This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
first and last rows are e x a c t b e c a u s e An easy i n d u c t i o n shows
over
p,
>
; 0
) G p-[ '* (2)
Here all the
-
using
E p,* ~ (0)
the s h o r t
E,(S0).
The
is p r o j e c t i v e .
five lemma,
now
that G p,* (0) |
F* (SO)
) Gp,,(1)
or G p'* (2)
>
Hom~
(0) ,F* (S 0) ) ,(S O ) (Gp ,*
is iso. In the case of UCT i, we now pass to d i r e c t
limits
and see that E,(X) is iso.
|
) F,(X)
In the case of UCT 2, we first o b s e r v e
tral s e q u e n c e for s p e c t r a l
EP'q(2) r sequences,
satisfies
that the spec-
the M i t t a g - L e f f l e r
and t h e r e f o r e
condition
-
F*(X)
25
-
= L
Because G p,, (0) = G p - l , , (0) 9 E p,, ~ (0) and E,(X)
= L i ~ Gp,,(0) P
we h a v e Hom~,~ (sO) (E, (X) ,F* (S O )) = L i m H o m ~ < P We can thus pass
to i n v e r s e
F*(X) is iso.
This
proves
Proposition two
t h a t w e can w o r k
theory
theories
E,
and
or
and t h a t
E,
is r e p r e s e n t e d
apply
to any o t h e r
We a s s u m e finite
Lemma
that
E
CW-complexes
F,
F*
lemmas object
lemmas.
For this p u r p o s e
category
[7, 8, 25]. We are d e f i n e d b y an o b j e c t
are s t a t e d (such as
is the d i r e c t
for
F,
limit
in w h i c h w e
assume
on this E
that the category,
in this
E,
cate-
but they
if w e h a v e of a g i v e n
an
also F
.)
system
of
E
18
For any o b j e c t is an
and see that
in a s u i t a b l e
homotopy
The n e x t two
o
17.
further
can do s t a b l e
gory.
(sO) (Gp,, (0) ,F*(S O ))
> Hom~,(S0) (E, (X) ,F* (S O ))
We n e x t n e e d we a s s u m e
limits
,
E
and a class
X
and any c l a s s
f 6 Ep(S p A DE
)
e 6 Ep(X) and a m a p
there
-
g:
S p A DE
9 X
Proof.
such
Take
2B
that
-
e = g,f.
a class
e E E
(X).
Then
there
is a
P finite that
subcomplex i,e'
= e.
E -p(Dx') ; h: DX'
so
e'
limit
may
is,
k*f
= e'.
in
E
such
p
there
of the
Dualising
that
),
(Dk),f
proves
Lemma
Lemma
(X')
such
k
is a f i n i t e
E
we
,
can
S- p A E
f
f
back,
obtain
Dk:
Sp A DE
= e'.
factor
h
and
in t h e
E - P ( s -p A E )
may
and we
complex
E
form
be
interpreted
such
that
as a c l a s s
a map 9 X'
We have
only
to t a k e
sP A D E
> X
.
18.
any o b j e c t
X
there
exists
form W =~
a map
g: W
> X
S p(8)
such g,:
is epi.
A
DE
(8)
that
E,(W)
is
9 S -p A E
in f
in
19
For
and
E E
by a map
DX'
g = i(Dk): This
e'
as a c l a s s
e'
be represented
is a c l a s s
(Sp A D E
a class
interpret
Since
DX'
That
and
P We m a y
--~ s-PE.
the d i r e c t
X' c X
9 E,(X)
an o b j e c t
of t h e
-
The allowing erators
the for We
construction class
e
27
-
is inunediate
in L e m m a
from Lemma
18 to run
over
18, by
a set of g e n -
E,(X). now
introduce
By a " r e s o l u t i o n
of
a diagram
following
of the
X
the
with
sort
of r e s o l u t i o n
respect
form,
to
with
E,"
we
we
need.
shall
the p r o p e r t i e s
mean
listed
below. x0 X = X0
Xl > X1
W0
(i)
The
x2 ) X2~ > X 3 ...
W1
W2
triangles X
Xr
/
r
X
r+l
Wr are e x a c t
(cofibre) (ii)
For
triangles. each
(x r).: is
r, E,(Xr)
) E , ( X r + I)
zero. (iii)
For
each
r,
E , ( W r)
(iv)
For
each
r,
W
is p r o j e c t i v e
over
E, (S ~ .
i.e.
the m a p
r
satisfies
UCT
1 or U C T
2,
-28-
E,(X)
|176
F* (sO)
> F,(X)
or
F*(X)
~
Ho~,(S0)
(E,(X) ,F*(S O ))
is iso. In order we introduce
Assumption
to prove
the existence
of such resolutions,
the following
hypothesis.
is the direct
limit of finite CW-complexes
20 E
E
for w h i c h
(i)
E.(DE )
(li) be,
DE
is p r o j e c t i v e
satisfies
for the theory
F,
or
and
F.
17, which
In practice requires
In practice 17 involves
E
(i)
the following
on
for given
it using Proposition DE
but none on
For each
over
p,
E*(S 0) .
F.
so the use of P r o p o s i t i o n two conditions.
sequence ~
E* (Ee)
HP(E
;E* (S0))
and
(ii) as a module
assumption
prove
is a ring-spectrum,
H* ( E ;E*(S~ is trivial,
this
we usually
The spectral
and
F*.
strong hypotheses
checking
E,(S~
UCT 1 or UCT 2, as the case may
In theory we can check E
over
is projective
-29 -
Examples. (i)
E = S,
the c o n d i t i o n s tion
the s p h e r e
are t r i v i a l l y
20 is v e r y (ii)
easily
are s a t i s f i e d
by any
The
X.
E = MO.
The conditions
complexes
It is w e l l
is s u f f i c i e n t plexes whose
of P r o p o s i t i o n to let is
E
MU.
and
q
=
are even.
It is s u f f i c i e n t complexes
which
is iso for
(v)
while
[23,
24]
K(Zp).
that .
HP(Mu;Muq(s0)) the s p e c t r a l
by any
X.
It
com-
= 0
unless
sequence
> MU* (MU) is free o v e r
MU*(S0).
run o v e r a s y s t e m of f i n i t e MU
in the s e n s e
Hp(Ee)
E = MSp.
of S. P. N o v i k o v case s h o w s
E
approximate
p < n,
run
any s y s t e m of f i n i t e
H p (MU;MU* (S 0) )
i,:
is
E
17
.
We h a v e
to let
limit
17 are s a t i s f i e d
Therefore
Again,
to let
~ Hsn(i)K(Z2) i
H* (MU;MU* (S O ) ) is trivia l.
= sn;
of P r o p o s i t i o n
whose known
run o v e r MO
limit
(iv)
p
E
E
directly. conditions
MO ~ V sn(i)K(Z2) i
Take
and of c o u r s e A s s u m p -
It is s u f f i c i e n t
o v e r any s y s t e m of f i n i t e (iii)
satisfied,
verified
E = K(Zp).
spectrum.
Hp(E
> Hp(MU) ) = 0
A simple
for
p > n.
adaptation
f r o m the u n i t a r y
t h a t the s p e c t r a l
of the m e t h o d
to the s y m p l e c t i c
sequence
H* (MSp;MSp* (S O ) )
that
> M S p * (MSp)
-
is trivial.
Again,
BU
-
HP(MSp;MSp*(S0))
The rest of the argument (vi)
B0
E = BU.
is as in
the spectral
= 0
unless
p
MSp*(S~
(iv).
Let us recall
every even term is the space
HP(BU;BUq(S0))
is free over
BU.
and
q
that in the s p e c t r u m We have are even.
Therefore
sequence H* (BU;BU* (S 0) ) ---> BU* (BU)
is trivial.
Again,
It is s u f f i c i e n t plexes w h i c h BU
to let
E
approximate
of the s p e c t r u m (vii)
BO
HP(Bu;BU*(S~
is free over
run over a s y s t e m of finite
as in
(iv) to the d i f f e r e n t
E = BO.
Let us recall
is trivial.
In fact,
can c o n s t r u c t
that in the s p e c t r u m
BSp.
I c l a i m that the
> BO* (BSp)
for each class
a real r e p r e s e n t a t i o n
begins with
h E Hsp+4(BSp(m)) tation of
spaces
sequence H* (BSp;BO* (S 0) )
character
com-
BU.
every eighth t e r m is the space
spectral
BU*(S~
Sp(m)
we
h;
The rest of the a r g u m e n t (viii) The reader will
of
Sp(m)
whose
we Chern
for each class
can c o n s t r u c t
whose
h E HSP(BSp(m))
a symplectic
represen-
C h e r n c h a r a c t e r begins w i t h is as for
h.
(vi).
C o b o r d i s m and K - t h e o r y with coefficients. find further examples
Assumption
in L e c t u r e
4.
20 allows us to use the m e t h o d of
-
Atiyah
-
[6]. The
require; will
31
next
lemma will
but we state
also
a diagram
allow
construct
it in a m o r e
us to c o m p a r e
of t h e
following
= X
x i X
X
>
I
0
angles,
are
W1
supposed
for e a c h
X
3
I
W2 (cofibre)
tri-
r.
We
also
- ~.. (x~_+ ~ ) suppose
given
a map
f: X
> X'.
21
Under of
given
and
zero
Lemma
X'
to be e x a c t
(x~) ,: ~.. (x~) is
We s u p p o s e
it
x2
!
~
W !
the t r i a n g l e s
so t h a t
i
1
!
>
t/i/i/ Here
form,
we
form.
o
X'
general
resolutions.
x i
0'
the r e s o l u t i o n s
with
the s e n s e
respect
that we
the p r i s m s
X
these
can
are maps
..
= X0
to
conditions E,
which
construct
of e x a c t
we
can
construct
admits
the
a map
following
(cofibre)
a resolution
over diagram
triangles.
l.>x., t.,x-l--x. w~
w~ I
w2. I
f,
in
so t h a t
-
In o r d e r respect which
to
all
E,,
suppose
to c o n s t r u c t
we need
only
X r'
the o b j e c t s
Proof
32
of L e m m a
the d i a g r a m
-
a resolution
apply
and
21.
Lemma
W'r
As
constructed
are
of
X
21 to the
with case
in
trivial.
an i n d u c t i v e up to the
hypothesis,
following
map.
X r
!
Xr Form
the
following
cofibre
X
x'f r r r
triangle.
) X' r+~
Z
Then we have
the
following X
commutative
r
Z
~
9 W w
square.
r
X'
r
Since we
(Xrf r), = 0,
can construct
r
E,(Z)
a map
W
Wr = ~
r
> E , ( X r)
is epi.
> Z
that
S p(8)
such
A
DE~(B)
W
By L e m m a r
has
the
19 form
- 33 -
and
E , ( W r)
commutative
We n o w h a v e
is epi.
> E,(Z)
the
following
square. X
W
I I r
r
X' ( r Here
E , (W r)
> E, (X r)
W' r
Form
is epi.
the f o l l o w i n g
cofibre
triangle ~ X
X
-
r
> X
r+l
W
r
This
triangle
(Xr) . = 0.
can be m a p p e d
This We h a v e
herits
completes
S p A DE
Assumption
20).
,
that
a resolution,
E , ( W r)
This p r o v e s
1 and U C T 2, u s i n g L e m m a
and
F,
or
F*
are d e f i n e d
Lemma
X
with
applying
respect
the f u n c t o r
to
E,, F,
or
for U C T
2
in-
r
(see
21. sequences
21 a n d the a s s u m p t i o n on a sufficiently theory.
as p r o v i d e d F*,
W
f r o m its
i, U C T
the s p e c t r a l
in w h i c h w e can do s t a b l e h o m o t o p y of
because
is p r o j e c t i v e
and s i m i l a r l y
We w i l l n o w c o n s t r u c t UCT
a n d we h a v e
the i n d u c t i o n .
constructed
the p r o p e r t y
summands
in the r e q u i r e d w a y ,
by Lemma
we obtain
that
large
Take
of E,
category
a resolution 21.
a spectral
By
-
sequence.
Now
0 ~ is
the
Since tral and fore
the
|
is
We h a v e the
as
UCT
the
1 or UCT by
of
|
the
~
T
spec-
resolution There-
sequence
is i n d e -
given
two
reso-
form V
the
W6
I
9
II
following ,>
II
V W0
II
II
> X2
> X 3 ..9
~lli~II
Wi
X'0 --
I
W2
II
II
X 0'
the
Ext.
Suppose
I
> XI
W0
=
or
E,(S0).
,F*(S~
spectral
1
II
X
of
projective (
...
> X2 i___> X 3 ...
W
X = X0
XV
~
over
El-term
!
> XI
I
can
this
resolution.
W0
we
the
Tor
E,(W2)
,
modules
HomE,(S0)
or
that
<
follows. X = X0
Then
2,
taking
required
to s h o w
choice
E , ( W I)
by p r o j e c t i v e
(S0) F* (sO)
the E 2 - t e r m
lutions,
E,(X)
is o b t a i n e d
applying
of
E , ( W 0) ~
satisfy
sequence
pendent
<
of
Wr
-
sequence
E,(X)
a resolution
34
2
diagram. w
X 1
V
"
X 1
/ !
II
WI V WI
X' 2
V
X" . 2 ""
- 35
We c a n n o w (i,i).
which
spectral
admits
for
maps
are
spectral
maps
and
algebra;
a third
to or f r o m
F,, iso
> X v X
"from"
of type spectral
the
for
first
F*
for r = 2 by the
therefore
they
are
.)
setwo
But
comparison iso
for
r. It r e m a i n s
sequences.
a direct
a "telescope" has
("To"
of h o m o l o g i c a l
X
resolution
comparison
comparison
finite
struct
21 to the m a p
a third
sequences.
these
theorem all
Lemma
We obtain
quence
both
apply
-
to d i s c u s s Given
limit
the p r o p e r t y
convergence
a resolution
X~
of
or i t e r a t e d
the
the
of
of t h e s e
X,
objects
we
Xr
mapping-cylinder).
can
con-
(by f o r m i n g The
object
X~
that E, (X~)
=
Li~
E.(X
r)
=
0
.
r In the
case
verges
in a p e r f e c t l y
therefore
Problem
F,(X)
of U C T
face
the
When
can w e
= 0
the
F,(X~) UCT
for e x a m p l e ,
the
satisfactory
manner
following
spectral to
sequence
con-
F,(X~,X0).
We
question.
22
or
F*(X)
This When
i,
answer
= 0,
assert
a special
affirmative,
implies
of UCT
1 or U C T
(for example)
and
spectral
in a s a t i s f a c t o r y
Unfortunately
case
we have
F , ( X ~ , X 0) =_ F,(X)
1 converges
E, (X) = 0
= 0 ?
is of c o u r s e is
that
the
way
the p r e s e n t
to
state
sequence
F, (X). of o u r k n o w l e d g e
2.
of
-
on P r o b l e m
22 a p p e a r s
we know
special
s, (x)
0
=
vary
Y,
F*(X)
of KT
sequences be
this
X
We
compute
limit;
we
we
well
is
23.
K(Z),
E,(X)
then
and
= 0,
functor
zero.
pause
We
Thus
to s h o w
in cases
consider
while
E
the o r d i n a r y
that
of the
so
then
of
Y
the
as we
with
zero
spectral
se-
that
our
which
spectral
are k n o w n
to
UCT
and
F
homology
space
2 for the
BU
are
case
in
the s p e c t r u m
of the
spectrum
and passing
BU.
Bu
by
to a d i r e c t
find
Whitehead's BU
f )Q
if
n
is e v e n
(o
if
n
is o d d
remark
(K(Z))
=
----n Now
owing
the
computation over
if
even
Hn (BU)
Ext
E = S,
converges.
=
By G e o r g e
if
Of c o u r s e
.
pathological.
which
considering
Again,
point
Example
can
satisfactory
is c o n t r a c t i b l e ,
therefore
can b e h a v e
somewhat
from
is a h o m o l o g y
1 always
At
X
= 0.
groups,
far
-
for e x a m p l e ,
that
E, (X A Y)
coefficient quence
cases;
implies
F, (X) = 0,
to b e
36
to the
Z.
favourable of
We
Ext
[31],
t'
this
.
is e q u i v a l e n t
bQ
if
n
is e v e n
O
if
n
is o d d
structure
over
this
of the
ring
to
. ring
reduces
BU,(S0),
to c o m p u t i n g
find
N o t e a d d e d in proof. now available.
A satisfactory
answer
to P r o b l e m
22 is
-
37
-
ExtP,q BU, (S O ) (BU, (K(Z)) ,BU* (S O ) )
I
Ext~(Q,Z)
if
p = 1
the result of H o d g k i n
We w i l l now make some
assume E
q
is
even
otherwise.
This agrees with
whose
and
exploration
was pioneered
comments
Assumption
objects F
20 and that
[5, 17].
on the situation
by Conner
that we have r e p r e s e n t i n g
satisfies
and A n d e r s o n
and Floyd E
and
F,
satisfies
We
[14]. that
the follow-
ing hypothesis. Assumption
24 F
is the direct
limit of finite CW-complexes
F
for which (i) (ii)
E,(DF DF
)
is p r o j e c t i v e
satisfies
(Compare A s s u m p t i o n verify
this assumption
UCT
over
E,(S0),
1 for the theory
20.)
and F,.
In practice we generally
by using P r o p o s i t i o n
17, as for Assump-
tion 20. Examples. (i)
E = MU,
even term is the space HP(Bu;Muq(s0))
= 0
F = BU. BU.
unless
In the s p e c t r u m
For the space p
and
q
BU
are even.
BU
every
we have Therefore
-
38-
the spectral sequence H*(BU;MU*(S0)) is trivial.
Again,
As in Example F
~> MU*(BU)
HP(BU;MU*(S0))
(vi) on A s s u m p t i o n
is free over
MU*(S0).
20, it is sufficient
to let
run over a system of finite complexes which approximate
to the different spaces
BU
of the s p e c t r u m
BU
in the
sense that i,: Hp(F is iso for
p ~ n,
(ii)
while
E = MSp,
eighth term is the space Conner and Floyd
)
> Hp(BU)
Hp(F ) = 0 F = BO. BSp.
It follows
BO
every
from the work of
[14] that the spectral sequence
Again,
> MSp*(BSp)
HP(BSp;MSp *(s O ))
The rest of the argument is as in With these assumptions the following results Proposition
p > n.
In the s p e c t r u m
H* (BSp;MSp* (S ~ ) is trivial.
for
for any
is free over
MSp*(S O ).
(i). (especially 20 and 24) we have
X.
25
We have TorP~ (S0) (E, (X) ,F, (S~
= 0
for
The spectral sequence of UCT 1 collapses,
p > 0 . and its edge-homo-
morphism E,(X)
|
> F,(X)
-
39
-
is iso. Compare Conner and Floyd these authors
state their theorem with the variance of UCT 3.
and use finiteness Proof. object
X.
assumptions. It follows
sP(8)ADEe(8)
g: W
> X
and are epi.
from Lemma 19 that given any
there exists an object
w = V and a map
[14, pp. 60. 63]; but
V V
W
of the form
sP(Y)ADFe (~)
such that both g,: E,(W)
> E,(X)
g,: F,(W)
> F,(X)
Arguing as in Lemma 21. we can now construct a reso-
lution of
X
with respect to
E,
which has the following
extra properties.
(i) Wr =V (ii)
The objects sP(8)ADEe(8) 8
Wr V Vy
have the form sP(Y)ADF e(u
9
Not only the homomorphisms (Xr),: E,(X r)
> E,(Xr+ I)
but also the homomorphisms (Xr),: F,(X r) are zero for all
> F, (Xr+l)
ra
Then the sequence 0 (
E,(X)
~
E,(W 0) <
E,(W l) 9
E,(W 2)
...
-
is a resolution Consider E,(W0)
of
E,(X)
-
by projectives
over
E,(S0).
the following diagram.
|
<
E,(WI)
F. (W0) <
<
40
|
<---
F . (W I)
<
-
E,(W 2) |176
9 F, (W 2) ... The homomorphisms
The lower row is exact by r are iso. Therefore the upper row is exact, and
construction.
TorP,~s0 ) (E, (X) ,F* (S0) ) = 0 We can now consider 0 (
0 <
E,(X)
|
F,(X) ~
(
for
p > 0
m
the following diagram. E,(W0)
|
F,(W 0) <
(
-41 -
<----- E. (W I) |
(
The
F. (W 1 )
upper
lower are
r o w is e x a c t
row
is e x a c t
iso.
situation, this
via
i.
that
so t h a t w e
G.
or
G*
G*
can
from
F.
more
and
~0
completes
given
and
the
and
~I
the p r o o f
F,
and
but
construct
i: E
of
or
a spectral
1 or U C T
and
which
is a
(It w o u l d
presum-
products
in h o m -
ourselves (E,G)
the d e t a i l s . )
satisfies
Lemma
sequence
for c o m p u t i n g
2; w e
suppose
satisfies
2.
G
enough
1 or U C T
sequence
two
a module-spectrum
a spectral
(F,G)
2.
given
E = MSp
G = F.)
theories
to U C T
of r i n g - s p e c t r a .
a spectrum
given
1 in this
suppose
> F
therefore
as in U C T
as in U C T
We
let us s p a r e of
to U C T
happens
data.
also
to s u p p o s e
E.
happens
F = BU,
of t h e o r i e s
construct
what
a map
the p a i r
from
the pair
can
exact,
The maps
This
(For e x a m p l e ,
cohomology,
21,
we
over
be sufficient and
is r i g h t
to ask w h a t
and
We suppose
We s u p p o s e
that
F
E = MU
module-spectrum
ology
is iso.
slightly
E,
(For e x a m p l e ,
ably
~
it is n a t u r a l
ring-spectra
E
|
construction.
we now know
we n e e d
F = BO.)
because
25.
Since
over
by
Therefore
Proposition
For
(S~ F* (sO)
also
Lemma
for c o m p u t i n g
21, G.
so t h a t or
- 42
-
Proposition 26 (i) E,
The spectral sequence for computing
coincides with the spectral sequence for computing
from
from G,
F,. (ii)
E,
G,
The spectral sequence for computing
G*
coincides with the spectral sequence for computing
from
G = F,
By specialising Proposition
26(i) to the
we obtain a result agreeing with Proposition
25; for of course the spectral sequence for computing from
G*
F,. Note.
case
from
F,
F,
collapses. Proposition
26 will follow almost immediately
from
the following lemma. Lemma 27
(i)
F. (W)
If
E.(W)
is projective over (ii)
is projective over
E, (S 0 ) ,
F, (S 0 ) .
If E,(W)
> G, (W)
|
is iso, then F,(W) |
G* (SO)
> G, (W)
is iso. (iii)
If G* (W)
> HO~,
(S0) (E, (W) ,G* (S O ) )
then
- 43
is iso. then G*(W)
) Ho~,(S0)(F,(W),G*(S0))
is iso.
Proof. (i) 25.
So if
F,(W) _~ E,(W) |176
E, (W)
projective over (ii) E,(W) |
is projective over
by Proposition
E, (S O ) ,
F, (W)
is
F, (S 0 ) . Consider the following commutative diagram.
F* (SO) |
F.(w)
F*(S0)'
(s~
(S 0)
OF, (S0) ~ , (S o ]
i@
v ) E,(W) |
~
G* (sO)
G,(w)
The left-hand column is iso by Proposition 25. the right-hand column is iso by assumption, and the top row is trivially iso. Therefore the bottom row is iso. (iii) G* (W)
Consider the following commutative diagram. ) HOmF, (sO) (F, (W) .G* (S O ) )
is zero by P r o p o s i t i o n n e e d to be c h e c k e d follows
> E , ( X r + I)
F , ( X r) 25.
> F , ( X r + I)
The remaining
are p r o v i d e d
by L e m m a
statements
which
27.
Proposition
for
p > 1 .
26
immediately.
Example.
This the r e s u l t
X
we have
( M U , ( X ) , B U * ( S O )) = 0
E x t p ,* MU,
F o r any
(S O )
follows
is t r i v i a l
immediately for
from Proposition
26,
since
- 45 -
Ext
*
~u.
(S o )
(BU, (x) ' BU* (S ~
"
The following result is required Lecture
for use in
3.
Lemma 28 If then
E, (E)
E = __BO, BU,
MU,
MSp,
is flat as a module over
Proof. vial.
MO,
E = MO,
The cases
In the cases
E = MU,
MSp
S
or
K(Zp)
E. (S O ) . S
and
K(Zp)
are tri-
we can apply the spectral
sequence H . ( E ; E . (S O ))
to show that E = MSp
E. (E)
>
is projective
this involves remarking
E,(E)
over
which is known to be trivial In the cases spaces
BU, BSp
B_~O,(BSp) (vi),
(vii)).
sequence
> MSp* (MSp)
(see A s s u m p t i o n
20, Example
(v)).
we apply this argument to the
to show that the modules
are projective
tive modules
t
E = BU, BO
in the case
that the spectral sequence
is trivial, by duality with the spectral H* (MSp;MSp* (S O ) )
E, (S0) ;
BU, (BU),
(compare A s s u m p t i o n
20, examples
We then remark that a direct limit of projecis flat.
This proves Lemma 28.
Note added in proof. I have been asked to say e x p l i c i t l y this point that UCT2 gives the following exact sequence. O-~Ext I*MU,(S o ) (MU* (X) 'BU* ( (S ~X ))--~BU* ) (X)-->H~ -
(S~
(MU*
at
"BU* (S ~ )) >O
- 46
-
L E C T U R E 2. THE A D A M S S P E C T R A L
In this lecture
I w a n t to discuss
s e t t i n g up an "Adams s p e c t r a l sequence" generalised homology
or c o h o m o l o g y
be t a k e n as p r o v i s i o n a l ,
SEQUENCE
the p r o s p e c t s
of
[i, 2, 15] u s i n g a
theory.
Everything
or as w o r k in progress,
is to
and no p r o o f s
w i l l be given. I shall assume t h a t we can w o r k in some s t a b l e category,
like those s u p p l i e d by B o a r d m a n
[7, 8] and P u p p e
[25].
I shall also s u p p o s e that we are g i v e n a h o m o l o g y or c o h o m o l o g y functor to use in our c o n s t r u c t i o n s . this functor takes v a l u e s
I will
suppose
in an a b e l i a n category.
As
that long as
w e are t a l k i n g g e n e r a l i t i e s , w e can t h e n s u p p o s e t h a t the f u n c t o r is covariant;
because
if it is c o n t r a v a r i a n t ,
r e p l a c e the a b e l i a n c a t e g o r y by its opposite. E,
for this h o m o l o g y
We w i l l w r i t e
functor.
I s u g g e s t that w e now a d o p t a c o n s t r u c t i o n s c e n t of those c o n s t r u c t i o n s
for
projectives
More p r e c i s e l y ,
and injectives.
p r o c e e d as follows. stable category.
Ext
Y0 (
Z0
Y1
Yl <
I s u g g e s t we
Z1
X,Y
of the f o l l o w i n g
Y2
remini~
w h i c h avoid using
S u p p o s e g i v e n two o b j e c t s
Consider diagrams
Y ~ Y0
we can
Y2 <
Z2
Y3
in our form.
-47-
H e r e the n o t a t i o n the t r i a n g l e s
Y ~ Y0
means
a homotopy equivalence;
are s u p p o s e d to be e x a c t
in our stable category.
(cofibre)
and
triangles
We r e s t r i c t a t t e n t i o n to the d i a g r a m s
such that E , ( y r) = 0: E , ( Y r + I) for each
r a 0;
this is the c r u c i a l
> E,(Y r) condition.
In this
case the s e q u e n c e 0
> E,(Y)
is exact. wish,
---> E,(Z0)
we can s u p p o s e w i t h o u t
get a s p e c t r a l
sequence;
s e q u e n c e we seek.
category
Y
X
"filtrations"
Y0
of
> Y.
... If we
that each
Yr
by a " t e l e s c o p e " ) .
i n t o such a f i l t r a t i o n of
Y
we
b u t this is not y e t the s p e c t r a l
However,
we can take all p o s s i b l e
filtra-
and c o n s i d e r t h e m as the o b j e c t s of a d i r e c t e d
(in the sense of G r o t h e n d i e c k ) .
ting proofsj
> E,(Z 2)
loss of g e n e r a l i t y
(replace
By m a p p i n g
tions of
E,(ZI)
We call such d i a g r a m s
is an i n c l u s i o n map
done,
~
I w i l l omit c e r t a i n d e t a i l s
(Since I am omitas to h o w this is
a l t h o u g h they w e r e g i v e n in the o r i g i n a l
lecture.)
F r o m each f i l t r a t i o n w e get a s p e c t r a l sequence, now take the d i r e c t this is the s p e c t r a l
limit of all these s p e c t r a l sequence
I suggest.
and we can sequences;
Let us call it
SS(X,Y;E,). I w i l l also o m i t some a r g u m e n t s definition,
in favour of this
a l t h o u g h they w e r e g i v e n in the o r i g i n a l
lecture.
At this level one s h o u l d a l r e a d y be able to set up
-
some formal p r o p e r t i e s suppose
48
of the s p e c t r a l
that we have a functor
to another,
and that both
(For examples,
TE,
Proposition mutually
If
determine
only
X
and
E,
for
and
F,
each other
see L e c t u r e
Y
filtrations
complex. quence
TE,.
are h o m o l o g y
strong
Do these y i e l d
complexes,
each
Yr
complexes.) approach
question.
Suppose
and that we c o n s i d e r
is e q u i v a l e n t
to a finite
the f i l t r a t i o n s ?
theory
E,
This
seis
has s u f f i c i e n t l y
the b e h a v i o u r
of our construc-
Do we have
SS(X,Y,E,) E,D
.
properties.
under S-duality.
(Note that
functors w h i c h
in the limit the same spectral
We can now c o n s i d e r tions
i,
then
~ SS(X,Y;F,)
true if the h o m o l o g y
finiteness
E,
4.)
are finite in w h i c h
for
(Compare L e c t u r e
in this way,
as if we did not r e s t r i c t
probably
4.)
,
as a f i l t r a t i o n
We can now raise the following that
functors.
25, or L e c t u r e
> SS(X,Y,TE,)
SS(X,Y;E,) (For examples,
category
is a h o m o m o r p h i s m
as a f i l t r a t i o n
26.)
For example,
are h o m o l o g y
i, P r o p o s i t i o n
every d i a g r a m w h i c h q u a l i f i e s
also q u a l i f i e s
sequence.
from one abelian
and
SS(X,Y;E,) because
T
E,
see L e c t u r e
Then there clearly
-
~ SS(DY,DX,E,D)
is a c o h o m o l o g y
This p r o b l e m
leads
to the construction.
?
theory d e f i n e d one to c o n s i d e r
on finite also a "dual"
-
We c o n s i d e r
diagrams
above,
the n o t a t i o n
and the t r i a n g l e s angles
diagrams
such
of the
Xl > X1 ~
W0
wl
X ~ X0
are s u p p o s e d
in o u r s t a b l e
-
x0 X ~ x0
As
49
W2
means
wish,
In this
E,(X)
is exact.
is an i n c l u s i o n
<
map
By m a p p i n g
E,(W0)
(replace such
a spectral
sequence.
filtration
(inversely)
to the
~
E , ( W I) <--- E , ( W 2)
(
. . Q
"filtrations"
X.
If w e
loss of g e n e r a l i t y X0
that each
x
r
by a "telescope"}.
a filtration
of
X
would
and t a k e a d i r e c t Does
of
this give
into
Y
be to v a r y
we get the
l i m i t of the r e s u l t the s a m e s p e c t r a l
as b e f o r e ? the s i t u a t i o n
t h e r e w e can d e f i n e
by r e s o l v i n g same.
attention
> E , ( X r + 1)
The s u g g e s t i o n
sequences.
Evidently algebra;
tri-
c a s e the s e q u e n c e
we c a n s u p p o s e w i t h o u t
sequence
(cofibre)
We r e s t r i c t
We c a l l s u c h d i a g r a m s
ing s p e c t r a l
equivalence,
that
r ~ 0.
0 <
a homotopy
to be e x a c t
category.
form.
x2 x2 ~ X3
E , ( x r) = 0: E , ( X r) for e a c h
following
M,
The proof
and w e w a n t there,
is like
Ext*(L,M)
t h a t in h o m o l o g i c a l by r e s o l v i n g
to s h o w t h a t the r e s u l t
as w e know,
is to r e s o l v e
L,
or
is the
b o t h of
-
them,
and show that that gives
50
the same r e s u l t as r e s o l v i n g
e i t h e r one.
S i m i l a r l y here;
tion of
and also a f i l t r a t i o n of
X,
one s h o u l d c o n s i d e r a filtraY,
and one s h o u l d
try to get a s p e c t r a l s e q u e n c e by m a p p i n g one to the other. Then one s h o u l d take a d o u b l e d i r e c t this gives the same s p e c t r a l filtering either d o w n any d e t a i l s
X
or
Y
limit,
s e q u e n c e as one o b t a i n s by alone.
I h a v e n ' t tried to w r i t e
about this.
If one can a t t a i n this
sort of m a n i p u l a t i v e
one o u g h t to be able to set up v a r i o u s the s p e c t r a l
sequences without
For example,
there s h o u l d be a p a i r i n g
SS(Y,Z;E,) w h i c h on the
and show that
formal p r o p e r t i e s
further assumptions
| SS(X,Y;E,)
ability,
on
of
E,.
> SS(X,Z;E,)
E= level is g i v e n by composition.
The next step w o u l d be to c o m p u t e the E 2 t e r m of our s p e c t r a l sequence. values
We are s u p p o s i n g
in an a b e l i a n category,
classifying
that w e can d e f i n e a h o m o m o r p h i s m
o b t a i n the r i g h t
takes
Ext
It is r e a s o n a b l e
by to h o p e
f r o m the E 2 t e r m to
The q u e s t i o n w o u l d be, w h e n can w e p r o v e
that this h o m o m o r p h i s m is an i s o m o r p h i s m ? one o b v i o u s l y needs
E,
so we can d e f i n e
long e x a c t sequences.
Ext**(E,(X),E,(Y)).
that
For this p u r p o s e
to choose the r i g h t category,
Ext
groups.
More precisely,
so as to
we need to
a r r a n g e a v e r y close c o r r e s p o n d e n c e b e t w e e n the a l g e b r a and the geometry,
so t h a t there is some a l g e b r a i c s i t u a t i o n w h i c h
-51 -
gives
us a legitimate
calculation
which
can be r e a l i s e d
geometrically.
At this point assume
cohomology, ring
all suggestions
that our functor (i)
The o r i g i n a l
of c o h o m o l o g y
approach has various (a) topologised
ring,
E*(X), ring
[24].
form an abelian
category,
> M
q < 0;
(a)
raise and
it.
classical
over the
[i, 2, 23, 24].
This
owing
modules
usually
to the existence
(b)
assert
remarks
modules
fail to
structure,
is not continuous. that
cohomology
by any p r e s c r i b e d
is a
of maps
of the m o d u l e
f-i
Similar
E*(E)
are t o p o l o g i s e d
but such that We cannot
case
We have to take account
Topologised
Eq(E)
operations
amount,
= 0
for
which
as well as ones
apply to our modules.
(b) mean that our c o n s t r u c t i o n s
lose a certain element
of finiteness
which
Both
and calculations is present
in the
case. (c)
omit,
E*(E).
we may have n o n - z e r o
lower d i m e n s i o n which
E*(Y)
w h i c h are i s o m o r p h i s m s
and continuous,
E.
asks us to work in as modules
In the g e n e r a l i s e d
of the topology
f: L
and
disadvantages.
and
over the t o p o l o g i s e d
E*(Y)
operations
groups
by a s p e c t r u m
formulation
E*(X),
Ext
for p r o c e e d i n g
is r e p r e s e n t e d
and consider
E*(E)
of the
although
By means
they were
of examples
(which I will now
given in the original
see that even in the c l a s s i c a l
case of ordinary
lecture)
we
cohomology
-
with
Z
coefficients,
52
-
approach
(i) o n l y w o r k s
under
finite-
P ness
assumptions
this
as
on
In the
generalised
object
Zr
in s o m e
sense
is
to c o n s i d e r
"free";
filtrations
in p a r t i c u l a r ,
applicable
to
E,
in w h i c h
snE
of
E.
Z
is b o t h
r
again,
case
in w h i c h should
modules,
this
in w h i c h
be
each "free"
and we
is
to
Zr
that we must
means E
X
.
a product
in p r a c t i c e connected
and
from
stick of
to the
suspensions
that we must
and
Z
is a
r
sum,
in w h i c h
n(i)
compelled
to p r o v e
by
topologised
"free" of
=
) ~
V
)
graded
properties
E = MU
(see L e c t u r e
unknown
for
E = MSp,
9
,
that
In o t h e r w o r d s , E*(Y)
modules
[24]
arranges
for t h i s
of 5).
admits
which
in d i m e n s i o n s
it is e s s e n t i a l l y
finiteness
i
or a s s u m e
Novikov
sn(i)E
i=l
as
"generators"
Although
ently,
from
a sum and
r
number
maps
in p r a c t i c e
Z
n).
Y
of
(E*(Z r) ,E*(X))
about
means
And
to t h e
countable
to k n o w
this
case
stick
see
have
Since we wish r
we may
E * ( Z r)
to t o p o l o g i s e d
[X'Zr]* --~ H ~
Z
case,
follows 9 We w i s h
should
Y.
less
have than
his w o r k
purpose
E*
which
are
The
corresponding
and d e f i n i t e l y
false
only n
a finite
(for e a c h
somewhat
differ-
relies
in the
on
case
properties for
are
a resolution
t h a t he
true
we
E = S,
are
-
although
the Adams
in some cases
spectral
53
-
s e q u e n c e works
from a double
covariant
in
Y,
dualisation.
but by taking
that trouble
The spectral E* (Y)
E* (Y).
This
(ii)
leads
we are taking
and Douady
E, (X)
and
classical
[15], and work
E, (Y) case
as modules
E = K(Zp)
owing to the fact that
module
over the ring
ring
E* (E)
approach between
suffers (i) and
functor
over the ring
(iii).
considering E* (E).
E, (E)
is not a field.
to g e n e r a l i s e In general
disadvantages,
a compromise
acceptance
This is
is then an i n j e c t i v e
but this fails
The way
In the
and this
or h a l f - w a y
ahead appears
the
house
to lie in a
of the idea that h o m o l o g y
is
than cohomology. (iii)
wholly
in homology,
its p r e v i o u s
from being
more w h o l e - h e a r t e d better
E* (E);
E, (S O )
retains
a
ask us to follow
this works quite well.
partly
to cases in which
is
to the next approach.
The next approach w o u l d
Cartan
(c)
sequence
c o n t r a v a r i a n t functor of Y, and then by taking ** EXtE* (E)(E* (Y),E* (X)) we are taking a c o n t r a v a r i a n t of
spectra
at least.
It may be seen from the examples arises
for these
My final s u g g e s t i o n
in homology,
with respect
and consider
to the c o a l g e b r a
is that we should work E, (X) , E, (Y)
E,(E).
the ground ring for our comodules
etc.
We use
as comodules E,(S 0)
The n e c e s s a r y
as details
- 54
are g i v e n in L e c t u r e this;
3.
in fact, we n e e d to a s s u m e
spectrum
and
E,(E)
now works
the c o a l g e b r a q
(X) = 0
Eq(E) gory.
= 0 Our
E,(E)
comodules
q < 0.
which we
comodules;
injectives".
E,(S0), the
E,(Wr)
the set of all f i l t r a t i o n s
for e a c h
r.
E
be
u:
E
constructed
A
Y
over
such
r
>
;
E
and
regain
cate-
that element
of
that
and
(E,(X),E,(Y))
X.
i:
the i n d u c t i o n
of
by
"relative
of
r.
21 of L e c t u r e
are c o f i n a l
F o r the s e c o n d ,
of the r i n g
in
we require
is an e x t e n d e d
> E.
geo-
X
for e a c h
c a n be c o n s t r u c t e d
SO
E,(Y)
a filtration
by Lemma
maps
which
can be constructed
E , ( S 0)
E , ( Z r)
it is
by c o m o d u l e s
filtrations
L e t the s t r u c t u r e E
and
cases we have
the p a r t of
we require
Such a filtration
f o l l o w i n g way.
E,(Y)
q,
E,(X)
c a n be c o n s t r u c t e d
Y
Everything
and a resolution
w e see t h a t s u c h
of
of
is p r o j e c t i v e
Such a filtration
a filtration
~,(E)
of r e s o l u t i o n
F o r the first,
Moreover,
Exts
latter play
Both sorts
metrically.
for
lost b e f o r e .
a resolution
over
28.
is true
f o r m an a b e l i a n
and c a l c u l a t i o n s
to c o m p u t e
to t a k e
are p r o j e c t i v e
i.
The
This
in t y p i c a l
for
for
is a r i n g -
E,(X),
large negative
In o r d e r
such that
comodules
are d i s c r e t e ;
constructions
sufficient
i, L e m m a
for s u f f i c i e n t l y
of f i n i t e n e s s
extended
The
E
E,(S0).
in L e c t u r e
much better.
we need some data
that
is fla t o v e r
the s p e c t r a m e n t i o n e d
E
Of c o u r s e ,
comodule
in the spectrum
Suppose we have
starts with
Y0 = Y-
Take
- 55
Z
r
=
EA
Y
r
and form the map
,
y
Then ~,
(E
verse E,(Z r)
r
~S0
iA
r
I > EA
is mono,
> ~,(E A E A Yr ) ,
Yr )
induced by
E
cofibre
E
A
is extended,
following
A Y
> E,(Z r)
E, (Yr) A
-
A
y
r
= Z
r
since it is defined
to be
and this has a one-sided A
Yr
1
> E
by the results
A
Yr"
in-
The comodule
of Lecture
3.
Form the
triangle. Yr <
Yr+ I
g r
Then
E,(Yr+ I)
induction. tions
> E,(Y r)
By adding
are cofinal
must be zero.
a few details,
This
completes
the
we see that such filtra-
in the set of all filtrations
We may say that at the present
of
Y.
time approach
(iii)
seems to be promising. The final convergence
step,
of course,
of the spectral
this question.
sequence.
w o u l d be to discuss I would
the
like to defer
-56-
LECTURE
3
HOPF A L G E B R A ~ND C O M O D U L E
In the c l a s s i c a l coefficients Hopf algebra, any space,
A.
so that we have
of the S t e e n r o d
if the h o m o l o g y
H.
c o h o m o l o g y with
is actually
an a c t i o n map
A* | H*
HOmzp(
we see that the
algebra
of a space
we have a c o a c t i o n map dition
case of o r d i n a r y
Z , the mod p S t e e n r o d algebra A* is a P and it acts on the left on the c o h o m o l o g y of
If we dualise by applying dual
STRUCTURE
H.
,Zp),
is also a Hopf algebra;
is locally
> A. | H..
unnecessary,
> H*.
finitely
and
generated,
(The f i n i t e n e s s
con-
but we do not need to spend
time on that here.) It is the object material mentioned eralised homology assumptions; to introduce,
of this
above g e n e r a l i s e s theory.
then we w i l l
two p r o p o s i t i o n s Finally,
we s p e c i a l i s e with
Z
to the case of a gen-
list the s t r u c t u r e
and list their p r i n c i p a l
c o m m e n t on the proofs which
to see how the
We w i l l begin by stating
Next we will give the d e f i n i t i o n s
case.
lecture
maps we p r o p o s e
formal properties.
of the s t r u c t u r e maps,
of the formal properties. relate
we use these
A.
to
A*
Then we give
to show that if
case of o r d i n a r y
coefficients, all our s t r u c t u r e P to those c l a s s i c a l l y considered.
and
in the g e n e r a l i s e d
two p r o p o s i t i o n s
to the c l a s s i c a l
our
maps
cohomology specialise
- 57-
It w i l l ing
in a s t a b l e
the
usual
category
c a n be
suppose
given
map
These
supposed
are
but
if the
1
E
to h a v e
reader
and
the
as if w e
we have
objects
E,
with
to this,
methods.
so t h a t w e
a unit
usual
are w o r k -
smash-products
by k n o w n
a ring-spectrum
diagrams
i A
S0 A E
in w h i c h
u: E A E ~
following
to w r i t e
"demythologised"
a product
the
convenient
properties;
statements shall
be
map
We are g i v e n
i: S O
properties;
our
that
> E. is,
are h o m o t o p y - c o m m u t a t i v e .
~ E A E
EA
EA
E ~ A
1 )
EA
1
E
"' )
E E A E
E ^ S o 1 A i.
~
> E
E ^ E
E A E
E
EA Here
is We
X
with
the u s u a l recall
En(X)
switch
that
coefficients
in =
E ~ map.
the h o m o l o g y
groups
E
by
are g i v e n
[Sn, E A
X] = ~n(E A X)
of a s p e c t r u m
.
E
-58-
The c l a s s i c a l MacLane
spectrum
eralised of
E
case is g i v e n b y t a k i n g
case
with
K(Zp).
The a n a l o g u e
is t h e r e f o r e
coefficients
E , ( S 0) = ~,(E).
Since
E,(E)
in E
products.
More precisely,
u: E A F
> G
three products,
E
E.
to b e
of
A,
in the gen-
= z,(E A E), The analogue
the h o m o l o g y of
is P we have various
is a r i n g - s p e c t r u m , suppose
given
Z
a pairing
of s p e c t r a .
Then we shall have
which
in the f o l l o w i n g
appear
the E i l e n b e r g -
to c o n s i d e r
commutative
diagram.
~.(E A X) | ~.(F A Y)
9,
> z.(G A X A Y)
| l"J
~.(X A E) | ~.(F A Y)
m
'It (': A i), I > ~,(X A G A Y)
i i
IIA !
z, (X A E)
Here
the p r o d u c t
as u s e d
~
as follows.
m ( f | g)
S p A Sq
in L e c t u r e
version
Suppose
f: S p
9 > ~,(X A Y A G)
is the u s u a l e x t e r n a l
(for example)
is a b a c k - t o - f r o n t
Then
| ~,(Y A F)
of
i, N o t e
9.
7.
homology
product,
The product
The p r o d u c t
m
is d e f i n e d
given maps
> X A E,
is the f o l l o w i n g
g: S q
> F A Y .
composite.
fAg > X A E A F A Y
IA,AI
v'
) X A G A Y
- 59 -
Since
it is i m p o r t a n t
in t h e i r
correct
By t a k i n g special
for us in this
order,
X = SO
or
we will Y = SO,
~
(E) | ~ P
m:
m
factors
as o u r b a s i c p r o d u c t .
we obtain
the f o l l o w i n g
~
(F ^ Y)
> ~ p + q ( G ^ Y)
| ~
> ~ p + q ( X A G)
q (X A E)
P m:
~
(E) | ~ q
In p a r t i c u l a r , uni t.
~, (E) ;
F o r any
Y,
the p r o d u c t
map
m: is the u s u a l 2
7,(E)
one,
is a r i g h t m o d u l e
factors
|
(F) . > ~p+q (G)
~,(E)
~,(E A Y)
7,(E
is a l e f t m o d u l e
over
2).
~, (E).
the m a p
F o r any
over
considered X,
~,(X A E)
The product
~,(X A E) | ~,(E A Y)
to g i v e
ring
7, (E ^ Y)
A Y)
I, N o t e
.
is an a n t i c o m m u t a t i v e
and c o i n c i d e s w i t h
(see L e c t u r e
m:
(F) q
P
in U C T
use
to k e e p
cases. m:
with
lecture
7, (X A E A Y)
a map A Y)
7,(X A E) | w h i c h w e also call
---~ 7,(X A E A Y),
m.
We h a v e p r o d u c t m a p s
and thus
m:
7,(E)
m:
~.(E
~,(E A E)
| 7,(E A E)
A E)
|
becomes
~.(E)
> ~, (E A E)
>
~,(E
a bimodule
s h o u l d be n o t e d t h a t the t w o a c t i o n s
of
A E).
over n,(E)
~,(E). on
It ~, (E A E)
60
are in general quite distinct; between
the g e n e r a l i s e d
means
this
is the main d i f f e r e n c e
case and the classical
we have only one action of these two actions
-
module over
~,(E)
in w h i c h
Z
on A,. The p r e s e n c e of P that the g e n e r a l i s e d case demands
a little more care than the c l a s s i c a l We now assume
case,
that
case.
~,(E A E)
(using the right
is flat as a right
action).
By using the
switch map T: E A E to i n t e r c h a n g e
the two factors,
lent to assume that ~, (E)
~, (E A E)
(using the left action).
restrictive, notably
> E A E
but it is s a t i s f i e d
we check
that it is equiva-
is flat as a left m o d u l e over This h y p o t h e s i s
is somewhat
in many i m p o r t a n t
cases,
the cases E = BO, BU, MO, MU, MSp,
(see L e c t u r e
i, L e m m a
S and K(Zp)
28).
With this hypothesis,
we w i l l see that
is a Hopf algebra
in a fully
satisfactory
any s p e c t r u m
~,(E A X)
is a comodule
~, (E A E). structure
X,
We will now make
sense,
~, (E A E) and that for
over the c o a l g e b r a
this more p r e c i s e by listing
maps we shall introduce,
and giving
their p r i n c i p a l
properties. The s t r u c t u r e maps
comprise
4: ~,(E A E) | ~, (E A E) two "uni t" maps
the
a p r o d u c t map > ~,(E A E),
-
61
-
nL: ~, (E)
> ~, (E A E)
nR: ~,(E)
) ~, (E A E)
a counit map ~: ~,(E ^ E) --~ ~,(E) a canonical a n t i - a u t o m o r p h i s m c: ~,(E ^ E) ~
~,(E ^ E)
a diagonal map > ~,(E ^ E) | ,(E)~,(E ^ E)
= ~E: ~,(E ^ E) and for each s p e c t r u m
X,
a coaction map > ~,(E ^ E) | ,(E) ~*(E ^ X)
= ~X: ~,(E ^ X) (The diagonal map action map
~X
~E
is obtained by specialising
to the case
.
the co-
X = E.)
It is important to note that in the tensor-product the action of
~,(E ^ E) @ ,(E) ~*(E A X), left-hand factor
~, (E ^ E)
action of
on the right-hand
~, (E)
the usual left action.) the tensor-product
is the right action. factor
on the (The
~, (E ^ X)
is
This is exactly what we need to use
notation in a systematic way.
The tensor-product
~,(E A E) | ,(E) ~*(E A X)
be considered as a left module over left action of
~,(E)
~,(E) ~,(e
on
~, (E),
~,(E ^ E);
~) x )
=
(~,e)
|
by using the
that is, x
can
-62
(i 6 ~,(E), The
e 6 ~,(E A E),
coaction
map
~X
X = E.
~,(E A E) | module
over
bimodules
Here
the p r e v i o u s
~,(E),
over
s p e c t to the a c t i o n s
the r i g h t a c t i o n
The diagonal
map
of the o t h e r
structure
of
~E
of
~,(E)
will emerge
can be t a k e n o v e r the i n t e g e r s .
is d e f i n e d
The p r i n c i p a l are as follows.
properties
The product
map
and has a u n i t e l e m e n t are h o m o m o r p h i s m s
a c t i o n of
~,(E)
on
le = ~ ( ( n L l ) Similarly,
~,(E)
on
@ e)
the r i g h t
action
the p r o d u c t
structure
is a s s o c i a t i v e ,
i.
of g r a d e d
~,(E A E)
on w h i c h
of t h e s e #
The maps
n L,
rings with
unit.
maps anticom-
n R,
e
The
is g i v e n by
(I 6 ~ , ( E ) ,
e 6 ~ , ( E A E))
~, (E)
~, (E A E)
of
re-
f r o m the p r o p e r -
map
c
as a r i g h t
maps with
The tensor-product
and
to
is a m a p of
ties g i v e n below.
mutative
apply
~,(E).
The b e h a v i o u r
~
~,(E).
two p a r a g r a p h s
c a n a l s o be c o n s i d e r e d
by u s i n g
factor.
over
.
the t e n s o r - p r o d u c t
~,(E A E)
~,(E)
the r i g h t - h a n d
x 6 ~,(E ^ X))
is a m a p of l e f t m o d u l e s
In p a r t i c u l a r , the case
-
on
.
is
g i v e n by el = ~(e |
(nRl))
(e 6 ~,(E A E),
I 6 ~,(E))
We h a v e en L = i,
en R = l, c~ L = n R, cn R = n L, EC = Et C 2 = 1 .
.
left
- 68
These properties and
c
-
determine the behaviour of
with respect to the actions of
r
nL,
w,(E).
n R,
e
In particular,
is a map of bimodules. The coaction map is natural coaction map is associative,
for maps of
X.
The
in the sense that the following
d i a g r a m is commutative. ~X
n,(EAX)
) w,(EAE)
~XI
|
z.(E)n,(EAX) 11 | ~x
~E Q1 ~,(EAE)
|
~,(EAX)
(Note that
1 | ~X
modules over
>~,(EAE)
is defined because
7, (E),
and
is a map of right modules can specialise
e~,(E) ~T,(EAE) |
~E | 1 over
~X
(E)
is a map of left
is defined because
n,(E).)
this diagram to the case
~, (EAX)
In particular, X = E,
~E we
and we see
that the diagonal map is associative. The behaviour of the diagonal with respect to the product is given by the following commutative z,(EAE)
| w, (EAE)
diagram.
> ~, (EAE)
~E
[~, (EAE) |
~. ( E )
~,(EAE) ]
" ~. (EAE) | (E)~*(EAE) |
[~,(EAE)
|
(E) z* (EAE) ]
64 -
Here the map
r
is d e f i n e d by
(e | f | g | h) =
(-l)Pqr
| g) | r
@ h)
where
f 6 ~ (E A E), g E ~ (E A E). It has to be v e r i f i e d P q that this formula does give a w e l l - d e f i n e d map of the p r o d u c t of tensor products the facts
over
~.(E),
stated above. The b e h a v i o u r
g i v e n by
but this can be done using
~E(1)
~EnL k =
of the d i a g o n a l map on the unit is
= 1 | i.
It follows
that we have
(nLk) | i , ~EnR ~ = 1 @
The b e h a v i o u r
(nRk)
of the d i a g o n a l
the counit is given by the f o l l o w i n g ~X
~ . (F, A X)
~.(E
^ X)
<
>
~,(E)
|
on
~. (E ^ X).
The map
|
r | 1
is a map of right modules
over
the following
diagram.
commutative ~E
7, (E A E)
~,(E).
^ X) left action of
is d e f i n e d b e c a u s e Similarly,
<
we have
) ~.(E A E) @7,(E) 7*(E A E) l|
~ , (E ^ E)
diagram.
n (E A x) 7, (E) *
Here the b o t t o m arrow is given by the usual 7. (E)
. 7,(E A E) |
.
map with r e s p e c t to
commutative
~ . (E A E)
~--"
(k s ~.(E))
(E) 7* (E)
-
65
-
H e r e the b o t t o m a r r o w is g i v e n by the r i g h t a c t i o n of
~, (E)
on
is a
~,(E A E).
The map
map of left m o d u l e s
over
1 8 e
is d e f i n e d b e c a u s e
e
~, (E).
The b e h a v i o u r of the d i a g o n a l w i t h r e s p e c t to the canonical anti-automorphism
c
is g i v e n by the f o l l o w i n g
c o m m u t a t i v e diagram.
~ , (E A E)
~E
~, (E A E) |
~, (E)
n, (E A E)
C
~ . (E A E) Here the map
C
q~E
~ ~,(E
~ . (E)
~ (E ^ E) *
is d e f i n e d by C(e @ f) =
(e 6 ~ (E A E), P
^ E) |
(-l)Pqcf @ ce
f E ~ (E ^ E)) q
.
It has to be v e r i f i e d that this f o r m u l a does give a w e l l d e f i n e d map of the t e n s o r p r o d u c t over
~,(E),
but this can
be done using the facts s t a t e d above. The f o l l o w i n g c o m m u t a t i v e d i a g r a m s
express
that
p r o p e r t y of the c a n o n i c a l a n t i - a u t o m o r p h i s m w h i c h in the c l a s s i c a l case is t a k e n as its d e f i n i t i o n .
~.(E
~,(E
A E)
^ N) | ~ . ( E ) ~ , ( E
~
^ E)
~(1|
--
~
~, (E)
~, ~ . ( N
^ N)
- 6 6
~ . (E ^
E)
-
> ~. (E)
--
~R
~*(E A E) #(c|
~r.(E ^ g) | It has to be verified defined maps
the facts
This
order
and
$(c@i)
do give well-
over
~.(E),
but this can
completes
the list of properties
of our struc-
stated
We also require
to show that certain
Lectures
~(l|
of the tensor product
be done using
ture maps.
that
i, 2).
ring-spectrum
Let
E;
F
z.(E A E) |
comodules
formal property
E.(X)
we might have
in
are extended
be a left m o d u l e - s p e c t r u m
diagram
.. (E)
above.
one further
for example,
Then the following
> ~.(E A E)
(see
over the
F = E A Y.
is commutative. m
7.(F)
> 7. (E A F)
~E | i[ ~. (EAE) |
~ (EAE) | ~,(F) ~. (E) * ~.(E)
The map
i | m
over
is defined
1|
because
~. (EAE) | m
~.(E)
~.(EAF)
is a map of left modules
~. (E) . We now give the d e f i n i t i o n
The product following
~
of our structure
is given by either way of chasing
commutative
square.
maps.
round the
-67 -
%)! ~,(E A E)
> ~,(E A E A E)
@ ~ , ( E A E)
[
{~ A l ) ,
~.
(For
~
ginning
and
(E
~',
of t h i s
E
A
A
see
the d i s c u s s i o n
lecture.) f: S p
then
Sp A
~(f @ g)
Sq
fag
following
We d e f i n e morphisms.
into
nL
and
~
E
suppose
given
) E A E;
composite.
IAi
SO
A
S0 A E
E A E nR
We define
> E A E A E A E
uAu > E A E.
as the
to b e E
iA1
the
and
c
> E A E
) E A E
left
and right
corresponding
~: E A
E
) E
and 9 : EA remains
E
factors. induced
homo-
to b e t h e h o m o m o r p h i s m s
induced by
It o n l y
at the be-
maps
E E
of p r o d u c t s
Sq
IATAI
) E A E A E A E
E
map
~ , (E A E)
In o t h e r w o r d s ,
) E A E, g:
is the
We have
which
(1AIJ) ~ >
E)
> EA
to d e f i n e
E ~X"
-68-
Lemma
1
If
~,(X ^ E)
then
7 , (E) ,
is f l a t as a r i g h t m o d u l e
m: ~ , ( X ^ E) @ , ( E ) ~ , ( E ^ Y)
over
> ~,(X ^ E ^ Y)
is iso.
Proof. KT 1
(see L e c t u r e
transformation for
Y = so;
Pass
to d i r e c t
This
is e s s e n t i a l l y
i, N o t e
between
12).
homology
therefore
the trivial
The map
m
functors
of
c a s e of
is a n a t u r a l Y
which
it is iso for any f i n i t e
is i s o
complex
Y.
limits.
We n o w d e f i n e h:
~,(X A Y)
to be the h o m o m o r p h i s m
> ~,(X A E A Y)
induced by
X A Y ~ X A SO A Y
The map
h
is e s s e n t i a l l y
IAiAl
> X A E A Y 9
the H u r e w i c z
homomorphism
in E-
homology. If
~,(X A E)
is flat, w e c a n c o n s i d e r
the
follow-
ing c o m p o s i t e .
~,(X ^ Y) We d e f i n e that
-I n,(X A E A Y) m > = m-lh.
~,(E A E)
X = E; ~y.
h
we
This
~,(X A E) | ~, (E) ~* (E A x )
In p a r t i c u l a r ,
since we
is flat, w e c a n s p e c i a l i s e
take the r e s u l t i n g completes The proofs
map
the d e f i n i t i o n
~
are a s s u m i n g
to the c a s e
for o u r c o a c t i o n m a p
of the s t r u c t u r e
of all the f o r m a l p r o p e r t i e s
maps.
are by
.
-
diagram-chasing.
In proving
we have to make our d i a g r a m for
h
and one for
coaction
m.
~X
(i | m) (~E | i)
prove
that
E,(F)
the following
of
~X'
of course
up out of two subdiagrams,
For example,
is natural
#F m =
-
any property
map is associative,
tary results;
69
in proving
we first prove for maps
of
that the
two more elemenX,
(which is the d i a g r a m
is an extended
one
and required
comodule).
to
We now set up
diagram. ~x
, (E A x)
~,(E A E) 8
~ (E A X) ~, (E) * i | h
~ . ( E ^ E ^ X)
~EAX
> ~.(E ^ E)
|
~, (E)
7 . ( E A E ^ X)
i[1 m |
~E01
ml ~, (EAE)
|
~n , (EAE) | ~, (E) T,, (EAE) | ~, (E) 7, (EAX)
(E) ~* (EAX)
Here the top square
is commutative
because
h
is induced by
a map X ~ S O A X iAl) E A X , and
~X
square
is natural
is commutative
F = E A X. iary
for maps
spect
the required
are proved
In proving to the product,
X.
by the second
This gives
results
of
Similarly,
the b o t t o m
result mentioned, result.
taking
The two subsid-
in the same way.
the b e h a v i o u r
of the diagonal
it is convenient
to prove
with
re-
a slightly
-
more
general
result
and
~,(A A B A E)
first.
70
-
Suppose
are all flat;
that
~,(A A E).
t h e n the f o l l o w i n g
~,(B A E) diagram
is c o m m u t a t i v e .
~, (AAX)
[7, (AAE) |
~. (E)
~, (AABAXAY)
| ~, (BAY)
7, (EAX) ]
> ~. (~l~N) | Here
[~,(BAE)
lower h o r i z o n t a l
m a p is the o b v i o u s
m a p sends
e | f 8 g | h
(-l)Pq~' (e | g) | ~(f | h) at the b e g i n n i n g
Next
observe
serves is.
(N) ~* ( ~ X A Y )
| ,(E) ~*(EAY) ]
the u p p e r h o r i z o n t a l
commutative
|
of this
lecture).
exactness,
since
the f u n c t o r
applying
of p r o d u c t s
This d i a g r a m
as b e f o r e
and the
into
(see the d i s c u s s i o n
in the same w a y that
product,
- separate ~, (E A E) |
it t w i c e p r e s e r v e s
is p r o v e d h
and
~, (E)
m.
pre-
exactness;
that
the r i g h t m o d u l e ~, (E A E A E) ~_ ~, (E A E) |
is flat.
So w e m a y
apply naturality
specialise
~,(E)
to the case
~, (E A E) A = B = E.
to the m a p A A B = E A E
we see t h a t the f o l l o w i n g
diagram
~3
E
;
is c o m m u t a t i v e .
Now
-71 -
~, (EAX) | ~, (EAY)
> ~, (EAXAY)
~X | ~Y[
~XA
Y
[~, (EAE) | ~.(E) ~. (EAX) ] 7, (EAE) @
Here
[~, (EAE)
|
~, (E)
map
(-l)Pq~(e | g) 8 ~(f | h). of the c o a c t i o n m a p w i t h
sends
This
e @ f | g @ h
diagram
respect
Finally we specialise
apply naturality
to the case
the r e q u i r e d
P>
commutative
to s h o w
rectly
to the c l a s s i c a l
groups
of a s p e c t r u m
E
X = Y = E
and
properties
does
.
formal
formulae,
case.
X
We r e c a l l
with
E -n(x) a Kronecker
as follows.
=
coefficients
[snAx,
E]
|
E
q
Suppose
(X)
.
:~ 7r (E) P+q
given maps
cohomology,
specialise
cor-
t h a t the c o h o m o l o g y
product
E-P(x)
involving
that o u r d e f i n i t i o n s
by
defined
homology
com m e n t .
We now turn to f u r t h e r
We h a v e
the b e h a v i o u r
diagram.
of the r e m a i n i n g
for any s p e c i a l
which will help
into
to the m a p
The p r o o f n o t call
gives
to the e x t e r n a l
X A Y = E A E We o b t a i n
, (EAXaY)
~,(E)
~, (EAY) ]
the l o w e r h o r i z o n t a l
product.
|
in
E
are g i v e n
-
f: S P A x Then
(f,g)
is the
IA g
sPAs q
) E,
IA ~
In p a r t i c u l a r , Since
these
are d e f i n e d
to s u s p e n s i o n ) , mology
groups
as follows.
Suppose
a- s P A E Then
af
E,
ag
U
> EAE
the c o h o m o l o g y
in terms
of m a p s
from
~ E .
groups E
to
E
l e f t on the h o m o l o g y
E*(X).
The precise
E*(E). (up
and coho-
definitions
are
given maps f: S q
IA f
> FAX,
g: srAx----> E
.
~ sPAEAX
aA 1
> EAX
,
is sPAsrAx
In this w a y become
.
is sp A sq
and
fA 1
> sPAXAE
we have
and
9 FAX
composite.
they act on the E,(X)
-
g: S q
following
~ sPAEAX
72
E*(E)
becomes
left m o d u l e s We w i l l
~g
over
show that
b y the c o a c t i o n
x E E,(X)
and
x i E E.(X)
.
a > E
a ring with
this
determined
Proposition
. sPAE
.
unit,
and
E,(X) , E*(X)
ring.
the a c t i o n map
~X x -- [ e i | x i i
~X" ,
of
E*(E)
Suppose where
on
E,(X)
a E E* (E),
e i E E,(E)
,
T h e n w e have:
2
ax = [ ( a , c e i > x i . i To p r o v e diagram.
this p r o p o s i t i o n ,
we
s e t up the
following
is
- 73-
a
~, (E A X) -
,(E
~. (E A X)
/
^ N ^ X).
a
)
~,(E
A
E a
X)
i
(UAI: ,~ ~,(E
Here
a
A E)
|
~,(N
is d e f i n e d
c~
^ X)
7, (E A X)
by e(e @ x) = ( a , c e ) x
It is e a s y proves
to s h o w t h a t the d i a g r a m
Proposition
m i n e d by the v a l u e s
of
an e l e m e n t
(z,x)
to ask for a c a l c u l a t i o n
on
in t e r m s
nearer
answer
~X"
this q u e s t i o n ;
than that which
a E E*(E),
where
of
e i 6 Eq(i)(E),
Proposition
This
There here
x 6 E,(X),
is a c h o i c e
I will gave
x 6 E, (X)
and
x i 6 E,(X).
is d e t e r it is
of the a c t i o n of
I actually
y 6 EP(x),
z 6 E*(X)
for a l l
reasonable
which
is c o m m u t a t i v e .
2.
In the c a s e w h e n
E*(X)
.
E*(E)
of f o r m u l a e
give one which in S e a t t l e .
seems
Suppose
~X x = .[ e. | x., l l l
T h e n w e have:
3
(ay,x)
= [ (-i) pq(i) ( a , e i ( Y , X i ) ) i
The in
~,(EAE),
on
~, (EAE) .
formula and
on the r i g h t m a k e s
(y,xl)
lies in
sense,
n,(E), w h i c h
because acts
ei
lies
on the r i g h t
-
To p r o v e
and
F)
~,(F ^ X)
as f o l l o w s .
f: S r
> F A X;
SPAS r
let
check
y,f
set up the
~.(E Here
8
a
E)
following
~.(E
is d e f i n e d
e 6 E
mutative.
q
(E). This
E
composite lay 9 FAE
.
.
>
> ~, (E A E)
2/
I~,(E ^ E A E)
1
A
A X)
>
~, (E A E)
by
It is e a s y shows
Proposition We w i l l
Sp ^ X
diagram.
Y*
(-i) p q e(y,x> to s h o w
that
this
diagram
is com-
that
y,x = [ i and proves
y:
be the
= (a,y,x>
8(e | x) = for
given
Y%
A E A X)
|
define
that
~. (E ^ X)
~,(E
first
9 TAI ) F A s P A x
(ay,x> We n o w
we
> ~,(F ^ E)
Suppose
iAf ) s P A F A X
Then weeasily
-
the p r o p o s i t i o n ,
y,: (for any
74
(-l)Pq(i)ei
,
3.
now discuss
the w a y
in w h i c h
our
constructions
- 75
specialise
to the c a s e
f r o m the d e f i n i t i o n s to t h e i r right with
classical
action
of
the left
either
side,
can b r i n g
~E
@,
on
because
follows
n
e.
and
The
coincides
acts as a u n i t on for i n t e g e r
t h a t in P r o p o s i t i o n
the K r o n e c k e r
clear
specialise
and
n,
the u n i t
it o u t s i d e
ei;
multiples
3 w e can b r i n g
and a f t e r t h a t we
product,
so as to o b t a i n
formula.
that
and
nR
~, (E A E) = A,
to the l e f t of
) H*,
~E
nL ,
(Y,Xi)
is the d u a l
Thus
= Zp
It f o l l o w s
(ay,x)
A* | H*
~,
counterparts
~,(E)
It is s u f f i c i e n t l y
K (Zp).
and so the r e s u l t
the f o l l o w i n g
It f o l l o w s
=
that
action,
of the unit. the f a c t o r
E
-
= [ i
(-i) p q ( i ) ( a , e i ) ( y , x i )
~X
is i n d e e d
the d u a l of the a c t i o n m a p
and
(specialising
of the c o m p o s i t i o n ~X
specialise
Since we have
to the case map
to t h e i r
X = E)
A* | A*
that
> A*.
classical
counterparts.
seen that (i 8 c)~ E = nLE
and ~(c | i)~ E = nRE it n o w
follows
that
c
specialises
,
to its c l a s s i c a l
counter-
part. It r e m a i n s the c l a s s i c a l ised
case
case w e h a v e
only
to p o i n t
o u t one d i f f e r e n c e
and the g e n e r a l i s e d introduced
case.
a left a c t i o n
of
between
In the g e n e r a l E*(E)
on
-
E,(X). H,
This does not specialise
which
is usually
the latter
considered
is a right action,
(y 6 H p
7B-
x E Hq
The connection Proposition
=
of
in the classical
defined
A* case,
on since
by
(-i) (P+q)r
a E A r) between
2 and 3. xa =
the two actions
may be read off from
We have
(-l)qr (ca)x
Thus the left and right actions automorphism,
to the action
(x E H , a E A r) q
9
differ by the canonical
as one might expect.
anti-
-
LECTURE COHOMOLOGY
bordism.
-
4 SPLITTING G E N E R A L I S E D THEORIES WITH COEFFICIENTS
S. P. N o v i k o v of the generalised
77
[23, 24] has emphasised
cohomology
theory provided
This is a representable
functor;
the importance
by complex
co-
if we take it
"reduced" , we have sun(x) It has been proved by Brown neglects
all the primes
spectrum
splits
BP(p)
coincides ) ~
primes
except
ducing
coefficients.
with
b
except
one prime
i
~ ~
[i0]
that if one
p,
then the MU-
> ~.
i
since
The business
Let to
Qp
p.
spectrum.
BP(p)
one may be formalised
prime
sn(i)BP(p)
the B r o w n - P e t e r s o n
the product
n(i)
a/b
as
.
and Peterson
V sn(i)BP(p) i
means
with
[x,snMu]
as a sum or product:
MU~
Here
=
The sum
is connected of neglecting
conveniently
and all
by intro-
be the ring of rational
Then we can form
numbers
MU*(X;Qp),
and we have sun(x; Qp) - ~
L n + n(i) (X)
,
i where Lm(x) and
L
is a suitable This
=
version
situation
[x,smL] of the B r o w n - P e t e r s o n
has been considered
spectrum.
by S. P. N o v i k o v
-
[24].
Potentially
theory
L*
it gives Lecture
78-
it is very profitable.
is just as p o w e r f u l
rise to the same 2).
However,
as
"Adams
MU*(
the groups
L*(X)
MU*(X;Qp);
groups
the ring of operations
algebra
L,(L)
calculations
similarly
(see Lecture
with
culations w i t h
L
3).
they involve because
ture of
L*(L),
MU*(MU),
etc.
L*(L)
and the Hopf
and easier
than cal-
split a c o h o m o l o g y
large elements
of choice.
propose
formulae
theory into summands,
To secure
these
ends
split the theory into summands The m e t h o d w h i c h
say
in helpful
e.
say
for the struc-
thesis.
of
When we
we should try to do and e n l i g h t e n i n g
I w o u l d even be w i l l i n g
to
larger than the i r r e d u c i b l e
I propose
operations,
It is
for the s t r u c t u r e
the following
issuing
are not
authors have not yet
as are available
so in a c a n o n i c a l way,
ical idempotents,
and by Novikov,
and i l l u m i n a t i n g
etc.,
have not yet been
The reason is that the split-
of this that these
I therefore
of c o h o m o l o g y
for the c o e f f i c i e n t
these benefits
in practice.
given such helpful
ones.
(see
For all these reasons,
tings given by Brown and Peterson,
formulae.
sequence"
MU.
fully r e a l i s e d
doubtless
for example,
are much smaller
should be smaller
Unfortunately,
canonical;
;Qp);
spectral
than the groups L*(S~
The c o h o m o l o g y
A,
is to take a suitable and c o n s t r u c t
Then w h e n e v e r
A
ring
in it canon-
acts on a module,
-
say
H,
H
-
w i l l s p l i t as the d i r e c t sum I will
theory.
79
first show h o w this thesis
cobordism.
For K - t h e o r y
tolerably complete
applies
to K-
it is u s e f u l as a tool in a t t a c k i n g I shall give a t r e a t m e n t w h i c h seems
and s a t i s f a c t o r y
I w i l l then turn to c o b o r d i s m Here the theory is s o m e w h a t
(Lemma 1 to L e m m a 9 below).
(Lemma i0 to T h e o r e m 19 below).
less complete,
cient to s h o w the e x i s t e n c e of c a n o n i c a l ism w i t h s u i t a b l e Let
R.
(1-e)H.
Not only is the case of K - t h e o r y s o m e w h a t easier,
but for t e c h n i c a l reasons
K*(X;R)
eH 9
R
K
summands
in cobord-
coefficients. be a s u b r i n g of the rationals.
be ordinary,
We w r i t e
but it is suffi-
for
functor; we w r i t e
complex K-theory, with coefficients K0;
BUR
Lim I
m e t h o d w i l l p r o v e more.
then
K(X;R)
K*(BUR;R). subgroup
in
is a r e p r e s e n t a b l e
for the r e p r e s e n t i n g
q u i r e some i n f o r m a t i o n on n e e d e d is that its
Let
space.
We re-
All that is r e a l l y
[21] is zero; but our
It is for this p u r p o s e that w e in-
troduce the first few lemmas. d
Let f: BU
> BU
of the space
be a p o s i t i v e integer,
and let
be the map o b t a i n e d by taking the i d e n t i t y map BU
and adding it to itself
the H - s p a c e s t r u c t u r e of
BU.
Lemma 1
If
d
is i n v e r t i b l e
in
R
then
d
times,
using
-
80
-
f,: H, (BU;R)
H, (BU ;R)
f*: H*(BU;R)
) H* (BU;R)
and
are isomorphisms.
Proof.
We w i l l prove that
as an inductive hypothesis, the Chern classes decomposable ment
in
Pn 6 H2n(BU;R)
such a
Pn
posable
elements.
Since
d
which
duality,
is a n o n - z e r o m u l t i p l e
f,
f*
is mono.
H,(BU;R)
f
dual argument
Indeed,
But we can find
of
cn
mod decom-
mod decomposables.
n
that
let
Lemma
R 1, R 2
f*
f*.
is epi; by
that
f,
is epi
i: BU
some m i n i m a l
knowledge
the P o n t r y a g i n product, so that
f,
and
is a h o m o m o r p h i s m
i. be two subrings )
2
R 1 c R 2,
shows
One needs
is an H-map,
We have an obvious map
If
ele-
the p r e c e d i n g p a r a g r a p h was w r i t t e n
correctly.
This proves Next,
Lemma
all
lies in the image of
n
and proves
as a ring under
the fact that of rings.
c
= dc
n
contains
is mono.
so as to dualise of
R,
the induction
A precisely and
in
f*c
f*
For any p r i m i t i v e
f*Pn = dPn"
Therefore
Suppose,
Then it contains
H2n(BU;R).
we have
is i n v e r t i b l e
This completes
is epi.
that the image of
cl, c2,...,Cn_ I.
elements
f*
the maps
BUR I.
of the rationals.
- 81 -
H , ( B U ; R 2)
i,:
9 H , ( B U R I ; R 2)
i*: H * ( B U R I ; R 2) ---> H * ( B U ; R 2) (i•
H,(BUxBU;
R2)----~ H , ( B U R I x B U R I ;
(ixi)*:
H*(BURIxBURI;
R2)
) H* (BU•
R2)
R 2)
are i s o m o r p h i s m s .
Proof. may assume Consider
If
R 1 ~ Z.
We n o w c o n s t r u c t
the p o s i t i v e
t h e m in a s e q u e n c e a map
fn: BU
integers
dl,
> BU, fl
BU
a construction of the
about sults
result (i • i),
about
coefficient
Lemma
about
i*
in
i.
theorem.
and a r r a n g e we have
n
Take the maps
or i t e r a t e d
for
B U R I.
n~
BU
i: BU
BU.
from Lemma
f r o m the K u n n e t h
(i • i)*
follow
This p r o v e s
Lemma
R 1 c R 2.
...
this
> BUR 1
is
We h a v e
= Li~(H,(BU;R2),fn,) follows
)
mapping-cylinder;
The map
f i r s t c o p y of
i,
BU
~...
9 i.
The result
theorem.
T h e re-
f r o m the u n i v e r s a l 2.
3
Suppose
BUR 1 .
R1 d
so w e
f
follows and
for
For each
d3, . . . .
9 BU
H,(BURI;R2) Now the
a model
f2 > BU
the i n j e c t i o n
d2,
is t r i v i a l ,
invertible
as in L e m m a
and f o r m a " t e l e s c o p e " gives
the r e s u l t
R1 = Z
T h e n the m a p s
-
(ixi)*:
are
i,:
K,(BU;R2)
i*:
K*(BURI;R2)
homeomorphisms
Proof. spectral
i*
respect
Let
P
; K*(BU;R2) R2)
The maps
with
-
> K,(BURI;Rs
K*(BURIxBURI;
isomorphisms.
82
> K*(BUxBU;
and
to the
be
2, the m a p the
proves
(i x i)*
i: BU
spectral
> BUR 1
sequences
the r e s u l t
is s i m i l a r ,
The
space
u: B U R 1 x B U R 1
about using
>
BUR 1 BUR 1
for
We r e t a i n
sider
set of p r i m i t i v e
the
the
set of e l e m e n t s
set may be
a
identified
The
tive
the
usual
in the
sense
and
the p r o d u c t
elements
with
that
the
map,
that in
and
and
let
onto
the t w o
R 1 c Rs
and
K(BURI;R2) ,
u*a = ~*a + ~*a.
set
of c o h o m o l o g y
con-
that
is,
This
operations
> ~ ( X ; R 2)
connected
+ y)
i*
let
X,
natural,
that a(x
for
X = BUR 1 .
sequence
be the p r o j e c t i o n s
such
for
> K * ( X ; R 2)
assumption
for all
an i s o m o r p h i s m
proof
an H - s p a c e ;
a: K ( X ; R I) are d e f i n e d
induces X = BU
i,.
is
B U R 1 x B U R 1 ---> BUR1
which
topology.
Consider
the s p e c t r a l
be
factors.
the
also
> K , ( X ; R 2) .
H*(X;K*(P;R2))
~,~:
are
sequence
By L e m m a
This
filtration
a point.
H,(X;K,(P;R2))
between
(i x i)*
R2)
= a(x)
+ a(y)
.
and
addi-
-
(If an o p e r a t i o n Such operations
is additive,
83
-
it follows
need not be stable.
This set is to be t o p o l o g i s e d K(BURI;R2);
in other words,
if it vanishes
to be c o n s i d e r e d
long as
R 1 c R 2.
dealing only with K(X;RI) A ( R 2)
~ ~(X)
to Lemma
is e s s e n t i a l l y
a
is close
of d i m e n s i o n
3 above,
independent
finite C W - c o m p l e x e s
| RI, K(X;R2)
~ ~(X)
to zero
n.
the set of operations
(This fact w o u l d be trivial
for the set of operations
it p r i m a r i l y
as a subset of
an o p e r a t i o n
in all C W - c o m p l e x e s
According a
that it is Rl-linear.)
X,
introduced
R I,
so
if we were
since
| R2.)
as the ring of c o h o m o l o g y
of
then we have
We t h e r e f o r e w r i t e above,
operations
and regard on
K(X;R2).
We define A(R)
= R + ~(R)
By m a k i n g the first s u m m a n d K(P;R),
the set
cohomology
A(R)
R
may be i d e n t i f i e d w i t h the set of
operations a: K(X;R)
which
act in the obvious way on
are defined
for all
X,
> K(X;R) natural,
and additive
(hence
R-linear).
Lemma
4
If
R 1 c R2,
we have a m o n o m o r p h i s m
I: A(R I) such that for each
a 6 A(RI)
> A ( R 2) and each
X
the following
-
84
-
d i a g r a m is commutative. K (X;RI)
i,
> K (X;R2)
aI K(X;RI )
i,
1 ~a > K(X;R2 )
This follows
from the preceding discussion
together
with the fact that i,: K ( B U ; R I) ---> K ( B U ; R 2) is monomorphic. Because of this lemma, construct idempotents
in
A(Q)
it will be sufficient to and then prove that they are
defined over some suitable subring of the rationals. over
Q
the idempotents
allows us to identify
are obvious.
K(X;Q)
But
The Chern character
with the product
H2n(X;Q)
.
n Let us define e
en
to be p r o j e c t i o n on the n
(h~
=
th
factor:
(0,0,...,0,h2n,0,.. -) -
n Then
en
is an idempotent
in
A(Q).
I now choose a positive
integer
d,
and seek to
construct a "fake K-theory" with one non-zero coefficient group every obvious.
2d
dimensions.
The required idempotents
Take a residue class of integers mod
d,
say
are
-
-
and d e f i n e
a E Zd ,
E
= e
This
85
s u m is c o n v e r g e n t
we use the C h e r n H2n(X;Q),
[ e n nee
E A(Q)
in t h e t o p o l o g y w h i c h
character
as above,
to i d e n t i f y
A(Q)
K(X;Q)
has.
If
with
then we have
n E
(h 0 , h 2 , h 4 , e e e
) =
(k~
4 ICe.
)
e
where
Theorem
if
n 6
[0
if
n ~
5
E
e
of r a t i o n a l s p ~ 1
k2 n = l 2h n f
mod
lies
in
A(R),
a/b
such that
where b
R = R(d)
contains
is t h e r i n g
no p r i m e
p
with
d. For example,
if
d = 2, R
is the r i n g of f r a c t i o n s
a/2 f . F o r the proof,
we n e e d to w o r k w i t h
a representation Go
of
A (R).
then where
Let
K(CP~;R)
n
be the c a n o n i c a l
line bundle
over
is the r i n g of f o r m a l p o w e r - s e r i e s
enn = We n o w m a k e the r i n g of f o r m a l which
a formal manipulation
power-series
are p o l y n o m i a l s
in
t.
in
~
Namely
:
with
= exp(t
coefficients
(I+{)
log(l+~)) t
= 1 + t~ + t ~2t - 'l ' {% 1.2
This
is true as a f o r m a l Now
consider
identity
ar/b r
p
such t h a t
integer.
find in the p - a d i c
integers
Set
moment.
p = ~m, Then
where
pd = 1
in the r i n g
ar ~r
p - 1
is a p - a d i c
~.
+
cited.
E n = [ enn = [ ~__r~n, nE~ r r
to s h o w t h a t the c o e f f i c i e n t s
any p r i m e
Q[t] [[~]],
tn (io~ (i+~))n n!
then n = [ n n
=
in
lie in
rood
Since
d
divides
a primitive
pU
R = R(d).
d; we w i s h
the i n t e g e r and
say.
makes
m
Take
to s h o w
that
p - i, I can
d th r o o t of is f i x e d
sense.
We w i s h
l,
for the
We h a v e
say
-
PeE n =
-
P~en~
~
e
89
n
=
1 + p~
+ P,~-l~2f~ "%
...
+
1,2
= Cm(~) , H e r e the b i n o m i a l
coefficient
b(t)
maps
Z
to
therefore Cm(~) which d
Z
t ( t - l ) . . . (t-r+l)
=
1.2
...
a n d is c o n t i n u o u s
it m a p s p - a d i c
are p - a d i c
integers.
for the
d
Een
Since
d -I
series whose Theorem
to p - a d i c
in
Take
~
[ ~ 1~m~d
integer,
coefficients
with
E n. e
-me
Cm(~)
this
are p - a d i c
So
coefficients we obtain
The solution
is
9
is a f o r m a l p o w e r integers.
This
proves
5. The p r o p e r t i e s
of t h e e l e m e n t s
E
6 A(R) e
follows.
Theorem
topology;
integers.
m = 1,2,...,d;
unknowns
= d -I
is a p - a d i c
r
in t h e p - a d i c
integers
is a f o r m a l p o w e r - s e r i e s
equations
say.
