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of the rules
functoriality
states,
is e x t e r n a l express
the
of
= £I)
®
,
a n d in-
composition. fact
there
is an i s o m o r p h i s m
is a g r e a t
Hom(V®V',V")
+ Hom(V, (V',V"))
< p((V',V"), similarly
(V,V'),
b y means of
((v®v')®v",-)
~
and one l a w says t h a t
(V, V"))>o
(c(V,V',V"))
p, c a n d s .
(v®v'(v",-))
~
axiom which
Another
expresses
(v,(v',(v",-)))
The o t h e r k i n d of axioms mous pentagonal
= d(V,V',V")
~
v®(v'®(v"®v"') ) ÷
a
Another in terms
(v,(v'®v",-))
are the c o h e r e n c e
expresses
.
axioms
the c o m m u t a t i v i t y
~
expresses of
p
:
(v®(v®v"),-)
examplified
d
.
by MacLane's
fa-
of
(v®v')®(v"®v"' ) ÷( (v®v')®v")®v"'
\
/
V® ( (V' ®V" )®V'") ÷ (V® (V' ®V") ®V"' (1.3)
A category
such t h a t
A
is e n r i c h e d
Hom(I,v(A,B))
composition
over
is n a t u r a l l y
V
if there
equivalent
is a f u n c t o r
to
Hom(A,B)
V(-,-):AOPxA
+ V
Also required
is a
map
V(B,C)®V_(A,B) lying above the c o m p o s i t i o n are t a b u l a t e d
of m o r p h i s m s
in 1.5 of [ Eilenberg,
in
+ V(A,C)
A
Kelly] .
as w e l l as m a n y c o h e r e n c e
-®-:VxA
is a functor
such t h a t
Provided
has,
A
by the a d j o i n t
and
-®A:~
+ A
~(A,-)
functor
has a r i g h t adjoint,
is,
commutes
theorem.
denoted
÷ A
for e a c h with
Analogously
[-,A] :V Op
axioms w h i c h
A tensor
A(A
limits,
, left a d j o i n t this
it f r e q u e n t l y
+ A
to
~(A,-) :A ÷ V
.
can u s u a l l y be shown to exist happens
(note the s w i t c h
that
~(-,A):A__ ° p ÷
in variance)
w h i c h deter-
mines a b i f u n c t o r
[-,A] :V °p x A c a l l e d a cotensor.
We h a v e
Hom(V,V(A,B))
(1.4)
Let
a function
S Sn
be a set a n d + S .
If
a m o d e l of or a l g e b r a n wS:S n + S for e a c h ~ such that for all
+ A
n
~ Hom(V®A,B)
be a cardinal
~ Hom(A,[V,B] ) .
number.
An
n-ary o p e r a t i o n
on
S
is
~ = {~ } is a class g r a d e d b y the c a r d i n a l s n of sets n for ~ is a set S e q u i p p e d w i t h an n-ary operation A morphism
n
n , all
~
n
Sn fn
f:S + T
of m o d e l s
of
~
is a f u n c t i o n
, the square mS
~
S
~T
~
T
1 Tn
I
f
commutes. (1.5)
a m o n g the e l e m e n t s
in n
are a s s u m e d
to be c e r t a i n
~. , iEn 1
where
realization
in any a l g e b r a
is the p r o j e c t i o n
operation
w
whose
and
to the ith coordinate.
is an n - a r y operation,
value on any a l g e b r a
t h e n there
The c a t e g o r y
of a l g e b r a s
most readily be formalized maps
include
(wi) ~
and morphisms
by building
all the f u n c t i o n s
In p a r t i c u l a r
m
A full
quotients example
inverse
limits.
category
of a v a r i e t y
algebras
in the variety.
V
If
V
under quotients.
is itself F
hence
F£V
.
object
Thus
in
W'
V
in w h i c h
understand
V
(1.9)
is free V ---+
Now suppose w ,
W
W'
,
in
W'
in
F
.
W'
the free
the closure
of a
is a q u o t i e n t
VEV
is unique
.
Since V
and
s h o w that
Henceforth V
of
W'
since every
The same a r g u m e n t w o u l d
and t h a t
of
so that
is a s u b o b j e c t
variety
but closed
to be a full sub-
of quotients
is also unique.
a containing
and
is an
contains
of a s u b o b j e c t
. Actually
V
groups
W'
it is a q u o t i e n t
and h e n c e
products
.
that it h a v e these limits
we m a y c o n s i d e r
that a product
[Linton]
of a v a r i e t y
and which
is a q u o t i e n t
is e m b e d d e d W
merely
trans-
and p r o d u c t s
a semi-variety
and equalizers
splits
of an o b j e c t
a semi-variety
W =
We d e f i n e
in
F
and
free a b e l i a n
to be a full s u b c a t e g o r y
argument
is a s e m i - v a r i e t y
of
can whose
is a c o n t r a v a r i a n t
is a n a t u r a l
u n d e r subalgebras,
of a torsion
in a v a r i e t y
to b e a s e m i - v a r i e t y ,
the free a l g e b r a s
An a l g e b r a
of algebras
is c l o s e d
of a q u o t i e n t
FEW' map
is a q u o t i e n t
the v a r i e t y
operation
If
This n o t i o n
are cardinals,
closed under subalgebras
under p r o d u c t s
It is a s t a n d a r d
the q u o t i e n t
of i.
t h a t w e are not r e q u i r i n g
and a subobject
a variety.
is free,
objects
m a y be found in [ L a w v e r e ]
subcategory
is a q u a s i - v a r i e t y
of the p r o d u c t
Details
in the variety.
closed
whose
and a morphism
a quasi-variety
Notice
S
m
The c a t e g o r y
full
b u t that they be the limits
(1.8)
functor
is a g a i n a variety.
We define
mi-ary
m = Zm. , i
n
of a variety which
of a n o n - v a r i e t a l
is an
and such t h a t
such functors.
subcategory
n o t quotients. under
set v a l u e d
~
is c a l l e d a variety.
is the s u m of t h a t m a n y copies
formation between (1.7)
Sn
a category
H o m (re,m) =
product preserving
(~i) , i£n
is the c o m p o s i t e S m ~ ~.smi 1
(1.6)
If
is an m - a r y operation,
w e will contains
.
t h a t the t h e o r y
m-ary operation
~
~
is c o m m u t a t i v e .
That means
and algebra with underlying
set
that for a n y n - a r y S , the square
n S n x m ~ Sm x n ~
Sn
n 6d Sm
commutes.
The
w
as a c t i n g on row vectors,
and
~
respectively
isomorphism
in the u p p e r
a n d the i s o m o r p h i s m
of this square
is t h a t
~
~
~
S
l e f t is m o s t e a s i l y
Sn×m
and
as transpose.
is a h o m o m o r p h i s m
Sm x n
as
described by thinking nxm
and
The i m p l i c a t i o n
with respect
to
~
mxn
of
matrices
of the c o m m u t a b i l i t y (or v i c e versa).
If this h o l d s
for all
on the u n d e r l y i n g (i.i0)
Suppose
and
are algebras,
V
operation,
~,~
the result
is t h a t each o p e r a t i o n
sets b u t is in fact a m o r p h i s m
in a d d i t i o n
that
(fi),
is a v a r i e t y
is a g a i n a m o r p h i s m . diate
that
that
~(V,-) I
closed
category.
~
~
FS
But since
(1.12)
Then
category.
FT
VE~
and it is imme-
(V',V)
It is easily
theorem,
seen
has an a d j o i n t
T h e n the r e q u i r e d
natural
-®V
.
trans-
as the c a n o n i c a l
closed
structure
- but
In fact,
let
for a c o m m u t a t i v e ~0
theory
be the c a t e g o r y
of al-
as above a n d w e w i l l W~
.
Then
W
-
see
has a
to
.
FT--~
on the sets
T h e n there
W(W,V)
~
w(w,v)
,
S
and
T
respectively,
and,
as
is an e q u a l i z e r
W(FT,V)
is a q u a s i - v a r i e t y ,
and
W
~
_W(FS,V)
',
vs
VT
and
Vs
and hence
W(W,V)
all b e -
V. Whether
o r n o t the theory
Here
structure
b e the c a t e g o r y
~ H]-modules.
regardless of K - l i n e a r
I
~
are the free a l g e b r a s
(1.7), b e l o n g
is c o m m u t a t i v e ,
is the e x a m p l e w h i c h
For
V
W has the s t a n d a r d s t r u c t u r e --0 ideal. In fact, let V~ and
closed
as
an n-ary
to s h o w that we have a s y m m e t r i c
is o n l y a s e m i - v a r i e t y
d a r d one. V
~
theory.
metric monoidal
and
and
V'
If
as a c o e q u a l i z e r
and in
long to
theory.
on Hom(V',V)
functor
on one generator.
FS
observed
a function
V
an a l g e b r a w h i c h we call and, b y the a d j o i n t
it into a c l o s e d
is an e x p o n e n t i a l
presentation
where
V' --+ V
~(V',V)
This m a y be d e s c r i b e d
in case t h a t
for the theory.
~
m a y e a s i l y be c o n s t r u c t e d
to a c o m m u t a t i v e
Even
Vn
an o p e r a t i o n
be the free a l g e b r a
w e can still m a k e
that
limits
and e q u i v a l e n c e s
corresponding
gebras
is again
preserves
let
(i.ii)
This d e f i n e s
Hom(V',V)
formations monoidal
for a c o m m u t a t i v e
a family of m o r p h i s m s
ion
is not m e r e l y
.
the c o m p o s i t e (fi) V' ~
Finally
q
in
V'®V is just
there
on the s e m i - v a r i e t y interests
of r e p r e s e n t a t i o n s
maps w i t h
symmetric
~ action
with
each g r o u p e l e m e n t
o b j e c t on one g e n e r a t o r
H
which means
K
which H
on K-vector
structure.
with
for
t h a t there
differs
spaces, according
For x~H,
x(v~v')
t h a t the f u n c t o r
K
as
f:V'--+ V
= xv®xv'.
Hom(I,-)
a field
otherwise
~(V',V)
a c t i n g by the i d e n t i t y map.
is a sym-
f r o m the stan-
be a group,
is c o m m u t a t i v e
monoidal
over
V Let
(xf) (v') = xf(x-lv ')
w e take the t e n s o r p r o d u c t K
us.
of
The theory of this v a r i e t y is a c l o s e d
it is p o s s i b l e
H
known
is, b u t
take the set and
v'£V ~ .
The unit o b j e c t
This
is not the free
is n o t the usual u n d e r -
lying set functor. (1.13)
Suppose
that
V
is a s e m i - v a r i e t y
a n d also a c l o s e d
symmetric
monoidal
category.
The v a r i e t a l
structure
free a l g e b r a
on one g e n e r a t o r
between
the internal
gives an u n d e r l y i n g
w h i c h we denote
and external Hom(V',V)
and that these
inclusions
between
The e x i s t e n c e
sets.
the h y p o t h e s i s an i n j e c t i o n
that
J
provided
functor
hom
J
these o b j e c t s
c
cohere w i t h the u n i t a r y of the r e q u i s i t e
be a c o c o m m u t a t i v e
ture a n d that b o t h the e x t e r n a l those u n d e r l y i n g
(1.14) in
of
< V(V',V)
- and
vast simplifications like,"There course
between
>
a map
In these notes,
awkward.
