

= X, H(1)

P'

P

in

for w h i c h P = H(P'). from

P'

if

are r e p l a c e d

respectively.

Equivalent

A s s u m i n g this (b).

g: X

and a lemma.

0

0, l,

L E M M A 1.

with

again by

we w i s h to prove

By the last e q u a t i o n we mean that

H(0),

are

assumption

Hence,

(a) will be given

is the set of all functors

there exists

by

m(A)

where

for some

g

P = Dr,

i.

DEFINITION. D(X)

or

By i n d u c t i o n a l

Df ~ Dg,

We need a d e f i n i t i o n

~

h(A),

If

~ Dfg.

c a t e g o r y w i t h two objects map

.

P = QR,

f: Z

An easier proof of

f = g.

P s Diy

There only remains

and

inductional

-

the same generality.

we shall continue

be d e f i n e d by

= Y, H(

>) = f .

- 90-

Then

>)

H (D

a map

x: 0

X

>

=

= Df. > 1

in

hence

,

Proof

Therefore, 2

--

f = g.

sketched.

by the lemma,

such that

H(D x) = D g .

It remains

to prove the lemma.

equivalence

Clearly,

We p r o c e e d by i n d u c t i o n

length of the proof that the given proofs For example,

there exists

on the

are equivalent.

assume that the last step in the proof of was a s u b s t i t u t i o n

P

rule,

say

_= Q

T (P) - T (Q)

By i n d u c t i o n a l Let in

H: 2 D(2).

assumption,

9 X Then

and P

H(P*)

= P.

Therefore,

H(Q')

= Q,

hence

= P, where

P*: T(A')

there exists

Q': A'

= T(H(Q'))

, B' >

B'

9 T(A)

For example,

the proof

T(A)

9 T2(A)

and with

h(T(A))

of the

T 2 (A)

to the axiom

seen to have

9 T(B')

= T(Q).

to point out that the converse

T 2 (A) ~

it is easily

have the same generality.

P': A'

lemma does not hold.

is not e q u i v a l e n t

Q

where

H(T(Q'))

We h a s t e n

T 2 ( A ) m(A)

and

= T(P'),

H(P')

above

P

T2(A)

> T2(A),

the same generality.

although

Indeed,

supposing

h(T(A))

m(A)

-- IT2(A)

.....

(7)

-

91-

we should have

n (T(A))

in

F(X).

not only

But then in

category

it w o u l d

F(X),

To see t h a t t h i s

but

free g r o u p g e n e r a t e d

deductive

For

sets,

T(A)

construction

idempotent

We s h a l l

see

l e m m a does h o l d

for the d e d u c t i v e

are known.

for the m e r e

existence

is a t h e o r e m

if and o n l y

and

m = 0

or

m = n

F i r s t we deductive

system

shall G(X)

the s a m e

but one relation

of the

a decision

terms A

A > B

this p r e s u m e s Note

if

constructions.

Given

of p r o o f s

and

is easy:

procedure B,

at l e a s t

that all maps

that a decision

if t h e r e e x i s t s

or

we have

t h a t the c o n v e r s e

of the f o r m u l a

f: X

X

set of t h e

idempotent

b y the e q u i v a l e n c e

D(X):

Of c o u r s e

in

A = the

constructions,

in o b t a i n i n g

up to e q u i v a l e n c e ? > Y

is c a l l e d

for i d e m p o t e n t

system

f i n d all p r o o f s

~ T(A)

construction.

= the underlying

constructions

later

We are i n t e r e s t e d

can w e

standard

T2(A)

A, etc.

rule to be s a t i s f i e d

on p r o o f s .

follow that

small

s y s t e m as for s t a n d a r d

additional

above

easily

in e v e r y

by

A standard .

= IT2(A)

is n o t so, t a k e for e x a m p l e

of c o u n t a b l e

nT.p = 1T2

u(A)

Tm(x) a map

procedure > Tn(Y)

f: X

> Y

n > i.

replace with

D(X)

the s a m e

by a Gentzen-type terms

and formulas.

- 92 -

However

has the following

G (X)

axioms

a n d r u l e s of

inference:

Df: X

when

P

A

" B r 6P) > T (B)

A

These

> Y

are d e r i v e d

rules

in

f: X

,

9 Y

9 B A

B 9

,

) T(B) (p)9 1 T(B)

T(A)

D(X).

9 T(B) (B)

~

P

A

In f a c t the p r o o f s

A A

in

> T (B)

, T(A)

> T 2(B) T(A)

T 2(B) > T(B)

> T (B)

show that we may define

r(P)

In v i e w of t h e s e

= h(B)P

definitions,

expanded

to a u n i q u e l y

converse

direction

(the e x p a n s i o n

of)

Proof.

I(P)

3.

= m(B)T(P)

e a c h p r o o f of

determined

we h a v e

PROPOSITION to

,

p r o o f of

.

G(X)

m a y be

D(X).

In the

the f o l l o w i n g .

Each proof

a proof

in

F o r any p r o o f

in

D(X)

is e q u i v a l e n t

G(X).

P: A

> B

in

G(~),

we

have

T(P)

= 1T(B)T(P)

-= m ( B ) T ( h ( B ) ) T ( P )

Next we shall

p r o v e by i n d u c t i o n

of

that

T

in

A

1A

~ m (B) T (h (B) P) = l(r(P))

on the n u m b e r

is e q u i v a l e n t

of o c c u r r e n c e s

to a p r o o f

in

G(X).

.

- 93 -

Surely

N o w assume

1 x ~ DIx.

IA ~ PA

in

IT(A)

There

- T(I A)

Again,

say

- h(A) l A

r(l A)

m(A)

- m(A) lT2(A)

remains

- T ( P A ) - I ( r ( P A ))

we have

h(A)

only

A,

for

then

G(X),

by the above.

the r e s u l t

- r(PA ) ,

- m ( A ) T (i T (A) ) = l(1 T (A) ) - I(PT(A) ) "

This

transitivity.

is h a n d l e d

by the

following: CUT E L I M I N A T I O N P: B

> C

to a p r o o f

are p r o o f s in

DfDg

in

G(X) ,

If

Q: A

then

PQ

) B

and

is e q u i v a l e n t

G(X).

Proof. occurrences

THEOREM.

We p r o c e e d in

by i n d u c t i o n

on the n u m b e r

of

T

Case

i.

Both p r e m i s e s

are i n s t a n c e s

2.

The

in the p r o o f

of

A, B, C. of

Dr.

Then

~ Dfg. Case

premise

uses

r.

last

step

of the s e c o n d

T h e n we have R B

A

Q-- B

R~

C

B

> T(C)

A

> T(C)

A

-_

~-~

B

B

>

A

) C

A

~ T(C)

C

- 94 -

since

r(R)Q

By in

inductional

=

(h(C)R)Q

- h(C)(RQ)

RQ

assumption

= r(RQ)

is e q u i v a l e n t

.

to s o m e

proof

G(X).

Case premises

A

use

S

T(A)

3.

I.

The

last

steps

in the p r o o f s

of b o t h

Then we have

> T(B) > T(B) T (A)

R

B T(B)

) T(C)

A

S

R

B T (B)

> T(B)

A

> T(C)

~ T (C)

T (A)

) T (C)

) T(C) ) T (C)

~

T (C)

since

I(R) I(S)

= m(D)T(R)

m(B)T(S)

-= re(D) m ( T ( D ) ) T 2 ( R ) T ( S ) _-- m(D)

T (m (D) T (R) S)

= I(I(R)S)

By in

inductional

.

is e q u i v a l e n t

I(R) S

assumption

to s o m e

proof

G(X).

Case premise second

uses

4. r

premise S

A

The

last

a n d the uses

) B

last

i.

B

R

step

in t h e p r o o f

step

of the

in t h e p r o o f

first

of the

Then we have

) T(C) R

A

>

T(B)

A

T(B)

> T(C)

>

T(C)

A

-

~

B

A

B

) T(C)

- T (C)

-

95

-

since

I(R)r(S)

= m(C)T(R)h(B)S

By i n d u c t i o n a l a s s u m p t i o n

-m(C)h(T(C)RS

RS

- RS .

is e q u i v a l e n t to a p r o o f in

G(X). The proof of the t h e o r e m is now complete.

The a d v a n t a g e of the s y s t e m D(X)

is this:

given a formula

A

G(X) > B,

over the s y s t e m we can find all

p o s s i b l e p r o o f s of this f o r m u l a in a finite n u m b e r of steps, by w o r k i n g b a c k w a r d s .

To give one a p p l i c a t i o n of P r o p o s i t i o n that

P: X

2(b); > Y

that is, that

in

D(X).

e q u i v a l e n t to some p r o o f o c c u r in the f o r m u l a f: X

let us p r e s e n t a new p r o o f

X

Q

6(X)

is full.

By P r o p o s i t i o n in

G(X).

) Y,

Q

3,

Since

m u s t be

P T

Df

Assume is does not

for some

> Y.

As a n o t h e r a p p l i c a t i o n ,

one m a y t r a n s l a t e

r e c u r s i v e d e s c r i p t i o n of the proofs r e c u r s i v e d e s c r i p t i o n of the m a p s follows that

F(X)

= F(1)

x X.

Lawvere, w h o also o b s e r v e d that the c a t e g o r y of finite o r d i n a l s

in

in

G(X) F(X).

the

into a It t h e n e a s i l y

This r e s u l t is due to Bill F(1)

is i s o m o r p h i c w i t h

and o r d e r p r e s e r v i n g maps.

We are now in a p o s i t i o n to o b t a i n the c o n v e r s e of L e m m a 1 for i d e m p o t e n t c o n s t r u c t i o n s .

-

COHERENCE Two p r o o f s

in

D(X)

Proof. to one

in

G (X)

Let

P: A

f

by

>

.

in

= X, H(1)

CONSTRUCTIONS.

D(X)

as

are equivalent.

is e q u i v a l e n t

the same g e n e r a l i t y ,

we

G(X).

in

Now,

P.

It c o n t a i n s

H: 2

))

obtained

= P.

G(X).

Define

= Y, H(

G(2)

H(P')

has the s a m e g e n e r a l i t y

has

) Y.

in

in

be a p r o o f

Df: X

Then

IDEMPOTENT

each proof

) B

be the p r o o f

-

the same g e n e r a l i t y

proofs

one a x i o m

P'

FOR

and t h e r e f o r e

H(0)

Let

with

Since

need only consider

exactly

THEOREM

96

) X

= f .

from

assume

Then there

by

P

by replacing

that

exists

Q: A

> B

a proof

*

Q

in

D(2)

Q*

is e q u i v a l e n t

H(Q*) P'

such that

H(Q').

~ Q'

It f o l l o w s

of

2.

Tm(0)

Proof. from symmetry,

=

But,

Q'

in

by Proposition G(2),

hence,

f r o m the l e m m a b e l o w

3,

also

that

hence

t

LEMMA

(P,Q)

= Q.

to some p r o o f

P = H(P')

proofs

H(Q*)

(Df,D), g

For

= H(Q')

- H(Q*)

idempotent

Tn(1)

in

constructions,

G(2)

consider

(I(P'),I(Q')),

any t w o

are e q u i v a l e n t .

We p r o c e e d b y i n d u c t i o n

we n e e d o n l y

= Q .

on

m + n.

the f o l l o w i n g

(r(P'),r(Q')),

Aside

cases:

(l(P'),r(Q'))

.

-

In the hence

first

case

97

f

-

and

g

are m a p s

0

>

i,

f = g.

In t h e assumption,

second

hence

The

We

case we have

P'

~ Q'

by

inductional

P ~ Q.

third

shall

case

is s i m i l a r

consider

the

to the

fourth

second.

case

after

the

following:

SUBLEMMA. then

B = T(C)

If

and

Proof.

A

This

of t h e p r o o f

of

is

is n o t h i n g

I, t h e r e

B = T(C)

ional

assumption,

hence

so is

and

A

> B.

is p r o v a b l e

is a l s o

provable.

induction

If the

to s h o w ,

last

so w e m a y

step

is a t h e o r e m .

C = T(D)

and

A

G (X)

length

in-this

assume

~ C

~ C

in

on the

T(A)

return

proof

it is

r.

By i n d u c t -

is p r o v a b l e ,

= B.

to the

fourth

case

in t h e p r o o f

of

2.

We have T (A)

) T(B)

Q' : T (A)

P'

~ B

is s h o w n b y

> T(C)

We n o w

there

> B

T(A)

Then

Lemma

T (A)

in

) B.

is a p r o o f

=- r(R)

and

Q'

proofs

P = I(P')

G(2),

where

In v i e w

of t h e

R: A

) B.

=- I(R),

and

P':

A

sublemma, By

Q = r(Q') > T(B)

of and

B = T(C)

inductional

and

assumption,

- 98 -

hence,

P - l(r(R))

= m(B)T(T(R)h(A)) - m(B)T2(R)T(h(A)) - T(R)m(A)T(h(A))

- T(R)

and

Q - r(l(R))

=

h (B)m(C)

T (R)

= h(T(C))m(C)T(R)

by the

the

idempotent

idempotent

rule

rule

(7).

comes

2.

We m i g h t biclosed \,

p,

monoidal

~, e,

8,

is t h e

only

place

where

in.

CLOSED

as w e l l category

Y)

This

- T(R)

CATEGORIES

start

with

a bang

as a 1 0 - t u p l e

(A,

and I,

define ., /,

where

is a c a t e g o r y , I

is an o b j e c t

9 : A_ x A_ k: A Z p p

x A P,

l,

> A_,

> A

are

~,

8, 7

p(A) : A

of

A,

/-

A_ x A ~

> A,

bifunctors, are

9 I

natural

) A,

isomorphisms:

l(A):

I

9 A

> A

> A. (B.C)

,

8 ( A , B , C ) : [A-B,C]

)

[A,C/B]

,

7 ( A , B , C ) : [A.B,C]

>

[B,A\C]

,

(A,B,C)

and

:

(A.B).C

,

a

-

such that the following

A.B

>

(A.I).B

) A.(I.B)

) A-B

,

(A.B). (C-D)

> A- (B. (C-D))

A- ((B-C).D)

>

(A. (B.C)).D

~

(A, I,

remarks

-, p, I, e)

were

Closed monoidal

categories

(A, I,

and Kelly

(1966).

(A,

., /, \ ,

Systems

and Categories

9

~

without

e, 8, 7) (I).

making

were d i s c u s s e d

were

However

and Kelly.

corresponding

categories

to residuated

(1963).

by M a c L a n e

Residuated

studied

changes,

from the paper by Eilenberg

.

" , \ , P, I, ~, 7)

in Deductive

one cannot

some other

>

Monoidal

studied by Benabou

conditions

categories

((A.B).C)-D

are in order.

"coherence"

studied by Eilenberg

studied

are identities:

>

The last two

and

maps

((A.B).C).D

categories

were

-

composite

Some h i s t o r i c a l

(1963) 9

99

strip off

as is clear

The deductive

system

had already been

in 1958 with a view of applications

to m a t h e m a t i c a l

linguistics.

With minor and Categories

changes

the proofs

(I) go through

in Deductive

to establish

Systems

the following

results: (i) together

with a functor

constructed deductive

A free b i c l o s e d

monoidal

6 (X): X

for any small category

system (2)

category

> U (F (X)) X

can be

using a suitable

D (X).

6 (X)

F(X)

is a full embedding.

-

D(X)

(3) system A

G (X) ,

> B

D(X)

(4) equivalent

may be replaced by a G e n t z e n - t y p e

allowing

in

I00-

one to find all proofs of a f o r m u l a

up to equivalence.

Two proofs of

A

> B

in

D(X)

are

if and only if they have the same generality.

To e x p l a i n in the paper

the concept

referred

to)

of g e n e r a l i t y

let us consider

(called

"scope"

two p o s s i b l e

definitions.

DEFINITION P: A

) B

where

X'

in

A.

D(X)

The G e n e r a l i t y

is the class of all pairs

is any small

category

functor such that there exists H(P')

of a proof

and

a proof

(X',H)

> X

H: X' in

P'

is a

D (X ' )

with

= P.

Here replacing

H(P')

is the proof o b t a i n e d

each object

by its image under

and each map of

X'

from

P

by

occurring

in

p,

H.

It turns out that when t e s t i n g w h e t h e r

two proofs

have the same G e n e r a l i t y we need not look at all small categories

X'

(sum of

disjoint

n

but only at c a t e g o r i e s copies

w i d t h of the given proofs. definition.

of

of the form

~), where

n

n~

depends

We make the f o l l o w i n g

on the

alternative

-

DEFINITION P: A

~

where

B

in

B.

D(X)

X' = n2

The @ e n e r a l i t y

for some finite

in

X

[A,B]

in

of a proof

is the set of all pairs

We note that compute

i01-

(3) and

F(X),

n

and

......

(4) t o g e t h e r

assuming

(X',H)

allow us to

of course that

[X,Y]

is known.

A functor is called

H

between biclosed monoidal

a strict map if it p r e s e r v e s

categories

the structure

exactly,

that is,

H(I)

= I,H(

9 ) = H(

Hp = pH, etc,

).H(),

H(~(A,B,C

)) = ~(H(A),

H(8(A,B,C) (f)) = 8(H(A),

Let

BMC

denote

categories,

the category

of exactly

A

functor

> L(B) L

is an absolute

with domain

BMC

consists

Tripleableness demand this:

isomorphisms.

and a functor

----> L(C)

> Cat

as m o d i f i e d by Par~,

Given two strict maps

C

reflects

etc.

construction

Precise

(2)

~ B

L (A)

of the s t a n d a r d

U

BMC

U: BMC

We may now ask w h e t h e r

conditions,

H(C)),

H(C)) (H(f)),

functor

(i)

in

H(B),

of small b i c l o s e d m o n o i d a l

in the sense of Beck's Beck's

(B, ...)

F.

the algebras

(UF, 6, -) Theorem.

H(B),

then the f o r g e t f u l

has a left adjoint

etc.

HI,H2:

K: B

Cat),

> C

coequalizer

is a c o e q u a l i z e r

(A, ...) such that

diagram diagram

then the pair

>

HI,H 2

(that is, for every has a

-

coequalizer

K'

We BMC

does

shall

and

shall

indeed

look

1 0 2

U (K') --~ K

not

give

satisfy

at a c r u c i a l

canonically.

here

the part

the

complete

Beck-Pard of the

proof

conditions,

argument

for

that but we

condition

(2).

Assume the

following

A~

that

x A

> B~

--

~

H

and

the

top

right

square

k

about

• A~

assigning the

functor

deal

these

x A

by the

A~

maps,

CS pp

let us

x C i

--

C __

the

two

squares

is an a b s o l u t e diagram. x ~

also

similarly

) C

denote

on the

coequalizer

It f o l l o w s to m a k e

it b y

also

canonical

• A~

with

the

\ .

that

the Then

functors

9

p

y?

transformations

> Ens 2

to e a c h m a p

functor

strict

the n a t u r a l

is to v i e w

followed

and consider

exactly.

We may

8: A ~

--

~

the b o t t o m

commute,

preserves

trick

given

> C Opp

is a c o e q u a l i z e r

is a u n i q u e

but what

are

2 Since

there

K

are

--

B L >) __

left commute. diagram,

• B

--

A

HI

K

diagram:

--

Since

and

HI,H2,

as f u n c t o r s , is the

functor

f: X x A

> Y

thus

functor

Ens 2

• Ens

/,

The

for e x a m p l e which,

> Ens

the p a i r

) Ens

to

and

when

x Ens

(X,Y) whose

yields value

at

-

(A,B,C)

is

103

-

( [A.B,C], [A,C/B] ).

We s h a l l n o w t u r n to a n o t h e r w a y of s t u d y i n g closed

categories

connection

by v i e w i n g

between

linear mappings

Gentzen's

in a l g e b r a

t h e m as m u l t i c a t e g o r i e s . method

Benabou

together with

a class

(1961).

also been

s t u d i e d by

being

any n o n - n e g a t i v e

are the i d e n t i t y m a p s composed

by

of a c l a s s

of o b j e c t s

..., A

..., A n

Among the multimaps

> A.

Multimaps

as f o l l o w s :

> B,

n

> B ,

integer.

IA: A

"substitution"

g: AI,

consists

of m u l t i m a p s

g: A1, A2,

there

have

out before

and C a r t i e r .

A multicate~ory

n

in l o g i c a n d m u l t i -

has b e e n p o i n t e d

I am t o l d t h a t m u l t i c a t e g o r i e s

The

f:

m a y be

Given multimaps

...r A,

...

> B

is a m u l t i m a p

f ( . . . , g , . . . ) : ..., A1, Substitution,

also called

T h e s e w i l l be s t a t e d

cut,

..., An, must

...

satisfy

as s o o n as w e h a v e

> B

~

four c o n d i t i o n s .

introduced

a conven-

ient n o t a t i o n .

We s h a l l u s e finite

sequences

capital

of o b j e c t s .

Greek

letters

to s t a n d

for

- 104 -

Thus a m u l t i m a p may be d e n o t e d by for the e m p t y sequence, b: ~

~ B

and

T

g: A

for w h i c h

> B.

n = 0.

We w r i t e

A multimap

may a l s o be r e g a r d e d as an e l e m e n t of are f i n i t e s e q u e n c e s of objects,

j u x t a p o s i t i o n is d e n o t e d by the c o m m a w h e n

~

or

T

#,T.

B.

If

their

The r e a d e r s h o u l d ignore

is the e m p t y sequence.

The cut

n o w takes the f o r m A

g>

A

~A,~

~,A,~

We s h a l l a v o i d the n o t a t i o n

f

B

) B

f(~,g,T)

for the r e s u l t i n g

m u l t i m a p as it tends to b e c o m e troublesome.

We now state

the four c o n d i t i o n s .

(i)

The f o l l o w i n g m u l t i m a p 1 A~ A

A

f:

f ~,A,~

#,A,T

(2)

is e q u a l to > B

> B

The f o l l o w i n g m u l t i m a p

is e q u a l to

g:

1A

A

(3)

> A ~,A,~

~; A ~,A,T

A

> A ) B

The f o l l o w i n g m u l t i m a p s

~,A,~ F,B,A > B > C F,#,A,~,A

are equal:

>

> C

,

A

F,BfA

)

F,#,A,~,A > A > C F,~~,A,~,~

B

> B

C

-

(4)

d

> D > C

The

~,C,8,D,T #,C,8,A,T

~,F,0tA,~

We may law,

following

call

and

The

multimaps

> B ~

B

F ,

(2)

A

identity

the c o m m u t a t i v e

reader

are equal:

> C > D

#,C,8,DrT

the will

class

laws,

> B

~,F,0,D~T

~,F,8,A,T

We denote [A;B].