7
(i)
E2 = E e
(ii) (iii)
E E = 0 e 8 [ E e
(iv)
e
if
e ~ 8
= 1 e
F o r any
x,y
in
K (X;R)
are as
-
we have a "Cartan
formula" E
Proof.
(xy)=
So parts
(i),
responding
equations
in
(iv).
lies in and
lie in
in
Let both
which
x
proves
part
in
and
y
Zd 9
d;
x
follow from the cor-
are obvious. x
or
to prove it w h e n
and
y
y x
to prove it for be the u n i v e r s a l
using the Chern character.
Since
R)
> K(BU• in
e l e m e n t in R)
(iv) and completes
in
Q) ~(BU•
R)
~(BUR;R) ;
by Lemma follows
3.
K* (X;R)
.
Let
The case in
the proof of T h e o r e m
is as in T h e o r e m
This 7.
to the results
(and indeed of
As above, we are s u p p o s i n g
x
then the
by naturality.
5 and 7 lead i m m e d i a t e l y
R = R(d)
We turn
Q),
are general
on the s p l i t t i n g of
integer
is a mono-
~(BU•
K(BURxBUR;
Theorems
required).
which
the result holds
be the u n i v e r s a l
result holds
(iii)
then the result holds
calculation
is m o n o m o r p h i c , y
(ii) and
> A(Q)
It is s u f f i c i e n t
K(BUxBU;
and
t : A(R)
A(Q),
K(X;R).
~(BU) ;
by an obvious
4,
so it is s u f f i c i e n t
external products. elements
AJ
The result is trivial w h e n either
K(P;R),
y
; (Esx) (Eyy) 8+7=~
By Lemma
morphism.
to part
90
K,(X:R),
if
given a p o s i t i v e 5, and
a
runs over
-91 -
Corollary
8
(i)
We have a natural K(X;R)
direct
_~ [ K
sum s p l i t t i n g
(X),
where K
(ii)
K
(iii)
xy 6 KS+ Y(x)
= E K(X;R)
.
is a r e p r e s e n t a b l e
(X)
If
(X)
x 6 K 8(x)
and
functor.
y 6 Ky(X) ,
then
.
(iv)
We have
_~',:~( s n )
(v)
;
R
if
~n 6 c~
0
otherwise
.
Define N
: K (x) by taking Then
~
(iii).
no trouble
Part
lation
(i) follows
For part
of
KI(S2).
in
(iv).
~(s2m;Q)
7 parts
and that we have
the axiom about d i s j o i n t
(for
For part
~ ).
Part
(iii)
unions
follows
(iv), make the obvious
_~ H2m(s2m;Q).
(i),
that a direct s u m m a n d
is an exact sequence,
about v e r i f y i n g
7 part
from T h e o r e m
(ii), o b s e r v e
K ) or w e d g e - s u m s
Theorem
a generator
is an isomorphism.
of an exact sequence
(for
(s2 A X)
the external p r o d u c t w i t h
Proof. (ii),
> ~+l
For part
from calcu-
(v), let the re-
N
presenting
space
into a map
for
BUR s
K c~
be
BUR c~ ;
) ~2BUR +i,
convert
the h o m o m o r p h i s m
and check as in part
(iv)
- 92
-
that this m a p induces an i s o m o r p h i s m of h o m o t o p y groups. It follows the r e p r e s e n t a b l e 2d,
from part
functor
K
(v), i t e r a t e d
(X)
d
times,
that
is p e r i o d i c w i t h p e r i o d
in the same sense that s t a n d a r d K - t h e o r y is p e r i o d i c
with period
2.
We t h e r e f o r e h a v e no d i f f i c u l t y e x t e n d i n g
it to a g r a d e d c o h o m o l o g y t h e o r y
K*(X).
Alternatively,
we can first take the s p e c t r u m BUR
, BUR
,
~+i
BUR
~+2
,
...
and then take the r e s u l t i n g c o h o m o l o g y theory. It follows f r o m p a r t theory
K0
that for
~ = 0
the
for
,
has products.
Let above;
(iii)
BUR
be the r e p r e s e n t i n g space
K
as
then w e h a v e m
BUR
II
BUR
c~
It is easy to o b t a i n the r a t i o n a l c o h o m o l o g y of the factors BUR
by i n s p e c t i n g their h o m o t o p y groups.
H* (BUR
sion
;Q)
2n,
is a p o l y n o m i a l where
r e s i d u e class
n
a l g e b r a on g e n e r a t o r s
of dimen-
runs over the p o s i t i v e i n t e g e r s
in the
e.
B e f o r e m o v i n g on to cobordism, result.
In fact,
Given
d,
we need one m o r e
we h a v e a m a p E0: BU
)
BUR !
where
R = R(d).
Let us d e f i n e
d i a g r a m is commutative.
E6
so that the f o l l o w i n g
-
H*(BUR;Q)
93
-
EQ
> H*(BU;Q)
H* (BU; Q) We r e m a r k E 0 (X) ,
that
where
in w h a t E
follows,
H*(X;Q)
really
arises
as
is the s p e c t r u m K (Q, 2n)
.
- ~ < n < + |
Thus
H*
should
Hp ,
while
H,
groups
H
be i n t e r p r e t e d should
be
interpreted
the
product
as a d i r e c t
of g r o u p s
s u m of
Let
p
todd be
as a d i r e c t
characteristic (i)
todd(~l
E H*(BU;Q)
class
which
~ ~2)
=
has
(todd
the
following
~i) (todd
~2)
properties 9
9 00
(ii) and
If
x E H 2 (CP ~)
n
is the
canonical
is the g e n e r a t o r
line b u n d l e
(so t h a t
eX-i todd Then we have
Lemma
the
following
n =
X
result.
9
There
is a c h a r a c t e r i s t i c T E K(BU;R)
such
that
class
ch
over
n = e x)
CP then
94!
E6t~ todd Here
R = R(d)
= ch x .
is as in T h e o r e m
5.
The m o t i v a t i o n
for t h i s r e s u l t is b e s t s e e n f r o m the p r o o f of T h e o r e m
Proof. which maps
to
Let
todd
todd' in
be the class
H* (BU;Q).
todd' (61 9 ~2) = for
61,
~2
in
K(X;R).
in
14.
H*(BUR;Q)
Then we easily
see t h a t
(todd'~l) (todd'62)
We also have
!
(E~todd) 6 = t o d d ' E 0 ~ for
6
in
K(X).
It is n o w e a s y to see t h a t
!
!
(E~todd, t o d d ) (61 9 62) = E ~t to od dd d
NOW
is c e r t a i n l y
augmentation
1
in
equal
K(BU;Q).
!
.E6todd ,E6todd ( t o d d 61) ~ t o d d 62) to
ch ~
Using
for some
the l a s t
9 T
formula,
of we
find that ch ~ (61 @ 62) =
(ch T (61)) (ch T (~2))
"
Therefore T($1 in
K(X;Q)
for any
F o r this p u r p o s e where lies
n in
X.
$2)
= T(~I)
We w i s h
" T(~2)
to s h o w t h a t
it is n o w s u f f i c i e n t
is the c a n o n i c a l K(CP~;R)
lies in
to c o n s i d e r
line bundle
t h e n the s p l i t t i n g
T 6 K(BU;R).
over
CP~;
principle
T (n), if
shows
T (n) that
K (BU;R). Next
Let
e
G(cpn;s)
let
S
be s o m e r i n g
be the m u l t i p l i c a t i v e
containing
the r a t i o n a l s .
g r o u p of e l e m e n t s
of
95-
1
augmentation
in
H*(cpn;s).
Then we can define
a homo-
morphism todd:
K(cp n ; s )
> G(cpn;s)
by todd(~ for
Here
~ 6 K(cpn).
nomial
in this
(1 + x) s
S
=
1 + sx
is d e f i n e d
by the u s u a l bi-
+
s(~,s-l,x 2 +
...
;
1.2
case the s e r i e s
morphism
agrees with
is finite.
todd'.
On
Passing
K(cpn;R)
the h o m o -
to i n v e r s e
limits,
we
g r o u p of e l e m e n t s
of
a homomorphism todd:
(Here
(todd~)S
series (I + x)
obtain
| s) =
morphism
in
1
H*(CP~;S).)
agrees with Take
> G(CP=;S)
is the m u l t i p l i c a t i v e
G( CP=;S)
augmentation
K(CP = ; S )
On
9
K(CP~;R)
this h o m o -
todd'.
an i n d e t e r m i n a t e
t
and take
S = Q[t].
Con-
sider todd(l This
is an e l e m e n t
power-series in
+ ~) t
t;
in
x
of
todd(l
+ t~ + t(t-l) 1.2 that
G(CP~;Q[t]),
with
coefficients
r~2
is,
which
+
"'"
)
"
it is a f o r m a l
are p o l y n o m i a l s
say todd(l
+ ~) t = 1 + Pl (t) + p 2 ( t ) x 2 +
But for any i n t e g e r have
=
n,
(i + ~)n
....
is a line b u n d l e ,
and we
96-
todd(l
+
e
~)n
nx
-I
nx n2X 2 nx + -- 1 + ~-[. + 3:
So for
integer
values
of
t
we
...
have
Pr(n)
n = (ru
Pr(t)
=
r !
thus tr (r+l) !
and t o d d (i + Consider series
in
~
coefficients prime the over
p
with
such
that
p-adic
(~ (n))
rational
actually
coefficients the
now
~)
of
lie p
in
9
tx
-I
tx A priori
R.
d
d;
are
p-adic
= ch =
is a p o w e r -
this,
we wish
and manipulate
ch(~(n))d
this
I claim
To p r o v e
mod
(TCn))
e
-
coefficients.
- 1
integers,
d
t
that
choose
to p r o v e
integers. as
follows.
T(dn)
tE~todd~ 9 t o d d ,(dn)
todd dEQ~ t o d d dn
the basic
remark
in t h e
proof
of T h e o r e m
5 is
a
that
We work
todd'E0dn s dn
Now
these
that
-
97-
dEon : [ (1 + {)P P where
p runs o v e r
primitive
Pn = m
m
for
d th r o o t of u n i t y
ch (T (n))
1 ~ m ~ d,
as in T h e o r e m
~
d
=
5.
and
m
is a
Thus we have
todd ( 1+ ~ ) p t o d d ( 1+ ~ )
p
=7
ePX-I
(eX-1)
p
(by the r e m a r k s
(i+$) P-1
~ =
above)
ch
P{ P Thus we have
=~ (1+~)P-lp{
(T (n)) d
P B u t for e a c h
p
the c o e f f i c i e n t s
of the p o w e r
series
(~+~)P-I are p - a d i c
integers;
and the d e n o m i n a t o r
TT p = P is i n v e r t i b l e . series
(T (n))
Therefore d
c-z)
d-I
the c o e f f i c i e n t s
are p - a d i c
integers.
This
in the p o w e r proves
that these
- 98-
coefficients
lie in
Finally, that
R,
as claimed.
since
the c o e f f i c i e n t s
d
of
is i n v e r t i b l e
T(n)
lie in
in
R.
R,
This
we deduce proves
Lemma
9. We now t u r n to c o b o r d i s m . Let MU*(X;R)
R
be a s u b r i n g
be c o m p l e x
is a r e p r e s e n t a b l e ing spectrum.
Lemma
cobordism
functor;
We r e q u i r e
of the r a t i o n a l s . with
we write
Let
coefficients MUR
in
R.
This
for the r e p r e s e n t -
the same i n f o r m a t i o n
as before.
i0
If
R 1 c R 2,
the m a p s
i,:
H , ( M U ; R 2)
i*:
H * ( M U R I ; R 2) ---~ H * ( M U ; R 2)
(iAi),:
H,(MUAMU;
(iAi)*:
H*(MURIAMURI;
) H , ( M U R I;R 2)
R 2)
> H,(MURIAMURI;
R2)
> H*(MUAMU;
R 2)
R 2)
are iso.
Proof.
Let
Y
n
be a M o o r e
(Y) = 0
Hn(Y)
for
=
T h e n w e m a y take
MU A Y
leads
to the result.
immediately
for
spectrum
with
n < 0 , n O n ~ 0 .
as a c o n s t r u c t i o n
for
MUR.
This
- 99--
Lemma
ii
Suppose
R 1 c R 2.
Then the maps
i,: K , ( M U ; R 2)
> K , ( M U R I ; R 2)
i*: M U * ( M U R I ; R 2) ---> M U * ( M U ; R 2) (iAi)*: M U * ( M U R I A M U R I ; are iso.
The maps
i*
phisms with respect The proof
and
R2)
(i A i)*
may be i d e n t i f i e d w i t h
(therefore
additive
by the f i l t r a t i o n to be c o n s i d e r e d as
R I c R 2.
operations
for all
ring of stable
and
n,
and Rl-linear).
topology.
operations
natural, This
According
is e s s e n t i a l l y
We t h e r e f o r e write
cohomology
This set
> Mun(x;R2)
X
just introduced,
3.
MU0(MURI;R2).
the set of c o h o m o l o g y
b: M u n ( x ; R I ) are d e f i n e d
topology.
is the same as for Lemma the set
R2)
are also h o m e o m o r -
to the filtration
We now consider
which
> MU*(MUAMU;
set is t o p o l o g i s e d
to Lemma
independent B(R 2)
and regard
operations
and stable
of
ii, the set R 1,
for the set of it p r i m a r i l y
of degree
12
If
R 1 c R 2,
then we have a m o n o m o r p h i s m
i: B(RI)
> B(R2)
as the
zero on
MU*(X;R2).
Lemma
so long
- I00
such that
for each
the following
diagram
This with
b 6 B(RI),
-
X
each
and each
n
the
is commutative.
Mun(x;RI) ,
i,
> Mun(x;R2)
Mun (X;R1)
i.
~ NUn (X;R2)
follows
from the preceding
discussion,
together
the fact that i,: MU~
is monomorphic. Because construct
> MU0(MU;R2)
(Compare Lemma of this lemma,
an idempotent
topy theory becomes
in
it will be s u f f i c i e n t
B(Q).
trivial.
But over
greater
generality
use later.
f: X
> MUQ
so that the following
diagram
stable homo-
than is needed
be a map.
now,
Then we define
for f!
is commutative.
f.
H, (X;Q)
Q,
to
We will give the next construc-
tion in slightly Let
4.)
> H, (MUQ;Q)
/ H, (MU ;Q)
Of course we can make by
K,,
or with
a similar
MU, MUQ
Now we define
definition
replaced
by
with
BU, BUQ.
H,
replaced
101
-
e: MU0(X;Q) as follows.
Lemma
If
f: x
) MUQ
is a map,
then
8 (f) = f !
-
is an isomorphism.
If we assign %
becomes
composition
in
This theory
) HOmQ (H, (X;Q) ,H, (MU;Q))
13
8
then
-
the obvious
topology
a homeomorphism. B(Q)
If
into c o m p o s i t i o n
to the
X = MU,
then
in the
Hom
lemma is a known c o n s e q u e n c e
H o m group, 8
carries
group.
of Serre's C-
[27]. I now choose
construct
a positive
a "fake c o b o r d i s m
integer
theory"
are p e r i o d i c with one m u l t i p l i c a t i v e dimensions. 6 B(Q)
Let
E0 6 A(Q)
to be the element
d,
and seek to
whose coefficient generator
be as above.
every
such that the following d i a g r a m
E.
\
) H, (MUQ;Q)
H, (MU; Q)
~H
H. (BU; Q)
H,(BU;Q)
2d
Then we define
is commutative. H, (MU; Q)
groups
EO,
>
H, (BUQ;Q)
102
-
(Here
@H
that
E
potent
is the Thom i s o m o r p h i s m is idempotent;
indeed
E
in homology.)
It is clear
is the most obvious
idem-
in sight.
Theorem
14
lies in rationals p ~ 1
Lemma
-
a/b
mod
B(R),
such that
d,
where b
as in T h e o r e m
R = R(d)
contains
is the ring of
no prime
p
with
5.
The proof will
require
A map
> MUQ
two intermediate
results.
15
f: S p
factors
through
MUR
if and
only if f!: K,(sP;Q) maps
K,(sP;~)
into
This Note
that if
spectrum, Lemma
K,(MU;R).
is the t h e o r e m of Stong and Hattori Sp
then
is regarded K,(S p)
is free for all
than as a
must be taken reduced.
X
be a connected r.
Then a map
spectrum f: X
K,(X;R)
into
K, (MU ;R).
such that
> MUQ
if and only if f !: K, (X ;Q)
maps
as a space rather
[16, 29].
16 Let
MUR
9 K,(MU;Q)
> K, (MU;Q)
Hr(X)
factors
through
-
Proof. f!
maps
First
(n-l)-connected duction spheres,
It is t r i v i a l
K,(X;R)
converse.
over
into assume
and
d.
that
result
15.
the f o l l o w i n g (i)
For
Now suppose
For
K, (X; R) into
into
is true
the
We p r o c e e d if
X
say
b y in-
is a w e d g e
Hr(A)
_~ Hr(X),
r < m
we h a v e
= 0,
a map
By
Hr(B)
of
a cofibering
= 0 .
j,: H r(x) ~_ H r(B) . holds
f: X
for
A
) MUQ
Then
fi:
(iii), we h a v e
A
and such
A
i
i'
that
> X
MUQ
sequence H* (X;MUR* (S O ) ) --> MUR* (X)
f!
maps
the f o l l o w i n g
g
MUR
B.
> N~JQ
diagram.
N o w the s p e c t r a l
to p r o v e
i) X J > B
we have
K, (MU;R) .
K,(MU;R).
then
We m a y n o w a s s u m e w e h a v e
The r e s u l t
given
We w i s h
factors,
is f i n i t e - d i m e n s i o n a l ,
r < m
H r(A) (iii)
X
f
properties.
i,: (ii)
t h a t if
(n+d)-dimensional.
The
by L e m m a
-
K,(MU;R).
A with
103
maps K, (A;R)
commutative
-
104
-
is trivial
(since the d i f f e r e n t i a l s
the groups
are torsion-free).
We deduce
i*: MUR*(X) is epi.
So
such that
g
extends
hi = g.
over
are zero mod t o r s i o n
and
that
9 MUR*(A) X;
h: X
say we have
9 MUR
Then we have f = i'h + k j
for some into
k: B
K,(MU;R).
- MUQ.
Then e v i d e n t l y
Now the s p e c t r a l
sequence
H, (X;K, (S~
> K, (X;R)
is trivial
(since the d i f f e r e n t i a l s
the groups
are torsion-free).
is epi. (iii), MUR. X
Therefore k
factors
k!
maps
through
This completes
and
and
that
> K,(B;R) K,(B;R)
MUR.
into
Therefore
the i n d u c t i o n
and proves
K.(MU;R).
By
f
through
factors
the result w h e n
is finite-dimensional. We now tackle the case of a g e n e r a l
X
K, (X ;R)
are zero mod torsion
We deduce
j,: K,(X;R)
maps
(kj) !
by
Xn H
r
such that
(Xn) = 0
for
i,: Hr(Xn) r > n.
= Lim M U R * ( X n)
MUQ*(X)
= Lim M U Q * ( X n) (n
K,(X;R)
into
Take a map K,(MU;R).
sequences f: X
Approximate
is iso for
r ~ n
We have
MUR*(X)
(since the usual spectral condition).
9 Hr(X)
X.
satisfy
9 MUQ
the M i t t a g - L e f f l e r
~such that
Then the c o m p o s i t e
f!
maps
-
X n in is such t h a t fi n
(fi n ) !
factors
f.
of
Proof above.
Lemma
(i)
The
g n 6 M U R 0 (Xn) . the e l e m e n t s
14.
following
Let
for a s u i t a b l e
gument dual satisfies
ch
choice
entirely
the a n a l o g u e The
gn
define
> MUQ
We equip
an of
be as ourselves
is n o t c o m m u t a t i v e .
> H. (BU;Q)
of
~K
we h a v e
9 ~H chz class
in c o h o m o l o g y follows,
of L e m m a
following
16 for
diagrams
.
and a h o m o l o g y
of the c a p p r o d u c t .
to the one w h i c h
(ii)
Since
> H,(MU;Q)
of a c o h o m o l o g y
is t a k e n in the s e n s e to w o r k
E.
diagram
ch #K z = t o d d
prefers
e: M U
16 to
ch
N, (BU;Q) -
(Here the p r o d u c t
Hence
remarks.
K,(MU;Q)
In fact,
K, (MU;R) .
16.
of T h e o r e m
formal
into
and t h u s g i v e a f a c t o r i s a t i o n
We a i m to a p p l y L e m m a
with various
> MUQ
K, (Xn;R)
is m o n o ,
L i m M U R * ( X n) (n
This proves
f
an e l e m e n t
M U R * (Xn) ---> M U Q * ( X n) element
-
> X
maps
through
105
class
The r e a d e r w h o
may write to v e r i f y
o u t an arthat
K*.) are c o m m u t a t i v e .
-
106
-
K,(MU;Q)
) K,(MU;Q)
H, (MU;Q)
- H, (MU;Q) E0!
(iii)
If
K, (BU;Q)
> K, (BU;Q)
H, (BU;Q)
> H, (BU;Q)
u 6 H*(BU;Q),
v 6 H,(BU;Q)
we have
!
E 0! ((E0u) NOW we wish
to c h e c k
that
ditions
of L e m m a
16.
we wish
to c h e c k
that
is
it is s u f f i c i e n t
iso,
9 V) = U ~ e: M U
So t a k e ElX
) MUQ
satisfies
any element
lies
to p r o v e
~KK,(MU;R)
(E0lV)
in
x
in
K,(MU;R).
that
#KElX
= K,(BU;R)
.
the c o n -
K,(MU;R); Since
lies
in
of
~)
But we have Ch~KClX
= todd
9 CHChelX
= todd
9 ~HE!Ch
= todd
9 E0l~HCh
x
x
(by d e f i n i t i o n
= E0! ( E ~ t o d d
9 C H C h x)
CK
107
-
-
6t o d d = E0!\todd = E0
" ChSKX
(ch T 9 ch
~K x)
(where
= E0!ch(T
~
is as in L e m m a
9)
9 ~K x)
= chE 0! (T 9 ~K x) Since
ch
is iso,
we h a v e ~Ke!X
But
T 6 K*(BU;R)
and
T . CK x E K , ( B U ; R ) . maps
K,(BU;R)
K,(BU;R) lies
the
Theorem
~!x
we have
K,(BU;R). lies
conditions
0!(~ 9 ~KX)
~K x 6 K , ( B U ; R ) ,
Again,
into
and
= E
in
so
E0:
Thus
16,
and
> BUR,
%K~!X
K,(MU;R).
of L e m m a
BU
lies
Therefore e 6 B(R).
so
in E
This
satisproves
14. The properties
of the e l e m e n t
E 6 B(R)
are
as
follows.
Theorem
17
(i) (ii)
~2 = ~ For
in
any
B(R).
x,y c(xy)
Proof.
Since
in =
B(R)
M U * (X;R)
(cx) (~y)
> B(Q)
we h a v e
.
is m o n o ,
E0!
part
(i)
- 108-
follows trivially
from the corresponding
To prove part
equation in
B(Q).
(ii), we have to compare the follow-
ing composites. P> MU
MU A MU
e) MUQ
MU A MU eAs ) MUQ A MUQ ~ We have to compare
(eu) !
with
(u(e A e)) !.
MUQ
.
If we compose
with the map H,(MU;Q)
| H,(MU;Q)
- H, (MU A MU;Q)
,
we obtain the two ways of chasing round the following commutative diagram. H.(MU;Q)
la
@ H,(MU;Q)
/
H.(BU;Q) e!|
1J
E0
commutative.
> H. (BU;Q)
,LEo:
~ H,(BU;Q)
1J
e,
) H, (BU;Q)
> H. (NU;Q)
~ H,(MU;Q)
Here the commutativity fact that
@ H,(BU;Q)
Z0 !@Eo !I, H,(BU;Q)
H,(MU;Q)
) H, (MU;Q)
of the central square arises
is additive;
from the
that is, the following square is
-
109
BU • BU
BUQ
(Here
x BUQ
K.)
This proves
>
~
) BUQ
13)
BU
w h i c h represents
addi-
that
(E~)! (using Lemma
BU
~
is the p r o d u c t map in
tion in
and
-
=
(~(~
A
~)),
that the following
, square
is h o m o t o p y -
commutative. MU
A
MU
U
) MU
> MUQ
MUQ A MUQ
In other words,
we have the formula (xy) =
for the external product, erator
in
MU* (MU)
MU* (MUAMU;
Q) .
y
and
y
are both the gen-
takes place in
Since
the equality
Q)
> MU* (MUAMU;
holds
MU* (MURAMUR; is iso,
x
and the e q u a l i t y
MU* (MUAMU; is mono,
when
(~x) (Ey)
the equality holds
are both the g e n e r a t o r
in
R) in in
R)
MU* (MUAMU;
R) .
> MU*(MUAMU;
R)
MU* (MURAMUR;
R)
MU*(MUR;R).
Since
when
Therefore
x
and
it always
-
holds.
This proves
Theorem
S. P. N o v i k o v cohomology values
operations
alternative
approach
by their
It might perhaps
be
and to see if this provides
an
Eo
(x) =
We now define Corollary
are characterised
6 MU2(Cp~). ew,
to
17.
MU*
on the g e n e r a t o r to examine
-
[24] has shown that m u l t i p l i c a t i v e
on
of interest
110
Mu* (X;R).
18
MU~(X)
is a cohomology
is a representable
theory with products,
and
functor.
The proof
that
MU~(X)
is exactly
as for C o r o l l a r y
fact that
MU~(X)
is a representable
8, using T h e o r e m
has products
17
is immediate
functor
(i).
The
from T h e o r e m
17
(ii). We write MU~(X).
MUR 0
for the r e p r e s e n t i n g
In order to lend credibility
is an acceptable
"Thom complex"
BUR 0,
that the following
we remark
a unique
"Thom isomorphism"
to the idea that
corresponding diagram
H, (MU;R)
~ --
H,(BU;R)
I v
i
factors
( -~
I
* > H,(BUR;R)
l
for MUR 0
to the space
~0-
H, (MUR 0 ;R) i, ~ H, (MUR;R)
H,(BUR 0;R)
spectrum
to give
-
This
follows
Theorem
immediately
6d,
-
f r o m the d e f i n i t i o n
of
~.
19
The c o e f f i c i e n t
(i) mial
1 1 1
ring over
R
with
ring
generators
~ , ( M U R 0)
is a p o l y n o -
in d i m e n s i o n s
2d,
4d,
.... (ii)
morphic
to
MU* (X;R)
product
of t h e o r i e s
iso-
MU~ (X) .
Note.
In p a r t
to be c a n o n i c a l , jection onto sufficient
(ii)
the s p l i t t i n g
b u t the i n j e c t i o n
MU~(X)
of
are of c o u r s e
is not a s s e r t e d
MU~(X)
and the p r o -
canonical;
this is
for the a p p l i c a t i o n s .
Proof. be its
is a d i r e c t
F o r any c o n n e c t e d
indecomposable
A,
algebra
let
Q (A)
Then
quotient.
> ~,(MUR)
: ~, (MUR) induces Q(E) : Q ( ~ , ( M U R ) ) with
Q(c)
9 Q(e)
= Q(e).
We h a v e
Q(Imc) Now x2, each may
x 3,... x
n
thus
Q (~, (MUR)) in d i m e n s i o n s
we h a v e choose
either
~_ Im(QE)
R-base
.
is R - f r e e w i t h 2,
4,
Q(e)x
a homogeneous
it to a h o m o g e n e o u s
) Q(~,(MUR))
for
n
= x
R-base
generators [20,
6,... n
or for
Q(e)x ImQ (e)
Q (~, (MUR)) .
30]. n
= 0.
Xl, For We
and e x t e n d
L i f t the b a s i s
-
elements remaining ~,(MUR)
in
ImQ(~)
basis
to elements
elements
hj,
and
gi"
But this subalgebra
gi
algebra
is precisely
find the dimensions
-
in
Imp,
and lift the
in any w a y to elements
is the p o l y n o m i a l Im~
112
h.. 3 by the
generated
the s u b a l g e b r a
is polynomial.
Then gi
generated
It remains
and
by the only to
of the generators.
We have ~,(MUR 0) | Q ~ H,(MUR0;Q) (by the remarks
above).
is a p o l y n o m i a l
algebra with
6d, . . . .
Now part
is free as a module base
for
represent
~,(MUR)
proof
over
elements
on the j
actually
factor
H*(BUR0;Q)
in dimension
2d,
dimensions
shows
Choose
that
4d,
over
Q.
~, (MUR)
a ~,(MUR0)-free
with the unit element
i)
and
by maps
f.: S n(j) 3 the map
th
above,
by counting
~,(MUR0).
g: V sn(J) 3 which
generators
(beginning
the basis
We now consider
But as remarked
(i) follows
The preceding
~ H,(BUR0;Q)
A
> MUR
MUR
0
.
~
MUR
is given by fjAi
It is clear Since
MUR 0
infinite
S n(j)
A MUR 0
> MUR A M U R
that
g
an i s o m o r p h i s m
induces
is connected
wedge-sum
and
n(j)
---> ~
is also a product.
u> MUR
.
of homotopy as
j
Therefore
> ~
groups. the
-
[X,MUR] ~ ~ J This proves p a r t
113
-
[X, sn(J)AMUR0]
.
(ii).
We h a v e now a c c o m p l i s h e d our o b j e c t of s p l i t t i n g MU*(X;R)
into a d i r e c t s u m of s i m i l a r functors.
that the f u n c t o r s MUR 0
MU~
and the space
and
BUR0,
K~,
I believe
t o g e t h e r w i t h the s p e c t r u m
are of some interest.
like to give f u r t h e r results
to p r o v e that
to B U R 0
for lack of time in w r i t i n g up
as
these notes
MU
is to
BU;
MUR 0
I would is r e l a t e d
I o f f e r the f o l l o w i n g in the d i s g u i s e of an
exercise.
Exercise
20
Show that P r o p o s i t i o n F l o y d theorem)
applies
25 of L e c t u r e
to the case
E = MUR0,
1
(the C o n n e r -
F = BUR 0.
Hints. (a) fore of
K
H p ( M U R 0;R) = 0
(MUR 0) = 0
K(MUR0;R).
map it into it to
for
~ ~ 0,
K(MU;R),
lift it into
class
u
K(MUR;R)
2d.
There-
is the w h o l e in
K(MU),
and r e s t r i c t
K0(MUR0).
This
class.
In c h e c k i n g A s s u m p t i o n
e x e r c i s e care in a p p r o x i m a t i n g
mod
K 0 (MUR 0)
the r e s u l t m u s t lie in
gives the n e c e s s a r y o r i e n t a t i o n
complexes.
p ~ 0
and
T a k e the o r i e n t a t i o n
K(MUR0;R);
(b)
unless
MUR 0
20 and 24 of L e c t u r e and
BUR 0
by finite
I,
-
LECTURE
In this certain
algebraic
of known results in B o u r b a k i
in algebra w h i c h
These results
on c o h e r e n t
[9, pp.
THEOREMS
I want to give an e x p o s i t i o n
theorems
topology.
-
5. F I N I T E N E S S
lecture
finiteness
1 1 4
62-63].
rings;
in the course of r e p r o v i n g
Novikov
[24].
Independently,
seem u s e f u l
one may find the latter interested
certain
in the
results
of S. P.
Joel M. Cohen b e c a m e
for a d i f f e r e n t
I am m o s t grateful
to C o h e n for sending me preprints
[12, 13].
topological
interested
in similar results
two papers
in
are slight g e n e r a l i s a t i o n s
I became
subject
of
application. of his
(So far as I know these papers
have not
yet appeared.) The f o l l o w i n g
results
of the sort of t o p o l o g i c a l
Theorem
1-5 will
serve as i l l u s t r a t i o n s
application which
I have
in mind.
1 (S. P. Novikov)
If
X
is a finite CW-complex,
finitely-generated
as a m o d u l e
then
MU*(X)
over the c o e f f i c i e n t
is
ring
MU*(S~ The m e t h o d s result,
which
Theorem
2
Let considered
I will give also yield
is slightly
X
stronger.
be a finite CW-complex.
as a m o d u l e
the f o l l o w i n g
over the c o e f f i c i e n t
Then ring
MU*(X), MU*(S0),
-
115
-
admits a r e s o l u t i o n of finite l e n g t h 0
> C n --~ Cn_ 1 --~...---> C 1 ---> C O ---> MU*(X)
by f i n i t e l y - g e n e r a t e d
--~ 0
free m o d u l e s .
S i n c e g i v i n g the o r i g i n a l
lecture
I h a v e h e a r d that
this r e s u l t is also k n o w n to P. E. C o n n e r and L. Smith; also be k n o w n to other w o r k e r s
in the field.
it m a y
I am g r a t e f u l
to L. S m i t h for s e n d i n g m e a preprint. I w i l l not q d o t e the r e s u l t s of C o h e n v e r b a t i m , w i l l r e w o r d t h e m to suit the p r e s e n t words
"almost all" to m e a n
Theorem 3
mod
"with a finite n u m b e r of exceptions".
X
be a s p e c t r u m w h o s e s t a b l e h o m o t o p y g r o u p s
are f i n i t e l y g e n e r a t e d ,
Then
H*(X;Zp) p
I w i l l u s e the
(J. M. Cohen)
Let ~r(X)
lecture.
but
and are zero for a l m o s t all
is f i n i t e l y - p r e s e n t e d
Steenrod algebra
A
as a m o d u l e over the
.
This r e s u l t can be u s e d to show t h a t u n d e r m i l d restrictions,
a space
Y
(as d i s t i n c t f r o m a spectrum)
h a v e i n f i n i t e l y m a n y n o n - z e r o s t a b l e h o m o t o p y groups. b e t t e r for this p u r p o s e
) G
is iso.
t o r s i o n s u b g r o u p of
G
is
S p e l l i n g this out, G
p-trivial
p .
if
it asks t h a t the
s h o u l d c o n t a i n no e l e m e n t s of o r d e r
p , and that the t o r s i o n - f r e e q u o t i e n t of d i v i s i b l e by
Even
is the v a r i a n t w h i c h f o l l o w s next.
W e w i l l say that an a b e l i a n g r o u p p: G
must
G
s h o u l d be
r .
-
Theorem
-
4 (J. M. Cohen)
Let groups
116
X
~r(X)
A-module
be a connected
are
p-trivial
H*(X;Zp)
spectrum whose
for almost all
can be presented
finitely many dimensions
stable h o m o t o p y
r .
by generators
and relations
Then the in only
in only finitely m a n y
dimensions. In particular, ~r(X)
= 0
for almost
case and Theorem
Corollary
r .
then
the theorem
The d i f f e r e n c e
if the groups H*(X;Zp)
applies between
~r(X)
may need
if this
are not
infinitely
many
in some dimensions. 5 (J. M. Cohen)
Let
Y
be a space such that
there are infinitely homotopy
all
3 is that
finitely-generated, generators
of course,
group
many values
~(Y)
of
H*(Y;Zp)
r
~ 0 .
Then
such that the stable
is not p-trivial
(and t h e r e f o r e
non-
zero) . This answers
a question
To prove these axiomatisation
of Bourbaki's
our assumptions, Corollary and sketch looking
definitions
12 onwards the proof
immediately
[26, p.219].
we will present
results.
We will
and general
of the results might
perhaps
after Example
given
a slight
first
theory.
we turn to the topological
for m o t i v a t i o n
beginning
results,
of Serre
above.
set up
From applications, Topologists
turn to the passage 14.
-
117
-
We s u p p o s e g i v e n a g r a d e d ring The w o r d
"module" w i l l m e a n a g r a d e d
The class
w i t h unit.
left R-module,
We s u p p o s e g i v e n a class
o t h e r w i s e specified. modules.
R
C
~
unless of p r o j e c t i v e
is s u p p o s e d to s a t i s f y two axioms*.
(i)
If
P -~ Q
and
P s C , then
Q s C .
(ii)
If
P s C
and
Q s C , then
P @ Q 6 C .
Examples. (i)
We d e f i n e
F
to be the class of f i n i t e l y -
g e n e r a t e d free m o d u l e s . (ii)
We define
with generators
to be the class of free m o d u l e s
in only a f i n i t e n u m b e r of d i m e n s i o n s . We define
E
to be the class of free
such that for each
n
there are only a f i n i t e n u m b e r
(iii) modules
D
of g e n e r a t o r s
in d i m e n s i o n s
(iv)
We define
n . O
to be the class c o n t a i n i n g o n l y
the zero m o d u l e . In w h a t follows the symbols
[
and
have the m e a n i n g s
just g i v e n to them.
2 and 3
we take
C = F ; in p r o v i n g T h e o r e m
we t a k e
C = D .
The a x i o m a t i s a t i o n
D
will always
In p r o v i n g T h e o r e m s
i,
4 and C o r o l l a r y
5
simply saves us from
g i v i n g the same proof twice over.
Definition
6
An R - m o d u l e
M
is o f ~ - t _ ~
5
if it has a pro-
jective resolution
* Note
a d d e d in proof.
It s h o u l d also be a s s u m e d that
0 E C.
118-
-
0
~
such that p.
M
<
Cr s
CO for
60, e x e r c i s e
...<--- C r <
C1 <
0 ~ r ~ n
.
...
(Compare Bourbaki
6.)
Examples. (i)
All modules
(ii)
A module
has a p r o j e c t i v e
is of
resolution
(iii) generated
are of
A module
C_-type - 1 .
~-type
by m o d u l e s
of
~
if a n d
in
F_-type 0
C
only
if it
.
is a f i n i t e l y -
module. (iv)
A module
of
[-type
1
is a f i n i t e l y -
presented module. (v) generated
A module
by g e n e r a t o r s (vi)
presented
A module
states
that
Theorem
F--type 0 . H*(X;Zp)
4 states
~ = ~
results
is o n e w h i c h
D_-type 1
can be
is o n e w h i c h
in o n l y f i n i t e l y m a n y
[-type
H*(X;Zp)
resolution
can be
dimensions
M
1 states
3
The conclusion
of
D_-type 1 .
is o f ~ - c o t y p e
such that
Cr 6 ~
be i n t e r e s t i n g
dimension
that
of T h e o r e m
w e w o u l d be d i s c u s s i n g
perhaps
about homological
1 .
is of
say t h a t
, for e x a m p l e , It w o u l d
of T h e o r e m
The conclusion
is of
that
it has a p r o j e c t i v e
dimension.
of
the c o n c l u s i o n
We could also
With
0
in o n l y f i n i t e l y m a n y d i m e n s i o n s .
Thus, is of
~-type
in o n l y f i n i t e l y m a n y d i m e n s i o n s .
using generators
and r e l a t i o n s
MU*(X)
of
for
if r > n
homological
to see if k n o w n
generalize
to c o t y p e
.
119
-
(perhaps
in
the
presence
particular,
is
cotype
will
? We
the
If w e "type" as
for
of
extra
analogue
not do
of
pursue
not
"~-type".
assumptions
Lemma
this
need The
-
to
7
(iii)
further
C
below
).
In
true
for
here.
emphasise
C
property
of
basic
on
, we
will
write
Definition
6 is
follows 9
Lemma
7
Suppose
given
an
0 ---~ L of
exact i y M
se~ence J--~ N
>
0
R-modules. (i)
type
n
If
, then
N
(ii) then
M
is of
i),
Lemma
then
M
type
is of n
If
M
L
type
(n - l)
n
and
M
is of
is
of
.
type
n
and
N
type
n
,
9
is
is of
of
type
Bourbaki
n
and
N
is
of
type
.
p.60,
special
n
type
exercise
case
see
6 a,
Bourbaki
c, p.
For
d. 37,
9.) We
0 <--- L
<
N
<
and
in w h i c h
for
of
significant
o
L
is
type
Proof.
of
of
L
(Compare the most
is
If
(iii) (n +
L
r ~ n
-
N
,
begin C~
one
with
<---- C 1,
<__..
<----
-
knows
how
-
to
C r = C r' O C r" ; s e e , and
C~
6 C
Note added in proof. has been obtained by
part
for
9
(ii). .<
C r!
"''< construct [ii,
r ~ n
Given
resolutions
<
"'"
<
"'"
a resolution C C r' 6 _
p.80]
If
, then
Cr 6 C
An affirmative answer Mrs. S. C o r m a c k .
to
of
this
for
problem
- 120
r ~ n
.
This We
are
given
proves
part
proceed
of
L
and
and
forming
then
N
.
If
with
By
C0 <
that
we
cylinder,
we
for
part
in w h i c h have
F s C
and
.
an
(iii), M
is
sequence
0
--~
---~
K
of
0
---~
K
!
by
type
~
N
n
.
M
---~
for r ~ n - 1
considering
the
N
is of
with
the
type
0
Compare
this
exact
sequence
By
Schanuel's
Leau~a
L
[18,
>
p.101]
L@ So we
Here
have
F
(ii).
type
n
suppose
an
exact
is of
F_~Me
F --~
type
=
Therefore
L
We
turn
, and
the
given
an
---~
0
have K
.
sequence
0 --~
now
we
N
and
is of to
result exact
M e K --~ M @ K type
the is
L --~ is of
n
by
general empty
sequence
for
0 n
type
part
case 9
,
(i).
Since
---~
L ---~ M
a resolu-
for
part
begin
F
i:
construct
projective.
exact
over
Cr s ~
proves
we
...
C r" = C r 9 C r'_ 1
and
This
~
map
can
and
r ~ n
r ~ n
prove
we
- - .
Cr
a chain
C" O = CO
Cr s ~ for
C I ~--...<
constructing
in w h i c h
case
i),
Suppose
(i).
cr
its m a p p i n g
To
(n +
.
C r" E _C
special
part
0
M ~-M
for 1
for
resolutions
0 <
r ~
(ii).
similarly
0<---L<---C
tion
-
by
part
M
is o f
(i). Since
n = -i
, we
may
-
0 ~ with
F s C
of t h e
and
K
composite
special
case
following
121
K ~
F ~
of t y p e
jq:
already
F
-
M
) Let
(n - i).
> N;
P
then We
considered.
0 P
has
can
be
the
type
kernel
n
construct
by
the
the
diagram. K P
> F
L
> M
[
The
sequence 0
> K ~
is e x a c t .
Here
K
has
so
type
n
by part
L
Lemma
has
P ~
type
L
(n - l) (i).
> and
This
0 P
has
completes
type the
n
,
proof
of
7. For
technical
reasons
we
need
the
following
corollary.
Corollar~
8
Suppose 0 ~
K ~ > C0
in w h i c h
Cr
an
inductive
Then
d ( C n _ I)
exact
sequence.
an exact
sequence
> C 1 ---~...---~ C n _ 1 ~
is of
Proof. As
given
type
The
result
hypothesis, is of
r
type
.
Then
M
is t r u e suppose
(n - i),
Cn
is of
for
type
n = 0
it t r u e and we
> M
for
have
> n
, by
0
.
7(i).
(n - I)~ the
following
122
-
0 So
M
tion
and p r o v e s
follows.
case
n
7
Corollary
(i).
The
"Noetherian"
of [ - t y p e
F--type
63 e x e r c i s e
=.
12d,
0
This
which case
are
that
we
of
p.
Although
applications,
it s e e m s w o r t h
of m o r e
subtle
the n th c a s e
modules
of t y p e
Theorem
9
conditions
are
(i) (n-l),
then
M
of t y p e (iii) 0
in w h i c h n.
) K Cr
given
. the
induc-
C
and
arises
~.
that The
as in w h i c h
"coherent"
of [ - t y p e
61 e x e r c i s e
1
7a and p.
it is n o t n e c e s s a r y describing
is t h a t
a hierarchy
in w h i c h
all
=.
n ~
0.
Then
the
following
all e q u i v a l e n t .
If C E C
(ii) of
are of t y p e
consider
all m o d u l e s
for the
Suppose
completes
[-type
in w h i c h
or b e l o w ) .
n
> 0
is e s s e n t i a l l y
(See B o u r b a k i ,
cases;
> M
n
8.
next question
is e s s e n t i a l l y
a r e of
by
The
all m o d u l e s
> C
> d(Cn_ 1)
is of t y p e
-
P
and
P
is of t y p e If
M
then
Suppose
given > Cn_ 1
is of t y p e
n
of
C
of t y p e
n.
is of t y p e
(n-l),
> Cn
is a s u b m o d u l e
P
n
and
P
is of t y p e
an e x a c t >... for e a c h
is a s u b m o d u l e n.
sequence > CI r.
> CO Then
K
> M
> 0
is of t y p e
-
(iv)
Suppose
Cn in w h i c h exact
C
6 C
in w h i c h
for
> Cn
each
an e x a c t ~
C1
r.
Then
sequence
---> Co we
> M ---> 0
can
extend
it to a n
We module
M
Every note
at the
to a v o i d i,
> Cn_ 1
)...
) C1
> CO
> M
> 0
C n + 1 6 C. (v)
n k
>...
-
sequence Cn+ 1
only
given
> Cn_ 1
r
123
we
that
of
type
n
in c o n d i t i o n s
right-hand
making
can
module
end
of
an e x c e p t i o n
suppose
given
of
(iii) the
of
the
is
type
and
(iv)
sequence
the
case
~.
is
the included
n = 0.
If
sequence d
Cn and
define
tion,
(ii). we
of T h e o r e m
Suppose
can
find
that
of
Co M
6 C
of
> C1
) CO
and
type
9. M
First is of
we
prove
type
n.
> M
) 0
that
(i)
im-
Then
by d e f i n i -
a sequence 0
with
>.--
M = C o / d C I.
Proof plies
> Cn_ l
K
> K of
(n-l);
>C
type
J
o
(n-l).
then we
have
Let the
P
be
a submodule
following
exact
sequence.
Since by
7
K (ii).
and
P
Since
0
>
are
of
j-ip
K
>
type is
j
-I
P
>
(n-l),
a submodule
P
>
j-Ip of
0
.
is of CO
and
type Co
(n-l) 6 C,
-
j-ip P
is of t y p e
is of t y p e
n
n
by
by
7
We p r o v e exact
>
in w h i c h
K
>
Cr
Since
Cr+ 1
8
assuming.
Hence
(ii).
(iii).
>...
n
> Cl
for e a c h
Z0
> C r)
Suppose
given
d > C O e> M
r.
interpretation
Let
an
~ 0
Zr c C r
is a s u b m o d u l e
of
CO
Assume n.
module
is of t y p e of
This
K = Zn
Cr
completes
We prove
be
>
r
C
by
>
7
the i n d u c t i o n . This
that
Z
9
proves
(iii)
implies
>...
> C1
assuming
exact >
0
Since (ii),
The
(n-l). 9
hypothesis
r-I
(iii).
are a s s u m i n g
n.
are
following
r
Then by
is of t y p e
a n d we
the
,
Zr
induction
(ii),
that
sequence.
.
Zr
is a sub-
is of t y p e proves
that
(iii). (iv).
Suppose
given
sequence Cn
d> Cn_l
in w h i c h
Cr 6 ~
type
Let
(iii),
Z
(n-l)
and we
is of t y p e
an e x a c t
>
Z0
as an i n d u c t i v e
We h a v e
0
> Cr_ I)
r = 0,n.
n = 0)
n.
Cr
for
if
is of t y p e
Zr
= Ker(d:
(or t r i v i a l l y
is of t y p e
r-i
n.
are
proves
implies
d> C n - 1
Cn
the o b v i o u s
Corollary
So
we
submodule
with
Z
This
(ii)
is of t y p e
Im(d:
Z0
(i), w h i c h
(i).
that
-
sequence 0
the
9
124
n. Zn
Zn
for e a c h be
is of t y p e
as
r.
Then
in the p r o o f n k 0.
Thus
>/C O
> M
certainly of we
(iii); can
find
> 0 Cr
is of
t h e n by an
-
125
-
epimorphism Cn+ 1 with
C
6 C. -
n+ 1
We module
M
This
prove
of
proves
that
type
(iv).
(iv)
n.
> Zn
implies
(v).
By definition,
we
Suppose have
given
a
an e x a c t
sequence Cn in which
C
an e x a c t
> C n _ 1 --->...---~ C 1 6 C
r
for
each
in w h i c h
C n + 1 6 C.
Cn+ I Continue M
~.
This
by
proves
an e x a c t
Here
C
can
>
0
extend
it to
> CO
applies
>...
> C2
The
induction C
and
> M
again > CI
> 0
to t h e
sequence
> Z0
> 0
constructs
shows
that
.
a resolution
M
is of
type
(v). that
(v)
implies
P c C
of
(i).
type
Suppose (n-l).
given
Then
we
have
sequence
C/P
is of
By
(n+l).
Therefore
9
We have
0
> P
type
n
(v), w h i c h P
seems
> C
> C/P
(by 7 we
is of
completed
It n o w tion.
(iv) w e
> Cl
(iv)
in
r
a submodule
tion).
(i).
Now
induction.
by modules
and
>...
> Cn
We prove C 6 C
By
> M
sequence C n + i ---~ C n
of
r.
> CO
the
are
(i)
---~ 0
or d i r e c t
assuming,
from
C/P
type
n
proof
of T h e o r e m
reasonable
by
.
7
to m a k e
is of
(iii).
the
the
defini-
type
This
proves
9. following
defini-
-
Definition
R
is
(n,C)-coherent
if the e q u i v a l e n t
stated in T h e o r e m 9 are satisfied 9 It is clear from 9
it is
-
i0
The ring conditions
1 2 6
(m,~)-coherent
for
(v) that if
R
is
(n,C)-coherent,
m m n.
Examples. (i) it is
The ring
is
(0,F)-coherent
if and only if
R
is f i n i t e - d i m e n s i o n a l
(left) Noetherian. (ii)
has n o n - z e r o that
R
We say that components
N ~ R n -N
R=
if it
in only finitely many dimensions,
Such a ring is
(0,D)-coherent;
the proof
so
is
trivial. (iii) coherence.
Coherence,
More precisely,
that every s u b m o d u l e i.
of F~type
C'
#
(This follows
exercise in
[,
Bourbaki
0
is
(I,F)-
9 (i) says in this case in
C
condition
is of F--type "C
is pseudo-
If
is exact and
C
P
in Bourbaki,
condition
This coincides w i t h B o u r b a k i ' s
coherent".
C.
as d e f i n e d
lla).
0
> C'
C"
satisfy
easily
> C
) C" ---> 0
this condition,
then so does
from Lemma 7; see B o u r b a k i p.
So in order to check the c o n d i t i o n
it is s u f f i c i e n t p. 63 exercise
to check it for
12a).
This proves
of our d e f i n i t i o n w i t h Bourbaki's.
C = R
62
for every (compare
the e q u i v a l e n c e
127
-
We w i l l n-coherence given the
a
passes
(graded)
following
either = F, rlt
we
assume
such
rn
that
R,
of
R n,
Rm,..., the
e,
R
over
find
R e.
If
an
and make
that
C
is If
set of e l e m e n t s a
n,m,...,p are c o n t a i n e d approximate C.
R e,
accordingly.
_C = _D,
on
We s u p p o s e
such
that
we
assume
we
can
in
that
find
R e.
Secondly,
as a r i g h t m o d u l e
an
This
sufficiently we
over
for e
assump-
closely
assume Re .
that,
With
w e have:
0 < n < =,
of the
and
Me
form runs
over
If
R
(n,C)-coherent.
is
the p r o o f that
R e .)
R
n > 0
and
Bourbaki below
is flat,
the R - m o d u l e s
R |
(ii)
(Compare
assume
cases
finite
can
depending
For
those
subrings
through
assume
we
of
ii
precisely
then
limits.
First,
Re
is f r e e
the p r o p e r t y
subrings
divide
~
n ~ 1
containing
dimensions
that
(i)
e
in
for
direct
for a n y
we
lie
assumptions
Theorem
the
that
in a s e n s e
for e a c h these
and we
set
tion ensures to
D,
r n in R
finite
R
assumptions.
or
r I , r2,...,
that
to s u i t a b l e
ring
F
rz,...,
any
now prove
-
where
the R e - m o d u l e s Re
p.
shows
M e,
is
that
for
than
Re
free,
are over
of t y p e
12e.
n = 1
n
runs
(n,C)-coherent
63 e x e r c i s e
rather
of t y p e
n.
for e a c h
A check we
need
only
as a r i g h t m o d u l e
-
Proof. R-modules
of
the
that
is
of
Me
For
part
form
R |
type
n;
128-
(i), w e Me then
begin
by
showing n.
are
of
type
there
is
an e x a c t
that
For
the
suppose
sequence
of
Re-modules Can with
Cte
) Can-I
in
_F
or
preserves
R |
) ' ' " ~ ,- C~ _D
as
the
> C~
case
exactness,
so w e
>''"
> R | eC~v
may
have
> Me
) 0
The
functor
be. the
following
exact
sequence. R O R e C ~n Here
the
as t h e
modules
case
may
R | be.
To p r o v e M
of
type
sequence
n,
are Thus
the
where
> R |
free
R |
and
e
converse, 0 < n <
is of
suppose ~.
Then
d
then
Ct 6 C each
= [, all.
such the
map
there If
number
that
the m a p s
in
F
type
n.
given we
or
an R - m o d u l e
have
an e x a c t
d > Cn_ 1
for d are
~ = D,
all
the
Re-module d
>...
each
t.
c a n be only the
of d i m e n s i o n s .
free
> 0
of R - m o d u l e s Cn
with
lie
e
) C1 Choose
a finite
rij
In e i t h e r
lie
by
the
in
bases
of
)
.
we
can Re.
Ct;
r... 13
in o n l y
R-free
.
in e a c h
elements
If
rij
in
a finite
find
an
e
Let
Ce t
be
base
to g i v e
Ca d e Ca > n n-i
) 0
a matrix
lie
case,
r.. 13
generated
by
number
elements
> M
R-free
represented
elements
restrict
) CO
e de ~ C1 > C~ . .
of
C t.
Then
129
-
The
original
sequence
_%
Cn is,
u p to
R
e
IGd e
is
free
(as a s e q u e n c e copies
of
d Cn_ I
the
n
>R |
as
-i
a right
the
of g r o u p s )
is
be
case
original
of
may
type
be.
exact,
was
define
n,
since R |
l|
for
each
part
(i)
M
By
(v)
for
9
e R |
over
Re, to
this
.
sequence
a direct
sum
of
de 9 C~ ---~ C~
exact,
the
sequence
> C~
> C~
M e = C~/dC~, e Ct
lies
preserves
> R |
in
and F _
or
exactness,
> R |
Me
e ~
D _
is as
the
an the
sequence
0
have
part
e. has Re,
R |
e
(i).
To prove ent
l@de>
de
can
Since
and we
proves
e
>...
M ~ This
9 R |
>...
sequence
We
R |
is
9 CO
sequence
exact.
Re-module
>...
isomorphic
C a d e> C e n n-I must
9 CI
module
Ca de Ca > n n-I Since
>...
isomorphism,
R |
Since
-
part Let
the Me
(ii), M
form
be
we
that
an R-module
M ~
is of
assume
R |
type
e
of with
(n+l).
Re
is
type
n.
By
Me
of
type
By part
n-coher-
(i),
n. M
-
is of the type type
n
(n+l).
is of type
-
We have shown that each R-module
(n+l).
9 (i), this is sufficient
Cor011ary
130
By the proof
to show that
that 9
R
is
of
(v) implies
(n,C)-coherent.
12 The ring
MU*(S 0)
is
(1,F)-coherent
MU*(S 0)
is a polynomial
but not
Noetherian. In fact, Z[xl,
x2,...,
Xn,...]
[20, 30].
Each
Noetherian
subring,
in T h e o r e m
Ii.
Corollary
on a countable
finite
p.
generates for the
63 exercise
a
Re
12f.)
13
dimensional
part
algebra which
any finite
N [ A r=0 r
is finite
coherent
and
the
in T h e o r e m Re
algebra
A
is both
but neither N o e t h e r i a n
In fact,
Example
these subrings
(Compare Bourbaki
(l,D)-coherent,
since
set of generators
of the generators
and we take
The Steenrod
Re
subset
ring
of
A,
[19],
(0,D)-coherent.
is a Hopf
subset
(l,F)-coherent
nor finite-dimensional.
of
A,
is contained and therefore
and any finitein a Hopf subboth
(0,[)-
We take such subalgebras
Ii; the whole subalgebra
and
algebra
for
is free over
Re
[22].
14 The stable
homotopy
groups
of spheres
form
(under
131
-
composition) nor
a graded ring w h i c h
is neither
(l,[)-coherent
(1,D)-coherent. We may now summarise
most classical
finiteness
g e n e r a t e d modules however,
do w i t h o u t
in algebra ring.
information
nels of maps
from one
about general
"good" m o d u l e
we can use the following
Corollary
results
on submodules.
algebra we can
submodules, (I mean,
The
provided
of course,
to another.)
are m o d u l e s Then This
that
R
of C - t y p e Ker f
is
In other
result.
1
(l,C)-coherent, and that
is of C - t y p e
follows
immediately
f: L
that > M
L
is an
9
(iii).
16
Suppose
that
R
is
(l ,C) - c o h e r e n t , and that
N
M ~
and
i. from T h e o r e m
L
that
ker-
15
Suppose
Corollary
finitely-
In our applications,
and in h o m o l o g i c a l
about kernels.
concern
The
are not Noetherian.
gives us finiteness
topology
we have i n f o r m a t i o n
R-map.
theorems
over a N o e t h e r i a n
condition
But in algebraic
words,
our g u i d i n g philosophy.
we have to use rings w h i c h
Noetherian
M
-
-
is an e x a c t of ~ - t y p e
triangle i.
Ker 1
f
by Lemma
7
contains
of course, a
that
the
the
any for
(reduced)
type
and
M
are
8 and
is of t y p e
assume
that
This
We a s s u m e theory
E * ( S 0)
is
the
is true,
that
with
class
E*
is
products,
and
(l,C)-coherent.
17
X
is a f i n i t e
a sphere,
A
and
to be
over
The
we
B
we
over
can
is true on one
the n u m b e r
find
is a
E* (X)
> X
> B
have
fewer
cells
X = Sn
I
generator. of c e l l s
for
E*(S n)
This
serves
in
X.
If
that
gives
the
than
X.
X
(For e x a m p l e ,
of
X.)
As o u r
induc-
E*(A)
and
E*(B)
are of
subcomplex
suppose
if
a cofibering
A
E*(S~
then
E* (S 0 ) .
E * ( S 0)
cofibering
over
CW-complex,
result
any p r o p e r
hypothesis,
of m o d u l e s
we
cohomology
ring
an i n d u c t i o n
i.
N
C = D.
over
tive
Thus
15.
C = F
is a f r e e m o d u l e
A
and
by Corollary
on o n e g e n e r a t o r .
The
take
1
free module
Proof.
in w h i c h
L
i.
is of t y p e
next proposition
of C - t y p e
is n o t
f
by C o r o l l a r y
coefficient
If
to s t a r t
is of C - t y p e
generalised
Proposition
module
1
in w h i c h
(ii).
For C
N
Coker
is of t y p e
-
of R - m o d u l e s
Then
Proof.
1 3 2
following
exact
triangle
-
133
-
E* (X)
E* (B) (
By C o r o l l a r y
16,
E*(X)
induction.
Of course,
is of type
|
E* (A)
is of type by T h e o r e m
This proves
Proposition
12 and P r o p o s i t i o n To prove T h e o r e m
For
Re
We will
of
nected
for
but
s p e c t r u m such that
is zero for almost
12).
of the sort
R @Re,
and let
spectrum.
as in
is p - t r i v i a l
= 0.
It follows
4 can be deduced,
from the special all
r.
Let
K(G)
G
be
be the
Then
N e x t let
p, = 0. ~r (X)
4.
must induce an i s o m o r p h i s m
> G
H*(X;Zp)
eral case of T h e o r e m H*(X;Zp),
(see C o r o l l a r y
and apply
is p-trivial,
p: G
H, (K (G) ;Zp) ,
then again we have
module
over a p o l y n o m i a l
sketch the proof of T h e o r e m
Eilenberg-MacLane
H*(K(G) ;Zp) = 0, p,
Me
ii.
an abelian group which corresponding
from
ii to reduce
of a r e s o l u t i o n
take such a r e s o l u t i o n
the proof of T h e o r e m
immediately
2, one uses T h e o r e m
we k n o w the e x i s t e n c e
required;
of type 1
17.
on finitely many g e n e r a t o r s
Me
the
17.
1 follows
the p r o b l e m to the study of a module ring
This completes
9 (v) a module
It is clear that T h e o r e m Corollary
i.
without
X
be a confor each
r;
that the genchanging
case in w h i c h
~r(X)
the
-
Next let that
F
H*(K(F);Zp)
-
be a free abelian group;
is of D-type
duce the same result K(G);
134
we consider
i.
one can show
This allows us to de-
for a general Eilenberg-MacLane
spectrum
a fibering K(FI)
and apply Corollary
> K(F2)
> K(G)
16 to the resulting
exact triangle
of
cohomology modules. Now we can prove the result just
n
over
n,
non-zero homotopy
as for Proposition
to the exact triangle suitable
groups.
fibering.
for a spectrum
X
with
This is done by induction
17, but applying Corollary
of cohomology
This completes
The proof of Theorem
modules
arising
16
from a
the proof.
3 can now safely be left to
the reader. To deduce Corollary
5, we suppose given a space
which contradicts
Corollary
5, so that
~*(Y;Zp)
Theorem
to the corresponding
spectrum.
4 applies
be a non-zero
class of lowest dimension
in
~ 0
Y
and
Let
H*(Y;Zp);
y then
PPfy = 0 for all sufficiently sible that
H*(Y;Zp)
large
f;
this makes
plau-
cannot have a presentation with rela-
tions in only finitely many dimensions, be proved.
it extremely
This contradicts
Theorem
and this can indeed
4 and proves Corollary
5.
135
REFERENCES
[1]
Adams, J. F.,
II
Une Relation entre Groupes d'Homotopie
Compte8 Rendue8 de l'Acad.
et Groups de Cohomologie",
des Sol,, Paris, 245; 24-26 [2]
(1957).
, "On the Structure and Applications of the Steenrod Algebra",
[3]
Comm. Math. Helv., 32; 180-214
"Vector Fields on Spheres", Bull
(1958).
~Amer. Math.
Soc., 68; 39-41 (1962). [4]
, "Vector Fields on Spheres", Annals of Math., 75; 603-632
[5]
Anderson,
(1962).
D. W., and Hodgkin, L., "The K-theory of
Eilenberg-MacLane Complexes", [6]
Atiyah, M. F., "Vector Bundles and the Kunneth Formula",
Topology, i; 245-248 [7]
Boardman,
[8]
(1962).
J. M., Thesis, Cambridge,
1964.
, "Stable Homotopy Theory", mimeographed notes, University of Warwick,
[9]
(preprint, to appear).
1965 onward.
Bourbaki, N., "Algebre Commutative",
Elements de Math~matiquej et Ind.
1290)
Chapters
27, Hermann, L .
1 and 2 of
(1961) o (Act. Sci.
136
[io]
Brown, E. H~
-
and Peters.no F0 P., "A Spectrum Whose
Zp Cohomology is the Algebra of Reduced pth Powers",
Topologyj 5; 149-154 (1966). [ll]
Cartan, H., and Eilenberg, S.0 Homological Algebra, Princeton University Press (Princeton Mathematical Series, no. 19) (1956).
[12]
Cohen, J. M., "Coherent Graded Rings",
(preprint, to
appear) .
[13]
, "The Non-existence of Spaces of Finite Stable Homotopy Type",
[14]
(preprint, to appear).
Conner, P. E., and Floyd, E. E., The Relation of
Cobordism to K-theoriesj no. 28) Springer, [15]
(Lecture Notes in Mathematics,
(1966)o
Douady, A., Seminaire H. Caftan ii (1958/59), exposes 18, 19.
[16]
Hattori, A., "Integral Characteristic Numbers for Weakly Almost Complex Manifolds", Topology, 5; 259-280
[17]
Hodgkin, L., "K-theory of Eilenberg-MacLane Complexes I, II",
, Izvestija Akademii Nauk S.S.S.R., Matematiceskaja,
[25]
Puppe,
(1967).
D., "Stabile Homotopietheorie
169; 243-274 [26]
31; 855-951
Serre, J~
Serija
I", Math.
Annalen,
(1967). "Cohomologie Modulo 2 des Complexes
d'Eilenberg-MacLane",
Comm. Math Helv.j
27; 198-232
(1953). [27]
, "Groupes d'Homotopie J
,
Abel~ens"
[28]
Stasheff,
,
et Classes
Annals of Math. j 58; 258-294 J. D., "Homotopy Associativity
Trans. Amer.
Math.
Soc.,
108; 275-292
de Groupes
(1953)
m
of H-spaces
(1963).
I",
138 -
[29]
Stong, R. E., "Relations A m o n g Characteristic Numbers ~",
Topologyj 4; 267-281 (1965). [30]
Thom, R., "Travaux de Milnor sur le Cobordisme,',
S~m~nalre Bourbak~ [31]
Whitehead,
i!~ no.
180
(1958/59).
Go W., "Generalized Homology Theories",
Trans. Amero Math. Soc.~ 102; 227-283
(1962) o
-
On H - s p a c e s
139
-
and infinite
loop
spaces
by Jon Beck
By a t o p o l o q i c a l (I) cal
For e a c h pair of o b j e c t s space,
(X,Y) (2)
and c o m p o s i t i o n
x (Y,Z)
~
For e v e r y
homeomorphism (3)
(X,(A,Y))
pactly with
~
spaces
unitary,
a category ~
(X,Y)
associative
which
in which:
is a t o p o l o g i -
operation
is an o b j e c t
(A,Y)
~ ~ and a n a t u r a l
):~ ~ > has a left a d j o i n t
spaces
taken
in m i n d
is itself
a topological
of r e s t r i c t i n g (X,(ytZ) ~
(E.Spanier,
assumptions
to a c a t e g o r y
(XxY,Z)
Annals
everything
A x ( ):~ ~ ~.
holds.
of Math.
category.
of spaces
Specifically,
The pre-
or something I have
com-
79
(1959),
142-197,w
but
can be p u s h e d
through
in the o r d i n a r y
spaces.
topological
maps.
(A,
the c o n v e r s i o n
of t o p o l o g i c a l
preserving
is a hom o b j e c t
Y ~ ~ there
functor
is a c t u a l l y
care a n d c o m p a c t n e s s
Another
understood
(A,(X~Y)).
of t o p o l o g i c a l
for w h i c h
generated
category
there
is a c o n t i n u o u s
space A the
of course
them
r ~
space A and o b j e c t
For e v e r y
caution
X,Y
is g e n e r a l l y
(X,Z).
The c a t e g o r y
like
c a t e q o r 7 I think
category
is that of spaces w i t h b a s e p o i n t s
We w i l l p r a c t i c a l l y
always work
in this c a t e g o r y ,
and b a s e p o i n t
which we denote
sim-
p l y b y Top. In this case p a i r i n g Topoloqical is n a t u r a l l y
preserving
x Top ~ Top
as A~K.
cylinder
over
if I is the u n i t
the p o i n t e d
rise
Top is also a c l o s e d to a d i f f e r e n t
X x Y / X • 0 + 0 x y, u s u a l l y is c a l l e d ues
For example,
space
interval
X, and m a p s
and X ~ Top,
I~X ~ Y are b a s e
then point
homotopies.
Of course, fact gives
spaces
written
I~K is the r e d u c e d
(3)
a pointed
in Top and p a i r i n g s
category,
pairing
written
X~Y where
is,
itself
a Top-category.
X , Y are b o t h p o i n t e d
This
spaces,
namely
functor
w i t h val-
X A Y.
topoloqical
A~LX,(A,Y)
that
as in
cateqory (2) ,(3)
if it p o s s e s s e s exist
a hom
for A ~ Top,
X,Y
~ ~.
-
140-
The point is that as soon as a category has a hom functor with values adjoints as specified above, in that category.
then the constructions
Let ~ be a Top-category
of algebraic
topology are available
in this sense and let X 0 be a fixed object
M. Usually ~ has some well known underlying-(pointed)-topological-space is chosen as the object which represents
in Top and
this via
in
functor and X 0
(XO, ):~ ~ Top. The tensor product
gives adjoint functors
( )ox o Top
/
> ~.
(Xo, ) Since cells and spheres are in Top we have objects en+l~Xo , sn~x 0 in ~, which are the (n+l)-cell and n-sphere in ~. Let us write e n + l ~ tions of completeness, lower-dimensional
CW-objects
exist in ~.
Modulo minor assump-
~ Y in ~! by adjointness,
(Xo,Y), the latter being the "underlying
development of CW-theory can be conducted supplied
instead.
Such are built up by glueing Z-cells onto
skeleta via attaching maps s n ~
the same as maps S n ~
sn~
in such a category.
these are
space" of Y. The usual The essential
fact to be
is that
~i(sn~K)
=
I
O,z, ii =< n,n.
This is true in all of the tripleable
or "theoretical"
examples of Top-categories
used
in this paper. As an example,
consider
phisms G ~ H form a space ue-wise, relations
the category of topological
(G,H)
1) = (a~go)(a~gl).
homomor-
for A ~ Top is val-
group generated by all symbols a~g modulo the
The discrete group z plays the role of X O. Given a
complex X with cells e n+l ~ X, the cells en+l~z tion of X ~ .
The continuous
~ Top. The group structure of (A~H)
and A~G is the free topological a|
groups.
X~Zgive
a group-cellular
decomposi-
Homotopy theory in this category can now be carried out in the usual man-
ner. Some of this has been done under the guise of the theory of simplicial groups. One application: group-theoretical
let O ~ I be the base point of the unit interval.
cone on G. The natural map G ~ I~G at the 1-end is an embedding of G
into a contractible
topological
near its neutral element, dle. Thus I~G/G
Then I~G is the
group. Under standard assumptions
the projection
is a classifying
on the topology of G
I~G ~ I~G/G is easily shown to be a fiber bun-
space for G. Later on we shall construct classifying
-
spaces
for o t h e r
types of H - s p a c e s .
141
-
L a c k of a ~ - p r o d u c t
in those
cases m a k e
the con-
struction more difficult. Another
algebraic
(no b a s e p o i n t s mutative (A,B).
are needed).
triangle.
The n-cell
is u s u a l l y
called
Notice bitrary
topology
F*(X)
those w h i c h
in this t o p o l o g i c a l fiber h o m o t o p y
Euclidean
[X,PL/O]
Pointed
objects
However, topological
is e
where
n
over
a fixed space X
is a m a p A ~ X, a m a p subspace
x X.
is a com-
(A,B)x of the u s u a l
Homotopy
topology
exists
equivalence
is w h a t
is R n • X and a m a p
~(X)
over X even w h e n X is a q u i t e R TM x X
ar-
R n x X / X is d i f f e r e n t i a b l e
F o r example,
there
is the g r o u p of d i f f e o m o r p h i s m s
s h o u l d b e an isomoro f S n-1
x X modulo
to D n • X, all /X.
o f spaces / X c o u l d be t a k e n as a b a s e c a t e g o r y zero sections),
W h e n X = 1 this p r o g r a m
spaces,
whose objects
category
(or PL)
in this p a p e r w e w i l l
counter-includes
in this c a t e g o r y
to the real c o m p o n e n t .
(those w i t h
usual properties.
of spaces
equivalence.
space/X
c a n be e x t e n d e d
The c a t e g o r y
in the c a t e g o r y
The m a p s A ~ B o v e r X form a c l o s e d
if it is so w i t h r e s p e c t phism
An o b j e c t
that d i f f e r e n t i a l
space.
arises
and concentrate
reduces
adhere
to the
topology.
and have
their
topology.
s t a n d a r d b a s e c a t e g o r y T o p of p o i n t e d tripleable
We r e c a l l
by a free-object
c a n be d e f i n e d
to o r d i n a r y
on c a t e g o r i e s
the case of spaces /X).
are d e t e r m i n e d
H-objects ....
for a l g e b r a i c
over T o p
that a t r i p l e a b l e
functor
(the d e f i n i t i o n
triple.
Then
(which a c t u a l l y category
is one
follows),
a n d for
t h e s e w e have: (4)
Theorem.
a pointed
Let T be a p o i n t e d
topological
category~
topological
more precisely,
g o r y h o l d and the tensor p r o d u c t A Q at l e a s t w h e n T is d e r i v a b l e We define
a pointed
w i t h oT = o and T c o n t i n u o u s , (XtY) ~
(XTIYT), T
~T
together > TT ~
Tq
T1 ~
(X,~) w h i c h
from a t o p o l o g i c a l
topoloqical that
triple
T =
is, e f f e c t i n g
with natural T
axioms
the c a t e g o r y
(1),(2)
is a s s e r t e d
of T - s p a c e s
for a t o p o l o g i c a l
to e x i s t
in
cate-
(3) does
exist,
theory. (T,~,~)
to be a f u n c t o r
for all X , Y
transformations TTT
TT
T~
D:id. )TT
>T
T: T o p ~ T o p
~ Top a continuous ~ T,
is
map
~:TT ~ T such that
142
-
commute.
A T-alqebra,
nuous m a p
or T-space,
such that the u n i t a r y
is a p a i r
-
(X,~)
and a s s o c i a t i v e
where
X ~ T o ~ and
laws hold:
XD X ~ - ~XT
Xp
XTT
~XT
XT
is c a l l e d spaces
the T - s t r u c t u r e
form a c a t e g o r y
The u s e f u l n e s s functors
give rise
precisely
Top
o f the space.
T
>X
W i t h an e v i d e n t
definition
of morphisms,
T-
.
of this c o n c e p t to triples
of those
~:XT ~ X is a c o n t i -
arises
from the
fact that,
T, and the c o r r e s p o n d i n g
spaces w h i c h
possess
the g e n e r a l
by composition,
categories
structure
of T-spaces
of v a l u e s
adjoint consist
of the right
adjoints. As an example,
consider
~:X~Z -- X be the usual Top with unit
adjointness
~ >~,
Then
~Z~
is then a p a i r
structure
maps.
functors
~ furnishes
Algebraically, any n points associative
(X,~) w h e r e
X with
for example, in X. T h e n
implies
P = ~ >
Z~ is a triple
in
X ~ Top and
Z~. ~:XZ~ ~ X is a u n i t a r y ,
associative
let
which
8 be a n y n - v a r i a b l e
the value
that
>x
all of the s t r u c t u r e
of 8 in X is
8 satisfies
every
loop
space
(B~X)~ ~ B~. As to w h e t h e r that q u e s t i o n
In the g e n e r a l responding
functor
loop
operation
spaces
on loops
[(Xo~ ..... Xn_l~)8]~.
all of the
identities
possess
and Xo, .... Xn_ 1
The
in X w h i c h
in general.
fact that
it s a t i s f i e s
~ is in
of loop spaces.
In p a r t i c u l a r ,
priori,
Let D:X ~ XZ~,
map:
Such a m a p
(B~)Z~ =
~ Top.
the c o m p o s i t e
NEn
the w o r l d
Z,~:Top
and m u l t i p l i c a t i o n
id. A Z~-space
the a d j o i n t
triple
w i l l be
case,
a Z~-structure,
there
investigated
if F : ~ ~ B
in A. There
has
by evaluation
are a n y Z ~ - s p a c e s in
that are not loop
spaces
a
(16).
is left a d j o i n t
is a c a n o n i c a l
of loops:
functor
to U : ~ ~ B,
let T = F U be the cor-
-
143-
x\ .)~,uT\,y/(, -
U
defined Z,Q,
b y B~ =
the v a l u e
T o p Z~ ~ T o p The
(BU,BEU), of
this
~:UF ~
canonical
at a s p a c e
adjoint
where
B is B~ c o n s i d e r e d
pair
(F,U)
adjointness
morphism.
In the c a s e
of
as a X ~ - s p a c e .
is t r i p l e a b l e
problem
fice
that a l e f t a d 3 o i n t
say
is the
functor
tripleableness it to
id.
is in g e n e r a l
if # is an e q u i v a l e n c e
very
~ for
difficult,
and we
# is e a s i l y
of categories.
shall
constructed
n o t go
into
The it.
Suf-
as the c o e q u a l i z e r
V
XF~ and
that
(X,~) of
the g r e a t e s t v (X,~)@~
~
~:X ~ XFU
relevant (5)
difficulty
is an i s o m o r p h i s m
and XFU ~
1
of T - a l g e b r a s ~
( X , ~ ) @ U T. The
simplicial
X(FU) n + l
s
attends
following
on s h o w i n g
this m a p fact
that
the a d j o i n t n e s s
is e s s e n t i a l l y
is u s e d
in s t u d y i n g
map
the c o m p o s i t i o n this m a p ,
and
is
later:
The a u g m e n t e d
with
ordinarily
face o p e r a t o r s
= X F ( U F ) i-I
b y ~ : i d ~ FU,
,
n )
-I
,
s :X(FU) n+2 ~ X(FU) n + I g i v e n i
~ (UF) n - i + l has
object
U,
I 4 i 4 n + I, a n d
by
~
= ~(FU) n+I, o
suitable
degeneracy
operators
induced
a "contraction" h
X(FU)n+2<
homotopy
equivalent,
n ~ -I
J
as a s i m p l i c i a l
object,
h n = X(FU) n + 1 D, a n d
to the c o n s t a n t
or
~ therefore
"discrete"
simplicial
X.
Finally, T-space
X(FU)n+I
h n e.I = ~.h 9 n , O 4 i 4 n, h n ~ n + I = id., n a m e l y
obeying
object
n
maps
as to the
topological
f:X ~ Y form a closed
nature
of T o p T,
if
(X,~) , (Y,8)
subspace
of the
space
o f all m a p s
are T - s p a c e s , (X,Y) , n a m e l y
their the
equalizer (X,Y) T
If A is a s p a c e ,
(A,Y)
> (X,Y)
If fT.8
is a T - s p a c e
~ (XT,Y) .
by means
of
the c o m p o s i t i o n
(A,Y)T ~
(AtYT)
~
(A,Y) ,
-
where
the
first m a p
structure (cf.
is a d j o i n t
of Y. T h e T - s p a c e
the e x a m p l e
c a r r y over
for the triple
to any suitable
shortly Another
ous triple tions
define, problem
morphisms
such as the
also
triples,
and r e s t r i c t
quotient
space
triples,
problem
this
over
letter
T will be u s e d
the c a t e g o r y
of sets,
it is
theory,
w h i c h we
of. That
the c o n t i n u -
But for other
construc-
A ~ T and the rest of the a l g e b r a i c the internal
is a t o p o l o g i c a l confusion
between
Lawvere
structure
of
theory. topological
to b o t h concepts.
is due to F.W.
is open,
thereon.
theories.
is evident.
to reveal
to refer
in g e n e r a l
and d i f f i c u l t i e s
disposed
topological
(ART) , the p r o d u c t
for w h i c h
(A~)T
This p r o b l e m
and triple
is e a s i l y
b y the Tof
from a t o p o l o g i c a l
space
it is n e c e s s a r y
to those
triples
same results
category
arising
there w i l l be a lot of d e l i b e r a t e
the same
of theory,
T is a triple
leads us to introduce
function
of c o n t i n u o u s
triples,
of "enriched"
the
is i n d u c e d
as a q u o t i e n t
on the quotient.
Z~. P a r e n t h e t i c a l l y ,
S ~ T form a t o p o l o g i c a l
topology
Since
But for t o p o l o g i c a l
can be d e f i n e d
the above
second
s h o u l d be p r o d u c e d
groups).
notion
when
-
(A,Y)T ~ YT and the
(X,~)
a T-structure
On the other hand, shall
A ~
of t o p o l o g i c a l
not k n o w n w h e t h e r in p a r t i c u l a r ,
to A ~
144
theories
The o r i g i n a l
(Proc.
NAS U S A
and
notion
50
(1963),
869-872). (6)
By a
T whose
(finitary)
objects
1+I+...+1
and is the c a r t e s i a n
with
to the theory,
pointed
power
(n)T
TM, w h e r e
continuous
(n,p)T ~
advanced
shall
is m e a n t
I, 2,
(m,n)T
a pointed
topological
category
... and in w h i c h m is the c o p r o d u c t is a t o p o l o g i c a l
(n)T is the space
space w i t h b a s e p o i n t
of n - a r y o p e r a t i o n s
family of c o n t i n u o u s
(1,n)T.
base point preserving
maps
(m,p)T~ concept
not bother
is a c o n t i n u o u s ,
over
O,
the hom o b j e c t
a more
b u t we
A m a p T ~ T' An a l q e b r a
Thus
theory
numbers
in T is an a s s o c i a t i v e
(m,n)T ~ Probably
topoloqical
are the n a t u r a l
(m times).
The c o m p o s i t i o n
pointed
a topological
the o b j e c t s
with
pointed, theory
product-preserving
A ~ n could
also be a t t r i b u t e d
that. I- and c o p r o d u c t - p r e s e r v i n g
T, or for g r e a t e r
clarity,
functor.
a T-spacelis
a
functor
T* ~ Top. T* is the dual or o p p o s i t e property,
such a functor
o f the t o p o l o g i c a l
is d e t e r m i n e d
category
b y the image
T. By the p r o d u c t - p r e s e r v i n g
of I e T w h i c h
is also d e n o t e d
by X
-
(and then n,
>xn).
(n)T ~ subject
identities,
equivalently,
commute.
"algebraic"
~
category
theories
discrete
topological
free-algebra
functor
Alg(Top) n
with maps
(n)T ~
For
(nS~
S ~ + ... + S ~
point
is suppressed). in the t h e o r y
More theories specified
of X , Y t h o u g h t
re-
of as func-
monoids,
that
are c a t e g o r i e s
of m o d e l s
the c a t e g o r y
is the
of
free
is left a d j o i n t
group
i.e.,
to the
A for-
is then
spaces
the
X equipped
make
furnished
by
(As the
b y the base
all of the n - v a r i a b l e
ope-
same as in the d i s c r e t e
X into a t o p o l o g i c a l
H-spaces,
arise
introducing
the d e f i n i n g
generated
on n g e n e r a t o r s .
are the
associative for
group
generator
exactly
clearly
o f the r e a s o n in w h i c h
spaces,
T on sums of O-spheres,
topological
and these
as above
of a l g e b r a s
Alg(Top).
(n)T are
groups,
from
Lie a l g e b r a s . . . . .
for the theory,
the a p p a r e n t
is, s t r i c t l y
of H - s p a c e s
triple
free d i s c r e t e
spaces,
and i n d i c a t i v e
and
arise
of t o p o l o g i c a l
topological
exists
is just the
(xn,x)
be the c a t e g o r y
space of m a p s m ~ n in the t h e o r y
the e l e m e n t s
(n)T ~
are those w h i c h
in the c a t e g o r y
of the r e s u l t i n g
of t o p o l o g i c a l
theories
rings,
manifestly
(nS~
b y T o p T.
Let Alg(Top)
interpreted
T. The
and this
Thus
The e a s i e s t
topological
to p o i n t e d
Maps
significant
but
is e x a c t l y
groups,
is free r e l a t i v e
equations
similarly.
topologies hold
group.
into
o n l y up to
homotopies.
Consider theory
of T are c o n t i n u o u s l y
all d i a g r a m s
is d e n o t e d
of sets.
m. The c a t e g o r y
group
Topological
type,
a theory
(n times),
of groups.
such that
theories.
groups,
(xn,x),
topological
transformatlon
of T - s p a c e s
The v a l u e s
> O, d e f i n e s power
map
T o p ~ Alg(Top)
~ Top.
cartesian
theory
is, the n - a r y o p e r a t i o n s
in the c a t e g o r y
for example,
rations
maps
Y
of t o p o l o q i c a l
specified
(nS~
to a family of c o n t i n u o u s
(n)T @ yn
The r e s u l t i n g
Examples
getful
is e q u i v a l e n t
is a n a t u r a l
a continuous
X
of any
that
f:X ~ Y of T - s p a c e s
(n)T @ X n ~
(7)
functor
b y actual m a p s X n ~ X.
A map tors!
the
-
(X n,X)
to v a r i o u s
presented
Thus
145
T has
the t h e o r y
of h o m o t o p y - a s s o c i a t i v e
(o)T = O, and its o p e r a t i o n s
H-spaces
of h i g h e r
power
(with strict unit). are g e n e r a t e d
This
by a binary
-
operation
x , y ~ xy w h i c h
(x y) z and x(yz). O-cells rious
of these
(3)T . . . .
(3)T,
l-cells . . . .
into
are,
(3)T whose
complexity, x ~ x
n
vertices
(1)T for example
, i-cells
in this c a t e g o r y required
are
containing
corresponding
to the va-
"powers",
(1)T is an i n f i n i t e - d i m e n s i o n a l
Maps
of course,
in
these h o m o t o p y - a s s o c i a t i v e
in fact
complicated.
H-spaces
2-cells
C W complex~
Top T o f c a n o n i c a l l y
to p r e s e r v e
the g e n e r a t i n g
homo-
1-simplex
and all of its c o n s e q u e n c e s .
More complicated is the t h e o r y homotopy ple,
(2)T, and l-cell
in c o n s i d e r a b l e
parentheses
are m o r e
topy-associative
in
-
to the u n a r y o p e r a t i o n s
of i n t r o d u c i n g
as p r o d u c t s
in
results
corresponding
ways
(2)T,
This
is a O-cell
146
is S t a s h e f f ' s
of H-spaces
which
associativities
in the
space
(4)A
theory A~
have
as above,
(cf. Trans.