An a d d e d a d v a n t a g e
Example.
spaces
and binary operations u n a r y operation, category. f,g
we will
V'
for
unit object
K.
K
canonical
map,
without
functor
V
and equations
the t h e o r y
are maps,
required
by
f+f'
: V~+
for this
hom
is
K
(2.1)
Let S
S
b e a set.
whose union
Let is
u =
S .
(S 8) be a
For
xES,
are c o n t a i n e d
to define
If
for
say things
Hom(
V
is n o t the
b u t as long as
of it. throughout V
denote
the c h a p t e r s the c a t e g o r y
It has the n u l l a r y
an a b e l i a n .
This
is to
unbearably
to be able to apply
uniformity)
field and
group as w e l l
O, u n a r y as a
It is w e l l k n o w n to be a c l o s e d
and
is d e f i n e d b y
Uniform
>
vanish.
continue
V'
are v e c t o r
spaces
(f+f') (v) = f(v)+f' (v)
the o p e r a t i o n
a n d the t e n s o r p r o d u c t
2.
sets of
V'
defines
are func-
For later use, w e
t h a t at the u n d e r l y i n g
questions
lcK
struc-
W h a t this m e a n s w i l l of
f u r t h e r mention.
I , for each
l £ K , (If) (v) = lf(v)
I>,
it means
a variety.
is commutative.
hom
do.
- of m o s t of the p r o p o s i t i o n s
(commutative)
seem
in m i n d is that
.
(that is b a s i c a l l y
is a l r e a d y
is
in this case the t r a n s p o s i t i o n
and a map
the p r o o f s
be a
Then
first m a p is hypothesis
in the future,
we m a y as w e l l take a d v a n t a g e
multiplication
In fact,
: V--+
milarly,
Let
over
The
~(B, (A,E))."
Here is the e x a m p l e w h i c h we w i l l
d e v o t e d to the theory. of v e c t o r
V
c Hom(<~
(2.5) b e l o w and to m a k e sense of the idea of a c o n v e r g e n c e
(1.15)
.
from
Nor does the q u e s t i o n
and
~(A,B)
is t h a t all c o h e r e n c e
w e s e e m to b e stuck w i t h one,
V
functions
Hom(V,V')
a rigid grounding
for functions
of the internal
But in all cases
- not to m e n t i o n
in
V '----+
Specifically,
H o m ( < IBI>,
maps
What additional
theway
from
~(A, (B,E))---+
statement.
isomorphism
sets that are very m u c h p a r t of their
that
this w i l l be u n d e r s t o o d
for a d o p t i n g
object
is not clear.
reference
it comes d o w n to a w e l l - k n o w n
a coherence
can be shown to follow
The m a i n p o i n t to k e e p
a pseudomap
map
b y the
.
and a s s o c i a t i v e
a n d the e l e m e n t s
in o u r theory.
is a c a n o n i c a l
m a k e the s t a t e m e n t s
reason
hom
for future
d e p e n d on the e x a c t
set level
,
functions
sets and c o m p a r e
The fact t h a t w e are s u p p o s i n g
Hom(
We w i s h to d e s c r i b e
coalgebra
stage in the theory.
is r e p r e s e n t e d
say that up to n a t u r a l
the counit m a p is an epimorphism.
at this
call an e l e m e n t
which
c Hom(
come p r o v i d e d w i t h u n d e r l y i n g
tions b e t w e e n
.
which will
n e e d e d to force the s e c o n d to be an i n j e c t i o n worth pursuing
<->
of
~
on
Hom(V,V')
and
. Si.
The
is the usual one.
Spaces.
cover
let
of
u*(x)
S, t h a t = U{UEU
is a c o l l e c t i o n
of s u b -
Ix~u }. Let u* ={u*(x) Ix~S}.
If
u
and
v
are covers,
such t h a t
x£ucv
that
implies
u£U
.
the e x i s t e n c e
called a star r e f i n e m e n t structure
U
on
w e say that
Then a uniform
S
of
u
of a
U.
refines
structure v~U
A pair
on
v
S
if for
such that
(S,U)
x6v~v
there
is a c o l l e c t i o n v*
consisting
w i l l b e c a l l e d a u n i f o r m spa~e.
U
refines
u
of a set
S
The covers
in
is a
u~u
of covers .
Such a
such v
is
and a u n i f o r m U
are c a l l e d u n i -
form covers. If
(SI,U_I)
and
if for all on
S
($2,U_2)
~2£U2
is c a l l e d
there
is
separated and
are u n i f o r m
contains both
x
y
For a t h o r o u g h
discussion
~I£UI
x~y
in
S
of u n i f o r m
spaces
I a n d pp.
28-29 of
X =
(S,U)
f:S 1
f-l(u_2)
there
F r o m n o w on w e w i l l
is a
suppose
including
tion g i v e n h e r e a n d t h a t g i v e n b y entourages, chapter
a map
which refines
if for
.
spaces
.
S2
is c a l l e d u n i f o r m
A uniform
u£U
str~cture
such t h a t no set in
all u n i f o r m
spaces
the r e l a t i o n b e t w e e n
see [ Isbell]
is a u n i f o r m
~
space,
, especially write
S =
are separated. the d e f i n i -
problem IXl
8 of
for the
underlying set. (2.2)
If
(S,~)
is a u n i f o r m
w i t h the same p o i n t
u~U .
This
tinuous
topology
uniform
spaces,
R ~
(see below)
(2.3)
If
uniformity
As u n i f o r m
is a u n i f o r m
i n d u c e d on
form t o p o l o g y uniform
.
topology
on
S'
spaces
in the i n d u c e d
S'
f'
by
(S,U)~
(2.5)
in
space
dense
space
all the sets
(S,T)
u*(x)
E v e r y u n i f o r m m a p induces
where
a con-
In p a r t i c u l a r
For instance,
t h e y are i n e q u i v a l e n t
S'
is a subset
the c o l l e c t i o n on
S'
two
as t o p o l o g i c a l
since the former
of
of all
S , there
uPS'
for
is the same as t h a t
(Sl, U_l) is c o m p l e t e
subset
uniformity,
[ Isbell]
or just b y
St
if
S'cS
there
is com-
is a n a t u r a l
u~ucU
. The u n i -
i n d u c e d b y the
if for e v e r y u n i f o r m function
f:S --~
II h o w e v e r y u n i f o r m
as a dense
U
and every
is a u n i f o r m
, chapter
is u n i q u e up to a unique
Lemma.
x
spaces b u t not conversely.
space a n d
(i.e. h a s the i n d u c e d uniformity) embedding
topology.
a topological
at
S .
uniform
It is shown
base
is not.
Namely
We say that the u n i f o r m
(2.4)
is a s s o c i a t e d
the same topology.
for the i n d u c e d u n i f o r m i t y
(S,U), e v e r y t o p o l o g i c a l l y
is
topological
m a y induce
b u t the latter
(S,U)
there
is c a l l e d the u n i f o r m
structures
(-i,i)
space,
Take as a n e i g h b o u r h o o d
one on the a s s o c i a t e d
distinct
plete
set.
S1
We d e n o t e
whose
space which
is
restriction
to
space can be e m b e d d e d
s u b s e t of a c o m p l e t e
isomorphism.
f':S'--~ S 1
space a n d t h a t this
this c o m p l e t i o n
of
(S,U)
is understood.
Suppose fl (SI,U_I)
(TI,V._l) is a c o m m u t a t i v e
d i a g r a m of u n i f o r m
~
f2
spaces
($2, ~ )
~ (T2,V2) such that
fl
is the i n c l u s i o n
of a dense
subset, e q u i p p e d w i t h the i n d u c e d uniformity, and
f2
is an i s o m o r p h i s m of u n d e r l y i n g sets
~ i has a b a s i s c o n s i s t i n g of sets w h o s e image u n d e r
topology of
~2
"
Then there is a unique map
f2
($2,U2)--+
is closed in the u n i f o r m
(TI,V_I) m a k i n g b o t h trian-
gles commute. Proof.
A l t h o u g h I have t r i e d to m a k e these notes l a r g e l y self-contained,
this a r g u m e n t
requires m o r e than a p a s s i n g a c q u a i n t a n c e w i t h the theory of u n i f o r m spaces to be understood.
For this I r e f e r to [ I s b e l l ]
.
Otherwise,
it can b e s t b e a p p r e c i a t e d b y ima-
gining the spaces to be m e t r i c and r e p l a c i n g C a u c h y n e t s by C a u c h y sequences, n We m a y suppose w i t h o u t loss of g e n e r a l i t y that tity m a p w h i c h m e a n s that w i t h r e s p e c t to and hs
{s } .
~2
There is
g
is uniform,
~{C~l
v Iv
Vev_{
~*
8
f i n i t i o n of C a u c h y net).
c o v e r of
ya8
{gsy},YaB
~2
hsEV
and thus converges to
to
hs
T hs
s£S 2
a C a u c h y net
as above
{gs }
s .
(2.6)
e>0
implies
Since
h
gs £V
V
Vc~*(hs).
of e l e m e n t s of
~2
"
~iE~l .
There is a
and converges,
Hence for all y>B,gs £v{*(hs).
~i
b u t we may argue as follows.
S1
constructed by choosing
w h i c h converges to it.
Then
and a uniform map Such an
f
f:X--~
X = (M,~)
In fact the covers
(S,~)
and any
f
u_EU
It follows there is a
such that the inverse image of
is c a l l e d a p s e u d o m e t r i c for
a c o r r e s p o n d i n g c o l l e c t i o n of
(T,VI)~.
T .
a u n i f o r m i t y c a l l e d the m e t r i c uniformity.
(M,~)
U
~2
is a u n i f o r m space in a natural way.
m e t r i c space
u .
N o w let
(T,V_I) and converges to a unique point, g~(s), of
determine
the 1-spheres refines
converges to "
(that is just from the de-
g~:(S2,U2)---+ (T,VI)
, 1.14 that for any u n i f o r m space
for
~i
is c l o s e d in that t o p o l o g y and e v e r y
is u n i f o r m to
from [ Isbell]
o v e r a basis of
{hs }
s~S 2
the net is e v e n t u a l l y in one of the n e i g h b o u r h o o d s of
{s }
(M,~)
is u n i f o r m
.
is C a u c h y in
A m e t r i c space
e-spheres,
h
Suppose n o w that
Then
But we h a v e just shown t h a t all such p o i n t s a l r e a d y lie in
by
is the iden-
and b y the h y p o t h e s i s we m a y also suppose
Thus
has a u n i f o r m e x t e n s i o n to
for each
"
is a g a i n a C a u c h y net in
.
as well.
This does not q u i t e show that g
~i
f2
T h e n g i v e n that ~i
is a C a u c h y net w i t h r e s p e c t to refines
Now
gs ~V , it follows that
The map
"
w h i c h c o n v e r g e s to
such that
in the u n i f o r m t o p o l o g y of
~i
~2
and
are c l o s e d in the u n i f o r m t o p o l o g y d e t e r m i n e d b y
a n d an index
Thus in every
S1
{gs }
such t h a t
that the sets in
hs
is a r e f i n e m e n t of
w e m u s t show it is w i t h r e s p e c t to
is a C a u c h y net in
Since
set
~i
T1 = T2 = T
X .
If
u
ranges
is c a l l e d a b a s i s of p s e u d o m e t r i c s
X .