-

> C

(1) a n d

(4)

105

9 B

> C

(3) the

associative

law for m u l t i c a t e g o r i e s .

of a l l m u l t i m a p s

appreciate

the

reason

A

> B

for u s e

by

of the

semicolon.

EXAMPLES 1. [A I,

...,

A

n

2. define

Thus,

Any

category

; B] = ~

a multicat

if w e p u t

n ~ i.

Any monoidal

category

becomes

a multicat

if w e

recursively [@;B]

=

[I;B]

,

[F;B]

=

[C;B]

-----> [F,A;B]

=

[C.A;B]

.

for e x a m p l e , [A,B,C;D]

3. if w e

for

becomes

define

Any

=

[(A.B).C;D]

(left)

closed

.

category

becomes

recursively [@;B]

=

[I;B],

[C,F;B]

=

[F;C\B]

.

a multicat

106-

-

Thus,

for example,

[A,B,C;D]

=

[C;B\(A\D)]

=

[I;C\(B\(A\D)) ] .

We also m e n t i o n given multicategory, categories M.

one application.

If

M

we may define M-categories,

for which

[A;B]

is

any

that is

is in some sense an object

of

This may be done in the same way as it was done by

Benabou

(1965)

for monoidal

Certain multicats monoidal

categories.

structures

are more easily

than into categories. multicats,

multicats,

Thus we may readily

left closed multicats,

and b i c l o s e d

introduced

multicats

into

define

right closed

by leaving

out u n w a n t e d

data

from the following.

A biclosed (M, I , . ,

/, \,

and

\

and

i: @

l: A,A\B mappings

monoidal

i, m, r, l)

are binary

where

operations

> I, m: A,B > B

multicate~or[

r: A/B,B

> A,

A.B,

>

T; C]

., /,

of

[r >

~; C] [r

,

A, B,

IF; A/B]

>

[F, B; A]

,

[F; B\A]

)

[B, F; A]

.

T; C]

M,

and

such that the following

and onto:

[~, I, ~; C] [r

is a multicat,

on the set of objects

> A.B,

are multimaps

are one-one

M

is a 9-tuple

induced

-

The

following

third

of the

cuts

s h o w how,

above

i >

I

mappings

~, I,

~, ~

_N

with

and with

E(f):

and

each

Let

are

induced:

> C

, F

E: M

> N

object

A

E ( A n)

Mult

the

first

> A/B

between of

f: AI, -----> B,

be the

M ...,

and

r>

A/B s B

A

) A

multicategories

an o b j e c t

E(A)

An

a multimap

> B

it p r e s e r v e s

category

and

of s m a l l

of

identities

multicategories

functors.

By a s t r i c t multicategories, structure

E(I)

monoidal functors

an e q u i v a l e n c e show

this

least

between

biclosed

which

preserves

= E(A)-E(B),

etc.,

E(i)

etc.

BMM be

> BMC

on o b j e c t s .

the

= i,

l

the c a t e g o r y and and

of c a t e g o r i e s ,

in d e t a i l .

monoidal

a functor

multicategories BMM

E

Thus

= I, E(A.B)

Let

map

we mean

exactly.

E (m) = m,

at

for e x a m p l e ,

F, B

each multimap

E(AI) , ...,

a n d cuts.

-

> C

A functor associates

~

107

Here

of

small

strict

maps.

BMC

> BMM.

but

biclosed There They

it is r a t h e r

is a d e s c r i p t i o n

are

obvious

establish

tedious

of t h e s e

to

functors,

- 108-

BMM

> BMC.

category

(M, I,

., /, \,

(A, I , . ,

/, \,

p,

multimaps

which

are n o t m a p s

example,

as

(A,

> A > A

A.I

> A

I,.,

I, u,

any biclosed

i, m, 8, Y)

r,

i),

as

follows.

monoidal

multi-

form

and define

p,

> B.C A,B,C

A,

Suppress u,

8,

all

B -1,

for

follows:

A A,I

A,B

Given

B,C

A.B,C (A.B) .C

> A.B A.B A,B ) C A > C/B

> C

BMC

Given

as f o l l o w s .

> A. (B.C) > A. (B.C)

> C/B

C/B,B

A,B

)

A.B

) C

) C

C

I

> BMM.

/,\,

A

B.C > A. (B.C) ) A.(B.C)

p,

I, u,

Define

A1,

any b i c l o s e d

B, y),

...,

form

A n

monoidal

(M, I,

> B

.,

/,

category \,

m,

as in E x a m p l e

r,

i)

2 above.

Let

i: @ m:

similarly

We

> A.B

mean

1

: A.B

B----> A

mean

B-I(IA/B) : A / B

now

whose

terms B

I

shall

M(X)

and

li:

1.

multicategory

A

mean

for

small

if

I

A,B

r: A/B,

and

)

are

turn

~, are

we

our

attention

can f o r m

the objects

terms;

A'B

of

and whose

>

I

8

> A.B

to BMM.

a deductive ~;

I, and

formulas

have

, ) A

9 B

Given

any

system A.B,

A/B,

the

form

A\B

-

AI,

A 2,

...,

Its

axioms

An

and

1 A.9 A

> B rules

> A

whenever >

~,~ ~,I,~

I; A , B

> C > C

;

relation

closed functor

of

U.

We deductive as

those

are

f: X,

...,

A-B;

X

~

;

can

and use

A,B A

showing

now

replace

M(X)

G (X) .

Its

terms

are

its

Here

Mf: whenever

and

and

) Y

f: Xl,

...,

Xn

) Y

>

I

>

9

F F,A

> C 5 ) C'D

r

) D

r

~,A,B,'~ r176

the

free

X

and

over

bia

left

Mult.

a Gentzen-type

Xn

@

- A ; B\A

equiva-

is t h e

formulas

axioms

;

B,A A

;

by

F

equational

by

) B

A\B

a suitable

that

...,

I

A,

'

;

> A > A/B

XI,

>

X

generated

be

M(X).

> Y

to c o n s t r u c t

F(X)

terms.

Xn

introduce it

are

in

> A;

BMM will

shall

B

follows:

> Y

n

C -~ C

> U (F (X)) Again,

as

A/B,B

that we

M(X)

and

M f : X 1 , ...,

multicategory

system of

inference

r ~,A-B,~

~ (X) : X

adjoint

Ai

cut;

~

on

monoidal

the

the

It is c l e a r lence

where

of

;

109-

rules

in ~

are

>C

the

of

same

inference:

_X , C

) C ) C

I

9

)

-

/

A,B

> \

> A

A

and

> A > B\A

A

> B @cArT @,A,B\A,T

Note

that, a s i d e

f r o m the a x i o m

> in

., /, \,

symbols

working

backwards,

in

eliminating

\

>

involves

s e r v e to i n t r o -

The notation

book.

Every

i.

) /

Surely,

one s y m b o l

r u l e of

\ ----> .

The o n l y n e w r u l e s

Here

is h o w the f i r s t

PROPOSITION to the e x p a n s i o n

Proof. of s y m b o l s

IA: A

Mf w h i c h

the r u l e s

left.

in K l e e n e ' s

A --~ D C,D > C'D F --~ C C,A > C'D F,A > C'D

whose

>

given

proofs

at e a c h

a

of it b y step.

is a d e r i v e d

G(X)

M (X).

Proof. and

half

> C ) C

w e c a n f i n d all p o s s i b l e

PROPOSITION

/

) C

on the r i g h t a n d the o t h e r h a l f

one on the

G(X),

-----> C

r

is f o u n d

formula

rule

@,A!T

B,A A

to i n t r o d u c e /

> B

> A/B

one of t h e s e

serve

-

A'

n o n e of the s y m b o l s duce

110

in

proof, > A.

A,

2.

A

in

when expanded

in

two are s h o w n

>

in

in

,

M(X): >

c

is e q u i v a l e n t

M(X)

G(X).

> A

A

., /

>

F i r s t w e show, that

>

A/B,B > A 0,A,~ ) B @ , A / B , B , T ---~ C

Each proof

of a p r o o f

are

by induction

on t h e n u m b e r

is a t h e o r e m

M(X),

in

is e q u i v a l e n t

G(X) to

C

-

Clearly

MIx:

> X,

X

C.D

) X

ix : X

since

6 (X)

> C.D

111-

is e q u i v a l e n t

is a f u n c t o r .

m a y be p r o v e d P

C

and we may and

assume,

Q -- 1 . D

equivalent

by

inductional in

thus

in

G(X) :

Q D ) D ) C.D ) C.D

) C C,D C.D

Expanding

to

assumption, we

M(X),

see

that

that the

P - ic above

is

to 1D D

) D

C,D

) C.D

Ic C

> C C,D

C,D > C.D > C-D

C.D

since

F(X)

=

> C.D

is a m u l t i c a t e g o r y .

is e q u i v a l e n t

to

[C.D;

>

C.D]

C/D

IC.D: [C,D;

C.D

C.D]

> C/D

Finally,

~

C.D,

Q D

) D C/D,D C/D

and we may and

assume,

Q - 1 D.

by

> C.D

C.D

) C.D

the

last proof

since

is o n e - o n e

m a y be p r o v e d

C,D

a n d onto.

thus

in

G (X):

P C > C

) C

) C/D

inductional

assumption,

that

P - 1

C

-

E x p a n d i n g in

M(X),

112

-

we see that the above

is e q u i v a l e n t

to

1C 1D D

C/D,D_ ) D C/D,D C/D,D > C C/D

since

~

F(X)

> C ) C

C/D~D ) C C/D --'-> C/D

is a m u l t i c a t e g o r y .

[C/D; C/D]

>

C\D

C/D

IC/D: [C/D,D;

C]

) C\D

Finally,

> C/D,

the last proof

since

is o n e - o n e and onto.

is t r e a t e d s i m i l a r l y ,

the i n d u c t i o n proof that

IA: A

> A

thus c o m p l e t i n g

is e q u i v a l e n t to a

G (X) .

) A.B

A,B

A

m a y be p r o v e d thus in

) A

B

A,B

Expanding

) C

C/D

is e q u i v a l e n t to

p r o o f in

C

in

>

G(X) :

- B

A.B

and u s i n g the above result, w e see that

M (X)

this is e q u i v a l e n t to 1B IA

B

> B

A,B

- A.B z

A,B

since

A'B

is a m u l t i c a t e g o r y .

A/B,B

with.

~

> A.B

F(X)

similarly.

A,B

>

A

and

B,B\A

>

A

are t r e a t e d

T h e r e remains the cut, w h i c h w i l l now be dealt

Let us,

for the moment,

a d d e d to the rules of

G(X).

assume that the cut has b e e n

,

-

1 1 3

-

CUT E L I M I N A T I O N THEOREM.

Every proof

G(X)

in

w i t h cut is e q u i v a l e n t to a p r o o f w i t h o u t cut.

Proof. G(X),

We a s s u m e that

e x p a n d e d into p r o o f s of

P

and

M(X),

Q

are p r o o f s of

and w a n t to s h o w that

the cut

A

Q - A

~tAr~

~,A,~

P )

B

) B

is e q u i v a l e n t to a p r o o f w i t h cuts of s m a l l e r degree.

By

the d e ~ r e e of the e x h i b i t e d cut we m e a n the t o t a l n u m b e r of o c c u r r e n c e s ~,

A,

~t

At

of the symbols

Case

We d i s t i n g u i s h

i.

P = Mf

and

Mg A

g

into

f,

in

since

>

Then

~(X)

,

> Y

~tAr~

Mh

> B

,

> B

X

o b t a i n e d by s u b s t i t u t i n g

is a functor.

The last step of

the m a i n o p e r a t i o n symbol of I

\

four cases.

Q = Mg.

~fX,~

is the m u l t i m a p in

C a s e 2.

it is

and

Mf > X ~,A,~

h

., /,

B.

Proof.

where

I,

A.

Q

does not i n t r o d u c e

For example,

then we c l a i m that

assume that

- 114 -

F,A F,I,A

R

R

> A > A

CtA, T P ) B 9 B

r

where

the

cut

this,

observe

on the

by

showing

>

I

> A

follows

the m a i n

A

degree.

I F,A

in v i e w

Hence,

our

~

B

r 9

To v e r i f y

F,I,A > A

of the

task

9 A

isomorphism

is e q u i v a l e n t

to

B

~

~

)

3.

>

,

the

the

The

F,A

associative

last

symbol

then we

cut on t h e

similarly

step

of claim

right

to C a s e

for multicategories.

I

F,I,A >

A

) A

~,A,~

r

from

operation

I

treated

>

'

F,A,A,T R C B 9 A FrI,A,A,~ > B FtI,A,A,T > B

where

> B ) B

> B

Case

Q

~

M(X),

~,F,I,A,~

this

it is

smaller

9 B

that

r

But

in

[F,A;A].

F,I,A >

has

> A 9 A

"inverses"

[F,I,A;A]

s

P

r

that

r,A F,I,A

must

right

FtA 9 A CtF,A,T CrFtItA,T

A.

of

law

P

for m u l t i c a t e g o r i e s .

does

not

For example,

introduce

assume

that

that

A Q > A

F,A,A,~

FrAtAf~

>

F,I,A,A,W

has

~

> B

smaller

2 above, u s i n g

) B

B

> B

degree. the

R

This

is

commutative

law

B

-

Case 4. introduce

115

-

The last steps of b o t h

the main o p e r a t i o n

assume they are

> I

and

symbols I

of

>,

r

)

P A.

and

For example,

then we c l a i m that

B

r r

>

> B

-----> B

This is an immediate [r

Q

consequence

of the i s o m o r p h i s m

[r

The reader may object that the examples u n d e r cases subcases

2 to 3 are h a r d l y

typical.

have already been t r e a t e d

Categories

in D e d u c t i v e

n + 1

objects

i,

..., n + 1

and

in

we mean the set of all pairs

is a finite direct H: X'

>

X

P'

M(X')

> n + i.

and only one n o n - i d e n t i t y

i, ..., n

in

Systems

we shall m e a n a category

multimap M(X)

all o t h e r

(I).

By a t e s t i n @ m u l t i c a t e g o r y with

However,

chosen

By the ~ e n e r a l i t y (X',H)

where

sum of t e s t i n g m u l t i c a t e g o r i e s

is a f u n c t o r with

PROPOSITION

H(P')

3.

of a proof

such that there exists

X'

and a proof

= P.

Equivalent

proofs

in

M (X)

have

the same generality.

This may easily be shown by i n d u c t i o n of the proof

that the given proofs

on the length

are equivalent.

-

PROPOSITION

116

-

The functor

4.

> U F (X)

~ (X) : X

d e f i n e d by

(x) (x)

=

x

6(X) (F) =

,

[Mf]

,

is full and faithful. The proof result

is similar to that of the c o r r e s p o n d i n g

for s t a n d a r d

constructions

COHERENCE are e q u i v a l e n t

T H E O R E M FOR BMM.

In view of P r o p o s i t i o n

show that two proofs same g e n e r a l i t y

involves

in

P

P,Q:

A ~

are equivalent.

X'

into

multimaps ...,

is the i n j e c t i o n X'

M(X)

in

2, it suffices G(X)

to

w i t h the

Look at all the axioms Suppose

the i-th axiom

" Xi,n (i) +i

be the d i r e c t sum of the t e s t i n g m u l t i c a t e g o r i e s

i,

Ki

in

the m u l t i m a p

whose non-identity

If

B

from left to right.

f i : Xi ,1 , ..., X.1,n(i) Let

Two proofs

only if they have the same generality.

Proof.

appearing

and will be omitted.

I

let

H: X'

HKi(J)

are > n(i)

n (i )

+ 1

.

of the i-th testing m u l t i c a t e g o r y .> X

be d e t e r m i n e d by

= Xi, j, HKi(

~) = fi

"

-

Now

generalize

P

fi

by

testing

P

the

i-th

is a s s u m e d

a proof

Q*

Proposition Therefore, P'

2,

Q*

That

LEMMA. multicategories

G(X)

any

for s o m e

X

and

be

assume

no o b j e c t

two p r o o f s

P

G (X') ,

Then

that

~ Q,

Let

in

proof

Q'

P ~ Q

will

a finite

in

X

occurs

and

Q

of

by

G(~).

follow

from

in a lemma.

> B

of

Since

we have

Now,

direct

A

= P. Q,

= Q.

shown

that

as

H(Q*)

be

replacing

H(P')

generality

is so w i l l

and

in w h i c h

Then

~ Q'

this

P'

same

such

H(Q')

-

multimap.

the

M(X)

also

~ Q'.

to a p r o o f

to h a v e in

117

is a f o r m u l a

more

A

s u m of t e s t i n g

than

once.

in

G (X)

> B

in

are

equivalent.

Proof. induction After and

on the

disposing P =

>

the

P =

last

roll

number of t h e

i.

step

F,~,~ ~ F,I,@,A

up o u r

of o p e r a t i o n

>

I,

Suppose of

Q

is

> C.D C.D

"

we

the

--->

Q _

in

and proceed

symbols

cases

9

step .

in

the

of

Then,

FtI,@

F-,I,%~

P

is

hence

F,8

~

C

also has

F,I,@ ~ - ~ > C

a proof

Q*

C

~ Q

~

C.D

C

in

Q = Mg,

I

say

I

,

> B.

following.

M(X)

F,@

by

A

P = Mf,

consider

last

-

Now we have

sleeves

preliminary

I, Q =

Case and

We

G(~).

> D

9

>

- 118-

By i n d u c t i o n a l

assumption

A*

pi

~ C ~ F,G ~ F,0,A .

.

Thus,

.

.

.

.

we wish

F,0

.

.

QW

to s h o w

A

F,I,0,A ;

these

,

.

FrO,A

[F,I,8,A;C]

Q,

proofs

in

M(X),

we

claim

D

A > C.D

that

C.D F~O--~ C A --~ D C,D--> F,I,0--> C C~A ~ C.D F,I,@,A -~ C.D

isomorphisms >

> C > C

~,@ > C [,I,0 ) C F,I,@tA

----> D > C'D > C'D

A--~ D C~D -~ C C,A - ~ C - D F,O,A--~ C.D r,I,O,A--> C-D

of the

F,0 F,I,0

that

f,0--~

In v i e w

put

N"

A ~-----~ D > C.D

> C

Expanding

we may

[F,I,8;C]

[F,8,A;C]

this

>

[F,0;C]

is t a n t a m o u n t

C-D

and

to s h o w i n g

that

g--~

I

F,I,0--~

F , I , 8 --~ C ..... C C~A--~ C.D F,0,A ~ C.D

F~O-.~

and

this

holds

Case and the

last

P =

As b e f o r e , both

i

C,A --> C . D > C.D

C

F,I,@,_A

F,0,A - ~

C.D

by associativity.

2.

step

Suppose of

Q

F,A P' ) A / B F,I,A ----> A / B

we use

premises

>

may

the

the

is

'

last > /

Q =

inductional

equivalently

be

step .

of

Then

F tI,A,B F,I,A

P

I

>

say

~ A ~ A/B

assumption inferred

is

to s h o w

from

that

F,A,B

~

A.

- 119 -

In v i e w of the monoidal

multicategory,

this

holds

~

3 and

a n d the

These

cases

the

cases,

p =

last

are

Case and

last only

show

the

4.

~--~ T,A

A

a biclosed

to v e r i f y

5.

Suppose

step

treated

of

I

that

F,I,A --> A / B > A/B A/B,B F,~,B--> A

> A

step

of

Q

Q

is

the

last

/

)

is

one of w h i c h

we

shall

) C ) C '

inductional

the

last step I

)

or

step

of

P

of

P

is

9

>

.

similarly.

Suppose

~,@,A/B,A,~ ~,I,0,A/B,A,~

Using

.

consider

A

I

as a b o v e ,

)

are t h r e e

sub-

as an e x a m p l e .

> B ~,I,8,A,~ ~,I,0,A/B,A,~

Q =

assumption

There

is

one w a n t s

) C ) C

to

that A

)

B

#,8,A,~

~,8,A/B,h,~ ~,I,8,A/B,A,~'

We omit

the e a s y

There have

one w a n t s

of

by a s s o c i a t i v i t y .

Cases I

isomorphisms

F,I,A--~ A/B A/B,B--> I r,I,A,B > A F,~,B > A

>

and

fundamental

already

ies

(I).

not

at all

>

C

> C > C

r

A {

~

C

> B ~,I,@,A,~; ) C ~,Z, 0 , A / B , A , ~ ----> C

verification.

are e s s e n t i a l l y

been

discussed

Unfortunately typical

the

of the

nine

other

in D e d u c t i v e five

cases

difficulties

cases.

Systems

and Categor-

considered that may

These

here

arise.

are For

120

-

example, X

have

cases,

assumptions

not yet been three

to t a k e warned

the s p e c i a l

the

lemmas

that

If

r

to

into

lemmas

The

these

the

remaining

reader

who wishes

matters

(Lemma

should

i), m u s t

be

be modified

as f o l l o w s :

>

B,

the m u l t i c a t e g o r y

To d e a l w i t h

to look

context

about

required.

one of t h e s e

in the p r e s e n t

is p r i m e

used. were

trouble

-

then

B

is p r o v a b l e

A

>

I

and

in

G(X)

r

~

and

B

are

theorems.

The We

say

that

contains

definition A

of

"prime"

is p r i m e to

no m u l t i m a p

~

f: Xl,

must

in

...,

also

G(X), X

be m o d i f i e d :

provided

> Y

with

some

X

n occurring

in a t e r m

of

A

and

Y

i

occurring

in a t e r m

of

4.

Just one

further

are

;

9

to g i v e

case.

the

Suppose

then we may

8

p

p = F

P--~ C

reader the

an idea,

last

steps

assert

that

a multimap

f: Xl,

and

D.

of

Y d

(Y = Yj)

in > D or

Q

t~ D F,@,A

...,

X

Y

through ZI,

and

> D

is p r i m e

Now

P

tl

9,.~

8

of

put

9

We

let us d i s c u s s

...,

to

> Y

n

must some Z

q

D.

be

For, in

> Y.