(unnecessarily)
as well
the h o m o t o p y
as m a n y
associativity
AMS
108
(1963),
275-292).
strict m u l t i p l i c a t i v e "higher
units
associativities".
generates
This
and
For exam-
an sl:
(x (yz)) t ((xy) z) t
/ " ~ - ~ - ~ - - - ~ x
(xy) (zt) The cell
structure
of
( 4 ) A ~ then
((yz) t)
x(y(zt) )
includes
a 2-cell
with
this
S i as its b o u n d a r y ,
and so
on. In o r d e r to k n o w
to be able
to m a n i p u l a t e
that e v e r y g r a d e d
topological
theory,
topological
or e v e n more,
spaces,
and that a r b i t r a r y
between
free
be tripleable,
the f o l l o w i n g
a ~
(i.e.
topological
can t h e r e f o r e
with
sequence
confidence,
o f spaces)
theories
are
it is e s s e n t i a l
generates
tripleable
be c o n s t r u c t e d
formula
theory,
the a d j o i n t
a free
over
graded
as c o e q u a l i z e r s
p a i r Top -- TOD T ~ Top is e a s i l y
from n o w on we c o n f u s e
topological
via their
functors.
free a l g e b r a
for the triple
in terms
of the
8.~T is i d e n t i f i e d
Theorem.
with
a.A ~ ~
theories
We w i l l
spaces
AT = ~ A n | (n)T / ( ~ i d e n t i t i e s ) . n~O i d e n t i f i c a t i o n s m a d e are g e n e r a t e d b y maps
cal r e a l i z a t i o n (8)
In fact,
they g e n e r a t e
The p r e c i s e
constructs
of m a p s
theories.
If T is a t o p o l o g i c a l
Top w h i c h
space
that
theories
these
much
the triples
find it u s e f u l
of o p e r a t i o n s
of finite
8 for a ~ A n , 8 ~(m)T,
with
seen
sets.
to in
to have
in the theory:
If a:m ~ n, then
as in M i l n o r ' s
geometri-
of a s. s. complex. Let
f:T ~ T t be a m a p of f i n i t a r y
pointed
topological
theories
such that
-
(n) f:(n)T ~
(n)T'
topy equivalence To p r o v e
is a h o m o t o p y
equivalence
this,
of A
topological sequence
note
(~in
that
the above
= finite
category
in Top.
on p a p p l i e d
sets).
to the spaces
Z
formula
to the
w e h a v e no use
G
is a h o m o -
F o r example,
gives
(~ - id.),
rise
A* | T w h e r e A * : ( S f i n ) *
Similar
for the a t o m i z a t i o n (n) (~o T)
= ~ A
theory of groups
cone
o
sequence
functors
exist
T d ~ T. The o t h e r
~ ~oA
in the cone
by induction
n4p. on t o p o l o g i c a l
spaces be w r i t t e n
(nT) w i t h
~ Top is
~ A* ~ Tf is a m a p p i n g
to c e r t a i n o p e r a t i o n s
of d i s c r e t e
the d i s c r e t i z a t i o n
the s i m i l a r
for any functor
space A* ~ Tf is c o n t r a c t i b l e
(n)Tf /
inclusion
g i v e s us the t h e o r y w h i c h has
sense
L e t T ~ T' ~ Tf be a m a p p i n g
t o p o l o g y o n X) and X ~ ~ X Cfor g o o d X). o
(7)). F o r
T h e n X f : X T ~ XT'
for AT m a k e s
of functors
that the
An |
of d i s c r e t e n e s s
L e t the two a d j o i n t s
tions.
for e a c h n)O.
of functors S f i n ~ Top. T h e n AT ~ AT'
One d e m o n s t r a t e s
The c o n c e p t
Actually,
-
for e v e r y C W c o m p l e x X.
T : S f i n ~ T o p and in fact is the ~ - p r o d u c t the p o w e r s
147
up to c o m p a t i b l e
(discrete
for t o p o l o g i c a l discretization
the o b v i o u s
y i e l d s A,
Xd ~ X
triples.
composition
theories. T ~ ~oT
of opera-
the t h e o r y of m o n o i d s
homotopies,
G
(see
, we h a v e
~ ~ G = G, the t h e o r y o f groups. o co
(9)
Proposition.
In the d i a g r a m
,L
G~
~
the h o r i z o n t a l
arrows
(8)). The v e r t i c a l on a n y c o n n e c t e d
For (n)G
I
are h o m o t o p y
arrows h a v e
tors
the h o m o t o p y
the p r o p e r t y
the
spaces
For the v e r t i c a l s
or of C W t r i p l e s
t h a t t h e y are e q u i v a l e n c e s
(n)A~ are u n i o n s
(by
when evaluated
monoid
theorem,
sonrwj
of c o n t r a c t i b l e
components,
the
let
and underlying
space
the s u b c a t e g o r y W_D ~ Top o f c o n n e c t e d
extension
Ewl
o f C W theories,
~ Top
free t o p o l o g i c a l
"preserve"
equivalence
C W c o m p l e x X.
T o p ~ Son(Top) be the u s u a l
transformations
.~ G
the h o r i z o n t a l s ,
as well.
of n a t u r a l
F,U remain
functors.
CW complexes.
adjoint modulo homotopy,
B o t h o f these Moreover, i.e.
func-
b y use of
-
is an a d j o i n t pair of functors, homotopy.
The same h o l d s
where
for groups.
148-
[W_~] d e n o t e s The natural
the c a t e g o r y m o d u l o h o m o t o p y or w e a k
forgetful
functor
to mon-
from groups
oids Mon [W_W_]~
Gp [W__o]
is a c t u a l l y an i s o m o r p h i s m b y the well k n o w n possesses over
a multiplicative
inverse up to homotopy.
Since b o t h c a t e g o r i e s
always
are t r i p l e a b l e
[W_o], this implies that the triple m a p A ~ G is an i s o m o r p h i s m w h e n r e s t r i c t e d
the c a t e g o r y (10)
fact that a c o n n e c t e d C W H-space
[W_o].
Discretization
process.
of the powers o f o p e r a t i o n s
Let T = (T,~,p) be a t o p o l o g i c a l
topological
to
theory
(m,n)Tfi n = ((nS~
of topological
triples
is an important
triple w i t h O T = O. L e t Tfi n be the p o i n t e d
m, w i t h c o m p o s i t i o n
law
(m,n)T |
(n,p)T ~
(m,p)T
g i v e n b y ~ | ~ ~ ~.~T.(pS~ The Tfi n c o n s t r u c t i o n
reflects
and is a b o u t the same p r o c e s s Of course,
triples
into the s u b c a t e g o r y o f t o p o l o g i c a l
as was a p p l i e d to an "algebraic
Tfi n can be c o n s i d e r e d
as a triple
itself,
triple"
theories,
in T o p in (7).
and this f i n i t a r y r e f l e c t i o n or
t r u n c a t i o n of T is an i n j e c t i o n Tfi n ~ T (cf. Linton,
La J o l l a C o n f e r e n c e
As an example,
consider
on C a t e g o r i c a l
Algebra,
the s u s p e n s i o n - l o o p s
has as its space of n - a r y o p e r a t i o n s
S p r i n g e r Verlag,
1966).
triple Z~. The f i n i t a r y t h e o r y
the space of loops on the sum
Si
+ ... + S
1
(Z~)fin of n
circles. (11)
Proposition.
The i n c l u s i o n
X(Z~)fi n ~ XZ~ is a w e a k h o m o t o p y e q u i v a l e n c e
(homotopy e q u i v a l e n c e
For the p r o o f we use the fact that the o p e r a t i o n to any e n d o f u n c t o r X'Ffin where
( )fin can a c t u a l l y be a p p l i e d
of Top: lim nS O ~ X
(nS~ F
the li T has to be u n d e r s t o o d /
if X is a C W complex).
in the right c l o s e d - c a t e g o r y ,
i.e. topological,
-
1 4 9
-
sense, namely as a quotient space of X X n | (nS~ x(zn)
We then have the diagram
>xzn
fin
X(ZE) fin
) X~E
X2fi n
--X~
where E is the contractible path space functor. The left column is essentially a fibration by means of a path lifting function which shifts the terminal segments of paths from "generator to generator" of the suspension. From the homotopy exact sequences, ~nX(E~)fin ~ ~ n XE~. Since loop spaces are G~-spaces there is a topological theory map G
~ (Z~)
fin"
By (8), this is a homotopy equivalence of CW theories. Thus in the following diagram of triples in Top, all of the arrows are homotopy equivalences, at least when evaluated on connected CW complexes. ---
bA
---
>G
AT ~
(En) fin En (12)
Theorem.
The triples A and E~ are naturally equivalent on the category of con-
nected CW complexes, that is, if X is such a space there is a natural homotopy equivafence of the "reduced product space" x~=
XA=
XZn.
(I.M. James, Ann. Math. 6_~2 (1955) , 170-197). The above fibration argument can be iterated to obtain the same result about the
triples
zk~ k , k~O. The i n d u c t i v e
step
in the proof of the
following
form x(Ek~ l~k+l) fin
X (EKil EKE)"
fin
x(Ek+ink) fin = X E(Ekn k) fin (13)
Theorem.
> X zk~In k+l
X zkilnkE > X zk+In k
The horizontal inclusions in the diagram
theorem has the
150
X(Z~k~k) fin
~
> X ~ EkQk
X Qfin
~
>X Q
are weak homotopy equivalences Q = l im
-
(homotopy equivalences
if X is a CW complex)~ here
Ek~ k.
The functor Q was introduced by Dyer-Lashof direct limit of direct limit preserving triples
(Am. J. Math. 84 (1962), 35-88). As a (the circle is compact), Q is itself a
direct limit preserving triple. The discretization Q ~ ~ Q is the natural map Q ~ AG, o the latter the free abelian group triple, and is not a homotopy equivalence.
Indeed,
Q ~ AG induces the Hurewicz homomorphism stably, and Q-spaces generally have non-trivial k-invariants. Here Q-space means an algebra over the Q triple: X
rl
> XQ
XQQ
X
XQ - -
X~
> XQ
~ X
Q-spaces more or less coincide with the homotopy-everything
H-spaces of Boardman-Vogt
(Bull. AMS 7_~4 (1968), 1117-1122). At least the functor h.e.-spaces ~ Top Q is evident, and we do demonstrate that Q-spaces are infinite loop spaces spaces X with ~ X abelian groups) o
((17) below~ we mean h.e.-
It would be desirable to have direct demonstrations
that the infinite objects of algebraic and differential topology, O,U,BO,PL,...
are
Q-spaces. The homotopy-finitary character of a topological triple has various consequences. Restricting to a suitable class of "linear" finitary triples,
it is possible to demon-
strate theorems of the type (14)
H,(XT) ~----(H,X) ( ~ T )
,
that is, existence of triples H,T on the category of graded vector spaces over a field rendering the diagram Top
vl ct
T
H*T
>
Top
;
> Vect
-
commutative
(non-naturally,
out in a later paper
enough to o b t a i n Dyer-Lashof
the c a l c u l a t i o n s
bases
it is p o s s i b l e
(15)
of M i l g r a m
to press
The p u r e l y c a t e g o r i c a l
leads to the c o n s t r u c t i o n
I hope to c a r r y this
5.1 gives H.(Q,Z/pZ)
to d e m o n s t r a t e
linear but have
triple-theoretic
(Ann. of Math.
8 4 (1966),
explicitly).
formulas
over a field and c o n s t r u c t i n g
in the c a r t e s i a n - c l o s e d
ring).
zk~ k, Q (which are not t h e m s e l v e s
their T h e o r e m
spaces w i t h e o a l g e b r a s objects
in a h e r e d i t a r y
It should be p o s s i b l e
(op. eit.,
"simplicial"
-
for c o e f f i c i e n t s
for the triples
linear a p p r o x i m a t i o n s ) .
151
like
techniques 386-403)
far
and
For triples w i t h
(14) , replacing
vector
H.T from theories w h i c h have hom
c a t e g o r y of coalgebras.
q u e s t i o n of w h e t h e r
the E,~ a d j o i n t pair
of u n i v e r s a l base spaces.
Tripleableness
is t r l p l e a b l e
w o u l d m e a n that the
functor TopE~<
is an equivalence. C W complex,
#
Top
The q u e s t i o n can be e x a m i n e d
it is of "descent
type"
XnEnZ _ net is a c o e q u a l i z e r
diagram.
is ~ull. For a ZQ-space
~z
XEnZ
~XZ
in several parts.
for this a d j o i n t pair,
> X~Z /
Restricted
that
If X is a c o n n e c t e d
is,
> X
to spaces
to be "effective",
for w h i c h
this d i a g r a m is a coequalizer,
letting B = (X,~) |
be the c o e q u a l i z e r
>B,
E~ the c o m p o s i t i o n
X ~ XE~ ~ BR wo1~Id have to be a h o m e o m o r p h i s m .
is u n r e a s o n a b l e
to expect this m a p to be better
this w e r e
so, B w o u l d be a c l a s s i f y i n q
sense of h o m o t o p y the structure is better,
theory.
But as a caution:
i n d u c e d b y the group law,
than a h o m o t o p y equivalence.
for the E Q - a l g e b r a if X is a d i s c r e t e
then B = O. Perhaps
but c o n n e c t i v i t y w o u l d be an a w k w a r d a s s u m p t i o n
Before r e s o l v i n g Ek~k-space
space
the d i f f i c u l t y ,
worsen
(X,~) w h e r e k is a n y integer
m a p x E k ~ k ~ X. R e g a r d the s i m p l i c i a l
space
Yet from e x p e r i e n c e
(X,~)
it
If even
in the usual
g r o u p and ~:XZ~ ~ X is
for c o n n e c t e d X the result later on.
it b y c o n s i d e r i n g
the g e n e r a l case of a
~ O! thus ~ is a unitary,
associative
structure
152
X zk(nkzk) n, with continuous
n>~O,
face operators
X zk(~kzk) n+l
~i
> X zk(nkzk) n,
O~n
I ~zk(~kz k) n-1 ,
i=O,
[ X zk(~kz k) i-1 (nknk) n+i-1,
l~i(n,
l
where ~ is the adjointness map ~kzk ~ id., and degeneracy operators
~i similarly defined
in terms of ~zid. - zk~ k. It is intuitively reasonable to replace the coequalizer
above
with Bk = geometrical realization of X zk(~k k), = [
~
n
• X zk(~kzk)n / ~-identities
n~O exactly as defined originally by Milnor BO
=
(Ann. of Math. 65 (1957), 357-362). For example,
X.
Using the d[stributivity of ~k over the realization
identifications,
we have a na-
tural map geom. realiz.
(X xk(nkzk)*~ k) ~ BkQk.
By an elaboration of (5), there is also a natural homotopy equivalence of X into the above geometrical (16)
Theorem.
realization,
hence by composition a map X ~ Bk~k.
Every zk~k-space
(X,~) has a k-classifying
map X ~ Bk ~k is a zk~k-map and a weak homotopy equivalence
space. Precisely,
the above
(homotopy equivalence
if X
has the homotopy type of a CW complex). It suffices to prove X ~ Bk~k is a homology equivalence.
When k=O this is X=X, and
when k>O iterated cobar constructions are applied. We can also de-loop in the limit~ (17)
Theorem.
Precisely,
Every Q-space
(X,~) has a classifying
space B which is also a Q-space.
there exist a Q-space B which is a functor of X and a natural map (X,~)--B~
which is a Q-homomorphism
relative to the induced Q-structure on B~ and is a homotopy
equivalence. Each zk~ k ~ Q is a triple map, so X has induced zkek-structures be the "classifying
spaces"
for these (16). Maps B k ~ Bk+ 6 ~
~k for k~O. Let B k
are easily obtained such
-
153-
that the d i a g r a m s
k
commute.
As B k c o n t a i n s no cells of d i m e n s i o n s
~O
< k, Bk -- Bk+s
is a h o m o t o p y equiva-
lence. For the p r o o f of tures
(17), let B = l i m ( B k + I n k) as k -- ~. B has c o m p a t i b l e
for all k ~ O b y the d i r e c t
Q-structure.
limit p r e s e r v i n g
B~ also has a n a t u r a l Q - s t r u c t u r e
rise to c o m p a t i b l e
zk~k-struc-
p r o p e r t y of zk~ k, h e n c e B has a natural
via the t r a n s p o s i t i o n
~Z ~ Z~ w h i c h g i v e s
kk X ~ -structures:
B~zk~ k ~ Bxk~k~ ~ B~. These
zk~k-structures
Bn = li~(Bk+e~k+e).
c o i n c i d e w i t h those d e f i n e d
U s i n g the "internal"
"internally"
b y the fact that
p o i n t o f view, X ~ B~ is seen to be a Q-map,
and it is o b v i o u s l y a h o m o t o p y equivalence. We c o u l d have d e - l o o p e d k times at once b y u s i n g ly d e f i n e d as the "telescope"
,
an example o f a "Q-space up to c a n o n i c a l h o m o t o p i e s "
results.
for the triple Q, and for other t o p o l o g i c a l
"homotopy-everything"
If B is e r r o n e o u s -
of
B i ~ B2~ ~ B3 ~2 ~ ...
useful c o n c e p t
B = lim(Bk+~s
structures,
Q-structures
T h i s m i g h t p r o v e to be a
triples.
In c o n t r a s t
are not t r a n s p o r t a b l e
to
along h o m o t o p y
equivalences. (18) D u a l i z i n g
the foregoing p r o d u c e s
in b o t h the c a t e g o r i c a l
and E c k m a n n - H i l t o n
Top, and an ~ Z - c o s t r u c t u r e By adjointness
"cotripleableness"
The c o m p o s i t i o n
on a space X is a counitary,
theorem,
or a simple m a n u a l
Although
~kzk, w a s p o i n t e d out b y Luke ~ d g k i n .
~Z is a c o t r i p l e
in
m a p u:X ~ X~Z.
D o e s the e x i s t e n c e
in the case o f loop spaces the
approach,
to a suspension.
we m e a n d u a l i z i n g
coassociative
t h e o r e m w a s m o r e or less useless,
is c a n o n i c a l l y h o m e o m o r p h i c
the c o t r i p l e s
senses.
imply that X is a s u s p e n s i o n ?
"tripleableness"
structure
striking p h e n o m e n o n ~
e v e r y s u s p e n s i o n c a n o n i c a l l y has such a costructure,
of a costructure general
a rather
in this instance
the dual
shows that e v e r y X w i t h an QZ-
This
fact, w h i c h also h o l d s
for
-
154
-
FUNCTORS BETWEEN CATEGORIES OF V E C T O R SPACES
by D. B. A. E p s t e i n * and M. K n e s e r * *
Let
K
and
L
of c l a s s i f y i n g f u n c t o r s
be fields. from
~(K),
We c o n s i d e r the p r o b l e m the c a t e g o r y of finite
d i m e n s i o n a l K - v e c t o r spaces
and K - l i n e a r maps,
any two c a t e g o r i e s
B
and
we h a v e a f u n c t o r f r o m ject of
A
to
the o b j e c t
identity map
1 B.
Any
B,
and for any o b j e c t B,
w h i c h assigns
For
B
B,
of
to e a c h ob-
and tD each m o r p h i s m of
A
the
functor.
The f o l l o w i n g results are s u b s t a n t i a l in
~(L).
f u n c t o r i s o m o r p h i c to such a functor
w i l l be c a l l e d a c o n s t a n t
on the results
to
improvements
[2].
Theorem 1
Let and let
K
F: ~(K)
be infinite.
a c t e r i s t i c and If
L K
Then
be a n o n - c o n s t a n t K
and
L
functor,
have the same char-
is infinite. is finite then T h e o r e m 1 is false.
* U n i v e r s i t y of Warwick, ** G o t t i n g e n ,
> ~(L)
Germany
Coventry,
England
For we
-
have
the
finite from due
forgetful
sets,
and
for of
finite
to A.
Borel.
We
shall
that
K
L
sets
to t h e
there to
infinite.
K
and
are many
of
functors
~(L).
This
assume
throughout
therefore
is
category
remark
is this
2
Let F: V(K)
> V(L)
integer
n k
be
0,
a functor
G
ii)
F
n
n
An:
is
iii)
with
one
constant
exists
• ...
V(K)
characteristic
• V(K)
> V(L),
additive
is
a subfunctor
V(K)
the
and
let
Then
for
each
= V(K) n
such
in e a c h
of
> V(L)
that
of
its
G
9 A
n
n
n
variables
where
> V(K) n
diagonal
functor
.
n
When
object
functor.
zero
a functor
is
F = |
Note.
have
a non-constant
V(K)
Fn:
i)
L
there
Gn:
are
~(K)
field
category
Theorem
gory
any
-
from
the
paper
and
functor
155
n = 0, and one
functors.
V(K) n morphism.
is d e f i n e d So
as
the
G O , A 0 and
cateF0
-
Corollary
156
-
3
K
can be e m b e d d e d
in a f i n i t e
field
extension
of
L.
Corollary
4
If polynomial in
K = ~,
the
functor.
Such
field
of r a t i o n a l s ,
functors
are
then
F
is a
completely
classified
of T h e o r e m
2.
The
4 follows
from
[2]. Corollary
easy
proof
[2] L e m m a
is g i v e n
F0 | k e r ( F V that
ponds
FV
We
Corollary
now
we m a y
n = 0 commence
isomorphic
assume without
(The c o n s t a n t
case
to loss
of g e n e r a l -
V : > F0
functor
in T h e o r e m
2.
That
the p r o o f s
of the
is
corresF 0 V = F0
.)
theorems.
5
Let sional
G
vector
be
space
homomorphism.
Then
which
all
contains
a finite with
5.
is n a t u r a l l y
> F0),
F0 = 0.
to the
Lemma
in L e m m a
7.2. Since
ity
3 is a c o n s e q u e n c e
over the
L.
to w h i c h
group
Let
smallest
eigenvalues
extension.
respect
angular.
an a b e l i a n
Moreover
and
8: G
>
extension
of there
all e l e m e n t s
8g
W
a finite AUtLW
field
for all
is a b a s i s of
8G
be
dimena
LI
of
g 6 G, of
L, is
WSLL 1
are u p p e r
tri-
-
Proof. Suppose
that
We use
for s o m e
multiplication. loss
l
is a r e p r e s e n t a t i o n hypothesis
to
W 1
We use Corollary n a
i.
3.
Since
space which
action
on the
find
basis
element
M|
then gives
Lemma
also
field
of
~ 6 L.
a vector
the
induction
the p r o o f .
Now
space
L1
triangular. embedding
of
is K
an L(via the
By L e m m a L
the
Each
for some
M
over
of
to w h i c h
2 implies
Gn ~ 0
for e x a m p l e ) .
respect
Then
that Theorem
~ 0.
Without
~ 0
apply
constant,
W.
scalar
@g.
completes
to s h o w
extension
to a f i e l d
is n o t
of
5 we
and an L laction
diagonal
K
in
and
let
of any position
L1 .
6
Let
G
algebraically vector
space
Suppose
and
that
g 6 G
be a n i l p o t e n t
closed
G
it is p o s s i b l e ment
is
is u p p e r
rise
We
This
is n o t
with
k 6 K
that
G.
W/W 1 .
F
@g
(@g)w = lw}
for
first variable,
a finite
for
suppose
lemma
dimension
an e i g e n v a l u e
M = Gn(K,...,K)
vector
may
and
on the
g 6 G,
{w 6 Wl
space
this
Therefore
be
we may
W ~ W 1 =
-
induction
element
Let
of g e n e r a l i t y ,
157
let
field. @: G
has n o to c h o o s e
acts
Let
W
> AUtLW
finite
cyclic
a basis
as an u p p e r
group
for
be be
a finite
triangular
be
an
dimensional
a homomorphism.
quotient W,
L
group.
so t h a t matrix.
each
Then ele-
-
Proof. multiplication dimensional a central tion
on
Let
e: G
for e v e r y
over
L.
series
for
i,
that
so w e
only
{i}
G,
G. l
acts be
x 6 Gi+ 1
can define
G
acts
G-module
W
which
G O
<
suppose
on
the
that
=
and
-
show
simple
Let
> AUtLW Let
and
We n e e d
158
W
G 1
<
...
<
we have
by s c a l a r
by
scalar
is
finite
G r
be
G
shown
by i n d u c -
multiplication.
representation. g 6 G.
and
6 L*
l(x,g)
Then
by
xgx
X(x,g)
-i -I g
6 Gi
= 8 ( x g x - l g -I)
or
8x 9 eg =
Taking
determinants,
unity,
where
(x,
)
unity all
form
of k th r o o t s a finite and
scalar
multiplication.
G
on
W.
We
recall
unipotent
if
(A-
for
enough
angular
is a k th r o o t
for
(upper
G
of u n i t y
in
By
group,
This
that
into L. and
Hence
Schur's
that i)
l(x,g)
of
all g 6 G.
of
a basis
that
It is o b v i o u s
cyclic
action
find
.
a homomorphism
x 6 Gi+ 1
large
see
k = dimW.
gives
the g r o u p
we
eg . 8xX (x,g)
a linear
map
is n i l p o t e n t ,
commutes
W,
with
respect
if
down
for
with
of the
> W (A-
the
A
is
lemma.
called
i) r = 0
to b e i n g
to w h i c h
of
is t h e r e f o r e
A: W
is e q u i v a l e n t
ones
= 1
the p r o o f
i.e.
into
the k th r o o t s
ex
ex
x,
and hence
X(x,g)
This
with
fixed
so
r.
triangular
L*, But
Lemma,
completes
for
of
able
to
is u n i t r i -
the d i a g o n a l ) .
-
Theorem
SL(V,K)
one.
Let
phism.
K
be
be
the
an i n f i n i t e group
e: SL(V,K)
i)
L
is i n f i n i t e ;
ii)
K
and
L
iii)
G
maps
unipotent
In fact,
have
if w e
but we
Proof. tained
of u n i t y trivial
must
be
elements L
mapped
let
dimKV
with
> 2.
determinant
a non-trivial
(dim V = r)
homomor-
characteristic;
to u n i p o t e n t
for
such
V,
that
then e
there
maps
to u n i t r i a n g u l a r
matrices,
this.
subgroup
of
SL(V,K)
multiplications [i] p.
into
elements.
38.
finite
by
So
is con-
the
r
SL(V,K)
th has
groups.
In p a r t i c u l a r
p.
the
infinite. K
have
characteristic
those
characteristic
to u n i p o t e n t
1 + xEl2(x
W
matrices
normal
homomorphisms
same
elements
for
of s c a l a r
are e x a c t l y
has
the
do n o t p r o v e
Every
in the g r o u p
Let
if
be
fix a b a s i s
a basis
unitriangular
L
and
of a u t o m o r p h i s m s
> AUtLW
exists
only
field
Then
iv)
roots
-
7
Let Let
1 5 9
6 K)
whose p,
elements.
form
order
then
Then
is a p o w e r
unipotent
The matrices
an i n f i n i t e
abelian
unipotent
of
p.
elements
of the
group
H
form of
are
So
-
exponent
p.
By L e m m a
so that
8H
consists
diagonal
entries
unity.
Hence
mapping
under
On the o t h e r S = i.
p,
eH
of
H
as in the
L,
closed
let
elements,
has
elements
iv).
angular
matrices
K
in
have w e may
characteristic
of e l e m e n t s
L
is not that
p,
%
is
characteristic are mapped
suppose
Lemma
SL(V,K)
AUtLW.
Let
L
to
It follows
that
i < j < k.
matrices
%(1 + ~Eik)
8(1 + ~Eik)
if w e w e r e
6, to d e d u c e
are s e n t
that unitri-
to u p p e r
Then
triangular
the c o m m u t a t o r
= 1 + IEik
is a c o m m u t a t o r
and is t h e r e f o r e
Without
is a l g e b r a i -
n o t be l e g i t i m a t e
We now a p p l y in
that
zero.
(i + IEij) (i + Ejk) (i - IEij) (i - Ejk)
of u p p e r
unitriangular.
. tri-
So
is u n i p o t e n t .
Changing
the b a s i s
is u n i p o t e n t
for all
e(l + IEij)
is u n i t r i a n g u l a r .
generate
of
is n o n - t r i v i a l .
of
K
The
roots
first p a r a g r a p h ,
and u n i p o t e n t
(which w o u l d
proving
angular
th
consisting
that if
W|
matrices.
are p
if the c h a r a c t e r i s t i c
of g e n e r a l i t y ,
matrices
of
for
elements. Now
cally
a basis
triangular
S
So we h a v e p r o v e d
unipotent
choose
to u n i t r i a n g u l a r
hand,
t h e n so has
loss
of u p p e r
the s u b g r o u p 8
-
5, we may
of an e l e m e n t
It follows,
trivial.
1 6 0
the g r o u p
i ~ j.
of
V,
we see t h a t
It follows Since
of u n i t r i a n g u l a r
that
8(1 + IEij)
for
i < j,
the e l e m e n t s
matrices
in
i+
IE.. 13
SL (V, K) ,
-
we
see
that
matrices,
6
and hence We
where
u..: 13
some
~.. 13
i < j,
K
with
into
and
L.
have
We
and
homomorphism
can n o w
> AUtLFV
L
minimal.
Theorem
7, e a c h
induced
i.
by
F
is t r i v i a l .
i,j:
V
> V~V
be
the
canonical
injections
p,q:
VeV
> V
be
the
canonical
projections.
have Fe =
has
0V@ v = ip e ip ~.
ivev,
we have
e.. 13
group zero,
1 is
homomorphism
dimension,
e
abelian
If T h e o r e m
space
Then
Then
characteristic
a vector
= jp + iq.
SL(V,K),
7 is c o m p l e t e .
deduce
of e v e n
has
on
of i n t e g e r s
of the d i v i s i b l e
that
of T h e o r e m
a pair
j - i
elements.
= 1 + ~i<jeij (x)Eij,
is n o n - t r i v i a l We p i c k
non-zero
t h e n by T h e o r e m
SL(V,K)
8
to u n i t r i a n g u l a r
to u n i p o t e n t
8(i + XEl2)
Since
It f o l l o w s
matrices
elements
be n o n - z e r o .
~.. 13
the p r o o f
false,
unipotent
> L.
must
-
unitriangular
therefore
is an a d d i t i v e K
maps
161
such
that
determinant
Applying
F
one and
Let
V
F V ~ 0.
Let
and Let
and
e2 = i.
remembering
0 = F(ip) 2 = F(ip).
be
Now
We
that
i v = pi pi.
Therefore
I F V = F(I V) which
F(ip)
F(i)
= 0
,
is a c o n t r a d i c t i o n . We
teristic we
= F(p)
assume
zero.
can d e f i n e
For exp N
from now
on t h a t
any n i l p o t e n t and
log
K
and
N
endomorphism
(i + N)
with
have
L
the
of
usual
characW,
power
-
series.
The
functions
tions b e t w e e n
the f o r m
G
K
where
and
N
inverse
endomorphisms
bijec-
a n d the set
x 6 K.
w h i c h maps
G
> L ( I < i,j < m)
Let
additive
from
K
> log
additive follows
8(1 + xN) g r o u p of
number
be a
matrices. defined
of
Let
by
to
= exp
of
L.
log
8(1 + xN).
homomorphism
matrices
the e x p o n e n t i a l
over series
K.
Now i n t o the
The
result
(which is zero
of terms).
9
Let x 6 K.
(m • m)
of
is a s u m of p r o d u c t s
is an a d d i t i v e
by e x p a n d i n g
after a finite
Lemma
8(1 + xN)
V
> GL(m,L)
be the f u n c t i o n %ij
Proof.
0: G
into unipotent
Then
homomorphisms
of
is a f i x e d e n d o m o r p h i s m
%ij (x) = 8(i + xN)ij.
There
and f u n c t i o n s
Xv: V
> V
be s c a l a r m u l t i p l i c a t i o n
exist endomorphisms e.: K
) L
of a d d i t i v e
by
Ai
of
F V ( I ~ i ~ r)
such that
e.
is a s u m of p r o -
1
ducts
give
be the g r o u p of a u t o m o r p h i s m s
N2 = 0
homomorphism
x
log
endomorphisms.
(i + xN),
with
8ij:
and
8
Let
V
-
the set of u n i p o t e n t
of all n i l p o t e n t
Lemma
exp
1 6 2
1
homomorphisms
and
F ( x v)
= E[=lei(x)Ai
.
163
-
Proof. tions
Then,
and
p,q:
F(VeV)
f a c t o r has maps
Fi
Let V~V
has
i,j:
V
> V
b e the c a n o n i c a l
FV~FV
Fp,
> V@V
be the c a n o n i c a l
as a d i r e c t
as its c a n o n i c a l and
-
injection
and the s e c o n d
projections.
summand.
The
first
and p r o j e c t i o n
factor
injec-
the m a p s
the Fj
and
Fq. We c o n s i d e r of e l e m e n t s F(I + xiq) choose
of the
K
form
a basis
for
in to
of
1 + xiq(x
is u n i p o t e n t
of F(I + xiq) of
the s u b g r o u p
for all
F(V@V),
AutK(V~V)
6 K).
x 6 K.
then each
is a s u m of p r o d u c t s
consisting
By T h e o r e m By Lemma
entry
7,
8, if we
in the m a t r i x
of a d d i t i v e
homomorphisms
L.
Now
Fp
9 F(I + xiq)
9 Fj = F(Xv).
The
lemma
follows. We now a p p l y L e m m a is the
f i e l d of r a t i o n a l
I ! > F(IV). F(I v)
a 6 L.
@ : > L
homomorphisms which Hence
term.
in
I,
for
on
with
such that N o w any a d d i t i v e
I : > al
in the m a t r i x
fpr some
of
coefficients entries
F(I v) in
the d i a g o n a l
has
the f o r m
entries
of
L
is a and
are m u l t i p l i c a t i v e
But e v e r y m u l t i p l i c a t i v e
is p o l y n o m i a l ,
Q
by
FV
FV|
I 6 @.
form
The d i a g o n a l
Q ~ > L.
acts
for e a c h
has the
Here
G = ~* c K*.
G
a basis
9 each e n t r y
function
zero c o n s t a n t
i a 0.
We can c h o o s e
By L e m m a
polynomial
phism,
numbers.
is u p p e r t r i a n g u l a r
homomorphism
5, w i t h
I : > Ii
F(I v)
homomorfor some
are all of the
-
form
I
i
,
where
of
the
entry.
L
described
It
It
=
{w
is
value
follows
in L e m m a
For
FiV
the
each
carries
to
see
F.V
into
contained
in
due
the [2]
to
D
and F:
are
fact
> ~
two
be
We
we
is
F ~
deviation MacLane
and
let
a functor
to
~ 6 ~ and
have
@.F. i
we
x D,
D
) C
x D,
N =
functors,
dim
FV}
then
F
some
which
material is
a
[3]. category
with
A
abelian
be
an
canonical
we
L.
repeat
have
~,
of
.
i
that
in
position field
linear,
such
objects
the
define
> W
arbitrary
on
extension
equal
completeness,
and
object, be
0,
V
depend
(the
0 all
e:
of
an
in
=
concerning
~
a zero
~
sake
Eilenberg
Let
Let
is
if
1
For
ducts
L1
i >
F.W.
1
may
that
integer
that
-
i
of
6 FVI (F~ v - ~ i ) N w
easy
notion
5),
164
F0
=
finite
0.
pro-
category. If
C
injections
and and
projections
i:
C - - C
such
that
FI(C,D)
pi
which
Lemma
i0
the
three
Fp
is
F(C to
= IC,
= ker
tion,
j:
qj
N ker
natural
= Fq. for
x D) _~ F C factors
1 D,
are
p:
pj
We
C
=
have
x D
0DC,
FP,
C
@ F 1 (C,D). Fq
qi
a direct
morphisms
@ FD
> C,
and
= 0CD. sum
> C1
The
q:
C
x D
Let
decomposiD
> D1 .
projections
1 - Fj
9 D,
9 Fq
- Fi
on 9 Fp.
-
The
injections
Lemma
are
obtain from
F1
an i n d u c e d ~2
= ~
is
and
inclusion.
can
defined
above
this.)
C_=k + l
:
of i n t e g e r s
of
>
functors,
(F2) I
respects
of
the
is a f u n c t o r process
to o b t a i n
ii for FI
which
> ~
the
i,
Lemma
(F1) I
~,
G: ~k
perform
fixed
to
a morphism
then we
functors
direct
sum
i0.
If
use
> F2
morphism
• ~
decomposition
some
Fj
-
ii
If
one
Fi,
165
such
that
variables,
Gi : = C k+1
Suppose
F
where
I =
1 g i. g j ]
k
.th l variable
on t h e
a functor
> A,
of
is
as
>
above,
{ i l , . . . , i k}
for
each
j.
for (We
A.
and we have is
a k-tuple
Then
we
define FJ =
where
J =
{i If
{l,2,...,n}, that IKI by
Pi~K be
induction
such
that
, . . . , i k , j}
and
C = C 1 • ... we
= Pi
the
(FI) J : C=k+2
denote for
number on
the
by
1 < j ~ k + 1
• C
1 ~ i. ~ j 3 FI(cl,...,Cn
for
n each
K
is
> C
C
and
of e l e m e n t s length
and
n
~K:
i E K
> A
Pi~K K.
of
a subset
the m o r p h i s m = 0
for
I =
such Let
to p r o v e ,
{ i l , . . . , i n}
that
) = I m ( E K ( - 1 ) IKIF(~K))
of
i ~ K.
It is e a s y
of n - t u p l e j,
.
c FC
,
-
where FI
K
runs
depends
-
o v e r all the s u b s e t s
o n l y on the l e n g t h
the o r d e r of the v a r i a b l e s
F (n)
is c a l l e d
Lemma
12
of
I
{l,...,n}.
is a d d i t i v e
Hence
and is i n d e p e n d e n t
of
of
We d e f i n e
= I m ( ~ K ( - i ) IK]F(~K))
the n th d e v i a t i o n
F,n,l~
of
C I , . . . , C n.
F (n) (Cl,.. . ,C n)
F
1 6 6
F.
.
We o b v i o u s l y
in e a c h v a r i a b l e
have:
if and o n l y if
(n+l) =
0
9
N o w w e t u r n to f u n c t o r s suppose 5).
that
F0 = 0,
A functor
F
space
eigenvalues
F1V
above if
(just b e f o r e
K
and
L
is the d i r e c t Theorem
Lemma
V
over
K,
are all e q u a l the s e c t i o n
have
(see j u s t b e f o r e
characteristic
therefore
to
li
.
zero,
assume
F
Lemma
I 6 Q, We h a v e
i if
the shown
functors),
then every
functors.
and we
of d e g r e e
and each
on d e v i a t i o n
s u m of h o m o g e n e o u s
2, w e m a y
that
functor
In o r d e r to p r o v e
is h o m o g e n e o u s .
13
If and
as w e c a n do
> V(L),
w i l l be c a l l e d h o m o g e n e o u s
for e a c h v e c t o r of
F: V(K)
F(n+l)
F
is h o m o g e n e o u s
n,
then
F (n) ~ 0
= 0 .
Proof. of a v e c t o r
of d e g r e e
space,
If
A
and
B
are c o m m u t i n g
then every eigenvalue
of
AB
endomorphisms is the
167
-
product
of
garding shown
an e i g e n v a l u e
F (k)
that
of
A
a functor
of
of
the
form
F (k) (X,...,X)
the
X ni. are
the
a subfunctor
of
is
that
+ n k = n.
nI +
...
that
F (n+l)
that
F (k)
~ 0.
Lemma
12 a n d
It
of
F (k) (V,...,V)
deduce
and
an eigenvalue i th v a r i a b l e
F (k) ( I , . . . , I , X , I , . . . , I ) ( X
eigenvalues ues
as
of
-
Since
= 0.
Then
Let
F (k)
so e a c h
n.
is
follows form
has that
...
n.1 >
1
be
the
each the
SV),
for
additive
equal
only,
B.
Re-
we
have
of
its
eigenval-
Xn l + n 2 + ' ' ' + n k .
F(Ve
k is
6 @)
of
it
Since follows
1 < i < k,
we
largest
integer
such
in
variable
by
each
to o n e .
1
Lemma
14
Let Then
F
F:
can be
copies
of
agonal
functor.
if
F (n)
e: F 1
> F2
then we ha~e
V(K)
> V(L)
embedded 9 A
n
,
This is
in
the
where embedding a morphism
a commutative
induced
Proof.
by
An:
~(K)
sum
is n a t u r a l of
functors
> G2
a
Consider
(n)
the
G
of of
> V(K) n
1
G 1
is
direct
> F2
[
B
homogeneons
diagram
FI
where
be
composition
in
F
degree
n.
(n - i)! is
the
di-
- that
is,
of d e g r e e
n,
168-
-
VeV
2
V
where
A
is
the
diagonal
expressed
as
the
composition
FV
FA>
which YV:
is e q u a l
to
9 F(2) (V,V)
(i - F i F p
- FjFq)FA
if
~:
of
F1
is e q u a l
is
= FA
o YV
- Fi - Fj. y
We
have
gives
rise
Fm 9
F (2 V)
FV
where
from
a morphism
So to
~(K) of
7V
~(L).
functors,
to
Moreover,
then we
have
a
diagram
F1
We
(2)
know
>
eigenvalues
YV
is
of
a monomorphism
. A2 .
F(2V)
we
may
Fm
. YV for
F2V
9 F2
(V,V)
that
of g e n e r a l i t y ,
the
F (2)
addition.
to
FIV
loss
is
F V @ F V ~ F (2) (V,V)
functors
> F2
commutative
m
F1 v + F1 v + F m
FV
a morphism
and
:~ V
(2)
(V,V)
= IFV + IFV + Fm
suppose are
each
that
equal V.
9 YV"
n > i. to
Hence
2n - 2 F
Then
Without all
and so
is e m b e d d e d
in
-
The
write
lemma
is n o w
F (2) = @ n - i T r=l r , n - r
has
degree
the
second
functor
r
in
the
of h o m o g e n e o u s the
degrees G:
G(r's):
~(K) r + s
respect
to t h e
to t h e
naturality Tr,n_ r
in
of t h e
in
T
Tr,n_r)
r,n-r
may
only
and writing
two
= V(K)
by
taking
variable.
By
embeddings
(r,n-r)
for
be
(F
embedded
(2)
)
in
may
embedded
in t h e
direct
copies
of
9 An.
Therefore
bedded
in
proof
F (n)
(n-l)!
of T h e o r e m
• A
n-r
copies 2.
can
s
add
induction lower
to
We
on
values
copies
define
with re-
using n,
we
the embed
of
Lemma
= F
of
n.)
with
n,
a
c F(VeV),
up
deviation
as a sum
r th d e v i a t i o n th
in
13,
(n)
.
it
follows
Hence
(r - i) ! (n - r - i) ! (<(n - 2) !)
of
r
it as
> V(L).
the
(r,n-r)
copies
be
Ft~,n, 9 A
We
F (2)
F(2)(V,V) must
the
From
n _ r )=
that
x V(K)
and
(A r
n.
n - r
regarding
variables
x A 9
on
degree
by
variable
(r,n-r)
and
done
(r - i) ! (n - r - i) !
(Tr,n_ r) that
is
the
> V(L)
second
variable
(We r e c a l l
V(K)2
first
induction
> V(L)
variable
functors.
Let
spect
V(K)
This
first
by
where
first
variable.
of
so t h a t
the
-
proved
,
Tr,n_r:
169
of
It
follows
sum
F (n)
of
that
T
r,n-r
. A2
(r - I)! (n - 1 - r)!
F c F (2) 9 An .
o A2 This
may
be
completes
emthe
-
170
-
REFERENCES
[13
Dieudonn~, J., "Sur la g~om~trie des groupes classiques", Springer, 1955.
[2]
Epstein, D. B. A.,
"Group representations and functors",
to appear in Am. J. Math.
[3)
Eilenberg, S., and MacLane, S., "On the groups H(~,n), If", Annals of Math., 60; 75-83,
(1954).
-
NATURAL
171
-
VECTOR
BUNDLES
by D.
1.
Let vector
manifold Cs on
M
map fibres
Cr
M
and
s ~ ~
is a f u n c t o r a
f:
)
N
a
makes
the
have
pact.
a countable
The
assumed
(In t h i s
to v a r y
from
the
bundle
New
examples
direct E
sums
) M
Given vector
of
k th
the
the
order
of
existing
bundle
vector (jk(v*)
VM
--~
to
each
> M
and
VN
natural Cs to
each
, which
is
linear
N
a manifold need
fibre
of
bundle bundle )*
1,
.
be
to
bundle
tensor
of
of
, we
bundle
(r = s -
Given
k-jets
boundary,
connected
operators
taking
no
or
will
combe
another.)
examples.
V
have
a vector
component
by
will
not
differential
etc.
a natural
VM
(r,s)
VN
tangent
generated
the
Vf:
but
one
be
is
~M:
f
,
can
, jkE
An
assigns
Vf
~
basis,
are
integers.
which
map
paper,
of
Examples
be
RESULTS
diagram
dimension
not
AND
bundle
Cr
M
commutative.
Epstein
V
vector
VM
will
A.
DEFINITIONS
0 ~ r ~
bundle
B.
and
(r = s - k). products,
a vector
sections
obtain
i)
a new
of
bundle E
natural
.
-
I.i. sional
real
Let
vector
VLin
be
spaces
and
172
the
continuous.
any
Cr
vector set
By
vector bundle
of
TE
[1]
TE
is
>
U
~
1.7,
bundle
~: M
category
linear
T: V L i n
be
-
maps.
is
E ---> M
dimen-
Let
polynomial. , we
can
follows.
The
, and
T(~-Ix)
T(~-ix)
finite
VLin
T
as
of
Then,
given
construct
a new
underlying is
point
the
fibre
x6M over
x
.
trivial, G:
U
Given
we
x V
any
have )
an
~-IU
differential
open
TG:
TG(u,v)
phism.
If
V
a new
natural
Definition
continuous,
F:
Cs P
map
• VM
is
by
U
=
.
Cr
.
We
give
which
E
is
bundles
TE
a topology
and
that
x T V ---~ U u s u T ( ~ - l u )
(TGu)
(v)
s T(~-lu)
vector
bundle,
, be
a
then
T V b y M ---~ T(VM)
bundle
say if, f: > VN
that for
an
any
p x M
(r,s) Cs
---> N
, defined
natural
manifolds , the
vector P, M
induced
by
F(p,w) is
, over
of v e c t o r
insisting
a natural
vector
of M
Cr we
isomor-
can
define
.
1.2
We
any
U
structure,
by
U
isomorphism over
defined
set
= V(fp)W
bundle and
map
N
is and
for
173
-
Definition
1.3
A natural if
dim
name
M
= dim
is d u e
Theorem
to
P.
.
to
dim
VM
said = dim
to b e
myopic
VN
(This
.
vector
be
(r,r)
bundle
is
continuous
and
myopic.
the
an
natural
is a v e c t o r
constant
vector
space
natural
W
bundle,
, wuch
vector
bundle
that M
where V
is
---> M
• W
.
V
,
1.6
restricted
V
to
(~,~)
be
C~
an
(r,s)
manifolds
natural
vector
natural and
vector
maps,
is
bundle.
isomorphic
Then to
a
bundle.
1.7
Let filtered
such
is
Freyd.)
there
Let
is
that
natural
V
Then
isomorphic
Theorem
implies
V
1.5
Let
unique
bundle
1.4
Theorem
Theorem
vector
N
Every
r < ~
-
by
V be (r,s)
an
(r,s)
natural
vector
natural
subbundles
0 = V_I
= Vo
= VI
bundle.
Then
V
~
that
i)
For
a fixed
ii)
For
each
manifold
integer
M,
VrM
= VM
for
i > 0, V i / V i _ 1 ~ - F i r
r
large.
, where
-
is the
tangent
functor
of d e g r e e
degree
i
means
to s c a l a r cation
m
of s u c h
.
from
that
>
-
Fi
is a h o m o g e n e o u s
VLin
Fi
to V L i n
sends by
scalar
li
.
.
continuous
(To say
Fi
multiplication
See
[i] f o r
the
has by
classifi-
functors.) Let
Then
(VjM) m
i
and
multiplication
iii) at
bundle
174
Vf
f,g: and
(VjN) n
M,m
Vg
> N,n
induce
if e i t h e r
have
the
j ~ i
the
same
i-jet
same map
or if
i = s .
2. C O N T I N U I T Y In this bundle the
2.1
point
f is c o n t i n u o u s
hoods
a natural
following
lemma
metric
space
space.
y ~ Y be
a point
For
Let
Let
each
at a d e n s e
U1,U2,U3...
be
f:X ~ Y be
xs
is a n e i g h b o u r h o o d
vector
provides
and each
and
a countable
with
a
a count-
a function
with
neighbourhood
V of x such
set o f p o i n t s
let Y be
fVc__W.
that
W Then
in X.
base
of open
neighbour-
o f y. L e t X i = { x ~ f x ~ U 1 or
We
The
that
a complete
property.
there
Let
1.2).
of neighbourhoods.
following {y,fx}
the p r o o f
set t o p o l o g y .
topological
able basis
Proof:
(see
L e t X be
Hausdorff
of
we begin
is c o n t i n u o u s
necessary
Lemma
the
section
claim
is open,
that by
X. is open. 1
the c o n d i t i o n s
let W 1 a n d W 2 be d i s j o i n t
f is c o n t i n u o u s In f a c t stated. open
at x}
it is o b v i o u s Let
f be
neighbourhoods
that
{xlfx~U.} 1
continuous of
at x,
fx a n d y re-
175
-
spectively
and
let V be an o p e n
f V c W 1. T h e n
f is c o n t i n u o u s
W 3 be a n y n e i g h b o u r h o o d an o p e n
neighbourhood
Then we obviously We a l s o
and
suppose
such
Then
f then
diction.
This
We w i s h
such
that
to s h o w
For
suppose
f
-1
W 2 meets U.
Baire
show
b y ~(p,w)
)w,
by
ip(m)
a neighbourhood proof
steps.
that
=
(p,m).
(W 3 N W 1) .
i and
of
the
that
x ~
X.. l
~P,M
Theorem
at x. F o r
and
this
conti-
is a c o n t r a -
lemma. (r,s)
natural
C s manifolds
• M) i
By r e s t r i c t i n g
:M ~ P x M is our
in f a c t a s s u m e take
fx
X t is
so fx = y. The
for all
is C r. H e r e
is C r w i l l
Then
neighbourhood
P
of P, we m a y
find
Wi a n d W 2 of
be c o n t i n u o u s
P • VM ~ V(P
= V(i
let
fV for e a c h n e i g h -
Category
P
The
can
the m a p CP,M:
defined
For
r X..
the h y p o t h e s i s
the p r o o f
only
we
for e a c h
t h a t V is a c o n t i n u o u s
We need
of V.
f V' c W 2 U
neighbourhoods
f must
from
x'
that
c W 3 n W i-
f x ~ U. for e a c h l
completes
bundle.
P a n d M,
then
follows
point
that W 2 c U i and
By the
of x s u c h
B y the h y p o t h e s i s
open
V meets
= N.X.. i 1
If x ~ X'
not.
of
defined
of x'
are d i s j o i n t
let X'
in X.
vector
fx'.
f ( V ~ V')
V o f x. H e n c e
Now
nuity
at e a c h
f V ~ W 1. B y the h y p o t h e s i s
bourhood
dense
neighbourhood
t h a t X. is d e n s e . 1
y respectively,
V o f x,
V'
have
claim
fx # y. T h e r e
of
-
place
attention
to
t h a t P = ~n.
in a n u m b e r
of
-
Lemma
first
m x VM
is c o n t i n u o u s
---~ V ( R
x M)
w 6 VM
lie o v e r
in t h e
variable.
Proof. ~: V ( ~
• M)
let
--~
Y
Euclidean and
-
2.2
#R,M:
and
176
be
Let R x M
the
space
the usual
be
one
point
n-l(R
metric
the
.
the
Y
.
Let
of t h e v e c t o r
compactification
x m)
on
projection
m 6 M
of
the
is h o m e o m o r p h i c
sphere
makes
Y
bundle
to a s p h e r e ,
into
a uniform
space. Let We must theses
f:
show
f
suppose
2.1.
tends
not.
f (ti)
Let
tends We
n(i) (i > there
0)
is a
properties
We
either
be d e f i n e d
to
be
by
first
prove
f ( t 0)
I ti+ 1 -t to
We
shall
{ti}i> 0
0 < and
Y
is c o n t i n u o u s .
of L e m m a
t O , f(t)
R --~
or
that to
a sequence
O
I <
w 0 ~ f ( t 0)
I t i -t
a strictly
increasing
, which
increases
sufficiently
function
~:
ii)
~(t2i)
R ---> R
as
such o
= tn(i)
V(~ • 1 M)
~(ti,w)
.
the
t
- E Y
hypo-
tends 9
to
For
that
I
with
sequence
of
integers
rapidly
so
that
the
= tO ,
~(t2i+l)
r
verify
:
i)
=
.
choose
C~
f(t)
Then
= r
.
following
177
-
The
left
hand
the right
side
hand
tends
side
to
tends
the
-
limit
V(~
x IM) W 0 , a n d
to a l i m i t .
But
this
so
means
w 0 = lim i #(tn(i),w)
= lirai ~ ( ~ t 2 i + l , w )
= lira i ~ ( ~ t 2 i , w )
=
which
is a c o n t r a d i c t i o n 9 Hence,
tO E i
f(t0)
.
Let
by Lemma
7t:
i
2.1, be
> i
V ( 7 t x IM)
is a h o m e o m o r p h i s m ,
V (Y-t
9
x IM)
is c o n t i n u o u s
f
translation since
of
t
s = t I - to at all
Proposition
vector
' we
points
0 s in bundles
by
in
see R
wl,...,w k .
Then over
that
#(s + t,w) ~
Then
inverse
.
is a c o n t i n u o u s
function
.
~(x,y)
be
we have
a basis a
Cr
for
x Rk
T
i n
=
= ~ ki=l
Yi[V(Yx)Wi]
the
fibre
isomorphism
mn Rn
given
an
.
2.3
Let over
it h a s
t
point
Now V(y s x iM)#(t,w ) =
Taking
by
at s o m e
) Vl n
i n
'
u
of of
Vl n
178 -
where
in
7x:
) Rn
Proof. w 6 VM
is t r a n s l a t i o n
By
induction
by
n
on
x
, we
.
know that
for each
, the c o m p o s i t e i Rn
is c o n t i n u o u s .
~n
Hence
~
~n
is c o n t i n u o u s . follows
that
manifold
, by
VR n
• VM
the
v(+) ,
~ > V ( ~ n x M)
T
acts
r ~ V ( R n x M)
composite
composite
the m a p Rn
Rn
x VM
This
isomorphism.
uous,
w,
sends
x to V ( u
of t h e t h e o r e m
= V(u
.
It
9
is a c o n t i n u o u s
as a t r a n s f o r m a t i o n
p(x,w)
V (Rn)
The
group
on the
action
is c o n t i n -
fixed
x
[3]
212,
since p(s,T(xl,Y)
and
) = ~(x + xl,Y)
is a h o m e o m o r p h i s m . is a
v(~ x)
Cr
Moreover,
isomorphism
of
VR n
for
.
By
p.
, the
map p: is
Cr
his
attention
that
.
T
(The a u t h o r
wishes
Cr
Cr
for
of t h i s
R. P a l a i s
for d r a w i n g
result.)
It f o l l o w s
isomorphism.
2.4
CU,W:
is
VR n
to t h a n k
to t h e r e l e v a n c e
is a
Proposition
~n x V R n ~
finite
U
x VW
dimensional
~ V(U
vector
x W)
spaces
U and W
.
179
-
Proof. be the c a n o n i c a l
Let
i: W ---~ U x W
injection
V W V(i)) is the i d e n t i t y . VW
over
0 .
to a b a s i s
Let
Let
By P r o p o s i t i o n
Let
V(U
i': i r ----> Rk
and
follows
and extend
of
V(U
• W)
=
this
over
0 .
isomorphisms ) V(U
x W)
by
(Xl,...,Xr,0,...,0)
from the commutative X
U • W • ir
for t h e f i b r e of
~2: U x W • ~k
be d e f i n e d
i'(xl,...,Xr)
VW
~ j ~ r)
Cr
9 W
Then the composite
be a b a s i s
for the f i b r e
2.3, w e h a v e
The proposition
x W) V(p)>
uj = V ( i ) w j ( l
~i: W • i r --~ V W
p: U • W
and projection.
wl,...,w r
Ul,...,u k
and
IUxW
diagram
i' ) U x W x ~k
U x VW
~
V(U
x W)
3. B A S E D M A N I F O L D S
Let
~,(s)
be the c a t e g o r y
b a s e poin t,
satisfying
necessarily
compact
Morphisms maps.
If
usually s p a c e of We give
in
suppress Cs
or c o n n e c t e d ,
~,(s)
(M,m)
preserve
and
(N,n)
from
Cs
manifolds
a x i o m of c o u n t a b i l i t y , and which
are
M to N
the c o a r s e
Cs
Cs
and a r e
not
Cs
based manifolds,
and write
C~(M,N)
, preserving topology
with
have no boundary.
the base point
the b a s e p o i n t s
maps
C~(M,N)
the s e c o n d
of
we
for the
the base points.
(on s o m e c o m p a c t
-
subset An
of
M
(r,s)
the
natural
to a f u n c t o r to t h e show
over
the
Definition
pair
manifold
the
induced
P
rise
TM
In d u e
we shall
course
is e q u i v a l e n t
equal
to t h e
bundles.
Cs
and
map
for
shall
if it is c o n t i n u o u s manifolds
M and N
map
is
> VLin
in t h e
Cr
and
if for
(N,*)
p x Mt
category
, for e a c h
P • *
) Ne*
of r e a l
vector
that
that
sense
We can
talk
every
CO
of
Cr
that
T
is
CO
for e a c h
functor if a n d pair
is
only
of b a s e d
, the map C~(M,N)
) Lin(TM,TN)
is c o n t i n u o u s .
Definition
3.2
We
say
dim M = dim N T
is l o c a l
T: M. (s) ---> V L i n
implies
that
if f o r a n y b a s e d
t
spaces
.
prove
We remark
is
Cr o
the origin.
in f a c t
Cs .
(M,*)
Cs
be the
preserving
automatically
each
P • T M ---~ TN
T: V D i f f ( s ) We
> VLin
manifolds
VDiff(s)
maps
functors
gives
, by putting
functors
T: M=, (s)
say t h a t
Let Cs
point.
be c l o s e ) .
obviously
VLin
such
vector
of b a s e d
Cs
and
~
V
should
3.1
We each
derivatives
the base of
-
bundle
~,(s)
study
of n a t u r a l
s
vector
T(V):
fibre
that
study
first
180
is m y o p i c
d i m T M = d i m TN manifold
(M,m)
.
if We and
say t h a t for
any
-
open n e i g h b o u r h o o d an i s o m o r p h i s m
U of m
TU ---~ TM
181
-
, the i n c l u s i o n of
local.
If
T
induces
.
We can a l s o talk of a f u n c t o r being
U in M
is local and
T: VDiff(s)
f,g: M,* ~
some n e i g h b o u r h o o d of the b a s e point,
then
N,*
---> V L i n a g r e e on
Tf = Tg: TM ---> TN
.
T h e o r e m 3.3
a) T
Let
T: VDiff(s)
---~ V L i n
Let
T: ~,~(s) ---~ V L i n
be c o n t i n u o u s .
Then
is local. b)
Then
T
c)
If
T: M.(s)
d)
E v e r y local f u n c t o r
---~ V L i n
Conjecture
VLin
is local,
VLi=~n has
T: VDiff(s) functor
3.4
every functor
T: ~,(s)
T: ~
)
> VLin
---> V L i n
are not c o n t i n u o u s functor
by c o m p o s i n g
the o b j e c t s of
VDiff(s)
and local.
functors which
some g i v e n f u n c t o r w i t h a non-
V L i D ---~ V L i n
Proof of 3.3.
is local and
is b o t h m y o p i c
N o t e t h a t it is easy to c o n s t r u c t
continuous
then it is m y o p i c .
.
Every functor
in R n
and continuous.
is local.
a u n i q u e e x t e n s i o n to a local m y o p i c T: M,(s)
be m y o p i c
[i] 1.2. c).
D u r i n g this proof we shall r e g a r d as b e i n g the o p e n u n i t d i s k s
, w i t h b a s e p o i n t at the centre.
Dn
-
Let
T
mt:
Dn
mt
is n e a r
1D
Fix
such
mt
) Dn = D
an Let
neighbourhood Cs
be
map
tative
-
continuous.
be d e f i n e d and
so
Tmt:
9:D,0
~
M,*
For
by
0 < t g 1 , let
mtx=
tx
TD ---> TD
.
For
is an
t
near
isomorphism.
.
of t h e b a s e
such
1 8 2
that
be a d i f f e o m o r p h i s m
point.
h ~ = id
on
h: M , *
Let mtD
o n to a
~
D,0
Then we have
.
be a
a commu-
diagram mt D,0
) D,0
mt I
Ih
D,0 In c a s e this Th
a)
above
follows are
since
~M:
Cs D,0
Given
with
and
let
T: ~
that
Cs
(T7) -I
small. that
T1
T
map,
, where
defines
.
In c a s e b)
that
T~
and
is local. d), w e
fix
for
a diffeomorphism of
the base
---> V L i n We put
point.
, we define
TM = TRm
.
If
we define 7:D,0
It is e a s y
T
To p r o v e
)
as f o l l o w s .
is a
To s h o w unique,
We deduce
n o proof. (M,*)
TD ~ TM
on to a n e i g h b o u r h o o d
N,*
r
so t h a t
It f o l l o w s
needs
---~ V L i n
~
M,*
is m y o p i c .
functor
Tf = T ( ~ I f ~ M 7 )
defined
c)
) M,*
T: M,(s)
me
T
based manifold
a local
f: M , *
= D,0
isomorphisms. Part
each
M,*
~ ~
to
) D,0
see t h a t
is e q u a l Tf
to
is w e l l -
a functor.
that
the e x t e n s i o n
and
T2
be two
of
T
to
extensions.
~,(9)
is
We define
an
1 ,
-
isomorphism to
a: T 1 ~
(Tz~M) (TI~M)-I
Corollary
of
V N
VNIM
.
If
V
3.3
the
VNIM
now
points 2.4,
aM:
TIM
) T2M
the p r o o f
is
follows local.
follows
of
M
proves
fact
. that
induces
an
T
V
is
that
from
The
natural If
M
of t h e
equal theorem.
myopic
is m y o p i c
that
the b a s e
bundle.
is an o p e n
Proposition
isomorphism,
vector
isomorphism
Cr
T
fact
by l e t t i n g
This
(r,s)
1.2).
, then
The
Cr
T
(see
inclusion
T = T(V)
is
b),
completes
be a m y o p i c
Proof. T
by p u t t i n g
This
is c o n t i n u o u s
, then
That
.
-
3.5
Let Then
T2
1 8 3
VN
subset
of
and
VM
local.
is o b v i o u s .
2.4.
By T h e o r e m
is i s o m o r p h i c
point vary
together
with
with
over
to
all
Proposition
is c o n t i n u o u s .
4. M Y O P I A
In this
Theorem
N VN
is an o p e n , then .
As
we p r o v e
4.1
Every M
section
natural
and
vector
closed
subset
M
is a r e t r a c t
M
varies
over
bundle
of
N
is b o u n d e d .
To
see
lection
of m a n i f o l d s
of t h e
.
this, same
is m y o p i c .
of the n o n - c o n n e c t e d Hence
all m a n i f o l d s
dim VM
V
let
VM
dimension,
manifold
is a r e t r a c t
of d i m e n s i o n MI,
If
m
of
,
M 2 , - . - , be a c o l and
let
M
be
- 184 -
their disjoint dim VM i .
For e a c h
of d i m e n s i o n
Lemma
union.
n
dim VM
integer
with
is an u p p e r
n > 0 , let
dim VP(n)
P(n)
bound
for
be a m a n i f o l d
maximal.
4.2
Let f,g: M and
Then
dim M 1 = n 1 , dim M 2 = n 2
1 U P ( n I)
let
, M 1 , P(n I) ~
f l M 1 = glM 1 .
Proof.
Then
f',g':
M 1U
M 2 U P(n 2)
Vf
We extend
f
Let
and
and
Vg
g
, M 2 , P ( n 2)
agree
over
M 1 .
to
P(n I) U P(n I) --~ M 2 U P(n 2)
(disjoint
unions)
by f i x i n g
a point
the t h i r d
summand
to this point.
in
P(n 2)
and s e n d i n g
Then we have the commutative
diagram
M I U P(n I) U P(n I)
/
P(nl~)
M I U
f'~M
M I U P ( n I) U P ( n I)
where summand second so
Va
~
is the c a n o n i c a l and
8
summand. and
I U P(n I)
-"
Both
V8
are
a
I-
inclusion
is the c a n o n i c a l and
2 U P(n 2)
M 1 U P ( n I)
avoiding
inclusion
8
have
isomorphisms
over
the third
avoiding
one-sided M 1 .
the
inverses
Hence
Vf = V g o v e r M 1 . F r o m the
lemma,
w e see t h a t w e c a n d e f i n e
a new
-
natural vector Moreover Let
V'
bundle will
V'
the n o t a t i o n
by
be m y o p i c
T = T(V) : ~,~s)
185
V ' M = V ( M U P(n) a n d so C o r o l l a r y
) VLin
of the p r o o f
and
let
of T h e o r e m
) IM .
3.5 a p p l i e s .
T' = T(V')
3.3, w e h a v e
.
With
a commu-
tative diagram
V ( D U P(n)
) V ( ~ U I), V ( M U P(n)
T VD
V~
and h e n c e
l
~
VM
a commutative
diagram
T'D
T
To _
the vertical
Corollary
3.5
composite
in the top r o w
are i n v e r s e both that
T'
is
and h e n c e
d i m TD = d i m T M
Theorem
T
Th )
maps
are
is t h e
T
~
VD
T'D
TM
C r , myopic
isomorphisms.
injective
Vh
T'~ > T'M ~
TD
In b o t h d i a g r a m s
) V ( h U i)_ V ( D U P(n)
T
TD
injective.
and local.
identity
It f o l l o w s
Hence
and
that
T'~ T~
so
V
is m y o p i c ,
the
and
and
t h e y are i s o m o r p h i s m s .
, and
By
Th
This
which
T'h are
shows
proves
4.1.
5. C O N S T R U C T I N G
Let We c o n s t r u c t
T: M,(s) an
(r,s)
NATURAL VECTOR
---> V L i n
BUNDLES Cr
be a m y o p i c
natural vector
bundle
V
functor.
using
T
.
-
If
M
is an n - d i m e n s i o n a l
and vector
space
T(M,m)
the
is
structure, ~:
~n
by
> M
on
over
insisting
-
manifold,
structure
fibre
1 8 6
VM
m
that
on to an o p e n
.
the underlying is
We
for
Um6 M T(M,m)
give
each
subset
point
VM Cs
of
M
set
, where
a differential diffeomorphism
, we
have
a
Cr
diffeomorphism ~: ~n given
by
~(x,w)
is t r a n s l a t i o n
= T~
by
x
To c h e c k defined, subset phism
we
of
let ~n
of v e c t o r
analogously
9 TYx
the
~n
, where
n
,
Yx"
differential
) M
We must
bundles
to
9 w
~ VMI~
~n'0
-'-> ~ n ' x
.
that
4: .
x T(~n,0)
be
show
over
structure
a diffeomorphism ~-I~
that ~-I~
is
(where
on
a
~
is w e l l to a n o p e n
Cr
isomor-
is d e f i n e d
~).
Now ~-l~(x,w)
where
y = -~-l~x
(~-l~x,Tyy
.
(x,xl)
>
from
definition
the
=
If
by
VflT(M,m)
is
Cr
= Tf:
is t h e We
3.1
f: M Vf:
same
now
Rn
The map
(yy~- I~Y x )x 1
Cs
is
.
Cs
is
Urn6M T ( M , m ) T(M,m) as
have
the a map
x Rn
>
~
Since
#-I~
that
> N
9 T(~-I~)
is
, we
9 Ty x
9 w)
mn
, given
T
is
Cr
by
Cr
, we
see
proof
that
Vf
.
define
> Un6 N T(N,n) T(N,fm)
proof
above
(in f a c t
.
The
that
~-I~
a functor)
is
Cr
.
-
~:
{myopic
C r functors
187
~,(s) 9
We
also 8:
have
the o b v i o u s
{(r,s)
natural
-
map
{(r,s)
by
taking
Theorem
5.1
the
vector
These maps
Proof. L e t us Then
e SV
vector
space
V
have
and
the
fibre
are
C r functors
have
structure
the base
inverse
It is o b v i o u s
V
vector
bundles}
bundles}
over
start with
natural
(also a f u n c t o r )
---> { m y o p i c defined
---> V L i n }
an
the
~,(s)
point.
bijections.
that (r,s)
8e = 1 . natural
same underlying
in e a c h
fibre.
same differential
> VLin}
To
structure,
vector
bundle
point
set and
see t h a t we
apply
e SV
V
and
Proposition
2.3.
6. T H E F I N E
Let For
T: ~,(s)
each based manifold
This
factorization
phic
to
6.1. convenient,
that
9 VLin
We may
be
(M,*)
we
> M
)
is n a t u r a l
T* ~ k e r ( T M
STRUCTURE
) T*)
and
have , so
the
Cr
functor.
factorization
. TM
is n a t u r a l l y
.
therefore
T* = 0 .
a myopic
assume,
whenever
it is
isomor-
.
188
Lemma
6.2
If
s = 0 , then
Proof. Let g
f:
In,0
--~
arbitrarily
We
suppose
~m,0
be
near of
zero
near
f
and
assume
Tg
to ~ 0 .
But
is a c o n s t a n t
T* = 0 .
such
in t h e
a neighbourhood enough
T
that
CO
this
Tf
the
g
3.3
is
We
such
local.
find
there
is
taking
T
fact
is
can
By
of
the
T
that
zero.
continuity
contradicts
b),
~ 0 .
topology,
on which using
By
functor.
g
, we may
that
T
is
local.
Theorem
6.3
There such
depending b)
the
tangent
For
the base
space If
from
based
the dimension is
isomorphic
M
to
--"
of
M,*
--~
if
t ~ k
By Theorem V D i f f (s) be
T0
the
results
N,* or
3.3 to
the
have
T
has the
degree same
Ttf
d),
we may
take
T
.
or
By
T
is
i ;
k-jet
then
VLin
some
;
k = s
inclusion.
[i]
for
= TM
if
functor. of
TrM
F i o ~ , where
f,g:
a continuous
By
of
VLi=~n ---~ V L i n
---> V D i f f ( s )
.
M,
Fi:
is o b v i o u s l y = 0
manifold
and
and
Proof.
I: V L i n
on
T i / T i-I
point
functor
each
only
c)
Cr
T1 c T2 c
that: a)
r
TO c
is a f i l t r a t i o n
at
= Ttg
to b e
.
a
Let T 6.1,
[2], w e
o I: ~ we may have
~ VLin assume
-
To
I = 8i>o
that
if
IW:
TW=
~. l>O
eral
not
Lemma
F i , where
Fi:
W ---> W
scalar
is s c a l a r
F i ( X W)
F.W 1
with
k < s
zero
k-jet
by
li
.
We
property I , then
have
Tf
f
, where
the by
sum decomposition
write
f
monomial 8 . Rn 1
0
will
in g e n -
is d i f f e r e n t i a b l e .
F.W 1
.
Suppose and
W 1
as
a finite
sum
of
degree
hence
f
by
order
(k + i)
is by
at
C s-k
(i + i)
loss
at
[ j / ( k + i) ]
T
generality
g
into
zero
~m ei
.
We
is
a
Rn
vanishes
that
g
that
at
Then k-jet,
k < s
. can
and
is c o n t i n u o u s ,
[(Tg) (x)] I ~ 0
of
with
by
of
for
3.
false.
if
, where
x.
is
i
I) > W2
which
, such
.
factorize
lemma
---> W 2 , 0
and
Since
g
the
co-ordinates
a polynomial
zero,
that
ZeiSi
in t h e
t =
, we write
1
l(k +
Rn
f =
a polynomial
without
We
k
by
vector
sends
where
F.W
f: W I , 0
, where
between
.
.l.~O
replace
8. 1
k = s
map
Tf
F iW2 c T W 2
x 6 T W = e.
We
Cs
a
Then
If
[Tf(x)] I ~ 0
> ~,0
be
zero.
if
x s FiW 1
approximate
assume
at ~ i=]
t = 0
in
exists that
order
has
multiplication
---> W 2 , 0
into
and
component
there
may
---~ V L i n
direct
by
f: W I , 0
FiW 1 c TW 1
such
the
be p r e s e r v e d
Proof 9 the
VLin
6.4
spaces,
if
is
-
multiplication
, but
Let
9j i=l
189
we
zero,
vanishes .
If
g
vanishes
can and
to
k = s
, we to
190 -
in
where
than Te
0
~
iN
0
~
i
TM,0
is a n e w v e c t o r ,
h ( x l , . . . , x n)
a monomial
8
each
entry
of w h i c h
is
in t h e
x.'s of d e g r e e g r e a t e r t h a n k (greater 3 k = s), a n d w h e r e e is a l i n e a r m a p . Now
i , if
F 3.~N
sends
We
to
have
Fj~m
for
each
a commutative
in
j .
Hence
0
[Th(x)] 1
diagram
In,0
0
h r
lq
where
q = k + 1
where
8
each
entry
is a l i n e a r being Since
know on
that I .
if
TS:
9
iN 0
k < s
and
map,
a power each
Ti N T liTh
taking
Fi
(Th
lq >
i , and
9 x) I = 0
Corollary
and
) of
is a p o l y n o m i a l
to t h e
above
9 x = Th
. T1
T8
k = s
by a scalar
10
depends
if
matrix,
X .
functor
[i]
in a p o l y n o m i a l diagram,
, and
we
see
9 x
.
9 x) 1
.
1.7,
fashion that
9 x
9 T(lq)
9 Th
components,
li(Th Since
q = i + 1
(possibly
=
Therefore,
IN,o
represented
9 Ti N
Applying
8
9 x) i = T8
this
llqT8
9 (Th
is p o l y n o m i a l
in
I , we must
have
is a c o n t r a d i c t i o n .
6.5
Defining
TJw
= e~ l = 1
F.W l
, we
obtain
a subfunctor
we
.
-
of
T
part
We may h
is
or
if
c),
assume zero. k
--
that
by L e m m a we need is
zero.
f + h
S(U
x U)
6.4,
Sh:
only
show
This
U,0
from
> W,0 Ttf
that
factors
6.3,
we need
c) b y p u t t i n g and
that
= Tt(f
the
only
k = 1 .
k-jet
+ h)
if
show
that
of
t ~ k
see
SU
x W)
> SW
that
+
~WxW
that we need
---> S ( W
~W
only
is i n d e p e n d e n t
is zero.
of
According
S(2) (iu,h) : S(2) (U,U) by a p p l y i n g
, since we
of d e g r e e
as
f•
>U•
follows
W ---~ S (2) (U,W) of f u n c t o r s
follows
show
S = T t , we
x h):
of T h e o r e m
9
U Writing
proof
b)
f,h:
We must s
the
because
Now
S(f
-
. To c o m p l e t e
prove
191
less
Lemma
know
that
than
t
6.4
this (see
h to
.
Now
[2]
9.1,
~ S(2) (U,W) to t h e
functor [2]
functor
is t h e
sum
9.4).
7. M I S C E L L A N E O U S
By n o w w e results
stated
1.5 and
1.6. We
5.1 a n d
T: V D i f f ( r ) that
in I.
first
Lemma ~
TO = 0 .
of g e n e r a l i t y ,
have
6.2,
completed
The
prove
only
VLin
with
We w i s h we m a y
to
only r > show
suppose
proofs
outstanding
Theorem
we need
the
1.5.
points
According
consider 0 .
of m o s t
By
Cr 6.1,
of t h e
are T h e o r e m s
to T h e o r e m
functors we may
assume
that
T = 0 .
Without
loss
(in t h e
notation
of T h e o r e m
6.3)
192
-
that
T = Ti
reduce W
.
There
But
(see
N)
each
Cr
loss
according
the direct
[2]
to T h e o r e m
as an
is a n a t u r a l (r - l,r)
integer f: 9
natural
and
) 9
T
We
choose
NxR ~
9 x RN
.
, we have
)
x 6 R
, y 6 RN
, where
-
terms ~X is
(fx,gx
9 y)
where
The exists
and
for a l l
Cr
~
is i n d e p e n d e n t We
choose
i
N~
,
For
diagram
> NR
V R ~ R x RN =
C r-I g =
maps
.
is
(x,~x
and
9 y)
for
(N • N)
We have Cr
and
gx
is an
(fx,f'(x) y)
diagram
f'(x)~(x) where
,
NR
r
NTf(x,y)
commutative
and
V
bundle.
is a n o n - s i n g u l a r
that
matrix,
bundle,
V
bundle
[NTf
know
We
7.1,
of c o p i e s
a commutative
r
fixed
between
is t h e t a n g e n t
trivialisations
In t h e s e
r
vector
[i]
vector
isomorphism
VR
which
number
natural
Vf[
(N • N)
and
(r,r)
map
=
6.3
to an
There
some
in s u p p o s i n g
T
V R -
Vf(x,y)
for
of g e n e r a l i t y
s u m of a c e r t a i n
corresponds
we can now
T (2) ( W , )
at t h e o r i g i n .
is an
matrix.
9.4,
space
considered N
with
T no
in
of t h e t a n g e n t
5.1).
where
then,
As
= 0 .
is t h e r e f o r e
is s i m p l y
(say
T i-I
i , by replacing
T = T1 . T
and
-
above
f: i --~
= ~(f(x)
leads R
to t h e
equation,
,
)g(x)
of
f
and
g
and
j
so t h a t
depends the
on
(i,j)
f
.
entry
of
-
~(f(x) k(x) x ~ be
)-iv(x) .
Let
0
.
a
is
h(x)
Then
Cr
non-zero = xr
h
is
function
f'(x) k(x)
= h(x)
193
for
for
x =
x a
cr-!
0
0
but
that
for
But
then
the
.
We
Cr
x
call
h(x)
and
not
such .
-
.
matrix
entry
= -x r
Let
near
this
f:
for ~,0
9
~,0
zero g(x)
, where
g
is
i
the
function
the
function
corresponding h(x)
to
, which
f
, has
is n o t
Cr
in
its
(i,j)
, and
this
In v i e w
of
entry
is
a
contradiction. We we
need
is
C~
only
Without
show
that
generality,
If
M
the
and
space
space
of
Lie
group
(under
of
replaced by
We by
jection without
M
need x N
with x N
loss
that
spaces,
the
space).
of
we
Theorem
6.3
to
yD~ff(~)
of
is
.
.
denote
by
differentiable
a finite
We
This
5.1,
T = Ti
origin, is
Theorem
denote
dimensional by
a finite
Inv
jim
dimensional
jets).
P • 0
> N,0
be
C~
.
We
have
to
N
are
be
deduced
composite
M
composing
at
i-jets.
• M,
restricted
ji(M,N)
composition
P .
i-jets
satisfying
assume
vector
origin,
P
T
when
we
a Euclidean
F:
the
3.1),
are
invertible
Let that
N
the
(in f a c t
of
1.6.
functor
of
preserving
C~
a
loss
the
is
Theorem
sense
manifold
show
prove
(in t h e
ji(M,N) maps
now
of
~
~
C~(M,N)
only
prove
, for the N
the
---~ L i n ( T M , T N ) this
general
injection .
We
generality.
when
can
M
both
case 9 M
therefore
can x N
M and then and
assume
the M
pro-
= M = W
,
194
We factor
C, (W,W)
C.(W,W)
,~ ) D i f f e o ( W
Ji(!,W)
~
=
to T h e o r e m
on the right with j: W
> W x W
q: W x W
Lin(TW,TW) ..
x W) ---~ Aut T(W x W)
is commutative,
(uf) (x,y)
(according
~
> W
Tj
Lie groups.
> Lin (TW, TW) .
6.3 c))
and
is p r o j e c t i o n
> W,0
a
) , 8 7
are defined
is induced by
is defined
and on the left with
Tq
as T
by c o m p o s i n g , where
8
on to the second factor.
is a n a l y t i c
7 and u' is
C~
follows.
since
it is a map of
are also analytic.
If
C = , then the c o m p o s i t e
P ---> C~(W,W) is o b v i o u s l y
~,8 and 7
is the injection of the first factor and
The maps
P • W, P x 0
and
(x,y + f(x)
We know that
The result
u
Inv Ji(Iw x W) ~
This d i a g r a m follows:
-
map b e t w e e n
> ji(w,w) finite d i m e n s i o n a l
manifolds.
-
195
-
REFERENCES
[I]
Epstein, D~ B. A., "Group Representations and Functors", (to appear in Am. J. Math.).
[2]
Epstein, D. B. A., and Kneser, M., "Functors between Categories of Vector Spaces",
[3]
[~o appea~].
Montgomery, D., and Zippin, L., "Topological Transformation Groups", Interscience
(1955).
196
-
SEVERAL NEW CONCEPTS: LUCID AND CONCORDANT THE C O N T I N U O U S
FUNCTORS,
PRE-LIMITS~
AND CONCORDANT
COMPLETIONS
PRE-COMPLETENESSt OF C A T E G O R I E S
by Peter Freyd
LUCID FUNCTORS
I
Given the c a t e g o r y special
a set-valued
El(T),
the c a t e g o r y
comma category Objects Maps
A functor
T: A
> S
is a s e t of
X 6 E 1 (T) lation, xi ETA i f: A i by
are m a p s
f: A
> B
S
of o b j e c t s S s S
that
a PETTY
a pre-initial in
and a map
a generating
set
B s A, y s T B
T.
(sometimes
set,
i.e.,
S
> X.
By trans-
{ <xi'Ai> }is there
is a m a p
The subfunctor
(In g e n e r a l ,
by
T" (B) =
(Tf) (x i) = y}.)
transformations
(T,V), V
{y 6 T B I ~ i , Note
any
Ai
given
generated
any c l a s s
- B
t h a t the c l a s s of n a t u r a l
functor,
if
s u c h t h a t for e v e r y
El(T)
(Tf) (xi) = y.
is all of
functor
we m a y d e f i n e T" c T
such that
A s A, x E T A .
(y,B)
has
t h a t for any
such
{ ( x i , A i)}
{ (x i,Ai> }
El(T)
there exists
such > B
to
is c a l l e d
there exists
then,
T, as the
of
(Tf) (x) = y.
" p r o p e r"'~ o u n ded, ") if there
we define
> S
of e l e m e n t s
(x,A),
(x,A)
such that
T: A
(I,T) , i.e. ,
are p a i r s
from
functor
is e m b e d d e d
in
-
KIV(A i) pi(~)
where
the
= n A (xi) . 1
i th
~
ni,B(f:
HAi
Ai
quotient
e
A
>
we
petty let
El(T)
the functor
> T = ni
if
is p e t t y
through
Let T
i.e.,
s~ch that Tu = T
the family of subsets
= {A' c A I
only s u b f u n c t o r s
a non-terminating of a n a t u r a l l y
U T
= T
are
e .
for each
6 ~ u, T o = T.
of
A.
Given
define
PK
Con-
the objects of
chain.
arising
functor
P: S
of
P
that is not
> S.
> B,
f: A
a chain of s u b f u n c t o r s
P (A)
is
(Pf) (A') = Tm(flA').
as follows:
for
to be such that
cardinality of
any
is petty.
for large
Then for
petty is the c o v a r i a n t p o w e r set functor
PK(A)
functors
this p r o p e r t y b e c a u s e
6 = sup ~(i).
An e x a m p l e
K
----> T
be the first ordinal such that
and c o n s t r u c t
each cardinal
A
ZI~
Conversely,
is n o t petty we may w e l l - o r d e r
We may c o n s t r u c t
from
where
the ordinals,
functors have
may
iff all chains of s u b f u n c t o r s
stationary,
~ (i)
s i 6 T 6 ( i ) A i. versely,
T
functors
(Tf) (xi) , is epimorphic.
S
in fact e v e n t u a l l y
i s I
of petty
functor of a s u m of r e p r e s e n t a b l e
ranging
Clearly,
the class of t r a n s f o r m a t i o n s
For petty
7H Ai
> B) =
T:
{T~},
~
map is d e f i n e d by
The c a t e g o r y
is locally small.
d e f i n e d by
-
projection
Consequently
be r e p l a c e d by a set. to
197
A' < K}.
In fact,
these
P, and all p r o p e r s u b f u n c t o r s
of
are the P
are
petty. A functor
T: A
) B
is called petty
if for e v e r y
198
petty set-valued A
T
S
>B
)S
functor
S .
tive terms
> S, it is the case that
is petty.