(2.7)
Suppose
S--+
T .
u£U
(S,~)
We say that
w h i c h refines
and F
(T,~)
are u n i f o r m spaces and
F
is an e q u i u n i f o r m set if for all
f-l(~)
for all
f~F .
is a set o f functions v~V
there is a single
It is easy to see that every finite set of
u n i f o r m functions is an e q u i u n i f o r m set. (2.8)
If, on the o t h e r hand,
o f sets of functions in
~
are e q u i u n i f o r m .
p s e u d o m e t r i c s for
V .
S --~
S
is a set and
(T,[)
is a u n i f o r m space, a c o l l e c t i o n
T determines a uniformity
U
on
S
such that
This is m o s t r e a d i l y d e s c r i b e d in terms of a b a s i s Let
the sets d
of
F(s,F,d)
= {s' [d(fs,fs') for
< 1
scS,F~,dEd
for all
f6F}
. T h e n the sets
{F (s,F,d)Is~S} are a cover of
S
which
pseudometrics
for this
sup{d(fs,fs')
If~F}
(2.9)
tions
S--+
F---+
T
T
.
structure
suppose
.
case
given a c o l l e c t i o n
so d e s c r i b e d
has the u n i f o r m i t y
in
[ Isbell]
, Chapter
(2.10)
If
X---+
is an i n j e c t i v e
is u n i f o r m l y
y
isomorphic
We note the o b v i o u s consisting
above
and e m b e d d i n g s
is a c o l l e c t i o n
by a pseudometric
pseudometric
and
in
where
X.
convergence
Xi
of u n i f o r m
possess
A basis
(d.F) (s,s')
S
and a set
of =
F
of
of sets of func-
structure
A special
on all o f
on
F
case S
.
We
is
~ = {S}
t h a t is con-
provided
is g i v e n the i n d u c e d
spaces
that
uniformity.
has a f a c t o r i z a t i o n
and monomorphic
the usual p r o p e r t i e s
(see [ Kelly]
of u n i f o r m
constructed
Pi:X---~
of
a uniform
uniformity.
as the e p i m o r p h i c
the e m b e d d i n g s
(2.11)
b e r of indices
vary.
u n i f o r m m a p we say it is e m b e d d i n g
a n d left c a n c e l l a t i o n
{X }
d-F
d
as a c o l l e c t i o n
to its image w h e n t h a t image
In p a r t i c u l a r ,
If
and
III.
under c o m p o s i t i o n
majorized
F
of subsets Z
describes
of u n i f o r m
fact t h a t the c a t e g o r y
of s u r j e c t i o n s
spectively.
~
a convergence
sidered
X
as
is g i v e n b y the f u n c t i o n s
T h e n the d i s c u s s i o n
F
structure
In t h a t case we can think of
w i l l call a u n i f o r m i t y in w h i c h
a uniform
.
We can instead
functions
determine
system
parts,
re-
of i n v a r i a n c e
).
spaces,
as follows.
be the c o r r e s p o n d i n g
every p s e u d o m e t r i c
on
Let
be a finite
i=l,
...
projective
, n
map.
X = HX
Let
d. l
is num-
be a
a n d let
d(x,y)
= s u p ( d l ( P l X , p l y ) , d 2 ( P 2 x , P 2 y) ..... dn(PnX,pny))
In the sequel we w i l l w r i t e
d = sup{di(Pi,Pi)}
m i t y d e f i n e d b y the p s e u d o m e t r i c s
so d e f i n e d
.
.
It is e a s y to show that the u n i f o r -
is the c o a r s e s t
for w h i c h the p r o j e c t i o n s
are uniform. (2.12)
Proposition.
discrete Proof.
uniformity. X
Let
{X }
has a u n i f o r m
q u i r e d to b e a u n i f o r m
and
V
set into a single p o i n t then (2.13)
f((x
)) = f((x'
form.
= X
Notes o n u n i f o r m
inverse
T h e r e are really g i v e n here.
and the t h i r d b y e n t o u r a g e s
(see [Kelley],
as the m o s t u s e f u l
if f
x
sets
That
= x'
factors
except
for things
The first
every such
for ~=l,...,n,
~ X ----~
of u n i f o r m
is b y families
~ i ..... ~ n
where
takes
through
is re-
is r e f i n e d
KV
f
three d e f i n i t i o n s
VI).
h a v e the
n u m b e r of indices,
of all
The s e c o n d
Chapter
X
image of that c o v e r
a cover.
that
that
and
a finite product.
cover of a p r o d u c t
be a finite
determines
in particular,
covers,
as w e l l
The
spaces
through
A uniform
1 .... ,n
in o t h e r words,
spaces.
of u n i f o r m
factors
T h e n the c o l l e c t i o n
otherwise
implies,
)), or,
Let
space.
The first is b y the u n i f o r m
vely geometric
X
cover b y singletons.
in the c o r r e s p o n d i n g
V £u , ~ = l , . . . , n
HX----+
cover of the product.
by a c o v e r of the f o l l o w i n g be covers
be a c o l l e c t i o n
T h e n any m a p
XlX...xX nspaces.
of s e m i n o r m s
is the m o s t i n t u i t i -
like t o p o l o g i c a l
groups where
each
neighborhood of
1
over the group.
This a u t o m a t i c a l l y d e f i n e s a u n i f o r m i t y - the c o n t i n u i t y of m u l t i p l i -
gives rise to the c o v e r gotten b y t r a n s l a t i n g that n e i g h b o r h o o d all
cation is e q u i v a l e n t to the e x i s t e n c e of star r e f i n e m e n t s - c a l l e d the c a n o n i c a l uniformity. A g r o u p h o m o m o r p h i s m is c o n t i n u o u s iff it is uniform.
Thus there is no dis-
tinction in t h a t case b e t w e e n t o p o l o g i c a l and u n i f o r m notions.
The d e f i n i t i o n b y p s e u d o -
norms seems the m o s t useful in these notes.
The d e f i n i t i o n by e n t o u r a g e s is p r o b a b l y
most useful in c o n n e c t i o n w i t h c o m p a c t n e s s arguments. canonical - and unique - ~ i f o r m i t y
A c o m p a c t H a u s d o r f f space has a
for w h i c h the e n t o u r a g e s are all n e i g h b o r h o o d s of
the diagonal. E v e r y m a p to a n o t h e r u n i f o r m space is c o n t i n u o u s any rate there is no "best" definition.
iff it is uniform.
At
It is i m p o r t a n t that all three are equivalent.
3. U n i f o r m Space O b j e c t s
Let
(3.1)
V
b e a s e m i v a r i e t y w i t h u n d e r l y i n g set functor
ture on an o b j e c t
V~V
we m e a n a u n i f o r m structure on
tions are u n i f o r m w h e n the p o w e r s o f (see [ Isbell] If ~I
and the
V
is a p r e u n i f o r m object in
V
let
denote the u n d e r l y i n g u n i f o r m space
V
has a p r e u n i f o r m s t r u c t u r e
the u n i f o r m c o m p l e t i o n of
consider
the case that
V
V
V n ~-q~
V
~
V~
Then
n ,
Vn
V
V .
V .
V
(3.3) that
N o w suppose that
< I V I>
are
(in
V)
is dense in
V~
(V~)n
(we omit the < > ) .
For any n-ary ope-
V
V~ n
~
V~
is s a t i s f i e d w h e n r e s t r i c t e d
But an e q u a t i o n is s a t i s f i e d e x a c t l y w h e n two maps are equal
V~ V
and
is c o m p l e t a b l e
has, then, an e x t e n s i o n to a m a p
and two u n i f o r m maps w h i c h agree on a dense subset are equal. w h a t is at issue is that
I< V > I
To e x p l a i n this more clearly,
is dense in
A n y e q u a t i o n r e q u i r e d to be s a t i s f i e d b y an a l g e b r a of to the dense subset
Both
U , w e say that
also lies in
is a variety.
and it is r e a d i l y seen that for any ~ , the m a p
V .
V .
provided
ration
are e q u i p p e d w i t h the p r o d u c t uniformity.
denote the u n d e r l y i n g d i s c r e t e o b j e c t o f
If
By a p r e u n i f o r m struc-
such that all the opera-
, C h a p t e r II.)
same u n d e r l y i n g set of
(3.2)
<->.
b e l o n g to the s e m i - v a r i e t y is c l o s e d
In the general case,
V
(by w h i c h we u n d e r s t n a d that the s i t u a t i o n is
e x p o s e d in section i, in p a r t i c u l a r that we have a symmetric m o n o i d a l structure).
We say that object
V
is a d m i s s i b l e p r o v i d e d for all p r e u n i f 0 r m o b j e c t s
U n ~ ( V ' , V ) of
~(V',V)
such that
< UnV(V',V)
>---+
< V (IV' I , IVI)>--+ is a pullback.
V' , there is a sub-
Here, of course,
Un(
Hom(
Un(
is the set of u n i f o r m maps b e t w e e n
these sets. (3.4)
If the theory o f the s e m i - v a r i e t y
V
has the i n d u c e d c l o s e d m o n o i d a l structure, is a -Th-algebra.
For
~ V = ~ :Vn
+
V
is the c o m m u t a t i v e theory it is easy to see that
Th
and
Un ~ V'>
is an o p e r a t i o n w h i c h w e have s u p p o s e d to be
10
uniform,
t h e n for any c o l l e c t i o n
f.:V'--~ 1
V,
i£n
(fi) ÷
Vn
~
V' is also a u n i f o r m morphism. Thus the a d m i s s i b i l i t y belongs
to
(3.5)
of
V
Th
e(f
),
V
the
d o w n to w h e t h e r
is not n e c e s s a r i l y
monoidal
structure,
and
are two s u b o b j e c t s
of
comes
is
~
the c o m p o s i t e
Th
operation
in
V(V',V)
or not t h a t p a r t i c u l a r
.
_Th--algebra
V
Even when
V1
But this
, of u n i f o r m morphisms,
.
Then both
at the u n d e r l y i n g lows that
there
is at m o s t one c a n d i d a t e of an o b j e c t
inclusions
set.
commutative
Since
V2
VoNV 1
for
V
~
VO
and
< V0 >
For suppose
=
Vo~VI ---+
set functor
Thus the s u b o b j e c t
has some ad hoc closed
Un ~(V',V)
such that
the u n d e r l y i n g
V O = VoNV 1 = V 1 .
and
< Vi >
V1
reflects
Un ~(V',V)
V0
as subobjects
become
equalities
isomorphisms
is unique,
it fol-
provided
it
exists. (3.6)
We say t h a t
missible,
V
completable
tegory o f u n i f o r m follows
objects,
that a m o r p h i s m
the u n d e r l y i n g
(3.7)
V
is a u n i f o r m o b j e c t and its c o m p l e t i o n
If
~
natural
closed monoidal structure
structure
Thus there
is,
thereby
it is not true,
unless
uniform
iff it is a f u n c t i o n of o n l y f i n i t e l y {U }
be a collection
uniformity
for e a c h
i
and
let
I)>
both
shows
dense
in
U
it
in
(3.4)
spares
every pre-
structure
is
and uniform.
that every o b j e c t
uniformity.
In fact,
to a d i s c r e t e
in
it
space
is
m a n y coordinates. Let
U = KU
the p r o j e c t i o n s V
t h e o r y w i t h the
are uniform.
be a p r e u n i f o r m
1
Un(
>
Hom(< IVl> ,
I-I , < - >
,< U
e q u i p p e d w i t h the
object.