YI'

C-D

otherwise,

X --

introduced

axiom

>

...,

with

X. i

into Yp

there in

8

the p r o o f

----> Z

is

- 121-

The

first

possibility

as a d i r e c t In the

sum:

second

assumed is n o t

the

case

components

all

among

Z., 3 c o m p o n e n t s of

that both By

and, X,

0

>

inductional

~

I

Substituting that both

and

>

Q

One

final

the

Proposition

A

in

the A

) B

coherence > B

are

M(X)

would

up

theorem

be

X. in t w o

) D

lemma

to

are t h e o r e m s

in

F Q,

G

9

~

into

P

I

F,G

and

Q,

; C

F,I

9

C

) C

we may

show

to

A ) D C,D CrA ) D ) C-D F,I,A > C.D > C.D

) C-D

) C F,A

details.

word

results.

once,

mentioned

are e q u i v a l e n t

I

We omit

A

D

proofs

F,G,A

M(X) .

than

Y

in c o m m o n .

a contradiction.

above

'

F

in

more

of

As w e h a v e

> D

> D

these P

and

nothing

A,D.

therefore,

the

definition

assumption,

IrA

'G,A

I

in

occurs

again

invoke

A p,, ~ 8

X

by the

can have

occur

of

the

Now we may

G(X).

Z. 3

that no object

disjoint

infer

is e x c l u d e d

about

2 tells

the usefulness us h o w

to e q u i v a l e n c e . tell

equivalent.

us e x a c t l y

to f i n d

of t h e

all p r o o f s

Proposition when

above of

3 and

two proofs

of

-

Since multimaps in in

M(X),

F(X).

F(X)

122-

are equivalence classes of proofs

we thus have a method for computing

[A;B]

in

This assumes, of course, that the sets

[XI, ..., Xn; Y]

in

_X are known.

REFERENCES For additional items see the references in 4. [i]

Beck, J., "The tripleableness theorem", manuscript.

[2]

Benabou, J., "Alg~bre ~l~mentaire dans les categories avec multiplication", 258; pp. 771-774,

[3]

C. R. Aoad. Sc. Paris,

(1964).

Benabou, J., "Categories relatives", Paris, 260; pp. 3824-3827,

[4]

C. R. Acad. Sc.

(1965).

Lambek, J., "Deductive systems and categories

(I)",

Math. Systems Theory, to appear. [5]

Par~, R., "Absolute coequalizers"

in this volume

-

123-

POSSIBLE PROGRAMS

FOR C A T E G O R I S T S

by

Saunders

i.

Communication number

of u n s p o k e n

matician ples,

should

INTRODUCTION

among M a t h e m a t i c i a n s

rules.

talk about e x p l i c i t

late this e x c e l l e n t

Category

theory

arises

field and its p r o s p e c t s

a generality

in s u i t a b l e disciplines; ledge.

are the many p a r t i c u l a r

Logic,

knowledge

and folklore

of further d e v e l o p m e n t

a flourishing

it exists

designed

a variety

and a

fields in develops;

of formal Indeed,

of d i f f e r i n g

full

category

On the other to e n c o m p a s s w i t h from special

Mathematical

concepts

has

historically

generalities:

General Topology,

in that

demand

of results

to help o r g a n i z e

as a generality.

Set Theory,

specialty.

is an a c t i v i t y

formal concepts

been a s u c c e s s i o n

to vio-

in a field w h e n the k n o w l e d g e

No one c o l l e c t i o n

sufficed

I propose

In the last six or eight years,

theory has b e c o m e hand,

programs.

or concrete exam-

today is both a s p e c i a l t y

w h i c h current M a t h e m a t i c a l

time workers.

theorems

that a Mathe-

rule.

Specialities

a new s p e c i a l t y

is g o v e r n e d by a

One of these specifies

and not about s p e c u l a t i v e

generality.

MacLane

know-

for long there has

Mathematical

A b s t r a c t Algebra,

-

Lattices,

Structures

makes

of Mathematical

it all the more

ities be developed vigorously. speaks

of Category

Theory

to Algebraic

Topology,

other potential

ideas and Mathematical

imperative

that good general-

and its Applications,

of this conference

emphasizes

Homological

But Category

and Categories.

The title of this conference

organization

Geometry.

-

(in the sense of Bourbaki),

Today the proliferation specialities

124

Theory,

applications.

and the

the applications

Algebra,

and Algebraic

as a generality,

has many

Here we shall explore

a few

of them. 2. Our vital should develop,

NEW GENERAL first question

CONCEPTS is: What other generalities

above and beyond those of Categorical

I refer here not to refined types of categories like categories,

but to such brand new notions

suited to the codification

of knowledge.

Algebra?

or systems as may be

I do not venture

to predict what the new notions will be, but only that their development

will require

of Mathematical

results 3.

sharp attention needing

of category

theory require

ing, and adaptation general use.

comprehension.

THE REFINEMENT

For their efficient

to the multiplicity

OF CONCEPTS

use as generalities,

the concepts

a long period of polishing,

to put them in the most effective

For example,

the notion of an abelian

perfectform for

category

-

required before

time,

adjustment,

it became

and a p p l i c a t i o n

much

longer s h a k e d o w n

abstract

of adjoint

period.

like that of adjoint

successful

codification

functor.

are the u n i v e r s a l

Still,

adjointness

flexible

constructions

illustrate

the interplay

The d e v e l o p m e n t

the same point,

an e f f e c t i v e

generality

(not an elaboration)

4.

CATEGORIES

scene presents

are a few w h i c h

categories

("vertical"

and

lead to the of this

and B o u r b a k i

defining

these two aspects

a suppleness

of

of h o m sets)

may

theory as

and a p e r f e c t i o n

of concepts.

STRUCTURE

of generality, added elements

a category w i l l often of structure.

The

too many and v a r i e d such elements.

seem likely to live,

are e q u i p p e d w i t h

"horizontal"):

(1948)

adjoint

of many other concepts

WITH A D D E D

almost

idea

to shake down into fully

vive b e t t e r w h e n they are more s u i t a b l y double

idea about

isomorphism

in the u t i l i z a t i o n

come e q u i p p e d w i t h suitable

Here

between

is

but the

that the use of category

requires

As an i n s t r u m e n t

present

(1958)

required

the n o t i o n

sources

of Samuel

and natural

six or eight years

form.

of the latter

The p r o K i m a t e

(universality

took another

To begin with,

did not d i r e c t l y

and the basic paper of D. M. Kan functors.

functors have

linear t r a n s f o r m a t i o n ,

1929 by Stone and von N e u m a n n idea of adjoint

to sheaf theory

really workable.

The basic notions a much

125-

though some may sur-

renamed.

two sorts

Ehresmann's

of c o m p o s i t i o n

They have a v a r i e t y

of realiza-

-

tions,

including

some recent

(not yet published) ~liqative

tive up to a c o h e r e n t

arrows

categories

consist

compositions

position

cluded as the special

sufficiently

complex,

clear w h e t h e r gories"

The com-

only up to

category

is in-

just one object.

of this idea is most suggestive,

so that it may be a w h i l e before

the s t r a i g h t - f o r w a r d

H where

generalization

but

it is

to "tricate-

Modifying

role is p l a y e d by those m u l t i p l i c a t i v e

for each object the functor

has a right a d j o i n t an e a r l i e r

m i g h t best be called

(an i n t e r n a l h o m functor terminology autonomous

lar, that their d e f i n i t i o n lying c a t e g o r y

Xl

is a s s o c i a t i v e

Gode-

is viable.

categories

each

calculus.

bi-

with

following

case of a b i c a t e g o r y w i t h

A fundamental

special

and

Benabou's

and two-cells,

so the m u l t i p l i c a t i v e

development

they

of objects

and of two-cells,

in a b i c a t e g o r y

associa-

isomorphisms;

five rules of f u n c t o r i a l

c o h e r e n t isomorphisms,

are c a t e g o r i e s

of arrows,

arrows,

The multi-

operation,

consists

with composition of objects,

of arrows

The further

product

a category

both of arrows

ment's well known

categories)

family of n a t u r a l

While

(morphisms),

tensor

by Paul P a l m q u i s t

of adj~nctions.

( = monoidal

an a b s t r a c t

are omnipresent.

applications

to the theory

cate~origs

equipped with

126-

case;

of sets.

of Linton's, cate@ories.

requires

A cartesian

it is a c a t e g o r y w i t h

> A >< X

X! Y:

> A | X ;Hom

(A,Y)).

such categories Note,

no r e f e r e n c e

in p a r t i c u -

to an under-

closed c a t e g o r y 9 is a finite p r o d u c t s

has a right adjoint;

in w h i c h

the lectures

of Law-

-

127-

v e r e at this c o n f e r e n c e i n d i c a t e the i m p o r t a n c e of this n o t i o n (the t e r m i n o l o g y ,

due to E i l e n b e r g - K e l l y ,

since t h e i r s o - c a l l e d c l o s e d c a t e g o r i e s With autonomous

categories

tensor p r o d u c t is n o t b o t h Xl

to the b i a u t o n o -

commutative,

the a b s t r a c t but where

> X ~ A h a v e r i g h t adjoints.

e x p e c t that t h e s e n o t i o n s

5.

comes

categories where

(known to be)

> A ~ X and Xl

do n o t e x i s t in nature).

one n a t u r a l l y

mous categories--multiplicative

is not fortunate,

still r e q u i r e

One can

a shake-down.

AXIOMATIC MANIFOLDS

C a t e g o r y t h e o r y o r i g i n a t e d in c e r t a i n q u e s t i o n s a l g e b r a i c topology,

and h i t h e r t o its a p p l i c a t i o n s

to c l u s t e r there and in a l g e b r a i c

geometry.

have t e n d e d

In the future,

this may n o t be the case; t h e r e m i g h t d e v e l o p e x t e n s i v e cations in other fields of M a t h e m a t i c s . m a y be the case in d i f f e r e n t i a l n o t yet u n d e r g o n e

I speculate

geometry,

appli-

that this

a s u b j e c t w h i c h has

the r a t i o n a l i z a t i o n e x p e r i e n c e d by t o p o l o g y

and a l g e b r a i c geometry.

For instance,

one m i g h t s e a r c h for

axioms on the c a t e g o r y of all C ~ - m a n i f o l d s gory e n l a r g e d

of

to c o n t a i n s u i t a b l e

or on this cate-

infinitessimal

objects

(as

in L a w v e r e ' s u n p u b l i s h e d w o r k on c a t e g o r i c a l dynamics).

6.

It is an e l e m e n t a r y

UNIFOLDS

observation

that c a t e g o r i c a l

m e t h o d s h a v e had no e f f e c t u p o n analysis,

classical

There m a y y e t be some n o t i o n s

open to such appli-

of a n a l y s i s

or abstract.

-

cations.

128-

One p o s s i b l e p l a c e is the t r e a t m e n t of t h o s e

p e r t i e s of f u n c t i o n s Xl,...,x m

f ( x l , . . . , x m)

of s e v e r a l v a r i a b l e s

w h i c h d e p e n d n o t on the e x p l i c i t

choice of the v a r i -

ables, b u t on the local g e o m e t r i c a l p r o p e r t i e s U with coordinates

X l , . . . , x m.

local p r o -

of a n e i g h b o r h o o d

These are, if you will,

proper-

ties of the s m o o t h m a n i f o l d in w h i c h U is a n e i g h b o r h o o d . ever,

the~e are m a n y such p r o p e r t i e s w h i c h are s t r i c t l y

and so n e e d no g l o b a l r e f e r e n c e to a manifold. n o t i o n may be that of a "unifold".

How-

local

The a p p r o p r i a t e

A C m- u n i f o l d

(a u n i f o l d of

class m) is d e f i n e d to be a set U t o g e t h e r w i t h a set F of r e a l valued functions

f: U---> ~, for w h i c h

i n t e g e r n, a list

Xl,...,x n

there e x i s t s a p o s i t i v e

of f u n c t i o n s in F and an open set

!

U

in _Rn all s u c h t h a t

u:

>

(Xl(U) ,...,x n

(u))

is a b i j e c t i o n

$

of U to U

for w h i c h the f u n c t i o n s

f in F are p r e c i s e l y the com!

posites words,

g ( x l , . . . , x n)

w i t h g a C m f u n c t i o n on U

to ~.

a u n i f o l d is just an open set in R n, b u t w i t h o u t

c h o i c e of any one c o o r d i n a t e

system.

In other the

Those local p r o p e r t i e s

of f u n c t i o n s of s e v e r a l v a r i a b l e s w h i c h are i n d e p e n d e n t of the c h o i c e of v a r i a b l e s folds.

can be n a t u r a l l y

For instance,

teristics

s t a t e d as p r o p e r t i e s

this is the case for the t h e o r y of c h a r a c -

of first order p a r t i a l d i f f e r e n t i a l

7.

of u n i -

equations.

A X I O M S ON THE C A T E G O R I C A L L E V E L

The a x i o m a t i c m e t h o d is u s u a l l y t a k e n to b e a w a y of s t u d y i n g a g r o u p or a t o p o l o g i c a l space by axioms v a l i d in any one g r o u p or in any one t o p o l o g i c a l ,, L

space.

An alternative

129

-

now open is that of considering holding

for the category

spaces. axioms

-

as axioms

those properties

of all groups or of all topological

This prospect was first envisaged by Lawvere for the category

Category

of Sets)

(The Category

and others.

many such systems--not

axiom systems

have been studied

just the one, noted above,

UNIVERSAL possible

ALGEBRA use of category

algebra has not yet been realized.

the basic idea,

as formulated

in Lawvere's

braic system is usually described tain unary, binary, fied identities. with a binary (the inverse), which together

ternary,..,

For example,

operation

satisfy

For each group-word

operations

in

Let us recall

thesis.

An alge-

satisfying

speci-

a group is a set G together

operation

the well-known

a unary operation

(the group identity) group axioms.

However,

are clearly not the only ones at hand.

W(Xl,...,Xn)

free group)

in n letters

x i (in the

there is the n-ary operation

> w(gl,...,g n) on G.

deals simultaneously

theory

as a set together with cer-

(multiplication),

and a nullary

these three operations

corresponding

of axioms

of all smooth manifolds.

The attractive

gl,...,gn:

for Mathematics).

It would seem that there might be

8.

universal

of all categories

as a Foundation

such "global"

on the category

(An E l e m e n t a r ~ Theor ~ of the

and for the category

of Cate@ories

Some additional by Schlomiuk

of sets

in his

The theory of groups

with all these operations.

really

By speaking

-

of m - t u p l e s

tive integers).

theory

This

as m o r p h i s m s

a category

is an i n v a r i a n t

It c o m p a r e s

inverse, abstract

by p a r t i c u l a r

group compares

generators

rewriting

the results

invariant

language

still remains

practise

of M a t h e m a t i c s

of C a t e g o r y categories

Theory.

years

(multipliof an indi-

of that group

The i m p o r t a n t

algebra

in this

task of

appropriate

FOUNDATIONS

that the c l a s s i c a l m e t h o d

set theory

as a f o u n d a t i o n

is no longer adequate

The device

in some G ~ d e l - B e r n a y s

a r r a n g e m e n t w h e n it was

operations

of group

to be done.

It is an open scandal Zermelo-Fraenkel

of the theory

description

to a p r e s e n t a t i o n

of u n i v e r s a l

applying

the n o n - n e g a -

just as the n o t i o n

and relations.

9.

> m, the

description

to the c l a s s i c a l

and identity)

n

(with objects

as the study of three p a r t i c u l a r

cation, vidual

-

of n - a r y operations

theory of groups b e c o m e s

of groups.

130

of

for all

to the p r a c t i s e

of h a v i n g b o t h

large and small

set theory was a c o n v e n i e n t

first p r o p o s e d by E i l e n b e r g - M a c L a n e

ago, but it no longer convenes

for functor

23

categories

(with large domain category)

or for the c a t e g o r y of all cate-

gories

of fibered c a t e g o r i e s

as used in the t h e o r y

bou's profunctors. in a G r o t h e n d i e c k

The a l t e r n a t i v e

U n i v e r s e has been e f f e c t i v e

w i t h the d e v e l o p m e n t tions

arrangement

of M a t h e m a t i c s ,

as to i n a c c e s s i b l e

to do w i t h the case,

or in Bena-

of categories

for g e t t i n g

but it i n t r o d u c e s

on

assump-

c a r d i n a l s w h i c h p a l p a b l y have n o t h i n g

and it leaves u n s e t t l e d

(as yet)

a variety

-

of q u e s t i o n s

of the p o s s i b l e

131

-

effects

resulting

from a shift of

universe.

W h a t should we conclude? v i d e d by one "monolithic" Principia Mathematica lithic character, lated w i t h i n of labor,

f o u n d a t i o n has b e e n

one system.

This p r o v i d e d

between Mathematicians form)

the system various

is i r r e t r i e v a b l y Mathematicians

lost;

s e c u r i t y pro-

lost.

and then Z e r m e l o - F r a e n k e l

that all w o r k i n g M a t h e m a t i c s

(usually in a "naive" within

The happy

who

First

had this m o n o could be formu-

a convenient

just "used"

division

the s y s t e m

and the L o g i c i a n who i n v e s t i g a t e d

classical

problems.

This p a r a d i s e

it is high time that o p e n - m i n d e d

set to work to c o n s t r u c t

young

a new one--perhaps

less

monolithic.

10 .

This brief the p r o f i t a b l e

survey does not p r e t e n d

current problems

as a generality. various

CONCLUS ION

specific

A much b e t t e r talks

in C a t e g o r y coverage

to cover all

Theory

and its use

is p r o v i d e d by the

given at this c o n f e r e n c e - - a n d ,

by the other new and d i f f e r e n t

ideas w h i c h

we hope,

lie in the future.

-

ABSOLUTE

132-

COEQUALIZERS

by

R o b e r t Pare~

The c o n c e p t of s p l i t c o e q u a l i z e r plays role in B e c k ' s t r i p l e a b l e n e s s this n o t i o n does have

theorem

(see

[i]).

an i m p o r t a n t Although

some j u s t i f i c a t i o n in the f a c t that

it is p a r t of a c o n t r a c t i o n of a s i m p l i c i a l

object,

it is

s o m e w h a t artificial.

In this p a p e r we c o n s i d e r an e q u i v a l e n t f o r m u l a t i o n of B e c k ' s

t h e o r e m in w h i c h

the n o t i o n of s p l i t c o e q u a l i z e r

r e p l a c e d by the m o r e n a t u r a l n o t i o n of a b s o l u t e A l t h o u g h the n e w c o n d i t i o n s w o u l d appear, be m o r e d i f f i c u l t

to verify,

p r a c t i c e is the a b s o l u t e n e s s

it seems property.

fact that we have no e q u a t i o n ,

is

coequalizer.

at first glance,

to

that all we n e e d in In m a n y cases,

but only a b s o l u t e n e s s

the

leads to

m o r e c o n c e p t u a l proofs.

In the first p a r t of this paper, we give the n e c e s sary d e f i n i t i o n s

and o u t l i n e

this p o i n t of view.

the p r o o f of B e c k ' s

theorem from

Then a p r o o f that the c a t e g o r y of semi-

groups is t r i p l e a b l e o v e r sets is s k e t c h e d to show h o w t h e s e new conditions

can be used.

-

133

-

The second part contains of absolute absolute

coequalizers.

coequalizers

Professor

characterizations

Then it is shown that

which

like to thank my d i r e c t o r

Lambek,

for his guidance

Definition

there exist

are not split.

I would

I. d Y1 -'L'>

several

of research,

and e n c o u r a g e m e n t .

Let A be a category

i:

and let

d Yo

*" Y

be a c o e q u a l i z e r

d i a g r a m in A. -

(do, dl;

d)

1

is called an a b s o l u t e and every

functor

coe~ualizer

G : A

> C,

if for every

category C

(G(d o) ~ G(d I) ; G(d))

is

also a coequalizer. e Let U : B

>

A be a functor.

do

Definition

2,

Let

Yo be maps

Y1

in

B.

dI d is U - a b s o l u t e

(do , d I) 9

if there e x i s t

U(d ~ )

in A such that

U(Y I)

--

Z and

> z

U(Y o)

d

~

U(Yo)

> Z is an absolute

co-

U (d I )

equalizer.

Definition each U - a b s o l u t e essarily

3:

B has U - a b s o l u t e

pair of maps

absolute)

in B.

in B has

coequalizers

a coequalizer

if

(not nec-

134

-

Definition if w h e n e v e r Yo

> Y

Y

1

~

in ~,

>

-

4:

U ~

U-absolute

Yo

is U - a b s o l u t e

the c a n o n i c a l

map

and h a s

Z

coequal izers a coequalizer

> U(Y)

is an i s o m o r -

phism.

Definition Y1

> Y

> Yo

by U implies essarily

5:

U reflects

being

Y1

mapped

> Yo )

absolute)

: A

> B.

to a triple algebras

and

is a c a n o n i c a l

:

map

U

adjoint

(T,n,~)

UT

coe~ualizers

an a b s o l u t e

if

coequalizer

is a c o e q u a l i z e r

(not n e c -

in B.

This

T =

into

> Y

F r o m n o w on let F

U-absolute

: B

pair gives

on A.

2A

> A

_ > A

# : B

Let A 2

have rise

(see

[6] or

be the category

its u n d e r l y i n g > AT

a left adjoint

making

[2]) of T-

functor.

There

the f o l l o w i n g

dia-

gram commute : r

Definition has

6:

U is s a i d to be t r i p l e a b l e

a left a d j o i n t ~ s u c h t h a t

the a d j u n c t i o n s

i

if > r

AT and

r162

> 1

are n a t u r a l

isomorphisms.

B

We can n o w

Theorem: has U - a b s o l u t e absolute

state Beck's

tripleability

U is t r i p l e a b l e

coequalizers,

coequalizers.

-:

-

theorem.