It is e a s y sentable
S: B
to see that it s u f f i c e s to c h e c k for r e p r e -
If one r e p l a c e s
the
this last c o n d i t i o n w i t h p r i m i -
" s o l u t i o n set c o n d i t i o n s "
as u s e d to b e s t a t e d
for the g e n e r a l a d j o i n t f u n c t o r t h e o r e m is obtained. n o w a d a y s , we
Of course,
replace that c o n d i t i o n w i t h the r e q u i r e m e n t that
the f u n c t o r be petty. R e t u r n i n g to s e t - v a l u e d functors, w e say that T: A
mS
~,B : P
is L U C I D
if:
(i)
T
(2)
For e v e r y
is petty. P: A
> S
and p a i r of t r a n s f o r m a t i o n s
it is the case that the e q u a l i z e r
~T
Ker(~,8)
is
petty. It s h o u l d be n o t i c e d that lucid is to p e t t y as coherent
is to f i n i t e l y generated.
The n e x t p r o p o s i t i o n allows
us to test for lucidity by r e s t r i c t i n g
P
to r e p r e s e n t a b l e
func tors~
Proposition
I. 1
A functor
T:
every representable
HA
the e q u a l i z e r
Ker(x,y)
Proof. we
is lucid iff it is p e t t y and for
>S
A
and t r a n s f o r m a t i o n s
x,y: H A
.> T
is petty.
Given petty
first use the p e t t i n e s s
of
P
and t r a n s f o r m a t i o n s
e,8: P
P
to o b t a i n an e p i m o r p h i s m
~T
- 199
ZH Ai
,P .
-
If
K9
I
I Ker(~,S) is a p u l l b a c k
then
the e q u a l i z e r
of
K"
is petty.
H At
>P
K'
~'~>T
is epi and
>Ker(s,6)
ZH Ai
But
~P
>P a ~ - ~ T
K' = 7K i'
.
where
It suffices K[
category
to the
category
orem is true, b u t the proof that
7K~
>K"
9 ~K(-~, ...,
ing from
in )
is epi. > K'
Proof.
x
and
x(l A)
to
of
functors groups
from an
G,
the the-
It is n o t the case
It is the case that K "l~
...in
sums of lucid functors
Given a pair of maps
y
is empty.
E TiA
, y(l A)
i
that
is the k e r n e l
aris-
.
is n o t in the same
T
e 'ive
is diffe:rent.
component
.
as
are lucid.
x,y: H A y(l A)
> ZTi,
if
the e q u a l i z e r
If, on the other hand, 6 TiA
then
is the same as the e q u a l i z e r H
is
1.2
Arbitrary
x(l A)
to show
is the e q u a l i z e r
of abelian
is epi w h e r e
HAil | " "eAin
Pro osition
> Z H Ai
. i
We m i g h t add here that for e' ive
K'
the e q u a l i z e r
a pair
of
x
and
of t r a n s f o r m a t i o n s
y
from
|
Again
this p r o p o s i t i o n
is true in the
e' ive
case,
of
-
but not the tors
the p r o o f .
correction.
Pr o P o s i t i. .o.n. __
under
and
fies
L
the
formation
isfies
L
Given
Since
I
it
2
condition
several able
closure
follows
holds
ways,
functors
Pro~0sition
of
of
lucid
1.7 to d e s c r i b e
property
on
functors
on
lucid
func-
~
)T 2
it follows is
that
is
for r i g h t
the easiest be petty.
since that
a subfunctor
Ker(e,~)
RIGHT
A
is
equalizers.
e, 8 : T i
Ker(~,S)
A category set
after
is :
subcategory
satisfies
T2 1
restriction
full
Proof. L1
other
until
1.3
The closed
-
We shall wait
The o n l y
true without
200
limits. that
This
satisfies
Ker(e,8) of T
satis-
and
1
satisfies
PRE-COMPLETE
being
T i
T
1
sat-
L z. l
if the s o l u t i o n m a y be
left-limits
rephrased
of
in
represent-
For example:
i. 4
PRE-COEQU~LIZERS
exist
if
representable
functors
are
lucid.
Proof. exist" set
that
{B
for
)C. }
-
We mean by the phrase any p a i r such
of m a p s
"pre-coequalizers
x, y: A
)B
there
that
i
PCE
(1)
A x -B
)C i = A
PCE
(2)
For
B
any
)X
Y -B such
>C i that
,
all
i.
exists
a
- 201
A
x
Y
> B----> X = A
-
there e x i s t s
X
) B B
~
a triangle
Ci for some
i.
X
H x , Hy : H B
The e q u a l i z e r of the t r a n s f o r m a t i o n is a s u b f u n c t o r
of
H B, g e n e r a t e d by the s e t
Conversely,
{B
g i v e n that the e q u a l i z e r
> Ci}.
of
H x, Hy
is p e t t y w e may r e v e r s e the a r g u m e n t to o b t a i n a p r e - c o e q u a l i z e r for
x, y. I
Proposition
i. 5
(Finite)
P R E - C O P E O D U C T S e x i s t in
ducts of r e p r e s e n t a b l e
Proof. exist" set
(finite)
~ Pj}
iff
(finite)
pro-
are petty.
We m e a n b y the p h r a s e
t h a t for any
{A i
functors
A
family
" (Finite) p r e - o o p r o d u c t s
{Ai} I
such t h a t for any
t h a t there e x i s t s ) X}
{A i
a
there e x i s t s
I• a
P.
> X
some
j
such that
Ai
~ Pj
all
i.
S u c h a set
X
is c l e a r l y s e e n to be n o t h i n g more nor set for
Corol!ary
HHAi
less t h a n a g e n e r a t i n g
9 l
i. 6
(Finite) petty functors
p r e - c o p r o d u c t s e x i s t in
are petty.
A
iff p r o d u c t s
of
202
Proof.
Let
{T i}
-
be a family of n o n - e m p t y
pet-
i6I ty functors.
~IEj. HAj 1 {k: I
For each
>~T i
is epi.
-~l j } "vI i
ZK~I HAk(i)
i
where
Let
k(i)
~ ~IEJiHAj
Proposition
let
Ej H Aj l K
6 J. 9 1
nite products demonstrated
Then there is an epi
their pettyness.
transformations.
K~
of
left-
K1
that fi-
The last c o r o l l a r y to verify
, (81,52>:
of the maps and
L Z. P
~i,82:
<81,B2>
Let >TI• T 2 P
>T 1
is the equal-
>P~
>T 2 and K1 >P~ >T 2 - l 2 2 case finite p r e - p r o d u c t s exist and hence
of lucid functors
The e'ive v e r s i o n
are lucid. A H
that any map
> E T i factors
subsum.
An observation, We'll
<~I,e2>
<~,~2>
of 1.2 is proved by o b s e r v i n g a finite
It remains
The e q u a l i z e r
In the @'ive finite products
are lucid.
petty and
The e q u a l i z e r
izer of the pair
B 6 B
in light of 1.3 to prove
of lucid functors
P
then finite
exist in A
are lucid.
It suffices
be lucid,
ing here.
functions
1.7
Proof.
through
Then
. I
limits of lucid functors
is petty.
be epi.
be the set of choice
If finite p r e - c o p r o d u c t s
T I, T 2
~T i
say
there exists
A a set
though e n t i r e l y
formal
is P R E - C O R E F L E C T I V E {Ai---->B} A i E A
in
is worth makB, if for every
such that for
-
every
A 6 A, A
>B
2 0 3
there exists
-
a triangle
A------~B
A• Proposition
1.8
m a y be fully e m b e d d e d as a p r e - c o r e f l e c t i v e c a t e g o r y of a f i n i t e l y ly left p r e - c o m p l e t e
left c o m p l e t e
iff A
category
i~
has p r e - e q u a l i z e r s
A
sub-
is finite-
and finite pre-
products.
Proof.
>clear <--=By the last p r o p o s i t i o n s
lucid f u n c t o r s
from
A op
Yoneda embedding embeds
is f i n i t e l y A
the c a t e g o r y of
l e f t - c o m p l e t e and the
into it by 1.4.
To r e m o v e the " f i n i t e l y " ' s we m u s t do m o r e work. ticularly,
we m u s t k n o w w h e n a r b i t r a r y p r o d u c t s
tors are lucid.
First,
We k n o w that a r b i t r a r y It remains
however, we c o n s i d e r
In the ~'ive case this is easy.
exact,
S,T
is p r o j e c t i v e
let
the r i g h t - h a n d side.
of lucid f u n c t o r ~
lucid.
lucid,
of lucid func-
sums of lucid functors are lucid
to show that c o e q u a l i z e r s
~--~
C
Given
S
(1.2). are
>T-->C--~ 0
be any t r a n s f o r m a t i o n ,
and we m a y lift to o b t a i n a t r i a n g l e
!%
Par-
- 204 -
>H A
K
L et
~
K
be a p u l l b a c k .
>~
is the k e r n e l
of
HA
>C.
S----~T But
is a l s o the k e r n e l
K
of a m a p
S 9 HA
and hence
>T
K
is
petty. F o r the s e t - v a l u e d
Proposition
1.9
If lucid
A
functors
has
T
petty.
>C
into
the d i a g r a m
L 2
T.
lucid,
for
C
of
then coequalizers
S
and
>T 8
.
we first display
symmetric
on the s y m b o l s
is
e,8: ~
C
ana transitive
The construction
typified
the w ~ r d
be
the c o e q u a l i z e r
For each word
where
S,T
of a r e f l e x i v e ,
formafions ple.
Let
To v e r i f y
equalizer
finite pre-coproducts
of
are lucid.
Proof. tions,
c a s e w e m u s t do more.
is f a m i l i a r
"~8"
and
"8~"
transformaC
is e a s i l y as the co-
p a i r of t r a n s to t o p o s - p e o we consider
by:
(~8) (Be) (Be).
We
let
K
be the l e f t - l i m i t W
of the d i a g r a m the l e f t - m o s t
(K~ = T) K
.
the word
c o p y of
Note that
is petty. W
for
If w e
w, K~ P1 >T
T, K ~ P2 ~ T w
under
the
sum over
the
stipulated
to the r i g h t - m o s t
hypothesis
all w o r d s
of
s
map t o copy.
propositions
~ = ~K
and W
consider
-
-
205
P1
the p a i r clear
of i n d u c e d
that
x, y s TB exists
P])
which
we
K
T
of
K
)
P2>
T
to the s a m e
that
P2
_
T, K
T,
> C = K
such
obtain
into
are s e n t
z 6 K--B
the i m a g e i.e.,
K
maps
I- C
~
and
element
in
that CB
T
it b e c o m e s for e l e m e n t s that
(Kpl) (z) = x, K ( p 2) (z) = y.
T x T
is
the k e r n e l - p a i r
of
there
Hence
T
> C,
a pullback P2
and
K
is p e t t y
mations ~, y
from
HA
is p e t t y .
K
; T
T
~-
(in fact, to
C.
We may
C
lucid). We wish
lift each
Now
to s h o w
let
E
be
the
~, [
that
be
transfor-
the e q u a l i z e r
to a t r a n s f o r m a t i o n
into
HA
HA
T
let
~
C
left-limit
T
> C
of KA
P
xlly 1
K
T P
By E
the h y p o t h e s i s ) HA
2
of the p r o p o s i t i o n ,
is the
desired
We may
collect:
equalizer.
I
E
is petty.
But
of T:
-
Propositlon
206
-
i.(i0)
If
A
of contravariant
has finite pre-products lucid-funcs
limits and arbitrary If
A
able functors
then the full category
is closed under finite
right-limits.
furthermore
has pre-equalizers
then represent-
are lucid and the yoneda embedding
as a full pre-coreflective
subcategory
travariant
I
lucid functors.
are closed under arbitrary
the yoneda embedding
displays
A
displays
of the category
We shall now find the conditions functors
left-
of con-
under which
left-limits
lucid
and under which
as a pre-reflective
subcategory. The first is the problem of finding when arbitrary products
os lucid functors
Proposition
are lucid.
i. (ii)
If products of representables
of representables
are petty and powers
are lucid then products
of lucid functors
are
lucid. Proof. sentable's {Ti} I
We have already
petty implies products
be a set of lucid functors
seen that products
of petty being petty. x, y: ~
mations.
We wish to show that the equalizer
For each
i 6 I
pi x, ply: HA--->
let u: K i T i.
>~
of repre-
>~T i of
b~ the equalizer
Then the desired equalizer
Let
transfor-
x, y of is the
is petty.
- 207 -
intersection
of the
K's.
d e f i n e d by
r i = uiP i .
the d e s i r e d
equalizer.
where
P(i,j)f
f, g
cid.
HI• HA
The c o m m o n Consider
the
common
and
ri:
equalizer maps
= r.3 "
equalizer.
P
>H A
of the
f,g:
Then
P
P
r.'s
is
l
>gi~iI-IA
the e q u a l i z e r is petty.
is l u c i d t h e n e v e r y p r o d u c t
be
of
Hence
of l u c i d s
if
is lu-
I We h a v e
sentables ducts.
s e e n t h a t the p e t t y n e s s
is d i r e c t l y
Powers
ther c o n d i t i o n pairs
P = HIK i
= r i , P(i,j)g
is the d e s i r e d
the p o w e r
Let
of m a p s
equivalent
to the e x i s t e n c e
of r e p r e s e n t a b l e s
are
of r e p r e -
of p r e c o p r o -
l u c i d if w e h a v e
t h a t for any s e q u e n c e that there exists
of p r o d u c t s
{(fi,gi):
a solution
B
set
the fur-
>A}i61
of
{A---~C. } ] j
as
follows :
(i)
B
(2)
For
fi
>A
>C
A---->X
]
= B
~
such that
= B J-~ A ---~ X
all
A B
i
>C. ] fi
,A
all i, j. >X
there exist
some
W e can f u r t h e r of sets of p a i r s {f.: B 1
>A}
reduce
of m a p s we h a v e
we d e f i n e
the c o n d i t i o n sets of maps.
an E X T E N D E D
j.
so t h a t i n s t e a d Given
PRE-COEQUALIZER
a set
to be a
- 208 -
set
{A
>C.} 3
such that
(i)
B
~i > A
(2)
For
A
= B ~
>~ >X
fk
= B
such that
A ---~ X
A
all
>A B i,k
-~ f~
all i, k,
>A
>X
there exists
>C Some
It s h o u l d be c l e a r is p r e c i s e l y
a generating
set of t r a n s f o r m a t i o n s
Theorem
j
t h a t an e x t e n d e d
s e t for the c o m m o n
j .
pre-coequalizer equalizer
for the
{Hfi}.
1.(12)
The f o l l o w i n g
are e q u i v a l e n t .
(1)
A
is r i g h t p r e - c o m p l e t e .
(2)
Left-limits
(3)
Products
(4)
Representable of l u c i d
(5)
of r e p r e s e n t a b l e
of r e p r e s e n t a b l e functors
functors
are
functors
functors
are petty.
are lucid.
lucid and left-limits
are lucid.
has p r e - c o p r o d u c t s
and e x t e n d e d
pre-coequal-
izers.
Proof.
(i) < = > (2)
>
(2)
by definition.
(3)
one n e e d o n l y r e m e m b e r
lucidity
(I.i).
the t e s t for
-
>
(4)
last p r o p o s t i o n .
(4)
>
(2)
by f o r g e t t i n g .
v e r s e we k n o w
for
5.
that
simple
are lucid.
cid f u n c t o r s are l u c i d
That
it s u f f i c e s
(1.6)
it s u f f i c e s
are
imply
imply
it s u f f i c e s
lucid.
to c o n s t r u c t
set u s i n g
that r e p r e s e n t -
that p r o d u c t s
equalizers
to show that p r o d u c t s
of lucid
of lucids
of lu-
functors
are l u c i d
to s h o w that p o w e r s
By our c o m m e n t s
after
of
the last p r o o f
for any set of p a i r s
{(fi'
solution
Since
For the con-
is easy.
(5)
(1.4) P r e - c o p r o d u c t s
are petty.
representables
(4) - - >
pre-coequalizers
and by the last p r o p o s i t i o n
a
-
(3)
Now
ables
2 0 9
gi >: B
)A} I
the e x i s t e n c e
of e x t e n d e d
precoequali-
zers and p r e - c o p r o d u c t s .
For e a c h
zer of {r:
I
fi' gi"
r, and izer of m E Mf,
f 6 Lr {A
let
K
{A
For each
be a p r e - c o p r o d u c t r let
{Pf
>Dm} Mf
>Cr (i)---->Pf } I "
of
> Dm}ms
>Cj}
i).
a map The
J6Ji
be the set of c h o i c e
E Ji } .
f E L r, r E K
(independent {{{{A
Let
> UJilr(i)
{C r (i)-"~Pf }i •
i,
A
r 6 K of
be a p r e - c o e q u a l i functions let
be an e x t e n d e d
There
For e a c h
{C r(i) } I"
exists,
then,
precoequalfor each
>D m = A--->Cr (i) --->Pf
>Dm
set
} rEK }
is the d e s i r e d
solution
set,
- 210 -
as f o l l o w s :
(i)
Choose
i, r E K, f q Lr, m E Mf.
T o show:
gi B,
>A
But
A
)D m = A
and
B
fi
(2)
-A
=
B
)A
gi
J>Cr (i) = B
fi
A
~X
>A
)X
r E K, f E L r, m
A
W e k n o w t h a t for e a c h
>D m.
,Cr(i)--->Pf---->D m
Given
B
to f i n d
>D m
-~A
> Cr (i)
9
such that
=
6 Mf
B--
gi
)A
>X
all
i ,
such that there exists
~ Dm
i E I
there exists
j E Ji
and
A -----~C j
Choosing
a single
that there
j
for e a c h
A
> C r (i)
i
we define
r E K
such
exists
all
i.
-
For the collection
[Cr(i)
and triangles
Remembering from
A
211
-
~X}
there exists
Cr (i) "-'>'if
that there
to
X
independent
of
that
m
A
>Cr (i)
~!m
>Pf
>X
Hence by the definition of Pf
there exists
.
is only one map under discussion
we note i.
f 6 Lr
and
is
{Pf---~D m}
>D
.
I
Theorem i. (13) may be fully embedded as a p r e - c o r e f l e c t i v e subcategory
(simultaneously pre-reflective)
category iff
A
is left
(and right)
of a complete
pre-complete~
Proof. <
clear. =>
The embedding
gory of contravariant
is, of course,
lucid functors,
Lucid
into the cate(A) ~
last theorem showed that the left pre-completeness implies the left-completeness demonstrated
of Lucid
its right completeness.
ly stated theorem we suppose that
T
(A~
The of
1.2 and 1.9
For the parentheticalis a lucid functor.
-
We seek a set ists
i
a
-
such that for any
{T > H A l }
and
212
triangle
T
>H B
A transformation
T*
is an element of the Isbell c o n j u g a t e ated at
B.
a generating precisely
carries
w o u l d be n o t h i n g more nor less than
set of elements
for
functor
is a
lim --->
T*
lim's~ into ~im's, hence
was c o n t r a v a r i a n t ,
NOW
the right p r e - c o m p l e t e n e s s
T*
is
lucid,
T*
.
Thus we w i s h to show
of a lucid functor
of representables,
(T
proof.
w h e n evalu-
And
that the Isbell c o n j u g a t e
Every petty
there ex-
is
T*
is petty.
conjugating
is a lira of representableso
a (l i m o f c o v a r i a n t
representables~
and the last t h e o r e m finish
a fortiore,
it
is
petty.
the
~
II THE R I G H T - C O M P L E T I O N
Let
B
be a r i g h t - c o m p l e t e
class or super-class
category
of r i g h t - c o n t i n u o u s
and
functors
~
any set,
from
(range not fixed). Let
C c B§
be the class of maps c a r r i e d
morphisms
by every functor
fractions
o b t a i n e d by f o r m a l l y a d j o i n i n g
in
~ .
B,
the maps
~AI, A
>AI>
>AI, A
~AI>
in
~
We recall that objects from
B
to
where ~
A
.
of
~
~
>A2>
be the c a t e g o r y
of
inverses of the maps
are the same as those of
are r e p r e s e n t e d
A---~A 1 6 ~ >A2, A
Let
into iso-
by pairs
subject to the c o n g r u e n c e if there exists
-
2 1 3
-
A-->A. all sides of w h i c h
are
in
C
such that
A 2 --~A 3 B---~AT-~A 3 = B given
)A2--->A 3 .
)B l , B
>B 1 )
Composition
and
is d e f i n e d
~ A I, A
- A I)
as f o l l o w s :
w e let
B--~B
be a p u s h o u t ,
noting
that
AI--->A 2
C
is in
and define
A~-~A 2 the c o m p o s i t i o n Proposition
any
such
that
any
G: B
>B
>A2,
A--->AI--->A2>.
2.1
B Given
as
B_,---~
is right-compl_ete, F: B - - - ~
~
in
~
there exists
~ _B ---> _E = F, F ; C
there exists
right-continuous.
and n a t u r a l
unique
is r i g h t - c o n t i n u o U s
transformation
unique extension
F: B ---~ E
F
and given
F
GIB
> G~ m
Proof. continuous.
Given
lira T r e m a i n s
collection C
all
T: D
{(TD--->
> X D, , X
to f i n d
such that
first that
~ B, D
the r i g h t - l i m i t
XD,
D, s u c h t h a t
(TD' wish
We observe
(lira
>
T ~
of
T: D - - - >
---> TD
X D, >
XL,
all
D'
with
X
> D
is r i g h t to s h o w t h a t
B - - - > B.
> X D, X
X ~
lira T ---> XL,
> B
small we wish
X----> X D > } D s D
B
Given
X ---> X D
any
in
) XD> in
XCD
, we
XIL), X ~
XL
in
> XL>
- (TD --~
C
XD, X
> XD>
-
all
D.
{X ~
We d e f i n e XD}.
first
Because
it is the case
that
X'
f: D' ~
D
-
as t h e r i g h t - l i m i t
each
functor
XD ~
X~
X each
214
~n
is r i g h t - c o n t i n u o u s
and
X ~
X'
are in
Xf (as m u s t e x i s t
~ X D, ---~ Xf X
of the g i v e n e q u i v a l e n c e s ) .
~
XI
Let
be a Xf --~
note
e_hat
the r i g h t - l i m i t family =
TD I ---~ X L
w e seek. (TD --~ map
To see
XD --~
XL} D
that
X ---> XD)
XL
in
C
C
> Xf' }.
{X
in
~.
TD' Hence
( l i m T ---~ XL, (TD ~
to be a
---~ T D ---> X L there exists > X~)
is the p a i r
X L, X ---~ XL)
that we have
such that
XL
We n o w h a v e
X
lim T ~
recall
XS f
and d e f i n e
such that
D' ---~ D
The p a i r
XD,
is in
of the f a m i l y
all
l i m T ---~ X L.
XfJ
X--~
{TD ---~ X---~
For
be s u c h t h a t
TD ~ ---~ TD ---~ X D ---~ Xf = TD I
pushout,
C.
---~ X D
let XDI --~
because
of the f a m i l y
TD ~
for e a c h
D
X L = T D ---> X
a > X
-
We h a v e
Any
functor
right
limits
tinuous. {A i
fami ly
F
shown
that
I! i
is a w e a k - l i m i t
lim T
from a right-complete
into weak-right-limits
We n e e d o n l y s h o w
9 B}
category which
is,
in fact,
in
L, B ~
Let L
L
be
the
B.
carries
right-con-
th a t a j o i n t l y e p i m o r p h i c
is c a r r i e d i n t o such. BI, B ~
--~ D
family
lim
the a s s o c i a t e d
of the
maps.
FL
XL~
215
-
is the w e a k - r i g h t - l i m i t
of the
FB} liAi
-
family
Given
a
FB
pair
f,g:
all
i
FB F ~
FB ~
X
such
there e x i s t s FL ~
continuous existence
and
to)
FA i ---~ FB
F L --~ FB ~
of the o r i g i n a l
is e q u i v a l e n t
observing
a map
X = f
epimorphism
that
u = vo
X
such
FL---~
family Hence
{A i --~ Fu = Fv
of
that sums e x i s t e a s i l y
(because
object
of c o - e q u a l i z e r s
in
B
let
B
joint
implies
and
B---~
by B
first is r i g h t -
B).
> Al ~
Let
(and
f = g.
For the
be given.
A2, A ---> A2)
B}
from
A (B ~
The
is p r o v e d
comes
FB q ;
that
X = g.
The r i g h t - c o m p l e t e n e s s
and e v e r y
X = FA i ~
be
and
a pushout
A2---~ A 3 in
B, n o t i n g
that
of the g i v e n p a i r A co-equalizer (B - ~
of
in
C
as
(B ~
(B ~
is in
C.
We m a y r e n a m e
A3, A 3 ~-~ A3> would
(A ~
the maps
A3, A ---> A 3), (B ~-~ A3, A ---~ A3>.
serve
and
as a c o - e q u a l i z e r
A3, A 3 ---> A3>
that the B - c o - e q u a l i z e r
of
(f,g>
in
E
for the
is an i s o m o r p h i s m . remains
We
a co-equalizer
B.
Given of
> A3
A3, A 3 I.~ A3 >
given pair since know
A
B
we
is c l e a r l y
define
F: B
> E
forced.
F(f/g)
= FB ~
Given
FA 1
its d e f i n i t i o n
(Fq)-1 -
AI, A ~ FA.
on the o b j e c t s AI> , g
in
The u n i q u e n e s s
X
-
is clear.
That
F
preserves
216
-
sums
and c o - e q u a l i z e r s
is e a s i l y
em
proved
along the
Given
lines of the last paragraph. I
a category
A
category of functors w h i c h
representables.
right-continuous define in
E
C
extensions
and the category
o b t a i n e d from
to
we
E
of c o n t r a v a r i a n t
Small(A ~
is
a free
to be the class of all
of r i g h t - c o n t i n o u s
C o n t ( A Op)
functors
on
A,
by functors
as the category of fractions
by i n v e r t i n g
is r i g h t - c o m p l e t e , full embedding;
right-complete,
the maps in
C, then
A
and every
> Cont(A ~
is a
right-continuous
may be e x t e n d e d
right-continuously
C o n t ( A Op) , uniquely up to n a t u r a l equivalence.
Proof. the
lim's
2.2
right-continuous E, E
as
carried into i s o m o r p h i s m s
Small(A ~
Cont(A ~
A ~
be the
S m a l l ( A Op)
observed,
If we define
as those maps
Proposition
appear
As U l m e r h a s
right-completion.
let
All b u t one s t a t e m e n t
last proposition. first observe
HA: ~ . . ~
~op
isomorphism. H A ---m T ~
As for the p r o p e r t i e s
that if
is in Hence HA = 1
(F--~
T, H A ---~ T)
(F --~
T --~ HA,
I}.
follow d i r e c t l y
HA ---~ T that
(T,HA)
there exists and
T
is in
~
A ~
is in
is an
such that C.
Given any map
we see that it may be renamed In p a r t i c u l a r
C o n t ( A Op)
then since
---~ (HA,H A )
T ---~ H A
- HA
of
from
the maps
in
as
C o n t ( A Op)
-
from to
HB
to
HA .
again
HA
That
217
are r e p r e s e n t e d
the c o r r e s p o n d e n c e
f r o m the s p l i t t i n g
of
to s h o w
given
that
T: D - - - ~
lim H T D ---~ H A
is in
The v e r y d e f i n i t i o n
n o t be
locally
say t h a t every
A s A
such that
is o b t a i n e d we wish
in
Cont(A~
that
it
it is. I
is t o o big.
It n e e d
completion.
functor
T: D 2
> A
P A I R OF D I A G R A M S
we
if for
function >
l i m (TD,A) <
D_I
By d i r e c t l y
if for e v e r y
C.
FUNCTORS
and
_D2
saturated
in
Small(A ~
to f i n d a s m a l l e r
is a S A T U R A T E D
lim (TD,A) ~---
is an i s o m o r p h i s m .
> T
li.~ T = A
insures
C o n t ( A Op)
HB
follows
HA
as a m a p in E
from
Cont(A ~
small
of
D1 c D2
the i n d u c e d
for
A ~
CONCORDANT
small
(DI,D2,T>
of
We w i s h
III ~
Given
> T
to s h o w t h a t
small.
AoP
is an i s o m o r p h i s m
suffices
For m a n y p u r p o s e s
in
is o n e - t o - o n e
A, D
It c l e a r l y C.
by maps
HA
The r i g h t - c o n t i n u i t y as f o l l o w s :
-
family
translating,
then,
the p a i r is
{TD---~ A } D s D --]
TD
TD , / ~ all
D
family
> D' s D I. {TD---~
There exists
A}D6Dz_
such that
a unique
enlargement
of the
-
2 1 8
-
TD
TD'
all
D
> D' s D2,
If e i t h e r
lim TD
or
1Lira TD
-Z
e x i s t in
A
then the
pair is s a t u r a t e d if and only if they both exist and are equal.
A functor carries
F: A
s a t u r a t e d pairs
(D I,D 2,T>
saturated
Proposition
:
> B
>
(D I,D 2,FT>
saturated,
3.1
A
right-concordant
Proposition
if it
into s a t u r a t e d pairs:
Right-concordant
If
is RIGHT C O N C O R D A N T
and
B
functors
are right continuous.
are complete
then
F: A--->
B
is
if and only if it is right-continuous. I
3.2
Representable
Products
functors
A~
of l e f t - c o n c o r d a n t
> S
are left-concordant.
functors
are left-
concordant.
Equalizers are
of maps b e t w e e n
left-concordant
functors
left-concordant.
If
A' c Lucid
(A ~
contains
the r e p r e s e n t a b l e s
and
219
-
if all f u n c t o r s embedding
A
in > A '
Proof. A
> A'
A oP
>
A'
are
left-concordant,
For the
.(-,T);
T h a t i t is s u c h is
t h e n the Y o n e d a
is r i g h t - c o n c o r d a n t .
last s t a t e m e n t w e n e e d
is r i g h t - c o n c o r d a n t (A,)op
-
S
is
only note
that
if a n d o n l y if left-concordant
the d e f i n i t i o n
for e v e r y
T 6 A'.
of the
left-concordance
(F ~
P ') 6 L u c i d
of
T! Lemma
3.3
For such that
A
pre-bicomplete,
the I s b e l l
conjugate
a nd for any l e f t - c o n c o r d a n t it is the case
that
Proof. sentables.
such that
We
Since
~HA. l
(F,T)
F'* ----> F*
T: A ~
~
first display
it is p e t t y w e
> F'
is onto.
P
HA
is an i s o m o r p h i s m
S
---> (F,T)
(A ~
(lucid or n o t ) ,
is an i s o m o r p h i s m .
F
as a
lim
can f i n d a s e t
For each
of r e p r e {HA. ~ l 6 I • I
F'} I
we
let
> HA i
~
F'
J be
a pullback.
Because
F'
is l u c i d
we know
that
P
is p e t t y
be s u c h
that
[KHAk
> P
(1.7).
is onto.
and Let
Define
A
has pre-products {H~ D --I
; P }Ki,j to be the
-
category k --~
whose
objects
i, k --~
j
sending
i 6 I
we
compose
HA k ">
for
to
>
Similarly lim
(D 3 ~
j.
we
and with
K. ~,j
We
k E K i,j-
A i, k 6 K i,j
Pi,j
k
-
I U U
for e v e r y
by
Similarly
are
2 2 0
to
~
HAi
lim
(DI---~ A)
may
and
find
maps
define
A k.
D 1 ---~ A
Given
determine
K--~
a map
i s D 1
A k --~
A i.
= F'.
D 3 ---~ A
such
that
A).
For every
i 6 D 1 HA
we
may
.
choose
>
F r
)
F
c(i)
s D3
and
i
HA
c(i) We
define
union D 1
of
D2
to be
the
category
whose
objects
those
from
D 1
and
whose
maps
and
D 3
i s DI,
and
k ~-~
j --~
together finally c(j)
the
union
D 3 --~
to
D2
no lim
assumptions (D 2 --~
A)
on =
new
maps
necessary k 6 K i,j.
of > A
the
We
i
>
define
one
disjoint from
for each
k ---~ i D z --~
embeddings
c(i)
F' ---~ F
those
c(i),
compositions
two p r e v i o u s
with
the m a p
i --~
are
the
going
to
it is e a s i l y
_A
;
c(i),
by
D 1 ~
A,
r(i).
With
the
case
that
F,
D I, D 2 an i s o m o r p h i s m .
the
for
extending A
with
D3,
are
is I
a saturated
pair
because
F'*
---~ F*
is
-
We c o n s i d e r define from
E
every
-
again the category
Small(A ~
as the family of all r i g h t - c o n t i n u o u s
Small(A ~
functors
2 2 1
which
from
functor
A, C in
are e x t e n s i o n s
as the maps
E
and
o b t a i n e d by i n v e r t i n g C
isomorphisms
conjugated.
when
consists
into i s o m o r p h i s m s
by
the category of fractions
the maps in
last proposition,
functors
of r i g h t - c o n c o r d a n t
carried
Conc(A ~
and
C.
By the proof
of the
of those maps w h i c h become
P r o p p s i t i o n 3.4
Conc(A ~ right-concordant A ~
E, E
uously
is r i g h t - c o m p l e t e , full e m b e d d i n g
right-complete,
to
Conc(A ~
Proof.
and every
> C o n c ( A Op)
is a
right-concordant
may be e x t e n d e d
; uniquely
right-contin-
up t o n a t u r a l equivalence.
As for 2.2.
Conc(A ~
Theorem
A
need not be
locally-small.
But.
3.5
If
A
is p r e - b i c o m p l e t e ,
then
Conc (A Op)
is locally-
e
small.
Proof. obtain
an i n d u c e d map
is the c a n o n i c a l where
Given any c o n j u g a c y - i s o m o r p h F'
F ---> F**.
> F** F'
the second map is the inverse
such that
F---~ F
> F'
F'
we > F**
> F** = F' ---> F'** ---~ F** of the i s o m o r p h i s m
- 222 -
F**
---~ F ' * * .
Lemma
3.51
(G only
if
>
G---~
FI,
F 1 ~
Proof.
= G--~
F--->
F 2 --
We
F**.
F**
FI> =
- (G--->
G
suppose
>
F2,
F2 ~
F 2)
that
G ~
the
pushout
F 1 ~
F ~
>
F 2 --~ that
the
Fn
>
of
induced
F 3 --~
map
F**
F 3 ---~ F * * ~
Fn ~
F 3
>
F4
>
F 1 ---~
Hence
G
(G ---> FI,
F --~
FI>
left
exercise.
an
The informs of G --~
us
(G,F**) F**
lemma
that as such
F**
=
isomorph.
as
F 3 ~
F**, is
- (G - - ~
easily
is n
=
such
F4 = G F2,
proves
Let
seen >
to F2
F ~
F2).
the
theorem.
(G,F)
as
defined
in
Conc(A ~
defined
in
Lucid
(A ~
, namely
there
exists
that
and
F**
F I ~
and
note
the
image
F3
that
1,2.
easily
if
F**.
first
Consider
F--~
F4 be
be
a conjugacy-
>
F4
The
and
converse
Indeed, is those
is
it
a subset maps
!
G
J F
where
F ---~
monomorphic.
F
1 J
is
'
**
a conjugacy-isomorph
and
F I ---~ F * *
is
-
Proposition
-
3.6
If closure
2 2 3
A
is small,
(A_~
in
Conc(A ~
of r e p r e s e n t a b l e s ,
Proof.
Given
in the left-closure
F E
(A ~
is the full leftand is reflective~
let
be its r e f l e c t i o n
of the representables.
conjugacy-isomorphism
F
> F
is a
and for ~ly c o n j u g a c y - i s o m o r p h F--~
there exists
then
F1
unique
F
> F
Hence
(G,F)
is defined
in
the same as
(G,F)
as defined
in
1
F
Conc(A ~
is c a n o n i c a l l y
(A Op ,S). I
Corollar[
3.7
If concordant
A
is small
functors.
Conc(A ~
is the category
I
We consider now a very special
Suppose case
Conc(A ~
SYMMETRIC
that
A
COMPLETION.
and could,
To verify
that a p r o d u c t
of r e p r e s e n t a b l e s
characteristic
property : ~01
where
1
(J) =
case.
is a p a r t i a l l y
~ Conc(A)
(~ Hi)
of
therefore,
the symmetry, ~H i
if j ~ i otherwise
In this
be called
we note
the
first
has the following
all
is the set w i t h one element,
is at most one map b e t w e e n
o r d e r e d set.
two products
0
i 6 I the empty set. of r e p r e s e n t a b l e s ,
There
-
hence
Conc(A ~
is p r e c i s e l y
representables. {JlJ ~ i
then
{ili > j
of s u b s e t s
3)
i a j
all
j 6 J - > i 6 I.
I
J>
we may define
is a D e d e k i n d pair. classes
of
only if
J c J'
J ~ I
in
are in n a t u r a l
n e e d n o t be e m p t y ,
one point.)
Conc(A~
namely
Given
We h a v e
are two c a n o n i c a l
continuous,
I.
a symmetric
i.e.,
f(j)
to
if and o n l y if
if for any D e d e k i n d
= inf f(i). I
correspon-
by
of
the if a n d to n o t e
it contains description
set of D e d e k i n d pairs.
and bi-concordant
extensions
t h a t the
(It h e l p s
b u t if n o n - e m p t y
Conc(A ~
the o t h e r r i g h t - c o n t i n u o u s .
(as a b i - c o n t i n u o u s ) CONTINUOUS,
I' c
obtained
B
i 6 I}
(J,I) ~ (J',I'>
as the o r d e r e d
a complete
all
and the e x i s t e n c e
is r e f l e c t e d
pairs:
if and o n l y if
and
We'll say that
W e may n o w n o t e
Conc(A ~
of D e d e k i n d
we
j 6 J
the s e t of D e d e k i n d p a i r s , objects
i 6 I},
J = {JlJ ~ i
Conc(A ~
I
Given
P A I R if
i s I
ordering
sup J
is a D E D E K I N D
all
natural
that
~ = ~'.
j ~ i
maps b e t w e e n
there
all
j ~ i
2)
dence with
of
i 6 I'}.
j 6 J, i 6 I =~m j ~ i
isomorphism
at m o s t
all
for all
(J,I>
of
if a n d o n l y if
i)
(J,~>
that
the s e t of p r o d u c t s
if a n d o n l y if
H I ,H i
Given and
HI,H i
-
i 6 I} = {JlJ ~ i
~ =
HIH i ~
a pair
HIH i ~
all
may de fine
2 2 4
A
> B
one
le f t-
T h e two c o i n c i d e
f: A (J,I)
> B
is U N I F O R M A L Y
it is the
case
-
IV.
Curiosity:
4.1
functors
from
It suffices
and p a i r of t r a n s f o r m a t i o n s x
and
y
it suffices Let
S
to
or from
S~
to
T
to show that for any petty x,y: H
is petty, w h e r e
H
; T
to show that subfunctors
be a s u b f u n c t o r
of
(-,A), define
{A' ~ > A}A,qS.
be a s u b f u n c t o r
Let
T
to be the set of e q u i v a l e n c e T.
Then
Pathology :
T
S (T)
are petty.
to be
is g e n e r a t e d by
relations
is g e n e r a t e d by
{A
of
on
(A,-), define A
Q(T)
i n d u c e d by e l e m e n t s
> A/-}_s
1
4.2
(Nunke's)
The f o r g e t f u l
to sets has a q u o t i e n t
to set, T + T
functor
(hence very petty)
(Hedrlin's) groups
T
A fortiore,
or r e p r e s e n t a b l e s
Then
T
that the e q u a l i z e r
is representable.
{Im(f) l f 6 TB, B 6 S}.
of
S
are lucid.
Proof.
of
-
WHEN DOES PETTY IMPLY LUCID?
All petty S
225
For has
T
that is not lucid.
the terminal
a quotient
from abelian groups
functor
from semi-
(hence r i d i c u l o u s l y
petty)
that is n o t lucid.
Proof. subfunctors as follows:
Nunke
finds
of the i d e n t i t y let
K
a strictly
functor.
be a n o n - l i m i t
ascending
Briefly, cardinal,
chain of
they are d e s c r i b e d let
SK c
HKZ
-
226
-
be the s u b g r o u p of K - t u p l e s w i t h s u p p o r t s m a l l e r let
A K = HKZ/Sk,
by
{x 6 ALl
let
L ~ K},
TK c I
let
than
K,
be the s u b f u n c t o r g e n e r a t e d
T = U T K.
T
is n o t petty.
I/T
then is the example.
H e d r l i n can fully e m b e d the b i g d i s c r e t e i n t o semigroups. C
In p a r t i c u l a r ,
of s e m i g r o u p s
T' c T
there e x i s t s
such that for any
A,B 6 C
category
a proper
class
(A,B) = g.
Define
as follows :
if
WA 6 C, A
if
V A E C,
T'
> T
> X
T'X = ~
T'
is n o t petty.
Let
1
1
T
be a pushout. A
P
is n o t
is PIL if all petty
Certainly, PIL.
lucid.
functors
The c o n v e r s e is true.
4.3
A
is
functors
petty,
>
if all s u b f ~ n c t o r s
Proposition
T
(A,X) =
PIL
P
We' ll say t h a t a c a t e g o r y from
A
to
S
of r e p r e s e n t a b l e s
are
lucid.
are p e t t y then
Indeed:
if and only if all s u b f u n c t d r s
of p e t t y
are petty.
Proof.
Suppose
T'
is a s u b f u n c t o r of
petty.
We c o n s t r u c t a n o n - l u c i d p e t t y
T, T' functor.
not Let
- 227 -
be a pushout~ But
T'
P
> T
T
J'
P
T + T~
is petty since it is a q u o t i e n t of
is the e q u a l i z e r
Proposition
T'
of the two maps
from
T
P.
to
4.4
If
A
is
PIL
then any category
of the form A'
(A,A)
(i.e.,
objects
A
; A', maps
A~'~
)
is
PIL
A" and any full s u b c a t e g o r y
Proof. subfunctors alent,
of a
PIL
is a
As we have noted,
of r e p r e s e n t a b l e s
it suffices
are petty.
via the usual a r g u m e n t made
PIL.
to verify that
This in turn is equiv-
familiar by Hilbert,
ascending
chain condition,
ascending
chains e x i s t i n d e x e d by all the ordinals.
is
PIL
by
A
F:
and > B
(A,A)
A' (A' l -)
A(A,-) o A(A,-)
B 6
n a m e l y that no strictly
(A,A)
may be e m b e d d e d
> A
subcategory of
A ~
then the functor in the functor
is the forgetful
of
A, A' 6 A'
functor.
then the
is a r e t r a c t of the
The i n c l u s i o n g e n e r a t e d by
by restrictions.
I
to an
is given by {x 6 SA}A6A,
If
S ~
A
represented
A(B,F(-))
lattice
lattice
Now if
A'
where
is a full
of subfunctors
of subfunctors the s u b f u n c t o r
, the r e t r a c t i o n
of of
is i n d u c e d
228
-
Proposition
4,5
Finite are
PIL.
is
and d i s j o i n t
A
is co-well p o w e r e d
and such that e v e r y map
as an e p i m o r p h i s m
PIL's
followed by a s p l i t - m o n o m o r p h i s m
then
PIL'.
Proof.
Just as for
The w r i t e r knows
category
A
A = Sets.
only one
the last few propositions.
a Cartesian
of s o m e t h i n g
product
discrete,
then every
PIL
not a c c o u n t e d
as a d i s j o i n t
of the form
of categories
of the last proposition.
J
To be more precise,
not to be o b t a i n e d
subcategories
K
union of
4.6
If
A
Cartesian products
I
Proposition
factors
-
But if PIL
(B,B)
satisfying B
for by
he knows
one
union of full where
B
is
the h y p o t h e s i s
is r e p l a c e d w i t h
B • A K,
the w r i t e r knows may be so
obtained.
The e x c e p t i o n a l distinguished
endomorphism.
one unary o p e r a t o r embedding
theories operators
is the category Because
into
(Fc,A)
and
PIL PIL
catogories. is lost.
yields
we know
a full of
that such
But allow just two unary
Indeed if
M
a
theory w i t h
(the e q u a t i o n s
can make it a p r o p e r embedding)
yield
of sets with
any a l g e b r a i c
and a set of constants,
of its algebras
the theory
A
is e i t h e r
the
-
monoid
on
two
generators
a,b
a 3 = a2b or
the
commutative a2 =
then
the
monoid
and
Sets M
is n o t generator
is
PIL
should
not
about
sets
V.
THE
Given a rank
one be
with
= b2 = may
c ~, be
PIL. it
expected
a5
a3 =
fully
Hence is
to b e
a,b,c a4
embedded
the
the
fact
in that
a set
that
Set M
We
need
a few
easy.
OF SETS W I T H
A
with
function
r:
that
adjoined.
transfinite
To
ENDOMORPHISM
an e n d o m o r p h i s m A--->
is, t h e
of
0*,
r
s:
where
ordinals
define
sequence
As+ 1 =
for
We
uniform
may
r(x) note
= sup that
be
Ay
=
A.
=
{slx r(x)
sAs
limit
may
understand
Define We
definition
for
case
we
subsets
0*
with
is
=
{(x)Ix
ordinal
define
A.
given
6 s A s}
,(~ Au =
l"~A s s
by
~sA
S< y
[I:~A~
6 As}. =
s
9>
x s As
and
>
A,
the
a maximal
first of
A
A0 = A
A
with
endomorphisms.
CATEGORY
ordinals", "~"
following
generators
three
semigroups
on
element
with
of
monoid
"extended
a4 =
category
free
define
ab 2 = b3e
ac = b c
the
we
=
-
with
ab =
( A , S e t s M)
facts
2 2 9
x ~ As+ I ,
the
M
-
=
0
<~>
x
has
no
r(x)
=
1 <'-->
x
has
a father
~(X)
=
2 <=~>
X
has
a grandfather
that
such
that
since
(Keep
= A
.
We
r(x0)
r (x)
r(s(x))
>
of
then
that
We
such
is
r (x)
< ~
but
lines
be
that
the
converse
S ( X n + I) =
s(x)
precisely
the
finite,
that
each
means
r (x)
= ~
ancester
s
of
-'~
set
(x 0)
all
of
ordinals):
=
constant). on
the
ancestral
means
with
length lines
unbounded
bounded
ancestral
exists
y
lines.
Lemma
5.1
For that
s(y)
any
= x,
Proof.
x
and
s < r (x)
there
s ~ r (y).
x 6 Ar(x)
=
B
n.
last
of e l e m e n t s
a bound
has
xn
the
even
the
and In particular
chains
periodic,
~
B >
and
descending
is
ordinal
s (Xn+ I) =
6 A + 1
there
an
grandfather.
empty. )
s(x)
(perhaps
great
all
that
~ ~ r (x) <
but
could
no
that
no
exist
are
means
all
but
there
lines
lines.
unbounded
then
therefore
ancestral
ancestral
obtain
grandfather.
= AB
such
x 6 As,
no
must
Aa
A~
qXl,X2,.. .
then
but
there
hence
note
(because
ancestral
are
and
that
qx0,xl,..,
that
a set
mind
r(x).
A
ancestors.
in
s <
implication
Note
is
= ~ ==>
=
infinite
A
A s = As+ 1
A s = A~.
If
-
r(x)
Note
sA
2 3 0
|
such
-
Proposition
231
5.2
If
A
and
B
are s e t s w i t h
distinguished
endomor-
1
phisms
(both
f(s(x))
called
= s f(x)
all
Proof. ancestral for
line,
If
Suppose
that
> rA(Y). that
thus
rA(Y)
rB(f(y))
there
> A, w e exist
of s e t s w i t h
A
respect
its
sum.
a set
A'
We
call
subsets
Hence
we
say
ancestral
line
that
< we
that
8
If A
A is
y
such
= sf(y)
= rB(f(x))
rB(f(x))
> ~,
> rA(x). 1
endomorphism
relation a subset
by
of
x - y
if
A, P U R E
if it
of a s u b s e t it.
A
is
may
In the c a t e g o r y
disjoint classes
is the
rB(f(Y))
choose
containing
=
>
f(s(y))
pure subsets.
the
inductively.
= rB(sf(y))
and
endomorphisms
A.
x.
an i n f i n i t e
The purification
minimal
of
S
distinguished
.
distinguished
decomposition
e <
an e q u i v a l e n c e
to
that all
argue
rA(Y)
rB(f(x))
with
of all p u r e
categorical
case we
~ < rA(x)
define
into
i t is c l e a r
It f o l l o w s
Hence
all
> rA(x)
an i n f i n i t e
that we know
n, m, s n x = s ~ .
be partitioned
containing
= 0
general
> e.
such
can c h o o s e
= 8, for any
a.
may
the i n t e r s e c t i o n
maximal
the
a set
closed with
is the
a
> ~
Given s: A
For
a map
= = .
rA(x)
rA(x)
= x,
rB(f(x))
if
we
and obtain
is s u c h
Given
s(y)
= f(x),
8
f
is
rB(f(x))
= ~
rB(f(x))
that
> r A(x).
> B
then
rA(x)
f(x) , y i e l d i n g
rB(f(x))
f: A
x,
apply
Note
is
s),
union
provide
only
an E S S E N T I A L
pure
the subset
EXTENSION
of
- 232 -
of
A'.
Note
then
the p u r i f i c a t i o n obtain
Lemma
that
of
A = B + C
given
A',
C
where
phisms
let
f: A'
A
> B
rA(x)
be
~ rB(f(x))
of p a r t i a l
It s u f f i c e s little
is
of
sets w i t h
that
all
an e s s e n t i a l
of
B
B
as
and
extension
of
A'.
=-~
if
sy = x, rB(Y) may by because
Theorem
5.1 rA(x)
of
of
A
endomor-
A'.
A map
if a n d
only
We use
Zorn's
maintain
the
if
map may be
the
inequality).
rank
a map
choose
f: A' U rA(x)
y s B
< rB(f(sx))~
{x}
> B
such
= y.
that
Define
inequality.
If
such
rA(x)
rA(x)
sy = f(sx) f(x)
= y.
a we
that
that
y s B
If
just
Accordingly
such
We may pick f(x)
on the
extended
x s A - A'
~ rB(f(x)).
= ~.
lemma
rank
any s u c h
and define
find
= such
< ~
then that we
rB(Y)
> rA(x)
l
5o 4
F o r the morphism
which
rB(f(sx))
= ~
to all
we may
= f(sx)
and
extension
lemma.
preserving
and d e f i n e
= ~
last
that
A' ~ A
distinguished
x s A'.
extensions
= f, sT(x)
rA(sx)
an e s s e n t i a l
to s h o w
(while
s x s A' flA'
compliment
B
may be extended
Proof:
show
as the
define
5.3
In the c a t e g o r y
set
A' c A, w e m a y
category
any s u b f u n c t o r
Proof.
We need
of sets w i t h
a distinguished
of a r e p r e s e n t a b l e
to s h o w
that
functor
endo-
is p e t t y ,
for a n y o b j e c t
A
and
-
transfinite
Sequence
of maps
the ordinal numbers, commutative
T c
is petty
let
let
gf~(i)
= hi"
T(B)
= {A ~
through
and a
be such that
Let i
~
the s u b f u n c t o r s
BJl~,g,
gf~ = f}.
be a g e n e r a t i n g qg: B
(i)
If
set.
> Ci
T
For each
such that
be an ordinal
such that
larger than {~(i)}. hi B 8 = A --~ C i ~ B B and
A ~
a triangle f= ( i ) ~
i~ (i)
A/ Conversely T
sequence f
ranging
of the t h e o r e m as follows:
sequence we c o n s i d e r
{hi: A --~ Ci}is I ~(i)
There exists
that
~ < 8
to the c o n c l u s i o n
d e f i n e d by
i 6 I
hence
{f~: A --~ B~},
that there e x i s s
the t r a n s f i n i t e
(A,-)
-
triangle
(This is e q u i v a l e n t given
2 3 3
suppose
is a s u b f u n c t o r
is not petty.
Let
of s u b f u n c t o r s
of
s T~+~ - Tu,
condition.) we shall
T
{Te} T.
of
be a strictly
transfinite
For each ordinal
and obtain a sequence
find a cofinal s u b s e q u e n c e u,8 s 0', ~ < 8
e
that v i o l a t e s
We shall in fact show more.
such that for every
(A,-) , and suppose
Given
of ordinals
choose the
{fe: A --~ B~} 0' c
there exists
0
a triangle
-
Given
the
ications Ca =
transfinite of
{xlZn
the snx s
endomorphism
of
B~ = C a + D~ lemmas
Lemma
image
Be).
is
(We r e c a l l s:
Be
D e = B e - C a. is
the
the
be
the p u r i f -
that > Be
Da
is
categorical
is
the
distinguished
a subobject.
sum.
The
next
two
result:
a transfinite
an e s s e n t i a l
cofinal there
of
fe"
Ca c Ba
5.41
Given Ca
let
+
imply
-
sequence
I m ( f e) }, w h e r e
where
jointly
234
is
sequence
extension 0' c
subsequence
0
of
{fa:
Im(f~)
such
that
A
>
there
for
C a}
exists
a,B E 0',
where
each
a e <
a triangle C~
Lemma
5.42
Given cofinal there
subsequence
exists
The first
find
between
a transfinite
De
0' c )
lemmas
a cofinal
of
that
such
{D a }
there
for
a,B E
result
rather
that
exists
0',
a
a <
D S.
imply
the
main
subsequence
the purifications
subsequences
0
sequence
of
in w h i c h
the
such
that
images
maps
exist
and
there then
between
easily. exist
We
maps
a cofinal the
complements
-
of
-
the p u r i f i c a t i o n s .
Proof on
A:
of L e m m a
x~ y
if
fe(y)
consequences
on
A,
such
that
all
the
factors
A/-
and
A--->
it clearly may
{ fe : A --~ are
Ca}
Thus we
of
A,
re(x)
Lemma
{A c
0' c
0
such
re ~ r 8
finite
that
(i.e.,
notation:
let
sequence
We wish
to
e
that
show
solve
are
a set
the
of
subsequence
We n o t a t i o n a l l y case. >
C
problem that
Each is
{A/-
) Ca}.
given
maps,
a sequence
fe
one-to-one
for the
one-to-one
given
we -
obtain the
e,B
6
~ rs(x)
I
be
a set,
of
function
a function
rank
and
in
fact
of e s s e n t i a l
find
all
I
defined
>
0*.
earlier. subsequence
8
i t is
the
case
x E
IAI).
We
{re:
from
IAI
a cofinal
e <
let
re:
function
to
0',
re(x)
that
the
a congruence
a cofinal
A/-
therefore, of
only
same.
already
it s u f f i c e s
for
are
find
the
where
induces
e
Ca}.
= rCe(X) ,
5.3 s a y s
Ca
to
can
are
a sequence
For each Where
>
f
There
we
is
assume,
is
inclusions.
extensions
such
suffices
notationally
Each
= fe(y).
congruences that
as
5.41.
and hence
assume, t h e r e f o r e ,
We
235
>
I
to the
there
is
a cofinal
each
e
produces
0 ~}
that
switch be
extended
subsequence
a transordinals. in w h i c h
r e ~ rs.
As on
I, w e
equivalence
before,
may
pass
to a c o f i n a l
relations
are
the
an e q u i v a l e n t
subsequence
same
and may
in w h i c h specialize
relation all
the
to
the
-
case
that
the
therefore,
functions
that such
Each if
re(x)
on
I
all
are
by
re
is
the
may pass
same.
a relation
preserving
We
<
Either
into
If t h e r e
a cofinal
is that
all
in the s u b s e q u e n c e .
it suffices case,
to p r o v e
therefore,
already
such
Either it is c l e a r and
that
functions case w e
are
Let
I' =
only
is
k 6 I on
a set
a cofinal
the s a m e
define
well-ordering
I m ( r e) c
I)
{ili < k}.
I
such
that
is w e l l
0*
is
in w h i c h latter
ordered
an o r d e r -
m,
{r~:
to a s s u m e
in w h i c h Im(re)
case
as
c
that
re(m)
it =
r e' = r e l I '
I' ~
that
0
follows:
~ 6 Im(r e)
I' = I - { m } ,
0}.
the
and
In any
sequence
is
0.
is
a set
or not.
of f u n c t i o n s subsequence
and we would to b e
the
element Then
of o r d e r i n g s
subsequence
for
(x e < y
ordinals.
dement
result
U e I m ( r e)
that
there
the
that
in w h i c h
Let
I.
a set
I "-~
to the
a maximal
it suffices
that
re:
subsequence
on
subsequence
subsequence
specialize
has
only
therefore
a cofinal
follows e
I
are
the e x t e n d e d
is
assume,
case.
to a c o f i n a l
or a c o f i n a l We m a y
there
that each
there
(or b o t h ) .
the
We n o t a t i o n a l l y
a well-ordering
assume
and
embedding
E Im(r e)
already
Again
-
one-to-one.
defines
< rs(y)).
and we
are
236
be
such
all
case
the
In the s e c o n d
(with r e s p e c t
is
first
in the s e q u e n c e
in w h i c h
that
U e r a(I')
appear
done.
first
In the
{re(k) } a set.
to the
is n o t For
a set.
each
-
function Because
f:
I'
UfVf
~ Uara(I')
= Uera(k)
there
a set.
cofinal.
We
may
notationally
sequence
(r
}
Vf
that
(re(k) Is E 0}
6:0
>
<
0
~}.
as
are
= sup(r(x)Ix such
6 De}.
that
cases
separately.
~ ~
=
f: I'
such
e,8 Now
0;
=
>
f}.
therefore
all
that
is s u c h set
6(a)
0
0'
that
relI'
define
for
~ <
f}.
is the
given
= rsII'
and
a function
= min(Slr~(k) ~
0'
> U6(~)r(I)
r6 (e) ~ r6 (~)
of p o s s i b i l i t i e s such
) = ~
P
they
are
all
and prove
We
that
subsequence
assume
treat
that
e.
and the
the
the
same.
such two
given
= x}.
there
it s a t i s f i e s
0'
For each
{ n I ~ x s D e sn(x) for
r ( D e)
or a c o f i n a l
all
exist
that
0'
may
define
a cofinal
(or b o t h ) .
r(D
of i n t e g e r ~
{D e}
is
a s
that
set
a subsequence
all
~ s
Given
there
case we
as the
subsequence
5.42.
first
Pe
e, d e f i n e Only
a
is
a cofinal
We
pass
the
to
such
conclusions
of
lemma.
Assume
the
=
{re(k)IreII'
{alreII'
a set.
Either
all
In the
e,
for
check
r ( D e) = ~
r (D~)
the
exist
assume,
6(0)
of L e m m a
that
sequence
to
Vf =
done.
Proof
0'
is n o t
must 0' =
that
follows:
It is e a s y
m%d we
Let
is s u c h
-
define
is n o t
that
237
and
for
case
then
any i n t e g e r
that
~x 6 D a
that n,
{D~}
is
and pair
sn(x)
= x <
given,
r(De)
of o r d i n a l s >
qx 6 D 8
=
~
~,8 sn(x)
all it = x
is we
-
wish f:
to s h o w
De ~
about
that
D E.
for
D E, w e
any pair
Clearly
transfinite
De
each
union
categorical
exists
a map
arbitrary
Ei
E i --~
point We
DE
= ~.
define
ElS =
{s n(x) In = 0 , i , . . . }
r (s n (x))
yields
a map
x 6 Ei
and
case we
repeat
Otherwise
n >
we
there
exists
sn(y)
= y
El/ =
= y,
be
that
the
sn(x)
using
Lemma
r ( D e) < ~ the
latter
e <
solution
all
5.3 n o w
the e. case
we
may
= ~ = x
= x,
choose
assumption f: E' 1 a map
case we
x
in
Ei
0}
>
a set
show
this
Partition
De
follows:
that into
its
that
Define
{D e }
We
as
such
that
f(si(x))
subsequence
r ( D e) < r ( D s ) .
such
D E.
that is
latter
E i-
integer
by
exists
In t h e
a cofinal
8
pick
there
y s D8
D8
assume
{r(De) Ie E
lemma
P e = P8 ) 9
>
define
< rD~(f(sn(x))}
Hence
any
positive
and
If w e
or not.
taking
an
an e s s e n t i a l
either
smallest
provides
Either
r(sn(x))
there
that
= sn(y). is
that
we p i c k
such
6 E'I
sn(x)
and
remaining
y 6 D8
Ei
above,
the
{x,s(x),...sn-l(x)}
to s h o w
that
r(Ei)
and
Disjoint
r ( E i) < ~
f(s n(x))
De
subobjects.
(rD8) s n (y) = ~) .
If
a problem
object.
If
note
exists
Given
it suffices
sn(x)
argument
x s El,
pure
a point
D s.
n
minimal
and
and
such
the
let
For
for
0
(finally
= s i(y).
In
-
longer
pairs.
and
we
there
about
i.
for
< =
Ei
is n o
each
f(x)
F El,. that
(because
its
sum
r(y)
of
ordinals
an i n d e c o m p o s a b l e
x s Ei
extension
of
but
into
D e = Uis is t h e
-
the p r o b l e m
sequences
partition
2 3 8
is
given,
or not.
provides minimal
such the pure
that
- 239 -
subobjects. there
is
D
a
such
that
and
f: E'
= U E i.
map
Ei
It s u f f i c e s
x
rDB (y) > r ( D ~ ) .
Let
E:l =
{sn(x) In = 0 , i , . . . }
function
Choose
f(sn(x))
y
i
any
be t h e
E i.
for e a c h
Pick
D8
6
that
D B.
>
>
to s h o w
= sn(y).
6
Dfl
Because
i
rD
(sn(x))
< rDs(f(sn(x))
lemma5.3
yields
a map
f: E i
-~ D B-
e In the
former
appear we
case,
{rDe(X) Ix 6 D e }
define
the
For each
e
we
there
there
The
that
map te,x: such
~ ---~ 0 that
because
t ,x D8
S~
t B
i.
by
t
such
De
Pick
that as
x E Ei
of for
E'l
that the
e
(sn(x)).
rD~ (x) Se are
=
imply
and hence all the same. Given pure
there
sub-
exists
exists
y s D8
{sn(x) In = 0 , i , . . . } , Then
rEi(sn(x))
lemma5.3
= rDB (f(sn(y)))
= tB,y(n) yields
a map
DB9 I The t e c h n i q u e s
of
the p r o o f
a
sequence
= te,x(n)
the
x 6 De
(n) = r D
follows :
there
= sn(y).
= D B(f(sn(x))) ~ And
~0X
they
to s h o w
SB, h e n c e
f(sn(x))
e, a n d
i n t o its m i n i m a l
Define
that
of s e q u e n c e s
a solution
in
,y.
0
on the v a l u e s
r E i x (sn (x)) = rD~ (sn(x))
= D 8 (sn(y)) Ei " >
by
=
of
It suffices
appears
For each 9
subsequence
each
of o r d i n a l s
of p o s s i b i l i t i e s
we partition
E i ---> D B
f: E'i "->
a set
D~ = U E i.
a set
: ~
a set
this p r o v i d e s
De,Ds,~ e = S 8 objects.
ar M
condition
are o n l y
is a c o f i n a l
We s h o w
t
obtain
is o n l y is a set.
sequence
{~e,xl x s Da} ~ that
there
above
can be u s e d
to
-
prove
the f o l l o w i n g
2 4 0
-
propositions
in w h i c h
are used
the
definitions: rD(x)
the rank
as d e f i n e d
r(D)
= sup{rD(x)Ix
tD,x
the s e q u e n c e
for lemma 5.3
E D} 0'
defined
by
tD, x(n)
= D(sn (x)) . the set of sequences
SD SD
SD2
iff for all
{tD,xlX
s D} there
t q SDI
exists
1
t' E $D2
,
t ~ t'. PD = {nlzx 6 D,
Proposition
sn(x)
= x].
5.5
Given distinguished if and only
objects,
DI,D 2
endomorphism
in the c a t e g o r y
there
exists
a map
of sets w i t h f: D 1 --~ D 2
if r(D I) < r(D 2)
or
r(D I) = r(D 2) = ~
and
P(D I) c P(D 2)
or
r(D I) = r(D 2) < =
and
S(DI)
It m a y also be p o i n t e d t: ~ --~ For any
0" limit
sequences s: A that
appears,
A8
sequences
by
w
u, let
A
ascending
sequence
be the set of a s c e n d i n g
to the o r d i n a l s
less
(st) (n) = t(n + i).
, as used whose
every
9 I
as follows:
ordinal
from
) A
out that
~ S(D2)
for the d e f i n i t i o n
initial
values
than
e.
Define
It m a y be e a s i l y of
are g r e a t e r
r,
verified
is the s u b s e t
than or equal
to
of
-
8.
Hence
corollary
rA(t)
= t(0)
and
241
-
rA(sn(t))
is that the terminal
subfunctor,
only on objects
=.
namely
Finally we note that the set need be.
Define
equivalent,
sequences
t N t'
an i n d e c o m p o s a b l e
indecomposable existence
all
object
of stable e q u i v a l e n c e all sequences.
t,t':
~
if there exist
t(i + n) = t'(i + m)
i.
An immediate
functor on the o p p o s i t e
c a t e g o r y has a n o n - p e t t y of rank
= t(n).
SD ~ 0
integers
that w h i c h
is larger
n,m
equivalent.
such that arising
from
The set
[S]
types may be used instead of the set of
It is amusing that the p r e - o r d e r e d objects
of maps)
of real v a r i a b l e
than it
to be stably
Any two sequences
are stably
is empty
of rank
is e q u i v a l e n t
analysis.
~
(the o r d e r i n g to the
family of
given by the
"orders of infinity"
-
242
-
THE C A T E G O R I C A L C O M P R E H E N S I O N
SCHEME
by J o h n W. G r a y
INTRODUCTION
This p a p e r is b a s e d on an a t t e m p t to find an analogue in c a t e g o r y t h e o r y to the c o m p r e h e n s i o n theory which
says,
essentially,
scheme of set
that g i v e n a property,
there
is a set c o n s i s t i n g e x a c t l y of the e l e m e n t s h a v i n g that property.
L a w v e r e has t r a n s l a t e d
this into a s t a t e m e n t about
adjoint functors (Sets,X)
(
> 2x
w h i c h are d e t e r m i n e d by substitution. valued functions on a category,
If,
on a set, one c o n s i d e r s
instead of 2-
set-valued functors
then he s h o w e d that there is a similar pair
of a d j o i n t f u n c t o r s xoP (Cat,X)
(
> Sets
w h e r e the functor f r o m right to left assigns the c o r r e s p o n d i n g
fibred c a t e g o r y over
It is n a t u r a l to ask if this extends being arbitrary
(split, normal)
r e v i e w and r e f o r m u l a t e properties adjoint
X
to
of comma c a t e g o r i e s
(
Cat X~ over in
w
> Sets fibres.
w i t h values X.
In w
we show,
we using
that there is such a pair of
functors (Cat,X)
F: X ~
with discrete
fibrations
fibred categories,
to
> Cat X,
-
and
in
w
we d i s c u s s
instance is not,
what
243
it w o u l d m e a n
of the c o m p r e h e n s i o n and the r e a s o n
sion scheme
scheme.
is that
is e q u i v a l e n t
-
for this
The
in this
to a s s e r t i n g
to be an
answer
context
is that
it
the c o m p r e h e n -
that
1B lim > where
1B
(B
~ Cat)
is the c o n s t a n t
is false,
unless
B
categories,
in a d e l i c a t e cally
the p r o p e r t i e s
take
to r e c a p t u r e on pro
of the
categories, way
and b i f i b r a t i o n s , of w h i c h
one can
and hence
of
the n o t i o n
should
not be s u p p o s e d
that
[CC].) Yoneda
this
as in L i n t o n
In p a r t i c u l a r , lemma.
were
2-adjointness, Cat,
which
in
altered
Thus,
an a m u s i n g
w
to the c o m p r e h e n s i o n
scheme
in
w con-
notion
categories (w
(It
adjointness and Kelly
version
determines in
to
hope
the basic
2-functor
with
of
It seems
the p r o p e r
or E i l e n b e r g
are c o m p u t e d
and some
w
scheme.
coincides
in
limits
then one m i g h t
introduce
of course,
and
is i n t r i n s i -
suitably
we d e s c r i b e
super
[AC]
one needs
which
of 2 - a d j o i n t n e s s
of 2 - c o l i m i t s we r e t u r n
Cat,
structure,
This d e t e r m i n e s
Cat
This
the n o t i o n s
adjointness
be found
above n o t i o n s
2-category
categories
in
~ 6 Cat.
of the d e f i n i t i o n
will
(w
enriched
to
a form of the c o m p r e h e n s i o n
in terms
of 2 - c o m m a
with value
to lie in a p p l y i n g
A discussion
if the
2-functors
struction
comma
of 2 - c a t e g o r i e s
that
account
appears
and i n t e r l o c k i n g
a 2-category.
reasonable
functor
*
is d i s c r e t e .
The d i f f i c u l t y of f u n c t o r
= B 6 Cat
w
and find
of the the n o t i o n
Finally, that
in
it does
-
work in considerably
244
-
greater generality
the basic tool being the calculation Cat-valued
for a categorical
interesting
and perhaps
hess for 2-categories the analogue
of 2-Kan extensions
we discuss
appears *
fact that I had already
1964,
In fact,
This was,
this is how I felt in
of course,
of
w
Actually,
this conference
how these constructions
Such is the power of the comprehension remark that in and adjointness of Tierney,
[BC], B~nabou promises are different
it seems
likely
to some of those presented is aware of the Yoneda-like
in of the
that I realized fitted together.
scheme.
I should also
that functor
categories
From remarks
that B~nabou has results in particular,
lemma in w
I first
discussion
for bicategories.
here;
during
of the
at Oberwolfach
but it was only after hearing Lawvere's
(almost instantly)
helped by the
f~r Mathematik
while on an NSF Fellowship.
scheme during
of
here is
found most of the constructions
the "basic construction"
comprehension
If one
scheme is a basic ingredient
a year spent at the Forschungsinstitut ETH in Zurich,
scheme.
then the entire theory presented
writing down the theory.
that
and that this in turn
for the comprehension
thought,
It is
that the form of adjoint-
should hold,
already rigidly determined.
described
of
of these
to be forced by requiring
that the comprehension
mathematical
the implications
foundations of mathematics.
significant
of equation
is forced by asking believes
above,
functors. In an appendix,
results
than indicated
analogous
he a p p a r e n t l y
-
245
-
PART I THE PROBLEM
w
CATEGORICAL
FIBRATIONS
For a r e v i e w of the G r o t h e n d i e c k fibrations
([SGA])
We reformulate Let
P:
E
let
EB = P
the n o t i o n s
) -I
and o t h e r
B
aspects
in the
be a f u n c t o r
(B)
with
theory
of c a t e g o r i c a l
of the t h e o r y ,
see
[FCC].
f o r m w e s h a l l u s e t h e m here.
and for e a c h o b j e c t
the i n c l u s i o n
functor
JB:
B E EB
IBI, >
E.
Definition
P: each
E
f: A
>
) B i)
B
i = 0,i
is an i - f i b r a t i o n ,
in
B, t h e r e
if,
for
exist
functors (i = 0)
'
I I
(i = i) f,
'
EA f*) ii)
natural
E B 'l
EB
~
EA
transformations !
@f: JA
) JB
o f,
i
,
@f: JB
o f
> JA
I
with P(m)
P(@f)
= f, s u c h t h a t g i v e n
= f, t h e n t h e r e
D
(@f)O
is a u n i q u e
f,D
E
An
i-cleavage
for
P
is a c h o i c e
any
m:
D
> E
in
E
with
factorization
D
f*E
> E
of the f u n c t o r s
and n a t u r a l
-
transformations.
To category
illustrated
It is c a l l e d
split-normal (iB) * =
(gf).
(gf)*
= g.f. for
a single
the
As
~i:
usual,
comma the
are
EB
= f'g*
identity
let
morphism
1
and
2
denote the
the
category
by
~
>
2
given
the
two
category
1
functor
functors
(F0,F I)
given F.:
by
B,
)
A,
1
%i(~)
1
is d e f i n e d
to be
=
i;
i = 0,1.
i = 0,i,
the
the
inverse
limit
[CCFM], defined,
SO =
F!
{E
~
[FCC] and
B
and
E2
>
B
~
F
i
In p a r t i c u l a r ,
are
~
B ~I
B2
w
there
0 , p2}:
B~~
(
induced
(P,B)
AI
(P,B)
9
and
(B,P)
functors
' Sl ,
=
{p2, E ~I }: E2
)
B
(B, p)
~
i
It w a s and
shown
only
if
in
[FCC]
there
is
that
P:
E
is an
i-fibration
if
a functor !
L0:
(P,B)
~
E2 !
LI:
E2
(B,P)
!
such
that 0
SoL o =
(P,B)
i
SIL l =
(B,P)
! i i
L0 (M
of
diagram
Ao (See
if
terminology,
0 with
-
(i B) . = E B
account
with
246
| N
equivalent
means
M
is
to a c h o i c e
| S0 left of
! !
SI
adjoint L,
and
rLI to
split
N).
A
cleavage
normality
can
be
is
.
- 247 -
described
in t e r m s
of e q u a t i o n s
satisfied
by
L.
In e a r l i e r
terminologies l-fibration
=
fibration
[SGA]
=
fibration
[FCC]
0-fibration
= cofibration
[SGA]
= opfibration
[FCC]
For
split n o r m a l
about cleavage
preserving
i-fibrations, functors
over
.
it m a k e s
sense
to t a l k
B; i.e.,
commutative
triangles F
/ B such that (P,B)
L0
> E ~-
(F,B,B 2 , B,B)]
F2
(~, B)
L~
commutes.
(See
tor w h i c h
assigns
normal functor
w
L!
~ E~
(B,P)
L}
> E~
for the n o t a t i o n . ) to
i-fibrations by p u l l b a c k s
g o r y of sm all
1
~ E~
(B,P)
B
the c a t e g o r y
over
B
The c o n t r a v a r i a n t Spliti(B)
(with s m a l l
fibres)
- is " 2 - r e p r e s e n t a b l e "
by
func-
of s p l i t - made Cat
into a
(the c a t e -
categories).
Proposition
There
Split0(B)
Proof.
are adjoint
<
> Cat B
Given
equivalences
Splitl(B)
a split normal
<
> C a t B~
0-fibration
P:
E
> B,
-
the c o r r e s p o n d i n g
functor
is
> Cat:
B
B It is i m m e d i a t e a natural
248
-
> EB:
f
> f.
that a cleavage p r e s e r v i n g
transformation
between
~
functor
determines
the c o r r e s p o n d i n g
cat-valued
functors. Conversely,
the functor
in the opposite
given by p u l l i n g back a u n i v e r s a l was p o i n t e d out by L a w v e r e denote by Cat
Ca~.
in
objects
(A,AI IA
IAi
and w h o s e m o r p h i s m s
are given by
Hom((A,A),(B,B))
= {(H,h) IH: A
is
(K,k) (H,k) =
P: Cat then a lifting
~ Cat: functor
is a split n o r m a l If normal
F: B
l-fibration
I [~ , Cat]l.)
A
Here
ICatl
> B
and
> A:
h: H(A)
> B}
~
If
(H,h) described
> H
,
showing
then the c o r r e s p o n d i n g
~ Cat, over
B
B B~
and
is easily
that
~
PG: ~ G
Cat, "- B
split
is the p u l l b a c k
EF
G:
(this
0-fibration.
0-fibration
w h i l e if
Cat
are
(KH,kK(h)).
(A,A) L
over
is
[ETH] ) w h i c h we p r o v i s i o n a l l y
(Later it w i l l be called
is the c a t e g o r y w h o s e
Composition
0-fibration
direction
~
F
Cat
*- Cat
then the c o r r e s p o n d i n g comes
split n o r m a l
from the p u l l b a c k
P
-
P
their
that we
> Cat
G
w
and
-
-
BO p
In
2 4 9
[CCC],
"i C a t
COMMA
a number
properties
are
(-) op
"> C a t
CATEGORIES
of
operations
catalogued.
We
on
comma
mention
categories
here
those
need. i)
Commutative
Ao
diagrams
~B~
.B 2
>
B
<
n !
Lo 8' ~0 )
induce
~ S '2
B'~
B '~I
F' ~
B'~
A'I
functors
]) {G O , H , G ~
"-
( L 0 , M , M ~ tMrLl)
(F0 ,F
L0
)
! (F aW ,F~)
[
• L1
A0 • A I over
A0 • A 1
arbitrary
and
morphism
L0 • Ll
respectively.
In g e n e r a l ,
given
an
e: over
-
250 -
(F 0,FI)
>
(F 6,F~)
L 0 • L 1 , the diagram f
Ao
Ao
u
(FS,F ) Lo R'
Q,
;
A
shows
that
there
<
B '2
is an i n d u c e d
~B'
<
I
a~
morphism
Q!
(F0,FI)
{P,
e,R}
A0
over
x
>
A~
A0 x A1 . ii )
Given
Fi:
Ai
, > B, i = 0 , i , 2 , 0
(F0'FI) defined
(F6L0,F~LI)
as
follows:
From
A1 (FI'F2) the
diagram
(F0,F 2 )
there
is a f u n c t o r
-
(F0,FI)
-
251
~i(Fi,F2)
(FI,F 2 )
(F0,F 1 )
Ao
A 1
B2
B2
B
8
we deduce
the e x i s t e n c e
of a functor
W
B
making
the d i a g r a m
TI (F0'FI)
> (F I F 2 ) Q /
~ (FI'F2) 1 ~,~
B~-,
)
Ba
B2
B ~o
Pl
So B2
B~I ---
)
B
~ Ai
(F 0 F I) -. commutative. commutative
It is easily diagram
checked
that this
W
also gives
a
252
-
-
(F0,FI)
P0
B3
A 0 F0 ~
and
hence
"o"
(FI,F2)
determines
B < B~0
2 ~ ~I >
B < F~~
A2
a functor
= {PoS0,BYw,RITI}:
(F0,FI)
• (FI,F2)
>
(F0,F2)
AI which
in
the
is
"associative"
Note.
~,
category
3
in
8, a n d
an
obvious
y
sense.
designate
the
indicated
morphisms
.
2 iii) as the
a functor
A natural m:
A
~
transformation B2
is
the
m:
same
F
thing
form =
i.e. t
{A,m,A}:
A
>
(F,G)
;
> as
G,
regarded
a functor
of
- 253 N
A
m
)
(F,G)
A• commutes. to
G
Thus
is
the
the
set
of n a t u r a l
1 1
iv)
If
functors
and
natural
F.: 1 if
A. 1
>
m0:
transformations,
A0 • A1
(F0,FI)
F
given
~ ,A0•
B
A
>
(F,G)
>
(A • A)
1
and
GO
~
then
there
( ~ ) : over
from
pullback N a t (F,G)
are
transformations
F0
(F0,FI)
by
the
•
mo• (Fo,F!) •
G.: 1 and
A. 1
>
ml:
F1
is an >
B,
i = 0,i > G1
induced
are
functor
(G0,GI)
composition
(G0 ,F0) x (F0 ,FI) x (FI ,GI)
I~OII
(mo ,ml ) ~
(Go,G1) In
[CCC]
satisfied
functor,
there
is
(hopefully)
by
these
In
[FCC],
then
the
a basis
for
the m a n y
relations
operations. it
is s h o w n
functor
PM:
that (M,X)
if )
M:
A
)
X
given
X by
is a n y
- 254 -
PM
> •
Xz
(M,X)
X
X
M
A
has a canonical
split n o r m a l
is a c o m m u t a t i v e
~
•
Furthermore,
0-cleavage.
there
diagram QM
A
(M,X)
X
Here This
QM
is left adjoint
is u n i v e r s a l
to the p r o j e c t i o n
(M,X)
> A .
in the sense that given any c o m m u t a t i v e
diagram A
Q
~
E
X
where
P
has a g i v e n split n o r m a l
is a unique
cleavage preserving
with
Q
HQ M
=
and
PH
=
PM
F r o m the d i a g r a m
"
0-cleavage,
functor
H:
then there
(M,X)
>
E
- 255 -
(M,x)
/ (M,X)
A
X~
X it f o l l o w s and t h a t
that the f i b r e of PM
corresponds,
PM
over
X 6
IXI
v i a the e q u i v a l e n c e
is of
w
(M~X~ to the
functor X
~ Cat:
X
F r o m the a b o v e o p e r a t i o n s checked
t h a t this g i v e s
>
(MFX~:
~(A
M > X) =
that assigns [G
~ X
0-fibration,
X
gives
(M--~) . it is e a s i l y
a functor
(M,-)
to a f u n c t o r
over
>
and t h e i r p r o p e r t i e s ,
Cat X
(Cat,X) where
f
(as in
6 Cat X
Conversely,
G: X w
) Cat forgetting
the operation
the c a t e g o r y that
it is a
a functor V
Cat X The following mapping
result
~ ~
(Cat,X)
is i m m e d i a t e
f r o m the p r e c e d i n g
property.
Proposition V
is left a d j o i n t
(Cat,X)
to
<
> Cat X
universal
-
Note. tion m a p s
in a m o r e g e n e r a l
perhaps,
and
context,
fits i n t o a m u c h
It w o r k s ,
framework,
for i n s t a n c e ,
In our c o n t e x t ,
in t h e c a s e of sets
manner.
>
is just c o m p o s i t i o n
(left)
A
) Cat,
Kan extension
easily described, X s
IXI
of
being
for .
F:
A
Cat X F.
Lawvere)
ZF(M):
) X
in the
induces
a functor
9 Cat A
with
M
scheme would
a b o v e be c a l c u l a t e d
ZF: C a t A M:
for a
2x
) Sets X O p
A functor
(following
(
(
~
catF:
For any
pair which
the comprehension
the c o m p r e h e n s i o n
t h a t the f u n c t o r
adjoint which
(this
t h a t one s o m e t i m e s
called
categories
(Cat,X)
which
Lawvere
functors
and in the case of small
following
w
like the a b o v e
(Sets,X)
require
of the a d j u n c -
SCHEME
has s h o w n
functors
richer
see
of h y p e r d o c t r i n e s ,
[CVHOL])
has a p a i r of a d j o i n t
p a i r of a d j o i n t
description
THE COMPREHENSION
In the t h e o r y
scheme.
-
For a d e t a i l e d
w
volume,
256
This
functor
we denote 9 Cat X
X
along
by
.
9 Cat F
has a l e f t
is c a l l e d
the
[AF]).
It is
(see
the f u n c t o r w h o s e
value
at
is g i v e n by [ZF(M)] (X) = lira ((F,X)
> A ~
Cat)
.
-
It s a t i s f i e s ,
for any
257
N: X ---~ Cat,
Nat(ZF(M),N) Now given
-
~ Nat(M,NF)
a composition H
A
F'
~ A'
~ X
L,
Cat
with
F = F'H,
then composition
with
N a t (M,NF')
H
induces
a function
N a t (MH,NF' H)
[[ Nat(ZF' (M),N) and hence,
taking
of a n a t u r a l
-> Nat(7~F(MH) ,N)
N = ZF' (M),
one deduces
transformation ZF (MH)
let
1A: A
the c a t e g o r y of
the e x i s t e n c e
) Cat I 6
denote ICatI.
)
ZF' (M)
the constant Then,
functor with value
for any o b j e c t
F: A
(Cat,X), ~.F(1A) : X ~
is an o b j e c t
of
Cat X ,
Cat
an d for a n y m o r p h i s m A
9 A'
X
of
(Cat,X),
one has
(since
I A . H = 1 A)
a morphism
9 X
258
)
ZF(I A)
ZF' (1A,).
-
It is easily
checked
that this gives
a functor
z (-) (i(_)) The analogue
(Cat,X)
:
of the c o m p r e h e n s i o n
> Cat X .
scheme w o u l d
say that
~ ~(-)(ic_ )) (Actually, Z(-) (i(_))
the c o m p r e h e n s i o n have a right
scheme
adjoint.)
is, we ar~ive at the p r o p o s e d
lim ) ((F,X) Or,
equivalently,
>
is the r e q u i r e m e n t
that
Since we know what
equation
A
1A ) Cat)
&
for any small c a t e g o r y
(F,X)
.
B,
IB> lim If
B
second
is discrete,
(B
Cat)
& B 6
ICatl
this holds,
which
accounts
instance of the c o m p r e h e n s i o n
the b e g i n n i n g
of this
this e q u a t i o n
is false,
ical c o m p r e h e n s i o n
section.
w
a number
B
at
is not discrete, there
is no categor-
in this sense.
II. THE SOLUTION
2-CATEGORIES
There are several ways We m e n t i o n
for the
scheme m e n t i o n e d
and t h e r e f o r e
scheme
PART
If
9
of them.
to d e s c r i b e
2-categories.
259
-
-
4.1
i)
The e l e m e n t a r y
In the s p i r i t theory
of a b s t r a c t
independent
position,
called
respectively. elementary addition
are
and
70 71
and
triple
elementary
is an i n t r i n s i c
descrip-
F
for domain,
and the w e a k is to s a t i s f y
of a b s t r a c t
categories
codomain
category
structures,
the a x i o m s ([CCFM],
and com-
for the
p.