I claim that
T h e n since
> )
I>)
a n d the h o m functors
commute with
so is
Evidently
the ca-
and u n i f o r m on
between preuniform
is finitary,
of d i s c r e t e
for w h i c h
To see this, < U n V ( V , U )>
products,
V
every p r e u n i f o r m
the d i s c r e t e
of u n i f o r m objects.
- the c o a r s e s t
is a u n i f o r m object.
(3.2)
the t h e o r y
f r o m 2.12 that a m a p f r o m a p r o d u c t
U
if it is addenote
F r o m the d e f i n i t i o n in
t h e n as o b s e r v e d
in
in that case no d i s t i n c t i o n
follows
product
Un V
for a c o m m u t a t i v e
induced,
a n d as o b s e r v e d
a uniform object when equipped with
Let
as the hom.
is b o t h a m o r p h i s m
of all the a l g e b r a s
is a d m i s s i b l e
Even in that case,
(3.8)
object,
We let
sets.
uniform
becomes
< Un V ( - , - ) >
is simply one w h i c h
is the c a t e g o r y
completable.
with
if it is a p r e u n i f o r m
is adimissible.
that
HU ~ U~
U
is admissible.
a n d the l a t t e r is a d m i s s i b l e
is the p r o d u c t
of the
U
) >
I)>
,
..... H o m ( < l VI>,
Moreover
is c o m p l e t e
being
Un(
U
so t h a t
a product
> )
is dense
in
U ~ = HU ~
of a d m i s s i b l e
1> U
from which
and
U
objects.
U
is
is completable. It is e v i d e n t
that
11
(3.9)
N o w let
U'
I
U"
be two m a p s of u n i f o r m o b j e c t s and
g i v e n the structure as a subspace of to show that
U
m a t t e r entirely.
is admissible.
U'.
U
be t h e i r equalizer,
E x a c t l y the same a r g u m e n t as above suffices
To show that it is c o m p l e t a b l e is, however,
It is easy to see that
U~
is a subobject of
U'~
n o t seem to b e any o b v i o u s reason that it is a r e g u l a r subobject.
another
b u t there does
If w e s u p p o s e d that
was a q u a s i - v ~ r i e t y - c l o s e d u n d e r p r o d u c t s and subobjects - that w o u l d settle it. BUt in the one e x a m p l e w e h a v e in which (see IV. 3.) - ~
~
is not a v a r i e t y - that of Banaeh spaces
is not a q u a s i - v a r i e t y either. What h a p p e n s there is t h a t any c l o s e d
subobject of a u n i f o r m o b j e c t is a u n i f o r m object.
Thus the a p p r o p r i a t e h y p o t h e s i s at
this p o i n t is unclear for w a n t of e x a m p l e s and we leave it as an open question. (3.10)
Proposition.
The u n d e r l y i n g functor
I I : UnV---+
V
creates products.
v i d e d the c o m p l e t i o n of an e q u a l i z e r is an a d m i s s i b l e p r e - u n i f o r m o b j e c t of functor c r e a t e s limits. tisfied provided
~
If
W
is the c a t e g o r y of all algebras,
is c l o s e d in
ject u n d e r l y i n g a c l o s e d
UnW
W
This last c o n d i t i o n m e a n s that if of the u n i f o r m space w i t h
I U'I
~ , the
this c o n d i t i o n is sa-
under all subobjects or at least if the
s u b o b j e c t of an o b j e c t in U n V U~UnV
, and
~ W , then
b e l o n g s to
< U '> c
I U' Ic V
Pro-
W
ob-
V .
is a c l o s e d subspace
.
In the sequel, w e will simply suppose that the c o n c l u s i o n of this p r o p o s i t i o n is a u t o m a t i c a l l y satisfied. (3.11)
Let
V£~
and
U~Un~ .
We let
[V, U]
denote
sest u n i f o r m i t y for w h i c h the map e v a l u a t i o n at
~(V, IU I)
v: [ V,U] --+ U
e q u i p p e d w i t h the coar-
is u n i f o r m for each
v£V.
E q u i v a l e n t l y we require that < [V,U] > be a u n i f o r m embedding.
is a pullback.
Then
IV,U]
~
is c e r t a i n l y a p r e u n i f o r m object.
~
U n(
,
Hom(< ~' I>, < H V , U ] >
N o w I c l a i m that
] >)
)
We b e g i n w i t h the fact that < UnV(U' ,U)> ---+ U n ( < U ' > ,< U > )
t < V( IU'I,IUI > is a p u l l b a c k in
S
.
Applying
'
H o m ( < V %-)
H o m ( < V > ,
H o m ( < V > , < V( IU 'I ,IUI > ) is as well.
N o w since
Horn(< IU' I>, < lUl>) we see that Hom(
H o m ( < V > , H o m ( < I U ' l > ,
is d i s c r e t e
Hom(
~ Un(
> )
12
H o m ( < V > , H o m ( < I U ' I>,<] U I> )) ~ H o m ( < ]U' I>, I>). since b y < V(V,UnV(U' ,U))>
>
Hom(
< V(V,V( I U'I ,IU I > - - - + is a pullback,
Hom( < V > , V ( IU '] ,[UI ))
it follows that so is < ~(V,UnV(U,U))>--+
< V ( V , V ( [U'~}UI)> If
f:V --+
(U' ,V_(V,U))>
Un V(U',U)
--+
[V, U] .
'
Un(
gom(
c V(U',U)
is a pseudomap,
is a p s e u d o m a p U '---+ ~(V,U) =
f
so
mines a u n i f o r m map
<[ V,U] >)
UnV(U',U)>)
then
I[V,U]
I> ) . f£ < V ( V , V ( U , U ) ) > I •
<
Since it also deter-
it gives a u n i f o r m p s e u d o m a p
That is, the image of the u p p e r map in the last square lies in In a similar w a y the image of the lower map lies in
U'
Un(
H o m ( < ~' I>, < I[V,U] 1> )
w h i c h gives the d e s i r e d result. (3.12)
This not only shows that
for a c o t e n s o r
Un ~ ( U ' , [ V , U ]
[ V,U ]
is a d m i s s i b l e but gives the d e s i r e d a d j u n c t i o n
) ~ V ( V , U n V(U',U)) . However, we still have to show that
[V,U] is c o m p l e t a b l e and that its c o m p l e t i o n is admissible. that
U
is complete.
Then for
U'
dense in
U" , we k n o w
F i r s t w e c o n s i d e r the case (essentially b e c a u s e we h a v e
a s s u m e d it) that U n V(U",U) ---+ is an isomorphism.
Applying
~(V,-)
Un V(U',U)
w e see that
V(V, UnV__(U",U) ) ---+ V(V,UnV(U' ,U) ) or U n V ( U " , [ V , U ] ----+ U n V ( U ' , [ V , U ] ) is an isomorphism.
This implies that
trary w e use the h y p o t h e s i s of an a d m i s s i b l e p r e u n i f o r m
V--object as well.
is u n i f o r m i z e d as a s u b o b j e c t of (3.13)
Proposition.
Proof.
In the square
Let
[V,U]
U
is c o m p l e t e as well.
(3.10) to infer that the c l o s u r e of
[ V,U~]
.
be e m b e d d e d in
U
is arbi[V,U~]is
To see that w e n e e d o n l y o b s e r v e that [ V , U ] M o r e generally, we have the following U'
<[ V,U] > ---+
<[ V , U ] > ----~ the u p p e r and r i g h t h a n d m a p are embeddings.
N o w if [ V , U ] in
.
Then
[ V , U ] is e m b e d d e d in
[V,U' ] .
< V>
Thus the c o m p o s i t e and t h e r e f o r e the left
h a n d m a p is one as well. (3.14)
One m o r e h y p o t h e s i s w i l l have to be made.
When
~(I,-)
r e p r e s e n t s the under-
13
lying set functor, the o p e r a t i o n s map.
More
map
A1
,
pseudomaps. (3.15)
B
extends
If
theory.
in
Moreover,
cations
if
AI---+
immediately
V
according
(i)
category
is a c a n o n i c a l
of v e c t o r
UnV
and
B
N o w we m u s t suppose
that there
spaces,
is simply
is s i m u l t a n e o u s l y
autonomous.
A closed
(iii)
.
a vector
in
is complete,
A(A,B) ---+ A ( A
it is a v a r i e t y w i t h a
(2.12)
field K
space
is a any
the same to h o l d for
map
the c a t e g o r y
o n groups
The fact that
limit of maps
of preuniform a uniform
. This
,B ). com-
objects
object
is the
is the same as a
such t h a t all
~calar multipli-
are continuous.
An autonomous
(ii)
B
space o v e r the d i s c r e t e
*-Autonomous
F r o m h e r e on, we revive
closed
and p s e u d o m a p s .
that the u n i f o r m
is a dense e m b e d d i n g
~
to the remarks
vecotr
group which
b e t w e e n maps implies
is the c a t e g o r y
4.
(4.1)
A2
A2
Thus the c a t e g o r y
same as a t o p o l o g i c a l topological
are u n i f o r m
to a m a p
It follows
Example.
.
is no d i f f e r e n c e
specifically,
mutative V
there
in the theory
By a * - a u t o n o m o u s
category
and call a s y m m e t r i c
monoidal
is m e a n t
G;
(_), : ~ op __+
An e q u i v a l e n c e
This is s u b j e c t
an o l d name due to L i n t o n
category
functor
Cateqories
d = dG
to one axiom,
~;
: G--+
G**
.
that the d i a g r a m G(G,G')
~
G(G'*,G*)
G(G**,G'**) whose h o r i z o n t a l
and vertical
a r r o w are the a c t i o n s
of
(-)*
on the internal
hom,
com-
mute. (4.2)
An i m m e d i a t e
one a split epi. mono which (4.3)
is:
equivalence T = I* .
G'
~
G °p
~ G
category which
G
of the former
(G®I)* ~ G*
.
G '--~
is full a n d faithful
and such that there
Hom(G~(G2®G3)*)
T G*
have c o r r e s p o n d e n c e s
with unit
and and G
G G' ~
G'**
.
In t h a t case,
is a i-i c o n s e q u e n c e is syrmaetric,
G'---+
G*
G--+
G'*
---+ G'
G'*---+
lemma
to h a v e a * - a u t o n o m o u s and tensor product
Since the tensor
G'~G--~
between
and b y the Y o n e d a
a n d thus is a l s o a split
I
T h e n we see that there
dence b e t w e e n m a p s respondence
is an i n s t a n c e
far less d a t a t h a n t h a t is r e q u i r e d
Hom(GI~G2,G~)---~
Let
arrow is a split m o n o and the v e r t i c a l
that b o t h are i s o m o r p h i s m s .
Suppose w e h a v e a m o n o i d a l w i t h a functor
the h o r i z o n t a l
But the latter
implies
In fact,
result
G*
---+ G'**
.
Since
maps
is s i m i l a r l y these
equipped
is n a t u r a l
d e f i n e ~(G',G) = ( G ~ G * ) * .
between
there
Putting
category. ~
together
GAG'---+ a
T and
correspon-
gives a cor-
(-)* is full a n d f a i t h f u l w e
14
Next define
G(G',G)
=
(G'~G*)*
.