U has

and U p r e s e r v e s

a left adjoint,

and reflects

U-

135-

-

dO Y I ~

Remark:

if

there

exist

lowing e q u a t i o n s

h

dl

Y-----~

YO

d Yo

is a split c o e q u a l i z e r

> Y hI

and

u

such that the fol-

> Y1

0

are satisfied: ddo = dd I dh

= i

hd

= doh I

Y

dlhl = iyo Equations izer,

(i) imply that

(1) 9

(do, dl; d)

thus we conclude that split c o e q u a l i z e r s

If we replace in the p r e c e d i n g

the w o r d

definitions

is a c o e q u a l are absolute.

absolute by the w o r d

split

and theorem, we obtain the origi-

nal form of the theorem.

Proof special

of theorem:

case of absolute

Since split

ones,

coequalizers

the s u f f i c i e n c y

are a

of the c o n d i t i o n

offers no difficulty. Now, that

B = AT

reflects,

suppose

that U is tripleable.

and prove

uT-absolute

that

and U T p r e s e r v e s

(YI' el)

> >(Yo,

8 o)

T be a U - a b s o l u t e

dI pair of maps equalizer

in A T .

and

coequalizers. do

Let

A T has,

We can assume

We have

the f o l l o w i n g

in A: do

Y1 ~-~---~ Yo L

~

1

d

> Y "

absolute

co-

-

136-

Consider the following diagram: T (d) O

T (d)

T(YI

~ T(Y )

j

T(dl)

01

1 eo

do

> T(Y)

>Y~ )

Y1

18

d

~Y

dI The upper and lower squares on the left commute s i n c e d o d I are T-homomorphisms.

Thus,

dSoT(d o) = ddo81 = ddl81 = deoT(d I) , but is a coequalizer; e

: T(Y)

and

(T(d o) , T(d l) ; T(d))

therefore there exists a unique

> Y making the square on the right commute. We shall prove that (Y,e) is a T-algebra.

Consider

the following cube : TT Yo)

J

TT(d)

T(Y )

T(d)

"~ T(Y.)

T(d) ~u(Y)

"Yo-

T(Y

TIe) ~ T(Y)

The top face commutes because of the 8o-aSsociativity

axiom.

One lateral face commutes by naturality of u, the other three by definition of 8.

Since

(TT(d o) , TT(d I) ; TT(d))

is a co-

137

-

equalizer mutes

TT(d)

-

is epi, and thus the bottom face also com-

(see [7], p. 43).

This proves the 8-associativity axiom.

To prove the 8-unitary axiom, consider:

"•) d

T

Yo ) T(d)

1

Y

d

~ I Y

/ T(Y)

The top face commutes

by the @o-unitary axiom, one lateral

face commutes by naturality of n, one by definition of 6, and the other trivially. commute.

Since d is epi the bottom face must also

(This diagram is actually a degenerate cube.) We have proved that

(Y,8) is a T-algebra,

and by de-

finition of 8, d is a T-homomorphism. Finally, morphism,

d = coeq (do, d I)

x, coequalizes

(do, d l)

in A_T for if a T-homo-

in A_T, it also does in A,

and thus there exists a unique z in A such that the following diagram commutes : do Yl--dl

d ~ Y~

/ /zY X~

It is easily verified that z is a T-homomorphism.

-

We have This

-

shown that A T has

construction

UT-absolute

138

also shows

UT-absolute

that UT preserves

coequalizers. and reflects

coequalizers. Q.E.D.

As an a p p l i c a t i o n t he c a t e g o r y i n t o sets,

of s e m i g r o u p s ,

is t r i p l e a b l e .

out for any a l g e b r a i c

of this with

theorem,

the u s u a l

The a r g u m e n t

we will

underlying

functor

can e a s i l y b e c a r r i e d

category. d

L e t Yl'

show that

Yo be t w o s e m i g r o u p s

and let

o

Y

> Yo

d 1--a l d o

b e two h o m o m o r p h i s m s .

absolute

coequalizer

structure

Assume

in sets.

that

Y

> Yo

We must define

respectively,

(dQ,d 0) 1

x Y1

....

1 d o and d~

a functor,

a unique

m

do

Y

X

m

d

> %

;~

d 1 are h o m o m o r p h i s m s ,

left c o m m u t e .

do)

> Y

m o

y

dmo(do,

of Y 1 a n d Yo

(d,d) ~. Y0 > x Yo

mI

fines

a semigroup

we have:

Y

o n the

is an

on Y, so t h a t d is a h o m o m o r p h i s m .

If m I a n d m o are th e m u l t i p l i c a t i o n s

Because

~- g

Since

the u p p e r

X-~-~-> X x X, f---~--> (f,f)

the t o p row is a c o e q u a l i z e r

= ddom I = ddlm I = dmo(dl, : Y x Y

> Y

and l o w e r

making

d I)

de-

diagram.

thus there

the s q u a r e

squares

exists

on the r i g h t

-

139

-

commute.

To see that m is associative,

consider

the following

cube :

iyo

.iyo

y2 ~

(d,d,d)

~ " ~ y2

(d,d)

(d ,d)

The top face commutes because faces

commute by definition

fore the b o t t o m must a semigroupo

It is easily

(d,d,d)

This

the lateral

is epi,

shows

that

there-

(y,m)

is

of m, d is a homomorphism.

verified

of semigroups.

the u n d e r l y i n g We conclude

of d, and

also commute.

By d e f i n i t i o n

the category

m ~ is associative,

d = coeq

It is also easy

functor preserves

that the category

that

to verify

and reflects

of semigroups

(do, d I)

in that

the right

things.

is tripleable

over

sets.

II. absolute

We shall now see several

coequalizers.

characterizations

of

-

140

-

d o

Theorem:

Let

d

Y1

~ Yo

> Y

be a c o e q u a l i z e r

d1 in A.

The

following (i)

(do, dl;

d)

statements

(do, dl;

d)

is p r e s e r v e d

(ii)

(d o , dl;

are e q u i v a l e n t :

is an a b s o l u t e

coequalizer,

i.e.

by all f u n c t o r s d)

is p r e s e r v e d

b y all r e p r e s e n t a b l e

functors (iii)

(d o , dl;

d)

is p r e s e r v e d

by

[Y, -] and

an i n t e g e r

n ~ 0 and ~h,

hi

(i ~ i ~ n)

[Yo , -]

(iv) in A such

that h

Y

h. I

~ Yo

> YI

verifying dd o = d d I

d

u(2)

dh

= 1

hd

= d

h

= d

i

d u(4) h 2

=

d

Y

~(I)

uC3)

h1 h2

~(5) h 3

o

a

d~(2n)h n where

~(i)

= 0 o r ~i. Proof:

(Jill

= iyo

.... (iv)

(i) : [Y, Yl ]

"- (ii)

"- (iii)

[Y,d ] .7 >

[Y,d 1 ]

[Y,Yo]

obvious. [Y,d]

>

[Y,Y]

maps

- 141 -

is a c o e q u a l i z e r Let

diagram

h s [Y,Yo]

in sets.

such that

is t h e r e f o r e

[Y,d] (h) = iy, [Yo,

Now

[Y,d]

i.e.

dh

=

onto. iy.

[Yo, d]

d o] > >

[Yo, Y1 ]

[Yo' Xo]

>

[Yo, Y]

[Yo ' d I] is also a c o e q u a l i z e r

[Yo,

d] (hd)

in sets.

= dhd

= d =

[Yo,

d] ( i y )

,

0

thus,

by the c o n s t r u c t i o n

of the c o e q u a l i z e r

in sets,

hd ~ 1y O

w h e r e '~ i s

the

equivalence

relation

generated

R = { (dox , dlX) I x 6 Thus

there e x i s t s

an i n t e g e r

by the

relation

[Yo , Y 1 ] } -

n > 0 and

~h. : Yo

> Y

1

(i = i, 2, 3,

.~

n)

such that hd = d

~(i)

h1

d ~(2) hl= d v (3) h 2 d~(4)h2 = d

d (2n)hn= where

9(i)

= 0 or 1.

(5)h3

iyo

We o b v i o u s l y

also h a v e

the r e l a t i o n

dd o = dd I 9 (iv) that

(do, dl; d)

preserved

~- (i)

: The e q u a t i o n s

is a c o e q u a l i z e r ,

g i v e n in

(iv)

imply

and they are o b v i o u s l y

by any functor. Q.E.D.

-

Remark absolute

i:

coequalizer

142

-

It can be shown that <

(d o , dl; d)

is an

is p r e s e r v e d by all

;- (do, dl; d)

full embeddings.

Remark 2:

It is not n e c e s s a r y

(do, dl; d)

is a c o e q u a l i z e r but

coequalizers

by the functors

It is n a t u r a l different sider

to assume that

only that it is m a p p e d

[Y, -],

[Yo, -],

to ask if absolute

from split coequalizers.

and

into

[YI' -]

coequalizers

The answer is yes.

"

are Con-

the diagram: do

iy ^ ; > Yo

Y1

> Yo

do (do, do;

iy )

is an absolute

coequalizer,

but it is split

O

<

> do

is a split epi.

Definition: zer is n - s p l i t

if there e x i s t

(iv) of the p r e c e d i n g

Above coequalizers n-split

exist n-split O<

n

maps d i

is verified.

is an example

of a 0-split

It is readily

coequali-

such that c o n d i t i o n

theorem

are 1-split.

implies

We

We shall say that an absolute

coequalizer.

seen that for

Split n>

(n + l)-split.

can show that for every coequalizers

which

integer

n > 0, there

are not i-split

for any

i
Indeed,

define

dO Yo

d > Y1

d~

> Y

in sets

as

0,

-

1 4 3 -

follows:

Y

o

= Y

1

d o = iy

= {1,2,3,...,

n + i}, Y =

, dl(i)

(i-l,

= max

i),

{i}

d(i)

Now

= i.

define

o

h

: Y

hk(i)

> Y = max

we have

the

1

b y h(1)

= 1 and define

(i + k - n ,

i).

following

results:

hk:

Y1

) Yo

h n _ 1 (i) = m a x

Then

(i-l,

by l)

and

n-k

=

hk

(hn_ I)

d I = hn_ 1 It f o l l o w s

immediately

that we have

the relations:

dd ~ = d d I

dh = iy

hd = dlh I =

(hn_l)n n-1

Thus

(do, dl;

d)

doh I

= dlh 2 =

(hn_ l)

doh

= 1

(h

..o, m)

Y

= o

n-i

)0

.

is n - s p l i t .

On the o t h e r (i = l, 2,

n

hand,

such

that

if t h e r e

exist

h'

l

h!i r

144-

-

h'd = d

d

~(2)

d we

can e s t a b l i s h

of the i m a g e s

the

h4 ~

= d

in g o i n g

following

of the m a p s

llm(dh') I = i least

n + 1

(do, dl;

d)

and

under

i.e.

is n o t i - s p l i t

example

~

2

= %

or

0

Im(h~) I

-1

to the f o l l o w i n g , b y at m o s t

for any

a more

i.

. the c a r d i n a -

But

thus t h e r e m u s t b e at We c o n c l u d e

m > n.

to M i c h a e l

to r e p l a c e

on the c a r d i n a l i t i e s

Im[h; ) l

IIm(iyo) I = n + l,

equations,

1

consideration:

can be c h a n g e d

I am i n d e b t e d preceding

h'

relations

f r o m one e q u a t i o n

lity of the i m a g e

~(3)

h'

(2m) h' m = 1Y o

IIm(d ,(i) h~) i ] Thus

v(1)

that

i < n. Barr

for s u g g e s t i n g

complicated

one.

the

-

145

-

REFERENCES [1]

Beck, J. "The Tripleableness

[21

Beck, J.

"Triples, Algebras,

(Diaser~ut~on, [3]

Theorem"

(Manuscript).

and Cohomology",

(1967), Columbia University).

Linton, F. E. J. "An Outline of Functorial Semantics", (Lecture Notes in Math, Springer - to appear).

[41

Linton, F. E. J. "Coequalizers in Categories of Algebras", (Lecture Notes in Math, Springer - to appear).

[5]

Linton, F. E. J. "Applied Functorial Semantics I",

(Ann~li d~ Ma~ema~ca, [6]

to appear).

Manes, E. G. "A T~iple Miscellany:

some aspects of the

theory of algebras over a triple", Wesleyan University, Middletown, [7]

Mitchell, B. York,

(Dissertation,

Conn.,

1967).

Theory of Categories, Academic Press, New

(1965).

-

146

GALOIS

-

THEORY

by

S t e p h e n S. Shatz

INTRODUCTION

This w i l l be a summary w i t h o u t proofs,

or w i t h b a r e s t

and d i s c u s s i o n

indications

- mostly

of same - of my

recent w o r k on Galois T h e o r y

for field extensions.

t i v a t e d by certain

applications

"obvious"

and these a p p l i c a t i o n s sort of theorems

suggest

classify

for an a r b i t r a r y

zt~s

coverings.

flat,

finite

yield

a description

scheme.

lines

for the

to c o n s t r u c t

connected

instead

a "fundamental

scheme,

so as to

of just c l a s s i f y i n g

For this one will need a Galois T h e o r y

coverings

of schemes.

of the B r a u e r

if it is the "correct"

those elements plication

the b o u n d a r y

one wants

the flat coverings

fine case)

of any such theory;

one needs.

Specifically, group scheme"

I was mo-

Any Galois Group

theory;

A blueprint

all isogenies

for

Theory will

(at least in the afone wishes

split by a given flat covering.

is to c l a s s i f y

the

to study

The t h i r d ap-

of a given group

for d o i n g this is given by the c l a s s i c a l

*The author w i s h e s to a c k n o w l e d g e the s u p p o r t of N . S . F . G . P . 7654, B a t t e l l e M e m o r i a l Institute, and S t a n f o r d U n i v e r s i t y during p r e p a r a t i o n of this manuscript.

-

s e p a r a b l e case,

147

-

and the h e i g h t one case as d o n e by S e r r e

A good Galois T h e o r y ought to put these t o g e t h e r

[3].

consistently,

as w e l l as e n a b l e one to h a n d l e the case of h e i g h t n > i.

To k e e p the e x p o s i t i o n short, w e shall d e a l only w Z t h the case of a finite, K/ko

Actually,

purely inseparable

generalization

field e x t e n s i o n

to the case of an a r b i t r a r y

fZnite e x t e n s i o n is easy, but r e q u i r e s m o r e space. to e m p h a s i z e t h a t t h e o r e m s in the field t h e o r e t i c treat the g e n e r a l

i~

We w a n t

s t a t e d b e l o w have b e e n p r o v e n o n l y

case - m o r e w o r k w i l l be n e c e s s a r y to

affine case.

THE GROUP S C H E M E AUT,

R I N G SCHEMES

Our p o i n t of v i e w is that any Galois T h e o r y s h o u l d be a r e l a t i o n s h i p b e t w e e n o b j e c t s morphisms.

Moreover,

closely as ~ossible

and

"groups"

it s h o u l d be f u n c t o r i a l ,

the m o d e l of the c l a s s i c a l

of their autoand f o l l o w as case.

It has b e e n r e c o g n i z e d for some time that, situations

for w h i c h o r d i n a r y groups

w h i c h groups o u g h t to p l a y a part, group scheme.

So, q u i t e n a t u r a l l y ,

Grothendiecks

a u t o m o r p h i s m functor,

if X, Y are schemes, is that f u n c t o r w h i c h

~: Y

> X

g r o u p of S - a u t o m o r p h i s m s

of

are i n a d e q u a t e and in

the c o r r e c t n o t i o n is a we are lead to c o n s i d e r Aut,

[1].

a morphism,

associates

in m a n y

Recall that then Aut

(Y/X)

to e a c h X - s c h e m e 9, the

Y x 9. X

T h a t is,

-

Aut

(Y/X)

148

(S) = A u t %

If Y and X are a f f i n e , spectively, of

and if ~ m a k e s

finite presentation, = Aut

type o v e r

(~/A)

A.

braic

k-group

K = k

say with

rings

then

Aut

~ a n d A re-

projective

then one trivially

is r e p r e s e n t a b l e

A-module

sees t h a t

by a group

scheme

if K / k is an a r b i t r a r y (K/k)

is r e p r e s e n t a b l e

Aut

(Y/X)

of f i n i t e finite

b y an a l g e -

scheme.

Observe, scheme.

(Y XXS).

~ a finite,

In p a r t i c u l a r ,

field extension,

-

Indeed,

however,

that

in the s i m p l e s t

(8), 82 = t, t ~ k 2 , ch

Aut

(K/k)

is a " l a r g e "

non-trivial

(k) = 2, o n e

case:

finds

an e x a c t

se-

quence 0 Now

> ~2 Aut

> Aut

(K/k)

defining

so m e r o l e

t u r n k, K i n t o f u n c t o r s

ing r e m a r k s

of this s e c t i o n .

a "k-algebra

> ~m

is a f u n c t o r i a l

ing k, K, and s h o u l d p l a y we must

(K/k)

construction

in the G a l o i s

as w e l l ,

in v i e w

F o r this, w e

scheme"

> 0.

Theory.

But

of t h e o p e n -

can c o n s i d e r

(and k, the same),

involv-

K as

say K, v i a

the e q u a t i o n K

(S) = A f f i n e = K ~k

(For k one s h o u l d any k - s c h e m e ; sentable

read:

and one

r i n g of

(Affine r i n g of S). k

(~) = A f f i n e

finds,

by r i n g s c h e m e s

(Spec K XkS)

trivially,

- more

exactly,

r i n g of S),

Here

S is

t h a t K and k are r e p r e k-algebra

schemes.

-

A simple verification Aut

(K/k)

> Aut

shows

(K_/k)

149

-

t h a t the n a t u r a l

is an i s o m o r p h i s m

injection - so the A u t d o e s

n o t h a v e to be c h a n g e d .

N o w the s u p p o s e d be a I-i l a t t i c e schemes But,

of K / k a n d the

this is n o t w h a t

those That

subringschemes there

ample,

arising

subfield

from subfields

above:

a ring scheme between of the l a y e r K / k

of

Aut

check.

(K/k).

02 = t , etc. ,

K | A = A | A (A)

For e x -

(as m o d u l e s ) .

such that

y

k a n d K, n o t a r i s i n g

(there are n o n o n - t r i v i a l

The g r o u p s c h e m e

One m o r e o b s e r v a t i o n : is c o n n e c t e d

the subring-

of the l a y e r K/k.

K = k(0),

< x, y > s K

should

We can o n l y c a p t u r e

yield'

is g i v e n b y

look at all p a i r s

we obtain

between

the r e a d e r m a y e a s i l y

(A)

There

subgroupschemes

our m e t h o d s

in the case d e s c r i b e d A, K

is clear:

correspondence

(closed)

are o t h e r s ,

for any r i n g If w e

inverting

structure

if and o n l y if K / k is a p u r e l y

2

= 0,

from a such').

Aut

(K/k)

inseparable

exten-

sion.

2.

Given want first

THE G A L O I S

a closed

to a s s o c i a t e w i t h attempt

subgroupscheme

H of

it a s u b r i n g s c h e m e

Aut

of K/k.

(K/k) , w e The n a t u r a l

at a f u n c t o r A ~

is n o t a f u n c t o r , ize"

CORRESPONDENCE

a n d has

it or u n i v e r s a l i z e

(K | A) H(A) the w r o n g

values.

it on the m o d e l

One must

of a l g e b r a i c

"localgeometry

-

150

-

[2], so as to d e f i n e Fix H

(A) = { ~ E K ~ A I for e v e r y A-alg.

in K | B is left fixed by

of

all of H This d e f i n i t i o n Fix H; m o r e o v e r

A~--~-> Fix H the p r o c e s s

B, the i m a g e

(A)

(B)}.

clearly yields

a functor,

is q u i t e o b v i o u s l y

the e x a c t ana-

(suitably c h a n g e d

for f u n c t o r i a l

log of the c l a s s i c a l p r o c e s s purposes),

Proposition.

For any e x t e n s i o n K/k,

Fix H is a sheaf in the f a i t h f u l l y loqy over k; in. fact, s qheme c o < r e s p o n d i n g

Fix H

flat~ ~ u a s i - c o m p a c t

to~-

is r e p r e s e n t a b l e by th e r i n g

to a u n i q u e

This field L is d e t e r m i n e d

the f u n c t o r

s u b f i e l d L of the layer K/k.

f r o m the e q u a t i o n

L = Fix H

(k),

and for e a c h A, one has Fix H

(A) = L ~ A.

O b s e r v e the r e m a r k a b l e

fact that the k - r a t i o n a l

p o i n t s of the s c h e m e Fix H d e t e r m i n e t h a t scheme.

N o w it may h a p p e n t h a t d i s t i n c t s u b g r o u p s c h e m e s H, h a v e the same Fix functors,

i.e.,

their associated

( = k - r a t i o n a l p o i n t s of the Fix functor) example,

in the o f t e n cited case

one finds that

Aut

(K/k), -

field.

~2

k

a ,

fields

are the same.

For

(8) = K, 8 2 = t , etc. , all h a v e k as a s s o c i a t e d

2,

T h e r e f o r e , we d e c r e e t h a t two s u b g r o u p s c h e m e s H, H

!

-

w i l l be called Galois

151

equivalent

!

Fix H

if and only if Fix H

(k) =

!

(k).

We w i l l w r i t e H ~ H

Fix then e s t a b l i s h e s classes

-

a mapping

of s u b g r o u p s c h e m e s

Given a (closed) associate with

following definition

case.

The functor

from the Galois e q u i v a l e n c e

of Aut to the subfields

of K/k.

, J

subringscheme

it a s u b g r o u p s c h e m e

our e x p e r i e n c e w i t h Fix,

Inv. S

in this

of

~ of K/k, we w a n t

Aut

(K/k).

j

and the c l a s s i c a l

is c o m p l e t e l y

(A) = {a E Aut A

to

In view of

definition,

the

natural:

(K | A) I for every A-alg. the image of a in Aut

B,

(K ~ B) B

when restricted

to S

(B) shall

be the identity.}.

Since Inv.

S

Aut A

(K ~ A) = Aut

(it is clearly

such)

Proposition.

Aut

is to show that they are

group scheme

Aut

(K/k).

Inv. S i s r e p r e s e n t a b l e (K/k).

If L is the smal-

b o t h Inv and Fix invert order,

doing this, however, Inv.

we w i s h

"mutually

to make

and w h a t

inverse".

some remarks

L.

Before

on the

L.

In the c l a s s i c a l Galois

of

of

of K/k such that S c L, then Inv. S = Inv.