2) and in
four axioms,
Ai(Aj(x))
= Aj(Ai(x))
b)
F(x,y;u)
~ >
F(~i(x),Ai(Y);Ai(u)),
i = 0,1
F(x,y;u)
~
F(Ai(x),Ai(Y);Ai(u)),
i = 0,1
F(x,y;u)
and
F (x' ,y' ;u')
~(y,y';v') d) first
of L a w v e r e ' s
a)
c)
The
A 0 , AI
sets of o p e r a t o r s
Each
there
there
2 - c a t e g o r i e ~.
Let
the s t r o n @
theory
of a b s t r a c t
and n o t a t i o n
categories,
tion of 2 - c a t e g o r i e s .
be two
theory
x = A0 x
three
axioms
and
F(v,v';f)
> x = AO x are
i,j = 0,i
and
F(x,x' ;v) and
and ~(u,u';g)
--~
f = g
.
symmetric
in the s t r o n g
and w e a k
structures
and lead to the t h e o r y
of d o u b l e
cate~qries
Ehresmann,
[CS].
says t h a t
in the p i c t u r e
The third
axiom
x I x ilv y
y'
V'
_---~ f = g
as in
-
where strong composition resulting
values
[CS])
from double
says that strong objects denote
and weak composition ii)
and weak horizontal,
The fourth asymmetrical
2-categories
In this paper we generally position
-
is vertical
are the same.
which distinguishes (Ehresmann
2 6 0
are weak objects.
2-categories.
We forgo any extended discussion
general
22
categories
and
is the 2-category
is built from
Note that corresponding
category,
and the "underlying"
categories,
restricted
"locally discrete"
ture is discrete. there is more
(meta)
2-category
Here,
2 0 = i, 2 1 = 2
by
to a 2-category
associated
strong structure
21 .
illustrated
naturally
universal
closed
The
in the same fashion that the basic theory of
abstract 22
of this theory here,
(But, see the appendix.)
idea is that it is a cartesian
built from
by juxta-
(.).
The basic theory of abstract
mostly because of ignorance.
axiom,
categories
strong composition
by dot
the
~,
there are three
the strong category,
category
I~I
which is the
to the weak objects. part of
~;
i.e.,
This
structure
is the
its weak struc-
Part of the point of this paper
"basic"
the weak
than is indicated
is that by this
analogy.
4.2 Cate@orical
Description
of 2-Categories
Suppose we are given an intrinsic
description
of
-
categories.
For the moment,
theory of abstract
of the basic
We denote
and the "set of objects"
functor
lacks
small
hom
since
i.e., with
Cat
A~
the
that is, categories Hom
functor
which
is itself
is r e a s o n a b l y
complete.
objects
of
Cat
by
a
Sets,
by .
attention
one m i g h t want
functors
x A
9 Sets closed
that con-
to c a t e g o r i e s
Hom
is a c a r t e s i a n
may speak of C a t - c a t e g o r i e s [CC];
together with
but that is not the p r o b l e m
HomA: then,
Cat
Cat----> Sets
If we r e s t r i c t
sets;
[CCFM],
some of the p r o p e r t i e s
for the c a t e g o r y of sets, cerns us here.
an object
of d i s c r e t e
I I: Sets
of L a w v e r e
theory and w h i c h
the s u b c a t e g o r y
Note that
-
we take this to mean the basic
categories
the axiom that there exists model
261
category
over
A
with
Sets, we
in the sense of E i l e n b e r g - K e l l y , A
together with a f a c t o r i z a t i o n
of
Cat
through
Cat
A~
and a " c o m p o s i t i o n
rule"
Cat A(A,B) w h i c h is natural tive,
x A
Hom A
> Sets
for any three objects
x Cat A(B,C)
and reduces
a functor)
0 ; CatA (A,C)
in all three variables,
has strict units,
(i.e.,
is strictly
to o r d i n a r y
associa-
composition
on
-
objects.
A
the objects of
and are frequently
morphisms (B~nabou
of
CatA(A,B)
[BC]).
objects of
A
(i.e.,
forget
CatA(A,B)
A, while
are called the 2-cells of
correspond
to the strong objects,
and the 2-cells
etc.,
A
structure
I~I ,
etc.
functor
F:
~Cat will denote
and >
l~I
script
ordinary categories
as a 2-category;
A 2-functor between 2-categories an ordinary
s
the 1-cells to
by underlined
and their underlying by
the
to the basic entities.
2-categories
CatA(-,-))
with its canonical
i.e.,
In
are the m o r p h i s m s
called the 1-cells of
We shall denote ~, B,
such a Cat-category.
In terms of the intrinsic description,
the weak objects,
letters
-
A 2-category is p r e c i s e l y
this description, of
2 6 2
~
l~I
thus
Cat
ICa~tl = Cat.
is a Cat-functor; together with
functors > Cat B (FA,FB)
FA,B: CatA (A,B) which commute with composition of
F
on objects of
CatA(A,B).
In this context, for which
CatA(A,B)
and reduce to the given values
a locally discrete
is a discrete category;
2-category
is one
hence
Hom A CatA(-,- ) =
(A~
Since 2-functors between ordinary
functors,
x A
) Sets(---> Cat)
locally discrete
2-categories
are just
we may and often shall identify categories
with locally discrete
2-categories.
There are a number of obvious gories.
.
Finite limits and colimits
constructions
clearly exist,
on 2-cate-
and there
-
are two kinds of opposite i)
opA; CatoPA(A,B )
Strong opposite s
and, of course, any functor
-
2-categories:
Weak opposite
ii)
263
CatAop(A,B ) = Cat A(B,A)
their combination
K: Cat
> Cat
will denote the 2-category
[CatA(A,B)]op
=
~176
which
.
In general,
is product
for
preserving,
with the same objects
as
A
KA
and in
which
= K[CatA(A,B) ]
CatKA(A,B) except
that we make the obvious
and, as above
~
=
(-)~
2-category" i) ii)
objects
F~A
=
(-)
XA
.
Given two 2-categories, "functor
XA
simplifications
A
and
B,
there is a
as follows :
are 2-functors
F: A
given two such 2-functors,
> F
and
G, then
CatBA (F,G)
is the inverse
limit of the diagram of categories
interconnected
pieces
made up of
of the form Cat (FB,GB)
r011 Cat (FA,GB)
Cat (FA, FB)
! (- ) FA, B I Cat(FA,GA)~
Cat(FA,GB)
Cat(GA~GB) [-)
> Cat(FA,GB)
GA,B
Cat (A,B)
-
for every
A
1-cells of
and B~
B
in
264
A .
-
It is easily checked that the
are the Cat-natural
transformations
of
[CC].
This functor 2-category is adjoint to direct products in the sense that there is an evaluation A • BA e v satisfying
2-functor
B
the usual universal mapping property.
adjointness
is enriched in
Cat;
actually in
This
2-Cat;
i.e. B
(B~) ~
Here 2-Cat refers to some
(possibly nonexistent)
of
It itself is then a 3-category;
(small)
i.e.,
2-categories.
is enriched in 2-Cat.
discrete then so is then if
B ~A ~
B~
Note that if
B
2-category
is locally
and if both are locally discrete
coincides with the usual functor category.
is locally discrete,
Finally,
then
4.3 Set-theoretical Description of 2-Categories
There are at least three ways to accomplish this. i) gories
Start with a set theoretic description of cate-
in which
Cat
or U-categories
is the category of small categories,
for some universe
U .
Then follow the des-
cription in 4.2. ii) 2-categories
Start with the elementary theory of abstract and define a 2-category to be a model of this
-
theory; tions
i.e.,
a set
satisfying
preserving
2 6 5
(or class)
the axioms.
-
equipped
with
A 2-functor
suitable
opera-
is t h e n a f u n c t i o n
the o p e r a t i o n s . iii)
See the d e s c r i p t i o n
Eilenberg-Kelly
w
[CC], p.
of h y p e r c a t e g o r i e s
in
425.
PRO 2 - F U N C T O R S
AND BIFIBRATIONS
Definition
A p a i r of f u n c t o r s A 9 P E Q is c a l l e d
a
i)
(1,0)-bifibration P
ii)
is a l - f i b r a t i o n
Let
EA = P-I(A)
PIEB
The
inclusion
EB
of
>
E
i) and ii).
iv)
v) preserving,
and
A s A
QIE A and
EA
Then
is a 0-
B 6 B . >
E
of f i b r a t i o n s
f~
or
for a b i f i b r a t i o n f*
A cleavage
whose
and in the
PIE B
and
The inclusion
QIE A
is a c h o i c e
existence
is c a l l e d
It is s p l i t - n o r m a l Q,
EB = Q-I(B).
functors
are m a p s
is a 0 - f i b r a t i o n .
sense
[FCC].
A cleavage the f u n c t o r s
and
for all
Q
and
is a l - f i b r a t i o n
fibration iii)
if
is p o s t u l a t e d
split-normal
for e a c h
of all
if
fibration,
P,
.
functors
of iii)
are cleavage
in
- 266 -
vi)
For any
f: A'
in B , f*: EA
and
> A in A > EA'
and
g: B ~ > B'
g,: E B
> E B,
are cleavage preserving. Note that as in the case of fibrations, can be expressed by equations
these conditions
involving the lifting functors.
A cleavage preserving m o r p h i s m between normal
(l,0)-bifibrations
{P,Q} and {P',Q'}
split-
is a commuta-
rive diagram T
E {P'Q}~
>
E'
~P''Q'} AxB
such that
T
structures. over
A
and
is cleavage preserving
The category of split-normal B
will be denoted by
Split(i,j) (A,B) only the
for both fibration (l,0)-bifibrations
Split(l,0) (A,B).
is defined analogously,
(0,1) case is interesting Split(i,i ) (A,B)
but
since
= Spliti(A
• B) .
Proposition There is an adjoint equivalence Split,l,O)~ (A,B)
Proof. {P,Q},
then
> Cat A~
Given a split-normal
~(P,Q):
A~
• B
> Cat
• B
(l,0)-bifibration is the functor whose
- 267 -
value
on
(A,B)
(f: A'
A EA EB = ~ EB
is
> A, g: B
> B')
is
f,g, = g,f*: extends
to a f u n c t o r The
construction
A EB
>
in an o b v i o u s
functor on w h i c h
a n d on
~'
EA' B' way.
is a s p e c i a l i z a t i o n
the w h o l e
of the b a s i c
theory depends,
to w h i c h w e
n o w turn.
Definition
Let from
A
to
A B
and
B
is a 2 - f u n c t o r F:
We F
This
forgo
~F
over
is s p l i t - n o r m a l construction
Cat~~
x B the
~' (F) =
of
However,
a construction NA • NB
this
as an o b j e c t
over an
and
I~I
of
it in two forms,
CatA(A',A ) • CatB(B,B')
and
i~l
~F
N
together )
with
F(A',B')
We
that
is of c r u t i a l
a category-
Cat
• B
~
-
between
form.
F: A ~
function
proof
to
category
equivalence
the
the construction
that
assigns
of 2 - b i f i b r a t i o n s .
notion
f o r m and a s e t - t h e o r e t i c first
which
whose underlying
category
h e r e and we g i v e
Observe regarded
) Cat
establishes
and a s u i t a b l e
I~FI.
theoretic
• B
(1,0)-bifibred
actually
definition
importance
A Op
shall d e s c r i b e
a 2-category
I~FI
A pro-2-functor
be 2 - c a t e g o r i e s .
c a n be
functors F (A,B)
- 268 -
In p a r t i c u l a r , or
B
specializing
and u s i n g
deduce
to the
the c a r t e s i a n
identity morphisms
closed
structure
of
of
A
Cat, we
functors 0B
F(A,B)
x CatB(B,B' )
> F(A,B') 0A
Cat~(A',A) w h i c h we r e g a r d forgo A
a formal
and
B,
diagrams F(A,B)
x F(A,B)
as o p e r a t i o n s treatment
and m e r e l y
) F(A',B)
of
A
and
B
of p r o - 2 - f u n c t o r s
point
out
on
F
We a l s o
as m o d u l e s
(to no o n e ' s
over
surprise)
that
like
x Cat~(B,B')
F(A,B')
• Cat~(B',B")
> F(A,B)
x CatB(B,,B- )
• Cat B
(B,B")
- F (A,B")
m~
commute.
The B a s i c C o n s t r u c t i o n . Then
~B
(A,X,B)
is the 2 - c a t e g o r y w h o s e where
objects,
C a t A ( A , A ' ) x ]i
gory.
A 6 A,
(A,X,B)
• CatB(B,B')
with
Let
codomain
rxl
and x id
B E ~
(A',X' ,B' ),
id • PX '~
F(A,B'),
objects
and
~ F(A,B)
F: A ~
x B
9 Cat
are triples
X 6 F(A,B). one g e t s
Given
two
two f u n c t o r s
x CatB(B,B' )
~B
> F (A,B')
ok
) Cat(A,A')
which
x F(A',B')
therefore
)
F (A,B')
have a comma cate-
We set
Cat~F((A,X,B),(A',X',B'))
.
=
(oB(PXI
x id),
0A(id
• [ X ~ ).
- 269
Composition If ~: F in
> G
-
is illustrated by the adjoining F
and
G
large diagram.
are two pro-2-functors
is a Cat-natural
transformation;
Cat ~Op • B , then for each
~A,B :
and if
i.e., a m o r p h i s m
(A,B)
F(A,B)
> G(A,B)
is a functor and the diagrams F(A,B)
• Cat(B,B')
F(A,B')
Cat(A',A)
• F(A,B)
> F(A',B)
G(A,B)
• Cat(B,B')
) G(A!B')
Cat(A',A)
! G(A,B)
> G(A'!B)
determines
a 2-functor
commute.
Hence
~
~: ~F where
~(A,X,B)
=
> ~G
(A,~A,B(X) ,B)
and
#(A,X,B), (A',X',B'): Cat~F((A'X'B)'
(A',X',B'))
>
Cat E ((A,~(X) ,B) , (A',~(X') ,B'))
NG
is the induced functor between comma categories whose existence grams.
follows
Commutativity
from the preceeding
with composition
as in w
commutative
i), dia-
is an easy diagram
chase. Besides this category-theoretic seem unable
(because of incompetence,
with a set theoretic d e s c r i p t i o n ((A,X,B),
of
Cat~F
~F
- are triples
(A' f X' 8 B')) (f,~,g)
description,
presumably)
later on.
Thus,
we
to dispense the objects
- that is, the 1-cells of
where
f: A
> A' in A ,
-
2 7 0
-
A
A
ii.
/
/~"
•
~
.~
.. ~
~"
v
A
v v
-I-I
~ i.
i.
,_
)
A
v
U X
=.
A A
).
,,
m
v v
4-1
X iii
s v v
4-1
) v
-i-I
~.
m
~/~
i--I
ell
-H
-
271
g: B ----> B' in B
and
The morphisms of
CatEF((A,X,B),
2-cells of
~F
from
~: F(A,g) (X)
(f,~,g)
(~,T)
where
T:
> g' in CatB(B,B')
g
~: f
-
> F(f,B') (X') in F(A,B').
(A',X',B'))
to
- that is the
(f',~',g')
- are pairs
> f' in CatA(A,A'), and the diagram
F(A,g) (X)
~ ) F(f,B') (X')
F(A,~) (X)I
[F(~,B') (X)
F(A,g') (X) ~'> F(f',B') (X') commutes.
The formula for composition of morphisms (h,~,k) (f,~,g) = (hf,~f 9 k~0,kg)
and of 2-cells is
(~',~')(a,T)
=
(~'~,~'T)
.
Proposition If F:
A Op
•
B
H:
~
A'
Cat,
> A,
K: ~'
> B
and
then
~F o (HoP • K)
[
A' x B'
)A•
is a pullback. Examples and Remarks. i) ~,: catAOP x B
As observed before, > Split(l,0) (A,B)
assigns to
is
- 272 -
• B
F: A ~
> Cat
projections
onto
locally discrete discrete;
the c a t e g o r y A
and
and
F
If
A
that
if
its c a n o n i c a l A
then
and ~F
B
are
N
is l o c a l l y
A~
x A
and
> Sets c C a t
,
A*
iii)
Let
~
be a 2 - c a t e g o r y w i t h
CatA:
~A
We define
with
not.
is a c a t e g o r y
HOmA: ~Hom A
Note
is s e t - v a l u e d ,
but otherwise, ii)
then
B .
I~FI
= ~Cat A
• A
, Cat
N
.
with
(6i)A: the canonical
~op
F~unA
projections 9
> A,
We shall
i = 0,i also write
Fun A -- IF unA1 Note
that
if
commutative
F: ~
) B
is a 2 - f u n c t o r
then there
is a
diagram F~unF
A• so t h a t
F ~ u( n- )
As
it p a r t i c i p a t e s
such,
(-) *
on
>BxB
can be r e g a r d e d
as an e n d o f u n c t o r
in t r i p l e s
analogous
Cat.
iv)
In p a r t i c u l a r ,
for
(_) (-) C a t Op
• Cat
> Cat
on 2-Cat.
to t h o s e
for
-
the c o r r e s p o n d i n g jections by are f u n c t o r s
6i:
2-category
~Fun
(A
)
X
B),
diagrams
273
-
~Cat' i = 0 , 1 .
F
>
> X'F
of
~,Fun Fun
X'
B ~: GX
(1-cells)
of
A'
X
where
and the pro-
The o b j e c t s
the m o r p h i s m s
A
Fun
is d e n o t e d by
G
is a n a t u r a l t r a n s f o r m a t i o n ,
composi-
tion b e i n g A
F'
F
X
A
- A" x"
x'
F'F
X
X" ~0'F
B
G'
G
and the 2-cells of
Fun
m A"
B
G'>~B, '
G'G
are d i a g r a m s
B ~ B ' G' where mations
a: F
> F'
and
T: G---~
such that the d i a g r a m GX ~X G X
commutes.
G'
are n a t u r a l
transfor-
(of n a t u r a l t r a n s f o r m a t i o n s ) X'F
Ix,
> X'F'
are
274
-
w
-
2-COMMA CATEGORIES AND SUPER FUNCTOR CATEGORIES
In order to p l a c e the c o n s t r u c t i o n s we are about to m a k e in t h e i r p r o p e r context,
it is n e c e s s a r y to e x p l a i n
the n o t i o n of a c a t e g o r y o b j e c t
A
in a c a t e g o r y
A .
Definition
A
A c a t e g o r y o b j e c t in
is an o b j e c t
A s A
together with a factorization Cat / /
N
Horn (- ,Ab I J
/ s
/ /
AoP
Hom(-,A)
Sets
A f u n c t o r i a l m o r p h i s m b e t w e e n c a t e g o r y objects f in A
such that
Ho~(-,f).
Hom(-,f)
lifts to a n a t u r a l t r a n s f o r m a t i o n
The c a t e g o r y of c a t e g o r y o b j e c t s
the p u l l b a c k Cat A~
AC
L A
i(_),i A~ > Sets
Yoneda We m a y assume,
as in
[CCFM],
is a m o r p h i s m
an i s o m o r p h i s m
Cat ~ Sets({4[}~
in
A
is then
- 275 -
the r i g h t serving fact,
hand
side d e n o t i n g
functors
for any
limits,
from
such
the c a t e g o r y
{41}~
"limit
to
Sets.
theory")
of limit preIt f o l l o w s
that
if
A
has
(in finite
then A C ~ A({4U~
In this r e p r e s e n t a t i o n assigns
to
structure
F:
becomes
Sets
its v a l u e
{~}op___>
of
is d e t e r m i n e d
I (-)*I
AC
is then e a s i l y
by an o b j e c t
the o p e r a t i o n at
deduced.
A 6 A
with
2,
which
F(2).
A category
The object
the f o l l o w i n g
structure:
i)
TWO m o r p h i s m s
~.: A
A,
)
i = 0,i
such t h a t
1
~.~. z ]
=~.
.
] l
ii) the same and
Let
T
I
for i = 0,i
~.x = I . 1
A'
).
A ~
A
be an e q u a l i z e r .
Then
~i = T~i
for
(It is
~i: A
> I
Let 8
:*
A
A"
0 )
> A'
1
i
and
1 A
~
A'
I
> A
be p u l l b a c k s . iii) ~0 Y = ~0 ~
Note:
and
There
is a m o r p h i s m
y: A'
> A
with
~iy = ~i ~ .
iv)
y{~0,A}
= A
and
v)
Y{~,Y~}
= y{u
{-,-}
denotes
induced
7{A,~
}
---- A
~
1
A" morphisms
> A into p u l l b a c k s
(or
-
276
-
out of p u s h o u t s ) . Cocategory about
2
in
[CCFM]
in the u n i v e r s e . any
objects
A ; and,
just say that
Hence
A2
in fact,
i.e.,
as an e n d o f u n c t o r
being
"functorial."
to d e s c r i b e
are d e f i n e d
on
a "comma
category"
construction
This
any c a t e g o r y
object,
but it is of c o u r s e
for c a t e g o r y
objects
in n o n - t r i v i a l
details,
[CCC].
see
Cat
in
for
Cat Cat
is n e e d e d
with
can be d e s c r i b e d
the u s u a l
starting
especially
functor
object
on f u n c t o r s
is all that
properties.
in
object
, its v a l u e s
structure
structure
object
is a c a t e g o r y Cat
The a x i o m s
is a c o c a t e g o r y
is a c a t e g o r y
(_)2
This
2
dually.
with
rich
categories.
For
Proposition
Fun
: 2-Cat
is a c a t e g o r y
2-Cat
N
object
in
(2-cat) (2-cat). Proof. i)
Let
transformation given
by
lA(u)
=
3: Id
> N Fun (-)
such that
T~(A) (a,~).
=
x~: ~
(A,idA,A) ,
be the C a t - n a t u r a l > F~unA
xA(f)
(A c a t e g o r i c a l
=
is the 2 - f u n c t o r
(f,idf,f)
description
of
and x~
is e a s i l y
N
given.)
Clearly
.3 = id
, so,
if we set
1
9 . = 6j ~3
1
and Id
is an e q u a l i z e r
for
6. = 36.
>
6. 1 Fun(_) ~
i = 0,1
.
F~un(_)
1
, then
;
- 277 -
ii)
Let P(
)
, Fun ~(-)
-
6 Fun be a p u l l b a c k Cat-natural
> Id
(of e n d o f u n c t o r s
on 2-Cat).
607 = 60e
by a d i a g r a m
similar
and
> ~Fun (-)
61y = 618
terms,
objects
(A,f,A')
f: A
> A'
PA
where are pairs
.
to the c o m p o s i t i o n
In s e t - t h e o r e t i c
of
Then there is a
transformation P(_)
such that
0
of
This can be d e s c r i b e d diagram
F~unA
~F
"
are of the form
is a 1-cell
of the form
for
in
A , so objects
{ (A,f,A') , (A',g,A") }.
We
set yA{ (A,f,A'), A 1-cell of
~A
(A'g,A") } =
is a d i a g r a m A
A'
m
A"
this
~
and
1-cell
~'
> B
m,~ /~
B'
~ '
Ik
gl
where
(A,gf,A") .
m"
are 2-cells
to be the one cell
~ B"
in
A .
We define
YA
on
- 278 -
m
A
A" of
Fun A
gives
, where
IFunAl
9 = k~
to
properties.
9 ~'f
a different
is e q u i v a l e n t
7
Finally,
.
)
B
>
B"
Note
2-category
being
of
satisfying
PA
I! ~
and
~A each A N
on
this
and
is
natural
iii)
If
A
N
Q(_)
P (-) is a p u l l b a c k ,
then
one
operation
looks
a fact the
which
stated
like
II
(o,~"). in
this
structure;
a functor
a 2-celi
that
7A
is t h e n
. v
~
easily
~
P(_)
.~ F u n (-) checks
that
a 2-functor
for
279
-
-
Definition Let [F0,F l ]
F.: A. 1 -~I
)
is the inverse F0
archy
is
(F0'FI)
(F0,FI) 2 = tions
a 2-comma = F0 B• F1
[F0,FI],
on 2-comma i)
, be
0,i
2-functors.
61
Fun B
> _B ~
is called
i =
Then
limit of the d i a g r a m ~0
40
[F ,F ] 0 1
B,
etc.)
F1 > ~B ~
41
9
category.
(The general
for sets,
(F0,FI) 1 =
hier(F0,FI),
we have various
As in w
opera-
categories.
Induced
functors [L 0 ,M, FunM,M, L l ]
{G O , • {G~,G} ~ ' ~
L A0
over
~0
over
L 0 x L1
• ~i
and
• ~i
L 0 • LI;
as in w ii)
except
There
exactly
iii) definition.
u:
,
A' NA'0 • N1
[F0,F I]
as
in
[F0,F I] that
B2
>
!
>
[F~,FI]
w
• ~]
with
[FI'F2] B3
The c o r r e s p o n d e n c e
[FoL0,F 1,LI ] ,
is r e p l a c e d
is an a s s o c i a t i v e
[F0'FI] derived
and
i
induces
a = {P,Q'a,R}: exactly
• L 0
Fun B
composition >
replaced
in w
by
[F0,F 2] by
iii)
~B
"
becomes
a
- 280 -
Definition If
F,G:
transformation
are
B
~
• ~A .
Cat-natural
m
induced
Note
that
Exactly
,
/
m0:
is n o t
as in
w
using
composition
9 F0
corresponding and
F1
ml:
>
[G0,G I]
to 2 - n a t u r a l )
G 1
transformations
.
Definition Let functor
A
category
are 2 - f u n c t o r s
and
B
A
to
cat un
is the 2 - c a t e g o r y B
Ostensibly,
the super
whose
objects
and such that
(F,G)
I is a pullback.
Then
be 2 - c a t e g o r i e s .
Fu~n(~,B)
from
a
one d e r i v e s
functors
A0 • AI GO
transformation
in general.
[m0,m I] : [F0,F I] over
a 2-natural
[F,G]
a 2-natural
transformation, iv)
then
is a 2 - f u n c t o r A
over
2-functors
[F,G] A
r~
this p u l l b a c k
is a 2-category,
-
but it is easily [F,G] ~
means
Composition
281
-
seen to be locally discrete.
the 2-functor
category described
is the induced m o r p h i s m
Cat(F0,FI)
Note that in
w
in the d i a g r a m
)
([F0,FI]
[F 1 ,F2 ]~A
in the usual
fashion,
one deduces
x Cat(FI,F2) %
Using this c o m p o s i t i o n following
the
result.
Proposition.
Fun(-,-)
:
(2-Cat)~
•
(2-Cat)
>
2-Cat
is a 2-functor. For future rendous like. G
set-theoretical
9: F i)
from
) G
ii)
A
to
we need to know in hor-
e x a c t l y what
B
Fun(A,B)
one deduces
that if
then a 2-natural
looks F
to an object
A E ~
, a morphism
F(A)
> G(A)
to a m o r p h i s m
(1-cell)
in f: A
and
transfor-
assigns
s
diagram
detail
From the above d e s c r i p t i o n
are 2-functors
marion
calculations,
> B of A , a
282
-
F(f)
F(A)
) F (B)
~A G (A)
G(f)
such that
a)
b)
for a composition
gf in ~
~gf = ~gF(f)
9 G(g)~f
for a 2-cell
one has
> f' in A , one has a
~: f
commutative diagram F(f) F (A) F
(
o
)
/
F (B)
F(f')
~A
~f/~f, G(f)
G(A )~ ( / ~ ~ ~ ~ ) / G ( B ) G(f') "I-e.
t
in
Cat B (F (A) ,i G (B)), the diagram
G (f) %oA -
~f
/
~B F (f)
I s
G (f')~0A
commutes.
~f,
(~ )
~OBF (f')
s176
283
-
If F
to
t: ~.
G
and
(i.e.,
~
are 2 - n a t u r a l
1-cells
) ~ in Fun(A,B) > ~A in 6
t A : ~0A o: f
~
) f' in A
-
of
transformations
Fun(~,B)),
assigns
such that
then
to each
from
a 2-cell
A s A
a 2-cell
N
for any 2-cell
, the d i a g r a m F(f) F (A)
It
F (~)/
F (B)
:l'Bl t~/ 1PB
F(f')
\
+~\tYJ /
G(f)
GCA) ~
G(~/
%oy\
j
+f,/~ G(BI
G(f') commutes.
This
commutativity
commutativity
can be e x p r e s s e d
either
as
of the cube
s G (f)s A
> ~OBF (f) BF (f) G f) ~!AJ
~f
--
> ~JBF (f)
G (o)s A G(o)~ A s
~f,
G(f')~A
(o)
SBF(O) ) ~)BF(f ' )
G(f')< G f')~A
~f,
> ~BF(f ' )
-
284
-
or of the square ~0f G f)~A
>
~B F (f)
tBF (e)
G(c)t A ~f,
G f')~A in
CatB(F(A),G(B)).
~BF(f ' )
Actually,
only the c o m m u t a t i v i t y
it is sufficient
of the top of the cube
but this does not so g r a p h i c a l l y fact,
used up all the available Finally,
natural
if
~: F
transformations
~:
of 2-cells
The prefix 2-subfunctor natural
means
equivalence
2-
that we have,
and F
~: G > H
> H
are 2-
is given by
~ ~f~0A
is simply will
refer
a monomorphism means
in
(st) A = sAt A to
Fun(~,B);
F~un(~,B)
an isomorphism
in
9 thus a
and a 2-
Fun(~,B).
Proposition B~
is a sub 2-category
of
Fun(~,B)
imbedding (-) (-) is a Cat-natural
f ,
= ~A~A
(~0) f = ~Bs The composition
for all
structure.
> G
then
(~)A
illustrate
to require
> Fun(-,-)
transformation.
and the
in
- 285
Exercise. In p a r t i c u l a r ,
Compare
show
-
A 22 ,
F u n ( 2 2 , ~)
and
F~unA .
F~unA ~ ~ 1 7 6
Examples. i)
[H,K]
(1,0)-bifibred
category
~CatB(H(_),K(_))
cular,
if
functor
"
and as a 2 - c a t e g o r y
I [~,Catl I ~ C a ~
3)
For any >
~
Fun =
5)
Fun(l,B)
6)
> If
(see
F: X
4)
~A: A
as a s p l i t
to
A0 • A 1 9
2)
F = X
is i s o m o r p h i c
> A0 • A1
~>
~ Cat, Cat
[Cat,Cat]
~
~ B
and
i.e.,
EF =
F~unA =
and for any
B
locally
for f u n c t o r s
= ~
In p a r t i • X
[~,~].
~
Fun(~A,B): are
[I,F].
[~,F]
, then and
induces
A
w
, the c o n s t a n t B
> Fu~n(A,B).
discrete,
between
then
locally
discrete
categories [F,G] 7) notions
The
object
object.
Then
that
looks
[CCFM],
p.
theory.
of
Cat
B =
[1,~]
B
(6) Cat
.
[F,G]
Thus,
and let
like the o b j e c t 17,
(F,G)
2-categories
in c a t e g o r y
terminal
=
are u s e f u l
let ~:
l: ~
~
9 and
of
Cat
> Cat
> Cat
is the c a t e g o r y
for c e r t a i n be the
be any o t h e r
in the u n i v e r s e
.
In the n o t a t i o n
9 = BCa t .
There
of
is a c a n o n i -
cal i m b e d d i n g B = and
B
[i,~]
is c o c o m p l e t e
[l,Cat,Fun,Cat,~] if and o n l y if t h i s
>
[Cat,~] functor
has a left
- 286 -
adjoint lira: [Cat,B] This is a f u n c t o r denoting F=:
shows,
D=---> B
The proof
for i n s t a n c e ,
such t h a t
the v a l u e s
lira
of
F
by
lim
F(A))
exists
that
w
A
F=
9
[Cat,B]
[Cat,B],
then
and
.
~
lim
>
IT
[Cat,B]
where
in
> B
= lim
was
[Cat,B]
lira
BA <
Note
F: A
from the diagram
[Cat,~]
(lim~.~)A
.
F(A) : D A B ,
(I_~
B
t h a t if
F = F=
we get t h a t in
follows
>
>
introduced
it is d e n o t e d
by
I] B
by G r o t h e n d i e c k
CatllB .
in
[SGA],
It w a s u t i l i z e d
by
D
Giraud
[MD]
and H o f m a n n
[CCEF], w h o
lim:> As s t i l l two o b j e c t s
of
Cat
another
[Cat,B] use,
Cat
>
note
is ful l
=
C
the f u n c t o r
B
that
if
, then [B,~]
(assuming that
also discusses
B
in the u n i v e r s e ) .
B
and
~
are
- 287 -
w
Given category
two f u n c t o r i a l
objects
morphisms
o n e can d e s c r i b e
the c o r r e s p o n d i n g
comma
category
the u s u a l p r o p e r t i e s . it w o r k s
2-ADJOINTNESS
(See
between
a p a i r of
adjointness
in t e r m s of
constructions
and d e r i v e
[CCC].)
In the p r e s e n t
case,
out as follows.
Definition
Let
F:
A 2-ad~unction
B
) A
and
U: A
>
be 2 - f u n c t o r s .
is an i s o m o r p h i s m [F,A]
~
>
[B,U]
B• of 2 - c a t e g o r i e s
over
B • A .
Proposition
If tions,
then
adjunction
~ #
is also a m o r p h i s m corresponds
between
Proof.
F
Since
[B,U] ~ ~ C a t A ( _ , U ( _ ) )
and
of s p l i t
to an o r d i n a r y
(1,0)-bifibra-
Cat-enriched
U .
[F,A] = s we have
A (F (-) ,-)
and
-
2 8 8
E~CatA (F (-) ,- )
\
Under
-
/ then
the h y p o t h e s e s ,
> ~ C a t B ( _ , U ( _ ))
corresponds
to a C a t - n a t u r a l
isomorphism CatA(F(-) ,-) ~ Remark.
In the case of o r d i n a r y
(F,A)
= EHom(F(_),_ )
crete
i-fibres,
crete
bifibrations
is a s p l i t
i = O,1. over
p h i s m of b i f i b r a t i o n s transformation
CatB(-,U(-))
and h e n c e
categories,
(l,O)-bifibration
A functor B • A
.
between
is e a s i l y
such dis-
seen to be a m o r -
corresponds
of the r e s p e c t i v e
two
with dis-
functors.
to a n a t u r a l Thus
an o r d i n a r y
adjunction (F,A) ~
\ /
(B,U)
BxA is always
(assuming
as a n a t u r a l
A
and
B
are
locally
In the case of 2 - c a t e g o r i e s shall
the
same
equivalence H o m A ( F ( _ ) , - ) _T__> H o m B ( - , U ( - ) )
as we
small)
see by e x a m p l e
these later.
conditions
. are not e q u i v a l e n t ,
-
w
The n o t i o n "functor
categories"
of l i m i t s and of
w
we e q u a l l y
example
well
tor)
the 2 - l i m i t
to be the r i g h t
it exists.
We w i s h
there
on the n o t i o n s
"adjointness",
>
BA
have
being
of
the a d j o i n t
imbedding
.
the c o n s t a n t
imbedding
: B N
functor
(resp.,
(resp.,
left)
to c a l c u l a t e
2-1im>A: is the u n i q u e
depends
5),
AAB = F u n ( T A , B ) and we d e f i n e
IN CAT
of the c o n s t a n t B
(see
-
2-COLIMITS
on one side or the o t h e r
In our case,
289
the 2 c o l i m i t
2-adjoint
2-colimits
~Fun(A'C~at)
of in
A~ Cat
func-
, when .
Thus
> Cat
(up to a 2 - i s o m o r p h i s m )
2-functor
such that
is an e q u i v a l e n c e
[2-1im A' CaCti
[Fun (~, Cat) A t] ~ ' ACa
/
\ Fun(A,Cat)
x Cat
Theorem
i) ii) then
Let
F: A
) Cat.
In g e n e r a l ,
2-1im) AF =
I
[IbF]
I
if 9
F: A
Then
2-1im)AF
) Cat
=
[I,F]
.
is a 2 - f u n c t o r
-
2 9 0
-
Corollary
For
IA: A
Note. 0-fibred makes
p. 17).
if
exists.
of
A
A =
2-1im, AF
the only proof
Cat
F .
This only r e a l l y
is c a t - c o m p l e t e
in the universe,
([CCFM],
We have
providing
separated
[~,F]
the t h e o r e m
Example
I know for ii)
(C~at,~) 0
denotes
(The Y o n e d a - l i k e
7.)
This proof
equivalence.
0-fibrations
lemma,
is an e x p l i c i t
the full s u b c a t e g o r y
mined by s p l i t - n o r m a l
A =
(See w
form in terms of the following
of the r e q u i r e d
tative d i a g r a m
by
is the split
and will only prove part i) for a small cate-
[1,A], A 6 Cat.
Let
2-1im>AF
the t h e o r e m can also be read as a s s e r t i n g
struction
Lemma
determined
is small and
has a nice c o n c e p t u a l whereas
, 2-1im A1 A ~ A .
says that
(See the appendix.)
into two cases gory
A
However,
the existence
i> Cat
The t h e o r e m
c a t e g o r y over
sense
> ~
over
con-
In the lemma, of
(Cat,A)
deter-
A .
Lemma)
[I,A], A s Cat
.
Then there is a commu-
-
291
-
F u n (J
A/
~
~Cat
/
rl
Cat
]
[I, - ] "
Cat
/ (cat ~ ) 0 where
the v e r t i c a l
Proof triangle angle
is t r i v i a l . category
w
also
= EF every
split
need
only worry
Fu~n(A,Ca~t),
on
that its
the
component.
that
if
F:
A
0-fibration over
morphisms
of the
of the
denotes
first
out
G
that
P
is a s p l i t
about
commutativity
while
0-fibration
~: F
If
The
w
pointed
, which
more
3, Note
of a c o m m a
[i,F]
is an e q u i v a l e n c e .
of Lemma.
is E x a m p l e
it was
functor
and
hand
right
hand
usual
projection
In E x a m p l e > Cat,
over
A arises
left
A
this
tri-
3,
then .
Further-
way,
so we
2-cells.
is a 2 - n a t u r a l
transformation
in
then [I,F]
[i-~]
>
[I,G]
A is a m o r p h i s m is a 2 - c e l l
in
over
A = i
Fun(A,Cat)
• A
.
then
Furthermore, t
determines
if
t: ~ ---> a natural
- 292 -
transformation
~: [i,~] over
A
whose component
~
[i,~] (~,a,A)
at the object
6
~I,F]
is given by ~(l,a,A) and
(t,A)
(l,~A (a) ,A) ----> ~i, ~A (a) ,A)
= (1'ta'A):
is a 2-cell of Conversely,
(cat,~) 0
-
suppose [I,F]~
T
[~A,G]
A commutes
(C~at,~0).
(i.e., is a 1-cell of
there is a unique
~: F
> G
with
x =
We must show that [I,~].
Define i
to be the 2-natural is the functor
transformation
~A: F(A)
i)
If
a 6 F(A),
ii)
If
f
on a m o r p h i s m
then
is a m o r p h i s m
~0A(f) = 9 (1,f,A) ; i.e., ~m
> G(A)
whose component
~A(a) in
= m(l,a,A). F(A),
> B in A , observe
to the morphism 1
/ F(A)
F(m)
A s A
such that
then
~0A = 3 I ~ , F ] A = YIF(A).
m: A
at
,
F(B)
that
To define assigns
-
in
293
-
[I,F], a morphism I
T (l,a,A)/
~(l,id,m)/ G(m)
G(A) in
[I,G].
We set
transformation
tion over
such that if
A , let
corresponding t A : ~0A
9 ~'
a
(~m)
Finally,
~(I,F
Note.
t
tion of w functors mations
~: F
> x' (X,a,A)
0-fibrations
over
corresponds A .
Thus,
On the other hand, T:
[I,F] > G
with components
II
This lemma says that, while
Fun(A,C~at)
such fibrations
the
.
ponds to split-normal morphisms,
transforma-
> ~0'A(a)
s (X,a,A) : T(I,a,A) =
is a 2-natural
whose component
transformation
II II s
~
is a natural
transformations
(tA)a: s
Then clearly
Then
be the 2-cell between
9 ~'
is the natural
A
(B)
[~,~]. > x'
s:
2-natural
~G ~
T (l,id,m).
=
~ =
t: ~
(m) a,B)
>
[X,G]
looks
Cat A
and cleavage preserving to all functors
it generalizes
A
between
the proposi-
the correspondence
over
corres-
between
and natural
like the Yoneda
lemma
(w
transforwhich
-
gives
a correspondence
2-functors
[Cat,F]
Proof
294
between >
-
2-natural
[Cat,G]
over
of the T h e o r e m .
over
represented
(Cat,~)
@
~
to e s t a b l i s h 0
and
• A .
isomorphisms
[Fun~(A,Cat)~ ,A ACat ]
[P,Cat]
cated,
Cat
We h a v e v e r t i c a l
[ [l,-] ,Cat]
so it is s u f f i c i e n t
transformations
• Cat
.
• ~]
[(Cat,~)0,-
an e q u i v a l e n c e
An object
of
@
as i n d i -
[P,Cat]
can be
by a d i a g r a m M
E
X
~ A
where
Q
is a s p l i t n o r m a l
[ (Cat,~%) ,-
• ~%]
looks
0-fibration,
while
an o b j e c t
of
like a d i a g r a m E
N
>
X
• A
A On o b j e c t s , gives
set
a bijection
l o o k like
~(M,Q)
= {M,Q}:
between
E
objects.
> X • A . Morphisms
This on e a c h
clearly side
295
M
E
-
- X
E
)
N
/\
and
X
x
A
//
E'
A
where
prx
= id.
Set ~(R,~,S)
This the
gives left
a bijection
is a p a i r
are natural ~(p,u)
=
=
(R,{~,Q},S
between
(p,~)
(p,a
x A).
1-cells.
where
transformations Then
.
Finally,
p: R
compatible ~
• A)
> R' with
a 2-cell
and ~
is an e q u i v a l e n c e
and of
o:
~'. 2-cate-
gories.
Examples. i) natural F:
A
Since
I is t e r m i n a l
transformation ) Cat
.
The
F
> 1A
induced
just
the
canonical
Since
any
, there
is a
functor
> 2-1im> A1 A
projection
~I,F] ii)
for
Cat
functor
2-1im>AF is
in
the
constant
> A imbedding
factors
S
via
on > S' Set
- 296 -
B
there
is an i n d u c e d
-
-
functor
[BA, A ]
> [Fu.sn
[lim>A, ~]
which,
Fun (A,B)
/
-
by the Y o n e d a
lemma,
)
[2-1ira)A, ~]
corresponds
to a 2 - n a t u r a l
trans-
formation
2-ii%A AS an e x a m p l e ,
A =
of R - m o d u l e s .
groups,
since iii)
since
if
F: A
(Rings) Op
Then
o v e r all rings,
and
is the c a t e g o r y
is the c a t e g o r y
of all m o d u l e s
lim_ F
is the c a t e g o r y
of a b e l i a n
is a t e r m i n a l
Sets
F(R)
2-1im F
while Z
lim A
c Cat
> Sets
object
of
(Rings) ~
is n o t c l o s e d u n d e r
.
2-colimits
then
2-1ira F =
[I,F]
=
(1,F)
f
and this
is d i s c r e t e iv)
only
2-1imits
if
in
t u r n o u t to be the c a t e g o r y If, were
in the d e f i n i t i o n required
A Cat
is d i s c r e t e . can a l s o be c a l c u l a t e d
of s e c t i o n s
of 2 - n a t u r a l
to be e q u i v a l e n c e s ,
of
~,F]
transformation,
over all
t h e n the c o r r e s p o n d i n g
and A . ~f
-
notion
of "2-limit"
long ago was
would
called
w
give
lim
THE
297
-
"cartesian
in
first
Let
A
which
[SGA].
2-COMPREHENSION
We m u s t
sections"
describe
SCHEME
the
2-Kan
extension.
Definition
2-adjoint
F:
> B
(when it exists)
by
~2 F
the
(left)
2-Kan
the
left
to 9 Fun (A,X)
F* = Fun(F,X) : Fun(B,X) is c a l l e d
Then
be a 2 - f u n c t o r .
extension
of
F
.
It is d e n o t e d
.
Proposition
Given 2-functor
whose
H: -A value
> -X , t h e n on any
72F(H):
B 6
NB
is g i v e n
> --X is the by
[Z2F(H) ] (B) = 2-1ira) ([F,B]
Corollary
For
1A =
(A
9 i
7 2 F ( I A) (-) =
Proofs. by u s i n g tion
The
the c o r o l l a r y
is p r o v e d
1
9 Cat),
[F,-]:
corollary
B
that
> Cat
follows
to the t h e o r e m s
by v e r i f y i n g
we h a v e
f r o m the p r o p o s i t i o n
in
the u s u a l
w
The p r o p o s i construction
still
- 298
makes
sense.
Thus,
-
define > A H
[~2F(H)] (-) = 2 - 1 i m ) ( [ F , - ]
T h e n we m u s t p r o d u c e #:
Fun(~,~)
over
objects K: NB
> NX .
2-natural
an e q u i v a l e n c e
[T2F,F~un(B,X)]
• Fun(~,~).
on b o t h
sides
I: Z2F(H)
.
component
at
(H,K)
X
.
But,
X
the
2-functor
by d e f i n i t i o n
, corresponds
>
KF
> ~
.
An
the
and
over
is a 2 - n a t u r a l
object
H,K
is a
of
transformation
I , observe
t h a t its
of
> K(B)
[72F(H)] (B) , any m o r p h i s m
[Z2F(H)] (B)
to a 2 - n a t u r a l
[F,B]
> A H > X
determined
by
in
a morphism
[F,B]
H: ~
[~2 F(H)] (B)
m: in
first describe
is a m o r p h i s m 1 B:
in
H
To u n d e r s t a n d
B 6 B
[Fun (A,X) ,F*]
[F~un(~,~),F*] ~:
over
> K
a given
of
transformation
,
To do so, we
over
An o b j e c t
[E2F,Fun(~,~)]
X)
)
X 6 X
.
m'
F(A)
transformation to the c o n s t a n t
assigns
H(A)
> X
to an o b j e c t
) X in X F(f)
<'_
B
m'
from
2-functor (F(A)
and to a m o r p h i s m
> F(A')
> B)
-
in
[F,B]
299
-
a diagram H(A)
H(f)
>
H(A')
X
in
X
. Now,
given
~: H----> K F
, define
I = ~' (~) : ~2F(H) to be the 2 - n a t u r a l
transformation
AB: [Z2 F(H) ] (B) corresponds on
<
above
to the 2 - n a t u r a l
> K
whose
component
> K(B)
.
!
transformation
is the c o m p o s e d
IB
whose value
diagram
H(A)
H(f)
> H(A')
KF (A)
KF (f)
_
(
)
K (B)
Conversely,
given
I: Z2F(H)
= ~(I): to be the 2 - n a t u r a l i) is an o b j e c t
transformation
For each object of
H
[F,FA],
> K
, define
> KF constructed
A 6 ~
, F(A):
so the a d j u n c t i o n
as f o l l o w s :
F(A)
morphism
> F(A)
- 300
eFA: H(A) is defined.
[7.2F(H) ] (FA)
We set ~A =
ii) composed
>
-
(H(A)
EFA>
AFA
[7.2F (H) ] (FA)
For a m o r p h i s m
f: A.
> A',
> KFA
r
. is the
square H(f)
H (A)
> H(A')
EF(fy
/ > Z2F(H) (FA')
Z2F(H) (FA)
IFA '
>'FA
/
KFA It is e a s i l y
checked
omit the m o r e Theorem
KF ( f Y
that
and
#'
lengthy v e r i f i c a t i o n
(2-Comprehension Let
~
X =
> KFA '
are 2-functors.
that
r
= ~-i
We
.
Scheme)
[l~],
X
6 Cat
.
Then the 2 - f u n c t o r s
7.2 (-) (1) [~
9
> ~
[ ,-]
have the f o l l o w i n g
property:
[7 2 (-) (i) ,~
(X,Ca~t) ] 9
There
~
(X ,Cat)
exist 2 - f u n c t o r s
r
such that
~.
[ [~
[I,-] ]
-
i) ii)
~
is(enriched)
Restricted
[72 (-) (i) ,~
-
left adjoint
X) ]
>
and hence on these
2-adjoint
to
~'
[[~
subcategories
), [I,-]]
~2(-) (i)
is left
[I,-] Restricted
further
to
(72 (-) (i) ,Cat X) defines
to
to the subcategories
~. = #-I
iii)
301
an ordinary
((Cat,X() , [i,-])
adjunction z 2 (-) (i)
Cat X
(Cat,~) [i, -] Lemma
> OP(ca~t X) c ~
Z2 (-) (i(_)) : [~ Proof of Lemma. F
as a functor
in
F
works as indicated. F: A
> X.
gories,
Since
The 2-Kan extension
is very sensitive An object of
A
and
X
an object
diagram
1A
to variances
[~
along and only
is a functor
are locally discrete
as 2-cate-
is the functor
72F (IA)
(F,-) : X i.e.,
of
~
X.
> Cat
A morphism
in
; [~
is a
-
302
A
-
M
> A
X
where
m: F ' M
mines
> F
a natural
is a n a t u r a l
transformation {M,m}:
whose
This d e t e r -
transformation.
)
(F,-)
(F' -)
component {M,m}x:
is the f u n c t o r
which
(F,X)
takes
>
(F',X)
the o b j e c t
FA
> X
of
(F,X)
to the o b j e c t
F'MA of
(F' , X);
i " e.
{M,m} x
8
(F,X) It is e a s i l y 2-natural,
-~ F A
is the c o m p o s i t i o n (F'M,X)
seen t h a t
> X
{M,m}
~>
is n a t u r a l ,
(F'X) and not just
so t h a t 7 2(M,m) (i A) = {M,m}
is a m o r p h i s m
of
~
X.
Finally,
is a d i a g r a m
M
X
a 2-cell
in
[~
303
-
where m
t: N
9 F't
= n.
{M,n}
to
{M,m},
whose
> M
is a n a t u r a l
This
determines
{N,n};
i.e.,
a 2-cell
component
v a l u e on
FA
transformation
a 2-cell
at
X
in
> X
in Cat X
such that
~
X)
from
{N,n}
is the n a t u r a l
from to
transformation
> {M,m} x
{N,n} x whose
-
(F,X)
in
is the m o r p h i s m
F't
F'NA
> F'MA
FA
X in
(F',X).
Proof
of T h e o r e m .
By the Y o n e d a - l i k e
>
[~,-]: Fun(X,Cq~at) and hence,
if we wish,
we m a y r e g a r d
[~,-] : ~ Using
(X ,Cat)
this and the p r e c e e d i n g
strictions
of the
The f u n c t o r s
are d e s c r i b e d 2-natural
as follows:
and
side
[~ q~at,~K ] 9
it f o l l o w s
t h a t the re-
~'
[[~
An o b j e c t o:
[Cat,X]
as i n d i c a t e d .
(X,Cat) ]
transformation
the r i g h t h a n d
~
c
w
it as a 2 - f u n c t o r >
lemma,
functors map
[7 2 (-) (1) ,~
(Cat,~)
l e m m a of
(F,-)
is a d i a g r a m
on the
K ], [3-,-]]
left h a n d
> K, w h e r e a s
side is a
an o b j e c t
on
- 304 -
A
The
functor
M
M
is d e s c r i b e d
M = {MI,M 2 }
where
MI:
A
A 6 A, M2(A)
6 K(MI(A)),
~
[I,K]
by a p a i r > X
while
is a f u n c t e r , f: A
for
M 2(f) : K ( M 1 (f)) (S 2(A)) W i t h the d e s c r i p t i o n , m: M 1
in
A,
.
transformation,
o:
(F,-)
> K,
define
~(o)
to be the
for w h i c h M(A) "= { F ( A ) , ~ F A ( i d F A ) }
If
> A'
> M2(A')
is a n a t u r a l
and for
> F. Given
object
m
of c o m p o n e n t s ,
g: A
> A'
, let
g
.
be the m a p FA
F(@)
) FA'
FA ' in
(F,A')
and set M(g)
Finally,
= {F(g) '~FA' (~)0 (~F(g))idF(A)
m: M 1
> F
Conversely, 2-natural ~X:
(F,X)
(A,f:
F(A)
is the given
transformation > K(X) ) X)
o:
identity. M, m,
define
(F,-)
> K
~'(M,m)
(F,X)
is
to be the
such that
is the f u n c t o r w h o s e v a l u e in
.
K ( f m A) (M 2(A))
on an o b j e c t and on a m a p
-
F(A)
305
-
F(~)
> F(A')
X
is
K(f'mA, ) (M 2 (g)) . The a c t i o n
ably more enriched
of
complicated. natural
and
general
be p u b l i s h e d
of the t h e o r e m
adjointness, of t y p e s
come out more
It s e e m e d
(A
>
dualization
comprehension
In any case,
than
as
that
1 A
Cat)
However,
gives
rise
then this along
the
- that = A~
desirable
. to this w i t h o u t
than getting
b u t t h i s m a y be o v e r l y
s e e m to be g e n u i n e
As w i t h
if o n e c o n c e n t r a t e d
as the a n s w e r
exhibited
in the
do n o t as y e t
0-fibrations,
approach
was more
scheme,
the p h e n o m e n a
scheme
which
naturally.
1
to me t h a t g e t t i n g
an a r t i f i c i a l
prehension
rather
find - for thi s 2-1im
of d u a l i z a t i o n
It is p o s s i b l e
statement
w a y one w o u l d
is a b i t m y s t e r i o u s .
of 2 - a d j o i n t n e s s
as b a s i c
might
behave
of w e a k d u a l i z a t i o n s
the p o s s i b i l i t y
names.
operations
elsewhere.
statement
special
and of the
) r
that these
on l - f i b r a t i o n s
neatest
id
The d i s t r i b u t i o n
to a n u m b e r deserve
~id,
Note.
ordinary
of this
is c o n s i d e r -
transformations
as the v e r i f i c a t i o n
indicated will
on m o r p h i s m s
The descriptions
o r as w e l l
~'
by the p r o o f
the
provincial.
of the c o m -
and n o t a r t i f a c t s
of the
-
306
-
notation.
APPENDIX
Some Remarks
on a C a t e g o r i c a l
The possibility, giving a f o u n d a t i o n the interesting
of M a t h e m a t i c s
indicated by L a w v e r e
for m a t h e m a t i c s
p r o b l e m of finding
axiomatization. uses.
Foundations
in c a t e g o r i c a l the correct
The answer of course depends
I would
in
[CCFM]
of
terms raises
form for such an on the intended
like to suggest here that there is in fact a
whole h i e r a r c h y expressability
of theories poses
and that their m u t u a l
a strong r e s t r a i n t
inter-
on the form of any one
of them. Specifically, where
an n - c a t e g o r y
(n-1)-categories. are categories, etc.
there
is the h i e r a r c h y
is a "category" Here
0-categories
2-categories
sets
(as suggested
"hom-objects"
are sets,
are as d e s c r i b e d
At the top are u-categories
simplicial
whose
whose
of n-categories,
1-categories in this paper,
"hom-objects"
by Epstein).
are
are
This sequence
should have certain properties. i) n-categories. n = 2
There
should be an e l e m e n t a r y
In this paper,
and the general
Nothing finitely
new happens
we have given
theory of a b s t r a c t this theory
case is easily d e r i v e d
until
~
from that.
and it is not clear
that this is
axiomatizable. ii)
There should be a basic
for
theory of abstract
-
n-categories. paper
there
However,
The case
a)
There
i-cells,
things
and a u n i q u e
Finite
limits
tor n - c a t e g o r i e s " there
here.
There
but t h e i r
n = 2, I c a n n o t to
Fun(A,-).
functor c)
d) theory; an
There i.e.,
n-categories themselves, iii)
associated
in,
are
"func-
Besides
this,
"hyper
functor
presented
tensor
products,
t h a t at the m o m e n t , is a d j o i n t
for
or 2 - a d j o i n t
of t h e s e h y p e r
theory.
of f u n c t o r s
from n-categories
n-categories.
is an o b j e c t w h i c h of
is a m o d e l
(small)
apparently,
is c e r t a i n l y
There
crutial.
n = 2
This
not.
and t h e r e
(n - i) m o r e
- | A
in the b a s i c
an n - c a t e g o r y
This
more
of
has two
2 0 = i.)
product.
as in the c a s e
are a n u m b e r
(n+l)-category
properties.
exist
is so c o m p l i c a t e d
discrete"
n = 2.
which
(Except,
to c a r t e s i a n
if
2n
I do n o t k n o w the d e s c r i p t i o n
There
for
is a p p a r e n t l y
certainly
e v e n tell
categories
to " l o c a l l y
2
sequence
constructed
structure
n-cell.
but
adjoint
are a l m o s t
is n e e d e d
n-category
and c o l i m i t s
is an i n c r e a s i n g
n-categories"
of w h a t
can be s a i d in g e n e r a l .
is f i n i t e l y d e s c r i b a b l e , b)
is n o t c l e a r y e t and in this
is a g e n e r a t i n g
i < n
-
n = 1
is an i n d i c a t i o n
certain
307
n
If we r e g a r d
different
theory
of n - c a t e g o r i e s
t h e n we also r e q u i r e
the
It is
ways.
list of d e s i r a b l e
requirements
the b a s i c
as a d e s c r i p t i o n
m-categories
n-categories.
n o t an e x h a u s t i v e
are some f u r t h e r
of the b a s i c
t h a t are e v e n
of a b s t r a c t in t e r m s
of
following.
s h o u l d be e x p r e s s a b l e
in t e r m s
of
-
n-categories
for
m < n.
n-categories
-
By induction,
simply e x p r e s s e d r e q u i r e m e n t t h e o r y of n - c a t e g o r i e s ,
308
this reduces to the
that the axioms of the basic
r e s t r i c t e d to "locally d i s c r e t e "
should imply the axioms of the basic t h e o r y of
(n-1)-categories. iv)
n-categories
m-categories
for
n > m.
in terms of 0 - c a t e g o r i e s
should be e x p r e s s a b l e Clearly,
In particular,
lation
Cat ~ Sets ({4}~
can be d e s c r i b e d
= sets, but one w o u l d h o p e that the
e l e g a n c e of the d e s c r i p t i o n w o u l d n - m.
n-categories
in terms of
increase with decreasing
t h e r e should be an a n a l o g u e of the rebetween models
of
(n-l)-categories
and n - c a t e g o r i e s . We suggest that a p r o p e r
f o u n d a t i o n of m a t h e m a t i c s
is an e l e m e n t a r y a x i o m a t i z a t i o n of the h i e r a r c h y d e s c r i b e d here. The actual
status of the p r o g r a m o u t l i n e d
r a t h e r m e a g e r c o m p a r e d w i t h its g r a n d i o s e
above is
intentions.
L a w v e r e has g i v e n a b e a u t i f u l d i s c u s s i o n of the i n t e r c o n n e c t i o n s b e t w e e n the cases
n = 0
and
n = i,
([CCFM]
that the basic t h e o r y of a b s t r a c t c a t e g o r i e s is i n a d e q u a t e
for the results
but it is k n o w n as p r e s e n t e d there
claimed.
In this p a p e r we h a v e tried to see w h a t h a p p e n s the case
n = 2
is included,
for 2 - c a t e g o r i e s of course, examples
of
by t r y i n g to d i s c u s s
if
constructions
b o t h in terms of sets and of categories.
And,
there is i n t e r p l a y g o i n g d o w n as e x e m p l i f i e d by the w
and the m a i n r e s u l t of this paper.
seems that there is a g l a r i n g
inadequacy
However,
in the b a s i c t h e o r y
it
-
of categories; 0-fibration
309
-
namely, we cannot construct the universal
(i.e.,
Cat
of w
in the basic theory.
needed is an axiom that looks like the operation theory.
Let
Cat
What is
Ux
in set-
be a model of the basic theory which is
full in the universe,
and for any A
6 Cat, let
A
be the
corresponding category in the universe that looks like
~.
([CCFM], p. 17.) Axiom There exists
P: Ca~
>! Cat with the following
properties:
i) I A: A ---> Cat
Given
~
6 Cat, there is an imbedding
such that I A
A
~
Cat
~
Cat
commutes. ii)
Given
F: A
>
B,
there is
H:
A
x 2
Cat
such that
Axaol~A A
A x ~
H
A x ~
> Cat
F
>
I IB B
> Cat
and Pr I
AxalI A
H --
2
1p f
> Cat
-
3 1 0
-
commute. p, iii)
If
Cat'
> Cat
then there is a unique
also satisfies
functor over
i) and ii),
Cat, N
Cat
> Cat '
Cat preserving
the structure Hopefully,
tion of saying EF
w
If not,
that for each
> A • B
described
this
is e n o u g h to give the basic construc-
this could be d e s c r i b e d F: A ~
• B
with properties
assume that once categories
theory by g i v i n g categories
there is a c a t e g o r y to those of
Cat.
We then
T h e o r e m and the P r e d i c a t i v e [CCFM],
it is p o s s i b l e
used in this paper w i t h i n
categorical
Cat(A,B)
by an axiom scheme
of these sizes are available,
Scheme of
struct the 2-categories
> Cat
analogous
using the C a t e g o r y C o n s t r u c t i o n Functor-Construction
in i) and ii).
formulas
category
for the objects,
and for the composition~
to con-
for the
-
3 1 1
-
BIBLIOGRAPHY [BC]
B~nabou, Jean.
"Introduction to Bicategories,
of the Midwest Category Seminar",
Mathematics,
Vol.
Reports
Lecture Notes in
47, Springer-Verlag
Inc., New York,
(1967). [CS]
Ehresmann,
Categories
C.
et Structures,
Dunod, Paris,
(1965) .
[CC]
Eilenberg,
S. and Kelly, G. M.
"Closed Categories",
Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag [MD]
Giraud, Jean.
Math. [GOD]
"M~thode de la Descente",
France, M~moire 2; VIII + 150 p.,
Godement,
Topologie
R.
Faisceaux, [FCC]
Inc., New York.
Alg~brique
Paris, Hermann,
Gray, J. W.
Bull.
Soc.
(1964).
et Th~orie des
(1958).
"Fibred and Cofibred Categories",
Pro-
ceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag Inc., New York. [CCC]
Gray, J. W.
The Calculus
of Comma Categories,
(to
appear). [SGA]
Grothendieck,
A.
Categories
Fibr~es
S~minaire de G~om~trie Alg~brique, Etudes Scientifiques, [CCEF] Hofmann,
K. H.
Paris,
et Descente,
Institute des Hautes
(1961).
"Categories with Convergences,
Exponen-
tial Functors and the Cohomology of Compact Abelian Groups", Math.
Zeitschr.,
104; 106-140,
(1968).
-
[AF]
Kan, D. M.
Lawvere,
-
"Adjoint Functors",
87; 295-329, [CCFM]
3 1 2
Trans. Amer. Math. Soc.,
(1958).
F. W.
"The Category Qf Categories as a Foun-
dation for Mathematics", on Categorical Algebra,
Proceedings of the Conference La Jolla 1965, Springer-Verlag
Inc., New York. [ETH]
Lawvere,
F. W. Lecture,
ETH, Z~rich, November 17, 1966,
(unpublished) . [CVHOL] Lawvere, F. W.
Category-Valued
Higher Order Logic,
(to appear).
[AC]
Linton, F. E. Functors",
"Autonomous Categories and Duality of
J. Algebra,
2; 315-349,
(1965).
313
-
NON-ABELIAN
-
SHEAF C O H O M O L O G Y
BY DERIVED
FUNCTORS
by R. T. Hoobler*
INTRODUCTION
Given an a r b i t r a r y G r o t h e n d i e c k sheaf of groups HI(x;G), G
G
in that topology,
the first c o h o m o l o g y
evaluated
algebraic
at
X , is well
geometry
stems
sheaves
> H Z ( X ; G ')
of groups
G'
sequence of pointed
set with c o e f f i c i e n t s known.
(for the given topology)
for
up to i s o m o r p h i s m
define
H2(X;G)
spaces.
HI(X;G)
sites for u n d e r s t a n d i n g
of
[6].
classified
and then e x t e n d e d
He
locally
homogeneous
spaces
this a p p r o a c h
classes classes
to
of gerbes of p r i n c i p a l
are of course numerous
prerequi-
this approach.
Since the b o u n d a r y m a p of sheaves of groups
extension
principal
local e q u i v a l e n c e There
theory.
g i v i n g a nine term exact
as a set of e q u i v a l e n c e
w h i c h are e s s e n t i a l l y homogeneous
descent
sets has been solved by G i r a u d
trivial
in
and a c o n n e c t i n g m a p
for a central
> G --~ G"
of
in
Its u s e f u l n e s s
H2(X;G)
adopted the point of v i e w that
G
and a
the c o n s t r u c t i o n
from G r o t h e n d i e c k ' s
The p r o b l e m of c o n s t r u c t i n g 61: HI(X;G '')
topology
6I
for a central
in the ~tale t o p o l o g y
plays
extension
a key role
*I w i s h to express a p p r e c i a t i o n for support e x t e n d e d by the N a t i o n a l Science F o u n d a t i o n through c o n t r a c t NSF GP 8718.
-
314
-
in the theory of Brauer groups of schemes having other applications, cohomology
the exactness
nine term sequence cohomology
Amitsur
this map
[12].
is well known
sheaf cohomology, case
although
topology non-abelian
Amitsur
abelian group c o h o m o l o g y
of
[5, C h a p t e r
G , a sheaf of groups,
Godement's
flask r e s o l u t i o n
[3] and
I, T h e o r e m
7.6].
is to give a set of
a functorial
"flask resolution"
can be p r o d u c e d which resembles of a sheaf of abelian groups.
c a t e g o r y of sheaves
sets with group action.
into "flask"
the p r o c e d u r e s resolutions we can
of pointed
sheaves.
a functorial
injection
The other axioms
of h o m o l o g i c a l
algebra
of m u l t i p l i c a t i o n
One
of such sheaves
then allow us to copy
and diagrams
(using group actions
in abelian groups)
in the
and so to produce
of "short exact sequences"
"diagram chase"
as
agrees w i t h non-
we define quotients
then gives
for
group c o h o m o l o g y
In order to get a "resolution"
of the axioms
for
since in an a p p r o p r i a t e
cohomology
from w h i c h
61
For group
it is m u c h harder
The essence of our approach axioms on a t o p o l o g y
sets.
is also easily d e f i n e d
this contains 3.3)
Its
a b o u n d a r y map
by Dedecker
However,
(see T h e o r e m
here.
[ii] and has even been
extensions
[5,10].
a non-abelian
of the c o r r e s p o n d i n g
and pointed
The b o u n d a r y map
cohomology
a special
properties
of groups
for n o n - c e n t r a l
Springer
is p r e s e n t e d
is T h e o r e m 3.2 which produces
and d e s c r i b e s
defined
we have d e v e l o p e d
theory part of w h i c h
culmination
[8] as well as
on sets
in w h i c h instead
to get the d e s i r e d
results.
-
315
-
A n u m b e r of e x a m p l e s are g i v e n w h i c h cations.
include the a b o v e a p p l i -
This a p p r o a c h to the p r o b l e m was s u g g e s t e d by
Grothendieck's non-abelian
[9] and A r t i n ' s
HI
[2] r e s u l t s
s h o w i n g that the
is an e f f a c a b l e functor.
Unfortunately
the d e f i n i t i o n s
and t e c h n i q u e s d e v e l -
oped here do not give an e x a c t nine t e r m c o h o m o l o g y associated G' G of
to an e x t e n s i o n of sheaves of g r o u p s
> G ---> G" and
sequence
G" = G/G'
where .
G'
The m a i n d i f f i c u l t y
H 2 ( X , G ') ---> H2(X,G)
b e t w e e n the v a r i o u s
is a normal
.
s u b g r o u p sheaf of is in the d e f i n i t i o n
A d e s c r i p t i o n of the r e l a t i o n s
c o h o m o l o g y groups a r i s i n g f r o m such a
s e q u e n c e can be given,
but it is b e y o n d the scope of this
p r e s e n t p r e l i m i n a r y r e p o r t on this approach.
S i m i l a r prob-
lems arise in G i r a u d ' s w o r k and are s o l v e d there w i t h the aid of the n o t i o n of t w i s t i n g by cocycles. central e x t e n s i o n dence H2
G'
9 G
H 2 ( X ; G ') ---o H2(X;G)
) G"
In f a c t for a
, he only has a c o r r e s p o n -
which makes
it u n l i k e l y
is the same as the one d e f i n e d here.
that his
It w o u l d be inter-
e s t i n g to find a u n i v e r s a l m a p p i n g p r o p e r t y for n o n - a b e l i a n cohomology
so that the v a r i o u s d e f i n i t i o n s
m o r e easily.
This exists for
HI
could be c o m p a r e d
and is the b a s i s of the
proof of T h e o r e m 3.3. We have a d o p t e d the n o t a t i o n and basic d e f i n i t i o n s of
[i] in this work.
T h o s e r e a d e r s f a m i l i a r w i t h the m a t e r i a l
on g e n e r a l G r o t h e n d i e c k t o p o l o g i e s late the s t a t e m e n t s
and results
in
[2] can r e a d i l y trans-
into the l a n g u a g e u s e d there.
-
The
increase
here.
in o b s c u r i t y
We w i l l
a v o i d all
t h a t all c a t e g o r i e s by u s i n g
316
-
does not
set t h e o r e t i c
are small.
questions
The reader
the t h e o r y of u n i v e r s e s .
be c o v a r i a n t ,
s e e m to j u s t i f y u s i n g
can
Finally
and g i v e n a c a t e g o r y
by assuming
justify
this
all f u n c t o r s
C , CO
it
will denote
will the
dual c a t e g o r y .
w
W e w i l l be S:
PRELIMINARIES
interested
Category
in the f o l l o w i n g
categories:
of sets a n d set m a p s .
S~
Category
s_G:
Category whose
of p o i n t e d
sets and point preserving
maps. objects
a right group
action not necessarily
(S,G), w h e r e
S • G
s 9 (glg2)
=
g l' g2 E G
~ S
are p o i n t e d preserving
and
s 9 1--s
and w h o s e m o r p h i s m s
are pairs
f(s
the p o i n t ,
satisfies
(s 9 gl ) " g2
8 s M o r G, w i t h
sets w i t h
9 g) = f(s)
9 8(g)
for all (f,8), for all
s s S,
f s M o r S', s 6 S,
g 6 G . GS:
Category
whose
objects
a left g r o u p a c t i o n n o t n e c e s s a r i l y (G,S), w h e r e (glg2)
G x S
9 s = gl
g l' g2 s G
" (g2
) S " s)
Category
preserving
sets w i t h
the p o i n t ,
satisfies and
and whose morphisms Gs_Gz
are p o i n t e d
1 9 s = s are pairs
of p o i n t e d
for all of m a p s
sets w i t h
s s S,
as above.
a left and
-
right group
action,
(S,G 2) 6 S_G gl s GI' maps
and
2:
(gl
9 s)
(GI,S)
" g2 = gl
6 GS,
" (s 9 g2 )
and w h o s e m o r p h i s m s
conditions
Category
.Ab:
-
(GI,S,G2) , w h e r e
s s S, g2 6 G 2
satisfying
3 1 7
analogous
of g r o u p s
Category
for all
are t r i p l e s
of
to the above.
and g r o u p h o m o m o r p h i s m s .
of a b e l i a n
groups
and g r o u p h o m o -
morphisms. The c o m p o n e n t s w i l l be d e n o t e d show w h e t h e r
by the same
it is a g r o u p
The d i s t i n g u i s h e d written Given
as
e
and the
r i g h t or on the set on w h i c h
obvious will
point
M s Gs_G, let
for o b j e c t s
of a m o r p h i s m
since
homomorphism
in a p o i n t e d identity
M G or G M
and SG
GM
and
.
Note
connecting
not be g i v e n
explicit
names
rely on the c o n t e x t
working
in.
~ ---~ Gs_G
gotten
left and r i g h t
these
in o r d e r
by a l l o w i n g
example
as
there
a group
from
fibred
products
(Ui---~U}i61
set map. be
1 . on the
be the p o i n t e d
are
w i l l be u s e d several
categories.
to s i m p l i f y category
These
notation.
we are
is a f u n c t o r
to act on itself
by
translation.
N o w fix o n c e and for all a G r o t h e n d i e c k Recall
will
acting
notation
to show w h i c h
As an i m p o r t a n t
Ms
that t h e r e
functors
always
G
let
Similar
forgetful
We w i l l
or a p o i n t e d
the g r o u p
and
or GsG
the c o n t e x t
set w i l l
denote
act. GS
s_G, GS,
of a g r o u p
left r e s p e c t i v e l y ,
MG in
letter
in
[1] that this m e a n s and a set
, where
giving
Cov T
in each f a m i l y
a category
of f a m i l i e s of m a p s
U
topology T
of m a p s
T.
with in
is f i x e d
T, and
-
which
(i)
If
~
(2)
If
{Ui~-~i U}
is an isomorphism,
, {Vij~iJ>ui } E Cov T , then
If
E Coy T , V ---~ U E Mor T
{Ui--~U}
T to ~ on
)V}
with values of
{Ui--~U}
F (U)
)
F (U)
) HF (Ui) I
HF(Ui) I
~
~
where various
factor
consisting
between
by
of
S (C), is
of presheaves
is exact;
• Ui2) U
F
where for
that is,
of * Pi
from the pro-
comes
i = 1 or 2 .
and m o r p h i s m s
by finite
in
products
and m o r p h i s m s
such that various
G, GsG _ , _S G , G~, or ~"
functors
The category
C , denoted
or final object,
respectively
functors
E Cov T ,
H F(Uil IxI
can be d e s c r i b e d
_C = Ab,
P(~).
that the objects
the "one point" or S
i th
on a category
of c o n t r a v a r i a n t
is the equalizer
j ection onto the
P(S),
in
H F(U i x U i ) I• 1 U 2
Note
with values
by
2(~)
such that for any
HF(Ui) I
T
is the category
the full subcategory
>
on
and is denoted T
, then
E Cov T .
of presheaves
with products
or ~
.
E Coy T .
The category
sheaves
{~} E Coy T
{Vij~U}
{U i x V U
P(~),
-
satisfy:
(3)
from
318
these values
~
S (C), of objects,
in
S(S),
diagrams
commute
Alternatively of
C
and
S
the give
-
identifications
of
P(C)
319
and
S(C)
set with group action objects functors
from
preserve
equalizers
inverse from
limits,
C,
Moreover, then
S,
i
P(S),
i: S(S)
and
to S, P(S)
to C,
S(S).
i.e.,
inverse
limits
to them.
in
P(S),
the f o r m a t i o n
commutes with finite Hence equalizers
C
S(S),
in
respectively. functor,
and so products
and the final object. may be computed
to functors
and
will often be used w i t h o u t
For instance
or S
which
finite
is the inclusion
---> P(S)
and
Thus
or S(S)
P(C) , or S(C)
These o b s e r v a t i o n s
referring S(S),
or S(S)
P(S)
and the final object c o r r e s p o n d
preserves
equalizers.
in
with group objects
and finite products,
P(C) , or S(C) if
-
P(S),
in
or S
of equalizers
inverse
limits
S(C) , P(C) , or
respectively
where
is any of the above categories. Artin's functor,
construction
#: P(Ab)
a left adjoint
~
S(Ab),
#: P(S)
[1] of the s h e a f i f i c a t i o n carries
~ S(S)
to
over d i r e c t l y i: S(S)
to give
9 P(S)
which
m
preserves
finite
it also gives
inverse
limits and the final object.
a left adjoint
#: P(C)
--~ S(C)
where
Thus C
is
any of the above categories.
We will give a brief d e s c r i p t i o n
of this p r o c e d u r e
fix notation.
Given of
U .
in order~to U 6 T , let
Thus the objects
{Ui--'>U } i61
of
and a m o r p h i s m
Ju
be the c a t e g o r y of coverings
JU
are coverings
~:
{Ui--~U}i6 1 --~
{Vj
~U}j6 J
D
is a f u n c t i o n
%o: I
) J
and maps
~0i: U i
) V~(i)
6 Mor T
-
such that
Ui
~ V~(i)
\/
320
-
commutes.
U be the corresponding that is, if
and
ob ~U = ob Ju
i.e.,
Note that
a+({U i
~+({U
9
Moreover,
~+: JU "-~ JV
=
{U i U• V
Now suppose
)V}
C
[1], and so
given
and
)U}) = {U i x V ~-V} U
>U})
V ~
For instance ning of let
w
v H~
C
i
: j0 U
is a left complete category
~,~.9 {U i {V. x Vj 3 1U 2
>U} I
such that direct
satisfying LI, L2, and L3 of
F 6 P(C)
C .
and
{U.--~U}
[i] exist.
at the begin-
s Cov T ,
P* (KF(U i) ~ I ~
~ F(U i x U 1 U IxI
))
12
is functorial,
Suppose we are given
) {Vj
>U}jx J
given
.
~-U};F) = Equalizer
)
lim --0 Ju
and
Since the formation of equalizers ~0(;F)
it is a one
~ ~V
might be any of the categories
For
=
U 6 Mor T ,
~+: ~U
(possesses arbitrary products and equalizers) limits over categories
)U})
is a connected directed category,
it satisfies L1, L2, and L3 of
we have a functor by
and otherwise
--
~U
preserves monomorphisms.
;U},{Vj
Mor -- ({U Ju i
Mor ju({Ui--~U},{Vj--->U})
element set.
partially ordered category;
)U}j s Mor JU .
is the product of
Since {Vj
>U}
with itself
.
-
in
Ju
that Pi:
' there
{Vj
maps
onto
{Vj
~ V. 32
i
~U}
such
JxJ
where
) {Vj
)U}j
comes
from the p r o j e c t i o n
J• i t--h factor 9
the
v H~
p2 ~ = ~
)U}
• V. i U 3
-
.--~U} --~ ~: {UI I
is
and
pl ~ =
3 2 1
This
iv ~-U};F)
gives
KF(V_.) J 3
P* ~ ~
HF(U.)
--->
a diagram
~ F(V_. • V. ) J• 91 U 32
iu
1 where
s
~ F(U_.
monomorphisms
induced
maps
• Ui )
9
u
i
= nF(~0 i) , 4" = ~F(~i ) , etc.,
canonical the
I
and
V
of the e q u a l i z e r s ,
between
iUg = ~*i v = ~*p~i v = r
the e q u a l i z e r s 9 V = ~*i V
2
and
i
are the
U
9,
~
Now
~U~
and so
~ = ~
_
Thus
This
~0({
vH0(
a functor +
: P(C)
Note
that
coming
from
F s P(C)
and
on
Define
; ): TO • P(C)
in
is a n a t u r a l
HIs176 = I~F(~~ {Ui
~~
= lim> H~
U and F
---> C .
be the f u n c t o r
Let
(+F) (U) = ~0 (U;F)
transformation
F(U)
--~
Artin
we
7~F(Ui)
i: I for any
introduce
the c o n d i t i o n
F s P(C) :
(+) : For all
};F).
, and so gives
U} 6 Coy T .
Following (+)
, --C .
is f u n c t o r i a l
---> P(C) there
.
v
};F) : J~
construction
are
{Ui---~U}is I 6 Cov T, F(U) --
> ~F(U i) I
+
- 322 -
is a m o n o m o r p h i s m . that for
Then copying Artin's
~ = ~ , (+F)
that if
F
E P(S)
satisfies
satisfies
(+)
proof for
(+), then
[i], we see
F s P(~)
(+F)
and
is a sheaf.
m
Let
#: P(~)
~ S(~)
easily v e r i f i e d
be given by
that
#
lim
#(F)
is a left adjoint to
Moreover,
since
preserves
finite products
over connected,
tion functor p r e s e r v e s
= +(+F).
finite
inverse
limits.
is a l r e a d y
sheafification
functor
adjoint
for any of the above values
i
#: P(C)
We will c o n s t r u c t in a larger one, However,
of q u o t i e n t s
to the reader.
which
the one
is left C .
by e m b e d d i n g
and r e p e a t i n g requires
For the r e m a i n d e r
of this
an o b j e c t
the process.
a number
of
~ , and
Pi: M1
is any a m b i g u i t y
which c a t e g o r y Let The kernel of
of
will
S(~),
the context will
is being c o n s t r u c t e d be a map
(a), KerP(u), u and e .
P(~),
will be the c a n o n i c a l
in notation,
~: M ---> M"
is the e q u a l i z e r
in
i t--h factor of a product.
the object
u, Ker
C
Ab, G, Gs_G, GS, _ _S G , or ~',
• M2 --'~ Mi
p r o j e c t i o n m a p onto the
and will be left
section
will be the trivial m a p or final object
there
of
w h i c h are easily proven
denote one of the c a t e g o r i e s
or
Since
is an induced
S(C)
resolutions
categories
concepts w h i c h are given below along w i t h some
of their p r o p e r t i e s
e
~
taking a quotient,
the d e f i n i t i o n
preliminary
there
as desired.
the s h e a f i f i c a -
point presheaf
to
a sheaf,
i
directed
and equalizers,
It is
in
S(~),
or Ker
Note that
Where indicate
in. P(C),
or
~
.
(a)
respectively,
iKer
(e) = KerP(iu).
323
-
s Mor and
S (C) or
only
image
if
of
object
e,
of
Mor
P(C)
is o n e - t o - o n e
~(M),
eP(M),
through
if the
notational if
e
M 1 e ) M2 ~
M3
if t h e
image
p(sG),
or ~G,
N
object
of
, then
or
or to
MG
0N(G),
action.
of
MG
acts
left g r o u p
denoted the
sheaf,
general, for For
some
G(U)
Ms
.
since
the
G 6 S(G), subgroup
a normal
or
if and
for all
P(G)
presheaf,
in
only
U 6 T
Ms
if
a subsheaf if
.
H(U) If
H is
M e ---- M"
M 6 S (S G) ,
If G
H
is a s u b g r o u p
under
S
M2
G, O N ( G ) , the
0 e ( M G),
subobjects
0 p ( M G) ,
with
determine
0~(G),
group
respect
and will
be
which
is a n o r m a l subgroup
is the
S (G) , P(G) , or G
P(G),
or
For
8 9
under
will
H
that
at
and
M
.
says
is e x a c t
or n o r m a l
- if
= #(ie p ( i M ) ) . This
similarly
or G
sub-
respectively.
of
in
of
context
subobject
a map
on
The
A sequence
N
are d e f i n e d
.
or S
Ms
of
Orbits
e(M) M"
kernel of
if
Thus
or S"
N • G
transitively
Let
G E S(G)
is n o r m a l
of
on.
~
the
.
smallest
M >~ ) M"
P(S')
the o r b i t
acts
normal
P(S),
write
SCS'),
actions
as a b o v e
group
, and
is a s u b o b j e c t
equals
0 e ( M G)
U 6 T
respectively.
image
U 6 T
is t h e
equals
equals
is t h e
all
factors.
we will
in
a monomorphism
~
S(S),
or o n t o
for
e(M),
of
in
purposes,
is m o n i c
for
image
is an e p i m o r p h i s m
or
which
eP(M) (U) = e(U) (M(U)) is o n t o
is of c o u r s e
~ (U)
M"
-
kernel
side
subgroup - in
of
respectively.
or s u b p r e s h e a f
is a n o r m a l a subgroup
H of G
subgroup object
of of
G
-
and
GI, G 2
a normal
are subgroup
subobject,
of
G 1 • G2
in
G
subgroup
normalizer
subgroup,
subobject
G of G
subobject
of
lim H
forms
G
containing
is a normal
object of
sheaf,
G
subobject
generated
normalizer
NG(H) , N~(H),
containing of
subobjects
object
H
of
under the m u l t i p l i c a t i o n
normalizer
normal
-
objects
then
G 1 9 G 2 , the subgroup
324
G . of
G
a directed
H
NG(U) (H(U)) iN G(H)
Let the or
NG(H) , be the largest
such that
This always
exists
containing
H
H
is a
since the set
as a normal
sub-
system by the above remark and so
subobject.
m N p(H) (U)
containing
U 6 ~ ,
Note that for any
or
G
NG(H) (U) , and for
G 6 S(G),
= N p (ill) 1G
if defines
M ~ s(sG),
P(s ~) or s_G, o: M s • M G
the group action on the right,
a subobject,
then the s t a b i l i z e r
or
is the largest
StMG(N),
that
of
presheaf,
over this system is a s u b g r o u p object of as a normal
as
by the image
map.
subgroup
H
N • H
As above, dition exists.
the subgroup objects
If
stabilizer
system,
of
N in M
and
) MS
j: H
H ~ MG
satisfying
StPG(N) such
.
this con-
and so the s t a b i l i z e r MG
is
> MG
N • H ~(ixj)~ MS iP 1
is a s u b g r o u p object of of
i: N
N in M, StMG(N),
subgroup object
is the e q u a l i z e r
form a d i r e c t e d
of
and
~ Ms
always
containing
the
N • G ---~ M S , d e f i n e d by the
-
restriction StMG(N)'
of
G , factors
StaG(N),
Stabilizers
or
of g r o u p
actions
right
we leave
to the r e a d e r
same way
since
acts
and left.
be a m a p
~: M"S • MG
M"
via
by of ~
will
Define
~
and
be d e n o t e d side
quotients.