There is a 1-1 c o r r e s p o n d e n c e b e t w e e n maps
G(G 2,G 3) = ~G2~G ~) *
G1 G2~G ~ G~G 2
w h i c h d e m o n s t r a t e s that berg-Kelly]
'
(G2~G 1 ) *
G2~G 1
~
G3
GI~G 2
~
G3
G(G2,-)
-eG It follows from 2 f o l l o w i n g p r o p o s i t i o n 3.1, that G
comes a u t o n o m o u s w i t h this d e f i n i t i o n of hom. is immediate. tor.
The e q u i v a l e n c e of G(G',G)
Note that this is i n d e p e n d e n t of w h e t h e r or not
T h a t seems to be m o r e or less e q u i v a l e n t to
ples b u t one.
[Eilen-
is left a d j o i n t to
, II.3 e s p e c i a l l y the m a t e r i a l
then be-
and ~(G*,G'*)
(-)* is a m o n o i d a l func-
I* ~ I , w h i c h holds in all the exam-
Thus the data at the b e g i n n i n g of this section suffice to determine a
*-autonomous category. (4.4)
O n the o t h e r h a n d suppose that the c a t e g o r y
~ , internal h o m functor
G(-,-),
u n i t o b j e c t I t o g e t h e r w i t h r e q u i r e d natural t r a n s f o r m a t i o n s and e q u i v a l e n c e s c o n s t i t u t e a symmetric c l o s e d category. G °p---+
~
T = I* .
Suppose also that there is a full faithful functor
a n d an e q u i v a l e n c e Hom(GI,~(G2,G~))
~ Hom(G3,G(G2,G{))
.
As before,
(-)*: let
T h e n there is an e q u i v a l e n c e Hom(G',~(G,T)) ~ H o m ( I , G ( G , G ' * ) ) Hom(G,G'*)
.
Similarly, Hom(G',G*) ~ Hom(G',~(I,G*)) Hom(G,G(I,G'*)) Hom(G,G'*) C o m p a r i n g these, w e see that
G* = G(G,T)
gether w i t h Hom(G',G) ~ Hom(G*,G'*)
.
and b y the Y o n e d a lemma (4.5)
Since
G* ~ ~(G,T)
Also, Hom(G',G*) ~ Hom(G,G'*) w h i c h to-
implies that
have Hom(G',G) ~ Hom(G*,G'*) ~ Hom(G',G**)
.
(-)* is an equivalence.
w h i l e e v i d e n t l y Hom(G*,G'*)
In fact, we = Hom(G'**,G**)
G' ~ G'** the internal c o m p o s i t e gives a natural t r a n s f o r m a t i o n (G',G) ~
(G*,G'*)
w h i c h f o l l o w e d b y the i s o m o r p h i s m above gives a m a p
(G',G)---+
(G',G**)
. It follows
from c o h e r e n c e that the d i a g r a m G(G',G)
+
G(G*,G'*)
" ~ G (G' ,G**) commutes. S i n c e the v e r t i c a l and diagonal maps are isomorphisms,
so is the h o r i z o n t a l
15
one w h i c h means that
(-)* is i n t e r n a l l y full and faithful.
Now we define G ' ® G = G ( G ' , G * ) * .
We see that there are natural i-i c o r r e s p o n d e n c e s b e t w e e n maps GI®G 2
~
G3
~(GI,G~)*
---+
G3
G~ ~
~CG1,G~I
G I --'+ G(G~,G~) ~ ~(G2,G3) Here we have freely u s e d the e q u i v a l e n c e b e t w e e n
G
and
G**
n o n o m o u s c a t e g o r y n o w follows from [ E i l e n b e r g - K e l l y ] , 1.3 (4.6)
.
.
That w e have a *-auto-
.
The m a i n p u r p o s e in these notes is to b e g i n with a good deal less than a *-auto-
n o n o m o u s c a t e g o r y and c o n s t r u c t one. This leads us to define the notion of a p r e - * - a u t o nomous situation.
This consists o f a c a t e g o r y
~ , two full s u b c a t e g o r i e s
~
and
D ,
an e q u i v a l e n c e of c a t e g o r i e s (-)* : C °p a functor
(-,-)
: Cjp x D--+
D
(I,D)
(ii)
D ,
and an o b j e c t
of a *-autonomous c a t e g o r y insofar (i)
>
I£~.
These are subject to the axioms
as they make sense.
In particular, w e suppose that
D;
Horn(I, (C,D)) ~ Hom(C,D);
(iii)
(CI, (C2,C~)) ~
(iv)
(C3, (C2,C~)) ;
Hom(C*,C') ~ Hom(C'*,C)
These are s u b j e c t to c e r t a i n c o h e r e n c e conditions w h i c h will b e i n t r o d u c e d as needed. (4.7)
Since
(-)* is an equivalence,
functor w h i c h w e t e m p o r a r i l y denote (C,D) ~
(D#,C *)
which, b y
may be r e w r i t t e n w e have, since
it has, up to a n a t u r a l isomorphism, (-) #:D °P--+ C .
(ii) implies that
Hom(D,C) ~ H o m ( C * , D #) .
(-)* is an equivalence,
Then
~iii)
an inverse
above may be r e w r i t t e n
Hom(C,D) ~ H o m ( D # , C *) .
Should it h a p p e n that
D
Similarly
(iv)
also belongs to
that
Hom(D,D) ~ H o m ( D * , D #) and Hom(D,D)
~
Hom(D#,D
*)
The c o h e r e n c e s a l l u d e d to above w o u l d r e q u i r e that if
f
~(f)
describes the m a p
D
+ C
%
and
g
I
then for 8(g)e(f) =
> 8(g)
C
f (fg)#
>
H o m (C,D)
>
H o m (D#, C * )
Hom(D,C)
,
(C*,D #)
describes
D
g
>
C'
,
~(f)8(g)
=
(gf)*
w h i l e for
Thus c o r r e s p o n d i n g to D
1
-~ D
1
~
D
1
~
D
g
f
)
D'
,
C
16
we have D* and
~(i)8(i)
identify
= i* = 1
D*
and
8(1)
while
D#
~ D#
e(1)
8(1)e(1)
(4.8)
C
If w e let
and
If w e let
in
(3.6.iii),
(CI,C;) ~
For
C,C'EC C* ~
Proposition.
If
observed
A,Be~
A,BEC UD.
or
A,BeD
(C3, (I,C[)) .
Suppose
~
D~
(Cl,I*) ~ C[ .
~ Hom(B*,A*)
{g~ : D ~
D~}
is clear,
f* : B*-~ A*
a family
of a factorization
and
under w h i c h 4.6(iv) C~
determines
~or all
~. Supposing
D-representation
system.
{C--+ D~} {C~
--
e~}
D~} u{C--+C*}w ' w h e r e
such that
m~h = g$
let a suitable {C --+ D}
family a
so is any larger family. {C--+ * } wC
D~_D has a C--generation w h i c h dualizes C
,
{fw:
~ D~
{C~
Of
D~}
C-generation In particular
C
~
and
h.e =f
be called a of
D .
If
so is the fa-
Thus we may suppose
to a C--generation.
to a D--representation.
and a C - g e n e r a t i o n
for all
is a _C-generati°n"
has a D--representation which dualizes
of
D~}
have the
~ D
and the dual notion
is a D - r ~ p r e s e n t a t i o n
a D-representation
{m~:C~
Namely suppose that every pair of families
h:D -+ C
this to be the case, C
from the remaining
such that every d i a g r a m
a unique map
of
follows
there is a family of maps
C commutes
one of the unstated coherence hypotheses
is an involution.
{e w :C w --~ D} ' C ~ ~C -- which collectively
C
C
C 3 = I , we get
If we let
this is just the duality while the other two follow as
for every object
and for every
properties
every
.
Then
of these morphisms
f£Hom(A,B),f
hypotheses.
mily
~ OlD
(C,C'*)
We wish to discuss a condition
(4.11)
and
,
Let
The n a t u r a l i t y
Cw-+ C}
CnD
in 4.7.
is that for
D~cD
Henceforth we can
we get
(C3,C~)
Hom(A,B)
(4.10)
.
agree on
(C,T)
(C',C*) %
Proof.
(-)#
I* = T , this may be summarized as,
Proposition.
(4.9)
D* ~ D#
and
O .
C2 = I
(3.6.i),
(-)*
a single functor
(CI, (I,C~)) ~ or, in view of
) D#
so that
Thus
(-)* : (CuD) Op that interchanges
~(i)
= i# = 1
by this isomorphism.
w e may think of them as determining
D*
)
Similarly every
Now fixing a map h : D - + C ,
both a s s u m e d to dualize p r o p e r l y
17
we begin with the diagram
in which
fw
e
is defined as
C
>
D
C
)
D~
~ew
and similarly
g~ = m~.h
. Dualizing and using 3.9
we g e t D* ~
h
then
~ + h*
has
a unique
~
C*
~
fill-in
which
we n a t u r a l l y
and the fact that it is an involution
of the
C~, f , g~
(4.12)
Theorem.
Hom(A,B) (4.13)
C*
e*
D* which
>
and
hypotheses
for
If we suppose that
V-functor
of
A,B£~UD. A
is a
and that the equivalence
h*
.
The naturality
of
4.6(i),
(ii),
(iii)
and
(4.10)
we h a v e
Thus we have a pre *-autonomous V
category,
we can ask that
a duality
situation.
(-)*
: C
+
D
be
a
be that of V-enriched home,
!(c,,c) ~ In addition,
by
m~ .
Under the
Hom(B*,A*)
denote
follow readily from the s~ne properties
Z(c*,c,*)
.
we can suppose that the isomorphism of
(4.6.ii) be V--enriched,
V(I, (C,D)) ~ V(C,D) In that case, we come to the notion of a V_-enriched pre-*~autonomous case the maps used in the C--generation and D--representation cessary, (4.13)
In that
in the paragraphs may,if ne-
be replaced by pseudomaps. The V-enriched versions of
to derive the analogue of a family
situation.
{C --+~ D}
and
(4.8) and
(4.9) go through without change.
(4.12) we must modify
In order
(4.10). This is done by ~gain supposing
{C-+ D~}
.
We observe that there is, for all
~(D,C)
"~
~(D,D~)
~,~,
a commu-
tative square,
_v(c~,c)--+ _v(c ,D~) We may now require that That is, given any object
~(D,C) be the simultaneous V
and commutative
V
>
V(C ,C)---+ one for each
.
V (q,D~)
V(C ,D~)
~,$, there should exist a unique map
Again argumenting
the representing
equalizer of all those squares.
squares
and generating
V--~ V__(D,C) inducing those squares. families
does not change the situa-
tion so we may suppose they are invariant under dualization. V(C ,C) ~ V(C*,C~)
and
V__(C ,D~) ~ V(D$,C~)
Then
V(D,D~) ~ V(D~,D*),
implies that V__(C*,D*) has the same uni-
18
versal mapping property (4.14)
Example.
pre-*-autonomous spaces w i t h
as V(D,C)
On the c a t e g o r y situations.