Clearly, remains

(A), the functor

is a s u b f u n c t o r

The functor

b ~ a closed s u b g r o u p s c h e m e lest subfield

(K/k)

group of K/L.

theory,

the g r o u p

In the more general

Inv. L is the

theory,

this w i l l

-

be i m p o s s i b l e w h i l e Inv.

since

Aut

152

(K/L)

-

is a group s c h e m e over Spec

L is a group scheme over Spec

k.

However,

L,

the n e x t

b e s t t h i n g is true, namely:

Proposition.

The n a t u r a l m o r p h i s m

Inv.

L | L

> Aut

(K/L)

is an i s o m o r p h i s m for e v e r y s u b f i e l d L of the layer K/k.

S u p p o s e we start w i t h a s u b g r o u p s c h e m e H of Aut

(K/k),

and form the a s s o c i a t e d s u b f i e l d L

of the layer K/k.

=

Fix

H

(k)

N o w w e f o r m the a s s o c i a t e d g r o u p s c h e m e Inv.

W h a t is the r e l a t i o n b e t w e e n H and Inv. closed immersion

H9

> Inv.

N o w by the u s u a l can e a s i l y show that

L?

Answer:

L.

"double c o m m u t a n t "

L c Fix

T h e r e is a

(Inv~ L)

i n e q u a l i t y w h i c h r e q u i r e s m o r e proof.

(k).

reasoning, we It is the c o n v e r s e

However,

i_~f L is a l r e a d y

the f i e l d a s s o c i a t e d to some g r o u p s c h e m e H, then one finds, upon application H 9

> Inv.

of the f u n c t o r Fix to the c l o s e d i m m e r s i o n

L, that L c_ Fix

Consequently:

(Inv. L)

(k) c_

For those s u b f i e l d s

fields to a s u b @ r o u p s c h e m e H of A u t L

=

Fix

(Inv. L)

clas9 c ~ 1 7 6 m a x i m a l member.

(k).

Moreover,

Fix H L which ~/k),

(k) = L. arise as a s s o c i a t e d w e have:

in the Galois e q u i v a l e n c 9

tO L, the ~ r o u p s c h e m e

Inv. ~ i s the u n i q u e

L.

-

3.

153

-

GALOIS THEORY;

COUNTING

W e m u s t n o w p r o v e t h a t for a g i v e n field e x t e n s i o n Z/k,

e v e r y s u b f i e l d L arises

g r o u p s c h e m e of

Aut

(K/k).

as a s s o c i a t e d A necessary

tion that L be the "fixed field" AUt

(~/k) ,

field to a sub-

and s u f f i c i e n t

condi-

of a s u b g r o u p s c h e m e of

is that

(*)

L = Fix

(Inv. L)

(k)

The p r o b l e m is c o n n e c t e d w i t h " s u f f i c i e n t l y many"

automorphisms

the e x i s t e n c e of

as we shall now sketch.

Be-

cause of the i s o m o r p h i s m Inv. equation

L ~ L ~>

(K/L),

(*) is c l e a r l y e q u i v a l e n t w i t h

(**)

k = Fix

Equation

Aut

(**)

is, in turn,

(Aut

(K/k)) (k).

equivalent

to the " m o v i n g p r o p e r t y " :

Let ~ be an e l e m e n t of e x p o n e n t one in K, t h e n t h e r e exists k - a l g e b r a A and an a u t o m o r p h i s m ~ s A u t A ~

Q A) such that

If an e x t e n s i o n K/k has the m o v i n g p r o p e r t y , shall say that it is a good e x t e n s i o n . f o r m of this work,

Now,

we

in the o r i n g i n a l

I p r o v e d that e v e r y e x t e n s i o n had an e m b e d -

d i n g in a good e x t e n s i o n by a fairly e l a b o r a t e procedure. ever,

I have since s h o w n that e v e r y p u r e l y i n s e p a r a b l e

K / k is good. state:

a

C o n s e q u e n t l y , we m a y now s u m m a r i z e

How-

layer

s e c t i o n 2 and

-

Theorem. extension,

154-

Let K/k be a purelY

then K/k is a good extension.

subgroupschemes

of

lence relation;

in each e q u i v a l e n c e

m a x i m a l member,

an d the c o r r e s p o n d e n c e s

Aut

cl

(K/k)

lattice

K/k~

Moreover,

in its class.

cl

of

Aut

Inv L

(K/k)

correspondence

sketched

of s u b g r o u p s c h e m e s

analog the statement: the famous groups? general

connection

above

subqroupscheme

member

actually

of

relation.

In each

(a field~),

and the

runs b e t w e e n

classes

of subringschemes.)

t h e o r e m of Galois Inv L ~ L ~ > between normal

Aut

Theory has as a n a t u r a l (K/L).

subfields

differences

W h a t about

and n o r m a l

sub-

occur b e t w e e n

the

cases.

alizer of H in G, d e n o t e d (%) = {u E G

of

of the t h e o r e m may be

Recall that if H is a s u b g r o u p s c h e m e

NG H

classes

that the s u b r i n g s c h e m e s

It is here that s t r i k i n g and c l a s s i c a l

the e q u i v a l e n c e

equivalence

and classes

The s u b g r o u p

there is a u n i q u e

and all the subfields

if one notes

there is a u n i q u e m a x i m a l

equiva-

(k)

the s t a t e m e n t

K / ~ are fibred by a n a t u r a l class

class

is the u n i q u e m a x i m a l

(Actually,

made more s y m m e t r i c

The set of all closed

(Inv L)

inverting F between

of s u b @ r o u p s c h e m e s

finite,

is fibred by a n a t u r a l

(H)-~-~ Fix H L ~

are i-i,

inseparable,

of G, the norm-

N G H, is d e f i n e d by the e q u a t i o n

(S) I for every

in G (U) n o r m a l i z e s

scheme U over ~, the image of the s u b g r o u p H

(U). }.

-

155

-

H e r e ~ is a scheme o v e r the same b a s e as 2, and G

(~), etc~

m e a n s p o i n t s of G w i t h v a l u e s

in S.

tion works

A s u b g r o u p s c h e m e H of G is n o r m a l

for c e n t r a l i z e r s .

in G w h e n and only w h e n

N

H = G. S

~

The same sort of d e f i n i -

We shall w r i t e H ~ G for

~

~

this ~

If K/k is a f i e l d e x t e n s i o n ,

and if L is a s u b f i e l d

of the layer K/k, we shall say that L is K/k - normal, only if for e v e r y k - a l g e b r a A, e v e r y a u t o m o r p h i s m t r a n s f o r m s ~(A)

into itself~

Observe

of a t o w e r of fields,

L of K/k is K / k - n o r m a l ,

k c K c

s 6 AutA(K|

that this is a

r e l a t i v e c o n c e p t - it d e p e n d s u p o n the field Ko give e x a m p l e s

if and

One can e a s i l y

~, w h e r e

b u t is N O T a/k-normal.

a subfield

The p r o b l e m is

that no longer does an a u t o m o r p h i s m m a p a g i v e n field e l e m e n t only on f i n i t e l y m a n y choices,

it m a y have i n f i n i t e l y many.

This m a k e s n o r m a l i t y v e r y s t r o n g l y d e p e n d e n t on the s t r u c t u r e of

Aut

(K/k).

Proposition.

Let K/k be a f i e l d extension.

A neces-

s a r y and s u f f i c i e n t c o n d i t i o n t h a t a s u b f i e l d L of K/k b e K/k-normal Aut

(K/k).

Sweedler,

is that

Inv ~

be a n o r m a l s u b ~ r o u p s c h e m e

W h e n K / k is a p u r e e x t e n s i o n , [5]), the n a t u r a l m o r p h i s m Aut

(K/k)

of

(in the sense of

(of ~ r o u p schemes)

> Aut

(L/k)

qiven by r e s t r i c t i o n of an a u t o m o r p h i s m is a s u r ~ e c t i o n of ~roupschemes with kernel Aut

Inv ~.

(L/k) = A u t

Consequently, (K/k)/Inv L~

156

-

Is such a large group After

all,

the c l a s s i c a l

-

as

Aut

(K/k)

case gives us a finite

order is e x a c t l y

the degree of the extension.

one w o u l d e x p e c t

some sort of finite

In the J a c o b s o n Theory, the Lie A l g e b r a

of k - d e r i v a t i o n s

that such an algebra termined [4].

[K: k] equals

pn, we find that

that the order

sociated

(pn)pn =

benius

But one knows

space of a u n i q u e l y height

de-

l, say G,

the field K over k.

As

Then one

of G is p r e c i s e l y Thus the @roup scheme

K/k of e x p o n e n t

as-

1 b~ the J a c o b s o n

[K: k] [K: k].

Let K/k be a p u r e l y

in w h i c h each g e n e r a t o r has

Let G (K/k)

[.

dim k L = n pn.

[K: k] [K: k].

Theorem. extension

of K, say

to obtain

to the e x t e n s i o n

m e t h o d has order

to work.

of i over K is n, w h e r e n is the n u m b e r

necessary

pdim k i =

B a s e d on this,

group scheme

of Frobenius

of generators

knows

group w h o s e

a large role is played by

is the t a n g e n t

finite group scheme

The d i m e n s i o n

necessary?

denote

the k e r n e l

m o r p h i s m on

Aut

i n s e p a r a b l e , pure

the same e x p o n e n t

of the r-fold power

(K/k).

r~

of th e Fro-

Then G (K/k) has order

[K: k] and i n t e r s e c t i o n

of the classes

groupschemes

Aut

of

of G (K/k) w h i c h

(K/k)

of Galois e q u i v a l e n t

w i t h G (K/k)

still classifies

in@ t ~ the p r e s c r i p t i o n s

above.

yields

the subfields When

sub-

a fibration of K/k accord-

r = i, G(K/k)

is_~-

-

c•

157

-

the g r o u p s c h e m e [ i e l d e d b[ the J a c o b s o n method.

Remarks

1.

The c l a s s i c a l t h e o r y can be put into this frame-

w o r k v e r y e a s i l y once we r e a l i z e that it is a c t u a l l y the s t r u c ture of K ~ K as K - a l g e b r a w h i c h is at stake in the u s u a l m o r e general)

Galois Theory.

The n o r m a l b a s i s

(and

t h e o r e m asserts

that K | K is a d i r e c t p r o d u c t of copies of K i n d e x e d by the u s u a l Galois group,

and this g r o u p acts on K | K as K - a u t o m o r -

p h i s m s by left translation.

2.

One can count Inv

scheme c o r r e s p o n d i n g to Inv ~), this c o u n t i n g process.

L

(the c o r r e c t finite group-

and do the u s u a l

things w i t h

A full e x p o s i t i o n w i l l be p u b l i s h e d

w h e n all the p r o o f s of the above are published.

4.

A CRITIQUE

The m o s t d i s t r e s s i n g t h i n g a b o u t the - h e o r y above is the use of classes of s u b g r o u p s c h e m e s . a c t e r i z e the m a x i m a l e l e m e n t s the g r o u p s c h e m e s

an A r t •

- Hasse)

Inv L in e a c h cla: s?

Inv L are e x a c t l y the K - s t a b l e

in each class, w h e r e K acts on k e r n e l of p o w e r s

H o w , an one char-

of F r o b e n i u s ,

Aut

(K/k), or a1

via a generalized

Probably,

groupschemes least on the (perhaps even

exponential.

I am e n d e a v o r i n g to answer this q u e s t i o n in some s a t i s f a c t o r y way.

-

158

-

REFERENCES [l]

Grothendieck, A., "Technique de descente et thGoremes d'existence en Seometrie Algebrique II", Sem. Bour-

baki, [2]

[1959-1960], Exp. 195.

Grothendieck, A., and M. Demazure, Seminaire

Geometrie

Algebrique de l' Institut des Haute8 Etudes Scientifique8, [3]

[1963-1964].

Serre, J.P., "Quelques proprietes des variet~s ab~liennes en caracteristique p" 739,

[4]

#

Amer

9

J. Math., 80; 715-

[1958].

Shatz, S.S., "Cohomology of Artinian groupschemes over local fields", Ann. of Math, 79; 411-449,

[5]

[1964].

Sweedler M., "Structure of purely inseparable extensions",

Ann. of Math., 87; 401-411,

[1968].

-

DERIVED

159

-

FUNCTORS W I T H O U T

INJECTIVES

by

H. B. Stauffer

1.

INTRODUCTION

Given a c o v a r i a n t where A is a small abelian category derived

satisfying

dir

of F.

in

_dir A

actually

using

functors

of

F

its right

category

~dir

extension

functor

of F are o b t a i n e d by dir

--. to A

a "right completion"

)B

This can be

a "unique"

functors

, and r e s t r i c t i n g

Saul Lubkin

injectives.

and allows

The d e r i v e d

taking the derived

F: A

and B is an abelian

by e m b e d d i n g A in a "larger"

w h i c h has injectives F

category

functor

AB 5 , one can c o n s t r u c t

functors w i t h o u t

accomplished

additive

, u s i n g the injectives ~dir

The c a t e g o r y

of A.

I would

is

like to thank

for his help and encouragement.

2.

-dir A

w

category

Let A be any category. We shall c o n s t r u c t a _dir A from A in the f o l l o w i n g way. The objects

of

are d i r e c t e d

-dir A

directed (Bj;

sets.

8jj,)j

systems

The m o r p h i s m s

are o r d e r e d pairs

is a set map and

fi: Ai

(Ai; eii,)i from

(Ai; ~ii,)i

(~,f)

~ B~(i)

in A over

where

to

~: I

is a m o r p h i s m

) J in

160-

-

for each

i ~ I

such that

in I implies

i' > i

there

j > r (i), % (i') in J such that the diagram commutes:

exists

fi

,

| .. ~_

Bq,

iifi A

An equivalence ((Ai; aii,)i

3

1

)j

%(i)

relation is introduced. , (Bj; 8jj,)j)

let

In

Hom~dir !

(~,f) ~- (r

I

only if there exists

(r

~"(i)

for all i and the diagrams

> r

r f

dominating both;

f )

|b

if and

i.e., commute:

f W!

l

(i) r 1

i ; Be,, (i)

Ai

(i)

fi,~T

B ~(i)

We shall refer to

as

(Ai; aii,) I

8r162 B~, (i)

(Ai) I where the meaning

is clear. Propo@ition

2.1

Let for

,

(A 2)

12

,

.

.

.

,

1 ~ h ~ k ~ n , let Mh, k c Hom~dir

be finite. in

(AI)I I

A-dir

(Bk ) ) I

(An )

~,k

= 8kl~

e

((Ah)~,

Then there exists isomorphisms

, 1 _< r _< n, and

In

8r:

A-dir

,

and,

(Ak)ik) (Br)i---~=(Ar) I

08 h c HOmA--dir ((Bh) I,

such that: (i)

set I, and

all

r

(Br)i are indexed by the same directed

161-

-

each morphism

(ii) by

(%,f)

where

= 11

Proof. (A 2 ) I 1

x

I

If t h e r e

and n o m a p s ,

12

2 = { (il,

if and

only

if

the o b v i o u s

i2)lil i I ~ i{

If w e h a v e

and

e I2'

Br (il,i2)

r

(A I)

(if'

systems.

and

I

1 set

directed i2)

Then

i 2 ~ i~}.

one m a p i2

of d i r e c t e d

two o b j e c t s

i2

and we

~ (A 2)

ii

are

by

c a n be r e p r e s e n t e d

f is a m a p

e If,

morphisms,

(%,f) : (A I)

M' h,k

we can construct

be g i v e n

(Br) I 1 x 12

and

in

' (if'

~

i'

2)

let

= A{ i r , r = 1,2,

with

are done. represented

then

we can

by first

choose

I

(BI)j

8-~--~(AI)__ ii

whose

image

indexed

and

objects

by a cofinal

(BI) I

=84 > ( B I ) j

finite

number

of

subset

where

by a c o f i n a l

We have

N

in

Then

the

image

(A 2) I2

of

set

{ili whose

12.

objects

are

we can choose

(@' , f' )0 e~

subset

(%,,,f,,) w h o s e

our

(@,f)0e~

i 0 ~ I is p r e c e d e d

(i.e.,

(~" , f")

are

12.

each

i ~ I

and

(%'",f",) N

~

{A~, 2 (j) IJ E J }

is finite) indexed

(~' ,f')

a

~ i o in I} image

Finally give

by o n l y

objects

we can

the d e s i r e d

choose (B2) I.

result:

(AI)I l (%'f)

.> (A 2) 12

(i, f ,,') ~

(BI)

(B2)

I The methods.

situation

with

n maps

c a n be h a n d l e d

by

similar

-

162-

Theorem 2.2 m

Let A be any category. (i)

Then

~dir is closed under lim (over directed sets).

(ii) The obvious covariant is a full embedding, F: A

)B

functor

and, given covariant

I~: A

functor

where B has !i~ (over directed sets),

exists a unique Fdir: A-dir

(up to equivalence)

)B

which preserves

and yields the commutative A--d.ir

(iii) A additive (iv)

covariant li b

)A -dir

there

functor

(over directed sets)

diagram: Fdi r

implies -dir A is additive 9

A small and abelian implies -dir A is abelian,

satisfies AB 5, and has enough injectives.

I~: A

) A-dir

is exact. Proof.

(A~)ieij ' jej conglomerate

In

A--dir li

(A~)

=

i~lj

, the directed system obtained by the of all objects and maps 9

F dir is defined by

Fdir(A i; uii,)i = li~ {F(Ai) ; F(uii,)} I use Proposition and abelian in computed

9

We can

2.1 to show that A-dir is additive (iv); for example,

"pointwise".

lim is exact

in B.

in (iii)

kernels can then be

The AB 5 property is immediate:

Since we have a set of generators

namely the objects of ~, there is a generator.

in A-dir This

I

-

ensures

the e x i s t e n c e

of e n o u g h

to

A e A

injectives.

_dir (Ai) I r A

Let us say that

object

163-

if and only

if

admits

~ final _dir (Ai) I is i s o m o r p h i c in A

(A).

Propgsition

If

2.3

(Ai, uii,)i

(i)

there

J ! io, Aj = A e Bj (ii) that

ajj,

admits

exists

i0 e I

for some

for e a c h

Proof.

If

such

there

(r

be the i n d e x of the i m a g e

that,

exists

for e a c h

a final

> (Ai,

e i i , ) i ----> (A) of

j' ~ j

such

.

(A)

object

A, t h e n

e A, a n d

> A 9 Bj,

(p,g) : (Ai,

at the c o m p o s i t i o n

object

(Ai, ~ i i ' ~ a d m i t s

isomorphisms

and its i n v e r s e

Bj

j > i0,

= 1 e 0: A 9 Bj

then we h a v e

a final

(~,f)

of the i s o m o r p h i s m s ,

.

and,

easily

object,

~ii,)i We

let i 0

looking obtain

the

results.

3.

Let tive

functors

F,

DERIVED

{Fn}n>0

n: F

)F ~

= F*: ~----~ B

with A small

AB 5, and F* a c o n n e c t e d be a n a t u r a l

FUNCTORS

and a b e l i a n ,

sequence

be c o v a r i a n t B abelian

of functors.

transformation.

addi-

satisfying

Let

-

Let

{A*:

is a r e s o l u t i o n category lifting

0

) A

1A in A.

) A~

Given

f: A

pushouts

f*: ~ A

is a f u n c t o r .

Theorem

Sn

-A

:

. . . [A*

of the s m a l l

of r e s o l u t i o n s

) B

of A

and resolution

We then have

) -B, n ! 0, g i v e n b y

f*(A*)

A* of B;

the i n d u c e d = A *lim e ~ A~ H n (F(A*)).

Sn(A)

3.1

(i)

n: F

are e q u i v a l e n t :

) F~

transformation

functor, ~: F ~

~

we get a resolution

The f o l l o w i n g

natural

) A'

all m a p s

of A, by u s i n g

functors

-

of A in A} be the o b j e c t s

~A with morphisms

) ~B

164

there 9 L

exists

is u n i v e r s a l

6: F

left exact

~ L, a c o v a r i a n t

a unique

natural

(i.e.,

given

left exact

transformation

such t h a t the d i a g r a m F 0

L commutes),

and F n + l is the r i g h t s a t e l l i t e (ii) The c o n n e c t e d

and,

if

functors there {f

n

G* = {Gn}n>0 with

exists

Y: F unique

}n>0 = f*: F*

sequence

is an e x a c t ~ G~

natural ~ G*

of f u n c t o r s

connected

a natural

such

transformation,

t h a t the d i a g r a m

:fo GO

F* is e x a c t ,

sequence

transformations

Fo

commutes;

of F n, n ~ 0;

of then

-

(iii)

Consider

165

-

the d i a g r a m

~dir

Fdir

A

Fn = Rn(Fdir) I

Then

= Rn(F dir)

0 I~

, n > 0 , where

f the

Rn(F dir)

(defined

are the right

on a c a t e g o r y (iv)

Fn

Proof. F* = {F n}

with

Sn

=

enough

, n

>

An

(exact)

: ~ ---~

lifts

n>0

F *dir = {(Fn)dir}n>0" . ~ d i r functor

F: A - - - - ~ B

Fdir:

~dir

lifts

to a n a t u r a l

defined

) B.

by

lifts

We c o n c l u d e

{A*:

~ (A) -----)(A~

resolution C--dAir w i t h

of

morphisms

n > 0, given by that

~

all maps

Tn(A)

sequence

to an

(exact)

) B

Similarly

connected

ndir:

i) }

and

functor

in B.

are

(iv).

~

.

be the o b j e c t s of r e s o l u t i o n s functors

.

.

= T n , n > 0.

these straightIf we

IA*

is

let a

of the c a t e g o r y of

(A) l i f t i n g

T n : -A d i r

- A*elicit~ H n (F dir (A*)) .

Rn(F dir)

) G dir

Using

(ii)---~(iii)

(AI)II

the i n d u c e d

n: F -----~G

F dir ,

(iii)r

sequence

a left e x a c t

transformation

by p r o v i n g

(A) in -dir} A

1 (A) in -A d i r , we have

immediate

connected

transformation

(i) ~===~ (iii)

of F dir

injectives);

to a left e x a c t

A natural

forward.

functors

0.

n dir(A.) 1 I = lii ~ { n ( A

observations,

0

derived

Clearly

) -B,

It is -C A c C--dAir

-

Conversely, the

if

finite 9

A*

maps

(A n ) in

sequence

)

e C-~Air

0

~ (A)

)

(B n)

(B)

) (B n+l)

.

can use ; (A ~

in A* 9

~ (B ~

I

I

Since

(B) I.