Let
G .
of
by inter-
properties
which
to d e f i n e
S(S G) , P(S G),
of
an induced be
G M''
which
in the the
QrP(e) S , or in
G M'' (~)
be these
G M''
action this
groups
on
QS
action
with
or S"
on
their
by
the a c t i o n
Let QS
action
respectively,
of p o i n t e d
be d e n o t e d
"
quotient
to be the co-
P(S'),
with
of
p(Gs_G ) or Gs_G
epimorphism
commutes
action
of the r i g h t
Qr(e)S,
S(S'),
l
the g r o u p
M" s S (GsG),
set c o m p o n e n t
with
or S G
the set c o m p o n e n t
e
let this
G M'' (e)
be
similar
determine
~ , and s u p p o s e
the left a c t i o n has
subobject
left are d e f i n e d
and will
be the c o r r e s p o n d i n g
Temporarily
G s''
in
M, Q r(~)S,
equalizer and let
on the
) M s'' be the m a p d e f i n i n g
respectively. of
is a n o r m a l
to prove
the c o n t e x t
, then
on.
M"
M G on M~
N
They have
We are now r e a d y ~-9 M---~
-
through
StMG(N)
changing
group
325
QS
sets.
"
of
Since
M~
,
G M'' (~) , PM" (~) , or "
Temporarily on
QS
and
let St
St.s ~''(~) (Qr(e)S)_ , St pGM'' (e) (QP(e)S) , or 9 StGM..(e) (Qr(~)S)" c~Qr(~) , GQ P(~) r , or
Define presheaf category group
of groups, M"
action
or g r o u p
is in) on
QS
GQr(~)
GM"(a)/St
together "
Note
to be the sheaf
with
that
if
(depending
the c o r r e s p o n d i n g M" s s(Gs_G),
of groups, on the left
then
-
Suppose
GQ_r(e) = #(GQP(i=)) . objects
of
M Gf!
el' ~2
of
H I, H 2
326
-
that
on
giving commutative
QS
I#
QS • Hi - ~ i = 1 or 2 .
M~
on
M~ .
H I N H 2 c_ M~ right of
.
Thus
H I 9 H2
acton on
QS
M~(e),
stand for
a1 = c2
of
Let
QS "
lim M Gl!
respectively.
in
together with their together GsG .
gives
Let
Sr
or
M~(u)
denote this
system of all
the above condition.
NM~(U(M G)) , N P ,,(uP (MG) ) , or G St
temporarily
StPM~(e ) (QP(u)S),
or
Qr (~ )G' QrP(~) S , o r
, Pr
or G
induced action on
Qr (~') ' QrP (~) ' or
Q' G Q' QS' and
to
for which such a group
If we temporarily
As above let
StM~(e ) (Qr(e)S),
M~r
M~
PM~(u),
over the directed
StM~(u) (Qr (~ )S ) ' and" define to be
M~(e),
satisfying
contains
when restricted
define a group action on the
Ul' a2
then it is
ai
and the group action of
can be defined.
,, MG(e)
NMG (~ (MG) )
Hi
is an epimorphism,
subgroup object of
subgroup objects Note that
by
on
QS
# x Hi
In particular
be the largest
by
Since
determined
diagrams
M~
# x Hi I
is uniquely
are subgroup
such that there are right group actions
MS" x H i ~
for
HI, H 2
QG
Qr (~)
Qr (~) G
respectively, QS "
Putting
this
in
s(GsG),
p(GsG),
temporarily
stand for these
327
-
right q u o t i e n t s ~: M" ---> Q
and their components
preserves
MG
is a normal
preserves
the right group action.
the right group a c t i o n ~
particular,
is onto if
G M''
acts t r a n s i t i v e l y or
G
then
Q
M~
and
on
QS
as a set with
QG "
functorial Q~(~)s(U) behavior
to
p(Gs),
M
is a normal
group
M, M"
in
preserves |!
MG(a).
.
In
, then
GQ
are in
S(G),
P(G),
subobject
S (G), P(G),
of
M"
coming
of the d e f i n i t i o n
set component. U s T .
coequalizers.
since
of
is
In p a r t i c u l a r The f u n c t o r i a l
is not as nice h o w e v e r
Finally
#
or G
of taking right q u o t i e n t s
for all
later.
also
M~
because
of the group c o m p o n e n t s
i , it p r e s e r v e s
~
to
or G S
~
left and right g r o u p actions
on the p o i n t e d = Qr(e(U))s
, then
on
If
Then
and if the
In any case
"
The o p e r a t i o n
will be d i s c u s s e d
M~
acts t r a n s i t i v e l y
from left and right t r a n s l a t i o n GQ
of
is r e s t r i c t e d
s(Gs),
in
is the q u o t i e n t
regarded
subobject
if
and the image of
__
respectively.
the left g r o u p action,
image of
Moreover,
-
#
Thus
and
is a left a d j o i n t Q_r(~)S = #(QP(i~) S),
and so e
is exact of
e
in
Or ---> M~ ---~ QS
S(S') , P(S') , or S"
under
the image of
(See P r o p o s i t i o n is trivial
)
2.2.)
if and only
on the image of
MG
where
in
)
Or
e
is the orbit
S(S') , P(S'),
In p a r t i c u l a r if the image of
or S"
the image of MG
.
M S in QS
acts t r a n s i t i v e l y
MS .
Left q u o t i e n t
objects
are d e f i n e d
similarly
and
-
will be denoted above.
by replacing
The above
statements
hold for left quotients. verification
w
In this of
with
with
l
in the notation
left and right
We leave their
AND N O N - A B E L I A N
section we first
interchanged
statement
and
some additional
us to define
a canonical
Several
examples
section
applied
"flask"
on
topological
The results
cohomology
theory
for groups
connecting
map Grothendieck
group of a scheme.
The remainder
the analogue
of the
enable
of the
[7].
last
the c o n n e c t i n g of groups
a non-abelian
and profinite used
which
of sheaves
spaces,
Then
~ la G o d e m e n t
to them enable us to define
on non p a r a c o m p a c t
the exactness
T
resolution
extensions
ALGEBRA
of quotients.
assumptions
are then given.
for central
to d e v e l o p i n g
HOMOLOGICAL
investigate
#, i, and the formation
we introduce
homomorphism
r
-
up to the reader.
FLASK RESOLUTIONS
properties
3 2 8
groups,
and the
in his study of the Brauer of this
section
3 • 3
lemma
is devoted
in our
setting. Definition
2.1
Given S' ~
M ~-~ M"
or S_G
if
monic
and
e
M' , M, M" E s(GsG), is a short exact
is a map
M" _~ Q_r(~),
in
S(s_G),
QP(e) , or
p(GsG),
sequence
or GsG
in
t
S(s_G),
p(sG),
or S G
Qr(~)
in
which
s(Gs_),
p(sG), is
p(Gs),
329
-
or
GS
respectively.
in
S(S_G) , P(S_G) , or S G
or _S G
which
M
if
e(M')
of sets
in
and
-
- M
for
is a m a p
of sets
>-
8(M)
in
sequence
S(Ab)
sequence
in
at
S , or
exact
Replacing
of a s h o r t
exact
M
in
S"
Qr(~)
at
sequence
r i g h t by
sequence
s(Gs),
of the e x a c t n e s s
is c o n t a i n e d
S ( S G) , P(S G) ,
S , QP(~)
is a s h o r t
of sets
of sets
S(S G) , P(S G) , or S G
P(s_G), or S_G .
The a n a l o g u e i
~
exact
MS" ~-- -Qr(~)
the d e f i n i t i o n
and an e x a c t
and
if
sequence
S(s_G),
left g i v e s
It is a s h o r t
is m o n i c
It is an e x a c t
-
(of sets)
p(Gs),
properties
in the f o l l o w i n g
or GS
of
.
#
two p r o p o -
sitions.
Proposition
2.2
l) S(S') exact
with
~
in
any U 6 ~
P(S')
in
QP(e) S . e (M')
r
If
in
r
If
transitively
~I
M v s S(G),
and
on
of sets
~
is
{U i
in
r
P(S').
iM ~
GM
and
acts
in the c e n t e r
8
8 for
and
exact
of
factors
in
with
MG
monic.
iM"
sequence
is an e x a c t
transitively
iM '-e > iM 8_> e
, then
iM"
M s , then P(S_G)
If
.
is a s h o r t
~
in
is an
U} I 6 JU
r
monic
sequence
iM ~ > iM"
monic
8 __ M"
i0 e(M~)
with
is c o n t a i n e d
iM' >~ >
) (y) =
M' -e - M
P(S')
is an e x a c t
G, GsG , G S , S G , or S"
there
S(s_G), then
M"
with
C = Ab,
with
sequence
then
, x 6 M"(U),
2)
sequence
in
S(C),
y 6 7~Mr
of sets
M' > ~ > M ~ monic,
sequence
is o n t o
or
If
and
through on MG
MS acts
is an e x a c t
-
Proof. ( ~
monic
there
eP (M') = is (M')) .
from the fact that
{Ui--->U} I 6 JU
is
Suppose
(the a r g u m e n t
similar).
represented *(Y) Pl
=
But now
by
*(y) P2
The rest follows
M 6 P(C),
y 6 IIM(Ui) I
(~s (x) I "
C = S"
for
and
(~i(M) (Ui)) (y) = I
being
-
The first p a r t of l) is s t r a i g h t f o r w a r d
implies
immediately
330
x 6
(+M)(U),
with
for the d e s i r e d
values
of
C .
for the other c a t e g o r i e s V x 6 lim H 0 ({U. >U};M) can be
Then
Y 6 HM(U i) I
with
6 ]I M(U. • U. ) I• ii U z 2
for some
(H~(M) (Ui)) (y) 6 I[ ~
{U ~--~U} 6 ~ i
~0 ({Vj i
"
is
>U. };M)
1
represented
(~0~) (x) which
by
p~(y)
gives
under
the d e s i r e d
e(M~)
is monic, factors
~I
8(~i)
through
Finally
m u s t s h o w that g i v e n
and
• U. ---->U. } ;M) I U 12 12
result.
is(M~)
QrP(~) S " {Ui~-~U}
If
It is clear that
m a p and
and Hence
the a d d i t i o n a l
x,y s M S(U)
sheaf of
(by definition)
8(U) (x) = e ' x 6 M S(U) '
6 Ju
( ~ * ) (x) .
under
the o r b i t
= ~P(M~).
is the t r i v i a l
(Ha(U i)) (e 9 g) = I
as desired.
H ~0 ({U.l P2*(y) s i26I
by
is the i n c l u s i o n
where
U 6 T_, then there is with
~ ~0 ({U. • U. ---~U. };M) i2EI ii U 12 12
is r e p r e s e n t e d
For 2), e
6
with
g 6 ~M~(U i) x 6 0 e(e(M~)) (U) hypotheses
we
8(U) (x) = 8(U) (y) ,
331
-
there
is
g 6 M~(U)
with U s _T
show that for any with
x = x
9 ~(U)(h),
g 6 IHGM(Ui) e 9 g
or
9 ((~is
e 9 a(U)(h) The p r o o f
using
p[ (i~i) (X)
Hence
)
(E~l) (g)
sheaf w e get
x
contained M'(U) where
e
"
(U i
i
6 ui
9 e(U) (g) = y
or
case
h = e
the c e n t r a l
as stated. assumption
to the o r i g i n a l
9 (~e(Ui)) ( g ) I
=
is
problem
and
{ui~i)u } (~I~l) (Y)"
9
2
= p[ (~) for s o m e
s JU
by the a b o v e
g s MG(U).
Since
as d e s i r e d .
|
observation. M
is a
2.3
Let sequence
{Ui~U}I
and so
Hence
(T~[) (X) u
is
let us
h s M G' ( u )
and
we can find a c o v e r i n g
(x), and so
g =
Corollary
N
IxI
First
( ~ I ) (X) = g
9 e
9 h).
there,
9
= g
.
There
in the f i r s t
Returning
with
(ui)
with
case using
the same.
= y
x s Ms(U)
h = e .
Then
(e(U) (h)))
the n o t a t i o n
--
then
of the o t h e r
and Hence
, given
= e = ~(U)(e
essentially
9 e(U)(g)
g 6 ~MG(U i)
respectively.
(g 9 e)
x
U 6 T
of sets
in
and S(S G).
in the c e n t e r
~(U)~
0e(MG) (U)
0e(MG) G = M G
Proposition
M' ;~ - M
of 8(U),
If
M'
8 > M" s S (Ab)
be a s h o r t and
a(M')
exact is
M G , then M"(U)
is e x a c t
in
SG
and
S"
"
2.4
#: P(C)
---> S(C)
preserves
monomorphisms
and m a p s
- 332 -
which ness
are o n t o for in
S',
C = G, GsG,
short
exact
S G, GS
sequences
of
S"
as w e l l
sets,
as e x a c t -
and t r a n s i t i v i t y
of g r o u p a c t i o n s .
Proof :
is c l e a r
that
are o n t o
for the a b o v e c a t e g o r i e s .
an e x a c t
sequence
#(eP(M'))
#
preserves monomorphisms,
lim
Since
This
also
sets
since
of t h e
shows
#
onto
~
#
preserves
preserves
is o n t o w h e r e
r i g h t g r o u p action. Returning Q_r(e)G , t h e r e
the commutative diagram below, f"(StM~(u ) (Qr(e)S)) Q rr
~ • f"
is an e p i m o r p h i s m
in
and so
c StN~(~ ) (Qr(U)S).
x StM~(s ) (Q=rr
~ Q_r(e)S
• f,, Q--r(~) S Thus
~ E Mor s(sG).
shows that for
x f" (St M
Qs
If
~(~
Qs
~ Pl
S
f" E Mor S (GsG), a similar d i a g r a m
~ E M o r s(GsG). and
) (Qr(~)S)) -
I
A similar p r o p o s i t i o n holds
which we leave to the reader to state
and prove. We are now ready to begin the construction canonical
"flask" resolution.
First we need a canonical
embedding of a sheaf into a "flask" definition gives the necessary Definition
sheaf.
The following
axioms.
2.6 Godement resolutions
can be constructed
C: S(S)
formation
with the following properties:
j: I C
) C
preserves
~
S(S)
in
if there is a functor
GRI:
of a
S(S)
and a natural trans-
finite inverse limits and mono-
morphisms. GR2 :
For all
e: M --~ M" E Mor S(S G), M" E s(GsG),
the canonical map isomorphism,
QrP(C(e))S ---> iCQ_r(e) S
and similarly for
is an
a E Mor s(Gs).
335
-
Moreover, GR3:
If
M1
iC
and
C M 1 = CM 2 GR4:
-
preserves maps which M2
are s u b s h e a v e s
if and o n l y
The maps
j (CM) : C M
if
of
M
M 1 = M2 .
- C2M
C(j (M)): C M --~ C 2 M
a r e onto.
have
, then Ce = e .
and
left
inverses
for all
M E S (S). GR5:
Let T
U E T
,
regarded
Su(S)
as s h e a v e s
the c a t e g o r y 4.12].
be t h e c a t e g o r y
Let
T_/U CU:
of o b j e c t s
Su(S)
Cu(M) (V) = CM(V).
Note
the m a p
C U (M)
inverse
for any
that GRI and GR3
functoriality
of
C
the c a n o n i c a l
map
in GR2.
Examples:
morphisms, the s p a c e canonical Xdi s
i)
of o p e n
resolutions
Let
X
subsets
of
{U i ~ U} E C o v ~ X
with
continuous
as above,
map.
and let
X
is n a t u r a l ) S (C)
of of
with
topology,
w
This,
several
in
M
the produce
examples
of
space,
inclusion maps Let
S X d i s (S)
for
Xdi s
f: X d i s
.
C
can be d e f i n e d .
a Grothendieck
and
II,
by
where
QP(c(~))S
U Ui = U .
Define Sx(S)_
[i, Chap.
be a t o p o l o g i c a l
if
the d i s c r e t e
U
on
has a l e f t
C- S (C)
We give below
topology
be d e f i n e d
which
at the b e g i n n i n g
in
E C o v T_/U , t h e n
7~0i,~0[Cu (M)
M E Su(S)
Godement
over
{Ui~-~U}
, and t h e d e f i n i t i o n
in w h i c h
the c a t e g o r y
If )
induced
---> Su(S)
imply that
is any of the c a t e g o r i e s
topologies
in the
of s h e a v e s
) X
topology
be s on
be the category
-
of sheaves
of sets on
X
is well known
that there
f*: Sx(S) --
SXdi s (~)
--~
gives
a triple
8: I
~ f,f*
tion maps
and
[7].
= Y~Udisf*(M)Y
value
=
) I
topology
trivial
on
SXdis(~)
where
are the respective
adjunc-
gave
j (M) : S
a section
) CM
of
M
is, of
over
This c o n s t r u c t i o n
[7].
where
on it, and so
.
at all of the stalks.
are either
triple
(f,f*) (M) (U) = f*(M)(Udi s)
~ y~uMy
originally
it
image functor
(f,f*,8,f,~f*)
the map restricting
Godement
Then
respectively.
The trivial
Moreover,
CM(U)
Xdi s
is a left adjoint
~: f'f,
has the discrete
-
to the direct
(C,j,k)
Udi s
course,
and
SX(~).
f.: SXdis(~)
336
to its
is the one
The v e r i f i c a t i o n s
or simpler versions
U
of GRI - GR5
of the arguments
in the
next example. 2)
Let
X
[2, Expos~ VII].
U
over
X
an
of
which
For each
the etale
to a fixed universe
) X if
T
is the full subcategory
is etale
(Case
U = U ~i(Ui ) (~i
[2, Expose VIII]
X-scheme
field.
U
s Coy _T
Following
This
belonging
structure map {U i ~ U }
be a prescheme,
X
a geometric
is the spectrum y E X , choose
k(y),
and let
GP(X)
geometric
points.
Let
site on
of schemes
such that the
(2) of
[i]), with
is n e c e s s a r i l y point
~
of a separably
a separable
of
etale)
X
is
closed
closure
~)
be the set of the c o r r e s p o n d i n g ~ =
~ ~ , the disjoint yESP (X)
union
of
- 337 -
the p r e s c h e m e s
Spec
be the c a n o n i c a l U • X = H X ys
(~)),
map.
y 6 X
Note
( H yis
that
~i ) =
, and
for
let
Hiy:
U ~ ) X 6 T
K ~ y6GP(U)
where
> X , since
~i = ~
U • ~ is etale over ~ w h i c h implies that each fibre over X Y 6 GP(X) is a f i n i t e d i s j o i n t u n i o n of copies of y . Let
Sx(S)
and
the etale above
sites
example,
(C,j,k)
ST(S)
=
over
be the c a t e g o r y X
and
the t r i v i a l
((][iy),(Hiy)*,
a, 8
are the r e s p e c t i v e
CM(U)
=
sheaf
on the etale
X
respectively.
triple
8,
of sheaves
in
maps.
(][iy)*M( ]; ( ~[ yi)). y6~(U) y i s -I (y)
As
in the
induces
S~(S)
(][iy) ,a (H iy) *)
adjunction
of sets on
a triple
in
S x (S)
where
For
U ~->
X 6 T ,
(H iy) *M
Since
is a
l
{Yi
) H ys
CM(U)
=
( H yis H ys
Essentially M
at
site over yi)}
X
and
is a c o v e r i n g
( H (~iy)*M(~i)) y i 6 ~ -I (y) by d e f i n i t i o n
y , where
=
(Hiy) *M(y)
M ~ = lim
family,
~ (Hiy)*M(~). ~6GP(U) = M~
M(X') , C ~
, the
being
stalk
of
the c a t e g o r y
T of p r e s c h e m e s y ---> X then CM(U)
X'
etale
[2, C h a p t e r
over
III;
X
with
i, E x p o s ~
) X'
a map
VIII].
If
V
over
> U s Mor
the m a p =
H
y--s
M~
~
CM(V)
=
~6GP(~(V))
(U)
from composing
( n M--) ~i6~ -I (~) Yi
the p r o j e c t i o n y 6 G P (U)
ys
(~ (V))
comes
T ,
338-
-
with the product
(remember
of the diagonal
Y = Yi )"
In particular
~: M ---> N E Mor S(S), u~: M ~ ~ thought of
N~
maps
C(e)
for all
of as sending
M~
.)
a section
(yD
M~i
for
is determined
y E GP(X) 9
H
by the maps
The map
j(M)
to its value
may be
in each stalk
M . Since
satisfied 9
C
GR1
and
j
come from a triple,
is immediate
is the equalizer
since the product
of the product
and
lim
GR4 is of equalizers
over
C~ I
C~
being
a connected
directed
izers and monomorphisms. II of
[i]
For GR2,
o: M~ • M G action
> M~
of
MG
~ M~ M s" • M G --~ Pl relation
x ~ y
recall
M3
on is
M~/~
if there
1.8 of Chapter
M e ) M" q Mor S G the right group
where
is a
N
is the equivalence
g q MG using
above of
Thus
for
of
with
x 9 e(g)
the d e f i n i t i o n
Qr(a~) s
= y 9 of
shows that
M ~ ) M" E Mor S(S G)
we
--~ (Qr(~)S) ~
coequalizer
of
CM G
and
CM~
that the
of
x CM G(U)
~(U)L CM3(U) Pl ~
is
~ Qr(u~)S yEGP (U) --
the above d e s c r i p t i o n
of the restriction
for
shows
V
equal-
e , then the c o e q u a l i z e r
and the d e s c r i p t i o n
see from the structure
CM~(U)
if
calculation
9 Qr(~)S
that
via
preserves
is Proposition
is the map defining
Now a straightforward
(M'y)S
GR3
category,
~ U E Mor T
map
that GR2 holds
This and 9
C M (U) --~ C M (V)
for right quotients 9
-
339
-
The argument for left quotients
is identical,
maps which are onto by Proposition {Ui~U}
2.2.
and
iC
Finally,
preserves
if
s Coy T_/U , we must show that
Cu(M) (V) = CM(V) inverse which section choosing
~
) ~ ~i*~[cUM(V)
is natural with respect
of
a point in the fibre over
ture sheaves induces
to
~[ ys
K( ~ yi ) I ~i6S p (Ui)
(the fibre is non-empty
3)
G
7KCM(~i)
~
if and only if
all sheaves
which has a left
G-structure
right translation
by
category
e-sets,
g-I
coming
for where
topology.
be the category
which
is
of left
I, Example
0.6].
U ~i(Ui) = U . I are representable. If M is a
g 6 G . e
M
Let
M(G)
~dis
via
be the
is the trivial group with functor
of topologies.
of sheaves
is
from functoriality
The forgetful
f: ~G -'~ ~dJs gives a m o r p h i s m Sdi s (S)
Then this
since
[i, Chapter
~G ' then the object representing
the corresponding
y s GP(U)
the category
Thus
of left
A
Hence GR5 is satisfied.
topology
sheaf in
can be defined by
,~ CM(V)
G-sets with the canonical
In this topology
y
for each
to
be a group,
{Ui~--~U}I 6 Cov ~
M .
are isomorphic.
and left inverse
Let
and
(U)
((Hiy)*M) (V x ~): HCM(V x Ui) U I U
IHCM(V uX Ui ) = CM(][VI uX Ui).
V
{~i } 6 Cov T_/U) since the struc-
since
of two such points
clearly natural
has a left
= IHM(V uX Ui)
of sets on
Let
S(S)
~
and
and ~dis
340
-
respectively.
Repeating
which
with
conforms
and
8
where and
there
a triple
Udi s
is
U
(C,j,k)
is the disjoint
CM(U) =
union
U s Mor ~G
of the diagonal
maps
M
CM
is
functorial in
~G
since
behavior
by
of
was defined
is a map in
s(sG).
CM
y6U H Qr(ey) S . catlons Hence
g-i
But
(MG)y Qr(ey)S
are untangled,
since
this and 2.2
f*M 6 Sdis(~)
followed
H (M~)y y6U on
set
Suppose
is induced
G .
The
of equalizers
and GR4 follows that
M a ) M"
of the coequalizer
where
~y
comes
of
from
shows that it is
, once the notation
is just the object
the only coverings
by the product
G-action
is clear,
(M~)y
= G .
The object
and the d e f i n i t i o n GR3
Y
is the composition
on the index
The d e f i n i t i o n
of
Y
= f*M(Udi s)
T , --dis = M(G) and G
and the left
by a triple.
H (M~)y • H ( M G ) y - - - - ~ y6U y6U Pl the group action
M
) CM(V)
shows that GRI holds. C
where
.
in
---> H M yi6~-1(y) Yi
H M g6G g
from right translation
Since
f, where
CM(U)
of its points
' CM(U)
Y representing
Now
M H M --~ H yEU y y6~(V) y
of the projection
functor
(f,f*,a,f,Sf*)
as a set.
H M = HOmTG( K G ,M) ys y ys y
V ~
=
image
morphisms.
regarded
(and using notation
is a left adjoint
to the direct
are the adjunction
U
For
the above process
it)
f*: S (S) --~ Sdi s (S) This defines
-
and i d e n t i f i -
representing
of G are disjoint
~(~)S"
sums of G,
-
s h o w t h a t GR2 choose ~U. I ~
is s a t i s f i e d .
a section U .
~: U
While
it does d e f i n e
cannot
s i n c e the
by
g
on
in e a c h f a c t o r
K G yEV y G
.
(0.6 bis)]
a l s o has G o d e m e n t
of a p r o f i n i t e
the one g i v e n
"flask"
In p a r t i c u l a r
resolutions
to r e s o l v e
short Let
and fix
U E T
the c a t e g o r y by
left
is a l s o
group
above with
V
translation
satisfied. the T a t e
[1, C h a p t e r
resolutions.
in
l, E x a m p l e
The construction
appropriate
restrictions. The next theorem
j .
from
G-map,
is n a t u r a l
topology which defines
groups
and
comes
as a
and so a m a p
which
T h u s GR5
cohomology
continuity
be c h o s e n
(yEVITG y ,M)
Y The G r o t h e n d i e c k
is e s s e n t i a l l y
usually
G-map
T[( K G ) I yEV• Y U ~
H~
G-action
as in 2), to v e r i f y GR5,
to the left
H G yEV y
IIHomT ( II G M) I zG yEV• Y' U ~
-
Finally, KU.
~
a map
341
of
summarizes
it w i l l
be u s e d
and t h a t t h e s e
exact
the properties
of
C
to s h o w t h a t w e g e t
resolutions
m a y be u s e d
sequences.
M E S (C)
where
.
a functor
Define
augmented
C = Ab,
_ G, G s_ G , S_G , GS,
S*(M) ( )
co-semi-simplicial
from
J~
objects
or _S" to
in
- 342 -
S*(M) ({Ui---~U}I):
M(U)
~
E M ( U i)
xU. ) E M(U i 1 U 12 IxI
p~
I
*
p:
Pl
f E M(U=
13
x U.
~I U
x
U
12 U
)
13
p* where p~:
1 ~
for
simplicity
write n+
j ~ n + 1
coming
U.
, for
which
x.-.x
excludes
U
the
the diagonal
Theorem
the map : Ui
x I U
) x...x U. U U in+ 1
1M(Uil
from
the
,
projection
.th 3-- f a c t o r . maps,
but
(The d e g e n e r a c y
these
won't
maps
maps
come
be n e e d e d . )
2.7
Let
C
I)
For
be a n y any
of
the
above
categories,
{Ui--~U}
E Cov
T
, there
n
(b) 1
and
=
pl,p
, I =
1
n , = n-I pl,p k p~_iPl ,
2 ,2 = ~ * ~ , -DI,D -
M E S(C). are maps
E CM(U i x...x U. ), n > I n-I 1 U U in_ I
that
n >
and
~
U. x U. x-.-x U. lj_ 1 U lj+ 1 U U in+ I
(a)
for
for
x..-x U. I U U in+ 1
9 E C M ( U i) ~ CM(U) and ~*" I n PI*: E CM(U. •215 U. ) --~ in lI U U in such
~*
E M(U. x-..x U. in iI U U i n ) --'~
. Pil'''''lj'''''In+l
from
we will
and
1
,
343
2) and o n l y
if
is monic. iC(e(M))
Let C(a)
in
M" 6 M o t S(C).
is monic.
Moreover,
a
is m o n i c
In p a r t i c u l a r
iCKer
if CM
j (M) : M
(e) = K e r P ( c ( a ) )
and
= C(a)P(cM). 3)
S(S'),
a: M - - ~
-
If
then
M'
iCM'
~-~ M
C(e)L
8 ~ M"
is an e x a S t
iCM C(8)_
iCM"
sequence
is an e x a c t
in
sequence
P(S') . 4)
If
M q S(S G)
,
_
particular acts sheaf
if
MG
acts
transitively of
G
, then
iC(#(iG/iH)) iCH >
on
then
transitively
iC(Ms). CH
If
iC(#(G/H))
on
H
is a n o r m a l
= iC(G)/iC(H) , and
> iCG ~
iCOMG(e)
= 0p (e). CM G
M S , then
is a n o r m a l subgroup
In
i C ( M G) subgroup
sheaf of
CG
,
so
is a s h o r t
exact
sequence
in
PCG). 5) Then which onto sheaf ment
there
Let
is a m a p
~ M" 6 M o r S(S G), M" 6 s(GsG).
8 : iC (Q-r (e))
is an i s o m o r p h i s m in
p(Gs).
of
M~
holds
, then
for any
M 6 S(C)
i)
is o n t o
natural
~(V)
Note
and
(~ • V)*:
~(V)
that
%
a(M G)
s Mor
set c o m p o n e n t s
is a n o r m a l
in
p(GsG)
p(GsG).
and
subgroup
A similar
state-
for left q u o t i e n t s .
N o w by GR5, inverse
QrP(c(a))
">
on the p o i n t e d
If m o r e o v e r
Proof..
shows
e: M
that
{U i
U} q C o v T_/U
C(M)(V) in
6 Mor C
~ i , ~ i*CU(M) (V) = C M ( U i U• V)
V
--~ and
for all
by
[i, I, 2.8].
E C M ( U i • V) has a left I U M . The n a t u r a l i t y in M V
(~0i,~0. ~
commutes
with
- 344 -
finite
inverse
limits
~,
~ C M ( U i) --~ I i n v e r s e of
= ~(U):
a left
CM(Ui" 1 uX'''Xu U.In) n-tuple
"~
= ~ n~-- (Uil
P~:
~ n C M (U i
E In
gives
(a).
below
commutes
x~In_l~i~01
Since
which
C~(U
i
1
IxI
~
gives
Let
U•
U U.i n )
ux'''• U i n )
is
for e a c h
Let n
Then
x...x U i n ) --~ U U
4.14]).
~( Uil
Now
.
Ux...xU U.I n ).
which
I
CM(U).
II,
IKCM(Ui uX Uil
( i l , ' ' ' , i n)
PI* n
in
[i, C h a p t e r
is a l e f t
P i*
inverse
~7 CM(U x Uil n+ I i U
x...x U ) U U in
is a s h e a f
the diagram
map,
(b):
U.
x...x 2 U U
)
in
P~-I~
n~i,~* CuM(U i x...x U. x U. x...x U. ) 1 1 2 U U Ik_ 1 U i U U in
CM(U. x U. x...x U. ) i1 U 12 U U in C M (U i In+ !
I n-1
CM(U i
• 2 U
C(~)
monic.
Fj = E q u a l i z e r
Let Let
1
LP i*
x.-.x 2
U
U
~: M - - ~
M"
fl,f2:
(L • M ~
x U. x...x U i ) U i U U n
n+ I
CM(U i In 2)
9 x. 9 9 x 1 U U Ulk-
Ui ) n
U
to
P2
L--~ M)
x U. x...x U. ) U. ik- 1 U i U U in be a map M
with
of ufl
be the graph
sheaves = uf2 of
with "
fj
Let ,
-
j = 1 or 2.
Since
C
C(fj):
is m o n i c
CF
fl = f2
"
Since
(C(~))
iCKer of
C2M
an d
iC
inverse CL
x
CM
limits
CF. 3
.
c(~)
B y GR3 t h i s g i v e s
j (M)
= KerP(c(e))
, and s i n c e
iC(~(M))
But
F1 = F2
has a l e f t
inverse,
is a m o n o m o r p h i s m . C
preserves
or it
Since
equalizers,
Moreover
C(e(M))
is a s u b s h e a f
preserves
maps which
are onto,
= C(~)P(cM), 3)
assertions group
Thus
(e) = Ker p ( C ( e ) ) .
CM"
.
2
in
C(j (M)) : C M
is a m o n o m o r p h i s m . iKer
finite
CL --~ C M
= CF
1
-
preserves
is the g r a p h of and so
345
follows
of 4).
s h e a f of
G
immediately
f r o m 2) as d o the f i r s t t w o
The c o n d i t i o n
that
H
is a n o r m a l
is e q u i v a l e n t
to s a y i n g
sub-
that
P2XPl x inv P2 H • G where
G • H • G --~ G inv
N o w as a set
Thus
CH
precisely
which
in
= iCG/iCH
= Qr (j) S ' j : H ---~ G subgroup
CH
of its a c t i o n
shows
is a n o r m a l
iC (# (iG/iH))
the s t a b i l i z e r
G
through
is the i n v e r s e m a p and t h e l a s t m a p
multiplication.
# (iG/iH)
factors
that
on
in
comes G/H
iC (# (iG/iH))
CG
subgroup
being
= iCG/iCH
from
sheaf
of
the i n c l u s i o n
of its a c t i o n
Hence
comes
CG
the argument in
P (GsG)
on
map.
CG/CH
subgroup below
But is in
for 5)
or e q u i v a l e n t l y
P (G).
5) iGC(Q_r(e)) Since
= iC(GQr(e))
P G C M ,,(C(e))
= iC(GM"),
.
since
f r o m the s t a b i l i z e r
.
H
= iC(GM" (e)/StGS,,(e) (Q-r(e)S))" the definition
of
346
-
, P
St~CM" (C (~)) ~Qr (C (~)) S) show that
GR1, and the onto statement of GR2
F
is contained in
iC (StGM,, (~) (Qr (~) S ) )
St~cM,, (C (a) ) (QP(c(a)) s) is onto.
(In 4),
C
-
ic (GQr (~)) --~ GQrP(c(~))
and that
applied to the first stabilizer equals
the second stabilizer which finishes the proof of 4). then shows that the map is onto.) subgroup presheaves an action on of
M~
in
iCM~
QrP(C(e)) S
GR1 and GR2 show that
whose actions on
Since
whose action on
M~
PicM~(C(e))
M~(a)
is the sup of the iCM~
induce
is a subgroup sheaf
induces an action on
iCM~(u)
4)
is contained in
Q_r(U)S , PiCM~(C(a)).
Moreover the argument above shows that iC(StM~(a ) (Qr(a)S)) u stPicM~(C(u)) (QP(c(~)) s) identified iC (Qr (~)G) iCM~(e)
CQ--r(a) S
QrP(c(a)) S "
with
~ QrP(c(a)) G
may not equal
where we have
Hence we get a map
which fails to be onto only because PiCM~(C(a)).
This together with the
map between the groups acting on the left and the isomorphism of the pointed set components gives iC (Qr (~)) If
u (MG)
- QrP(c(a))
6 Mor p(GsG)
is a normal subgroup of
a normal subgroup of
iCM~ .
and the map is onto in
Thus
p(Gs2)"
which is onto in M GI! , then
p(Gs).
C(a)P(cM G)
iCM~ (~) = PiCM~ (C (e))
i
This gives a reasonably good picture of the functorial properties of
C .
does the same thing for
The last result of this section
Q-r "
In particular we will need a
is
347
-
non-abelian
analogue of the
-
3 • 3
lemma w h i c h will follow
by combining the results below. Theorem 2.8 i)
Given M' >e : i J'I J N' > ~ -
e
Qr(J')~'~
where all of the sheaves
8 > i" -- J"
N , ~ ~ N"
Qr (j)
are in
S (Gs_~)c, suppose that
j, j' , a, -a 6 Mor S (SG) , the other maps are in
S(S'),
m
~, a, j', j, and are exact in
j"
S (S')
are monic in and
M~
s(Gs)
acts transitively on respectively, 2)
~ M'
Consider
S(S').
N S'
then
the first two rows
acts transitively
Then the third row is exact in G N'
S(S'),
and
~'
!
on
Moreover,
MS .
if
N~
, ~' 6 Mor S(S G)
~
where all of the sheaves are in
> M
8
) M"
,N
8
)
N"
S( GsG ), -~, _~, ~' 6 Mor S( S G ), m
, and if
becomes a map in
NG
S(S G).
or
is monic.
N ,
N~ _~ Qr(e)S
or
is restricted
to
e(N~),
then
- 348 -
If
9 = r,
and
~I:
J(M) G "->
If
9 - s
~[:
GJ(M)
J"(M") G
and
9' , j, j",
Moreover
~(x)
= e
.
and
x E ~ N ~ ( U i)
i)
there
Since
is m o n i c ,
ment
of
that
{U i
a(e
sheaf 9 N~
For
and
E Mor -
S (sG) .
= ~(y),
there
with
~' (x) = ~ ( x )
and
with
x
Thus
the
=
n' (y
9 g)
=
of
with
Then
shows
so
j'(e
as d e s i r e d
assume
~(x)
above
, and
that Then
is
{U i
= e
,
if
= e 9 j (g).
Taking
a refine-
gl s ~ M G ( U i )
9 gl ) = x Q_r(j ')
such
since is a
transitively
on
x, y E Q-r (j')S (U)
- U } I E JU
~' (y) = ~0~(y), ~(~' (x 9 g))
that
~([(x))
~(x)
acts
given
with
{Ui--~U}
since
N G'
~i
in t h e
{Ui~-~i U } I 6 C o v T But
a refinement
x = e
with
9 g = e .
is
7' (x) = s
9 g
"
in t h e c o n c l u s i o n .
e
the rest
~'
be i n t e r c h a n g e d
there
g E ~ M G ( U i)
= e
Hence
-~ Q r ( ~ ) S
9 g E ~ ( M ~ ) ( U i). I if n e c e s s a r y , t h e r e is
~U}
- gl)
is m o n i c .
is
Qs
U E T_ , x E Q r ( J ' ) s (U)
(and t a k i n g
.
S( GS) , a n d
right may
is onto,
with
necessary) j"
and
Let
~'
Q_r(j") S ~_ Q_r(~)S
then
are interchanged
Since
so b y 2.2
then
B, -B E Mor
is o n t o ,
left
if t h e y
Proof:
and
is onto,
'~ G j " ( M " )
hypotheses
8, -8 E M o r S (S G ) , and
j' , j, j",
' x, y E ~ N ~ ( U i)
and
= ~(e)
~' (x 9 g) = ~0~(x)
g E K N ' ( U i) I u = ~(~' (y 9 g)). 9 ~' (g)
m
~(y)
9
~' ( g ) ,
and
so
x = y
.
The
other
case
is s i m i l a r . 2) preserve
are,
The
hypotheses
except
for
7',
on t h e g r o u p 8, and
8
actions
required
the maps for the other
349
-
hypotheses
and
the
follows
result
the
N s • MG
conclusion from
x N~
-
to m a k e
a diagram
~
sense.
chase
For
in t h e
6XMG. N~ _
N s x MG
. = r
diagram
,
below.
x MG
• Pl xp 3
Pl
N s x N~
P~
~
6
Ns
,
N~
~
f
,
Qr(j)s All
unlabeled
projection tion
8
M ---~ N
if
argument
using
of
L
coequalizer
from
L
> M
group
actions
is a l s o
~"
is the
and
Ms x NG
Pi
denotes
By defini-
Since
>> N
2.2
and
.th l-- f a c t o r .
the
is an e p i m o r p h i s m ) .
Proposition of
onto
is o n t o ,
> M
Q
the
~ Qr(j")S
coequalizers.
j'' (M")G
coequalizer
Qr(j)s
Pl
product
are >
~
come
the
and
~I : J (M)G (the
x NG
maps
from
~
the
~ Ms
a coequalizer
coequalizer A
straightforward
structure
where
of
of
the
is a s h e a f
NG
of
Pl groups
acting
coequalizer
on
of
any
sheaf
of
sets
MS x NG • L
shows
that
~ MS • L
for
Q x L any
is the L 6 S(S).
PlxP3 m
Thus the
6 x MG hypothesis
tative.
Now
coequalizer.
• N G'
and on
are
also
NG
shows
that
a diagram
chase
shows
Hence
Qr(j '') -
S
coequalizers.
the w h o l e that
~
Qr(~)
-
-
~
since S
Finally
diagram
is c o m m u -
is a l s o
a
-
N~
-
is onto.
> Qr(J')G For
350
9 = s , the r e s u l t
follows
f r o m the d i a g r a m
below 9 GM• G M • N S • N G bl~ p 2 ~
Ns • %
Qs
that
above
-Pl
~
COHOMOLOGY
This Hn(U;M),
section
M 6 s(GsG),
including
to a c e n t r a l
It c o n c l u d e s definition
with
agrees
---~
~, ~",
chase
as a b o v e t h e n s h o w s
COEFFICIENTS
to the d e f i n i t i o n
the u s u a l
sequence
of c o e f f i c i e n t
theorem one
of
of its p r o p e r t i e s
of a 9 t e r m c o h o m o l o g y extension
, and
8
WITH NON-ABELIAN
a comparison with
Rs
8, n • N ~
as d e s i r e d .
is d e v o t e d
N;
~
and a d e s c r i p t i o n
the e x a c t n e s s
associated
Ns
A diagram
is a c o e q u a l i z e r
~ GM • N~
i ~s
show that
are c o e q u a l i z e r s .
w
mology
~
S • N~
The a r g u m e n t s GM • 8
!
GM • N s
which
sheaves.
says t h a t our
for w e l l
known
coho-
theories 9 As b e f o r e w e
resolutions
fix a t o p o l o g y
can be c o n s t r u c t e d ,
in w h i c h
(T,C,j),
Godement
for the e n t i r e
-
Let
section.
M s s(GsG).
351
-
The c a n o n i c a l
resolution
of
M
is the complex:
M
where
do
> CM
e
dI -9 CQs
Q~ (J0) of course
d i s Mor S(S')
fix
if
i
is even.
(thus g i v i n g
Let
9 . .
Q~(J2 )
the left g r o u p action
U 6 T
w h i c h we w i l l derive).
~
Jl = J(Q~(J0 )), etc.,
J0 = j(M), preserves
section
~ CQ_r(j I)
Qr(Jl)
and the right g r o u p a c t i o n of this
d2
0)
Z n (M) s GsG
and
if
i
is odd
For the r e m a i n d e r
ru: s cGs_GI be d e f i n e d
GS_G by
dn Zn(M)s
= Ker S. (CQ. (Jn_l)s(U)
and
G Zn (M)
and
G Q. (Jn_l) (U)
a set w h e r e shows that GZn(M)
being
the l a r g e s t
Qs
for
n
odd w h e r e as
Qr(j_1) e .
For
the sheaf of p o i n t e d C Q r ( J n )s " n
even.
Define
and
Zn(M) s
as
Proposition
c Qr(Jn_l)s(U)
2.2
and
for
to be i n t e r p r e t e d
let
sets w h o s e
Then by r e s t r i c t i o n
Q" (Jn-i)G(U)
Zn(M)G = Qs
odd,
0e(sCQs
n
Z n (M) G
even and
is always n
of
stabilize
on
Zn(M)s = 0 e ( G Q r ( J n _ l ) ) ( U )
= GQr(Jn_l) (U)
)
which
depending
Zn(M)s = 0e(Qz(jn_l) G) (U) n
subgroups
respectively
= s or r
with
~ CQ. (Jn)s(U))
as
M
0e(CQr(Jn) G) 6 S(S_G)
right group
)) 6 s(Gs)
component
and be
is
in a similar way for
we have a m a p
-
!
: 0 e (CQ- r('3n_ 2 ) G ) (U)
n-I
odd for
n
is odd
then
say ,
Hn(M)
is,
n
in
H n (M) S
= Hn(S)
and
image
If
GM
acts
on
"
a forgetful
for
n
which
Note
that
functor,
if
just
n is in G S
6 Gs_G
odd.
Thus
if
and
M 6 S (Ab) , the
dI CQs
0) (U)
{x}
denote
Finally
observe
elements.
in
Hn(M)s
transitively
M S(U).
n th
d2 > C Q r ( J l ) (U)
the
equivalence
that
H n (M) S
The neutral
>
elements,
~n_I(CQ. (Jn_2)s(U))
on
M s , then
H n(M)~
H 0 (M) s = Ms(U)
H 0 (M) S
Thus
if
M 6 S(G) , H 0 (M) = M ( U ) ,
H I (M) S
are quotients
acts
since
e
,
N Zn(M) S .
on
Qs
of
is a p o i n t e d
of
(since
9 9 9
class
acting
Z 1 (S) S = Q s Qz(J0)s ) , and
Zn(M)
for
= Z n (M)s/ ~'n-i (CQ r("3n-2 ) G (U)
let
F
the g r o u p s
acting
S_G
= Q. (nn_l)
. = r
dO
neutral
the
is in
complex:
x 6 Zn(M)s
are
CMG(U)
of t h o s e
transitively acts
on
transitively
C M s (U)Now
H0(
even
via
> C M (U)
set w i t h
on
which
>
Hn(U;M)
Hn(M)s
of t h e
e
where
~ Z n (M)
= G Z n ( M ) / S t G Zn(M) (Hn (M) s) .
homology
x
for
-
0e(GCQ--s
Define
. = Z
G Hn(M)
If
zn-l:
even.
where n
I
and a map
352
)S:
s(Gs)
sheaves,
> S"
9
): S(G)
if
GM(U) then
> G acts H~
.
Moreover,
faithfully 6 Mor
defines
If w e r e s t r i c t
t h e n we c a n e n l a r g e
H0(
p(Gs),
H ~ (M) S = 0e(GM) (U)
GS
on
a functor
the c a t e g o r y
the range
category.
given
8: M ---> M"
Ms(U)
or
since
8
in e i t h e r
Thus 6 M o r s(Gs)
is o n t o case
of
in
- 353 -
8(StGM(U) (0e(GM)) (U))
acts
trivially
The f u n c t o r i a l i t y more
complicated.
for
8: M
~ M" 6 Mor S(G) HI(M) s
2)
H2(8)
3)
H I (8) , H 2 (8)
In e i t h e r
defines exists
8 (M)
2) or 3)
S"
exist
>
_
j " :- -0-.
M" (U)
in
In g e n e r a l
816
81~ 0 = ~"C806
(and so
since and
8(M)
takes this of
case HI(M) S
J !)
observations
)S: S(G)
if
--~ S"
of
"
J~
M"
.
~
") (u) CQ_~(jo
6 M o r S"
in
.
s(Gs_ G)
Q_r(Jl
) (U)
11
l - Qr(j].)(U) ,,
since and
a map
In e i t h e r
of
2.5
= StGCM(J0 ) (Qs M"
.
as required. 0 (CM(U))
Qs
2) or 3) ,
by P r o p o s i t i o n
= J0(a)
stabilizers
) (U)
~ !>
(U)
ci] 2j
in the c e n t e r
H l (8) 6 M o r GsG -GQs
9 x = x 9 Hi(B)(g).
and so it induces
StCMG(J0 ) (Qs
into
in 3),
However,
~ H I (M")
is c o n t a i n e d
and
diagram:
.~0 Q_~(jo ) (U) ~ S(S').
,
and
M 6 S(Ab)
CQs
ST
C8 I) is a map
in
GsG
11 "
Mor
stabilizers
following
is onto,
Hi(B)(g)
) (U) ~
Mor s(sG),_
in 3),
8
on the f o l l o w i n g
H I (8) = 81: H I (M) El
is a little
in the c e n t e r
x 6 Hi(M"),
_~0 Qs
CM" (U)
H2(M)
H2(8) (H2(M) s') -c H2(M") S' ' and
is b a s e d
CM(U)
HI(
if
Jo. M(U)
and
the
a functor
in
0e(GM") (U).
suffice:
is c o n t a i n e d
g 6 Hi(M),
The proof
HI(M)
For our p u r p o s e s
i)
given
of
on
Thus
CB
Moreover,
in
is the s t a b i l i z e r which
are the
-
groups
acting
H I (M") s
on
H I (M) S 9
is i n d e p e n d e n t
t r u e of the a c t i o n
of
Since
C81
81
and
so
Qs
I!
and
independent the a b o v e 8: M
of
--~
M"
I!
GQs
M"
6 Mor
, then
be f a c t o r e d
are o n t o
S(S_G)
and
M"
on
H 2 (8 ) is o n t o
in 3)
in the c e n t e r I!
Qs
in
S"
followed
of
is 2.5 and
Note
is c o n t a i n e d exist
this is
in 2) and
the proof 9
8 (M)
on
I!
Qs
contained
to f i n i s h
as a m a p w h i c h of
in
on
we can a p p l y P r o p o s i t i o n
and
S(G)
H 1 (M) 6 Ab
it acts on since
and its a c t i o n
again
The first
that
if
in the c e n t e r since
8
can
by a m o n o m o r p h i s m
.
exactness
statement
is n e e d e d
for the
theorem.
Proposition
3.1
Let in
)
of
) (U) E A b
81 (Qs (j 0 ) )
H 1 (8)
into the c e n t e r
comparison
of the side
of the side,
argument
-
The a c t i o n
Qs
Qs (J 0 ) s S (A_~b) w i t h
354
S (S G)
M' > e > M ~
with
M'
M"
M 6 S (G)
be a short Then there
60: H 0 ( M '') ---~ H I ( M ')
transformation
in
exact
sequence
is a n a t u r a l S"
giving
a
sequence
1 which at
,
H0r is e x a c t
H 0 (M"),
in
SG
and e x a c t
Proof: where
Hor
the m a p s
at in
60
H0r H 0 (M') S"
The p r o o f
at
and
in
GS
H i (M').
is b a s e d
are the o b v i o u s
H 0 (M), e x a c t
H r
ones:
on the f o l l o w i n g
diagram
- 355 -
M' (U) ~
cM' (ul
a
>
>
cN(u)
~rO
first
I
a 1
row
is e x a c t
exact
comes
from a short
acts the
sequence
by
2.8
of
g.
CM"(U)
exact
if
SG
6 Mor
shows
that
_. s , and
in
60: M"(U)
Thus
by 2.2. in
II
Qs
sequence
~i~0
2.2
Define
sets
i
~
SG
!
since
are
u s u a l way.
in
exact
transitively, columns
~
) (U) >--~Q_~(j0)(U)
short
is m o n i c
~
~
M" (U)
~
0
Q_s The
~
M(U)
of
) (U)
The
second
row
is a
third
row
by 2.7.
The
sets
S(S_G)
in
s(sG). the
Since
third
Z 1 (M')S
~I
!
Qs
r o w and
all of
= Qs (J0') (u).
~ H I ( M ') 6 M o r
x 6 M~ (U) = H 0 (M"),
and
S"
in the
choose
y 6 CM(U)
m
with
8 (y) = j'~ (x).
z s Z 1 (M') s immediate
that
is n a t u r a l
where
we and
hand,
Ul (z) = n 0 (y). is i n d e p e n d e n t
60
f 6 Mor
have
are
= e
Let
, there
60(x)
of t h e
~ >>
N' >
B :_: N"
a ~ N
and in
SG
left quotients
x E H 0(M''), t h e n 6~
M' > a ) M
S(G)
exactness
forming
if
81 (n 0 (y))
is a u n i q u e
= {z}.
choice
It is
of
y
and
for m a p s
f'
already
with
Since
= 6~
6~ ')
f" at
is t h e
induced
map.
and
H 0 (M) .
row,
if
H ~ (M')
in t h e t h i r d = 6~ (8(g)
for
M"
x, x'
9 x).
6 H~
We Since
g s G H0 (M)
On t h e o t h e r S = M~(U),
-
then
we
-8(y)
= j"(x),0 -8(y')
y,
can
. y-i
6 M(U),
Exactness action the
choose
in
61
:
at
of
CMG(U)
third
column.
H I ( M " ) --~
next
y'
on
so
-
6 CM(U)
= J0"(x') "
and
S"
The
Y,
356
8(y'
H I ( M ')
C M s (U)
with
Since
~ 0 (y,
. y-l)
9 y- 1 ) = e
9 x = x'
follows and
T 0 (y) = T 0 (y')
from
the
and .
as d e s i r e d .
the
transitive
exactness
in
S"
of
| theorem
H 2 ( M ')
provides
for
a boundary
a central
extension
map of
sheaves
of g r o u p s . *
Theorem
3.2
Let of M
sheaves .
of g r o u p s
Then
8~
transformation
z)
M' --a .- M ~
is
M"
with
u (M')
a homomorphism,
81 : HI (M")
i --~ Ho (M') ~
exact
in
2) pointed
sequence
SG
H I (M)
at t h e
B~
HI (M")
sequence
in t h e
and
is a n a t u r a l
there in
S"
center
such
of
that
H ~ (M")
H~(M) ~ at
exact
contained
H o (m ~
of g r o u p s other
a short
---> H 2 (M')
~0 H~(M') ~ is an e x a c t
be
the
H~(M") first
three
terms
and
terms. 61 ---> H 2 (M')
is a n e x a c t
sequence
of
sets.
* Using a slightly different definition of Q_(u), the exist e n c e of 81 for a s h o r t e x a c t s e q u e n c e of s ~ e a v e s of g r o u p s c a n be s h o w n . However, with this definition o n l y i) - 3) ( s u i t a b l y m o d i f i e d in t h e n o n - c e n t r a l case) of t h e e x a c t n e s s properties below can be proven.
357
-
3)
H I (M")
~I
sets with n e u t r a l 4)
~--~ H 2 (M)
elements,
is,
neutral
for all
g 6 H 2(s')s
the maps
with
x
if
e
is b a s e d
J0 _"
B
!
by 2.8 sets
in
2 ~ Q_r(j! ) (U)
row is an exact
r o w comes (regard 2S
J
5
!
third
i1
QZ (j 0 ) (U)
-~
CQs (j~) (U) >----> CQz(jo ) ( u )
row is a c e n t r a l
M" (U)
l
]' , Qz(jo)(U) J!
Qr(jl) (U) >
sequence
extension from
short
M" = Q_r(e))
by 2.2.
diagram
l t
Jl
'1
on the f o l l o w i n g
~ CM (U)
C~
Qg(j0 ') (U) >
, then there
ones:
~ M(U)
J0
The first
action;
9 ~*2(g) 6 H 2 (M) s! "
The p r o o f
M' (U) >
c M ' (u)
sequence
group
82 (x) 6 H 2 (M")~
are the o b v i o u s
I
is an exact the right
of
! (~,2)-I(H2(M)s).
~I (HI(M") S) =
under
sequence
g s H2(M') G , x 6 H2(M) S , we have
Proof: where
is an exact
H 2 (M")
elements
B,2 (~) = B,2 (~ . e,2 (~)) , and is
i.e.
H 2 (M') ~--~ H 2 (M) 8 ~
of sets w i t h that
H 2(M')
-
of
and
Moreover,
,,[
TI
1
Qr (j ] } (U) of groups,
CM"(U)
exact
'~I
by
sequence
CM' (U). of sets
so is an exact ~i'
and the second The in
S(S G)
sequence
Bl 6 aor s(Gs2 ) ,
of
-
81 6 M o t S(S2 ) the r i g h t short
acting
is onto. short
sequence
exact
sequence
the
the
first
rows
~
~I'
in
S(S2 )
r o w is a
6 M o r GsG
I
-
and
the
'
~I 6 M o r S_G
the f i f t h r o w c o m e s
from a
(since by e a r l i e r
sequence
as do the
remarks.
of sets
in each of the t h r e e
GS
the s h e a v e s
the first,
are the
in
S_G
columns
last two a r r o w s
Finally
H 2 (M') 6 A b
of g r o u p s
1
row respectively,
cocyles.
Moreover,
on the r i g h t
form exact
for c o n v e n i e n c e
with
acting
on the c o r r e s p o n d i n g
third
0 and
in the two c o l u m n s by 2.2.
--
'
on
. define
in
column.
act t r a n s i t i v e l y
defining
2S
transitively,
two a r r o w s in
acting
by 2.7 the f o u r t h
2.2 it is an e x a c t
of sets
Since right
that
of g r o u p s
a 2, 82 6 Mor s(Gs2 )
first
sequences
in
act
of sets
and
So by P r o p o s i t i o n
exact
of sets
2.8 s h o w s
~r (j~) 6 S(Ab))
Moreover,
Thus
on the r i g h t
Then
-
and the s h e a v e s
act t r a n s i t i v e l y .
exact
groups
is onto,
3 5 8
H 1 (M') S
we will
, H 2 ( M ') S
on the sheaves
of sets
the sets
in t h e s e
the last two a r r o w s sequences
identify
or
left,
in
S"
H 1 (M'),
H I (M')G
'
H2(M
')
G
,
etc. Since clear
that
60
8 6 M o t _G
and
is a h o m o m o r p h i s m .
T I!0 6 M o r S_G
chase using H I (M') 9
II J0'
Exactness
at
gives
H I (M)
T 0 6 M o r _SG
A straightforward
exactness
follows
') (U)
If
t h e n w e can c h o o s e
x, y 6 ZI(M) s
for
{x},
{y}
in
diagram 2S
at
since
for all
= 8,1({y}),
g 6 Qs
of sets
B 1 (x 9 ~I (g)) = 81 (x) 8,1({x})
, it is
respectively
and
x 6 Zl (M) s "
representatives such that
-
Bl(x) Then
=
there
{x} =
{y}
is
g 6 Qs
if
I!
B l (Y) = J l (x).
alter
not
6 M o r G s_ G
CQs
class and
=
(j '0) (U)
S"
at
and
H 1 (M") .
of
9 -
3)
the
since
~i , B2
with
B2~ I (y')
B1
in
S
9
and
z 6 Z 2 (M') s
with
Altering
choice
class
the
of
z
since
on
')(U)
of t h e
side
it a c t s
alters
y
change
way.
with
CQs
on.
by an e l e m e n t {z}.
remark
immediately in
factors Suppose
is o n t o = 82 (x).
B 2 (y) = 82 (x) , t h e r e 9 a~({g}) Finally
of
above
follows
{x) 6 H2(M) S
Then,
{x}
Hence
Altering in
The exactness
about
the
gives
of
action
of
exactness
in
by u s i n g
the
exact-
II
is t r i v i a l 9
represents
Thus
is onto.
in the u s u a l
immediately
Jl._
Since 4)
x
CQs
on
{z}.
action
of
II
ness
B
9 al (g)"
is e x a c t
cohomology
so d o e s n ' t
9
of
the
and the
cohomology
row
is a u n i q u e
is i n d e p e n d e n t
Jl n0 (CMG(U))
CQs
and
Y 6 CQs
fifth
61({x})
of
does
the
there
Let
the
SG
x = y
) H 2 (M')
, choose
a2 (z) = ~I (y) "
Zlal
with
) (U)
Since
is t r i v i a l ,
y
B 1 , T 0 , ~"0 6 M o r
61: HI (M")
x 6 ZI(M") S
--
"j" 7TI 1
-
9 a~({g}). Define
Thus
-B,
since
Bl(Y)
359
is =
S"
through
the
first
part
x 6 Z 2 (M) s = 0 e ( G Q r (jl) ) (U) with
in Let
S"
B,2({x})
=
, there
is
Y = ~i (y')"
g 6 Qr (jl) (U)
{y} 6 H 2 ( M ) ~
we m u s t
Qr(a2)
relate
{B2Cx)}
y' 6 C Q s
9
0) (U)
Since
with
x
as d e s i r e d 9
this
6 H2CM")~
cohomology
9 a 2 (g) = y B theory
to
9
-
the usual ones. If M 6 P(G) v ZI({ui---~U};M) = (g Eixi ~ M(Uil
3 6 0
and
{U.--->U} 6 Cov T , let
uX U.12) IP~(g)
= Pl*(g) 9 p~Cg)
6 IH3MCUil uX U.12 uX U.13)},
V ~I ({Ui...>U};M) = Z 1 ( { U i ) U } ; M ) / N
and define if there is
where
gl N g2
h 6 ~M(U i) with gl = Pl9 (h) " g2 " P2* ( h - l ) . I comes from the projection map onto all but the
p[
As usual
-
.th
i-- factor.
Given
~: {Vj---~U}
) {Ui---~U} 6 Mor Ju
'
there is an obvious
induced map v ~ : ~I ({U.---~U};M) ~ H 1 ({V. ~U};M) which depends only on i 3 the domain and range [i0, P r o p o s i t i o n 1.2]. Let v Then it defines a functor ~I (U;M) = lim H I ({U i >U};M).
~1 ( U ; ) :
P(G)
.) S"
which together with
a different cohomology ~I(u;M)
theory.
v H 0 (U;M)
In particular
for
is the set of "locally trivial principal
spaces for
M
in the topology over
give M 6 S(G), homogeneous
U ."
T h e o r e m 3.3 l)
Let
M 6 S (Ab) .
Then
FU: S(Ab)
--~ Ab
derived functor of
H n (M)
is the
evaluated
Fu(A) = A(U)
for all
A 6 S(A_~b).
2)
~I(U;M)
is n a t u r a l l y isomorphic
Proof:
i)
indeed a resolution it suffices
n t-~h M
at
to
HI (M) S "
Since the canonical resolution in the usual sense for
to show that
CM s S(Ab)
where
is
M 6 S(Ab),
is flask
-
[1, II, C o r o l l a r y must
4.4].
Theorem
n+ 1 i , 7. (-i) Pi i=l x s Ker
2.7 w h e r e
has
the b o u n d a r y cohomology.
=
i
map
p l , p [ = p*i_iPl,
), t h e n U. in+ 1
U
Theorem
for
i >
1
2.7 shows
where
and
direction.
in the o t h e r
= x .
Hence
that
in the r i g h t
~
pl,P~(X)
is
If
pC : ~ CM(U i x...x U. ) l-i in 1U U In goes
dn
just
.
7. (-I) Pi(X)" i=2
Pl*
, we
defined
S*(CM) ( { U i ~ U } )
(d n) ~ H ICM(U i • i n+ I U n+l
p~(x)
trivial
-
{Ui~-~i U} 6 C O V T
Given
s h o w that the c o m p l e x
before
3 6 1
hand
term
H CM(U i x...x U. ) in+l i U U in+ 1
x = ml,P~(X)
Moreover, n+ 1 ip 7 (-1) ,p[(x)
=
i=2
1
n+ 1 =
(-1)ipi_l* (x)Pl,(X)
7
as d e s i r e d .
= d n-I (-Pl*(X))
i=2 2) Since so
M
For
and
~0 (U;QrP(j))
CM
M 6 S(G),
consider QrP(j)
are g r o u p s
= ~~
is an easy a r g u m e n t
M-
j
- CM
satisfies
by d e f i n i t i o n to p r o v e
the r e s u l t s
of
> Q_r(j). (+)
and
# .
It
of P r o p o s i t i o n
3.1
for a short e x a c t s e q u e n c e of p r e s h e a v e s in P(S G) using v ~0 (U;M) and H I (U;M) i n s t e a d of H 0 (M) and H I (M) (see or
[i0]
for the d e f i n i t i o n
[5, C h a p t e r
I, T h e o r e m
~I (U;CM)
and
H I (CM)
the a b o v e
sequence
of
60
3.1]).
are t r i v i a l
shows
and m o s t
Thus for
of the a r g u m e n t
if we can s h o w t h a t M 6 S(G) , then
that
~I (U;M) _~ G C M ( U ) \ ~ 0 ( u ; Q P ( j ) ) s
~_ G C M ( U ) ~ Q r ( J ) s ( U )
~_ HI(M) s
- 362 -
as d e s i r e d 9 But
for a n y
{U. 1
%n
HI({ui---~U};CM)
such
that
= pl,p~(x)
= Pl*(p 1 ,(x)) PI*(x) CM(U)
E Cov T
= e , for g i v e n
p~ (x) = p* (x)
x = pl,p~(x)
)U}
x s
6 ~ C M ( U i). I
~ CM(Uil IxI
9 p~ (x) , T h e o r e m
2.7
x U. ) U z2 shows
that
9 pl,p~(x)-I Thus
Pz*(Pl *(x))-I
9
,
H I (CM)
Moreover
-j - C(CM) (U) ~ ) Qs
(U)
v
9
x N e
since
is c o m p u t e d
But
from
j 6 M o r S(G)
has
a
-r
left
inverse
3 : C (CM)
) CM
b y GR4.
Hence
C (CM) (U)
is
m
a semi-direct
product
straightforward (Ker j) (U) Since
Ker
H I (CM) S = e
of
argument
) C(CM) (U) j
is a s h e a f , as d e s i r e d .
(Ker j) (U) shows
that
) Q
(j) (U)
this
shows
and
(CM) (U), a n d a
the c o m p o s i t e is a set i s o m o r p h i s m . that
~
is onto.
Thus
-
363
-
REFERENCES
[1]
Artin, M., "Grothendieck Topologies", mimeographed notes, Harvard University, Cambridge, Mass.
[2]
(1962).
Artin, H. and Grothendieck, A., "Cohomologie etale des Schemas", S~minaire de G~ometrie Algebrique, Institut des Hautes Etudes Scientifique,
[3]
[4]
Dedecker, P., C. R. Acad.
(1963-64).
Sci. Paris, V. 258; ii17-
ll20,
(1964), V. 259; 2054-2057,
4139,
(1965).
(1964), V. 260; 4137-
Eilenberg, S. and Moore, J. C., "Adjoint Functors and Triples", Illinois J. Math., V. 9; 381-398,
[5]
(1965).
Garfinkel, G., "Amitsur Cohomology and an Exact Sequence Involving Pic and the Brauer Group", Dissertation, Cornell University, Ithaca, New York,
[6]
(1968).
Giraud, J., "Cohomologie Non Abelienne", mimeographed notes, Columbia University, New York, New York,
(1965).
[7]
Godement, R., Theorie de8 Faisceaux,
Hermann, Paris
[81
Grothendieck, A., "Le Groupe de Brauer I; Algebres d'Azumaya et interpretations diverses", Seminaire Bourbaki, No. 290,
(May 1965).
(1958).
-
364
-
Grothendieck, A., "Sur Quelques Points d'Algebre Homologique", Tohoku Math. J., 119-221, [i0]
(Series II) V. 9;
(1957).
Hoobler, R., "A Generalization of the Brauer Group and Amitsur Cohomology", Dissertation, University of California, Berkeley, California,
[11]
(1966).
Serre, J. P., "Cohomologie Galoisienne", in Mathematics, No. 5) Springer-Verlag,
[12]
(Lecture Notes (1965).
Springer, T. A., "Nonabelian H 2 in Galois Cohomology", Proc.
Sympos.
164-182, AMS,
Pure Math.,
(1966).
(Boulder, Col., 1965) V. 9;
365
FONCTEURS
-
DERIVES
ET K-THEORIE
par Max Karoubi
On e x p o s e obtenus gique de
en a p p l i q u a n t
est
~valeurs
caract4ris~e
techniques
paraitront
c i t e r que
dans par
des s e c t i o n s
un f a i s c e a u les a x i o m s
globales
le f a i s c e a u
F
plus
r4sultats d'alg~bre
complets
prochainement
Hn(X;F),
homolo-
accompagn~s
[4].
on s a i t q u e n ~ 0,
la c o h o m o -
F
variable,
suivants= r(x;F),
(1)
du faisceau
Hn(X;F) si
connues
cet e x a m p l e ,
H 0(x;F)
groupe
certains
Des r 4 s u l t a t s
demonstrations P o u r ne
cet a r t i c l e
des
~ la K - t h 4 o r i e .
leurs
logie
dans
f.
(2)
= 0, Y n > 0 A toute
est f lasque.
suite exacte
de f a i s -
ce aux 0 est a s s o c i 4 e 9 .-
Nous
des
des
aboutir
"i F
exacte
~
F"
~ 0
avons e s s a y 4
d'adapter
additives
en g r o u p e s
n > 0, de telles
cat4gories.
de ces f o n c t e u r s
a pu e t r e
(D4finition
certains
~ la K-
des
1.2).
cas,
(Th4oreme
a celle On a p u
foncteurs
K n,
la p 4 r i o d i c i t e
A
demontr4e
> -.-
generalement,
axiomatique Dans
> H n (X;F)
ce f o r m a l i s m e
et, p l u s
de B a n a c h
~ une d e f i n i t i o n
(3)
de c o h o m o l o g i e
> H n - I ( x ; F " ) ~ n- >1 H n ( X ; F ')
categories
cat4gories
ainsi
F'
une s u i t e
> Hn-I(x;F)
th4orie
~
2.3).
366
-
-
Cat article est divis4 en deux parties. premiere gories"
nous
d4veloppons
en nous i n s p i r a n t
pl~tement
continus
conde partie nous donnons blable
la n o t i o n
d4finissons
une c a r a c t 4 r i s a t i o n ~ calla
de Hi lbert.
axiomatique
CATEGORIES
Dans
comla se-
flasques"
et
de la K - t h ~ o r i e
~ valeurs
EN GROUPES
la
de cate-
des o p 4 r a t e u r s
les "cat4gories
de la c o h o m o l o g i e
I
de "suite exacte
de la th4orie
darts les espaces
Dans
sem-
darts un faisceau.
DE B A N A C H
SUITES E X A C T E S
Definition
I.i
Soit
M
un groupe
est une a p p l i c a t i o n des p r o p r i 4 t 4 s
de
M
ab41ien.
dans
m +
Une q u a s i - n o r m e not4e
x ~-~
su__E M
llxll j o u i s s a n t
suivantes : lIxll = 0 ~ = ~
x = 0
(I)
IIx + y[l ~ ilxll + [ly[l
II-xll On appelle
Le groupe
un espace m ~ t r i q u e
translation
d(x,y)
IIxll
(3)
groupe q u a s i - n o r m 4
muni d'une quasi-norme. naturelle
=
pour
= llx-yll.
(2)
M
un groupe
ab41ien
M
est alors de m a n i ~ r e
la distance
R4ciproquement
invariante tout groupe
par ab~lien /
muni d'une distance
invariante
si on pose
Iixll = d(x,0).
quasi-norm~
complet pour
par t r a n s l a t i o n
Un groupe la distance
de B a n a c h
est q u a s i - n o r m e est un groupe
d e f i n i e par
la q u a s i - n o r m e .
- 367 -
Exemples. groupe de Banach.
Un espace de B a n a c h est ~ v i d e m m e n t Iien
est de meme d'un groupe
conque muni de la q u a s i - n o r m e
suivante
famille d ~ n o m b r a b l e
a b ~ l i e n quel-
(dire "discrete"):
iixll = 0
si
x = 0
il~ll = 1
si
x ~ o
Un espace de F r e c h e t dont la
un
t o p o l o g i e est d~finie par une
de s e m i - n o r m e s
Pi
est aussi un g r o u p e de
B a n a c h pour la q u a s i - n o r m e
llxll = [2 -i Tousles Banach
sorites
se d ~ m o n t r e n t
p o u r r a par exemple un s o u s - g r o u p e
d~velopp4s
i(x)) pour les espaces de
aussi b i e n pour les groupes
faire
fermi.
les a p p l i c a t i o n s
Inf(l,p
le q u o t i e n t
Si
born~es
M
et
de
M
de Banach.
d'un groupe de Banach par sont deux groupes
N dans
On
N
de Banach,
forment un groupe de
Banach pour la q u a s i - n o r m e
llfll = sup x~0 Definition
1.2
Une c a t ~ @ o r i e additive
llf (x)II iixli
t
o~
Hom
en groupes
(M,N)
de Banach est une c a t ~ o r i e
est muni d'une s t r u c t u r e
de groupe
T
de Banach de telle M, N
et
[
de
on ait l ' i n ~ @ a l i t ~ n_eed~pendantug~de
sorte u~e,
quels que soient
et les morg.hismes
ilvoull ~ Cllvilxlluil, C M, N e t P .
u: M
les objets > N, v: N
; P,
~tant use c o n s t a n t e
368
-
Exemples. est sans doute vectoriels pact
X.
L'exemple
(r~els ou complexes) si
E
s'identifie
pr~banachique
en groupes
et
de Banach.
T = ~(X)
en K - t h ~ o r i e des fibres
F sont deux fibres vectoriels, de Banach des sections
HOM(E,F). dans
important
de rang fini sur un espace com-
~ l'espace
fibr4 en h o m o m o r p h i s m e s cat~gorie
le plus
celui de la cat4gorie
En effet,
Hom % (E,F)
-
Plus g4n~ralement,
le sens de
Hom
(M,N)
toute
[3] est une cat~gorie
Enfin une simple cat~gorie
est aussi un exemple,
du
additive
en
~tant muni de la q u a s i - n o r m e
T
discrete.
Remarque l'application
l-
Dans une categorie
d~finie par la c o m p o s i t i o n Hom
(M,N) x Hom (N,P) T
en groupes
de Banach,
des m o r p h i s m e s
> Hom
T
(M,P) T
est continue.
Remarque ~.
Comme pour
les espaces
convient d ' i d e n t i f i e r
deux q u a s i - n o r m e s
lorsque
equivalentes.
celles-cisont
que aux categories Si
D
F
p.s = Id E.
un m o n o m o r p h i s m e
objets
directs
La meme remarque
quelconque
s: E
et si
> F
On voit a i s ~ m e n t que
(resp.
un epimorphisme)
sont les fleches
sont les objets
abelien s'appli-
E
et
on appelle m o r p h i s m e d i r e c t de
la donn~e de deux fl~ches
telles que
phismes
0
sur un groupe
de
D.
on
de Banach.
est une c a t e g o r i e
sont deux objets de dans
en groupes
de Banach,
d'une
et s
On notera
E
p: F
(resp.
> E p) est
et que les mor-
cat~gorie
dont les
(s,p): E
F
~ F
369
-
une telle
fleche.
Supposons
gorie en groupes
de Banach
additive
pleine
T
appelle
T-filtration
i 6 I
ensemble
directs
f9 =
de
(si,Pi):
Si E
d'indices
maintenant
que
et consid4rons
N. de
-
E
quelconque, > E
soit une cat~-
une sous-cat~gorie
est un objet de
la donn~e
E.
D
d'objets et de
satisfaisant
Ei
D
on de
T,
~-morphismes a l'axiome
sui-
vant:
F i. tration
de
Si
Ei
et
E, il existe
Ej
sont deux objets
un troisi~me
tration qui rend commutatif
objet
Ek
de la filde la fil-
le diagramme E
E.
E.
3 Soient
{E i}
dira que la filtration
et
{E'.,}~ deux filtrations
{E i}
est moins
de
fine que la filtration
!
{Ei,}
si, pour tout indice
et des morphismes
directs
i, on peut hi. i
trouver
qui rendent
un indice
commutatif
le
diagramme E
hi'i E l9 --
On dira que les deux est plus
fine que
filtrations
l'autre
~
On
E.
Ei,
sont ~quivalentes
et reciproquement.
si l'une
9
1
I
- 370 -
Diagramme une cat6gorie
en groupes
quelconque
dens
mutatif
~-pr~s
~
diagramme E
~
F
Commutatif
D.
Nous dirons
~-pr~s
llg'f- k.hll <
dens
d'une
dent que de
D
f
k
une c a t 4 ~ 0 r i e
g j: E
additive
E) v ~ r i f i a n t
Soient F
F. 3 > F
~
x-filtration
F 2.
objet
joignant
dens
Hom(E,F).
> H
>
F
signifie que l'on a l'in~galit4
sur la cate~orie
un
de ce
Hom(E,F).
une s o u s - c a t ~ o r i e
f: E
(E,F)
(f,g)
IIf -gll < z
est com-
1.3
Soit
et
A
dire que le d i a g r a m m e
est c o m m u t a t i f
Definition
D
un d i a g r a m m e
que le d i a g r a m m e
on a
G
~,
A
si, pour tout couple d'objets
E
de
Soit de n o u v e a u
de Banach et soit
dens ce d i a g r a m m e
tration
e-prAs.
et pour tout couple de m o r p h i s m e s
Par exemple,
!
~
un
E
pleine
de
de Banach et soit
~.
Une
(i) est la donn~e, E i, i E I, sur
les axiomes
un objet de
D-morphisme.
de la f i l t r a t i o n 3
en groupes
de
pour tout objet
E
(I
ne d~pen-
suivants:
i, F
Alors, F
T-fil-
un objet de
~z
et un
> 0, i_~lexiste T-morphisme
tels que le d i a g r a m m e
(i) On dit aussi que
x
est s o u s - c a t 4 g o r i e
D
id4ale de
D.
- 371 -
F
soit commutatif F 3. et
f: E
~ ~-pres. Soient
> F
un objet
E.
,
3
E
un ob~et de
u nn ~-morphisme.
de la filtration
9, F
Alors, de
E
un ob~et de
Vc > 0, il existe
et un
T-morphism e
1
tel
> F
gi: Ei
que le diagramme
P.
F
1
E.
1
soit commutatif F 4. tration (E 9
~ c-pr~s. Si
E. 9 F. z 3
E de
et
F
E 9 F
sont des objets est ~quivalente
de
D,
la fil-
~ la filtration
F) k .
Exemples
i. et soit
~
peus alors vante:
Soit
H
la cat~gorie
la cat~gorie consid~rer
pour tout espace
H
des espaces comme
des espaces
de dimension
~-filtree
de Hilbert
de Hilbert
E, E i
finie.
de la mani~re
On sui-
sera la collection
-
de ses
sous-espaces
l'injection onale.
si
X
finie
2.
Soit
de B a n a c h m u n i exemples:
A
la c a t 4 g o r i e
Soit
(M I, ..., M n,
des m o d u l e s ...)
parmi
M 1 • ..0 Xn,
par
[I(A)
Ll-somme, A-modules. Banach groupes
...)
dit,
•
(i.e.,
choisis
(resp. triviaux
infinie
Nous ~
--.) parmi
ou
sera D.
A.
de
[(A)
fini d ' o b j e t s
de
i(A).
. o .
f o r m ~ des
que
ZllXnll < + ~ pour
suites Ce s o u s - e n -
.
la " q u a s i - n o r m e " 9
On d~signe
s o n t de t e l l e s
les h o m o m o r p h i s m e s categorie
d~finir
E i E Ob~,
les
fini
born4s
en g r o n p e s
une c a t ~ g o r i e
~'-filtr~e,
Les o b j e t s
un n o m b r e
fini sur
le s o u s - e n s e m -
une
allons
qui a
T
et s o i t
d'objets
les o b j e t s
~tant
un g r o u p e
ilxY{I~ cilxllxilyll ;
de type
de B a n a c h
dont
soit
4quivalente
(E 1 , .-., E n, suite)
libres
telles
les m o r p h i s m e s
quelconque.
cat~gorie
~T(X)
ou un a n n e a u discret)
comme • Mn
la c a t ~ g o r i e
de B a n a c h
on
que
vectoriels
Itx II = N x I II + ... + IIx n It + "'"
Ceci
compact
HT(X),
telle que
un n o m b r e
s e m b l e e s t en f a i t un g r o u p e L1 , ~ s a v o i r
de
fibres
une s u i t e
Ll-somme
leur
...,
des
de B a n a c h
choisis
ble du p r o d u i t (Xl,
id~ale
orthog-
de l'unite)
un a n n e a u de B a n a c h
une a l g ~ b r e
On d ~ f i n i t
la p r o j e c t i o n
e s t un e s p a c e
d'une multiplication
~tant
> E ~tant
(resp. h i l b e r t i e n s ) .
i CA)
Mi
s : Eo l 1
une p a r t i t i o n
la c a t e g o r i e
de d i m e n s i o n
x =
> Ei
(en u t i l i s a n t
d~signant
finie,
E
est une s o u s - c a t ~ g o r i e
HT(X))
les
Pi:
Plus g 4 n ~ r a l e m e n t ,
~T(X)
-
de d i m e n s i o n
canonique,
voit aisement
3 7 2
de Ei
H, K,
x' R
de de
en
~tant une sont
~tant o.o, L
les s u i t e s
(pour c h a q u e d'objets
de
- 373 -
T
En particulier,
9
H 9 K 9 -.- 9 L .
direct de
s'identifie
Horn (Ei, Ej) de B a n a c h F =
chaque objet
o..)
changer
les objets
objets
F I, ..., Fn, .oo, L .
.
un d e u x i e m e
(i, j),
ferm~ de l ' a n n e a u
Soit m a i n t e n a n t
objet de
D.
Quitte
..., L, on peut s u p p o s e r que
...
les
sont aussi choisis parmi
Hom(Ei, A.
un facteur direct de
pour tout couple
a un s o u s - g r o u p e
H, K,
Donc
de la suite est facteur
De plus,
A = E n d ( H 9 K 9 ... 9 L)
(FI, ~176 Fn,
H, K,
Ei
Fj)
est pour
Considerons
les m&mes
raisons
a present
une m a t r i c e
C e t t e m a t r i c e peut etre
interpret~e
infinie f = o~
fji E Hom(E i, Fj).
d'apr~s
la d i s c u s s i o n
A 9 ... e A 9 ... f
est
la somme D'autre est
L1 part,
Ll-bornee
de
pr4c4dente
dans
"Ll-born~e"
(fji)
si
A • A ... f
la m a t r i c e
• A x ...
se p r o l o n g e
N0-exemplaires
et si,
comme une a p p l i c a t i o n
f
de
A
"permutante"
existe une b i j e c t i o n
telle que
dite
~
d~signe
le symbole
"~-permutante"
mutantes.
Enfin
existe une suite vers
f
pour
de Kronecker.
si elle est somme
la m a t T i c e fr
est dite
de m a t r i c e s
la q u a s i - n o r m e
L1
si elle
ligne et sur chaque en d'autres
ou
de
dans elle-meme.
il y a au plus un ~ l ~ m e n t non nul: k: ~ ----> ~
On dira que
en une a p p l i c a t i o n
est dite
sur chaque
.
de
colonne,
termes
il
fji = 6k(i)ifji
La m a t r i c e
f
finie de matrices zl-permutante
Z-permutantes d~finie
est per-
s'il
qui converge
pr~c~demment.
Les
-
morphismes
de
D
-
sont alors les matrices
qu'on vient de d4crire. nition des morphismes H, K, ..., L
374
On v4rifie ais~ment que cette d~fi-
est ind4pendante
de
~
On d~finit 4galement
comme la quasi-norme
d~finit.
Celle-ci
quasi-norme,
groupes de Banach. D
la quasi-norme
d4pend ~videmment
la categorie
est
Soit maintenant
T1
Definition
Soit > F
un
A Pour cette
~'
la sous-cat4gorie
~
(EI,...,En,...)
Alors un calcul facile T
et que la cat~gorie
D.
1.4
une cat~@orie ~-morphisme.
~-filtree et soit
Les deux assertions
suivantes
son t alors ~quivalentes: (i) tration de diaqramme
F
D
Pour des raisons qui apparaitront plus loin,
la cat~gorie
Propositio n e t
f: E
est 4quivalente
~'-filtr~e.
on notera
du choix de
dont les objets sont les suites
x'
qu'elle
est bien une cat~gorie en
nulles a partir d'un certain rang. montre que
et
d'une fl~che
LI(A)
n'en d~pend pas.
D
L1
(pour le produit des
de la fl~che
mais sa "classe d'4quivalence"
pleine de
du choix des objets
qui ont servi ~ d~finir la quasi-norme
que l'on obtient ainsi une cat~gorie matrices).
zl-permutantes
Vz > 0, i_!lexiste ~n ~ et un ~!Q/~F~_~
fj: E
Fj ) Fj
de la filtel ~
l_~e
- 375 -
F
fj~F
sj
soit commutatif { c-pr~s. (ii) tration de
We > 0, il existe un objet et un morphisme
E
f.: E. ~ 1
1
de la fil-
Ei
tel u ~
> F
le
dia@ramme E
Pi
F
E.
21.
e-pr~s.
soit de meme commutatif
v~rifiant
Un morphisme ~uivalentes
(!) ou
l'une des conditions
(i_~) est dit compl~tement
Remarque.
Cette d4finition
de celle des op4rateurs
compl~tement
dans les espaces de Hilbert
continu.
est ~videmment continus
inspir4e
(ou compacts)
(cf. exemple 1 des cat4gories
fil-
tr~es). Soit
T
une sous-cat~gorie
alors d~finir une categorie quotient suivante:
les objets de
phismes sont ceux de continus.
En d'autres
D
~/~
id~ale de ~/T
D.
de la mani~re
sont les objets de
modulo les morphismes termes on a
On peut
9, les mor-
compl~tement
376
-
Hom~/T(E,F) d~signant continus
= Hom0(E,F)/K(E,F),
le s o u s - g r o u p e de
E
dans
Definition
F.
La c a t ~ g o r i e
Hom
T
et
T'
dont nous
!
deux c a t e g o r i e s
additif
__si l ' a p p l i c a t i o n
(~M,~N) T
est ~ v i d e m m e n t laissons
la
1.5
Un foncteur
Serre "(2)
universel
~/~
compl&tement
au lecteur.
Soient Banach.