This
there
is no r e a s o n n o t to e x t e n d D
to be the c a t e g o r y
C-which
A linearly means
space
tient modulo
open),
A linearly
of c l o s e d
linear
subvarities
where
is a p o i n t a n d
Linearly
space
there
as h a v i n g
is,
first,
form a topological
of
Lefschetz
V
transformation
also
finite
to a space of the f o r m
Ks
compact,
KS .
t h a t the c o n t i n u o u s
naturally Here
equivalent
S.K
pies o f
and K .
s i m p l y take and,
pactness. compactness
T.K + S.K
s t a n d for the direct
See [ L e f s c h e t z ]
pp.78-82
the set of c o n t i n u i n g the d i s c r e t e
One observation K
to the linear
T.K
of course,
t h a t if
As w e l l he s h o w e d
.
for details.
linear m a p s w i t h
That
Since the quo-
of a p r o d u c t
of
if e v e r y c o l l e c t i o n
intersection
v + V' property
a n d p r o v e d all the e l e m e n separated
s u c h space Here
since
to a
is isomorphic is l i n e a r l y KS
÷ KT
are
o f the duality.
respectively
To get the r e q u i r e d the usual v e c t o r
quotients
compact
K
linear m a p s
is the s t a t e m e n t
sum o f an S-fold,
in t h i n k i n g
t h e n linear c o m p a c t n e s s
T h e n linear c o m p a c t n e s s f r o m finite
S
spaces.
a T - f o l d of co(-,-):c°PxD
space
+ D
structure
topology.
that may be helpful
is finite
which
.
from a l i n e a r l y
s h o w e d that e v e r y for a set
compact
linearly.
is a set of the f o r m
the n o t i o n
is to
uniformity.
an o p e n set to a point,
compact
(topologically) so is
0
are c l o s e d u n d e r products,
linear
Lefschetz
in
V) w i t h the
defined
spaces
at
it is a subspace
is l i n e a r l y
subvariety
The s e c o n d
of l i n e a r l y
base
vector
category but
the d i s c r e t e
topologized
(when y o u i n d e n t i f y
space
dimensional
a *-autonomous
to a larger category.
is the same as s a y i n g
A continuous
is closed.
C = D = finite already
considered
space w h i c h
spaces
compact
are at least two r e a s o n a b l e
D °p- w e take the c a t e g o r y
a subspace
intersection.
separated
to
(a linear
V'
tary p r o p e r t i e s .
subspaces.
the s t r u c t u r e
topologized
has a non-empty
and closed
in fact,
is d i s c r e t e
this
spaces.
v
spaces,
, a l l spaces
(vector)
such a s u b s p a c e
that point becomes
is,
is a v e c t o r
t h a t the o p e n sub
discrete
v
must be equivalent compact
of v e c t o r
they are isomorphic.
The first is to take
the usual duality.
take For
and hence
fields
about
linearly
is e q u i v a l e n t
m a y b e t o u g h t of as the t r a n s f e r
to a r b i t r a r y
ones.
compact
to o r d i n a r y
spaces
is
topological
of the n o t i o n of
com-
C H A P T E R II.
E X T E N S I O N S OF STRUCTURE
i.
(1.1)
The m a i n goal in these notes is to c o n v e r t a p r e - , - a u t o n o m o u s
*-autonomous category. and
The Setting.
D
T h a t is, g i v e n a V--category
A
w h i c h d e t e r m i n e an e n r i c h e d p r e - , - a u t o n o m o u s
subcategory
w h i c h contains
D
and can b e e q u i p p e d w i t h a *-autonomous
structure e x t e n d i n g the g i v e n s t r u c t u r e on
C
and
D
.
The r i g h t degree of a b s t r a c t i o n has p r o b a b l y not b e e n r e a c h e d here.
pose of e x t e n d i n g the s t r u c t u r e on
that
~
e q u i p p e d w i t h subcategories
situation, we w i s h to find a full
and
(1.2)
G c A
situation into a
A
C
and
is a c a t e g o r y of u n i f o r m objects
For the pur-
D , I h a v e found it e x p e d i e n t to suppose in a closed c a t e g o r y
V
w h i c h is a semi-
variety. In this chapter,
objects, ~ an o b j e c t
and
D
IcC
and a functor
nomous situation. and for e v e r y
objects in uniform
(-,-):C__ °p x D - - + D
{A--+ D~}
w h i c h give a V - e n r i c h e d p r e - * - a u t o That every o b j e c t of
A
denote the full s u b c a t e g o r y of
UnV
of maps - or even of p s e u d o m a p s - w i t h each
The i n c l u s i o n o f
A--+ UnV
It is clear from the d e f i n i t i o n of
have a n e m b e d d i n g into a p r o d u c t in subobjects and h e n c e u n d e r limits. But if w e have
D
A
~ .
D~D
Given
A£A ,
is called a D__-
has a left adjoint.
as the full s u b c a t e g o r y of objects w h i c h
that
A
is itself invariant under products and
Thus w e need o n l y verify the s o l u t i o n set condition.
U£UnV ,
A~A
and a map
and m o r e o v e r lies in
A .
Thus all the algebras o f
exceed that o f
c o n s i s t i n g of all
A .
Pro~sition.
Proof.
(-)*:c°P---+D,_ _
e v e r y o b j e c t of the form [V,D ] can b e e m b e d d e d in a p r o d u c t of
We n o w let
r e p r e s e n t a t i o n of
U
the c a t e g o r y of u n i f o r m
W e m a d e the f o l l o w i n g a d d i t i o n a l hypotheses:
V{V, D6D
D .
is such a category, U n V
V__-objects w h i c h can b e e m b e d d e d in a p r o d u c t of o b j e c t s o f
a family
(1.3)
then, V
are full s u b c a t e g o r i e s e q u i p p e d w i t h a V--enriched d u a l i t y
U
U---~ A , the image has c a r d i n a l i t y
and all p o s s i b l e maps of
U
A
~
that of
w h o s e c a r d i n a l i t y do not
to such algebras c o n s t i t u t e a solution
set. (1.4)
For
A£A
c o n s i d e r the d i a g r a m AoP
G
>
(Un!) ° p
[-,~]
!(-,A)
+ I !(-'A) V
with
~
inclusion,
v a r i a n c e exhibited, V(U,A) ~ ~(~U,A)
~
T
the a d j o i n t above and
[-,A]
is left a d j o i n t to
V
T
and
V(-,A).
~
are the identity. The a d j u n c t i o n
6
With the I~
gives
w h i c h is the c o m m u t a t i v i t y r e q u i r e d to a p p l y [Barr,73], T h e o r e m 3 and
20
conclude
that the image of
is a c o t e n s o r e d (1.5)
[-,A]
lies
in
A
.
This
It is easy to see that w e get a f a c t o r i z a t i o n
the s u r j e c t i o n s
and
have a surjection
is c l e a r l y
the cotensor.
Thus
V--category.
M
the e m b e d d i n g s
A'----+ A
s y s t e m on
A
by taking
(with the induced uniformity).
and an e m b e d d i n g
B - - + B'.
We w a n t
E
Suppose
to be now we
to c l a i m that
V(A, B) -----~ V(A' ,B)
1
!
V_(A,B')---* Z(A' ,B') is a pullback. above s q u a r e
This
is so iff for e v e r y
is a p u l l b a c k
in
S .
V~V
Using
the functor
the e o t e n s o r
Hom(V,-)
adjointness,
applied
to the
we get a c o m m u t a -
tive square
Hom(A,[V,B])-----+ Hom(A',[V,B])
w h i c h w e need to k n o w B'
, [V,B]
1
1
Hom(A,[V,B']) ~
Hom(A' ,[V,B'])
is a pullback.
is a s u b s p a c e
of
[V,B']
According
to
(I.3.13)
when
so that we h a v e a d i a g o n a l
B
is a s u b s p a c e
fill-in
of
in any c o m m u t a -
tive s q u a r e
AI - - . - - ~
A
Iv, B l - - . [ v , which D
is e x a c t l y w h a t
a n d the r e m a i n i n g
(1.6)
Example.
is required. structure
Let
~
of
K
vector
fulfill
for the p a i r
spaces
of v e c t o r
C, D
that is,
the k e r n e l
of
so t h a t the k e r n e l -- and w i t h particular to
this A
the f o l l o w i n g
.
In fact,
contains A - ~ D ix
,.the s u b c a t e g o r i e s
if
a finite ... x D~n.
A
the spaces
The
Not e v e r y
latter
for every o p e n s u b s p a c e
~
dimensional
is a linear
analogue
we say t h a t a space
BoA,
is a finite
there
A
.
Wel con-
CeC
is a power topolo-
linearly
topologized
in
and every
product
discrete
KD
space D 6D
of maps
A-+ D ,
is finite dimensional, dimensional.
In
space and no such space
a D_-representation
to the p r o p e r t y
Namely,
K
h a v e linear
of the k e r n e l s
It is e a s y to see that a space does h a v e
b e i n g t o t a l l y bounded.
in
is e m b e d d e d
intersection
and
C = D = finite
T h e n every
it every o p e n s u b s p a c e -- is c o f i n i t e
is n o t true of any infinite
property which
o v e r the field
For the first w e take
Since
~
of 1 . 4 .
spaces
topology).
D- representation.
however.
then e v e r y o p e n s u b s p a c e
.
(with the d i s c r e t e
gies so does a n y t h i n g w i t h a
A
all the c o n d i t i o n s
and h e n c e h a s a D--representation.
has a D - r e p r e s e n t a t i o n
belongs
Thus the c a t e g o r y
be the c a t e g o r y
sider two p o s s i b i l i t i e s dimensional
B' ]
of a u n i f o r m
is l i n e a r l y
number o f elements
iff it has space
totally bounded a I,
..., a n
iff
such
21
A
is in the linear
wording
that
of the h y p o t h e s i s
b y the f o l l o w i n g Proposition.
A separated completion
Proof.
A
If
dimensional.
A--+ ~ A / A
cause
A
A/A
I claim
and h e n c e
mensional form
A0cA
subspace
A0
also t h a t
where
V
whence
B/B 0
is d i s c r e t e
the l i n e a r l y
of discrete
Now if
compact
of the l i n e a r l y
C
ones.
i.e.
full s u b c a t e g o r y .
temporarily If w e w i s h
designate
we w i l l
dual of
A
A#
want
shows
in
A
that
.
B
it m a p s
is to t a k e
that
A
to give = 0
be-
space
in the
compact.
U
~
to e a c h
is e m b e d d e d
D
0
V
in
finite diB
is the
contains
is l i n e a r l y
injectively
T h e n for any
an o p e n
topologized
to
A/A 0
and
.
to be the d i s c r e t e
spaces
and
iff it is l i n e a r l y
a subspace
of a p r o d u c t
in that case
A
consists
of the
the d u a l i t y
C~ V(C,A)
Duality.
on
CUD
to all o f
We are t r y i n g to d e f i n e
C-+ A
C-+ A
this means we r e q u i r e
the internal
T
.
there
V(A,T)
A
, w h i c h we A # - + C*
V(C,A)--* ~ ( A # , C *)
that
V(A,T)
A~
.
w h i c h means C*
.
More-
in such a w a y that the ~#I ~ ~(A,T)
the c o a r s e s t
~(C,A),
~ V(V(A,T),V(C,T))
is to say t h a t w e e q u i p
a dual of
a pseudomap
equipped with
a n d each e l e m e n t
or at least to
s h o u l d b e one
h o m as w e l l
This r e q u i r e s
A
Accor-
uniformity
the c o r r e s p o n d i n g
is uniform. with the
ele-
Another way
coarsest
unifor-
such t h a t
A#
is u n i f o r m
for all
C~C
.
A@
coordi-
of l i n e a r l y
and h e n c e
of
Then
Thus
to e x t e n d
under
mity
0
compact
a D--representation.