Proposition

>

I

. . .

H n (F d i r (A*))

=

, we

use

Proposition

at

resolutions

I

The

situation

reduces

to

may

looking

over

A @ B., j > i 0 in I. It is e a s y to s e e 3 resolutions map into resolutions o v e r A. Hence,

we

2.1 on

~ . (A') ~ . . . Io I1 W e o b t a i n the i s o m o r p h i c

> (B I)

I

~l i m F (Bi)n I ) = l iI m) H n ( F ( B n ) )

Hn(

on

, we

; (A n + l ) i n + l

0

166-

are

that

2 3

such

done.

REFERENCES

[1]

Buchsbaum,

D

A.,

9

Ann. of Math., [2]

Freyd, York,

P.

J.,

71;

Abzs

"Satellites 199-209,

and

Universal

Functors"

(1960).

Categories,

Harper

and

Row:

New

1964.

MacLane,

[4]

Mitchell,

B.,

New

1965.

York,

S.,

Homology,

[3]

Springer:

Berlin,

Theory of Categorizs,

1963.

Academic

Press:

8

-

SIMPLICIAL

167-

DERIVED FUNCTORS

by

Myles T i e r n e y

1o

and W o l f g a n g

INTRODUCTION

In this note we sketch where)

a theory of d e r i v e d

eralization

of the classical

functors

and a p r o j e c t i v e

After the c o m p a r i s o n w

In w

notions cotriple

class

kernels",

introducing

of derived ~

in

rived functors is abelian 0-level

addition

E

of E i l e n b e r g

if

A

has

concepts

functors

the theory

then our p r o c e d u r e

is additive, and Moore

functors

class

limits

and s t a t i n g

of w

yields

the

of D o l d - P u p p e

in

with other

if there is a

of Barr

as models,

then we o b t a i n

[4].

finite

are d e f i n e d

agree w i t h those of Andr~

functors

project-

the given p r o j e c t i v e

derived

the p r o j e c t i v e

derived

that kernels

this theory one can

In p a r t i c u l a r ,

that realizes

of w

gen-

is abelian~

the d e r i v e d

the cotriple

With

the basic

functors.

A

[2] ; if we choose

ized)

B

except

and absolute

)B

we show how to compare

then we obtain

A

ones.

E: A

and

theorem,

that is a simple

procedure,

ives are replaced by relative arbitrary

(details w i l l appear else-

functors

are replaced by "simplicial

derive

Vogel

class,

and Beck

then the de-

[I]o

(suitably

Also,

if

relativ-

[3], and if in

the relative

theory

-

We Evrard

remark

theory

t h a t the m e t h o d

[5] in the a b s o l u t e

us t h a t b o t h

case.

of

w

Michael

h e and M a x K e l l e y h a v e

was

also used by

B a r r has

informed

also considered

this

(unpublished).

2.

2.1

168-

DERIVED

FUNCTORS

Definition

Let

A

be

a category,

and

fo

X

J '

)

Y

fn a

sequence

kernel

of

of

n

(fo,

+

1

_ -,

_

A-morphisms fn)

for n >_ 0 .

A sim~licial

is a s e q u e n c e

k~

) K

| I

X

kn+l

of

n + 2

0 < i<

j

A-morphisms < n +

1

,

and

satisfying universal

fikJ = f J - l k i with

T h a t is, if h o

H

! J h n+l

X

respect

to

for this

property,

-

is any o t h e r 0 < i < such

sequence

kih

= hi

Obvious simplicial

morphisms

2.2

and

lat e r

exists

yields

for

a unique

h: H ~ K

of a co-

of d e r i v e d

functors

are u n i q u e

up to iso-

class.

simplicial

if t h e y e x i s t ,

the n o t i o n

a theory

to an i n j e c t i v e

Clearly,

fihJ = f J - l h i

0 < i < n + 1 .

dualization

kernel,

respect

prove

satisfying

j < n + 1 , then t h e r e

that

with

169-

kernels

and c o n c e r n i n g

existence

one c a n

easily:

Proposition

If exist

A

has

finite

for any s e q u e n c e

limits,

(fo

_

We review briefly in

A

of

A.

, for d e t a i l s W e say t h a t

see

_

, fn )

the n o t i o n

[4].

f: A

_

then simplicial

Let

~A'

P

a n d any

n

kernels > 0.

of a p r o j e c t i v e b e a class

is P - e p i m o r p h i c

class

of o b j e c t s if for all

X s P , A(X,f) is s u r j e c t i v e . A c A

We c a l l

there exists

~A(X,A')

P a ~ro~ective

a P-epimorphism

A sim~licial out degeneracies)

: A(X,A)

o b j e c t of

in a d i a g r a m

A

class

e.. X

~A

augmented

if for e a c h with

over A

X r P.

(with-

- 170-

3 ~

d o

n-i X

: ,

n

"X

I i l

n-i

~n ~

D o oh

Xn_ 2- _ _ X 1 ) ~n-i

1

n-1

1

n

,A

Xo 9

~O

in

A, w r i t t e n

X~A,

such

~i n-i i

The

are c a l l e d

~j = n

that

~j-1 n-i

i

0 < i < j < n

and in the

face o p e r a t o r s ,

future we will

n

denote

a diagram

s u c h as

o

~n )

x

X n

j

n-i

I

~n n

simply by

X

0

We w i l l

to

such

n. as

n )X n 1 ' a g r e e i n g

n

~i n

also often

if this

~o

and

X

omit

does n o t

always

that

i runs

the'subscripts

lead to c o n f u s i o n .

from

in m o r p h i s m s If f: A

)A',

~o

)A

o v e r A and A',

and X'

)A'

are a u g m e n t e d

then a m o r ~ h i s m

~

over

m

f X

~X'

A

)A'

simplicial

f, w r i t t e n

objects

-

consists of

~'n:

~-morphisms

171-

for n > 0 satis-

X-----~X~ n

fying

n ~n = ~n-I

n

for 0 _< i _< n , where we put f-i = f" ~O

A, and

If P is a projective class in

X

~A

is an augmented simplicial object over A, then we say ~o X ~A is P-projective if each X n ~ P . If A has simplicial kernels, then we have a factorization (2.3)

~o n+l

n

0

1

~A

Xn+len+l~ ~ /i+ i ) xn%

/ ~

~'-i---

Xe l ~

%+1

for

KI

kin Kn-------~Xn_ 1

where

)/~~

i

~n-i

is

n > i, if we put m

a simplicial X

= A.

X

kernel ~o )A

of

Xn_ 1.

~Xn_ 2

is said to be P-exact

--1

~O

if ~o o ~ro~ective

exact.

and

e

n > 1 n

are

resolution of By 2.2, if

be so resolved.

is a

P-

P-projective and

P-

P-epimorphic.

X

;A

--

A

A

if it is

has finite limits then each

A E A

can

Moreover, such a resolution is unique up to

simplicial homotopy equivalence as results immediately from

-

172

-

the following comparison theorem. 2.4

Theorem 0

Let

P-exact.

X ~ )A

be a P-projective

Then any morphism

fz A

.)A'

and

X' ~* ~A'

be

can be extended to

a morphism

A

over f.

Furthermore,

homotopic,

'

any two such extensions

That is, if

f, then there exists

f

T,~z X

)X'

are two extensions of

~-morphisms

hiz X ----) n n Xn+ 1

0

9

i

such that

~o

n+l

h o

n

=

n

~n+l h n n+l n = ~n

i hj =

and

~

hj-I 3i

i 9 j

h j ~i-i

i 9 j + 1

~j+l hJ+l . ~ j+l hj

are s i m p l i c i a l l Y

9

173

-

By u s i n g

similar

-

(but easier)

techniques,

one

can

O

show

that

a P-projective

degeneracy

operators.

resolution That

sJ:

is,

Xn

X ~ )A

there

)Yn+l

0 < j < n --

i

sJ-i

~

i

i < j

J

i = j, j + 1

s

s j ~i-i

In g e n e r a l

the i d e n t i t y

fied,

as we s h a l l

but,

E: A

Given

2.4,

when

B

)B

a projective

us denote mension Y

by n )Y

n

see,

First,

the chain Y

n

,

n- 1

resolution

and if

to d e r i v e A

Y

has

complex

is abelian.

a functor

finite

limits

whose

object

category,

component

~

operator

.

Now

~ o. A

of

A

if

A c A

and define

A----~B m

D

for n > 0 by s e t t i n g - H

n

let

in di-

n X

and

i

(-i)

L E." n

L E(A) n

A

is a s i m p l i c i a l

boundary

i

is n o t satis-

when

in an a d d i t i v e

and w h o s e n ~ i-0

i < j

can be a c h i e v e d

it is clear h o w

P.

is

a P-projective

sis j - sJ+is i

degeneracies)

kY is

i > j + 1

is a b e l i a n

class

(with or w i t h o u t

n

--

that

~

9

pseudo-

exist

n

such

has

(k E X)

0

choose

-

The definition resolution

is independent

chosen

3.

jective

3.1

174

A

class

P.

of the

(up to isomorphism)

and functorial

COMPARISON

Let

-

in

WITH

A

by 2.4.

OTHER T H E O R I E S

be a category with

finite

limits

and a pro-

We have

Lemma 2 ~

Let X

c P

A

r

A

If

.

the augmented

Y

is P-exact,

set

(without degeneracies)

simplicial

_A(X,Y) has a contraction.

>A

then for each

>A(X,A)

That is, if we set Y-I = A, then there

are

functions h for n > -i

n

:

A(X,Yn) --

such that if

) A ( X , Y n + I)

f- X

>Y

--

~i hn n+l

and

(f)

= hn-i

8n+l n+l hn

Now if

E: A - - - - ~

choose

as models

subcategory

of

whose

3.1

A

the resulting

and 4.7

of

[1],

(~if)

0 < i < n ---

(f) = f

is abelian,

Denoting

, then n

is an arbitrary

functor where

for the Andre theory

objects

homology

[i]

are the p r o j e c t i v e s theory by Hn(

the full in P.

,E), we have by

-

3.2

175

-

Proposition

There

are n a t u r a l L

Let

G--

P = P~ , where existence

shown

E ~ )H n

(~,r

P~

isomorphisms

(,E)

be a cotriple

in

A

such that

is the class of ~ - p r o j e c t i v e s

of the c o n t r a c t i o n

any P - p r o j e c t i v e plicial

n

(in A and E)

resolution

~-resolution

[2].

given in 3.1 implies in the sense of

in the sense of

there that cotriple h o m o l o g y

[2]

w

The

easily

that

is a sim-

, w

and it is

may be c o m p u t e d with such

resolutions 9

(Simplicial ~ - r e s o l u t i o n s

generacies

[2], but these play no role in the argument.)

in

are assumed

to have de-

Hence we obtain

3.3

Proposition

If

@

is a cotriple

in

A

with

P = P

, then there

- -

are n a t u r a l

isomorphisms L n E ~--~->H n

(,E)

Since most classical examples

of cotriple

of e a s i l y compute

constructed

these

simplicial

3.3

" theories

provides

resolutions

appear as

a large class

from w h i c h

to

theories.

and pullbacks,

simplicial

cohomology

cohomology,

Assume n o w that object

n > 0 --

9

object

in

A

it has A

is abelian. all finite

Since

limits 9

(with or w i t h o u t

A If

has an 0X

degeneracies)

is a then

-

the normal complex

NX

176-

of X is the chain complex given by

(NX)~

= X0

/-~ (NX)n = i> 0 ker

and

=

8n

no

I

i {8 " .LX

(NX)

If P is a projective 4.1

.~Xn_l}

n 9 0

n

class in

A, we have

Proposition If the augmented simplicial object

exact in the sense of

w

X 8~

is P-

t h e n the augmented chain complex

O

NX

~) A

for each

is P-exact in the sense of Eilenberg-Moore P s P

...

[4], i~

the sequence of abelian groups

)A(P,_

(NX)n)

AIP, X )

)A(P,_ (NX)n_ I)

)

...

)AIP, A ) - - ~ 0

O

is exact.

Let us denote by S (A) the category of genuine s implicial objects over degeneracies isfied. write over In

A.

That is, an

X z S (A)

has

and all the usual simplicial identities

Morphisms C(A)

Ao

commute with

are sat-

faces and degeneracies.

for the category of positive

Then,

faces and

chain complexes

the normal complex is a functor

[3] it is shown that N has an inverse,

We

N:S(A)----~C(A).

i.eo there is a

-

functor

K: C(A)

) S(A)

there are n a t u r a l

177-

C c C(A)

and for

and

X c S(A)

equivalences ~C: C~----~NKC

#X: K N X ~-----)X

It follows

from the c o n s t r u c t i o n

we assume

(as we now do)

of coproducts n > 0 i ff C --

Now

to w r i t e

we can still, iterated

~X" KNX

that commutes with generacy

operator

n

[3] that if the formation

P for each

e

X

if

has only pseudo-

where

generacies.

If

satisfy

s

i

s

j

of Dold-Puppe, generacies. degeneracy to

In essence, X

degeneracies,

f. X---+Y

Y X

-- s

i

i <

shows

faces and de-

to have only p s e u d o - d e meaning

they

j, then this ~ is the above it commutes with

c o m m u t a t i o n with

to prove Lemma that

~ is n a t u r a l with

degeneracies,

and then, of course,

is e n o u g h

a

last p s e u d o - d e -

c o m m u t i n g with

has genuine s

define

and the This

is also assumed

j+l

a fixed ord-

)X

in eachdimensiono

In any case,

~X

by c h o o s i n g

all face o p e r a t o r s

to morphisms

generacies,

ies of

(KC)

then

P for each n > 0.

operators,

er in w h i c h

applied

given in

n

degeneracy

respect

K

that P is closed under

and retracts, ~

of

(#X)

w h a t we have p r o v e d

n

3.17 of

faces

all de-

and the last

[3], w h i c h when

is an i s o m o r p h i s m

for n > 0.

is that the p s e u d o - d e g e n e r a c -

can be r e p l a c e d by d e g e n e r a c i e s

satisfying

all the

178-

-

simplicial Barr

identities.

(unpublished)

mation

by means of a direct,

E: A

(relativized)

>B

is the following.

resolution

T h a t is,

be an a r b i t r a r y

m e t h o d of D o l d - P u p p e

functors

jective

4.1.

fact has been p r o v e d by inductive

approxi-

process.

Let

rived

A similar

~

C

~

Then the

functor.

for d e f i n i n g

(0-level)

For A r A choose

de-

a P-pro-

in the sense of E i l e n b e r g - M o o r e .

is P - p r o j e c t i v e

and P-exact

as defined

in

Then set

L'E(A) n

= H

n

By the m e t h o d of w ial r e s o l u t i o n The derived

X ~~

functors

(kEKC)

choose

and assume of w

L E(A) n

a P-projective

simplic-

it has p s e u d o - d e g e n e r a c i e s .

are given by

- H

n

(kEX)

but we have (+x)

9

n

for each n 9 0. that of A.

NX ~a~.A Since

Hence,

since

#X induces

complexes

in

n

P is closed

is an E i l e n b e r q - M o o r e

it follows

P-projective

an i s o m o r p h i s m

k (EKNX)

of chain

x

n

) k (EX)

B, we obtain

isomorphisms

by 4.1

resolution

-

(4.2)

Here

L'E(A) n a little

phisms

care

are n a t u r a l

Finally, and

B

are

must in

~

)L E(A) n

be t a k e n

n 9 0 - -

to s h o w

that

these

isomor-

A.

consider

abelian

179-

and

E

the c a s e

E: A-----)B w h e r e

is a d d i t i v e .

If A

E A

A

and

O

X ~ ;A

is a P - p r o j e c t i v e

resolution kEX

as c h a i n ly,

complexes

chain

in

equivalence

B. and

in the

sense

of

w

then

= EkX

Letting~

and ~

isomorphism

denote,

of c h a i n

respective-

complexes,

we

have (4.4)

kX ~

k(KNX) ~

N(KNX)

~

NX

.

O

Since

NX

)A

is

a P-projective

resolution

in the s e n s e

of

O

Eilenberg-Moore,

it

Thus

functorscf

the

derived

Naturality

follows

is t r i v i a l

from w

in this

(4.3)

coincide case.

that with

kX those

)A of

is also. [4].

180-

-

REFERENCES

[1)

M. Andre,"M~thode

Simpliciale en Alg~bre Homologique

et Alg~bre Commutative",Spa/nger

ie~t~re Noted in

Mathematied, 32, (1967). [2]

Mo Barr, J. Beck,"Homoloay (to appear in Spa~ngz~

[3]

A. Dold, D. Puppe,

and Standard Constructions," Lecture Noted.)

"Homologie nicht-additiver

Ann. Inst. Fourier, ii; 201-312, [4]

S. Eilenberg, J. Moore, logical Algebra",

[51

M. Evrard,

Functoren".

(1961).

"Foundations of Relative HomoAMS Memoir, 55, (1965).

"Homologie dans les Categorles non Ab~liennes",

C.R. Acad. So. Parid, 260~ 749-751, 1044-1051, (1965).

-

181-

ACYCLIC MODELS AND KAN EXTENSIONS by Friedrich Ulmer

Introduction The aim of this note ils to point out the relationship

be-

tween the technique of acyclic models and the standard procedure in homologlcal

algebra to compute derived functors by

means of proJectives, l)

This leads to a useful generalization

of acyclic models reflecting of a functor

G

the fact that the derived functors

can be computed more generally by G-acycllc

resolutions. Let

C

and

proJectives,

A

be categories,

and let

small category,

J : M--~C

[M,A]

and the restriction

Rj

adJoint

~[C,A]

: [M,A]

that for every extension of

abelian with sums and enough be the inclusion of a full

the objects of which are referred to as models.

Then the functor category

Ej

A

t : M-~A t .

has also enough proJectlves

[~,A]-~[M,~]

, T~-~T'J

called the Kan extension. the functor

Roughly speaking,

Ej(t)

algebra to compute the left derived

l)

Ej

: [M,~]--~[~,~]

: C-~A

the technique

models turns out to be the standard procedure

sion

, has a left

functors

Note is an

of acyclic

in homological of the Kan exten-

by means of projective

resolu-

The method of acycllc models was introduced by EilenbergMacLane [6]. Barr-Beck [2] gave a different version by means of cotrlples.

-

182-

tlons. 2) Eilenberg-MacLane T. , T. : ~--~A

[6] showed that two complexes of functors with isomorphic augmentations

homotopically equivalent, "representable" for and

~.M-~T

IM

sion of

T'J

T'J

are

Tn ' Tn

are

provided the functors and the complexes

T.M -~T_IM

are exact for every model

-~0

show that a functor restriction

n~ o

T_l--~@_l

T : C--~A

is "representable"

Is projective in

is again

T

(i.e.

above homotopy equivalence between

[M,A]

T@

T_l.J

and

We

iff the

and the Kan exten-

Ej(T.J) = T ).

In the f o l l o w l n g way 3)., The complexes projective resolutions of

M aM .

--~0

and T..J

Thus the

T @ can be obtained and

T_l.J .

T..J

are

Hence there is

a homotopy equivalence

h. : T . . J ~ T . . J

whlch is compatible

wlth the augmentation

T_l.J ~ T 1.J .

Applying the Kan exten-

sion

Ej : [M,~]'--~[~,A]

yields a homotopy equivalence

2)

I observed this after I had received a first draft of a paper by Barr-Beck in 1967. Their improved version of acycllc models in [3] w and the fact that cotriple derived functors are the left derlved functors of the Kan extension make fairly obvious that acyclic models and Kan extensions are closely related. In a recent paper of Dold, MacLane, Oberst [5] a relationship between acyclic models and projective classes was pointed out. As in Andr6 [1] and Barr-Beck [3] the Kan extension does not enter into the picture of [5] and all considerations are carried out in [C,~] , the range of the Kan extension. It appears that the use of the Kan extension establishes a much closer relationship between acycllc models and homological algebra than the one in [5]. Moreover, it gives rlse to a useful generalization of acyclic models which cannot be obtained by the methods of [5].

3)

This was also observed by D. Swan (unpublished).

-

Ej(h.)

: T.~

T.

183

-

because by the above

Ej(T..J) = T.

and

T,, hold. This procedure also shows that the n-th homology of value of the n-th left derived functor of i.e.

LnEj(T_I.J ) ~ Hn(T.)

Hn(T ) : C

)A

Hn(=,T_l'J)

holds.

Ej

Moreover,

at

T.

is the

T_l.J ,

one can show that

is isomorphic with the n-th homology functor

: C--~

of Andr~

Grothendieck AB4) category.

[1] P.3, provided

A

is a

(Andr4 denotes the category

It is well known that the left derived functors of

by

N .)

Ej

can be computed more generally by

This suggests defining a functor sentable" if

Ej(T.J) = T

and

Ej-acyclic resolutions.

T : C-~A

as "weakly repre-

LnEj(T'J ) = 0

for

This weaker notion of "representability",together acycliclty,

n>o with

no longer guarantees a homotopy equivalence between

the complexes

T.

and

T.

as above, but only that they have

isomorphic homology and that every existing chain map f. : T . - ~ T .

compatible with the augmentation induces a homol-

ogy isomorphism. T.'J ~ ~.'J

Moreover,

can be extended to a chain map

of the Kan extension topy equivalence functors

every chain map on the models

Ej : [M,A]--~[C,A]

.

T.--> T.

The lack of a homo-

can be compensated by conditions

F : A-->~'

, guaranteeing

is still valid and that

Ff.

that

by means

on additive

Hn(F-T .) ~ Hn(F'T .)

: F.T.---~FT. induces a homology

isomorphism. Morphism sets, natural transformation and functor categorles are denoted by brackets

[-,-]

, comma categories by

- 184-

parentheses U-categorles,

(-,-)

.

We assume that all our categories are

where

U

is a sufficiently

large universe,

use the same terminology as Verdier in the appendix of naire geometrie algebrique

1964".

and we "SEmi-

We mostly refrain from men-

tioning this universe except when we also consider categories belonging to a larger universe. abelian groups are denoted by

The categories of sets and S

and

Ab.Gr.