K(E,F)
ferm4 des m o r p h i s m e s
la solution d'un p r o b l & m e formulation
-
~: de
T
> x'
en groupes
est dit "de
HomT (M,N)/Ker~,
est~ une b i j e c t i o n
born~e
de
dans
ainsi que son inverse.
~
Le foncteur est dit born~ si l ' a p p l i c a t i o n ~, : HornT (M,N)
est born6e.
> Hom T, (~M,~N)
Pour pouvoir p a r l e r m a i n t e n a n t de categories vante: Banach;
il nous
faut introduire
les objets de un m o r p h i m e
B
de suites
la "cat~gorie"
sont les cat4gories
~:
x
) T'
de
exactes
B
B
en groupes
suide
est un foncteur de
Serre.
Definition
1.6
Soi t T'
8
-
(2) Cette
terminologie
cation naturelle fibration
,.-
Iso
,, -'~
est justifi~e (M,N)
de Serre dans
classiques.
X T
"%-
par le fait que l'appli-
> Iso
,(~M,~N)
le cas des cat4gories
est une de B a n a c h
-
une suite d'objets dite exacte
3 7 7
-
et de morphismes
de
B.
Cette
suite est
si le noyau de l'application Hom
(E,F)
> Hom
,,(XE,XF)
T
est l'ensemble continus
des
T-morphismes
pour une filtration
de
de
!
E
dans
[
compl~tement
par la "cat~@orie
image"
8 (~') (3) . Soit que.
T
une categorie
On peut munir
dentes.
la cat~gorie
La premiere d'un objet
la seconde
(dite discrete)
nul 0
E
de
T
le second)
(resp.
~
de Banach
de deux
E
la filtration
de
x.
la s o u s - c a t ~ g o r i e
Dans
quelcon-
filtrations
consiste
l'objet
de la cat4gorie
(resp.
x
(dite grossi~re)
filtration
l'objet
en groupes
~vi-
~ prendre
comme
seulement. E
Dans
se r~duit
le premier
cas
id~ale est 4gale
~
T
0).
Application. 0
>
est bien exacte, la filtration
P
Avec T
>
la cat~gorie
discrAte
9
9
9
cette def~n~tlon,
(resp.
D
>D/T
T
(resp.
la suite ~
D/z)
grossi~re).
0
~tant munie
R4ciproquement,
de si
on a une suite exacte 0 la cat~gorie
> ~, T"
8
s'identifie
> 9
> T"
~ la cat~gorie
> 0 quotient
T/e(T').
(3) La cat~gorie image 8(x') est la sous-cat4gorie pleine de dont les objets sont isomorphes aux images des objets de x' par 8. On d~montre en fait que la filtration de T par 8(T') est unique (~ e q u i v a l e n c e prAs).
- 378 -
II
CARACTERISATION
Definition
cat~@orie
D
une c a t 4 ~ o r i e
est dite
D
DE
LA K-THEORIE
2.1
Soit
9 :
AXIOMATIQUE
tel q u e
>D
flasque les
en groupes
s'il existe
foncteurs
~
de B a n a c h .
La
u__nnf o n c t e u r
born~
et
r 9 Id D
soient
isomorphes.
Exemples. flasque.
La c a t 4 g o r i e
En effet,
il s u f f i t de p o s e r
(somme h i l b e r t i e n n e meme que au
la c a t ~ g o r i e
31
existe
x
une c a t ~ o r i e
~
= E 9 ... 9 E 9 ...
E).
On d ~ m o n t r e
2 des c a t e g o r i e s
categories
filtr~es
(R,0,0,
en groupes
(d~pendant
uN > D
T
e s t une c a t ~ @ o r i e En effet,
s o (E) =
de
est
de
filtr~es
fl a s q u e .
a l o r s une s u i t e e x a c t e
0 D
(exemple
~(E)
de H i l b e r t
2.2
Soit
o~
de M 0 - e x e m p l a i r e s
w i) e s t une c a t ~ g o r i e
Theoreme
des e s p a c e s
> T'
de B a n a c h .
canoniquement
Ii de
> 0
flas q u e .
on c h o i s i t
D = T1
au
eo = le f o n c t e u r
w i) et
(cf. e x e m p l e
2 des
d4fini par
.o.).
Application.
Le t h 4 o r A m e
d'affirmer,
p a r des r a i s o n n e m e n t s
"r~solution
flasque
canonique"
2.2 n o u s
standard,
permet
ainsi
l'existence
d'une
379
-
0
Uo
> x
de route cat~gorie que si
x
a
> Xl
1 > x2
a2
en groupes
est une categorie
-
de Banach
x i.
d~finit
T
quotient
n i~me
Xn/an_l(Xn_l).
est une cat~gorie pseudo-ab~lienne
D4finition
associ4e
Soit
x
Kn(x),
S~x).
le cas o~
Dans
ce__~s groupes 2 dans
~
on
la categorie
[3]
w
que si
~
x
la c a t ~ g o r i e
~.
e_nn groupes
n k 0, le groupe x
de Banach.
de G r o t h e n d i e c k
est une cat4@orie
sont p ~ r i o d i q u e s
de p~riode
O_nn
de
pr4banachique,
8 dans
le cas r4el et
le cas complexe.
n~cessite Definition
l'introduction
de ce th~or~me
des alg~bres
est d ~ l i c a t e
de C l i f f o r d
(cf.
et [3],[4]).
2.4
Une th~orie
de l__acqhomologie sur
la donn~e de foncteurs des groupes 6n-I
d~finis
[3],
2.3
La d 4 m o n s t r a t i o n
J~
le sens de
le cas g 4 n ~ r a l
on a d~sign~ par
une c a t ~ o r i e
d~si~ne par
II est ~ n o t e r
Dans
(cf.
> ...
dans
comme
Rappelons
additive
et T h ~ o r ~ m e
de
Xn
T.
pr~banachique
il en est de meme des cat4gories la s u s p e n s i o n
an_l>
> ....
Fn
ab41iens :
F n-I ( T" )
B
de ! a c a t ~ o r i e
et d ' h Q m o m o r p h i s m e s ~_ F n ( T )'
pour toute suite exacte 0
> T'
> x
est par d~fini-
> ~"
> 0
B
dans
naturels
la
-
On s u p p o s e suivante
que
F n(~)
= 0
3 8 0
si
-
x
est f l a s q u e
et que
la s u i t e
est e x a c t e
F n - i (T)
Th~oreme
> Fn-l(~") ~,n - i> Fn (T')
> Fn (~)
> Fn (~")
.
2.5
Ii e x i s t e isomorphisme le__~s g r o u p e s
une t h ~ o r i e
pros
sur
Fn(T)
B
de la c o h o m o l o ~ i e
telle ~
coincident
F0(~)
avec
et une seule
= K(T).
les g r o u p e s
De plus,
Kn(T)
d4finis
haut. Le d e m o n s t r a t i o n tr~s
~vidente.
Bass
dans
foncteur
On d o i t
d~riv4"
tiellement
~
~
~,:
~: Pour
les o b j e t s
donc
aussi
de
de f l ~ c h e s
de
de
T1
> T'
> ObT' ~(~)
Les
par
e s t le " p r e m i e r
de S e r r e e s s e n -
deux categories
les r a i s o n n e m e n t s ,
est b i j e c t i f . suivante:
~l
pour
fl~ches
flasque
8(E)
On a alors
=
de
la r e l a t i o n
~, par
K
introduit
en
on
On a s s o c i e
les o b j e t s associee
de ~
D(~) ~),
!
~i(~)__ = 0.
d~fini
entre
(cat~gorie
est c o m p l ~ t e m e n t et
K1
un f o n c t e u r
simplifier
la c a t ~ g o r i e
sont
du g r o u p e
K I.
~
ObT
n ' e s t pas non plus
le f o n c t e u r
maintenant
surjectif
que
que
du f o n c t e u r
de Banach.
supposera alors
se s e r v i r
[i] et r e m a r q u e r
Consid4rons
groupes
de ce t h ~ o r ~ m e
(E,0,0,
D(~)
o.o)
les c l a s s e s
d'~quivalence
continu
un f o n c t e u r ~
sont
(pour la
suivante: x-filtration)
~vident
~ ~(~) sur
les o b j e t s
et par
-
8(f)
avec
~(~)
d'ailleurs de telle
filtrant
0
~:
pour
0
....
0
0
....
(4).
Le f o n c t e u r
On a alors
Remarque.
d~duit
> S1T
de S e r r e
(4) Si
de
> 0
essentiellement
surjec-
= Kn(D(~))
Kn(~)
> Kn(T)
sont n a t u r e l s
Ce t h ~ o r e m e
"restriction compact
la suite
par celle
la s u i t e e x a c t e
homomorphismes
est un e s p a c e
est
exacte
~ui e s t i n d u i t p_a_ ~ !e f o n c t e u r
foncteur
induite
9
, __nn pose
Kn-l(x) -----> Kn-l(T')6n-~-
~n-i
0
2.6
tout f o n c t e u r > ~'
o~ t o u s l e s
....
~(~) f
et T h e o r e m e
9
n ~ 0
0
a la s u i t e
Kn+l(~) si
0
la f i l t r a t i o n
> ~'
Pour tif
-
= f, sur les m o r p h i s m e s .
sorte qu'on
Definition
i
381
exacte
des
fibr4s"
et
Y
> Kn(T'), ~
~'
s'applique ~T(X)
un s o u s - e s p a c e
de c o h o m o l o g i e
~' = 0 on r e t r o u v e b i e n Kn+I(T) = K n ( s l x ) . groupe
l,exception >
n ~ i, de
D(~).
en p a r t i c u l i e r > ~T(Y) fermi.
en K - t h ~ o r i e
(~ i s o m o r p h i s m e
o~
au X
On en topologique
pros)
le
-
Kn-l(x)
> Kn-I(Y)
382
> Kn(x,Y)
-
> Kn(X)
> Kn(y),
n k 1 ,
sans ~voquer les alg~bres de Clifford ou la p~riodicit4 Bott.
Ii r~sulte une construction
relativement ~14mentaire
la K-th~orie en tant que th~orie cohomologique compacts.
de
Bien entendu la p 4 r i o d i c i t ~
de
sur les espaces
des groupes
Kn
(th~or~me 2.3) n'est pas 4vidente avec ce point de vue et doit etre d~montr~e
s~par6ment.
-
383
-
BIBLIOGRAPHIE
[I]
Bass,H.
"K-theory and Stable Algebra",
Math4matiques [2]
Godement,R.
[3]
Karoubi,M.
de I'I.H.E.S.
5-60.
(1964).
Th~orie des Faisceaux. "Alg~bres de Clifford et K-th6orie",
Annales Scientifiques
[4]
No.22!
Publications
p. 161-270.
(1968).
Karoubi,M.
S~minaire
de itEcole Normale Sup~rieure.
sur la K-th~orie
(~ para~tre) .
- 384 -
RELATIVE
FUNCTOR~AL
ADJOINTNESS
SEMANTICS :
RESULTS*
by F. E. J. Linton
INTRODUCTION Central are the Godement an adjoint
to any e x p o s i t i o n construction
functor
ciated
of algebras
construction
of operations relations
over a triple,
and "underlying"
[6] of the clone
analogues.
among
category
functions
is replaced by an arbitrary
category
V, one expects
exposition Kock
at the level of
[7] have,
however, patible
to develop
both
indeed, these
symmetric
gone
closed
functors,
S
[2].
V
co-
of sets and
or m o n o i d a l
Bunge
add the assumption on
and
tuples
and their
far in this direction;
structure
the asso-
a satisfactorily
V-categories
treatments
monoidal
with
adjointness
these constructions
If the usual base
from
construction
of tripleary
[8], along w i t h the various
available
arising
the E i l e n b e r g - M o o r e
adjoint pair of "free"
the Kleisli
triple
[4] of the triple
situation,
[3] of the category
of the theory of triples
parallel [i] and
unfortunately, of a com-
whenever
they
* Research s u p p o r t e d by N~ grant NSF-GP 6325, W e s l e y a n Faculty Research Grant 5427-143, and Battelle M e m o r i a l Institute, Seattle Research Center.
-
3 8 5
-
deal w i t h the case that the base c a t e g o r y
In w h a t rudiments
for m o n o i d a l assumptions
equivalent)
V, the other
however,
of will
V
V
at all; moreover,
case.
(equalizers) when
V
difference
in the closed m o n o i d a l
therefore
of
w
An a w k w a r d
is one
no symmetry
Just
as in the w o r k
in e i t h e r
case,
that
in order to get any one m u s t k n o w
L A = V(A,-):
V
> V
kernels
(as it a u t o m a t i c a l l y
case).
These
assumptions
be in force t h r o u g h o u t the paper,
first p a r a g r a p h
V
of triples,
is closed,
left r e p r e s e n t e d e n d o f u n c t o r
preserves
(and, w h e n
closed-
one must assume,
has d i f f e r e n c e k e r n e l s
that each
for
is closed.
we shall p r e s e n t the
V-theories
are n e e d e d in e i t h e r
[i] of Bunge,
theory
thereforet
of two s e p a r a t e b u t p a r a l l e l
closed monoidal,
V
follows,
V
shall
e x c e p t in the
and in Lemma 2.
consequence
metry is the m i n o r n u i s a n c e
of our refusal
that the cotriple
to use sym-
theory
cannot
be o b t a i n e d simply by c a r r y i n g
out the triple
duals of all the
i n v o l v e d -- there are no such
V-categories
duals,
in general.
merely
reverses e n o u g h
exposition theory.
This is n o real difficulty,
to o b t a i n
left to the reader; before
arrows
in the e n s u i n g
a valid exposition
This e x e r c i s e
immediately
theory on the
in judicious
however:
triple
of the cotriple
arrow
reversal w i l l be
a good time for h i m to e n g a g e w
theory
in it is
one
-
Throughout the e l e m e n t a r y
them:
see
[i] and
functors,
notions
along w i t h
the V-category ness b e t w e e n
transformations
to this s e t t i n g
are p r e s e n t e d
the
"free"
the E i l e n b e r g - M o o r e over a V-triple
and "underlying"
also appear in B u n g e ' s w o r k
around
is found in w
V; however,
of the
construction
and the
of
V-adjoint-
V-functors.
These
[i], at least in the
the a r g u m e n t p r e s e n t e d here
a split d i f f e r e n c e
kernel
w h i c h seems n o t to have b e e n p u b l i c l y
phenomenon
r e c o r d e d even
(Lemma i) in the
V = S.
The jointness, hitch.
over
The i n d i s p e n s a b l e
of V-triple
of algebras
case of m o n o i d a l
case
categories,
[5].
p r e s e n t work,
centers
and n a t u r a l
relevant
The d e f i n i t i o n
matters
familiarity with
of closed and m o n o i d a l
[2] for full information.
adjointness in
-
this work, we assume
notions
and categories,
3 8 6
V-triple
which
analogue
is the concern
of the s t r u c t u r e - s e m a n t i c s of w
goes through w i t h o u t
The main tool is the fact that,
tors b e t w e e n
V-categories,
a functor
ada
unlike most ordinary AT
P: X
satisfying
(*)
UT o p = U at the level of o r d i n a r y V-functor V-functor
X
functors,
> A, admits
where
U
is a fixed
at most one e n r i c h m e n t
c o m p a t i b l e w i t h the v a l i d i t y
level of V-functorso
A general
func-
of e q u a t i o n
form of this
fact,
to a (*) at the accompanied
- 387 -
by a m a n a g e a b l e
c r i t e r i o n for s u c h an e n r i c h m e n t to exist, is
r e c o r d e d as L e m m a 2, and is u s e d all t h r o u g h the r e m a i n d e r of the paper.
A s s u m i n g f a m i l i a r i t y w i t h the
V-cotriple
of the results of the p r e c e d i n g s e c t i o n s , gory g e n e r a l i z a t i o n of L a w v e r e ' s triple,
e x p o s e s the
V-cate-
as y e t u n p u b l i s h e d t r i p l e - c o -
(Kleisli category) - ( E i l e n b e r g - M o o r e c o n s t r u c t i o n )
adjointness
V-triple
w
analogues
t h e o r e m p r e s e n t e d in his
A description,
as in
T
V-valued
c a t e g o r y of
in terms of
at Battelle.
[8], of algebras V-functors
~, w h i c h was o r i g i n a l l y
here, has b e e n o m i t t e d ,
lectures
f r o m the K l e i s l i
i n t e n d e d for i n c l u s i o n
for the r e a s o n that n e i t h e r an a d e q u a t e
b a c k g r o u n d in c o n t r a v a r i a n t
V-functors n o r any V - v a l u e d Y o n e d a
lemmas are yet a v a i l a b l e w i t h o u t use of s y m m e t r y s m a l l n e s s of the d o m a i n
over a
(cf.
(cf.
[2]) or
[1], T h e o r e m 4.7), r e s p e c t i v e l y .
It is h o p e d to r e m e d y these o m i s s i o n s e l s e w h e r e .
w
,t
A L G E B R A S O V E R A V-TRIPLE
J
T =
A V-triple of a V - f u n c t o r n: i d A
(T,n,~) A
Tz A ) T
) T, ~: TT
and
on a V - c a t e g o r y
consists
V-natural transformations
s a t i s f y i n g the f a m i l i a r triple iden-
tities o
A
nT -- W
o
Tn -- id T ,
IJ o ~T -- W o T~ .
-
A
T-algebra
A-morphism
structure e : TA
e
> A
388
on
A
6 obj
e o UA =
the s e t
set
of T - a l g e b r a
of all t h o s e
as u s u a l ,
an
for w h i c h o n A = id A
The u s u a l
is,
A
~ o Te
maps
f:
A-morphisms
. (B,8) , t h a t
(A,~)
f: A
Tf
TA
,
making
> B
the
is, diagram
- TB
A
~
B
f
commute,
can be v i e w e d
of the p a i r
of
as t h e
difference
kernel
(or e q u a l i z e r )
functions A o Ca,B)
A 0 (A,B)
*~
A 0 (TA,B)
(TA, 8 ) A 0 (TA,TB) When its
T
is a V - f u n c t o r ,
difference
V-object
kernel,
of T - a l g e b r a
fore h e n c e f o r t h
this
if
can be
makes
a lovely
any,
maps
assume
diagram
V
from has
(A,a)
difference
V-object
AT((A,e) , (B,8)),
frequently
to b e
difference
of the
the
kernel
to
lifted
candidate
(B,8). kernels
abbreviated
lifted
diagram
A(e,B) A (TA, B)
A (A,B)
A (TA, TB)
to
We
V, and for a there-
and define as
the
A~(u,8) ,
- 389 We shall write
UT (A,~), (B,8)
UT
, or simply
0
for the
(monic)
V-morphism u T 9 A T ((A,u) ,(B,8))
= A ~ (e,B)
A(A,B)
canonically
associated with this difference
Proposition
1
Assume
that
V is a closed category with difference
kernels
that are preserved by each
V(X,-):
V
> V
(a)
Let
T
commutative
is a monoidal
be a V,triple
There i s p r e c i s e l y J(A,u) : I
left--represented
V
(resp., that
difference kernels).
kernel.
cate~or y with
on a V-category
one V-morphism
AT((A,e) , (A,u))
>
the triangle UT
ATCCA,~) , CA,~))
)
J(A,e),(A,e)~
(A,A)
/ ~ A I
(b)
There is preqisely
L (A'e)
: A T (8,7)
(resp.
M (B, 8 ) AT (A,e) , (C,7) : (B,7)
one map
V(AT (~,S) ,A~ (~,7) )
(s,S), (c,7)
|
AT
(~,8)
functor
AT
A.
-
commutative
390
-
the L
(A,e)
VCATC~,@),AT(~,7))
(Bt@)t(CPY))
A~(8,7)
V (id,U T)
A(B,C)
B,C
V(A(A,B) ,A (A,C))
> V(AT(a,@),A(A,C)) V(uE,id)
M(B, 8) (resp.
(A,u), (C,7)
Az(S,~) e A~(~,S)
UT | UT1 A(B,C)
U
A (A,C))
| A(A,B) ~A,C
(c) T-algebras
With the structur~e ~
and the V-objects
AZ((A,u) , (B,8))
betwee_._~n them form a V-category . . . . the passage make Proof.
(A,~)
(a)
[
b~
> A
a
TAA
A (A,A) "
(b) ,
_of T.-mor~hisms
A T , while the V-moz]~hisms
It suffices A(A,A)
(a) and
A(~,A)
V-functor
U T- A T
) A .
to show that the diagram ) A(TA,TA)
~ A (TA,A)
UE
- 391 -
commutes.
Since
TAA o JA = JTA,TA , however, it's merely a
matter of knowing that which is elementary
A(TA,~)
o JTA,TA = A(u,A) o JA,A ,
(see [2], Chapter I, diagram (9.10) and
Chapter II, diagram (8.10)). (b)
In the case that
V
is monoidal, it suffices
to prove that the perimeter of the diagram A T C C B , 8 ) , CC,y)~ @ AT((A,u) ,(B,8))
A(~,C)~T I
~
A'B I
~I A(TB'TC~@A(TA'TB) M= | __
~ I MTB
A(B,C) @ A(TA,TB)
I ,
~ .
i~/ ~ /
/ J ~
MT
J
tor. A
Indeed, square
Squares and triangles
is a V-category.
I
/
I I~B ,~
V A (A,C)
A(B,C) ~ A(TA,B>//
A (TA 'C) "~ commutes.
~/ %e/
id~A(TA.,)7
% V
" ~
I ,
4~
~
commutes because
T
is a V-func-
II, III, IV, and V commute because
Region i) is obtained from the difference
-
kernel
definition
of
ii)
kernel
definition
> A(B,C),
commutes,
because
of
difference
kernels
squares
is a bifunctor, iii)
while
is a reflection
V-category. obtain
(c) V-category
of
v), vi),
and vii)
the commutativity
sponding
for
diagrams
a monomorphism, lished in conditions
VF 1
(b) and
is a
commute b e c a u s e
V(-,-)
i), ii),
and
this d i a g r a m is just w h a t
one w o u l d
diagram
if
VC n
candidates
V
were
of
|
to
V(-,-),
VC n') (resp.
A, the fact that each
of j ,M)
VF 2')
to remove
of
[2] for on
AT
of the corre-
T U(A,u), (B,B)
of the diagrams
(these commutativities (or
and
proof.
(resp. j ,L
closed m o n o i d a l
(b) are a consequence
VF 2
T
rule in a
and the commutativity
(a) and
on the
of the composition
The conditions
(a) and
because
of squares
from the m o n o i d a l
p r o v i d e d by parts
to estab-
of the d i a g r a m
II commutes
structure
V(X,-)
I and III commute by defini-
the adjointness |
that
that it suffices
pentagon
from the previous
all occurrences
ensures
of properties
using
from the difference
the assumption
pentagons
(Incidentally,
one set about,
the
iv),
similarly,
by tensoring w i t h
of the p e r i m e t e r
AT(-, - ) , w h i l e
V-functor;
closed,
And indeed,
commutes;
> A(TA,TB).
V
lish the c o m m u t a t i v i t y
tion of
and therefore
AT((B,B) , (C,7))
In the case
n e x t page.
by t e n s o r i n g with
it is obtained
TA, B o UT: A~((A,e),(B,8))
preserves
-
AT((A,u), (B,8))
UT: AT((B,8),(C,7)) region
3 9 2
is
estab-
are e s s e n t i a l l y [2], so that the
the
-
3 9 3
A
-
rO
~
or
.
.-.
[~
.....
~
.....
g
~
~
~
'-
.~
g
~
~
~
.~
.--.
v
/
.<~
~
I
~
v
.~
-,-I
-,.-t ,~1
F-v
,[H
U
A
0
>~
,~
>-
k~ v
- 394
V-functor is known
assertion
to be made a
monoidal,
similar
each outer vertex VC n'
UT
of
picture
VC n'
to the similarly
Since
the
UT
arguments
following
in case
lemma, w h i c h
appears
case
for
A.
is
Now
it, UT
are
as w e l l
maps
the along
from the last outer vertex the p e r i m e t r i c
AT
V
AT; w i t h i n
and so does each square
is closed, a (left)
when
located inner vertex,
Full details,
TO construct
the classical Lemma
V
for
VC n'
last inner vertex is a monomorphism, gram must also commute.
V-natural; the square commutes because the base is
A(-,-)
A(A,Sn B) = A(A,id B) = idA(A,B)
n
is
is a bifunctor;
.
The proof of b) resides in the diagram
/ A (TA, TB) I
~
A(A,B)
(TA,TB) A (~A,TB) /
AtTB
A (TA, S) A (TTA, TTB) A (TA,B)
I
ACTTA,' ~
A (TTA,TS)
TTA, B ~
[a(Ta,
A (TTA,TB) I
A (TTA, B)
A (TTA,TB) ) A (TTA,B) A(TTA,8)
A (TA,B)
~ A {ljA,B )
396
-
-
The pentagon commutes because
u
is V-natural.
hand diamond commutes because
T
is a V-functor,
square, because
8
is a T-algebra structure
= B o UB ) , and the commutativity follows
from the bifunctor Relation
on
the central B
(8 o T8 =
of the right hand diamond
character of
c) follows easily
, while
~A o Tn A = idTA
The left
relation
A(-,-).
from the triple identity
d) results
from the commuta-
tivity of both squares in the d i a g r a m T A(TA,B)
--
TA,B
A (TTA, 8)
; A(TTA,TB)
A(nA'B) 1
> A(TTA,B)
,LA(TnA'TB)
A(A,B)
-
> A(TA,TB)
> A(TA,B)
TA, B
A(-,-)
,
A (TA, 8)
the first c o m m u t i n g because because
I A ( T n A ,B)
T
is a V-functor,
is a bifunctor.
the second
The proof of Lemma 1 is
thus complete.
Recalling now that structure FT(A) =
on
TA, w h a t e v e r
(TA,u A) .
~A" TTA ~ ; TA
is a T - a l g e b r a
A 6 obj A, let us write
Lemma 1 asserts
that the pair of V-morphisms
A (~A,B) A (TA,B)
; A (TTA,B) TTA 8 B )
TTA'B~ whose
A (TTA,TB)
difference kernel is, by definition, UT : A~((TA,~A),(B,8)) F ~(A) , (B,8)
9 A(TA,B)
- 397 -
fits,
in fact,
kernel
A (A,B).
A(A,UT(B,8)) which
in a split e q u a l i z e r There
structure stitute
--~ ; AT((TA, UA),(B,8))
(see
is, however, structure
equally
on
F~
mutativity
[5])
relation
simple
of Lemma
factors,
A T (FTA,F~B)
these
Referring
this back
with
and
the con-
F T.
It
V-functor to the d i a g r a m
identity
indicates
b),
com-
that the map
---> A(TA,TB)
by a map we shall
---~ A (TA,TB)
FT
UT
1 to e s t a b l i s h
A(A,B)
= A~(FTA,(B,8))
isomorphisms
between
to describe
of the upper p e n t a g o n TA,B:
to endow
for w h i c h
directly.
used in the proof
UT:
[i] or
of a V-functor
a V-adjointness
uniquely
result isomorphisms
= A(A,B)
can be used
d i a g r a m with difference
call
FT A,B
, through
: A z (F~A, FTB)
A(A,B)I ~
AT,B A
(TA, TB)
By the use of Lemma 2 below FT
is a V-functor
and the n a t u r a l i t y
is negligible;
(associated
trans formation s
satisfies
the proof
of
the identity
the proof that
that
T = UTF T
of the above isomorphisms
A(A,UT(B,8)) being easy
(see w
~: id
--~ > A(FTA,(B,8))
front adjunction
is the
~ ; T = uTF T, while
V-natural
the back
UTc T (A,u) = a), we have e s s e n t i a l l y
adjunction completed
-
Theorem
-
1
The to the
3 9 8
V-functor
V-functor FEA =
FT:
UT:
A T --~ A
A -- - A T
is
determined
front
adjunction
a d ~ u n c. t i o .n . T .
as follows:
.has
is
n: i d - - - >
the e f f e c t
9 >
A(TA,TB)
T = UTFT;
e T(A,~)
the b a c k
= ~: F T A ---~ (A,~);
and ----.theadjunction~ 9 isomorphism 9 is t h a t i n d u c e d registered the p a i r defined
in L e m m a
1 that
of maps w h o s e
A(A,B)
difference
b y the
is d i f f e r e n c e kernel
fact
k e r n e l , of
AT(FTA, (B,S))
is
to be.
Remarks on
V-adjoint
(TA,~A)
UT T = TA : A(A,B) F T A , F T B o FA, B ,B The
(right)
on L e m m a
A, a c o t r i p l e
category
V A~ v T(X)
vT =
I.
If
v v v) (T,n,
by the
following
T =
(T,n,~)
is o b t a i n e d
is a V - t r i p l e
on the f u n c t o r
means :
= X o ToP (X: A ~
((V)x) A = X(n A)
---> V, A 6 obj
A)
as a s s e r t i n g
that,
((V)x)A = X ( ~ A ) f
Lemma each
T-algebra
1 may b e (B,S),
V of the T - c o a l g e b r a
interpreted
the o b j e c t s
obtained
A(A,B)
by a s s o c i a t i n g
for
are the v a l u e s with
the
functor
RB = A(-,B) : A ~ > V the n a t u r a l t r a n s f o r m a t i o n v v v 8: R B > T R B w h o s e c o m p o n e n t s are 8A = A(TA,8) o TA, B: RB(A)
= A(A,B)
> A(TA,TB)
---> A(TA,B)
= RBTA =
(~(RB))(A).
-
It must,
399
of course, be v e r i f i e d
transformation;
the c o a l g e b r a
-
v B
that
really is a n a t u r a l
structure,
however,
is then
fully g u a r a n t e e d by Lemma 1.
This remark the e x t e n s i o n machinery where
proved,
can be extended,
as soon as the n e c e s s a r y
for c o n t r a v a r i a n t
as well,
and the converse
as n o t e d
V-valued
Yoneda
V-functors
to
Lemma
(needed else-
at the end of the introduction)
has
b e e n cons tructed.
w
V-STRUCTURE AND
Although V-categories V-functor, what
may,
an ordinary in general,
follows
continue
is unique.
lemma,
functor
to m e e t i n s t a n c e s the
To isolate
for w h i c h
P: X
9 V
between
carry s e v e r a l e n r i c h m e n t s
if any, we h a v e already met
m i n i m a l side conditions, P
V-SEMANTICS
to a
(in F T) , and shall in in which,
subject
to
V-functor structure,
if any,
of
the ideas, we state
the usual s t a n d i n g h y p o t h e s e s
the following on
V
are
re laxed.
Lemma 2
Lent and
V: V
> A
X, Y, A be
be
V-categories,
V-functors.
Vy,z: V(Y,z) --~ A(vx,vz)
Then a functor
U: X ~
that each
is ~ m o n o m o r p h i s m
no__~t a s s u m e d m 0 n o i d a l , that each as well).. ~r
Assume
let
V(X,Vy, z)
A
V-morphism
(and, i_~f V
is
is a m o n o m o r p h i s m ,
P: X ---> Y, s a t i s f y i n g
400
V at the ,L._r
ment
level of ordinary
functors,
9
_
to a V-functor
P = U
o
it admits
such
only if)
the dotted
admits
at most one enrich-
I
eL
satisfyin9
moreover,
(2. i)
(2.1)
at the level of V-functors;
an e n r i c h m e n t
if
(and, of course,
arrow in each d i a g r a m
X(W,X) --> V(PW,PX) UW,X1 IVpw,PX A(UW,UX) = A(VPW,VPX) c__~ be filled in by a V
-
~
rendering9 __the square
PW,X
commutative. Proof. satisfying to the side in
[2].
An e n r i c h m e n t
(2.1) involves conditions
Since each
precisely
named
Vpw,p x
one such s y s t e m of fillings ment.
So much
Now assume such have
of
VF n
PW,X
fillings
(resp.
in, h e n c e
VF n')
there
"only if"
available.
I
= A(VPW,
X (W,W)
>~ PW ,W
if
V
is monoidal,
(n = 1,2)
can be at most
the diagrams
UW,UW)
in, s u b j e c t
at most one such e n r i c h -
(and the
are indeed
I
and,
to a V-functor
such
is monic,
for the uniqueness maps
P
V (PW~tPW)
assertion).
Then we
-
401
-
•
X
~<
~
~
.~
s
X
~.
v
~< .,.4
s X v
0
-
in e a c h of w h i c h westwards
each
small
m a p of type
V
4 0 2
region
It f o l l o w s
as is r e q u i r e d
P
Lemma
2 is thus
A-valued
(e.g.,
the
functors,
[8]) w i t h
corner commute,
The p r o o f
3: T' --~
to a n o t h e r
(T,n,u)
T =
V-analogue
of
of the f a m i l i a r
for t r i p l e s
building
on
A
the c l a s s i c a l T
lemma.
situation,
from a V-triple
to be a V - n a t u r a l
compatible
with
and
on the u s u a l d e v e l o p -
the aid of the p r e c e d i n g
a V-triple map
~ T
t h a t the p e r i m e t e r s
adjointness
By a n a l o g y w i t h
9 : T'
hand
complete.
structure-semantics
ment
and the n o r t h -
lower right
to be a V - f u n c t o r .
We n o w d e v e l o p
adjoint
commutes
f r o m the
is a m o n o m o r p h i s m . for
-
the u n i t s
T'
=
we define (T' ,n' ,~.)
transformation
and m u l t i p l i c a t i o n s ,
in t h a t the d i a g r a m s
T''
~
T
T'
id A
T'T'
both
commute.
sense
and h e n c e p r o v i d e s
Thus,
(defined by AT(A,a)
=
definition
on
an o r d i n a r y
A T : AT
o
A ~
,
A z
=
functors.
V, the n a t u r e
of the
> TT
TT
in the u s u a l
(A,a o ZA)
at the l e v e l of o r d i n a r y
~ T
is a l s o a t r i p l e m a p
U T'
assumptions
-.~
V-structure
functor (f) = f)
~
A T'
satisfying
UT
(2.2)
Using
the s t a n d i n g
of
U T'
on
A T' , t h e p r e c e d i n g
and
AT
a n d the lemma
-
4 0 3
-
guarantees a unique V-functor structure for (2.2) holds at the level of V-functors,
At
such that
provided only the
perimeter of the diagram UT
AT((A,a),(B,8))
T ~ tB
; A(A,B)
~
; A(T'A,T'B)
TA, B
UT
~A (id, ~B
A (TA, TB) ~
A
(T'A,TB)
A(id, B) I A(A~'B)
commutes. of
A(~,B)
[A (id, ~)
~ A(TATB)
A(~A,B ) ~A(T'A,B)
Now the left hand pentagon commutes by definition
A T, V-naturality of
~
guarantees
that the upper right
hand square commutes,
and either leg of the lower right hand
square is
So the perimeter does commute,
A(TA,8).
and we
have proved the first part of Proposition
2
Each functor
A T'
A T : AT
(3: T' --~ T
a
V-triple map) is in a unique way ~ V-functor satisfyin@ UT'A T = U T
at the level of V-functors.
Moreover,
the
familiar relations ArT,
=
A T
,
o
expressin@ th__~efunctionality
AT
id T ,
A
of triple semantics,
even at the level of V-functors when maps and
T
is a V-triple.
= idAT
T
and
3'
are valid are V-triple
404
-
Proo f.
The v a l i d i t y
level of V-functors assertion
hand, U: X
of the d i s p l a y e d
is a simple
consequence
Let us then gather
together,
on the
A, along w i t h
V-category
them -- forming a category
all
V-adjoint
> A
A-valued
e q u i p p e d with
at the
of the uniqueness
all
V-functors
on the one hand,
V-functors
making
between
all
V-Trip(A)
a V-functor
front and b a c k a d j u n c t i o n s F), w i t h
relations
of Lemma 2.
V-triples between
-
U
V-triple maps
-- and, on the other
(that is, F: A
all
V-functors
> X
V-adjoint
and V-natural
(on the right)
their domains m a k i n g
to
commutative
triangles X
X'
>
A as morphisms sition
- forming a category
2 then asserts
= T]
I
>
IT: T'
> T]
I
> [AT: AZ
a contravariant
V-Trip(A)
to
V-Adj (V-Cat,
V-triple
the o t h e r d i r e c t i o n V-adjoint
tions
AT
[(T,n,~)
The
n: id A
[UT:
Propo-
> A;FT,n,c T], AT'],
functor -- V-triple s e m a n t i c s
- from
A) .
structure
is easier:
V-functors w i t h > UF,
A) .
that the passages
constitute
are
V-Adj (V-Cat,
~: F U - - ~
functor
if
U: X
(V-natural)
(also contravariant) > A
and
F: A
front and b a c k
in
> X adjunc-
id X , there is no trouble to see
-
that F':
(UF,n,U~F) A
> X'
P: X
> X'
to
and h e n c e
U':
and
e'
for w h i c h
X'
> A
V-functcrs,
U'
a V-triple
with and if
o p = U, it is w e l l transformation
is a t r i p l e m a p
(UF,n,Ue F ) .
and
, respectively,
an o r d i n a r y n a t u r a l
that actually
(U'F' ,n' ,U'E'F')
If
of V - a d j o i n t
n'
is a V - f u n c t o r
> UF
V-natural,
pair
adjunctions
k n o w n h o w to c o n s t r u c t Tp: U'F'
-
is a V - t r i p l e .
is a n o t h e r
f r o n t and b a c k
405
from
In fact, h o w e v e r ,
map, b e i n g
~p
is a
given explicitly
as
the c o m p o s i t i o n U'F'
of
U'F' n
V-natural
contravariant
> U'F'UF
= U'F'U'PF
transformations. triple
The
structure
U' ~ ' P F
> U ' P F = UF
functoriality
functor
of the o r d i n a r y
then guarantees
that
the
pas s age s (U;F,n,e)
~
P constitute
A)
to
In T h e o r e m tics
and
V-triple
end, w e o b s e r v e V-adjoint =
first
For,
( UTeT)F~ A =
is q u i t e
,
,
f u n c t o r -- V - t r i p l e
structure
-- f r o m
V-Trip(A).
2 below,
structure
V-functor
(T,n ,u)
again.
> Tp
a contravariant
V-Adj(V-Cat,
(UF,n,U~F)
we shall prove
are a d j o i n t
t h a t the
V-triple
(UT;F~,n,e~) obviously
UTF ~ = T, n = n, and (UTeT)(TA,UA)
= UA).
structure
other
than
seman-
T o this
of the
f r o m the
U ~ e TF~ = u This
V-triple
on the right.
arising
nothing
that
V-triple Z
itself
(for this
identification
recall
is o n e of
-
406
-
the a d j u n c t i o n maps for the a d v e r t i s e d a d j o i n t n e s s b e t w e e n V-structure
and V-semantics.
Let The w e l l k n o w n defined by
(U;F,n,e)
be as b e f o r e ,
"semantical
CX =
(UX,Uex),
the level of o r d i n a r y
The o t h e r is d e s c r i b e d as follows.
comparison
and let
functor"
Cf = Uf, s a t i s f i e s
functors.
T = r
9
(UF,n,U~F).
X
> AT
UT o r = U
8
at
We use L e m m a 2 to p r o v e
ProRo~ition
The u s u a l s e m a n t i c a l a r i s i n g f r o m the V - a d j o i n t --
with
r
x
_
~
9 = >
comparison
functor
V-functor situation
.
jl L
.
i.
r
(U: X
~ AT
X
> A; F,n,~)
L
(UF,n ,UeF), admits _a u n i q u e e n r i c h m e n t
to a V - f u n c t o r
AT
UT
at the level of V-functors;
o
r =
moreover,
L
V-functor ~
(2.3)
U
r
is the o n l y
l
(2.3), and i n addition, w e h a v e
r o F = FT
(2.4)
~r = id T
(2.5)
and
Proof.
The f i r s t p a r t of P r o p o s i t i o n
3 is proved,
u s i n g L e m m a 2, b y s h o w i n g t h a t the e x t e r i o r of the d i a g r a m
- 407 -
U
~
A (UX,UY) ~
LUF..
X(X,
UFUY)
UX,y
A (id,Ucy)
~ X (FUX, FUY) X(Ex,Y) ~
IX(id,Ey) X (FUX,Y)
A (UX, UY)
commutes.
"-'-A(UFUX,UX)
A (UEx,id)
But the two upper triangles commute by the definition
of the V-functor structure of a composition of V-functors; remaining triangle commutes because
~
is V-natural;
the
and the
other regions commute for even easier reasons. Since the ordinary semantical comparison functor the only functor for which
~
is
(2.3) holds at the lowest level, the
second uniqueness statement now follows from the first. Since and since both
(2.4) is known at the level of ordinary functors ~F
and
FT
are V-functors to
portion of Lemma 2 establishes the validity of of V-functors.
Finally,
since relation
ordinary triple-structure-semantics,
A T, the uniqueness (2.4)
at the level
(2.5) is well known for
it is true here as well.
To give the reader two strategy options for proving Theorem 2, and to make our exposition more complete, we state
- 408 -
Lemma
3
Let
(U: X
situation , let
3'
b e a V-triple
be the V-structure P: X
> A T'
> A; F,n,e)
triple
of
on
A, and let
(U~F,n,c).
V-functor T =
(UF,n,U~F)
Then ~iven _a V-functor
satisfying U T'
at the level of V-functors, 9 : T'
b_~e a V-adjoint
> T, so!vin~
o p = U
(2.6)
th e unique ordinary
triple
map
the e~uation
p = A~ o ~ of ordinary ~emantics valid
functors
(available
b y ordinary t r i p l e - s t r u c t u r e -
_adJointness ) , is in fact a V-triple
at the level of V-functors Proof.
already
observed
~
With this
Theorem
is, of course,
just
~p
and
(2.7)
is
w h i c h we have
The validity
is a by now familiar lenuna b e h i n d
map,
as well.
is a V-triple map.
level of V-functors
proofs
(2.7)
of
consequence
us, n o t h i n g
(2.7)
at the
of Lemma
can p r e v e n t
2.
two
of
2
(Adjointness
of V-structure
The contravariant V-triple-semantics transformation
i ~
and V-semantics)
fun ctors V , t r i p l e - s t r u c t u r e
are adjoint
and
on the right, w i t h one a _ ~ u n c t i o n
by the system of V-semantical
comparison
- 409 -
, and the other given ~
V-functors
_the i d e n t i f i c a -
tions
T =
(UTFT,n,UT~TFT)
First Proof systems
of V-semantical
tifications deliver
in Outline.
(2.8)
First v e r i f y
comparison
are natural;
(2.8)
9
functors
that the
and of iden-
then r e l a t i o n s
(2.5)
and
(2.8)
the adjointness.
Second Proof correspondence V-functors
between
adjunctions;
functors
there remains
the i d e n t i f i c a t i o n s
w
LAWVERE'S
(2.8)
and the class of all (2.6).
one of the
suppose
ADJOINTNESS
the base c a t e g o r y
V
to be
apply d u a l i t y
to obtain
Nevertheless,
diligent mimicry
of this paper will p r o d u c e V-cotriples
about the V-category
of ~ - c o a l g e b r a s ,
A~
that
the other.
shall assume known about
structure-semantics
That delivers
that the family of
constitutes
constitute
for cotriples.
of the first sections
one-one
only the easy v e r i f i c a t i o n
we cannot blindly
part results
3 provides
KLEISLI-EILENBERG-MOORE
Since we never symmetric,
satisfying
and the i n f o r m a t i o n
comparison
Lemma
V-Trip(A) (~',T)
P: X --~ A T'
the adjointness, semantical
in Outline.
adjointness
9
relations
counter-
all that we
on a V-category and about the appropriate
to
A,
-
V-coadjoint
V-functors
Bearing V-triple
with
a new
Clearly
V-cotriples
semantics
on a V - c a t e g o r y
versus
functor.
410
V-cotriples.
in m i n d , Given
V-adjointness
on one c a t e g o r y
to a V - c o t r i p l e
and
T
~T.
are all p a i r s
a V-triple
we a l l o w all
as
p a r t of a f u n c t o r
(whose o b j e c t s
on
the
X: A
V-functors
on a n o t h e r
on
AT
from a V-triple
is,
A
as w e s h a l l V-Trip
a V-category
as m o r p h i s m s
--~ A'
(T,~,~)
A T, w h i c h we
f r o m the c a t e g o r y with
the
> A; F T , n , E T ) .
This passage
(A,T)
A, w h e r e
T =
V-category
is a V - c o t r i p l e
briefly
actually
V-triple
(UT: A T
situation
(FTU T , ~ T , F T n U T )
let us r e c o n s i d e r
the
A, we h a v e p r o d u c e d
shall designate
see,
-
(A,T)
i9
CA' ,T' )
such t h a t
T ' X = XT, ~,X
and
=
Xn:
u,X = X~:
to the a n a l o g o u s l y on v a r i o u s
X
T'T'X
defined
V-categories
(3.1)
---> T ' X = XT, = XTT
> XT = T'X)
category
V-Cotrip
and s t r i c t l y
of all
V-cotriples
V-cotriple-preserving
V-functors.
To d e s c r i b e from
(A,T)
to L e m m a X: AT
2.
to
the e f f e c t
(AL,T'),
we s h a l l
To b e g i n with,
> A ,~'
satisfying
on a m o r p h i s m once
we d e f i n e
again
X: A
--> A'
have recourse
an o r d i n a r y
functor
-
411
-
XU T = U~'~
,
1
~ F T = F ~' X , X T Namely, X(A,u)
if =
(A,e: TA
(XA,Xu:
r
=
> A)
T'-X
.
is a E - a l g e b r a
T'XA = XTA
> XA).
that
X(A,u)
relations
(3.1)
and the d e f i n i n g p r o p e r t i e s
the ~ - a l g e b r a
Xf: XA
> XB
A, let
is in fact a T ' - a l g e b r a -- all he needs
In much the same way,
whatever
in
It is left to the reader
to decide
maps,
(3.2)
map
is actually
the reader f:
(A,~)
of a l g e b r a s t r u c t u r e
can e a s i l y verify that, >
a ~'-algebra
is
(B,S), map,
the A ' - m o r p h i s m
call it
~f,
from
N
X(A,e)
to
X(B, 8).
It should then be clear that relations
(3.2)
functors.
are at least s a t i s f i e d
In fact,
properties.
~
is easily
Now an a p p l i c a t i o n
to a V-functor s a t i s f y i n g level of V-functors, the v a l i d i t y V-functors
as well.
morphism
from
in s e e i n g
that
is a functor
at the
the only functor h a v i n g
of Lemma 2 enriches
and another a p p l i c a t i o n
it now follows (AT,~ ~)
to
YX = YX
(A,T)
~
(A,T',~T').
a functor
EM:
Since
at the level of not b e i n g
there is no p r o b l e m
id(A,T ) = idAT
i ~ (AT,~ z) ,
V-Trip
at the
is in fact a V-Cotrip
xl> constitute
uniquely
(3.2)
of the third e q u a t i o n
that
and that
X
these
of Lemma 2 guarantees
of these e q u a t i o n s
The v a l i d i t y
and that
level of o r d i n a r y
the first of the e q u a t i o n s
of the second
i n t e r f e r e d with,
X
) V-Cotrip.
, the passages
-
The purpose adjoint
to
412
-
of this section is to describe
the left
EM, Consider,
V-category
A.
with objects
therefore,
Let
KZ(T)
those of
A
A
to
a V-triple
T =
(T,n,u)
be the usual Kleisli and
K/(T)-morphisms
TB.
We make
KZ(T)
on a
category
from
A
for
to
B
T, all
A-morphisms
from
into a V-category
by setting
KZ(T) (A,B) = A(A,TB),
and using the adjointness
iso-
Kl(~) (A,B) = A(A,TB)
~-- *- AT(FTA,F~B)
(3.3)
morphisms
to find
(uniquely)
units
such a way that the
and composition
(iso-)
morphisms
A a (V-fully the units I
faithful)
rules
for
KZ(T)
in
(3.3) make the passage
~ FTA
V-functor
IT: KZ(T)
~ AT .
Explicitly,
are jA > A(A,A)
and the composition
A(A, nA ) m A(A,TA)
,
rules are
KZ(T) (B,C) | KI(T) (A,B) . A(B,TC)
= KZ(T)(A,A)
.
.
Kl (T (A,c)
.
! A(A,TB)
I TB,TC @ id A(TB,TTC) or, if
V
@ A(A,TB)
is closed,
M
> A(A,TTC)
A(A,~c )
, A(A TC)
,
413
-
-
Kl(~) (B,C) ..... > V(KI(T) (A,B) ,Kl(~) (A,C))
lJ
A (B,TC)
I
TB,
V (A (A,TB) !IA(A,TC)) U (id'A (A'~c))l
TC
A (TB, TTC)
V (A (A,TB) ,A (A,TTC)) LA
By Lemma 2, the passages A~
> A
A(A,B)
> A(A,TB)
= KI(T) (A,B)
A(A,n B) constitute
a V-functor
The V-fully faithfulness serves as
(right)
back adjunction structure
of
of
V-adjoint
V-adjoint
rE, with
to
(i F )- 1 (ETFT) .
if
(U: X
there is a canonical
UT = u T
with the full image and the
- indeed,
transformations
making the Kleisli
V-functor
and
F: A
the associ> X
V-isomorphic
Moreover,
!' is obviously
idE.
is one of the adjunction V-category
adjoint on the left.
V-triple structure
as a covariant A)) ~
> X.
is a
~: Kl(T)
is clearly
V-triple structure
(V-Adj(V-Cat,
uT
again.
> A; F,n,e)
V-triple map induced by T's
T
T = (UF,n,UEF)
Kl(T)
V-category of
This shows that the system of
def
front adjunction
is just
V-functor situation, with
satisfying
UTI T =
ITf T = F T .
It follows that the V-triple
(uT;fT,n, (IT)-1 (~TFT))
ated V-triple,
TfT = F
shows that
IF
On the other hand,
satisfying
> Kl(T)
fT. A
construction
Alternatively,
functor > Trip(A)
,
and
viewing
414
-
it has of
V-triple-semantics
AT
- the E i l e n b e r g - M o o r e
et al - as left adjoint,
Kleisli
V-category
construction
As e a r l i e r Clearly,
inducing
X:
AT----> A 'T'
a unique
A X
(A,T)
,
~fT
=
fT 'X
,
KI T
=
i ~'AX
,
turns
Thus, V-adjointness K-~
V-Trip
passage
cotriple
the mimicry
Theorem
another
actually winds
KZ(T' )
F ~') ,fT' n' u T' ))
a up
satisfying
from
uT ~
fT
9
V-category becomes
along w i t h
the
a functor
mentioned
Kleisli-type
the above
considerations,
at the head of this section,
functor
KL:
V-Cotrip
3
V-Trip
inducing
to
to the Kleisli for
vary.
out to be a C o t r i p - m o r p h i s m
we must dualize
procedure
KL: EM:
(3.2),
A
> V-Cotrip. Actually,
obtain
T'
(A',T'),
>
u T'AX
(KI(T),(fTuE,(IT)-](~EFT),fTnuT))
l(c
>
A X: Ks
=
actually
(KZCT') ,(fT ' u T ' ,(I T ' ) -
we wish to let
satisfying
Xu T
Furthermore,
as right adjoint.
X:
V-functor
construction
and the above described
in this section,
each T r i p - m o r p h i s m
V-functor
-
V-Co,rip > V-Cotrip~
> V-Trip
has as
(right)
> V-Trip.
by to
-
Proof Sketch.
X.
-
We content ourselves with presenting the
front and back adjunctions. V-category
415
Let
9
Form the Kleisli
have the V-triple
~
V-category
Kl(~)
on
be a V-cotriple on the KI(~)
gory of
T~-algebras on
as was seen before.
V-functor
Ks
@
(X,~)
X
to
A.
@
X
~,
There is then the V-functor
makes both triangles commute,
let AT
T
= (Kl(~ ~),T ~)
of T-algebras in
whose Kleisli T: Kl(~ T)
to
= ((KI(~))~,
(A,T).
A.
V-category we now form.
> A
making both triangles (next page)
We leave to the reader
the verifications that, using these adjunctions, EM.
T~)
be a V-triple on the
commute, indeed, giving a V-Trip morphism from
adjoint to
to the V-cate-
~ (Kl(~))
~
Form the V-category
This bears the cotriple
KL(EM(A,T))
There is then
from
EM(KL(X,~))
For the back adjunction, V-category
We
It can then easily be seen to be a
V-Cotrip morphism from
~
~.
coming from the V-adjoint pair
of V-functors at the left in the inset diagram. the V-semantical comparison
for
KL
is left
-
416
-
A
Incidentally, the same sort of thing can be done replacing canonical E'~ has
EM by
E-~: V-Cotrip
> V-Trip, assigning to
V-triple on the V-category K-~
as left adjoint.
delivers the proof.
X~
of
(X,~)
~-coalgebras.
Once again, purely
the Then
formal mimicry
-
4 1 7
-
REFERENCES
[l]
Bunge, Marta, "Relative Functor Categories and Categories of Algebras", J. Alg.
[2]
(to appear).
Eilenberg, Samuel, and Kelly, G. Max, "Closed Categories", pp. 421-562 in Proo. Conf. Categ. Alg. Springer, Berlin,
[3]
(1966).
Eilenberg, Samuel, and Moore, J. C., "Adjoint Functors and Triples", Ill. J. Math.,
[41
(La Jolla, 1965),
9; 381-398
Godement, Roger, Th6orie de8 Faisceaux.
(1965). Hermann, Paris,
(1958).
[5]
Kelly, G. Max, "Tensor Products in Categories", J. Alg., 2; ]5-37 (1965).
[6]
Kleisli, Heinrich, "Every Standard Construction is Induced by a Pair of Adjoint Functors", Proo. Amer. Math. SoQ., 16; 544-546 Reviews,
[7]
(1965).
31; #1289,
Review by P. J. Huber, Math.
(1966).
Kock, Anders, "On Monads in Symmetric Monoidal Closed Categories", Preprint Series 1967/68 No. 14 (April, 1968), Mat. Inst., Aarhus Univ.
-
[8]
Linton,
F. E. J.,
in Seminar
4 1 8
-
"An Outline of Functorial
Semantics",
on Triple8 and Categorical Homology Theory,
(Springer Lecture Notes
in Math.,
Vol.
80)
(1968).
-
419
MINIMAL SUBALGEBRAS
-
FOR DYNAMIC
TRIPLES 1
by E r n e s t Manes
This p a p e r is a p r e l i m i n a r y project dedicated
to the p r o p o s i t i o n
and compact t o p o l o g i c a l each other. accessible gorists)
to t o p o l o g i c a l
that the w o r k
prerequisite
origin
"adjoint"
of the main
[Ma] or
theorems
(as o p p o s e d
d o e s n ' t s e e m to come up.
[Eb] and the references
the paper
[Mb].
is a k n o w l e d g e
of triples It is h o p e d
ing s e m i g r o u p
non-classical as c o m p a c t
is e n t i r e l y to include
of t/le a l g e b r a
exotic
of uni-
groups;
"(IT,I~)"
algebra on one generator.
IResearch s u p p o r t e d by Harvey M u d d College.
as
of triple-
though
infinitary
(as d e f i n e d in 2.1)
of all d e r i v e d u n a r y operations;
The
as just a
accurate
transformation
for
that the m e a n i n g
is clear w i t h o u t k n o w l e d g e
algebra", w h i c h
such
there
in sets such
in such a case think of a "T-algebra"
sufficiently
to cate-
of the a l g e b r a we study here.
for r e a d i n g
may be found in
examples
a lot to learn from
dynamicists
algebra in the language
"universal
algebra
The author has tried so h a r d to make this p a p e r
the d y n a m i c a l
theory;
that u n i v e r s a l
dynamics have
The reader is r e f e r r e d to
versal
report on a larger
the e n v e l o p is the set
is the free
-
i.
Let p 6 E,
{Lp}
function
E
be
{~}
DYNAMIC
-
MONOIDS
an a b s t r a c t
denotes
induced
420
by
the
monoid
{left}
with
{right}
unit
e.
For
multiplication
p.
i.i D e f i n i t i o n
E right rive
1.2
ideal (i.e.
is
a dynamic
I
such
all
monoid
that
if
I, q u a
E
possesses
semigroup,
left m u l t i p l i c a t i o n s
of
I
a minimal
is l e f t
are
cancella-
injective).
Definition
A c E a n d if f o r ~ p x = 6q
is
all
a division
p,q 6 E
for all
there
set in
E
exists
x 6 E
A
if
is n o n - e m p t y
such
that
~ 6 A.
i. 3 T h e o r e m
Assume and
a maximal
E
that
division
possesses
set
a minimal
A.
Then
in
E.
the
right
following
ideal
I
statements
are valid.
and
a.
A
is
b.
There
up = p
a left ideal exists
(p £ I);
c.
I N A = IA
d.
I
a dynamic
is
a left
monoid).
u £ I N A
such
(in p a r t i c u l a r , and
I N A
is
cancellative
that
6u = 6
(~ £ n)
uu = u). a group.
semigroup
(and h e n c e
E
is
-
Proof.
that
6ax
=
6u =
6
with
6px
=
with
upxy
and
By
the
up
= p
6e =
=
uq
and
= p
and
4
is
a 6
(6 E
h).
is
4,
left
As h,
u 6
as
up
= uuq
c.
I4 c
ideal.
6p~y
=
6upxy
I
I N
p 6
a subsemigroup
Let
I,
4
so
I N 4.
There
exists
y 6
=
xp6xp6y
inverse =
xp6u
I
=
with
xp6x
=
z6
I
let =
exists
6
p = uup
x 6
I
xp6y
=
xu,
x 6
with
I
I N
then
xp6z
A
That
in I
a with
p6x
To
see u;
= u
a right p = up
as
a typical
Since
=
u.
q 6
is
u
with
with
4.
=
idempotents
since
is
x.
6q.
uu
exists
I
element and
then
is
6
I4.
two-sided
x = xp6y x
ideal
there
xup6y
also
u =
of
then =
unit.
a
xp6z
left =
xp6xp6Z
xp6. d.
u,au
p6
about
=
y £
p.
4
is
that
6uq
I
x 6 E
exists
=
h.
such u £
exists
there
there =
I N 4 6 E
Then
In p a r t i c u l a r
uq
If
There
of
x 6 E
ax.
I
fact
=
u =
=
=
maximality
exists
upxI
h.
uI
the
There
6 E.
Clearly p 6
from
Define
a general
ideal:
I.
p,q
6 £
-
clear
h).
Let
of
then
a
Let
For
I)
right
b.
(6 £
maximality
minimal
is
A).
uq.
(p £
This
6
(8 £ 6q
a.
421
I N =
4
there
6-16up
=
Let exists
6-1aup
=
a,p,q
E
6 E
I N
6-lap
I
=
with A
ap =
with
6-1aq
aq.
6u =
= q.
As
au.
The
Hence
proof
is
c o m p le te.
1.4
Definition
E
is
com~actible
if
there
exists
a compact
Hausdorff
-
topology a ll
on
E
with
respect
422
-
to w h i c h
Lp
is c o n t i n u o u s
for
p 6 E.
i. 5 D e f i n i t i o n s
For sets
X,F,
the
cartesian power
on
X.
be
For
toPOlOgY
any E - v a l u e d
if for e v e r y
monoid
if
E
F Y ; E, E Y
on
xF
i n d u c e d b y the d i s c r e t e function
{F Y > E Rp) E: p 6 E}
~uasicoE~actible
topology
topology
F Y ) E
the s u b s e t
the f i n e - ~ o w e r
has
of
is
topology
define
EY
E F.
is a
a minimal
E
to
right ideal
and
is c l o s e d in the f i n e - p o w e r
F E °
on
I, 6 H i e r a r c h [ T h e o r e m
E a c h of the f o l l o w i n g those
beneath E
is c o m p a c t i b l e .
b.
E
is q u a s i c o m p a c t i b l e .
c.
E
has
d.
E
is dynamic.
a minimal
E
making
a nd c o m p a c t n e s s , p E I, pE I
continuous
E
has
is c l o s e d
is a m i n i m a l map
right ideal
a im Z!es b.
Proof. on
on
E
implies
it.
a.
topology
four conditions
each
a minimal
since
Lp
defined
by
T
closed
F
Y
By
right
is c l o s e d , Let
division
be a c o m p a c t
continuous.
Lp
r i g h t ideal. f
Let
and a m a x i m a l
~ E.
so
Hausdorff
Zorn's
ideal pE = I.
Consider
set.
Lemma
I.
For Hence
the
-
423
-
f
Irx
(E,T)
(E,T)
'>
F
(x E r)
(E,T) As
E¥
hence
is the
image
is c l o s e d
in
the
b implies and
let
a 6 I.
a p x = aq
(~)
Let
& > ~ E
be
finite
subset
exists
x 6 E
for e v e r y (ioe.,
If
be
the
of
4.
with
that
is c l o s e d
I
x £ E.
be
of d i v i s i o n
sets
There ~px =
such 7Rq
of
the
exists
Hence
~
a maximal
is
division
c i m L~ lies d. .........
Examples the i m p l i c a t i o n s verses.
in
exists
set.
be
F c
b e.
YRq
closure
Zorn's
a There that
~ 6 EY~P
yRq = 7 R p R y By
set.
shown
some
with
so t h a t
F
We h a v e
fine-power
a division
so t h a t
~ = UAu.
Let
with
agrees
By the h y p o t h e s i s , 7Rq ~ E 7 ~ y £ E.
~
there
that
is in
and set
(6 E Ae).
A
ideal
is a d i v i s i o n
function.
exists ~q
right
aEE = I = aE {a}
and
E F.
on
a minimal
then
inclusion
(E,T) F
in
topology
Hence
subset
~ = yRpRx)
This says
Let
p,q £ E
a chain
finite
f, E 7
fine-power
C.
for some
Let
of
on
F.
of
EYRp. for some
Lemma,
there
set.
This
i. 8, 1.9
is
and
the h i e r a r c h y
1.3d.
The p r o o f
i. 10 b e l o w theorem
show
is c o m p l e t e .
that none
1.6 h a v e
true
of
con-
424
-
i. 7 T h e o r e m
Let
E
be
0) , a n d p r o v i d e cation.
E
Then
E
let
~
x = sup{l~: with
X~ =
inf{Iy,x} l~ =
I 6
17~
=
which
binary
has
all s u p r e m a
infimum
a minimal
fine-power If
A £
F
= inf{~y,p} IyR x.
Let
~' ~ =
= inf{l¥,l'y,q} ~
its
is
the
P}.
and
inf{Xy,x}
with
{0}
be in
17~
lattice
monoid
(including
multipli-
is q u a s i c o m p a c t i b le.
Proof. and
any
l~.
F.
of
There
l~.
E 7.
F
7
> E,
De fine
exists
p £ E
exists
q £ E
with
i n f { I 7 , A " ~}
As
~ = 7 R x.
Let
l~ = i n f { I 7 , 1 # }
Therefore =
ideal.
there
so t h a t
l" 7 ~ .
Hence
closure then
l' ~
~ inf{Ay,q}
right
X"
is
The p r o o f
arbitrary,
is c o m p l e t e .
1.8 E x a m p l e
A quasicompactible Let
E
and
B =
be
the d i s j o i n t
[0,i]. or
complete
lattice.
Since rays
ar~y c o m p a c t
topology
As
and
1.7,
topology
sets on
in
mu~t
[0,i],
on
p ~ q
E
A =
{p 6 E: p L x = 0}, w h e r e
not
compactible.
compactible.
intervals
if a n d o n l y
if
A =
A
with
as n u m b e r s ,
is n o t
respect
hence
E
compactible.
x = inf(B),
it
or
is a monoid.
to w h i c h
be equal
[0,i~ ~
(p,q £ A
is a q u a s i c o m p a c t i b l e
containA
t h a t is n o t
of the r e a l
p ~ q
p 6 A, q 6 B)
are c l o s e d
usual
unioD
Defining
p,q 6 B
monoid
closed
to- the Since
follows
that
E
is
[o
- 425 -
i. 9 E x a m p l e
A monoid
with
division
set
that
infinite
set
and s e t
tion
f
or
let
I =
: x E X}
=
{x {~ }
x
is
{~,4}
is
There
exist
To p r o v e
E =
denote is
the
f,g,h
that
the
function
E.
It is e a s y
of
E7
but
that
and X
f
is the
is
a submonoid
corresponding right
Let
6 E X.
set w i t h
~ ~
Let
x E X
that
x ~ f = 6f,
6g = ~fh = x ~ f h = x~g,
is n o t q u a s i c o m p a c t i b l e , and
check
let
f E Xx
that
f7
be
is in
For
such
Then that
with
x~ ~ ~.
x ~ g ~ ~g,
~fh =
~,
a contradiction. let
any
the
X x.
function.
were
6.
an func-
of
constant
ideal.
be
identity
~ E E
such
a maximal
Let
Suppose
x7 = x to
ideal
set.
£ E
then E
E
a minimal
a division
But
{f E xX:
injective}.
a division
~fh = ~g.
right
is n o t q u a s i c o m p a c t i b l e .
is n o t
x E X
a minimal
X --~
E
function
fine-power
be not
in
closure
f7 ~ E 7.
i. i0 E x a m p l e
A dynamic Let
X
Set
E =
be
an i n f i n i t e
{f E xX:
X -
{x0,x I}
of
X X.
A But
were
set with
induces
is
{x0,~ I }
{x0,x I}
is
a division
(what w e
x 0 ,x I
U
mean by
set. set.
of
right
so
E
By
elements
X-
{ x 0 , x I} E is
ideal which
is d y n a m i c .
1.3b,
For
division
distant
{x0,xl,lx}.
is a m i n i m a l
division
no maximal
a bijection
left cancellative,
a maximal
ff-lg = g
f
t h a t has
and x0f = x 0 = xlf}
I =
semigroup
monoid
let
,,f-l,, b e i n g
~ =
{x 0}
f , g E E.
clear).
If
set° of
X.
onto
a submonoid qua Suppose or
~ =
{xl }"
If
f ~{ I,
gE
I,
426
-
fg = g.
Finally,
if
-
f E I, g ~ I
then
fg = g
a left c a n c e l l a t i v e
monoid.
on
{x0,xl}.
i. ii T h e o r e m
Let
E
c l o s e d in the
be
fine-power
Proof. power of
closure
F
the n
Hence
of
Let
F ~-~
E 7.
Let
there e x i s t s
E
6 F
X~L a = l Y L a ~ ( X ). = a(lo~)
F o r all
= a ( l o y ) p ( l o) ;
p(1)
= p(lo )
with
is a f i n i t e
subset F).
This shows Let
that
lo 6 F.
Xo~L a = loyLa~(l we have
It f o l l o w s
~L a
For each and
)
a(~oy)p(l)
f r o m the h y p o t h e s i s
I 6 F.
$ = y%(lo
E
=
EYLa.
I 6 F
for all
Therefore
= y~(lo)La.
6 E
~;
is
be in the fine-
F
If
(~ 6 F). of
F
~
with
p £ E
p(1)
let
a 6 E.
closure
there exists
for all
and
I~L a = I ¥ ~ L a = I Y L a ~
is in the f i n e - p o w e r
that
topology
E¥
Then
Therefore
E
~L a = 7 L a R p ( l o )
The p r o o f
) £ E 7.
on
is
comp le te.
1.12 C o r q l l a r [
Every
group
We r e m a r k left cancellative to p r o v e For
let
ideal
in
E E
that
E
is a g r o u p if a n d o n l y if
and possesses
is q u a s i c o m p a c t i b l e be
E.
is q u a s i c o m p a c t i b l e .
left c a n o e l l a t i v e There exists
a minimal using and
p 6 I.
right ideal
i. ii,
let Since
I
1.12 m u s t
E
is
(so t h a t apply).
be a minimal pI = 7, p u = p
right for
427
-
some
u £ I.
so that from
-
By left c a n c e l l a t i v i t y ,
I = E.
B u t then
[L, II.2o18]
that
pE = E
E
u
is the unit of
for all
p 6 E.
E,
It follows
is a group.
The ~ o l l o w i n g t h e o r e m p r o v i d e s m a n y e x a m p l e s of quasicompactible,
n o n - c o m p a c t i b l e monoids.
i. 13 T h e o r e m
A countably infinite
Proof. and that
T
a b e l i a n g r o u p is n o t c o m p a c t i b l e .
To b e g i n with,
suppose
X
is a c o u n t a b l e set
is a c o m p a c t H a u s d o r f f t o p o l o g y on
X°
Letting
S
b e the t o p o l o g y g e n e r a t e d by c h o o s i n g a p a i r of s e p a r a t i n g open sets for e a c h t w o - e l e m e n t s u b s e t of p a c t and
S
is H a u s d o r f f ) .
X, S = T
(as
It follows t h a t
T
T
is com-
m u s t be s e c o n d
c o u n t a b le.
Now, and s u p p o s e topology
A
T.
let
A
be a c o u n t a b l y i n f i n i t e a b e l i a n g r o u p
w e r e c o m p a c t i b l e v i a the c o m p a c t H a u s d o r f f Then
T
is s e c o n d c o u n t a b l e and a d d i t i o n is
s e p a r a t e l y continuous. Hausdorff m e a s u r e on
By the t h e o r e m of
t o p o l o g i c a l group. A
with
i n v a r i a n c e we h a v e p r o o f is complete.
h(A)
= i.
Let
h
[W], A
be the u n i q u e H a a r
By c o u n t a b l e
the a b s u r d e q u a t i o n
is a c o m p a c t
a d d i t i v i t y and
h ({0})
• ~ =" i.
The
-
1,_14
-
Example
Let For
4 2 8
there
remove
x
be
exists
a point,
one-point
a set.
a compact
Then
Hausdorff
discretify,
compactification
induced
cartesian
= Prxp
for
all
power x 6 X;
E = Xx
and
topology).
this
compactible.
topology
restore
topology.
is
proves
X
the p o i n t Let
If
on
E
is
with
have
p 6 E ~
(e.g., the
the
then
~pr
x
continuous.
I. 15 T h e o r e m
Let minimal
E
right
ideal
cancellative. a. is
For
be
E
the
following
p 6 I
which
the
b.
{Iu
: u 6 I
and
c.
If
u,v
are
6 I
qua
and let
I
be
semigroup
is
left
statements
unique
are
u 6 I
a
valid.
with
pu = p
pq = puq
= pu
since
That any
a group
If
partitions
idempotents
pu = p
so that
idempotents
u = v.
uu = u}
then
I
if
into
Iu N Iv
groups. is
u = v.
Proof.
is
in
Then all
monoid
an i d e m p o t e n t .
nonempty,
are
a dynamic
and Iu
is
semigroup (eog.,
see
let
q = u.
p 6 I u N Iv
q 6 I
This
with
proves
then
uq
(a).
= u; If
pu = p = pv
a group
doesn't
require
S
a right
unit
such
that
The
proof
is
with
[L, I I . 2 . 1 8 ] ) .
then
u,v
E I
and so
cancellability, pS = S complete.
(p 6 S)
- 429 -
I. 16 E x a m ~ ! e
A monoid with idempotents infinite
(and so w h i c h
set.
Define
in j e c t i v e
and
X - im
submonoid
of
trivial
a minimal
xX°
to c h e c k
is n o t
E = f
is
is
I
{f £ E
t h a t has
Let
f = ix
countably I =
ideal
dynamic).
{f E xX:
Define that
right
X
be
f
or
no an
is
infinite}.
E
: f ~ IX}.
It is
a minimal
right
is
a
ideal with
no
i d e ~ o t e n ts.
i. 17 D e f i n i t i o n
E
is the
Let
E
set,
ES,
the b i n a r y
be
a monoid.
of
The
left
all u l t r a f i l t e r s
compactification
on the
set
E
Of
with
multiplication
U • V =
{A c E
: ZV 6
V V v 6 V.~U 6 U • U v c A}
.
i. 18 T h e o r e m
Let Then
E
be
a monoid with
the
following
a.
E8
is
a monoid with
of
E
and
the u n i t
ultrafilter b.
statements
for
induced by
The m a p
left c o m p a c t i f i c a t i o n
ES.
are v a l i d . unit
p 6 E, ~
~
(where
denotes
e 6 E
is
the p r i n c i p a l
p) .
E E~ > E~
sending
p
to
~
is a m o n o i d
homomorphism. c. it the
EB, free
with
its u s u a l
compact
space
on
compact Hausdorff E
generators)
topology becomes
a
(making
-
compactible d.
For
U 6 E8
and
p
a.
Let
U,V,W 6 EB.
is
an u l t r a f i l t e r .
U.
is
trivial.
E
V =
{p 6 E
Define V,
V
A 6
A.
(U-
For each
Define
~ = U VwW w6W
v E Vw,
c = vw.
E
There
topology A c E is
of
ES,
with
obvious
that
recall
LU
is
£
Define
• V.
of
V.
the
U • V.
c 6 @.
U £
U
the
A £
U •
(v
of
(b)
and
(d)
are
that
{A =
: A c E} {V 6 EB
~V 6
U • W £
The
U.
Uv c
with
uv c
B w.
w 6 W,
Then
B w-
• w).
trivial. a base
: A 6
V}.
V Y v 6 V ZU 6
V.
of
associative
some
is
whenever at
For
must
v 6 V.
U • V
£
proof
for
is
the
U,V 6 ES,
Let
U.
W 6 V.
To p r o v e
U v c A. This
It
proves
complete.
i. 1 9 T h e o r e m
Let Hausdorff {p 6 E
: ~
E
be
topology is
a compactible T.
Let
continuous}.
D
be Let
monoid any
U½.
when
definition
Zu
with
we
• V
Let
For
E Vw
Hence
continuous
U
V E
V vv
Let
exists
U • V 6 ~. that
~v w
where
U
clearly in v i e w
with
A N B E
q W E W V w £ W ~B w 6
V • W.
The proofs must
• W.
. U=
To begin
A ~
E - A ~
w E W
Uc = U v w c B w w c A,
(c) w e
that
V)
. p=
That
A};
: p v ~ A};
shows
U
Suppose
U . Up ~
{p £ E
This
let
BwW c
: YU £
U =
U 6 U.
law,
E,
U • V
A,B
-
monoid.
Proof. show
430
via
the
submonoid
DB - ~
E
be
compact of
the
unique
431-
-
continuous
extension
(i.e.,
is the u n i q u e
in
U~
T) °
Then
~
Proof. = U~
• V~}.
commutes.
Let
F = DS, w h i c h
point
of
map of
E
D
into
to w h i c h
U
E
converge s
is a m o n o i d h o m o m o r p h i s m .
Let
U £ D8
p £ D.
By 1.18d,
the c o n t i n u o u s
1.20
of the i n c l u s i o n
Since > D8
E
>
LUg,
completes
F =
{V 6 D
: (U • V)
is c o n t i n u o u s ,
DB
~ £ F.
maps
and set
E
Since
~Lu~,
F
the d i a g r a m
F
is the e q u a l i z e r
is closed.
of
Hence
the p r o o f .
Theorem
8 the c a t e g o r y
is a f u n c t o r
f r o m the c a t e g o r y
of c o m p a c t H a u s d o r f f
of m o n o i d s
monoids with
into
continuous
left m u l t i p l i c a t i o n s .
Proof 0 We m u s t s h o w That and
f8 let
Vf £ H e n ce
Vf
that
E l
f
> E2
El8 f--~ E 2s
preserves A 6
A l f c A.
Let
is a m o n o i d h o m o m o r p h i s m .
There exists
~V £ V,Vv £ V , q U v 6 U.
A £ Uf
• Vf.
a monoid homomorphism.
the u n i t is clear.
(U • V) f.
a nd for all
be
v £ V,
U,V £ El8
A1 6 U - V
U v V c AI.
(Uvf)Vf =
The p r o o f
Let
with
Then
(UvV) f c A l f c A.
is c o m p l e t e .
-
2.
Let
Z =
432
-
DYNAMIC
(T,n,U)
TRIPLES
be a t r i p l e
i n the
category
of sets.
2.1 D e f i n i t i o n
The s t r u c t u r e the set of n a t u r a l to =
T 1
q b ~_
equipped TT --~
For ~g = X Xq_ the s e t
of
~, w h i c h w e
transformations
with
the b i n a r y
denote
f r o m the i d e n t i t y
multiplication
ET ,
is
functor
g • h
T.
(X, ~)
XT - ~
{~g:
monoid
a T-algebra
X.
and
The e n v e l o p i n g
g £ ET
define
semigroup
of
is
(X,~)
g 6 E~}.
2.2 T h e o r e m
submonoid
The
following
ao
ET
b.
For every
of
Xx
C.
For e v e r y
statements
are valid.
is a m o n o i d w i t h algebra
unit
n.
(X,~), E
and the s u b a l g e b r a
of
is b o t h
(X,~) (X, ~) X
a
generated
by
{Ix}. epimorphism of
of
(IT, 1 ~ )
by the Y o n e d a d.
ET
on to
algebra
onto E
E
(x,~)
(X,~),
(x,~) (whe re
g:
~
~g
is a m o n o i d
a n d an a l g e b r a h o m o m o r p h i s m ET
is i d e n t i f i e d
with
correspondence). If
are i s o m o r p h i s m s .
(X,~)
=
(IT,I~),
the h o m o m o r p h i s m s
of
(c)
IT
- 433
Proof. generalities
Let
about
(X,~) X
is d e f i n e d
unique
homomorphism
function
IX
is
be a T-algebra.
(X,~)
T-algebras, by
-
the s t r u c t u r e
~ (X) .pr x = PrxT. ~ IT ~&~
(X,~) x
IxT. ~ (X) .
Reviewing map
~ (X)
(x 6 X)
sending
some of
and the
1 to the c o n s t a n t
The d i a g r a m
(xX)T p r x T • XT
IT~X~/~Xx Prx ~ X and
the Y o n e d a
lemma shows
Hence
(i x)
easily
from the d i a g r a m s
that
= Im ~ = E(X,~ ) .
X(g.h) >
X
g~ = ~g
for all
The r e m a i n i n g
XT
details
g 6 IT. follow
> X
!
k/ X
and
IT
1 the s e c o n d
of w h i c h
iTq
.!g shows
~
ITT ~
IT
'~ IT how
to r e c o v e r
g
from
(lu) g.
-
2.3
all
x £ X,
(X,~)
be
<x> = xE.
a T-algebra,
For
<x>
=
Then
E = E ( X , ~ ).
(ixPrx>
=
for
Pr x = E pr x
xE.
2.4
Theorem
Let let
I
be
of
set
E.
is a s u b a l g e b r a
If
I
is a r i g h t
(p> c
The
E = E(X,~ )
and
following
E,
of
I
is
a right
EI
ideal
a.
The m a p
of
Rp I
Since
E
Ip c
I}
is
is
> EI
is e a s i l y
E
then
for all
which
checked
a subalgebra,
is a T - h o m o m o r p h i s m .
{p 6 E:
in
I.
homomorphism.
P =
of
I
I-restriction
I I --~
subset
If
Proof. the
a T-algebra,
E. b.
p 6 I,
be
are valid. a.
ideal
(X,~)
a non-empty
statements
P
-
Remark
Let
=
434
Hence
a subalgebra
sends
to b e
the
to
p
a T-
inclusion
the p u l l b a c k
of
Since
E.
i x 6 P,
-- E ,
b. algebra have
2.5
for
of
It is e a s y (X,~) x
p 6 I
then
that
to c h e c k EA =
{~:
= pE E =
that
if
A
p 6 E}. {p~.
is
any s u b -
Using
q 6 E} c
2.3 we I.
Co.rollary
Let
(X,~)
be
a T-algebra,
set
E = E(X,~ )
and
let
-
I c E.
Then
a l g e b r a of
I E
is a m i n i m a l
435
-
(i.e., m i n i m a l non-empty)
if and only if
I
sub-
is a m i n i m a l right ideal of
E.
2.6 T h e o r e m
The f o l l o w i n g s t a t e m e n t s a.
Every n o n - e m p t y ~ - a l g e b r a has
b.
E~
Proof. group ET
E
of
right ideal.
a implies b.
By 2.5,
(IT,I~)
has
(X,~) °
M~
the e n v e l o p i n g semi-
a m i n i m a l right ideal.
By 2.2d,
E. a.
By r e v e r s i n g
a minimal subalgebra
e m p t y ~-algebra, to
has
is i s o m o r p h i c to
(IT,Iv)
a minimal subalgebra.
has a m i n i m a l
b implies b",
are e q u i v a l e n t .
there e x i s t s
the a r g u m e n t of "a implies If
M.
(X,~) from
a homomorphism
(X,~).
is a m i n i m a l s u b a l g e b r a of
is a non(IT,Iu)
The p r o o f
is complete.
2.7 D e f i n i t i o n
A u n i v e r s a l m i n i m a l T - a l g e b r a is a ~ - a l g e b r a
M
s a t i s f y i n g the f o l l o w i n g three p r o p e r t i e s . UMI.
M
UM2.
If
is a m i n i m a l T-algebra.
a T - h o m o m o r p h i s m of UM3.
is a m i n i m a l T - a l g e b r a then there exists
N
Eve ry
M
onto
N.
T - e n d o m o r p h i s m of
M
is an a u t o m o r p h i s m .
It is clear that if a u n i v e r s a l m i n i m a l
M
exists,
-
it is i s o m o r p h i c
436
-
to any a l g e b r a s a t i s f y i n g UMI and UM2.
we s h o u l d speak of the u n i v e r s a l m i n i m a l algebra. s u b a l g e b r a of the p r o d u c t of a r e p r e s e n t a t i v e
Hence
Any m i n i m a l
set of all
m i n i m a l a l g e b r a s w i l l s a t i s f y UMI and UM2 and hence w i l l be the u n i v e r s a l m i n i m a l
a l g e b r a w h e n it exists.
2.8 D e f i n i t i o n
T
is a d y n a m i c
triple if
E
is a d y n a m i c monoid.
2.9 T h e o r e m
The f o l l o w i n g c o n d i t i o n s
T-algebra
a.
T
b.
Every n o n - e m p t y
Proof.
m i n i m a l right ideal of
minimal
EM =
M
{Rp:
f
s t r u c t u r e map of
of
Suppose
Let
T - a l g e b r a contains
a minimal
2.6 is h a l f (IT,I~),
the work.
and let
M
Let
E
be a
w h i c h q u a s e m i g r o u p is left
is a m i n i m a l
is onto.
p 6 E}.
In the d i a g r a m
E
(IT,Iu), M
algebra.
Of course
are e q u i v a l e n t .
a u n i v e r s a l m i n i m a l T-algebra.
a implies b.
be the e n v e l o p i n g s e m i g r o u p
i s o m o r p h i c to
T
i s dyn ami c.
and there e x i s t s
cancellative.
on
a l g e b r a by 2.5.
admits M f-~ M
Since
If
M, there exists
is
a h o m o m o r p h i s m onto e v e r y is a T - h o m o m o r p h i s m .
As r e m a r k e d in the p r o o f p 6 E.
E
MT-~ g 6 ET
M
of 2.4b,
is the
with
~g = ~ .
437
M
Mg
Mg
M
both squares I° 13a
M
commute.
has
> MT
'~
= URxf = ufR x =
(uf)x,
so
~
MT
Hence
an i d e m p o t e n t
{
f~
• M
~
M
= ~f
u.
For
f = Luf
for all x £ M
which
p 6 E.
we have
proves
By
xf =
that
f
(ux) f is
injective. b implies (IT,I~).
Let
M
satisfies
UMI,
UM2,
M
is a m i n i m a l
a.
be
Let
E
a minimal M
right
be
the e n v e l o p i n g
subalgebra
is the u n i v e r s a l ideal.
Let
of
E.
minimal
p £ M.
L
semigroup Since
algebrad
of
M By 2.5 e
is a 7P
endomorphism x £ IT.
of
(1T,l~) 1T
since
In p a r t i c u l a r ,
a T-isomorphism.
As
~.pr x = Prxp
is a T - e n d o m o r p h i s m
E~
is i s o m o r p h i c
to
for a l l of
M, h e n c e
E, E T is d y n a m i c .
The p r o o f is c o m p l e t e .
2. i0 D e f i n i t i o n
T
is d i s t a l if
E T is a group.
2. ll T h e o r e m
The
following
a.
T
b.
(IT,I~)
is the u n i v e r s a l
c.
(IT,Iu)
is a m i n i m a l
Proof.
conditions
on
are e q u i v a l e n t .
is distal.
Let
E
be
minimal
algebra.
algebra.
the e n v e l o p i n g
semigroup
of
(IT,Iu).
-
a implies b. ideal, proof
E
438
since
is the u n i v e r s a l
-
E
is already
a minimal
right
m i n i m a l algebra in view of the
of 2.9. . . . . a. c i m ~ lies
with
a right unit, E
2.12
Definition
E
is a m i n i m a l
is a group.
A T-algebra x,y 6 X there exists
Since
(X,~)
right ideal
The proof is complete.
is homo eneous
a T-endomorphism
if w h e n e v e r f --~
(X,~)
(X,~)
with
xf = y.
2.13 D e f i n i t i o n
A T-algebra
(X, ~)
@ene rator if there exists algebraic
is p a r t i a l l y
a triple
$
_--
free on one
(S, n
I
,~' )
and an
functor S$
~
> ST
S
such that
(IS,I~')~ =
(X,~).