A£A.
s u c h that c o r r e s p o n d i n g
this u n i f o r m i t y
is l i n e a r l y
0
If we h a v e a m a p
m e n t of
to d e s c r i b e
NA
of a p r o d u c t
to b e i n g
to b e the o b j e c t
V ( I A # I , Ic* I)
subspace
that is e q u i v a l e n t
is its i n t e r n a l h o m into
dingly we define
of
o p e n and shows
any n e i g h b o r h o o d
since
D
to each p s e u d o m a p
eventually
is
an o p e n sub(vector)
consisting
A
of
to h a v e the d u a l i t y b e V__-enriched
that corresponding over,
BoA
Extension
Let
A #.
is c o f i n i t e
combine
spaces.
T h e first task is to e x t e n d
a large
A cA
A-+ A/A
T h e n a space h a s a D - r e p r e s e n t a t i o n
to h a v i n g
topologized
HA/A
that
This
and
2.
(2.1)
.
For as n o t e d e a r l i e r spaces,
contains
and also l i n e a r l y
is finite d i m e n s i o n a l for
iff
compact.
suppose
U DB 0 = BNA 0
totally bounded
In fact its kernel
0
is t h e n a c l o s e d
is a n e i g h b o r h o o d
The o t h e r c h o i c e
topologized.
A
first
(see [ L e f s c h e t z ] ) .
U = BAY
of
This is e v i d e n t l y
is l i n e a r l y
, A/A 0
The maps
is an embedding.
of
is l i n e a r l y
then every o p e n s u b s p a c e
dimensional.
image of the s u b s e t of
To see the c o n v e r s e open subspace
bounded,
Any neighborhood
The c l o s u r e
space
compact.
totally
in all others.
spaces
and al, ..., a . Of course this is just a ren is finite d i m e n s i o n a l . The a n a l o g y is h i g h t e n e d
topologized
is finite
which
is the inverse
compact
linearly
A/A
is separated.
in the product.
B
A/B
is l i n e a r l y
is l i n e a r l y Thus
a map
nate a n d
that
fact.
its u n i f o r m
which
s p a n of
Since
A#
, [ V(C,A) ,C* ]
can h a v e o n l y a set o f
uniform
covers,
only a
22
set of
C
cluded
need be
among
used
from any one
the objects
of
C
A# among
these
maps.
This
map
A
used)
.
By t a k i n g
C = I
(which m a y
always
be
in-
we get
~ [Z(I,A),I*] =[ IAl, T]
is an i n j e c t i o n
since
the underlying
map
in
V
factors
as
injections
IA#1
Thus t h e r e
(2.2)
is a f a m i l y
This
IBI
means
) V(A,T)
.
= V(A,T)---+
{C
that
}
such
to m a p
Second
we
V(IAI,ITI) ~
that
A#
is e m b e d d e d
B---+ A # , w e
need
I[ IAI ,T]
require
in a p r o d u c t
two things.
B--+ [V(C,A),C*]
I BI
I -
K[V(C
First,
, for e a c h
C6C
we
such
,A),C~]
need
.
a map
that
~ V_(A,T)
l[v(c,A),c* ] I --- v(v(c,A) ,V(C,T) ) commutes, ~(B,C*)
the right such
hand
map being
composition.
This
is e q u i v a l e n t
to m a p s
~(C,A)
÷
that
V(C,A)
~
V(B,C*)
V (V (A,T) ,V(C, T) )
commutes. (2.3)
Thus
isomorphic second
datum
morphism. for
to
to
C
map
C *-+ C #
~(C,T) required
The
take
coherence
C
CeC
take
we
the
need
itself
the
< C*>
of
V(C*,C'*)
is t r i v i a l . so t h a t
first
composite
~(C',C)~
is a m a p
required
we may
for
s o we c a n
the
~ < C#>
canonical
for w h i c h
On t h e o t h e r identity
gives
we
hand
.
But each
is
isomorphisms. take
among
The
the duality
iso-
the candidates
a map
c#--+ [ v(c,c) ,c* l which (2.4)
composed Now
with
let
DeD
D*
is i s o m o r p h i c
may
follow
get
each
D ~---~ " ~C*
the .
unit
I--+ ~(C,C)
We k n o w
that
to a s u b o b j e c t with
the name
, and hence
of
gives
there HC*
~ D*
.
C*
.
Thus
is a C - g e n e r a t i n g .
Since
I---+ V, (D C) _
D#
C# - +
C* N C #
family
D#---+ I[ [ V ( C
,D),C~]
of the corresponding To go t h e o t h e r
way,
{C --~ D}
we
maps require
such
is a map,
that we
C~---+ D
to
for a l l
C£C
V (C, D*) ---+ V (D,C*)
for which
we
take
the duality
isomorphism. #
We h a v e write
A*
now established
instead
of
A#
.
that We n o w
() turn
.
#
is an e x t e n s i o n to t h e
of
functionality
()
and will
of this
henceforth
operation.
23
(2.5)
Proposition.
Proof.
For any
C£C
, c o m p o s i t i o n of maps gives a m a p
!(C,A)----+ V(A*,C*).
This is the same as a map
A*
~ [ V(C,A) ,C* ]
w h i c h w e have. Corollary. Proof.
For any
A , B £ A , c o m p o s i t i o n of maps gives a m a p
V(B,A)---~ V(A*,B*)
We r e q u i r e a map
A*
~ [ V(B,A) ,B* ]
We c e r t a i n l y have a m a p V_(A,T)~ Moreover,
for any
C6C
V(V(B,A),V(B,T))
.
w e have
A* - ~
[ Z(c,A) ,c*]
--~ [ V(B,A)~V(C,B) ,C* ] [ V(B,A) ,[ V(C,B) ,C* ] ] and h e n c e there is the r e q u i r e d map the fact that (2.6)
IV,-]
A*--+ [V(B,A),B*]
.
Note that this a r g u m e n t uses
p r e s e r v e s embeddings.
This shows that
(-)* : A --+ A
is a V-functor.
There is nothing to guarantee
that it is an e q u i v a l e n c e and it is in fact u n l i k e l y that it is always so. Since
I e C , c o m p o s i t i o n gives a m a p
A*---+ [V(I,A),T] IAI = !(I,A)---+ !(A*,T)
=
>** I
and the b e s t y o u can u s u a l l y h o p e for is t h a t the above map morphism. w h i c h gives
It is always a m o n o m o r p h i s m . {A**--+ D**~ ~ D~}
.
I A!--+
I A** I be an iso-
For there is a D - r e p r e s e n t a t i o n
Thus w e h a v e
IAI
~
I A**I --+ ~ ID I
{A---+ D } is a m o n o m o r -
p h i s m and h e n c e the first m a p is. (2.7) in AA£A Let
Let IA** I since
AA
A A c A ** .
{A---+ D }
so that w e have an isomorphism, (2.8)
denote the s u b o b j e c t of
b u t such that
AA
A**
We h a v e
I AAI
~ I AI
be a D - r e p r e s e n t a t i o n of A A---+ A
is uniform.
q u a s i r e f l e x i v e if
The c o n d i t i o n that
A
such that
IAA I
has the induced uniformity.
IAI
by the inverse of the above inclusion.
A .
Then each
We say that
A A = A**
is the image of
There is no q u e s t i o n of
A
A--+ D
gives
is p r e r e f l e x i v e if
A**----~D**~ D AA--~ A
is
and r e f l e x i v e if b o t h of these hold.
be q u a s i r e f l e x i v e
is e q u i v a l e n t to the a s s e r t i o n that
~(I,A) ~ !(A*,I*) or that e v e r y p s e u d o m a p sarily unique -- o f
A .
A*--+ T
is r e p r e s e n t e d b y e v a l u a t i o n a t an e l e m e n t -- neces-
24
(2.9)
Proposition.
Suppose
(i)
T
is c o s m a l l
in
(ii)
T
is i n j e c t i v e
A
(see proof);
in the V--category
~
with
respect
to the class of
embeddings; (iii)
~
is c l o s e d
under
finite
h a v e the u n i v e r s a l Then every p s e u d o m a p Proof.
sums and
mapping
f : A*---+ T
D
under
properties
is r e p r e s e n t e d
We can f i n d a family of p s e u d o m a p s
finite p r o d u c t s
for p s e u d o m a p s
by evaluation
and these
as w e l l as maps.
at an e l e m e n t of
{C--+ A} such that the h o r i z o n t a l
A
.
arrow
in the d i a g r a m
A*
~ EC*
T
is an embedding. that say
T
be c o s m a l l
C[ x
d u c t in _C ° p - - +
Then we have
.
Since
f# : ~C* -
m e a n s t h a t such a m a p
... x C*n " ~
D _
a map
Since
_D
x(C 1 + whose
Example.
are l i n e a r l y
tinous
l i n e a r map,
B 0 = A 0 NB
jection
choices
where
A0
C
is e a s i l y
spaces,
and
D
Since
(-)*
:
For if
B
the former
the h y p o t h e s e s
subspace in
A
is e m b e d d e d
f .
till t h e n e x t section.
is e m b e d d e d
m u s t b e an o p e n
b y an
C 1 + ... + Cn---+ A
seen. to r e p r e s e n t
of
(2.10)
, in fact for any choice
is an o p e n s u b s p a c e
and b o t h are discrete,
.
is r e p r e s e n t e d
under the p s e u d o m a p
is p o s t p o n e d
of v e c t o r of
topologized. its k e r n e l
~
is a l s o their p r o -
so w e h a v e
t h a t the lower p s e u d o m a p
of the d u a l i t y
In t h e e x a m p l e
spaces
1.6,
it follows
The image of that e l e m e n t
(2.10)
of f i n i t e l y many,
this
s u m in
The h y p o t h e s i s
(CI+...+C)*
are the ones g i v e n o r i g i n a l l y
fied for b o t h p o s s i b l e
f .
S
... + C n
discussion
a product
finite p r o d u c t s
, that is their
components Further
through
extends
> ~C*
T c C
which
"'" + Cn)* ~ C1* x ... x C*n
i for
under
... + Cn{ ~ (C 1 + A*
From duality
factors
is c l o s e d
C1 + C2 +
is an equivalence,
> T
in
A
B0 c B .
Since
and .
are satis-
for w h i c h the B---+ K
is a con-
T h e n as we showed
B/B0---+ A / A 0
in the latter.
in
is an in-
Then we have a
diagram
B ----+ B / B 0
I A and t h e n o r d i n a r y A/A 0
is discrete.
vector
space
~ K
I ~ A/A 0
theory provides
the r e q u i r e d
A/A0----+ K , c o n t i n o u s
since
25
3.
(3.1)
As with the duality,
Extension of the Internal H D m
we wish to extend the functor
(-,-)
: C °p × D ÷ D
to a
functor denoted ~(-,-) Although we are for convenience is not in general.
: A_Op × A ÷ A .
using notation suggesting
It is not generally
symmetric,
(3.2)
To b e g i n with we require uniformity
for every pseudomap
C ÷ A
A(A,B)
Chapter
III is devoted to finding
on which it is well-behaved.
give it the coarsest
that
it
does not always have an adjoint and
when it does, the tensor is not always associative. a nice subcategory
this is an internal hom,
~(A,B) I
and every pseudomap
have the weak uniformity
[ V ( C , A ) ® V(B,D), (C,D)].
= ~(A,B),
the V--valued hom.
such that the pseudomap
A(A,B)
B ÷ D .