The phrase "Let

be a category with direct limits" always means that direct limits over U-small index categories.

D

is not necessarily small.

prove that this specific

has

However, we some-

times also consider direct limits of functors where

A

F : D ->A

,

Of course we then have to

limit exists.

Our approach is based on the notions of generalized representable functors

(cf. Ulmer

(of. Kan [8], Ulmer text.

[ll]) and Kan extensions,

[13]) which prove very useful in this con-

We first recall these notions and some of their prop-

erties. (I)

Let

M

and

pair of objects

A Ae A

be categories, and

alized representable functor

MeM M-,A

A

with sums.

To each

there is associated a gener(cf.

[Ii]).

It is the com-

posite

A@[M,-]

where

[M,-]

: M->S

: M-~S--~A

is the hom-functor and

A@

: S -~A

is

- 185 -

the

left adJoint

assigns

(2)

of

to a set

Yoneda

Then there

J~

[A,-] the

lemma, cf.

: A-~S A-fold Let

[Ii].

.

Recall

sum of

A .

that

A@

: S-~A

4) be a functor.

t : M-~A

is a b i s e c t i o n

[A@[M,-],tJ --~-- [A,tM] natural

in

A ,

M

is an isomorphism

and

of abelian

The functors

Proof.

a pair of adJoint

{M,A]--> [M,S],

r

the usual Yoneda

t .

A~

(for

and

[A,-]

: A-~S

induce

[M,S]--~[M_,AI, S~'>A|

~>[A,-].r . lemma

the b i J e c t i o n

groups.

: S--~A

functors

is additive,

If

and

Thus we obtain by adJolntness r-t

and

s-{M,-]

and

).

[A(~[M,-],t] =~ [[M,-I , [ A , - ] ' t ] ~_ [A,tM] Q.E.D.

(3)

Lemma.

natural each

t, t'

transformation

M~M

~(M)'T(M) = formation.)

4)

Let

: M-~A which

be functors

is obJectwlse

and

split,

is a morphlsm

~(M)

id

(Note that

need not be a natural

~

t-~t'

i.e.,

there .

: t'M-~tM

~:

a

for

such that trans-

Then every diagram

For the dual notion of a corepresentable functor M-~A we refer to Ulmer [ll]. Note that a corepresentable f-unct--or is also covarlant. The r e l a t i o n s h i p b e t w e e n generalized r e p r e s e n t a b l e and corepresentable functors is entirely different from the relationship between covariant and contravariant hom-functorS.

-

186-

-7

t

f f f f

A|

>t'

can be completed as indicated.

In particular,

Jan, this shows that the generalized projective

relative

in

which are obJectwlse

[M,A]

Proof.

Since

~(M)

[A,tM]-9 [A,t'M] from the Yoneda (4)

Remark.

to the class

A

representable

is abel-

functors are

6~ of short exact sequences split exact. is split,

: tM-~ t'M

is epimorphic. lemma

if

the induced map

Thus the assertion follows

(2).

A(~ [M,-]

It is obvious that (3) remains valid if

is replaced by a direct sum

~(Ai~[Mi,-] ) . A

(5)

Another variation

abellan and

AEA

projective,

need not be obJectwlse preserves wise

~

objects

Ai

M--~A

functors Let and by (6)

and

A

are proJectlves are projective

Theor@m

in

A~[M,-]

in

[M,A] A .

is

W~ : t -~t'

the hom-functor

This shows that

[A,-]

and llke-

, provided

the

Using constant

one can show that the converse is also true.

be a morphlsm in r~

A

then the eplmorphism

split, because

eplmorphlsms.

( A i @ [Mi,-] )

If

of (B) is the following:

M .

Denote by

d~

its domain

its range.

[11].2~15.

with direct sums ( M

Let

M

and

need not be small).

be categories, Then for every func-

-

tot

t : M-gA

187-

there is a direct limit representation t = llm t d ~ @ [ r s , - ]

The objects of the corresponding M 9

phlsms of

The morphlsm set

ment if either empty (cf. Proof: erty of

:x' = Idd~

or

For every

[~, ,~! ]

consists

o<' = Idr .

mor-

of one ele-

Otherwise

it is

[12] 2.19).

We give an outline, @

index category are the

making use of the adJointness

, the dual of [12] 2.21 and the usual Yoneda X eM

and

Ya A

llm ----> td~,| llm

,

tX

=lim

lemma.

there are natural isomorphisms ~--

Im

~r~,X], [td~,Y~ ~

This shows that

prop-

d~[r~,X],

[-,X],[t-,Y] i ~

td~ @[r~,X]

[tX,Y]

.

J

Q.E.D. (7)

R e m a r k ~ ) Moreover, Y

: M_~

one can show that the Yoneda functor [M_,A]

, M x A

shares with the usual Yoneda embedding

5)

the property that it is

The properties (2), (5) and (7) show that generalized representable functors are a reasonable substitute for homfunctors in an arbitrary functor category [M,A] . In the following we omit "generalized" and call a functor A | [M,-] : M - ~ A representable. In order not to confuse Eilenberg-MacLa~e's notion of a "representable" functor with ours, we use quotation marks for the former. We will show later that the two notions are closely related.

-

dense

(cf. Ulmer

otherwords,

[12]) or adequate in the sense of Isbell.

every functor

presentables

t e [M,A]

(Y,t)

(8)

(K an_extension

Definition

functor

is the direct

In

limit of re-

over the canonical index category which is the

comma category

t : M-CA

188-

.

be functors. Ej(t)

: ~-~A

Let

[8]).

J

and

: M-gC

The right Kan J-extension such that for any functor

of

t

is a

s : ~ *

there is a blJection

[Ej(t),s] ~ [t,s.J] , natural in

s .

Dually,

~JCt) : C_-~A

the left Kan J-extension

t

of

is a functor

such that there are natural blJections

[s,~J(t)] ~ [s-J,t] It follows immediately up to equivalence.

that

Ej(t)

and

We are mainly concerned with

short we call it the Kan extension. to state the dual theorems faithful, Ej(t) (9)

EJ(t)

for

EJ(t)

ere determined Ej(t)

and in

We leave it to the reader .

If

J

is full and

it follows from (i0) below that

is an extension of Lemma.

Let

of a representable

t .

E (t).J ~ t , i.e. J This explains the terminology.

J : M .,->C be a functor. functor

A@[M,-]

Ae[JM,-]

The Kan extension

: M_-~ A_ is

: C_-~A

-

Likewise

189~4. (Ai~ [M1,-])

for an infinite sum

the Kan extension

is

[JMi,-] ) : C--~ A

(Ai|

Similarly, is

[aM,-]

Proof.

the Nan extension

: c--,s_

of a hom-functor

[M,-]

: M-~S

.

For every functor

s : C-~A

the Yoneda

lemma (2)

gives rise to biJectlons [ A ~ [ J M , - ] , S ] -~ [A,sJM] ~ [ A ~ [ M , - ] , S ' J ] which are natural in

S

The second half can be established

.

similarly. Q.E.D. Assume that

E (t)

exists for every functor

t : M-~A

J

Then the Kan extension the restriction direct limits.

.

~

[~,A]~

Ej

: [M,A]--~[~,A]

[M,A], s ~ s . J

This is also true if

.

is left adJoint to Thus

Ej

preserves

Ej is not defined every-

where. (i0) Assume

Theorem. A

lim td~ | , )

Let

J : M-,C

has arbitrary sums.

Then

t : M-,A Ej(t)

exists and

~a(t) is valid.

and

= lira t d ~ |

-]

be functors.

: ~-~A

exists iff

190

-

(Ii)

Corollary

limits,

(Kan [81~.

If

then for every functor

-

M

is small and

t : M C}A

A

has direct

the Kan extension

exists. Proof of ( i 0 ~

By (6) there is a representation

t = lit t d ~ |

.

Assume that

lim t d ~ [ J r ~ , - ]

From (9) it follows for every functor Wlim t d ~ [ J r ~ , - ] , L-~

=~/ lim (m

s : C--~A

exists. that

~ d ~ @ [ J r ~ , - ] , ~ "%" I--

<

Hence ly, if

Ej(t)

exists and

Ej(t)

exists,

(12)

Let

subcategory

it follows

that

Ej(t)

Converse. in

is the limit

.6)

J : M--~

be the full inclusion of a full small

(referred to as models) and let

category with sums and enough proJectives.

A

functor

P@[M,-]

is projective

in

Choose for every functor

A

.

: M-+A

be an abelian

As shown in (5), a

representable

6)

.

then by reading the above isomorphism

the opposite direction, lim t d ~ @ [ J r ~ , - ]

Ej(t)--- lim t d ~ [ J r ~ , - ]

is projective

iff

t : M-+~

P and

The theorem shows that the existence of Kan extensions is equivalent with the existence of certain direct limits in A whose index systems are not sets but proper classes. From-this it is fairly obvious that not every functor admits a Kan extension. For a counterexample see [13] p.6. Sometimes special properties of M or t guarantee the existence of E~(t) . For instanc--e, if t is a small direct limit of representable functors, then one can show, as in (i0), that Ej(t) exists, provided the range of t has direct limits.

-

every

Me M

tlve.7)

an epimorphism

191-

PM--~ tM , where

By means of the Yoneda lemma

PM

is proJec-

~PM~[M,-],t]

~ [PM,tM]

the family of epimorphlsms determines a natural transformation ~(t)

: M~M(PM@[M,-]) - > t

which can be easily shown to be

eplmorphlc (use the Yoneda lemma (2)). In

[M,A]

Thus

t

is projective

iff it is a direct summand of a sum of projective

representables.

If

t

is the restriction of a functor

T : C--~A , then this family and the Yoneda isomorphlsms BM@[JM,-], mation (13)

~(t)

=

determine also a natural transfor-

: H,.we( P M e [ J M , - ] ) - - ) T

Theorem.

Let

T : ~ >A

.

be a functor.

The following

are equivalent: I)

T

is "representable"

in the sense of Ellenberg-MacLane.

(We use quotation marks to distinguish thelrnotlon of representability from ours~ ll) T.J is

is projective in T , l.e.

Moreover,

[M,A]

and the Kan extension of

Ej(T.J) = T .

the restriction

Rj : [~,A]--P[M,A]

morphism between the "representable" functors projective functors in Kan extension

[M,A] .

Ej : [M,A]~-e[~,A]

tion of a "representable"

7)

T.J

sets up an iso[~,~]

and the

The inverse is given by the .

(This shows that the no-

functor (Eilenberg-MacLane)

and a

If A is the category of abellan groups, one can choose P~ to be the free abellan group on tM It is instructive for the following to have this example in mind. It links our approach wlth the one of Eilenberg-MacLane [6].

-

representable

192

-

functor (Ulmer [II]) are closely related.

ally the functor

T

is also projective

in

Actu-

[~,A] , but thls is

irrelevant in the following.) Proof.

Eilenberg-MacLane

sentable"

call a funct0r

T : ~-~

if the above natural transformation

~(t)

: ~ (PM@[JM,-])--~T M is a direct summand of

admits a section

T

,$ (PM ~[JM'-])

E (PM

"repre-

[M,-]) --- PM Ej-Rj

sum

on itself.

~(PM@[JM,-]) M direct summand T

of

~M ( P M @ [ J M

By (12) the restriction

ly, if

T-J

Kan extension

is projective and ~(t)

Since

J

is a full Inclumaps the

Obviously the same holds for a J

~])

T.J

'

i.e.

Ej(T-J) = T

is projective.

Ej(T.J) : T

: ~(PM@[M,-])--~T-J

?~

(in other words,

: [C,A] --> [M,A] --> [C,A]

valid.

epimorphism

.

(of. (9)) and

sion, the composite

~

: ~(PM|

is

Converse-

holds, then the

splits and" so does its .

It follows easily

M

from this and (12) that the restriction functor Kan extension

Ej

sentable" functors

Rj

and the

set up an isomorphism between the "repreC-~ A

and the projective functors

M~A

.

Q.E.D. From this we obtain immediately the following: (14) Theorem.

Let

T, : ~ - ~ A

be a positive complex 8) of

functors together with an augmentation

T.-~T

i .

The follow-

ing are equivalent: 8)

This means that Ing we abbreviate

T , = o for negative n . In the follow"positive complex" to "complex".

-

l)

The functors

Tn

are "representable"

augmented complex

ll)

T.'J

T.-J--~T 1 . J - ~ 0

is a projective resolution of

Ej(Tn'J) = T n holds for (15)

Corollar F.

satisfying T.

193-

i)

If

n~o

T*-~T-I

such that

T1-J

holds, then

Every chain map

T.

(i.e.

T. ~ T . .

Ej

at

By (14)il) the complexes

T..J ~ T..J .

and Is

is com-

T_l.J % T 1-J . 5.) Is the value of t , where

Hn(T.) ~ LnEj(t)

projective resolutions of equivalence

(and

T.

T.--~ ~.

provided its restriction on M

the n-th left derived functor of

Proof of (15).

and

.

T_I.J ~ T_I-J

the n-th homology of

T_l.J ~ t ~ T 1.J

and the

is exact.

patlble wlth the augmentation isomorphism Moreover,

n~ o

is another augmented complex

are homotopically equivalent.

a homotopy equivalence,

for

). T..J

and

T..J

are

t , and hence there is a homotopy Applying the Kan extension yields

Clearly the restriction of every chain map

f. : T.~-~T.

on

provided

f.J

T 1.J~T

1.J .

yields again

M

is a homotopy equivalence

f.J : T.J ~ T.J

is compatible with the augmentation isomorphism Applying the Kan extension

f..

Hence

f.

By standard homologlcal algebra

Ej

on

f.J

Is also a homotopy equivalence. LnEj(t) ~ H n E j ( T . - J ) = HnT *

holds. (16)

Remark.

The method of acycllc models can be duallzed by

replacing the representable

functors

A~[M,-]

by corepresent-

-

able functors

[-,A] - [-,M]

by Injectlve functors intro.).

left

"corepresentable" ~ : T

[.T=Id

injections

and projective functors

[-,I].[-,M]

Kan extension

has enough InJectlves.

l.e.

-

( 16~

Thereby the right Kan extension

placed by the

morphlsm

194

A functor

InJectlve, cf. [ll] Ej

has to be re-

(cf. (8)).

T : ~--~

Assume

is sald to be

(In the sense of Eilenberg-MacLane) ~-[-,IM]-[-,M]~_ _

.

EJ

P@[M,-]

(As In (12)

[TM~IM}M,

~'

Iff the

has a left inverse

~- ,

Is associated wlth a family of

9 , where

IM~ ~

Is InJectlve.)

Wlth

this, one can easily duallze (13)-(15) and also the considerations below. (17)

Remark.

Of course the notion of acycllclty Is self-dual. Roughly speaking, the above shows that the

method of acycllc models Is the standard procedure In homologlcal algebra to compute the left derived functors of the Kan extension by means of proJectlves. left derived functors of

Ej

It is well known that the

can be computed not only wlth pro-

proJectlves but more generally by Ej-acycllc resolutlons. 9) Thls leads to a useful generalization of acycllc models.

For

the considerations below, one can drop the assumption that has sums and enough proJectlves and assume instead that the Kan extension and its left derlved functors exist (for instance, if A

is a Grothendleck AB4) category (cf. [7]) or If

M

consists

of the proJectlves of a cotrlple, cf. [3]).

9)

Recall that for n > o .

t~[M,A]

Is called Ej-acycllc If

LnEj(t) = 0

-

(18)

Definition.

presentable"

A functor

iff

A

-

T : C --~A

Ej(T-J) = T

Clearly a "representable" If

195

and

is called

"weakly re-

LnEj(T-J ) = 0

for n > o

.lO)

functor is "weakly representable".

is AB4), then every direct summand of a sum

(A i @ [ J M i , - ] ) Ai ~ A .

is "weakly representable",

where

Further examples are A-representable

M jaM

and

functors in the

sense of M. Andre [1] p.ll. (19)

Theorem.

presentable" and

T.-e~_l

Let

where

Then t

: C--~A

such that

T 1.J ~ T 1.J

T..J--~T_I.J ~ 0

is a functor isomorphic

tended to a chain

T.-->T.

Ej : [M,A] -->[C,~]

.

1

is valid and the augT..J ~ T 1 - J - ~ O holds for

T 1.J

T..J--~T..J

or

are

n ~o ,

T 1.J .

can be uniquely ex-

In other words, the restriction of chain

f. : T . - ~ T .

vided its restriction on

lO)

to

T.-~T

by the Kan extension

[T.,T.]->[T..J,T.-J]

chain map

and

Hn( T. ) ~= L n E J (t ) ~= Hn(T . )

Every chain map on the models

maps

be complexes of "weakly re-

functors together with augmentations

mented complexes exact.

T.,T.

is a blJection.

Moreover,

induces a homology isomorphism, M

every pro-

is compatible with the augmenta-

There is a useful modification of the notion "weakly representable". Let ~ be a proper class of short exact sequences in [M,A] in the sense of MacLane [9] p.367, for instance, the class of obJectwise split exact sequences. Then the condltion LnE~(T.J) = 0 can be replaced by ~ L ^ E ~ ( T J) = 0 , w n e r e ~-L~ denotes the left derived functors with respect to ~ . If the complexes T. and T. in (19) below have the property that T.J-*T~J-90 and T~J--,T.IJ-~0 a r e ~ - e x a c t , then the statements (19)-(22) also hold if L~E~ is replaced by ~-L,Ej.

-

tlon

T -- l-J

Proof.

5_1

~

oJ

-

9

The first half is standard homological algebra

see Grothendieck

[7], RShrl

[T.,T.]-->[T..J,T..J] tlon isomorphism (cf. (8)). tion

196

[10])9

The restriction map

is a blJectlon because it is the adJunc-

[T.,T.] ~ [Ej(T..J),T.]~-

Hence for every chain map

Ej(f.J) = f.

striction of

f.

holds, where on

M .

Ej-acyclic

f.J : T . - J - g T . - J also exact and (20) are

Remark 9

[T..J ,T.-J]

f. :

T.-~T.

the equa-

f.J : T . . J - e ~ . . J

The mapping cone

Kan extension of the mapping cSne of sists of

(e.g.

f.J

is the re-

Cf~ of

f.

is the

II)~ The latter con-

objects and is exact because by assumption

is a homology isomorphism.

f. : T.--> T.

Hence

C~

is

induces a homology isomorphism.

It is clear from the above that

homotopically equivalent iff

T -J

and

T.

T -J

and

T.

are.

In

general there need not be a homotopy equivalence between and

T..

T.

In practice this lack can be compensated by the

following: Let

F : ~--~'

be an additive functor,

assume that the objects and

C ~E

.

isomorphism,

ii)

Then where

Ff.

TnC

and

Tn C

: FT. > F T .

f. : T.--~T.

A'

abelian, and

are F-acyclic

for

n ~o

still induces a homology

is as in (19).

This can be

For the definition of the mapping c~ne and its elementary properties, we refer to Dold [4]. Recall that a chain map g~ between complexes induces a homology isomorphism iff the mapping c6ne of g~ is an exact complex.

-

197

-

proved in the same way as above by means of the mapping cone technlquel2),If either Jects 13) or

A

~

is AB4) and has enough F-acycllc ob-

has sums and enough proJectlves,

construct a resolution

P.(t)

of

t

then one can

consisting in each dimen-

sion of a sum of representable functors which can be embedded into a diagram T.J ~ T-I 9J

P.(t)

~__

0

One can choose F-acycllc

Pn(t)

=

> T..J

t

~ --

T -i .J

0

0

~(QM@[M,-])

where

Q, is either

(first case) or projective (second case).

the values of

EjPn(t ) : ~--,A

Clearly the complex

are F-acycllc for

EjP.(t)--~Ej(t)

and acyclic on the models.

Moreover nDo

.

is "weakly representable"

Thus we obtain from the preceding

argument homology isomorphisms

Hn(FT.) = ~ H n (FEjP. (t)) =

H n (FT.) The theorem below, which we state without proof, gives further information about this problem. (21)

Theorem 9

Let

T : C -~A

be a "weakly representable"

12)

In the examples, the objects proJectlves.

TnC

and

Tn C

are usually

13)

i.e. every object in 6 is a quotient of an object such that LnF(Q) = 0 for n ~ o .

Q

-

functor and let

F : A --~A'

right adJoint.

198-

be an additive functor with a

Assume either that

gories with enough F-acyclic

A

and

A'

are AS4) cate-

objects or that they have sums and

enough proJectives. Then TC

F.T : C ~ A ' are F-acyclic

L.Ej(F-T-J)(C) (22)

for

objects

CeC

Let

of functors TnC , Tn C

CeC

.

and

T.

~T_I

from

C

and

MeM

.

TIM

other assertions [email protected]_I (23)

L F(TC)

and to

, cf. [14] (19).)

T*-'~T-I

A

as in (19) such that the

~ T iM

are F-acyclic

, where

for

n~o

T_I "J =~t =~ @ -I .J

denot~ the Kan extension.

in (19) also hold for

and

Clearly the

FT.--~FT_I

and

.

Remark.

The isomorphism

Hn(T.) ~ H n ( T . )

also be obtained from Andre's computational homology functors

H.(,-)

AB4) [cf. [I] P.7). H.(,-)

be augmented

Then it follows from (19) that

H n ( F T.) ' ~= L n Ej(F't) ~= H n(F@.) E~ : [M,A']--9"[~,A']

iff the objects

(This follows from the equations

~ L.F{Ej(T.J)(C))=-

Corollary.

complexes

and

is "weakly representable"

Ho(,-)

Thus an augmented

device for his , provided

A

is

One can show that the functors

: [M,A]--~ [~,A]

in particular

: [M,A]--~[~,A]

in (19) can

coincide with = Ej(-)

(cf.

complex of functors

L.Ej(-)

[14], ( 4 ) T.-~T_I

: [M,A]--->[~,A]y this volume). satisfies

Andre's condition in [i] prop. 1.5 Iff it is acycllc on the models and

Tn

is "weakly representable"

prop. 1.5 in [I] implies that

for

n~ o .

Hn(T.) ~ Hn(-,T_I'J)

Hence

~ H n ( T .) .