2° 14 T h e o r e m
Let
(X,~)
following statements
be a m i n i m a l
T-algebra.
Then the
are equivalent.
a.
(X,~)
is p a r t i a l l y
free on one generator.
b.
(X,~)
is h o m o g e n e o u s .
°
Proof. functor
of 2.13.
generator hence
of
h
Let
Let
and let
x,y 6 X
(IS,I~').
is minimal,
Using p a r t i a l with
is all of
X
f
with
be the algebraic 1 6 X
be the free
if = x, ig = y.
is onto so again,
fh = 1 X.
Zf = 1
f
Z 6 X.
a T-endomorphism
the e q u a l i z e r
Hence
- a/%d Since
for some
there exists
By minimality,
so
~
There exist $-endomorp_hisms
- f,g
freeness
lh = Z.
-
a implies b.
T-endomorphisms
(X,~)
439
of
fh,
is injective,
ix
f-lg
is the d e s i r e d endomorphism. b i m p l i e s - aa"
Since
for the rest of the paper, we only to d y e d - i n - t h e - w o o l subcategory
{ (X,~)}
of
is tractable
because
A
triple
$0
Let
A
S T.
h a r d to show that is the i n c l u s i o n reasoning know
that for all
=
(X,~)
A
sending
x
is then clear.
to
[(l,i)
> S ~.
ST
algebraic
comparison
of
$
it is not
~
ST ]
where
of "a implies b" we
That
a unique
lim[(l,i)
~> S ~]
is complete.
2.15 Remark
The proof
of 2.14
; S
functor
Using the sort of
there exists
The proof
be the full
A ~
of the semantics
y.
recognizable
and so has Linton structure
in the proof
x,y 6 X
A
functor
be the unique
(IS,I~')# = ~ i m functor
language
Let
Using the d e f i n i t i o n
that appeared
automorphism
The
is small,
S~ ¢-~ S T
• S$o
lapse into
categorists.
i n d u c e d by the r e f l e c t i v i t y functor
this result is not crucial
"b implies
a" shows
that if
i
- 440 -
(X,~)
is m i n i m a l
erator
2.16
where
then
n
is the
(X,~) n set
is p a r t i a l l y
of h o m o g e n e o u s
free
on one
components
gen-
of
(X,~).
De f i n i t i o n
T ideal which
is h o m o g e n e o u s is
a group.
if
ET
Notice
possesses
that
T
a minimal
is d y n a m i c
if
right T
is h o m o g e n e o u s .
2.17
Theorem
Let T
T
be
The
following
conditions
on
are e q u i v a l e n t .
one
ae
T
is h o m o g e n e o u s .
b.
The
universal
minimal
s e t is h o m o g e n e o u s .
co
The
universal
minimal
s e t is p a r t i a l l y
free
on
generator.
Proof. in
dynamic.
the e n v e l o p i n g
Then
M
is the
there
exists
endomorphism
a i m p l i e s b.
Let
semigroup
(1T,l~)
universal
p 6 M of
proof Hence
there
exists
of 2.9 M
is
minimal
with
p x = y.
be
a minimal
which
T-algebra. But
is If
right
ideal
a group. x,y 6 M
is a T -
M.
b implies Then
of
M
there
a.
Let
M
be
a T-endomorphism exists
a group.
p 6 M
In v i e w
as above. f with
of 2.14,
with
Let
x,y 6 M.
xf = y.
f = ~. the p r o o f
By
Hence is
the p x = y.
complete.
-
3.
441
-
EXAMPLES
3.1 D e f i n i t i o n
is com~actible
if
ET
is compactible.
3.2 T h e o r e m
Let triple
be any triple
corresponding
the category and let
E
ET8
S T ® j~
2.4b, all
E R~._ E
El8
p 6 E l . By 1.19, E
(onto b e c a u s e
since
Lp
is even
El
since
of
E.
of X =
Then
(IS,I~) $
is
monoid epimorphism
of
quotient
of
of
it is also a functor
As in the p r o o f
T ® |8, ~
also
ET
ET8
E
1.20
for
is compactible,
for all
(B
E
monoid quotient
by 2.2c so that by
of
of
is continuous
is a continuous
is dense;
by" since
Hence
semigroup
a T ® |8-homomorphism
is a m o n o i d q u o t i e n t
a continuous
X.
generated
is in the T - e n v e l o p i n g
p 6 E l . By the d e f i n i t i o n
for all
Let
and products).
is a dense s u b m o n o i d
(Ix> T
B
(noting that there is no
to "subalgebra
is closed under s u b a l g e b r a s
E I =
of
a continuous
E = T ® |8
ambiguity with respect
El
semigroup
be the
E.
Proof.
B
$
subcategory
of compact T-algebras.
and there exists
onto
in sets and let
to any B i r k h o f f
be the e n v e l o p i n g
compactible of
T
p E E). El8
preserves
from sets to compact
But
is outoness
spaces).
The
of
-
proof
442
-
is complete.
3.2 is p o t e n t i a l l y a p o w e r f u l s t r u c t u r e t h e o r e m for a large class of u n i v e r s a l m i n i m a l algebraic problem
is to d e t e r m i n e
subsemigroups
E8 × E8
of
for an a r b i t r a r y m o n o i d
algebras.
The a s s o c i a t e d
the n a t u r e of the c l o s e d
w h i c h are e q u i v a l e n c e relations~ E.
3.3 T h e o r e m
Let
•
be a t r i p l e
an a l g e b r a i c f u n c t o r
in sets such that t h e r e exists
S T ~ ) S~
(this includes all the
of 3.2, but c o n c e i v a b l y there are others).
Then
•
$'s
is com-
pactible.
Proof.
Let
o p i n g s e m i g r o u p of of
(X~)
X
X.
X =
(IT,Iu)
As
and let
E = ~, E#
E
is a c l o s e d s u b s e t
and so b e c o m e s a c o m p a c t H a u s d o r f f
p 6 E, Lp
is a T - h o m o m o r p h i s m and, hence,
e n d o m o r p h i s m of
E~.
The proof
be the e n v e l -
space.
For
is a c o n t i n u o u s
is complete.
3.4 T h e o r e m
Let ated triple
E
be any m o n o i d and let
(whose a l g e b r a s
is i s o m o r p h i c to
E.
T(E)
are r i g h t E-sets).
Hence,
be the a s s o c i Then
ET(E)
any m o n o i d can be the s t r u c t u r e
m o n o i d to a triple.
Proof. just
E.
The free
~ ( E ) - a l g e b r a on one g e n e r a t o r
= the o r b i t of
1x
in
is
E E = {Rp: p £ E} ~ E.
-
The p r o o f is
443
-
complete.
3,5 E xam~ le
Let dynamic, (2.6).
E
be
the m o n o i d
b u t e v e r y E - s e t ha s By 2.7,
of i°16
Then
T(E)
a minimal E-invariant
there is n o u n i v e ; s a l
minimal
is n o t
subset
E-set.
3.6 T h e o r e m
Let an a l g e b r a i c minimal sense,
T
any t r i p l e
functor
T-algebra
to a m i n i m a l E T - s e t . concerning
about minimal
Proof.
in sets.
monoid
For e a c h
in s o m e
algebras
reduce
to
T-algebra
(X,~)
define
(X,~)
the a c t i o n
It is e a s y
which
every
actions.
> X
x, g ~
have
there exists
sends
Hence,
minimal
X × ET
sends
Then
S T ~-~ E T - s e t s w h i c h
all q u e s t i o n s
questions
to be
be
to c h e c k
that
T-homomorphisms from 2.2c
that
.
is w e l l - d e f i n e d
to e q u i v a r i a n t
and 2.3
completes
~
x~ g
maps.
on o b j e c t s For
x 6 X
<x> T = X E ( x , ~ ) = xE T =
and we
(X>E(T)
the p r o o f .
3.7 T h e o r e m
Let an a l g e b r a i c
T
be
functor
any t r i p l e S T ® |8
in sets. ~
>
Then
compact
there e x i s t s
ET-sets which
- 444 -
sends every m i n i m a l
c o m p a c t T - a l g e b r a to a m i n i m a l
compact
E~-set.
Proof.
If
X
is a c o m p a c t T - a l g e b r a ,
make
X
an
E T - s e t as in 3.6, and leave the t o p o l o g y alone.
Since each
~g
b e i n g con-
is continuous,
s i d e r e d discrete,
the action is c o n t i n u o u s of course).
If
= (<x~T>|8 = <<X>E(T)>|8 = (X>E(T)
(E T
x 6 X, we h a v e ®|8
<x>T ® J8
which completes
the
proof.
The f o l l o w i n g e x a m p l e
arose in c o n v e r s a t i o n w i t h
J.F. Kennison.
3.8 E x a m p l e
Let
be the triple
T
class of a l g e b r a s unary o p e r a t i o n s
X
c o r r e s p o n d i n g to the e q u a t i o n a l
equipped with binary
Ul,U 2
operation
m
and
s u b j e c t to the e q u a t i o n s XUlXU2m = x xymu I = x xymu 2 = y
A T - a l g e b r a amounts to b e i n g a set fied b i j e c t i o n
X × X-
m;>
X.
A
• X, t o g e t h e r w i t h T ®|8-algebra,
a speci-
then,
amounts
to b e i n g a compact H a u s d o r f f space
X, t o g e t h e r w i t h
fied h o m e o m o r p h i s m
The cantor set b e c o m e s
X × X ; m;>
minimal
T ® ~-algebra
2 ~
by
2
{1,2,.o. }.
as follows:
5 = i, [ = 0. Define
X.
Let
a homeomorphism
let
2 = {0,1}
a specia
and d e f i n e
be the p o s i t i v e
integers
~N × ~
by
m>
2~
-
(x i) (Yi)m = (XlYlX2Y2...). be an even integer. (yi) 6 ~
for
1 ~ i ~ n; for then
n = 2
For
(Yi)
(x i) 6 ~
and let
by i n d u c t i o n of
< (xi)>T
on
(~,m)
n, that given
agrees w i t h
(yi) 6 (<(xi)>~>iB
and
n > 2
= <(xi)>T
is minimal.
(yi)
®18
The proof
for
is clear: (Xl...)
(Xl...)
m =
(x I . .. )
(Xl''")
m = (XlXl...)
(x I . ..)
(x I...) m =
(xlx I...)
(xl...)
(xl...)
(XlXl...)
n > 2, let
exist (i~
some e l e m e n t
(yi) £ ~
-
Let
We show,
any
for all
445
(yi) 6 ~ .
~).
for
B u t then
m =
.
By the i n d u c t i o n
(ai) , (b i) 6 ( (xi)>T i~
(xlx I...)
with
hypothesis,
ai = Y2i-l'
(a i) (bi)m 6 < (x i) >T
bi = Y2i
agrees w i t h
1 ~ i ~ n. In p a r t i c u l a r ,
E
T
acts m i n i m a l l y
on the cantor set,
by 3.7. 3.9 T h e o r e m Let
E
be any monoid. E8
× E ~
U, p~.--~
Then E8
ES, w i t h
action
,
U • ~
is the free c o m p a c t E - s e t on one igenerator. Proof. continuous
there
for all
Since
U • p = U~,
p E E.
Let
X × E
~
= ~8 > X
is i n d e e d be another
- 446 -
C o m p a c t E - s e t and let
x E X.
equivariant map sending f
e E E
Let
E
to
x
f
E~
be the u n i q u e
(i.e.,
be the u n i q u e c o n t i n u o u s e x t e n s i o n of
E
) X
f
pf = xp). to
Let
ES:
> ES I
D
I
X
Since the m a p
x~
is e q u i v a r i a n t , 1.19. f.
> xp
the proof b e i n g e n t i r e l y
The u n i q u e n e s s
The proof
is c o n t i n u o u s for all p E E,
of
~
similar to that of
follows f r o m the u n i q u e n e s s
of
is complete.
It f o l l o w s from 3.6 and the proof of 2.9, t h a t any minimal
closed E-invariant
m i n i m a l c o m p a c t E-set.
subset of
E8
is the u n i v e r s a l
- 447 -
REFERENCES
[Ea]
Ellis, Robert,
"A Semigroup Associated With a Transfor-
mation Group",
Trans. Amer.
Math.
Soc.,
94; 272-281, __J_
(1960). [Eb ]
Ellis, Math.
[L]
Robert,
"Universal Minimal Sets", Proc., Amer.
Soc., Ii; 540-543,
(1960).
Ljapin, E. S., "Semigroups",
Amer.
Math.
Soc.
Trans.
Math. Mon. , (1960). [Ma ]
Manes, Ernest, A Triple Miscellany: Theory of Algebras University,
[Mb]
dissertation,
Construction
to appear in the proceedings
Zurich Conference
Wesleyan
of Compact
of the 1966-67
on Categorical Algebra.
Wu, Ta-sun,
"Continuity
Amer.
Soc., 13;
Math.
of the
(1963) .
Manes, Ernest, A Triple-Theoretic Algebras,
[w]
Over a Triple,
Some Aspects
in Topological
452-453,
(1962).
Groups",
Proc.
- 448 -
CATEGORIES
OF SPECTRA AND
INFINITE
LOOP SPACES
by J. Peter May
At the Seattle culation where
of
H,(F;Zp)
was m e a n t
and
F(n)
equivalences
then,
step towards
Since
all primes
p, as a Hopf algebra
Dyer-Lashof
algebras.
is somewhat
lengthy,
ings.
The c o m p u t a t i o n
H,(2nsnx;Zp),
which
Milgram
[8], yields
H,(~nsnx;Zp) of
required
on n-fold study,
spaces
result,
generalizes
and
the c o m p u t a t i o n H*(BF;Zp),
over the Steenrod
of
for
and
while not difficult,
in time for inclusion
in these proceed-
a systematic
and infinite
study of
loop spaces.
I have also been able to compute
as a Hopf algebra
for all connected
monoid
and I was not able to write up a co-
presentation
As a result of this
p,
This c o m p u t a t i o n
I have c a l c u l a t e d
The calculation,
herent
operations
for odd primes
of an n-sphere.
as a preliminary
a cal-
is the topological
H*(BF;Zp).
homology
I presented
as an algebra,
F = lim > F(n)
of homotopy
conference,
X
over the Steenrod
and prime numbers
algebra,
p.
those of Dyer and Lashof
explicit H,(QX;Zp),
descriptions
This [3] and
of both
QX = li~> 2nsnx,
as functors
H,(X;Zp). An essential
first
a systematic
categorical
and infinite
loop spaces.
step towards
analysis
these results
of the notions
The results
was
of n-fold
of this analysis
will
449
-
be presented
here.
These
include certain adjoint functor
relationships
that provide
H.(~nsnx;zp)
and
between
spectra.
and
i,
of
These categorical
(bounded)
easier
they are sufficiently
these categories
spectra
~-spectra.
these cate-
that these
to work with topolog-
We shall
in a sense to be made pre-
are equivalent
for the purposes
categories
of
(bounded)
We extend the theory to unbounded
in the last section.
theory and as topology, have useful concrete
here is quite simple,
that there
applications.
is a natural
We shall
space to its ordinary homotopy our results with other
to
indicate two
In the first, we observe
epimorphism,
from the stable homotopy
struct a collection
both as category
but it turns out nevertheless
of these at the end of the paper.
coupling
and the
but it is not clear that
theory to the standard
The material
spaces,
~-spectra,
large to be of interest.
remedy this by showing that,
moti-
categories,
is to propagandize
ically than are the usual ones,
spectra and
to maps
considerations
spectra and
are considerably
of homotopy
H,(X;Zp)
spaces
It is clear from their definitions
categories
cise,
of
of certain non-standard
main purpose of this paper gories.
reason that
are functors
relate maps between
vate the introduction I
the conceptual
H,(QX;Zp)
and that precisely
-
realized by a map of
groups of an infinite
groups.
In the second,
information,
of interesting
loop by
we shall con-
topological
spaces and
-
maps;
the other
possibility
information
of p e r f o r m i n g
1
4 5 0
by itself gives no hint of the
this construction.
THE C A T E G O R I E S
in
In order to sensibly loop spaces, framework
it is n e c e s s a r y
AND H O M O L O G Y
study the h o m o l o g y
to have a precise
in w h i c h to work.
tion to p r e s e n t
-
of iterated
categorical
It is the purpose of this sec-
such a framework.
We let
T
denote
spaces w i t h b a s e - p o i n t
the c a t e g o r y
and b a s e - p o i n t
of t o p o l o g i c a l
p r e s e r v i n g maps,
and
we let ~: HOmT(X,~y)
) HomT(SX,Y )
denote the standard
adjunction
loop and s u s p e n s i o n
functors.
We define
B i = ~Bi+ I 6 T gi = ~gi+l
jects
of n-fold
B = {Bil0 ~ i ~ n}
and maps
6 T; c l e a r l y
0 ~ i ~ n.
homeomorphism
the c a t e g o r y
Ln, to have objects
We define
B 0 = niBi
and
such that
gi = n g i + 1 E
B 0 = ~iBi
go = ~agi
for all
of p e r f e c t
sequences) .
For all
Un:
~
n < ~,
) T Un B
by and
~-spectra
n, we define
UnB = B 0 Ung
such that such that
go = nigi
B i = ~Bi+ I E T
such that
the category
the
for
to be the c a t e g o r y w i t h ob-
g = {gili k 0} and
relating
loop sequences,
g = {gil0 ~ i ~ n}
t = i
B = {Bili k 0}
(i.i)
and
are n-fold
and m a p s
T; c l e a r l y
i ~ 0.
We call
(or of infinite forgetful
Ung = go"
L~ loop
functors Of course,
loop spaces and maps.
if We
-
say t h a t a s p a c e if
X = U=B
map
f 6 T
X £ T
is a p e r f e c t
Un.
{~n-isnxl0
letting spect
loop s p a c e
and we say that a
loop m a p
if
f = U=g
g ~ i~o
the f u n c t o r s
the case
infinite
B £ i~
infinite
We s e e k a d j o i n t s
Clearly,
-
is a p e r f e c t
for some o b j e c t
for some m a p
Qn x =
451
Qn x
For
n = ~
n < ~
~ i ~ n}
and
Qn f
, we
Qn:
T
>
In,
1 ~ n ~ ~ , to
, define
and
Qnf =
are o b j e c t s
first define
Q X = li~ ~nsnx, w h e r e
{~n-isnfl0
and maps
~ i ~ n}.
in
a functor
in.
Q: T
For ) [
the l i m i t is t a k e n w i t h
by re-
to the i n c l u s i o n s ~nu-i (Isn+Ix) : ~ n s n x ..-> ~ n + I s n + l x
For
f: X ~
is c l e a r define Q~f =
that
Y, w e
define
QX = ~ Q S X
a functor
Q~:
Qf = l i ~ ~nsnf: and
T--->
Qf = nQSf. i~
by
QX
~ QY.
It
We can therefore
Q~X =
{QSiXli > 0}
and
{Qsifli > 0}.
For each
n,
1 ~ n ~ =
~n: H ° m T ( X , U n B ) Proof. posites snx
~nx
Observe
, t h e r e is an a d j u n c t i o n
" H°min(QnX'B)"
first
that
the f o l l o w i n g
t w o com-
are the i d e n t i t y . s n u - n (isnx ) • sn~nsnx
un ( i n n s n x ) > snx,
~-n ( i s n ~ n x )
~nn ~nsn~nx
X £ Y
(1.2)
X £ T
(1.3)
(l~nx) ~
~nx,
- 452 -
In fact, in
since
(1.2)
natural
= u(inz)
and the proof
transformations
iT
• Sf
. sn~-n(isnx)
T, ~n (l~nsnx)
proves
Tn:
~(f)
> UnQ n
of
n < ~;
~ 0}. Q = U = B
• B
if
n = ~;
= ~-n(isn x)- x
~(X)
= lim u-J(IsJx) : X---> (1.2)
0 ~ i ~ n
since factors
) UniX
(1.6)
¢
U~Q=X
for all
follows
from
(1.7)
B 0 = UnB = SnB n.
that
(1.2)
is just
For
n < ~;
if
(1.4)
(1.5)
n = =.
the f o l l o w i n g
two com-
n.
U n ~ n (B)
UnQnUn B
if
= QX
#n (Qn x)
QnUn~n x
and
= nnsnx
(1.3)imply
identity
Un B Tn (UnB)
for
~ i ~ n}:
and
Qn X QnYn(X))
n < ~,
and
if
Tn(X)
For
Now define
> B
= {li~) ~ J u i + J ( 1 B 0 ) l i
are the
) i~
; this
= isnx
is similar.
• ~Z
QnUn B
~(B)
posites
(1.3)
f: Y
by
= {fln-i~n(1B0)I0
that
= ~n~-n(isnx)
~n: QnUn
~n(B)
We c l a i m
for any m a p
> Qn x, x £ Y
(1.6)
~ UnB , B E in
(1.7)
by a p p l i c a t i o n (1.3)
applied
n = =, o b s e r v e
to
that
of
~n-i
X = B n, T~(X)
as the c o m p o s i t e
x ~-i (Isx)) ~sx n~- (sx) nQSX = Qx T~(X)
= u-iT=(six)
It f o l l o w s
that
~-iT~(six)
= u - i ( a T = ( S i+Ix)
Observe
also
for all
• u-I (isi+ix))
i > 0
= U- (i+l) T~ (si+1 X) .
that
aJ~(si+Jx)
: ~Jsi+Jx
) ~JQsi+Jx
since
= Qsix
-
453
-
is just the natural inclusion obtained from the definition of
Qsix
as
lim ~Jsi+Jx. >
We therefore have that:
@~(Q=X) i " Q=T=(X) i = li~> ~J~ i+j (IQx)
• li~) ~ksi+k~-(i+k) T=(si+kx)
= lim~ nJ~ i+j (lQx)
• ~Jsi+J -(i+j)T~(si+Jx)
=
~J T~ (si+Jx)
li)m
U~@~(B)
• T~(U~B)
IQS ix
=
= limm ~J~J (IB0)
= li~> ~JuJ(IB0)
• u -j (1SJB0) = li~ IB0 = IB0
In both calculations,
the second equality is an observation
about the limit topology. formulas
(1.2) and
@n(f) = @n(B)
The third equalities
(1.3) respectively.
• Qn f
if
f: x
~n(g) = Ung • Tn(X)
if
g: Qn x
It is a standard fact that ~n
since the composites If
@n
B 6 in, we define
H,
) B
is a map in
T
(1.8)
is a map in in
(1.9)
is an adjunction with inverse (1.7) are each the identity.
H,(B)
= H,(UnB), where hom-
in any Abelian group
as a functor defined on
do not specify a range category.
follow from
Finally, define
~ UnB
(1.6) and
ology is taken with coefficients regard
• lim) u-k(IskBQ)
~.
We
in, but we deliberately
Indeed, the problem of
determining the homology operations on n-fold and
(perfect)
infinite loop spaces may be stated as that of obtaining an appropriate algebraic description of the range category.
It
- 454
follows easily from ~n (X). : H. (X)
) H. (UnQnX)
is adjoint to sense,
free objects
in the category
in .
It is therefore
H. (Qn X)
in
~ UnB
to be a functor of range category. ~ = Zp
in
T
free in the sense that
I have proven
if
induces a map
B E in
> H. (UnB)
an epimorphism.
since
~B).-
H. (QnUn B)
All of these statements
of u n s t a b l e algebras over the Steenrod
the
Qn x
is
The category is the appro-
Zp-coefficients.
play the role analogous to that of
and their fundamental
ogous to that of
of
realize
~ H. (B)
algebra
priate range category for cohomology with K(Zp,n) 's
in terms
are analogs of well-
known facts about the cohomology of spaces.
Products of
is deter-
that go into the definition
In this sense, we can geometrically
enough free objects
> B
is g e o m e t r i c a l l y
~n (f)*" H. (Qn x) ---> H. (B)
of the homology operations
then any
~n(f) : Qn X
H. (Qn x)
f* = Un#n (f)*~n (X).- H. (X)
the functor.
H. (X), with
and have computed the
Ln, and the functor describing
m i n e d by
Qn
in a w e l l - d e f i n e d
By the previous proposition,
f: X
Since
are,
that this is the case if
map
is a monomorphism. Qn X
in the appropriate
functor.
(1.5) of the proof above that
Un, the objects
natural to expect values
(1.2) and
classes play the role anal-
H.(X) c H.(Qn x) .
By use of Proposition
i, we can show the applica-
bility of the method of acyclic models to the homology of iterated loop spaces. natural transformations
The applications
envisaged are to
defined for iterated loop spaces but
-
not
for a r b i t r a r y
spaces•
categorical.
Let
A
category
A
denote , and
F: S
the
let >
A
M
be
be
the
category
of sets.
functor
R:
objects
and
f:
X
T
T
transformation
call
that
a natural is the
by
>
R(X)
=
A
to b e
in
T.
by
)
is the define
× R(M)]
where
r £ R(M).
Define
l(X) (~,r)
representable
~: R
ring
on m o r p h i s m s ,
and R
S
functor,
= F[ U H o m T ( M , X ) M6 M
R
by
such
transformation.
let
Let
where
is any
(f • ~,r)
I: R ~
natural
on
if a nat-
= R(v) (r). M
if t h e r e
that
I - ~: R
With
these
a
Reexists >
notations,
lemma.
R:
~: H o m T ( X , U B ) A
T
) S
~: M ---> UB, tion
~
> S
l'n =
IU: R U
a functor
let
be
an a d j u n c t i o n
representable
S = R • U:
Define by
a natural
n(B) (v,r)
r £ R(M). defined )
) HomL(QX,B)
L
)
A.
by Then
M. S
Define is
QM.
Proof. • U
be
and
{ Q M I M 6 M}
r e p r e s e n t a b le by
n: R
functor,
T
category,
2
let
QM =
A-module
following
Let and
free R:
is p u r e l y
a commutative
objects
transformation
the
over
any
of m o d e l
is s a i d
identity
we have
Lemma
R
denote
~ 6 HomT(M,X)
ural
needed
a set
A
then
argument
of m o d u l e s
R(f) (~,r)
) Y,
The
-
temporarily
If
)
4 5 5
RU = S
=
Write as
above since
= s~(~) [R~ -I (IQM) (r) ] = R [ U ~ ( ~ )
transformation
(~(~) ,R~-I(IQM) (r)) I'
for
for
R.
the n a t u r a l We
for transforma-
have
~'n(B) (~,r)
•
~-I(IQM) ] (r) = R(~) (r)
R
456
Therefore, l'(n~U)
if
~: R
satisfies
> R
= IU • ~U = l: S Of course,
and if
Tj
denotes
if
k~ = i: R
> R, then
> S, and this proves
the result.
is an a d j u n c t i o n
¢
the product of
j
factors
cJ: HomTj(X,UJB)
> HomLj(QDx,B)
(X E T j , B E t j ).
Thus the lemma applies
R: T j
~ A
and
RuJ:
Returning singular
ij
to
remark above,
to the usual
and C a r t e s i a n
products
{Am} , the standard
n < ~
and the m o d e l
models
[4] is a p p l i c a b l e
n-fold
and p e r f e c t
2
and
We conclude
t
interest. classical
that
For example, groups
loop spaces.
A.
in
objects
and, by the i~
(tensor With
T, we have these spaces
{QnAm } c in
that the m e t h o d
OF C A T E G O R I E S
i
The
are
of acyclic of
loop spaces.
is a r e a s o n a b l e
it is not obvious
in
chain complexes).
U~Q~A m = QAm;
The w o r k of the p r e v i o u s category
on
be the
to the study of the h o m o l o g y
infinite
COMPARISONS
> A
1 ~ n ~ ~
functors
set of m o d e l s
are c o n t r a c t i b l e
acyclic.
for
of singular
if
C,: T
with coefficients
related
UnQnA m = ~nsnA m
therefore
let
in ---~ A
lemma applies
M =
to functors
) A.
functor,
C,Un:
T, then
is also an a d j u n c t i o n
to topology,
chain complex
as in the lemma
OF S P E C T R A
section
shows that the
object of study c o n c e p t u a l l y ,
but
is large enough to be of t o p o l o g i c a l
it is not clear that the infinite
are b/mmotopy e q u i v a l e n t
We shall show that,
to p e r f e c t
infinite
from the point of v i e w of
-
-
homotopy
theory,
g o r y of
(bounded)
a-spectra.
proceed
by stages
through a sequence
restrictive
i
457
is in f a c t e q u i v a l e n t
categories
is a map. sequence
By a m a p of m a p s
grams a r e h o m o t o p y
Bi
of s u c c e s s i v e l y
is a s p a c e a n d
> B'
g: B
gi:
w e s h a l l h a v e to more
we shall mean a sequence
where
~ 0},
care-
of s p e c t r a .
By a spectrum, B = {Bi,fili
To d o this,
to t h e u s u a l
commutative,
Bi
>
~Bi+ I
of s p e c t r a w e s h a l l m e a n a
> B~ l
Bi
fi:
such that the following
dia-
i k 0.
g B
fi
±
1
1
inclusion
spectrum
the c a t e g o r y be a m a p
in
I S
~gi+l
if e a c h
fi
of
let
objects
if e a c h
S.
fi
W e say t h a t
for e a c h S.)
spectra
of
al = I N aS
i k 0.
We say t h a t
is a h o m o t o p y S
r e t r a c t of
mBi+ 1
(Thus, B 6 S
objects
be the f u l l
a r e the i n c l u s i o n
(2.1)
a r e the
for a l l
i.
mS
com-
if
W e let
be the
m-spectra, of
I
A spectrum
spectrum
I
m-spectrum
let
subcategory
n-spectra.
in
is n o t a f u l l
is an We
a map
actually
I
is an
We obtain
by l e t t i n g
equivalence.
whose
B £ S
is a n i n c l u s i o n .
w i l l be s a i d to be a r e t r a c t i o n mation
~B' i+l
s u c h t h a t the d i a g r a m s
full s u b c a t e g o r y we
~
category
of i n c l u s i o n
m u t e on the n o s e subcategory
1'
(2.1)
fi
~Bi+ I We c a l l th e r e s u l t i n g
> B'. 1
Bi
and
whose B 6 al
is a d e f o r R
denote
the
- 458 -
full s u b c a t e g o r y spectra. B 6 i
of
Clearly,
~I i
we may take
objects
of
i
the d i a g r a m s
whose
objects
are the retraction
is a f u l l s u b c a t e g o r y fi = 1
and then any map
w i l l be a m a p (2.1).
of
in
i
R, s i n c e
in
R
between
by t h e c o m m u t a t i v i t y
Thus we have the following
if
of
categories
and
inclusions i c R c
nl c
aS
and
I c S .
For e a c h of t h e s e c a t e g o r i e s are m a p s gi
in
C, t h e n w e say t h a t
is h o m o t o p i c
is a
(weak)
gi'
to
homotopy
topy e q u i v a l e n c e . and a q u o t i e n t
clusions
Now each
functor
C c D have
classes
of
(2.2)
and
sions
since
H: C
if
gig'
if e a c h
C
> HC.
in
C.
HC
g ~ g'
> HD and
E C
Note
preserving
C, t h e n
to
g'
gi
is a
HC
g
category of
homoHC
HC
are
are homotopy
t h a t e a c h of t h e inin t h e s e n s e t h a t in
in
if
(weak)
D.
if
We therefore
and these are still
g ~ g'
> B'
We say that
The objects of
B
i
has a h o m o t o p y
is h o m o t o p y
functors
g,g':
is h o m o t o p i c
a n d the m a p s
in
C, if
for e a c h
C
of m a p s
g ~ g'
induced
T
equivalence
the s a m e as t h o s e of equivalence
in
g
(2.2)
D, t h e n
inclu-
g ~ g'
in
C. The following needed
in o r d e r
categories
Definitions
definitions,
to o b t a i n p r e c i s e
d u e to S w a n
comparisons
[ii], w i l l b e
of o u r v a r i o u s
of s p e c t r a .
3
(i)
A category
C
is an
H-category
if t h e r e
is an
-
equivalence such that fg
relation f ~ f'
is defined.
quotient
such that quiring
C
fg
) D
C
f ~ g
> HD
is a functor.
is h o m o t o p y
C
in
Let
transformation maps in
n(C): D
natural
and
T
of functors natural n.
equivalence
are n a t u r a l
T(fg)
to re~ T(f)T(g)
H-category,
T
0.
Clearly,
determines
be prefunctors.
n: S
) T
• C'
in
C.
T
a functor
~: T
is a c o l l e c t i o n of
> T.
equivalences
is said to be a
such that for each
n: S Hn: HS
a natural
such that
of prefunctors,
T(f) n(C) ~ n(C')S(f)
n
• S
of functors preserving,
A natural
if there exists a n a t u r a l
of p r e f u n c t o r s
are h o m o t o p y n.: S.
in
~(C) n(C) ~ IS(C)
transformation
H-category.
is h o m o t o p y p r e s e r v i n g
if
of p r e f u n c t o r s
and
an
and
C E C, such that
of p r e f u n c t o r s
~ 1T(C)
D
C is also an
> D
> D
f: C
a natural transformation S
C
> T(C),
natural equivalence
n(C)~(C)
and a
HT = T,H.
of p r e f u n c t o r s
for each m a p
If
if and only
S,T:
S(C)
transformation
HC
This amounts
T(f) ~ T(g)
such that
(iii)
C.
T: C
implies
preserving
T.: HC ---> H0
whenever
on objects and maps,
C E C
for each
is d e f i n e d
in
and
is a function,
we say that a p r e f u n c t o r if
fg ~ f'g'
be any c a t e g o r y
T(I C) ~ IT(C)
whenever
implies
H: C--~ HC.
T: C
HT:
g ~ g'
on its h o m sets
We then have a q u o t i e n t c a t e g o r y
Let
A prefunctor
-
, c a l l e d homotopy,
and
funotor (ii)
459
> T
of functors.
Hn
A
determines
• HT
and,
if
transformation
n,H = Hn;
then
C E C.
if
and,
n is a if defined,
-
(iv)
If
S: D
preserving
prefunctors
is a d j o i n t
to
S
460
> C
-
and
between
T: C
H-categories,
if there exist n a t u r a l
prefunctors
~: TS
each
D 6 0
the c o m p o s i t e
topic
in
to the identity map of
C
composite
~ (TC)
> 19
TC.
tors,
> HC
then
functors,
and
S.: HD
If
S
and
information
in this sense.
sidered,
restrictions,
let
C
locally
finite c o u n t a b l e
the full
> Bi+ 1
CW-complex, (in each
S) Bi
Bi
of
R
>
categories
is e q u i v a l e n t
to
HOmH0(T.A,B)-
of spectra. to
S
~I
attention
is e q u i v a l e n t
~S.
of
S
C.
Bi
is a
c o m p l e x and each Observe
that if
W
is B
type of a c o u n t a b l e is h o m o t o p y
In fact,
equivalent
con-
whose
w h o s e objects are those spectra
W
to
To state the
such that each
simplicial
to
on the types of spaces
to
{Bi,f i}
S
0
are a d j o i n t
the full s u b c a t e g o r y
then every object of
is h o m o t o p y
> HC
I, and that
has the h o m o t o p y
to an object of
in
the
in the sense that no homo-
is simplicial.
subcategory
such that each
in
denote
are those spectra
SB i
theory
compares
objects
u(fi):
I
C £ C
are a d j o i n t p r e f u n c -
T.: HD
that
of
is homo-
and for each
T
Under r e s t r i c t i o n s
it s i m i l a r l y
> SD
is h o m o t o p i c
and
T
such that for
is lost by r e s t r i c t i n g
spectra and maps of spectra ~S
SD
our v a r i o u s
implies
for the p u r p o s e s of homotopy topy invariant
we say that
#. = ~.T.: HOmHc(A,S.B)
We can now c o m p a r e theorem
> ST
> TC
are h o m o t o p y
transformations
S~(D) T(SD) : SD
with adjunction
The f o l l o w i n g
T: i c
• TT (C): TC
the identity m a p of
> D
if
equivalent
{Bi,f i} 6 W, then
to a locally finite
simplicial
- 461
complex
B1 f
B~ 1
by
i > ~Bi+ 1
equivalences to
{B',f'}
> ~B.~+
B. <-----> B! 1 1
f~ 1
is the composite
determined
1
and if
~(f.~), then
~(f[)
to
by chosen homotopy
is a s i m p l i c i a l
is homotopy
{Bi,f i}
and therefore
Theorem
i] ; if
°
> B. 1
mation
[9, T h e o r e m
-
equivalent
approxito
B'i,~i ~"} 6 C
4
There
is a homotopy
preserving
prefunctor
M: S
> I
such that (i) tors J:
n: 1S
I
There exists > JM, with
> S
inverse
is the inclusion.
equivalent
to the identity (ii)
MJ:
is a map
in
by
= 6(JB),
~(B)
a natural
I
I
if
~: JM
of
is n a t u r a l l y
HS.
~: MJ
is a natural
then
J.M.
is a functor,
B 6 I, and if
of prefunc-
> IS, where
Therefore
functor
> I
equivalence
6(JB):
JMJB
>
is defined
11
> JB
transformation
of
functors. (iii) relationship
n
¢.(f) g:
>
M.
f: A
> J.B,
an adjoint
prefunctor
Therefore is an adjunction, and
¢~I (g) = J.g
where
• n.(A),
B.
(iv) serving
and
> HomHI(M.A,B)
= ~.(B)M.f,
M.A
J
between
¢.: HomHS(A,J.B)
establish
and
prefunctor
(iii) with respect (v)
M
By restriction, ~S
-~ ~I
induces
which
to the inclusion
By restriction,
M
a homotopy
satisfies ~I
induces
pre-
(i) through
> ~S. a homotopy
462
preserving through
prefunctor
~S n C
(iii) w i t h r e s p e c t Proof.
Assume
nj: Bj
Let
) MjB
MjB,
on
i
~j : MjB
B~ J
(i) and
Define
as follows. Mjf,
and assume
(i)
> ~S N C.
and prove
j ~ i, and
no = 1 = t0
and
M
satisfies
R N C
B = {Bi,f i} 6 S.
by induction that
which
to the inclusion
Let
MB = {MiB,Mif } E I
constructed.
> R N C
We first c o n s t r u c t
(ii) simultaneously.
MoB = B 0 .
-
Let
j < i, have b e e n
further
that
have been c o n s t r u c t e d
such that (a)
~jnj = i: Bj
(b)
n~j
Define
> Bj
• M3-1 . f = f.3-I Mi+IB
Clearly
• ~ j-1
) Bi+l,
inclusion,
Mif
nj£j ~ l: MjB and
to be the m a p p i n g
~(fi ) • S~ i" SMiB the s t a n d a r d
and
let
> MjB
nn j • fj -i ~ M 3-1 f • nj_ I •
cylinder
of the m a p
ki: SMiB
> Mi+IB
and define
Mif = u-](ki):
is then an inclusion.
;
Consider
denote MiB
> ~Mi+IB-
the d i a g r a m
S£ i SMiB ~
~ SB i Sn i
ki=~ (Mif) [
[~ (fi)
Mi+IB
(
> Bi+ l ni+~
Here
hi+ I
tained
and
by the standard
(a) is s a t i s f i e d ~i+l
£i+i
" u(Mif)
for
are the inclusion properties j = i + 1.
= u(fi)S~i , and
and r e t r a c t i o n
of m a p p i n g
cylinders,
It is s t a n d a r d
n~i+ I • Mif = fi~i
obhence
that follows
by
-
application tained
of
by a s i m p l e
j = i + 1 maps
n(B):
B
= ~(JB):
g: B
> B' Mjg
> JM_B
MJB
n~gj3 = M j g
(d)
RMjg
and
if
in
S.
found
I
> B
by
Define
for
hi:
if
= Mj_if'
SMiB
× I
(b) for
and c o n s t r u c t s in
S.
If
M
such that
on maps.
Let
and a s s u m e (with
with
× I
and
u(fi)S~i,
Mi+ig[x't]
Mi+iglY] It is t r i v i a l
equality
of m a p s
g £ I;
> ~Bi+l' '
MiB
u, w e see that t h e r e
> B' i+ 1
from
to c h o o s e [x,t]
y 6 Bi+ 1
in the c a t e g o r i e s
hi
and
for
exists to
to be the c o n s t a n t [y]
for the
in the m a p p i n g
and s i m i l a r l y
S
with
' St l~SM~g (fi)
Mi+IB'.
images
of
cylinder Define
by
= ~[SMig(x),2t],
0 ~ t ~ 1/2
[hi(x,2t
1/2 ~ t ~ i.
=
if
• Mj_ig.
Write
Mi+l g: M i + I B ----> Mi+IB'
tinuous.
proves
M 0 g = go
j ~ i
Applying
and w e agree g 6 I.
6 SMiB of
is ob-
(b) and w e can d e f i n e
£!3 " M j g ~ gj£j
g 6 [.
gi+lu(fi)S£i,
(e)
£(B) : JMB
(c) and the d e f i n i t i o n
a homotopy
Mi+IB
on o b j e c t s
I, fi£ iMi ' g ~ flgi~ i ~ ngi+Ifi~i:
homotopy
This
We n e x t c o n s t r u c t
in
• nj;
• Mj_If
equalities
(x,t)
and
M
" ni
etc.)
(c)
by
" fi = M i f
of the diagram.
m B.
have been
-
~ni+l
is a m a p
be a m a p
= n(B'),
Then,
chase
£(JB)
~(B)
!
Now
and thus c o n s t r u c t s
B 6 I, then
that
u-l.
463
- i)],
[gi+l (Y) ] •
to v e r i f y
Now consider
that
Mi+ig
the f o l l o w i n g
is w e l l - d e f i n e d diagram:
and con-
-
464
-
(Mif)
~ i+!
SMiB
~ Mi+1 B ~
SMig
> Bi+ 1
Mi+] g u (Mif')
SMiB'
~ > M ! + 1 B '~
gi+l B i+i '
q. l+l
Since and
ni+l (Y) = ' ~i+1
simple
[Y]'
!
~i+lMi+ig
This p r o v e s
(c) for
merely
observe
since
~(Mif) (x) =
course,
of
a functor.
h i(x,t)
that
Mg
p-i
is a m a p since
(d) for
> Mi+IB'
I
t.
j = i + l, commutes,
I, a n d MJ:
is in
by e x p l i c i t
to this
in
g
for all
square clearly
and a p p l y
Z: M i + I B
(a) and a
If the m a p
To p r o v e
(ii) of the t h e o r e m If
from
= gi+1 p (fi) S&i(x)
j = i + i.
[x,0],
is o b v i o u s ,
" ni+l
is e a s i l y v e r i f i e d
t h a t the l e f t - h a n d
(d) p r o v e s
the p r o o f
square.
= gi+l~i+l
since
= Mi+Ig
then follows
" Mi+l g ~ gi+l~i+l
computation
square.
Of
(c) c o m p l e t e s >
I
is c l e a r l y
is any m a p w h a t e v e r
such
Zni+ 1 ~ n i + l g i + 1 , t h e n
Mi+ig ~ nl+l$~+lMi+ig It f o l l o w s
M: S
>
Z
i > 0. (b), and
of
hi,
class
and f r o m this
is a p r e f u n c t o r .
g ~ g': B
Mig ~ Mig
~ nl+igi+l~i+ 1 ~ £ni+l£i+ 1 ~ £ .
t h a t the h o m o t o p y
of the c h o i c e
if
" gi+l
c h a s e of the r i g h t - h a n d
I, t h e n
that
' ni+l
> B'
in
S,
• ni$ i = nigi~ i ~ !
Now
M
of
Mi+ig
is i n d e p e n d e n t
it f o l l o w s
is h o m o t o p y
easily
that
preserving
since
then n 1 g i!~ i = M i g
(i) of the t h e o r e m
follows
|
• ni£ i ~ M i g
immediately
|
from
, (a),
(c). (iii)
To p r o v e
(iii), w e m u s t
show that
the f o l l o w i n g
-
two composites
are h o m o t o p i c
465
-
to the identity map.
(f)
JB n(JB)
) JMJB J~(B)
_ JB, B E
(g)
MB Mn(B)
~ MJMB
~ MB, B 6 S.
By
(a) and
map.
For
~ (B) = ~ (JB), the composite (g), note that
By the u n i q u e n e s s applied Since
~(MB)
proof
to the case M~(B)Mn(B)
this proves
~(JMB) n(JMB) above
(f) is the identity = 1 ~ n(B) ~(B) : JMB
for the h o m o t o p y
g = ~(B), we have
~ 1
M~(B)
by the fact that
that the composite
M
class
of
~ ~(JMB)
> JMB.
Mi+ig = ~(MB).
is a prefunctor,
(g) is homotopic
to the iden-
tity. (iv) S
and
Since
aS
I, it suffices
and
for
B E aS, and this follows gj : nBj+ 1
> Bj
if and
inverse
it suffices
~i
simplicial
is simplicial,
By Hanner
map
[5, C o r o l l a r y
simplicial
complex
by K u r a t o w s k i
an ANR.
complex
MiB
SMiB
[7, p. 284], Mif
if
Mjf.
MB q R N C with
~ B = B
is a locally
> Bi+ l
neighborhood
finite
u (Mi_if),
is the m a p p i n g
every c o u n t a b l e
is an absolute
if
show that
inverse to
and that each m a p Mi+IB
of
fj, then
i, starting
u(fi)S~i:
Since the image of
to
to show that
on
since
3.5],
MB q ~I
(b) w h i c h
is a h o m o t o p y
By induction
of the simplicial
and,
(a) and
n O = 1 = ~Q, we see that each
countable and
Again,
B E aS N C.
from
> MjB
are full subcategories
(iv) to prove that
is a homotopy
njgjn~j+ 1 : nMj+IB (v)
~I
n i,
cylinder
[i0, p. 151]. locally retract
finite (ANR)
the loop space of an A N R is is a closed
subspace
of the
-
ANR
~Mi+IB , Mif
MiB
[6, p. 86], and therefore
a d e f o r m a t i o n retract of
~Mi+IB
I
It is also conceptually
fine a functor spectrum of
satisfactory in
~ I
X, zi x = six
> Y zg
z: T
is a map in
Y
This proves
• Y
in v i e w of the following
to maps
by letting and
fi
I.
UB = B 0
in ZX
I.
We can de-
be the suspension
~- i ( i s i + , x )
=
Let
and
it is clear
U = Ui:
Ug - go"
is the identity functor.
If
.
Zig = Sig;
T, define
is in fact a map in
the forgetful functor, UZ: T
is
is not only large and convenient.
o b s e r v a t i o n relating maps
that
[i0, p. 31].
MiB
MB £ R N C, as was to be shown• The category
g- X
-
has the homotopy extension property w i t h
respect to the ANR
that
466
I
• T
be
Observe that
With these notations,
we have the following proposition. Proposition
U: HomI(ZX,B) Proof• inductively by i > 0.
If
f0 = i, fl = f0' and
= i(Rifi
• UB, define Now
u-i(isix):
> B.
Since
fi
for
ZX
X
• ~isix.
~: 7U
~(B)
Clearly
. fi
1 )
n~i+l (fi+l)
• fi) = fi~i(fi), ~(g) = @(B)Zg.
fi: B0 ___> ~iBi
fi+l = n i f
Define a natural transformation
= i(fi+l)
= g.
is an adjunction.
B = {Bi,f i} 6 I, define
~(B) = { i(fi) }: TUB
g: X
) Homy(X,UB)
11
if by
• ~-I (Isi+IB0)
is a map in U~(g)
I.
For
= u° (f°)Z0g
is easily v e r i f i e d to be Therefore
~(ZX) = i: ZX
> ZX; since
-
we obviously that
have
ZU(Izx)
= I: ZX
we compare
The f o l l o w i n g
conceptually homotopy variant
Theorem
theorem
to
I
theory
to
information
and m a p s
>
ZUZX = ZX, this
4, it a l s o
Theorem
i
shows
to the c a t e g o r i e s that
L
is n i c e l y
implies
I, ~I, a n d related
and
is eq%livalent for the p u r p o s e s
~I
in the s e n s e
in
L; c o u p l e d
shows
that
of h o m o t o p y
of w e a k
t h a t no w e a k h o m o t o p y
is lost by r e s t r i c t i n g
of s p e c t r a
for the p u r p o s e s
with
i N W
attention
the r e m a r k s
in-
to s p e c t r a preceding
is eq~livalent
to
R N W
theory.
6
There
is a f u n c t o r
formation
of f u n c t o r s
inclusion,
such t h a t (i)
LK:
L:
i
is an a d j u n c t i o n
is t h e
-. L
(ii) Lg'
in
If
(iii)
Let
in
and a natural
identity
• n(A)
I, t h e n
B E ~I, t h e n
B 6 R N C; t h e n
a n d if
> LB'
Proof. inclusion,
i
K: L
>
functor
I
transis the
and
for
Lg
g: L A
> B.
is w e a k l y
n(B) : B
homotopic
> KLB
is a
equivalence.
topy equivalence LB
if
>
) HomL(LA,B)
L-I (g) = Kg
g ~ g'
L, and
weak homotopy
Lg ~ Lg':
with
I
) KL, w h e r e
n: i I
L: H o m I ( A , K B )
to
-
~U = i. Finally,
R.
4 6 7
Let
g ~ g':
in
B
n(B) : B ---> KLB > B'
I, t h e n
i.
B = {Bi,f i} E I.
w e can d e f i n e
in
is a h o m o -
Since
each
LiB = li~) ~iB i+j' w h e r e
the
fi
is an
li/nit is
- 468 -
taken with respect to the inclusions Clearly in
~Li41B = LiB , hence
I, define
sense since of maps
in
n: 11
> KL
I.
clusion;
q(B)
= hi(B).
Now
Lq: L
functor
of
i
is evident,
are easily verified This implies
> Bi
~i = lim riJ: LiB map.
is a map
by the definition Lg £ t.
Define
be the natural
in-
~qi+l (B) ° fi
~JBi+ j
its image under
Since
j > 0.
We can therefore Obviously
Suppose further
that
and omit the inclusion maps
> KLK
and
to prove
(iii).
ri+jfi+ j = i, we have define maps ~iqi: B i B 6 C.
£J+tBi+j+ 1
> Bi
is the
Then we claim
Let us identify
£Jfi+j
is
> Bi, define maps
As in the proof of
is now an ANR.
LK
r io = i, r i I = r i, and
if
in
nK: K
(i) and it remains
by
~Jfi+j
and
ri: ~Bi+ 1
> LiB.
The fact that
to be the identity natural
inductively
> B i.
ni~ i ~ i: LiB
4, each
> B'
; the limit makes
since
of the limit topology.
ri,j+l~Jfi+ j = riJ.
that
> ~j+IBi+j+l"
(ii) of the theorem is a standard consequence
= rij~Jri+ j
identity
LiB
a map in
B 6 R, with retractions
ri,J+l
> L.B' 1
LiB
hi(B) : B i
is obviously
> LKL
riJ: ~JBi+ j
g: B
~Li+Ig = Lig, hence
by letting
transformations. If
~JBi+j
= ~J+Igi+j+l~Jfi+j
Clearly
of the definition the identity
If
LB £ i.
Lig = li~, ~3gi+j: sJfi+j~Jgi+j
~Jfi+j:
for all
(v) of Theorem £JBi+ j with i
and
from the notation.
j Then
the inclusion ~JBi+ j x I U £J+IBi+j+ I × I c ~J+iBi+j+ 1 x I is that of a closed homotopy
extension
subset
in an ANR, and it therefore
property with respect
to the ANR
has the
~J+IBi+j+ 1 •
-
In particular, tion retract
by of
homotopies: kij:
Bi
is a strong deforma-
and we assume given homotopies
> ~Bi+l,
kij:
-
[10, p. 31], each ~Bi+l,
ki: ~Bi+ 1 x I
4 6 9
ki: 1 ~ r i rel B..
nJ+IBi+j+ 1 x I
> nJ+IBi+j+l
1 ~ nJri+ j rel ~JBi+ j , in the obvious
(kij,t = n3ki+j,t).
hij:
let
hil = k i = ki0
hij: n3Bi+ j x I
hi0
induce
,
fashion on
j t we
can
> ~3Bi+ j ,
hi,j+ 1 = hij
be the constant
, and suppose given
sider the following
1
m
1 ~ riJ rel B i , such that
To see this,
k.
We claim that, by induction •
choose homotopies
The
l
on
homotopy,
hij
~JBi+ j × I. let
for some
j > 0.
Con-
diagram:
(~JBi+jxI U ~J+IBi+j+ Ix~)x0
- (~JBi+jxI U ~J+iBi+j+ 1 xl) xI
I~3+I Bi+j+ 1 ~.
j +1
i, %
(~j+l
Bi+j+ 1 xI) x0
> (~j+l Bi+j+l xI) xl N
The unlabeled hij(x,s,t)
arrows are inclusions,
= hij(x,st)
if
~(y,l,t)
= hij(nJri+j(y ),t)
verified
that
hij = kij fore obtain
~ij
and
hij
x £ ~3Bi+ j ; and if
is defined by ~ij (y,0,t)
y E nJ+IBi+j+1
is well-defined
It is easily
and continuous
on the common parts of their domains. Hi,j+ 1
•
and that We can there-
such that the diagram commutes.
hi,j+l (x,s) = Hi,j+] (x,s,l) .
It is trivial
= y ,
Define
to verify that
470 -
hi,j+ 1
has the d e s i r e d
li~ hij:
LiB
× I
from
to
ni£ i .
1
properties.
• LiB
Now
is d e f i n e d
Finally,
if
and
is c l e a r l y
g ~ g':
B---~
B'
a homotopy in
I
and
B E R N C, t h e n Lig = L i g n i ~ i = nigi~ i ~ , This
completes
the p r o o f
We remark Propositions In fact, Q~
n ,g i,£ i = Lig 'ni~ i ~ Lig ' ,
of
(iii)
relationships
1 and 5 and of the t h e o r e m LZ:
functor
0 .
and of the t h e o r e m .
t h a t the c a t e g o r i c a l
the c o m p o s i t e
ia
are c l o s e l y
T
•
i
of
related.
is p r e c i s e l y
, and the a d j u n c t i o n ~:
of P r o p o s i t i o n
• H o m i(Q X,B)
HomT(X,U=B) 1 factors U -I
HornT (X,UKB) The v e r i f i c a t i o n
as the c o m p o s i t e
~ Homl (~X,KB)
of t h e s e
statements
L
(U~ = U)
_~ H o m i (Q~X,B) .
requires
only a glance
at
the d e f i n i t i o n s .
3 We shall of the p r e v i o u s promised
INFINITE here
summarize
section
for
applications.
extension
collection
It is c u s t o m a r y loop s p a c e If
X
if
is g i v e n
X
is the as an
SPACES the
infinite
We then make
of o u r r e s u l t s
an i n t e r e s t i n g
LOOP
implications loop s p a c e s
spectra
of c o n n e c t i v e to say t h a t initial
H-space,
and g i v e t h e
a few remarks
to u n b o u n d e d
space
B0
about
and p o i n t
cohomology X 6 T
of the w o r k
the out
theories.
is an i n f i n i t e of an ~ - s p e c t r u m
it is r e q u i r e d
that
B.
its p r o d u c t
-
be h o m o t o p i c
to the p r o d u c t
valence
> aB 1 .
X
an infinite m a p of
loop map
~-spectra
4 satisfies
~B
the identical
If
if
a map
and
infinite
~g
M:
= go •
aS
equi-
is said to be go
> nl
of a of T h e o r e m
We t h e r e f o r e
see that
loop spaces and m a p s are o b t a i n e d to i n c l u s i o n
is any infinite
implies the e x i s t e n c e
f 6 T
is the initial m a p
The functor
= B0
f: X---> X'
f
-
induced from the h o m o t o p y
Similarly,
g.
we r e s t r i c t a t t e n t i o n
471
a-spectra
loop map,
of a c o m m u t a t i v e
and m a p s
if
in
I.
then T h e o r e m 6
d i a g r a m of i n f i n i t e
loop maps g>
X
Y
I 1 X' such that infinite
f'
g'> Y'
is a p e r f e c t
loop spaces
and
(3.1)
infinite
g
and
g'
loop m a p b e t w e e n p e r f e c t are w e a k h o m o t o p y
equi-
valences. If
X
is an infinite
loop space of the h o m o t o p y
type of a c o u n t a b l e
CW-complex,
of B o a r d m a n
and V o g t
[i, p. 15] that there
map
~ Y
Y
g: X
such that
is the initial
this fact w i t h theorem,
and
(iii)
of T h e o r e m
equivalence
~S ~ W.
4, the remarks
loop and
Combining
preceding
6, we see that if
loop map b e t w e e n
CW-complexes,
in
from a r g u m e n t s
is an infinite
is a h o m o t o p y
space of a s p e c t r u m
(v) of T h e o r e m
is any infinite of c o u n t a b l e
g
then it follows
f: X
spaces of the h o m o t o p y
that ~ X' type
then there is a h o m o t o p y c o m m u t a t i v e
- 472
diagram that
of infinite
f'
loop maps,
is a p e r f e c t
homotopy
infinite
loop map and
in
g
(I), such
and
g'
are
equivalences. Therefore
homotopy
theory
nothing
is lost for the purposes
if the n o t i o n s
maps are replaced maps,
of the form given
by those of p e r f e c t
and similarly
strict a t t e n t i o n
of infinite
for homotopy
to spaces
of w e a k
loop spaces
infinite
loop spaces
theory provided
of the homotopy
and and
that we re-
type of c o u n t a b l e
CW-complexes. The promised topy groups
of infinite
of P r o p o s i t i o n say
Y = B0
comparison
i.
) LMB
n(MB) : MB
in
> LB Qy
a = Qn 0 (MB),
if
Y
in
I.
homo-
is now an easy c o n s e q u e n c e is an infinite
B E ~S, then that p r o p o s i t i o n
¢.(LMB) : Q.LoMB
by
loop spaces
In fact,
where
of stable and u n s t a b l e
i, and T h e o r e m
loop space, gives
6 gives
a map
a map
Define maps
a ~ QLoM B
8 ~ LoMB < Y
S = ¢. 0 (LMB),
and
y
Y = no (MB).
y
is c l e a r l y
a
since
t
a weak homotopy Q: Y
> T
valences. LOMB, nn(QX) fore
equivalence,
is easily v e r i f i e d Since
8,
¢.,0(LMB)
is an e p i m o r p h i s m
= K nS(X) , the p(Y)
K S(Y) infinite
and therefore
~th
= y:ls,~,:
) ~,(Y). loop map,
to preserve w e a k h o m o t o p y
• T.(LeMB)
If
stable homotopy
It is clear
> E,(y) that if
X E T, then
group of
gives f: Y
X.
There-
an e p i m o r p h i s m ) Y'
then
p(y,) (Qf), = f,p(y) . ~S(y)
equi-
is the identity map of
on homotopy.
~,(Qy)
so is
> En(Y')
.
is any
-
4 7 3
It should be observed loop spaces
since the composite
not be an infinite > Y
and
B0 = Y
which
giving
f
loop map.
g: Y
topological
allow c o m p o s i t i o n tions m o t i v a t e
of maps,
suspension
the use of
loop maps B
giving
with
g.
loop spaces
a classifying
t
One
to be
space argument
is awkward.
in the d e f i n i t i o n
application of
of the category F (n), where of maps
topological
that
infinite
but this
in the computation
F = i~
fine
infinite
These
to
condi-
of homology
i.
convenience let
loop maps need
of a map of spectra
and using
The following be used
of infinite
the range of a map of spectra
this by requiring
monoids
from a c a t e g o r i c a l
In fact, g i v e n
is simultaneously
can get around
of infinite
) Z, there need be no spectrum
and the domain
in section
that the notions
and maps are not very useful
point of v i e w
f: X
-
monoids
flnx
i.
S : F (n)
y: ~nx x ~(n) is, with
H*(BF), Let
the limit
under
of our results, illustrates ~(n)
composition by
identified
the technical
= HomT(sn,s n)
is taken w i t h respect
) F(n+l) .
> ~nx
w h i c h will
and
of maps.
y(x,f) with
F (n)
F
If
to
are
X 6 T, de-
= ~-n(un(x)
HomT(S°,~nx),
and
° f), by the com-
posite nnx×~(n)~n×l> This defines
N
HOmT(sn,x) xF(n)COmposition ) H o m y ( s n , x ) an operation
B = {Bi,f i} 6 aS, and let verse and
to
fi"
Define
gn:
~nBn
> B0
of gi:
homotopy
~(n) ~Bi+1
on
nnx. > Bi
equivalences
in the obvious
~-n
> ~nx.
N o w let be a homotopy fn: B0
inductive m a n n e r
in-
> ~nBn and define
- 474 -
Yn = g n y ( f n Observe since
that
fails
to d e f i n e
the a s s o c i a t i v i t y
condition
course,
Yn
x i): B 0 × ~(n)
Yn
ativity
coincides
with
is t h e n r e t a i n e d .
y
~(n)
×
fn+l×l>
a n+l
an o p e r a t i o n
of
(xf)g = x(fg)
on
~nB n
Now consider
~nBn
B0
> B0 .
if
diagram:
!~ n f ~ Bn+l
/
[l×S
Of
~nB n
l
ll×S
B0
B 6 i, and a s s o c i -
Y ~ ~n
Bn+ 1 x F (n)
on
is lost.
the f o l l o w i n g
Y
x NF (n)
~(n)
_n+l g > BO
fn+l x 1 B 0 x ~(n)
> ~ n + I B n + 1 × ~(n+l)
The l e f t - h a n d
triangle
y
on
is n a t u r a l
y(1 x S) = y
and s q u a r e
n-fold
commute
loop maps,
hence
trivially.
~nfn Y = y(~nf
is h o m o t o p i c
chosen have
to
retractions,
gn+l~nfn, then
Yn = Yn+1 (i × S)
Y = lim > Yn: B0
x ~
~ B0 .
natural
For
maps,
and the d i a g r a m
we h a v e an o p e r a t i o n is a m a p
in
i, t h e n
B 6 R .
• Sf)
and the
Thus
if
.
gi
are
B 6 R
we
and w e can d e f i n e
naturally
R.
and if
gn __ g n + l ~ n f n
not t r a n s f o r m e d on
x i).
since
~ - n ( ~ n ( x ) f ) = ~ - ( n + l ) ~ ( ~ n ( x ) f ) = p-(n+l) (~n+l (x) gn
Clearly
Since
by m a p s
in
B E i, the
× ~
R, the m a p
f's
trivializes. Y: B0
the r i g h t - h a n d
and
g's
Therefore,
triangle Y
is
is n o t
are t h e i d e n t i t y for e a c h
> B0
and
if
h: B
h 0 (xf) = h 0 (x) f
for
x 6 B0
B £ i, > B'
and
f £ ~.
- 475 -
Stasheff
[unpublished]
has g e n e r a l i z e d work of Dold and
Lashof
[2] to show that if a t o p o l o g i c a l
space
X, then there is a n a t u r a l way
quasifibration
X
> XXMEM
quasifibration
M
> EM
of the h o m o t o p y
equivalences
B £ i
and
therefore on
) BM.
form
YxEEF.
M
operates
to the c l a s s i f y i n g As usual,
of spheres.
Of course,
on a
to form an a s s o c i a t e d
let
principal
F c ~
consist
By restriction,
Y = B0, we have an o p e r a t i o n
of
F
on
Y
this c o n s t r u c t i o n
if
and w e can is n a t u r a l
i. Boardman
inclusions between
and Vogt
infinite
homotopic
[1] have p r o v e n
U c O c PL c Top c F
sum
(on
to the c o m p o s i t i o n
that we can pass to (spaces h o m o t o p y
i
that the s t a n d a r d
are all i n f i n i t e
loop spaces w i t h
tures given by W h i t n e y
F.
> BM
monoid
respect
to the H-space
F, this s t r u c t u r e
product
its product. product since map.)
of
~BI, w h e r e
composing
if
of
F
on
~ £ B0
F
spaces
and of the spaces I shall 7, : H. (B) ® H. (~)
GXFEF
and
~f = ~
on
by
F
G
to
the loop
f E F
the trivial
the g e o m e t r i c
on its various
of
Observe
under
for all
to u n d e r s t a n d
sig-
sub H-spaces
BGxFEF.
show e l s e w h e r e ) H. (B)
F
is not e q u i v a l e n t
any map w i t h the trivial map gives
of these operations
BG.
is the i d e n t i t y
B £ t, then
It w o u l d be of i n t e r e s t
nificance
of
to) each of these sub H-spaces
operation
(In fact,
We n o w k n o w
operations
The same is true for their c l a s s i f y i n g
that the r e s u l t i n g
struc-
is w e a k l y
used above).
and obtain n a t u r a l
equivalent
loop maps
that, w i t h
gives
H, (B)
rood p
coefficients,
a structure
of Hopf
- 476 N
algebra where
over
over
operations
operations for
H, : i
>
types
known
information,
algebra
algebra,
?
1.
i.
operations
these
commutation
formulas
extend
our r e s u l t s
of s e c t i o n
2 to u n b o u n d e d
denote
the c a t e g o r y w h o s e
~Bi+ 1
are s e q u e n c e s
such
{gill 6 Z}
tor
C: S
defined
if
i < 0.
For e a c h to be
on o b j e c t s
with
the i m a g e
{gill 6 Z}
inverse
of
Its o b j e c t s such
range how
that
~
These cate-
these
by
defined
and m a p s
C
The maps
completion
CiB = B i for
if
i a 0 C
in
func-
i > 0 and
and
Cif = 1
is an i s o m o r p h i s m
forgetful
functor D
of
~ S
• S. define
is of p a r t i c u l a r
~.
for all
E Z}
{gill > 0} 6 S
are s e q u e n c e s
B i = nBi+ 1
Let
and
subcategories in
to
{Bi,fili
i < 0.
that
on maps.
way
spectra.
an o b v i o u s
the e v i d e n t
under
is a n a t u r a l
for
Cif = fi
similarly
of our p r e v i o u s l y
interest.
We have
i < o, w i t h
i < 0, and d e f i n e d
of c a t e g o r i e s
[3].
are all that is
B i = n-iB0
g = if
CiB = n-iB 0
and
map
gi = ~gi+l • S-
of the
coupled with
are s e q u e n c e s
is the i d e n t i t y
and
for
objects
{Bi,f ill > 0} ~ S )
in
and
H* (BF) . that t h e r e
fi : Bi
in terms
and,
we observe
that
algebra
by specifying
commute,
B ~ SS),
is also a K o p f
and L a s h o f
Finally,
such
for
The a p p r o p r i a t e
is d e t e r m i n e d
of h o m o l o g y
H, (B)
is d e f i n e d
by D y e r
on
afortiori,
of the S t e e n r o d
which
introduced
to c o m p u t e
(and,
as in s e c t i o n
are all n a t u r a l
three
required
B 6 i
the o p p o s i t e
the D y e r - L a s h o f
homology
gory
for
H, (B) = K, (B 0)
algebra over
H,(F)
i
{B ili E Z } and
and
gi = ~gi+l
for
- 477 -
all
i.
Clearly
all of the results of s e c t i o n 2 r e m a i n v a l i d for
the c o m p l e t e d categories. Our results s h o w that any r e a s o n a b l e by w h i c h w e m e a n any c o h o m o l o g y B 6 ~
theory d e t e r m i n e d by a s p e c t r u m
N ~, is i s o m o r p h i c to a c o h o m o l o g y
s p e c t r u m in
T ~ ~
c o h o m o l o g y theory,
theory d e t e r m i n e d b y a
and that any t r a n s f o r m a t i o n of s u c h t h e o r i e s
d e t e r m i n e d by a map
g: B
in
> B'
v a l e n t to a t r a n s f o r m a t i o n
~
is n a t u r a l l y e q u i -
A ~
d e t e r m i n e d by a map in
T ~ ~.
Recall
that Hn(x,A;B)
= HomHT(X/A,Bn)
defines the c o h o m o l o g y t h e o r y d e t e r m i n e d by (X,A). where
C a l l s u c h a theory c o n n e c t i v e P
is a point.
Any i n f i n i t e
Of course,
loop space
Y
if
Hn(p;B)
H-n(p;B)
determines
B ~ ~
on
= 0
CW pairs for
n > 0,
= E 0 (~nB 0) = Kn(B0).
a connective
(additive)
c o h o m o l o g y t h e o r y since, by a c l a s s i f y i n g s p a c e argument, w e can obtain
CB 6 ~
K 0 (Bn) = 0
such that
for
n > 0;
B0
is h o m o t o p y e q u i v a l e n t to
a c c o r d i n g to B o a r d m a n
such c o h o m o l o g y t h e o r y is so o b t a i n a b l e homotopy equivalence CQ~X
determines
CnQ~X = QSnX n > 0.
for
of i n f i n i t e
a connective n > 0, and
and d e t e r m i n e s
loop spaces.
cohomology
B £ i, then
C~ B: C Q ~ B 0
> CB
determines
[1], any Y
up to
since = h s(X) n
i, t h e s e t h e o r i e s p l a y
to be of interest.
and
X 6 T, then
= nn(Qx)
role a m o n g all c o n n e c t i v e c o h o m o l o g y t h e o r i e s , their properties might prove
If
theory,
H -n(P;CQ~x)
In v i e w of P r o p o s i t i o n
and V o g t
Y
i~
a privileged
and an a n a l y s i s of Observe
that if
a natural transforma-
- 478 -
tion of cohomology theories
H * ( X , A ; C Q = B 0)
if the theory d e t e r m i n e d by
CB
is e p i m o r p h i c
) H*(X,A;CB)
is connective,
on the cohomology of a point.
and,
this t r a n s f o r m a t i o n
- 479 -
BIBLIOGRAPHY [i]
Boardman, spaces",
J. M. and Vogt, R. M. "Homotopy Everything HMimeographed Notes, Math.
Inst., Univ.
of Warwick,
(196 8). [2]
Dold, A. and Lashof,
R., "Principal Quasifibrations
Fibre Homotopy Equivalence 3; 285-305, [3]
[4]
Dyer, E. and Lashof,
R., "Homology of Iterated Loop Spaces",
Amer. J. Math., 84; 35-88,
(1962).
Eilenberg,
S. , "Acyclic Models", A m e r .
S. and MacLane,
Hanner,
O., "Some Theorems
[7]
Kuratowski,
Milgram,
Milnor, plex",
H. , Dimension Theory, Princeton
C., "Sur Les Espaces Localement Connexes et n"
Fund. Math., 24; 269-287,
(1935)
R. J., "Iterated Loop Spaces", Annals of Math.,
84; 386-403, [9]
Retracts",
(1961).
P~aniens en Dimension [8]
on Absolute Neighborhood
(1950).
Hurewicz, W. and Wallman, Univ. Press,
J.
(1953).
Ark. Mat., i; 389-408, [6 ]
Ill. J. Math.,
(1959).
Math., 75; 189-199, [5]
of Bundles",
and
(1966).
J., "On Spaces Having the Komotopy Type of a CW-com-
Trans. A.M.S. , 90; 272-280,
(1959).
[i0] Spanier, E. H., Algebraic Topology, McGraw-Hill, [ll] Swan, R. G., "The Homology of Cyclic Products", 95; 27-68,
(1960).
Inc.,
(1966).
Trans. A.M.S.,
-
HOMOLOGY
4 8 0
-
OF SQUARES AND FACTORING
OF DIAGRAMS
Paul Olum i. Introduction Our purpose mappings
here is to present
of spaces which
tions and which topological.
extensive
some w e l l - k n o w n
A'
CPo
,
more
fo
~o --~
-----~Y
~
@ "i
holds
Y'
11
everywhere.
We want
to know under what
there will exist a map h: X + Y, shown by the dotted
such that the diagram with h present will continue
commutative
or, at least,
h will be said to "factor" In a systematic a precise
definition
it be a factorization. for the existence tions.
in §4 below;
(of solid arrows):
~i
circumstances
than the
the use of the method by applyresults
) A
. i
where commutativity
as nearly diagram
so as possible. (i.i).
of what w~ require
We would then develop
and all proofs
however,
A portion
an obstruction
of the definition
theory
of these obstruc-
but will defer the
to a later work.
What we shall do in the present pal consequences
first give
of this h in order that
of h and study the properties
account
to be
Such a map
treatmen%' of this p r o b l e m we would
We will not do this here,
systematic
applica~
will appear in a later work.
Let us look at the diagram
arrow,
to other categories
We shall illustrate
applications
in the study of
seems to have a number of useful
should generalize
ing it to derive
some technique
discussion
is give the princi-
of a factorization
and indicate
of this work was done under NSF grant GP-7905.
-
481
-
the groups in which the obstructions tomary) main properties.
lie as well as their (cus-
For the applications
we have presently
in mind this is all that will be needed. 2. Some properties
of a factorization
All of the spaces in diagram
(1.1) are taken to be path-
connected and to have a base point *; all mappings are to preserve base points.
and homotopies
Apart from this we make only the
following assumptions : (2.1a) CW-complex (2.1b)
Each of A', A, X', X has the homotopy type of a and the base points are non-degenerate. Either ~l# : ~l (Y') ~ ~l (B') or ~l# : ~l (B) ~ ~l (B')
is onto. By way of notation,
a homotopy £ : X × I ~ Y will be called
"rel ~l T! or "rel ~o !! if r(~l×l ) or ~o r is stationary. homotopies
rl, r 2 : X x I ~ Y with rl(X,1 ) = r2(x,O),
Given two we write
£1.£2 for the homotopy which results from r I followed by r 2. We adopt also the following convention: (2.2)
A homotopy of homotopies,
be assumed to be proper,
e.g., £1 ~ r2 will always
that is, to be stationary on X×0 U X×l.
As indicated in §l, we will not give the definition of a factorization here, but the following
two theorems contain its
main properties : Theorem 2.3
A factorization
of (1.1) gives rise in a canon-
ical wa 7 to a mapping h : X @ Y and four homotopies
Al: fo = hal
A2: go ~ h~l
rl:
r2: gl ~ $o h
f l = ~oh
such that in the diagram
-
A'xl
~o xl
482
-
AI ) Axl ~ . ~
~o
Y-
_)Yr
(2.5) ~oXl X
'
xI ~
I
~°ixl
Xxl I--
) B
~B'
FI
¢i
all mapping squares are properly homotopy commutative .(in the sense of (2.2)).
(We shall say the factorization induces
e: Vl(X) ~ vl(Y) if h induces this e.) For the next theorem,"fibratio~'means bration,
regular Hurewicz fi-
that is, the homotopy lifting property for any space,
with the lifted homotopy stationary wherever the original one is. If the spaces A', A, X', X are CW-complexes ~o' ~i are cellular,
and the maps, ~o' al'
then fibration may be taken to mean weak
(or "Serre") fibration. Theorem 2.6 For each of the conditions
listed below, if (i.i)
can be factored then the factorization can be so chosen as to make the accompanying properties hold in addition to those ~iven in Theorem 2.3.
For any combination of the conditions this can be
done so that the correspondin~ properties hold simultaneously: (a)
~i cofibration:
fo = h~l and A 1 is stationary;
F 1 and
F 2 are rel ~i and the homotopy ~ F 1 ~ ~IF2 is rel (~i×i) (b)
~o and ~i cofibrations:
(c)
~o fibration:
A 2 is rel ~o
fl = ~o h and F 1 is stationary;
A 1 and A 2
are rel ~o and the homotopy Al(~oXl) ~ A2(~o×l) is rel ~o (d)
~o and ~i fibrations:
Remark 2. 7 by ~o" ~l'go;
F 2 is rel ~i"
We can replace ~o' ~i' fo in (a) and (b) above
this is clear from the symmetry of the diagram;
similarly for $o' $1' gl instead of ~o' ~l' fl in (c) and (d).
-
4 8 3
-
But we can not add these to the theorem since, ~i and ~i are both cofibrations
for example,
if
it need not be true that fo = hal
and go = h~l for the same h. Remark 2.8
There is, as one would expect,
notion of the homotopy logues of Theorems this here;
of two factorizations
an appropriate
of (i.I),
2.3 and 2.6 for this notion.
and ana-
We shall omit
it will be found in the later account promised
in the
introduction. 3. Cohomology
and homotopy
of squares;
obstructions
Let S 1 denote the mapping square ~o' ~l' ~o' ~l and S 2 the mapping
square ~o' ~l' 4o' 41"
we shall need the cohomology ficient group)
groups Hk(s1;
G)(where
and the homotopy groups Vk(S2).
the most important exact sequences
Vk(~o)
G is a coef-
For our purposes
property of these groups is that there are
(we omit coefficients):
(3.1) Hk( l) (3.2)
As a setting for our obstructions
Hk(ao) 4# , Vk(~l)
I) J,Hk+l( l) t irk(S2 )
~ )Vk_l(~o )
and the same with ao' ~i replaced by ~o' ~i and ~o' E1 replaced by 4o, 41. Definitions
of these groups and proofs
and (3.2) are due to Eckmann-Hilton;
of exactness
see [2,Chap.9].
It is easy to see that the homotopy groups ~k(S2) groups
at the base point in Y (i.e.,
any homomorphism
for (3.1)
Vl(Y) operates
e: Vl(X) ~ Wl(Y) induces
Vk(S2)
are local
on Vk(S2))
and
as a local group
in X and hence in the square SI; we denote this induced local group in S 1 by 8*Vk(S2).
As indicated
nition of the obstructions,
in §i, we shall omit the defi-
but the following
theorem gives all
-
4 8 4
-
the information we need about them here: Theorem 3.3 For n ~ 3, the n-th obstruction~en to a factorization of (i.i) inducing a given e: ~l(X) ~ Vl(Y) is a subset (possibly void) of Hn(sI;
e*~n+l(S2) ).
It has the following
properties: (i) O e ~ ne if and only if ~n+l ve is non-void (ii) Suppose Hn(S1; n.
e*~n+l(S2)) = 0 for all sufficiently larse
Then there is a factorization of (1.1) inducing e if and only
if 0 e ~ n
8
for all n > 3.
To complement this theorem we need conditions which will imply t h a t ~
is not void.
Proposition 3-5 below gives some
sufficient conditions which are adequate for our present needs. We require some notation for this.
Let ~ ( e o , ~ o ) be the free
product ~l(X')*Vl(A ) modulo the normal subgroup N generated by the set [~0(a)~o(a -I) product.
I aeVl(A')],
i.e., the "reduced" free
The maps ~i' ~i together clearly define a homomorphism :
(x)
Then we have Proposition 3,5
Suppose that either (a) or (b) holds for
diagram (1.1): (a) (~l,~l)# in (3.4) is an isomorphism; el#(~2(A'))
~l#(~2(X')) and
together generate v2(X ).
(b) ~o#:Vl(Y) ~ ~l(B) is an epimorphism;
@#:vi(~o )-9 ~i(~l)
is an isomorphism for i = 2 and an epimorphism for i = 3. (Recall also 2.lb.) Then there is a unique e: Vl(X) ~ Vl(Y) for w h i c h ~ 3e is
-
4 8 5
-
non-vold.
Remark 3.6 For the homotopy p r o b l e m (see Remark 2.8) the @
e Vn+2(s2)) and there are
obstructions lie in the groups Hn(S1;
obvious analogues of Theorem 3.3 and Proposition 3.5.
4. Examples We give three examples to illustrate the application of the material above. l) Our first example is a theorem of James.
For this we
recall that a loop is a "non-associative group", M with .multiplicalion and a two-sided identity,
that is, a set and such that the
equations xa = b, ay = b
a, b in M
admit one and only one pair of solutions x, y in M.
The follow-
ing is Theorem 1.1 of [4]. Theorem 4.1
Let X have the homotopy type of a CW-complex
and let Y be a connected H-space.
Then the homotopy classes of
maps of X into Y form a loop with multiplication inherited from Y. Proof.
The diagram is the following special case of (1.1):
*
~ i
*
~ X
>YxY f ~ Y
~
-~}i;*
where f and g are given maps, ~ is the multiplication in Y and Pl is projection on the first factor.
It is clear that what has
to be proved is the existence of a factorization h of this diagram, unique up to homotopy,
and the same for (4.2) with Pl re-
placed by the projection P2 on the second factor.
-
4 8 6
-
Obviously ~ induces an isomorphism Vq(Pl) ~ Vq(p) for all q and therefore
(by (3.2)) ~q(S 2) = 0 for all q; similarly for P2"
The existence of the factorization now follows from Proposition 3-5 and Theorem 3.3.
The uniqueness
follows similarly from the
analogQus results for the homotopy of two factorizations;
see
Remarks 2.8 and 3.6 above. The other theorems of [4, §4] follow in the same way from similar diagrams. 2)
Our second example is a result of Hilton [3].
We con-
sider maps of path-connected spaces: (4.3)
F
i
)E
p
~X
where each of F,E, X has the homotopy type of a CW-complex,
i is
a cofibration and pi = *; here * is a non-degenerate base point in X.
Let W be a 1-connected space and suppose there is given a map f: (E~F) ~
(W,*).
Denote by Pi : Hn÷i(x;
Vn(W)) ~ Hn÷i(E'F;
the cohomology homomorphisms
induced by p.
contains the main theorem in Hilton
Vn(W)) The following then
[3,P.77]:
Theorem 4.4 (a) Suppose Po is an epimorphism and Pl a monomp~phism for all n ~ 2.
Then there is an h: X ~ W such that
hp ~ f tel F. (b) Suppose P-I is an epimorphism and Po a monomorphism for all n ~ 2.
Suppose h, h'
: X ~ W satisfy hp ~ h'p r e l E
Then h ~ h'. (An immediate consequence of (a) here is a well-known result
-
proved by several authors
487
-
(Ganea, Hilton, Meyer, Nomura) to the
effect that if (4.3) is a fibration with X
(m - 1)-connected
(m ~ 2) and with the homotopy groups of F = ~W zero outside a band of width m - l, then the fibration is equivalent to one induced by a map h: X ~ W; see [3, P. 81].) Proof.
The diagram for 4.4 (a) is the following special case
of (1.1).
F
) *
)W
)*
)*
;*
(4.5) E By Prop. e
:
3.5(b),
l(X)
P
since W is l-connected, ~ e
l(W) is
necessarily trivial.
is non-void,
where
The vanishing of all
obstructions follows at once then from Theorem 3.3, the hypotheses of (a) above and the exact sequences
(3.1) and (3.2), so that
diagram (4.5) has a factorization h.
By Theorems 2.3 and 2.6
(since i is a cofibration),
hp ~ f rel F.
Part (b) follows in corresponding fashion from the analogous results for the homotopy of two factorizations;
see Remarks 2.8
and 3.6 above. The other theorems in [3] follow similarly from appropriate specializations of diagram (i.I). 3) Finally, we look at a theorem of Dold [i] on fiber homotopy equivalences: Theorem 4.6
Let ~ be a map of one regular Hurewicz fi_bra-
tion into another over the same Nase
- 488
(4.7)
-
F
~IF
) F'
y
a
~ y,
B
)g
We suppose all spaces path-connected the homotopy-type morphisms
of CW-complexes.
vj(F) ~ vj(F')
and also that Y and Y' have If ~IF : F ~ F' induces
iso-
for all j, then ~ is a fiber homotopy
equivalence. Proof.
The diagram is now
*
~ Y
~Y
)Y'
(4.8)
Since we may identify Vj+l(~) duces isomorphisms
= vj(F),
vj(~) ~ vj(~')
Vj+l(~' ) = vj(F'),
~ in-
for all j, and therefore
(by
(3.2)) all homotopy groups of ~hm right hand square vanish. existence exists
of the factorization
then by Prop.
3.5(b)
The
u' as shown by the dotted line
and Theorem 3.3.
By (c) and (d) of Theorem 2.6, G' may be so chosen that ~'
= ~', ~'~ ~ 1 rel ~ and ~ '
the assertion
~ 1 rel ~'.
This is precisely
of the theorem.
Remark 4.9
If Y and Y' are CW-complexes
then by the same argument it is enough to suppose
and ~ is cellular
(see the remarks preceding
Theorem 2.6)
that ~ and ~' are weak fibrations
here.
-
4 8 9
-
References
[i]
A. Dold, Uber fasernweise Homotopie~quivalenz Math. Zeit. 62(1955),
[2]
von Faserr~umen,
lll-136.
P. J. Hilton, Homotopy Theory and Duality, Gordon and Breach, New York, 1965.
[3]
P. J. Hilton, On excision and principal fibrations,
Comment.
Math. Helv. 35(1961), 77-84. [4]
I. M. James, On H-spaces and their homotopy groups, Quart. J. Math. 0xford (2), 11(1960),
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann...
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann...
OX FO RD LO G IC G U ID E S
Series Editors D.M. GABBAY A.J. MACINTYRE D.S. SCOTT
OX FO RD LO G IC G U ID E S Availabl...
Category Theory Steve Awodey Carnegie Mellon University
in memoriam Saunders Mac Lane
Preface Why write a new textbo...
OX FO RD LO G IC G U ID E S
Series Editors D.M. GABBAY A.J. MACINTYRE D.S. SCOTT
OX FO RD LO G IC G U ID E S Availabl...
OX FO RD LO G IC G U ID E S
Series Editors D.M. GABBAY A.J. MACINTYRE D.S. SCOTT
OX FO RD LO G IC G U ID E S Availabl...