-

This shows that Andre's eralization

199

-

computational

of acyclic models 9

the notion

replaced by "weakly representable". this method to improve

category

[~,A]

"representable"

Barr-Beck

their original

in [4],but all considerations

method is actually a gen-

[3] w

used

version of acyclic models

were carried out in the functor

without using the Kan extension.

sentation in [3] w

being

Their pre-

made me realize the relationship

between

acyclic models and Kan extensions.

Before we discuss their ap-

proach, we sketch how the technique

of acyclic models works

when

M

(24)

Let

of

~

is not small. M

be a full but not necessarily

and let

proJectlves.

A

be an abellan category with sums and enough

Denote by

groups 9 the underlying large universe of objects

V

M , C and

the category of abellan

sets of which belong to a sufficiently

A .

Let

A

U

functors

from

A

to

which take U-small direct limits in inverse limits.

rect limits. Jectlves. PeA

I : A

Note that

~

The proJectlves is projective.

>A

is exact and preserves U-dl-

has V-direct

Define a functor

T : C-->A

: ~)(PM~[JM,-])--91"T IM

limits and enough pro-

are V-sums of hom-functors Using the V-completion

can proceed as before 9 and (12)-(22)

~(t)

(cf. intro.) and the sets

be the category

of contravarlant

The Yoneda embedding

where

A_~b.Gr.

which contains

COntu(A~149 Ab.Gr.

small subcategory

[-,P] 9

one

carry over to this case.

to be "representable" has a section (of.

if

(12)).

A

,

-

functor

T : C-->A

M J~ C T_~ A ~

extension.

is "representable" Iff the composite i

(I'T'J) = I'T

200-

is projective in [M,~]

holds, where

Ej . [M,~]

and

~ [C,~]

9

is the Kan

Two complexes of "representable" functors

T.,~. : C--*A

with augmentations

T.~> T_I

and

are

T.~T_I

homotopically equivalent, provided the augmented complexes

T..J-->T_I.J ~ 0 ~_l'J

and

is valid.

@ . . J ~ T_I-J ~ 0

Every chain map

is

T.--~@.

T_I'J ~ T_I.J .

is isomorphic with the value of where

T 1-J ~ t ~ T 1.J .

Jectlves.) T : ~-~

(Note that

Hn(T.) at

I.t,

As before, the notion of a "representable" functor can be replaced by the notion of a "weakly repre-

and

LnEj(T.J)=

or

for

n ~o .

and

Hn(T .) ~ LnEj(t)

0

Then

T~

Ej(I.T-J) = I-T and or

T.

and

with the augmentation isomorphism

f. : T.--) T.

also a homology isomorphism, provided C~ C

and

0

induces a homolM

is compatible

T_I.J ~ T_I.J .

is an additive functor, then

F-acycllc for

LnEj(I.T.J)=

I'Hn(T.) X Hn(I'T.) ~ LnEj(l't)

Likewise every chain map

A'

Ej(T'J) = T

still have isomorphic homology

ogy isomorphism, provided its restriction on

(25)

Moreover

[M_,~] has enough pro-

The latter can mean either

F : A~

homotopy

LnE J : [M_,~I-~ [C,~]

sentable" functor.

hold.

a

J~

is compatible with

equivalence, provided its restriction on the augmentation isomorphism

T

are exact and

TnC

If

Ff. : F T . - , F T . and

Tn C

is

are

n~ o , etc.

Barr-Beck [2] gave a version of acycllc models in which

-

the model category

M

201-

For the sake of complete-

is not small.

ness, we briefly recall their approach. in category note by

~

(cf. [3]) and

J : M ~ ~

T. : C - , A

Tn

triple.

and T.

GC , where

M

TnG

where

C e~

iff

.

A complex of funcis called

T.-~T_I

T..G -~T_I-G

has a contraction.

~

Tn

~ : G - ~ idc

admits a section is the co-unit of the co-

Barr-Beck proved in [2] that two complexes of repreT. , T. : ~ - ~

sentable functors tions

De-

is called representable iff the canonical nat-

ural transformation U- : Tn--~ T n G 9

be a cotriple

be an abelian category.

with an augmentation

acyclic on the models The functor

6

the inclusion of the full subcategory con-

sisting of the objects tors

~

Let

T-I

-~

T-I

T*-~ T-1 and

in [3] w

T.

with isomorphic augmenta-

are homotopically equivalent if

are acyclic on the models. 14) have isomorphic homology.

that for the latter (i.e.

T.--> T_I

In particular

Later they pointed out Hn(T.) :~Hn(T.) ) their

notions of representability and acyclicity can be considerably weakened.

Denote by

associated with

G

H.(-,T)G

: ~--~

the cotriple homology

and a coefficient functor

T : C--*A

Their modified definition of representability is: and

Ho(-,Tn) G :

Tn; of acyclicity:

Hj(-,Tn) G = O

for

J ~o

T.M ~ T _ I M

~0

14)

It should be noted that this homotopy equivalence cannot be obtained by the method outlined in (24) if one takes J : M--*~ to be the full inclusion of the objects of the form GC , where C ~ ~ . To obtain the homotopy equivalence one has to define J : M - ~ C in a more elaborate way which we omit in view of the more general situation we discuss below.

-

is an exact complex for ly weaker because hold

15)

202

M~M .

-

These definitions

Ho(-,T.G)~ = T.G

for every functor

and

T : ~-~A

and

connection between this (generalized)

are obvious-

Hj(-,T-G)G~ J 7o

.

To make the

version of acycllc models

and classical homological

algebra we first recall that

Ho( ,-)~ : [~,A]

is the composite

~ [~,~]

Rj : [~,A]--~[M,A]

with the Kan extension

(of. [14] this volume). the composite

of

Rj

Moreover

with

L.Ej

0

of the restriction Ej : [M,~] --~[~,A]

H.( '-)G : [~'~] --~[~'~] : [M,A]-~[C,A]

.

This shows

that the modified notions of acyclicity and representability Barr-Beck

[3] w

is

of

coincide with "acyclic" and "weakly repre-

sentable" as defined in (18). models is essentially

Hence their method of acyclic

the standard procedure in homological

algebra to compute the left derived functor of the Kan extension

Ej : [M,A]--~[~,~]

by means of

Ej-acyclic

resolutions.

We leave it to the reader to state theorems analogous

to

(19)-(22).

15)

Clearly the same is valid for every direct summand of T-G.

-

203

-

BIBLIOGRAPHY

[1]

Andre, M., Me~thode ~impliciale

et alg~bre commutative,

en alg~bre homologique

(Lecture notes in Mathematics,

#32), Springer, 19679 [2]

Barr, M. and J. Beck, "Acyclic models and triples", in;

Conference on Categorical Algebra, pp.336-343, Springer, [3]

(1966).

. "Homology and standard constructions",

(to

appear in Lecture notes in Mathematics.) [4]

Dold, A., "Zur Homotopietheorie der Kettenkomplexe",

Math. Annalen, [5]

140; 278-298,

(1960)9

, S. MacLane, U. Oberst, "Projective classes and acyclic models", in; A. Dold, Heidelberg and Eckmann (eds.), Report8 of the Midwest

Seminar,

Category

(Lecture notes in Mathematics, #47);

78-91, Springer, 1967. [6]

[7]

Eilenberg, S

and S

Am. J. Math9

75; 189-199,

(1953).

Grothendieck, A., "Sur quelques points d'alg~bre homologique"

[8]

MacLane, "Acyclic models"

Tohoku, Math

J., 9; 119-221,

(1957)

Kan, D., "Adjoint Functors", Trans. Amer. Math9 Soc., 87; 295-329,

(1958).

-

[9]

204

-

MacLane, S., Homology, Springer, 1963.

[i0] R~hrl, H., "Satelliten halbexakter Funktoren", Math.

Zeitschrift, 79; 193-223, (1962). [ii] Ulmer, F., "Representable functors with values in arbitrary categories" [12]

J. of Algebra, 8; 96-129, (1968)

. "Properties of dense and relative adjoint functors", J. of Algebra, 8; 77-95,

(1968)9

. "Properties of Kan extensions", Mim. Notes,

[13] E.T.H.,

(1966).

[14]

"Kan extensions, Cotriple and Andr6 (co)homology" in;

Lecture

notes

in mathematics,

this

volume.

-

205

-

I

D E R I V E D C A T E G O R Y AND P O I N C A R E D U A L I T Y

by

M.

The n o t a t i o n s functors

Zisman*

for d e r i v e d c a t e g o r i e s

are those of H A R T S H O R N E

Z, d e s i g n

[3].

Capital

locally c o m p a c t H a u s s d o r f f spaces;

and d e r i v e d letters X,Y,

A is a r i n g

w i t h unity given once for all; we w r i t e A x, A Y, ~or the categories

of sheaves of left A - m o d u l e s

over X,Y.

not s u p p o s e the ring A to be c o m m u t a t i v e , add the h y p o t h e s i s commutative

ring.)

(If we do

we g e n e r a l l y m u s t

that A is a A - f l a t A - a l g e b r a w h e r e A is a In the f o l l o w i n g a sheaf always means

a

s h e a f of A - m o d u l e s .

i.

THE F U N C T O R f!

i.i t

Let f: X sheaf over X. way:

> Y be a c o n t i n u o u s map,

We d e f i n e a s h e a f f!F o v e r Y in the f o l l o w i n g

G i v e n any open set U in Y, then flF(U)

all s e c t i o n s

and let F be a

s 6 F(F-Iu)

the r e s t r i c t i o n

flK: K

* I n s t i t u t Henri Poincar~,

is the set of

w h o s e s u p p o r t K c f-l(u)

is such that

> U is a p r o p e r map.

Ii Rue P i e r r e Curie,

Paris.

-

It is e a s y

to c h e c k

that

i. 2

>

flF(U)

a sheaf.

Examples

1.2.1. fiF =

Fc(X,F)

F with

If the

where

compact

subset

putting

space

Fc stands

Y is a p o i n t , for the

then we have

s e t of all

sections

of

support.

1.2.2. open

-

the presheaf U

is in f a c t

206

If

X i n t o Y,

(F Y)

f: X

"

- Y

is the

inclusion

then

flF is the e x t e n s i o n

of an

FY

of F to Y,

= 0 for y ~ X. I) Y

1.2.3. then

Of c o u r s e

f! is n o t h i n g

but

if f: X

the u s u a l

> Y is a p r o p e r

direct

image

functor

map, f,.

1.3

Consider entiable

manifold

the u n i q u e numbers. the

map

constant

the

of d i m e n s i o n

the c a s e w h e r e

sheaf

known

R over

) ~

following

>

X

n, Y is a p o i n t ,

f r o m X to a p o i n t ,

It is w e l l

0 has

for a w h i l e

that

a n d A = R the

the d e R H A M ' s

is a d i f f e r f: X

> Y

field

of r e a l

resolution

of

X ~o

;

fll

;

...

>

~n

> 0

property:

I) G i v e n a s h e a f F o v e r w r i t e F x f o r the s t a l k

a space X and a point of F o v e r x.

x 6 X,

then we

-

Given

any o p e n

207

-

set U in X we g e t Rqf! (~Pu) = Hq(U'~P)c = 0

for any q > 0 and any p > 0.

More

generally,

if f: X

> Y is a c o n t i n u o u s

and F a s h e a f o v e r X, w e say t h a t F is f - s o f t

if t h e

map

follow-

ing r e l a t i o n holds: R qflF u = 0 for any q > 0 a n d any U o p e n In o t h e r w o r d s , s e t U c X.

1.4

F is f - s o f t

if F u is f ! - a c y c l i c

(Confer G O D E M E N T

The c o n d i t i o n

(i) qo

for any o p e n

[2] f o r the n o t a t i o n

F ). U

A(A)

We say t h a t the c o n d i t i o n f: X

in X.

A(A)

holds

for a m a p

> Y iff f is of f i n i t e

> 0 such that

(ii) T h e r e

cohom0!0@ical

dimension

(i.e.

there exist

R q f ! F = 0 for any s h e a f F a n d any q ~ qo).

exist

a finite

) A

) no

resolution

of the c o n s t a n t

sheaf

A over X 0 where

the s h e a v e s

The maps

for w h i c h

~P are f - s o f t

following A(A)

three

9

...

> ~r

> 0

and A-(right)-flat.

lemmas

assert

that there exist

holds.

1 . 4 . 1 Lemma. dimension

) ~i

If f

!

is of f i n i t e

a n d if f is n o t h e r i a n ,

then

A(A)

cohomological holds

f o r f.

-

1.4.2 Lemma.

2 0 8

-

Let A

> B be a ring homomorphism

(which send the unity of A into the unity of B). holds

If ~(A)

for f, then ~ (B) holds for f. 1.4.3 Lemma.

and compositions.

~ (A) is closed under change of basis

Moreover,

if 5 (A) holds

for g.f,

it holds

for f~ Using these lemmas we easily for any map between any change

(topological)

of basis,

1.5 Theorem

see that ~(A)

manifolds

and then, using

for a lot of maps.

(Poincar~ Duality)

Let X and Y be locally compact Haussdorff and let f: X condition to natural

) Y be a continuous

~(A) holds for f.

(*) R Hom

(For a proof,

see

Hom D ,

Suppose

f!

:

D + (AY)

spaces

that the /

a unique

(up

> D § (Ax) pro-

isomorphism

(Of!F',G')

where F" is any object

map.

Then there exists

isomorphism ) functor

vided with a bifunctor!al

but

holds

~ - R Hom

in D(AX)

[4] expos~

(F',fIG ")

and G" any object

4.)

Since H~

Hom"

in D+(AY). is nothing

we get, taking the 0-cohomology: I

HOmD(AY ) (Rf!F'oG') so that the f u n c t o r

f"

I

is

in

~ HomD(AX ) (F',f'G') some sense

right

adjoint

to

Rf

I "

Q

1.5.1. following way.

The description

of f! can be done in the

-

Since

the category

whose

objects

sheaf

in the complex being

homotopy

D+(AY)

to the category

below complexes injective

of maps b e t w e e n

that G" is such a complex. 1.4

-

is e q u i v a l e n t

are b o u n d e d

classes

209

Given

of sheaves,

(and whose maps

complexes),

each are

we may suppose

a resolution

fl" as in

(ii), we put (fiG')

The complex

(U) = Hom" (f!~ u,G

)

of presheaves > fIG'(U)

U is in fact a complex

of sheaves,

the complex we are looking

for.

1.6

In order to make u n d e r s t a n d a b l e theorem

1.5 is called

now compute

both

a Poincar~

duality

sides of our formula

X is a topological

manifold

the reason why theorem,

we shall

(*) in the case where

of dimension

n, Y is a point,

and where we take G" = A. 2)

Let ~ sheaves.

design

If x is a point

the cohomology

of a complex

in X, a c c o r d i n g

to GODEMENT

of [2],

4.1, we have: ~q(f!A)x

= lim ---~ H q Hom'(f !n" U 'I'A)

2) Recall that a complex of sheaves over a point is simply a complex of A-modules. Moreover, we write F for the complex which has 0 in any degree @ 0 and has F in degree 0.

210

-

where

I'A is an injective

through X.

the filtered

-

resolution

of A, and w h e r e U runs

set of all open n e i g h b o r h o o d s

Since X is a manifold,

we are allowed

of x in

to suppose U = R n,

and then we get

~q(f!A)

A0 if q @ -n if q = -n

x =

so that the c o m p l e x of sheaves

f IA is q u a s i - i s o m o r p h i c

to the

c o m p l e x w h i c h has 0 in any degree ~ -n and has the usual orientation

sheaf 0 of X in degree -n.

c o m p l e x F" is simply a sheaf F.

Suppose

The formula

now that the

(*) gives

an

isomorphism H q Hom" (RF i.e.

a

spectral

c

(X,F),

I'A) ~ Extq+n(F,0)

sequence

E p'q = ExtP(Hcq(X,F) , A) ~ ExtP+q+n(F,0)

In particular,

if A is a field, we have the usual

isomorphism Hom(H~(X,F),

2.

PERFECT

COMPLEXES

(Confer

If we w a n t to w a l k formula

for sheaves,

tions on the sheaves

A) ~ Extn-q(F,0)

9

OF SHEAVES

[4] expos~

9)

in the d i r e c t i o n

we must introduce

some

used in the theory;

of a L E F S C H E T Z

finiteness

condi-

-

211-

more precisely we first define a perfect complex of A-modules as a complex which is bounded, each degree and afterwards subcategory

of

Db(A x)

free and O f finite type in b denote by Df.f.Z.(AX) the full

whose objects

sheaves which are quasi-isomorphic whose sheafs

are those complexes

of

to bounded com~!exes

are flat in each degree;

(f.f./. stands for

"finite flat length") 9 2.1 Theorem Let F" be a complex in locally compact Haussdorff A(A) holds for the map two conditions

Df b.f./.(A x)

where X is a

space such that the condition

nX: X

> point.

Then the following

are equivalent:

For any Doint X in X, (i)

The local inverse system

U

- RFc(U,F

) is essen-

tially perfect. (ii)

The local direct system

U

; RF(U,F')

is essentially

perfect. (U runs through the set of all open neighborhoods Recall that the conditions

(i) and

(ii) mean that,

given x 6 U, there exist two open neighborhoods x, contained

V and V' of

in U, such that the natural maps RFc (V,F") RF(U,F')

factor

of x in X.)

~ RFc (U,F') -

-

RF(V',F')

(in the derived category D(A) of the category of

-

A-modules)

through

a perfect

212

-

complex.

2.2 D e f i n i t i o n A complex of sheaves the p r e c e d i n g

satisfying

t h e o r e m will be called

the conditions

a perfect

complex

of (of

sheaves). Perfect of sheaves) ([i] and

complexes

satisfying

[4]).

look like sheaves

the B O R E L - W I L D E R

(or complexes

equivalent

conditions

In fact we have the

2.3 T h e o r e m

Let X be a spac e as in 2.1 and F" be a p e r f e c t complex (i)

of sheaves

Given

over X.

an[ compact

subset

K of X and an[ open subset U

of X such that U D K, then the canonical RF(U,F') factors

in D(A)

through

~ RF(K,F')

a perfect

(ii) Given two open subspaces

factors

in D(A)

c

(U,F')

through

complex.

U and V of X such that U c V

and U compact , then the canonical RF

morphism

morphism

> RF

c

(V,F')

a p!erfect complex.

213

-

3.

-

A FEW PROPERTIES OF f!

(Confer [4] exposes 9, i0, ii) 31

The dual com lex Let X be a space as in 2.1, we define the dual

complex Dx(F') of a complex of sheaves F" by the formula D x(F') = R ~ m " where~m

(F',~IA)

is the inner Hom in the category A x (see for

example GODEMENT

[2], 2.2).

a natural morphism

F"

>

3.1.1 . . .Theorem . . comple x F" is perfect 9

from F" into its bidual.

Suppose the

Then the complex DxF" is also perfect

(In particular, ~iA:

DxDxF"

(Bidualit[ Theorem).

and the natural morphism

perfect, so is

As in the usual cases, we get

F"

> DxDxF"

is an isomorphism.

if the constant sheaf A over X is

confer 1.6 for the case where X is a

manifold 9 3.2 Consider a map for f.

f: X

) Y

such that A (A) holds

In both cases

(i)

G" 6 0b D f.f./.(AY), b F

(ii)

G" 6 Ob D+(AY), F" 6 Ob Df6.f./. (AYA), flF" 6 D b(AYA )

6 Ob D+(AYA )

-

we are able to construct

where

~ stands

product

that

-

a natural

morphism

for the left derived

and f* is the usual

(and last)

214

theorem

gives

functor

inverse

image

a condition

of the tensor

functor.

The next

on f in order to involve

r is an isomorphism. 3.2.1 Definition.

A map f: X

product

if for each point

borhood

U in x, an open n e i g h b o r h o o d

logical

space

making

diagram

U

a

x 6 X there exist an open neigh-

Z p r o v i d e d with

the following

> Y is locally

~

V of f(x),

an h o m e o m o r ~ h i s m

and a topou

~ v x z

commutative

> V

•

Z

V

3.2.2

Theorem.

Under

a)

f: X

b)

the condition

c)

for any y 6 Y, the constant

the following

assumptions

- Y is loca!l [ a p r o d u c t A(A)

holds

for f sheaf A over f-l(y)

is

p e r f e c t , W e have (1) If F

is an object

in D .f.s

(AY)

then f!F"

is an object

in D b f.f.s

x)

(2) In both cases

is perfect)

(resp.

i s perfec t)

(i) and - (ii) of 3.2 the m o r ~ h i s m r

is an isomorphism.

(resp.

fIF'~f*G"

9 f l (F'~G')

-

3.2.3

How to define

struction in case constructions

-

~ (in case

(i) is quite different;

(ii); the conhowever both

agree when F" and G" satisfy conditions

and

(ii) together).

a)

Because

flF'~?

is left adjoint to

since flF" is bounded, F'G" b)

215

(i)

R~m'(f!F',?)

it is enough to define a morphism 9 R ~m"

(flF',f! r

But f* is left adjoint to Rf, and then we just need a

morphism 9 Rf, R ~ 0 m

G.

c)

"

(flF',f'" (F " |L

Now the Poincar~ Duality theorem

instead of Hom')

(using the functor ~ m "

gives an isomorphism Rf

R ~ o m " (flF',fl (F'~G'))

!

and so we seek a morphism G"

d)

> R ~ 0 m " (Rflf!F"

,F' G " ) ,

or, using once more the adjunction between ~ and R ~ 0 m ' ,

a morphism

Rf,fl ' G" e)

>

at this stage we very happily remember the adjunction

between Rf! and fl:

in fact this gives a natural morphism !

~: Rflf"

9 Id

-

216

-

so that we finally define ~' by the formula ~' = ~(F')

~ Id(G')

9

REFERENCES [i]

BOREL, A.

"Poincar~ Duality in Generalized

Manifolds", Michigan Math. J., ~; 227-239, [2]

GODEMENT, Theorie des [aisceauz,

Hermann

scientifiques et industrielles, [3]

HARTSHORNE, Residues and Duality, Mathematics,

[43

VERDIER-ZISMAN,

(1957).

(Actualites

#1252),

(1958).

(Lecture Notes in

#20), Springer Verlag,

"Seminaire sur la f o ~ l e

(1966). de Lefschetz,',

(multigraphi~) Publication IRMA Dept de

Mathematique Faculte des Sciences a Strasbourg.

Offsetdruck:Julius lklt'z. Weinheim/Ikrgstr.