+
Second we will
(C,D)
is uniform
More abstractly,
determined by all
CeC, D ( D and
This makes sense for the underlying map in
we require
A(A,B)---+ V
is
V(A,B) --+ V(V(C,A)®~(B,D),V(C,D)) which is the transpose
under adjunction to composition V(C,A) ®V(A, B) ®Z(B, D) --+ ~(C,D)
Since there is an epimorphic D~)
family
{f~
: C~--+ A)
and a monomorphic
family
{g~ : B ÷
, there is a m o n o m o r p h i s m V(A,B)--+ ~!(C
, D~)
.
This map factors V(A,B)--+ nZ(Z(c
,A)®!(B,D~) ,Z(C ,D~))-+ H V ( I ® I , ! (Cm,D~)) ~ ]IV(Cw,D ~)
where the second map is induced by the names of the is an injection as well. {D~}
of objects of
~
Hence for some and
D
f~
and
(not necessarily
respectively,
A(A,B)
g~
.
Thus the first map
the same)
is embedded
families
{C ) and
in
H[Z(Cm,A)~V(B,D~), (C ,D~)] (3.3) Proof.
Proposition. Since
(-,-)
If
C~
and
DED
: C °p x D + D
, A(C,D)
is canonically
isomorphic
is assumed to be a V_-functor,
to
(C,D)
.
there is for each
D~D, a natural map
which expresses
the fact that
v(c',c)
÷ V((C,D),(C',O))
(-,D)
is a V-functor.
Similarly,
there is for each
C6C , a map V(D,D') Each of these maps lies alone ordinary and replacing
C
by
V(C' ,C)®V(D,D')
C'
+ V((C,D), (C,D'))
.
composition of functions.
Putting these together
in the second we get a map
÷ V((C,D), (C',D))®!((C',D), (C' ,D')) ÷ !((C,D), (C',D'))
the second map being composition.
By the cotensor
adjunction,
,
this gives a map,
26
(C,D) -+ [ V ( C ' , C ) ® V ( D , D ' ) , (C',D')] for any
C'c~ , D'£D
.
This implies that
(C,D)--+ A(C,D)
is uniform. To go the o t h e r
way w e o b s e r v e that A(C,D)--+ [V(C,C)~V(D,D), (C,D)] m u s t b e uniform. maps of
C
and
We may compose this with the map induced b y names of the identity D r e s p e c t i v e l y to get
A(C,D)
÷ [ I®I,(C~)] ~
(C,D), the latter isomor-
p h i s m coming from V(A,[ I,B]) ~ V(I,V(A,B)) ~ V(A,B) from w h i c h uniform. V(C,D) (3.4)
[ I,B] ~ B
b y the Yoneda lemma.
This implies that
A(C,D)-+
(C,D)
is
It is easy to see that both of these maps lie over the identity m a p on
. Proposition.
Proof.
The b i f u n c t o r
A(-,-)
: A_°p x A + A
is
(or lifts to) a V_-functor.
We m u s t s h o w that there are natural maps V(A',A)--+ V(A(A,B),A__(A',B)) V(B,B')--+ V ( A ( A , B ) , A ( A , B ' ) )
for all
A,A',A,B'{A .
We do the first, the second b e i n g similar.
T h e n we require a
map
A(A,B) ~ [V_(A' ,A),A(A' ,B)] w h i c h means first a map
V(A,B)-+ V(V(A',A),V(A',B))
and second for all
CE~, D £ D ,
A(A,B)--+ [V(A',A),[ V_(C,A')eV(B,D), (C,D)]] by
(I.3.13).
The first is just c o m p o s i t i o n and the second comes from A(A,B)--+ [V_(C,A)®V(B,D), (C,D)] -~ [V(A',A)eV(C,A')eV_(B,D), (C,D)] [ V ( A ' , A ) , [ V ( C , A ' ) ~ V ( B , D ) , (C,D)]]
(3.5)
Proposition.
(c ,D~)]
Suppose
A(A,B)
is e m b e d d e d in the p r o d u c t
9[V(C
. Then there is a c o m m u t a t i v e d i a g r a m
_A(A,B)
+ ~[V(C ,A),_
(Cm,B)]
~[V(B,D~),(B,D~)]----+ K[V(C ,A)®V(BrD~),(Cm,D~)] e,~ The upper m a p and left h a n d m a p are embeddings. Proof.
From
(3.4) w e h a v e a map for all V(C
w h i c h transposes to
~,
,A)--+ V ( A ( A , B ) , A ( C
,B))
,A)®V(B, D ) ,
27
A(A,B)--~ [V(C ,A) ,A(C ,B)] and is the
~
component
of a map A(A,B)--+ n[V(C
Similarly,
there are maps
which transpose
Cotensoring
,A)
~
component ~
the one factorization
C£C
i.
If
is embedded 2.
For any
Corollary
3.
There
There
H[V(C
is a canonical
Finally
the pro-
embedded
~
in
in a product
,A),
(C ,D)]
~[V(A,D~),
from
~[V(B,D~),_ (C,D~)]
; if
D£D,
.
A(I,A)
I(I,A) I .
~ A .
For some family
D~]
while
(I,A)
{D~}
of objects
is canonically
of
em-
(I,D~)] .
map
~(A,B)--~[ We have,
follows
.
K[V(A,D~),_
is a canonical
The last property
systems.
isomorphism
IAI = V(I,A)
in the isomorphic
Proof.
we get
(Cw,D~)]
,A)®V(B,D$), (C ,D~)]
is embedded
A £ A , A* ~ ~(A,T)
is canonically
4.
is a V-functor,
and the other is analogous.
, A(C,B)
By hypothesis,
Corollary
_~[~(C
of factorization
in a product
Corollary
bedded
,D~)]
,A) ,A(C ,B)]--+ n [V(C ,A)®V(B,D~), (C ,D~)]
Corollary
, A
[Z(B,D~),(C
,B)-~
g i v ~ a map
properties
Proof.
A(C
of a map to
from cancellation
A(A,D)
Z(A(Cw,B), (C ,D~))
,A) ,A(cm,B)] ---+ [!(Cm,A)®Z(B,D~),
~[!(C This gives
V(B,D~)-+
and using the fact that the cotensor
[Z(C
duct over all
~,~
to
~(C
This is the
for all
,A),A(C ,B)]
Im , B]
(3.4), the canonical
map
Iil = Z(I,A) -+ V(A(A,B),~(I,B))
Z V(A(A,B),B)
which has the transpose
i(A,B) ---+ [ IAi ,B] (3.6)
Proposition.
Suppose
A
and
are reflexive.
The first is thus a split mono and the second a split epi. of the first and hence
and
~(B,D)
Thus
~ ~(D*,B*)
~ V(B*,A*)
Ce~
A(B*,A*)--+
and
.
In particular
.
Thus we have
D£D
[V(D*,B*)®!(A*,C*)
~ A(B*,A*)
V(A**,B**)
is also a split mono from which
V(A,B) for
V(A,B)-+~*,A*)-+
A(A,B)
lence.
are isomorphisms.
we have
Then
Since
instance
(-)* is a V--functor,
B
Proof.
it follows
we have
an equiva-
The second
V(C,A)
is an
that both ~ V(A*,C*)
, (D*,C*)]
[ V(C,A)e!(B,D), (C,D)] so we have
~(B*,A*)-+
A(A,B)
and similarly
in the other direction.
Note that
(-)*
28
is not in general
A
an
functor.
In fact if
is q u a s i - r e f l e x i v e
B
and not reflexive,
the m a p B ~ A(I,B)
.--+ A(B*,T)
= B**
is n o t uniform. (3.7)
Let
CeC,
De~
is the p s e u d o m e t r i c
,
S
a s u b s e t of
d e f i n e d on
~ = }(C)
(d,S),
is a c o l l e c t i o n
S£~,
uniformity sets in
d on
#
is separated.
in
For if
such t h a t
which
d(fx,gx)
(-,-)
#(C)
C
~ 0 .
there
in
sists of s o m e t h i n g requirement
%(C)
is an
is s o m e t h i n g
is a c a n o n i c a l
%
is s e p a r a t e d
there
structure
uniformity
of p s e u d o m e t r i c s
If the u n i o n of the
t h e n this
union,
is g i v e n
for every p s e u d o m a p
sets or all
.
for each
for
CE~
of u n i f o r m
~(C)
sets or the like.
that there is a natural
d
of the functor a
conver-
f : C ÷ C'
In practice,
finite
structure
hence an e l e m e n t
o n the values
has the s t r u c t u r e
f(S)~S'
separated)
is a p s e u d o m e t r i c
if there
(C,D)
such t h a t
.
all singletons)
is an e l e m e n t of the dense D
~(C,C')
there
then the c o l l e c t i o n
includes
and if, m o r e o v e r ,
to s u p p o s i n g
which
C
in
like all c o m p a c t
(d,S)
.
on the sets
$
such that
S'~(C')
amounts
of
I x~S}
convergence if
is a c o n v e r g e n c e C
. Then
not n e c e s s a r i l y
Since
of
D
a pre-(i.e,
there
.
on
, defines
We say that the u n i f o r m
of subsets
gence o n the sets
(e.g,
f ~ g
fx # gx
: C__ °p x D ÷ D
S(#(C)
D
C , t h a t of u n i f o r m
is dense
x~S£~
family
on
a pseudometric
: sup {d(fx,gx)
o f subsets
a pseudometrie
d
and
by
(d,S) (f,g) If
C
V(C,D)
and
usually
con-
The second
transformation
[--+ H o m ( ~ ( C ) , # ( C ' ) ) At a n y rate
like a V - e n r i c h m e n t .
it is e n o u g h
to g u a r a n t e e
that
~(C,C')----+ V((C',D), (C,D)) for any
DeD
.
The s i m i l a r V__(D,D' ) ---+ V_((C,D) , (C,D') )
for any (3.8) US
CeC
We n o w let where
~(Ci)
nerate
~(A)
Dually
because
# (A)
for each
n set in
HD
exists
the inverse
denote
{B ÷ D
}
Si
{C --+ A}
in this manner,
if
the family of a l l subsets
i ~ l,...,n,
If a set
(3.9)
Proposition. ~(A)
{A(A,B)---+ Proof.
.
Let
If
uniform convergence
A,B~
on
.
.
represents Then
(A,B)
A#(A,B) ~(A)
with
A
of the f o r m
SIU...
C.-+ A of a l is s u f f i c i e n t to ge-
C £C
generates
such that
B
A
.
is e m b e d d e d
in
B .
A(A,B) A
of
is a p s e u d o m e t r i c .
same p s e u d o m a p
{ C - - + A}
of p s e u d o m a p s
generates
represents
momentarily,
of p s e u d o m a p s
we say that the f a m i l y
{ C ~ - + A}
(C , D~)}
Let,
is the image under
is a c o l l e c t i o n
, t h e n w e say that the p r o d u c t
gence on
image of a p s e u d o m e t r i c
has the u n i f o r m i t y
and
{B ÷ D~}
of u n i f o r m c o n v e r -
represents
B
then
.
denote
V(A,B)
e q u i p p e d w i t h the u n i f o r m i t y
T h e n a b a s i s of p s e u d o m e t r i c s
on
A#(A,B)
of
consists
of
29
(d,S)
,
d
a pseudometric
(Ti), Ti£~(Ci),
on
A , S£~(A)
fi£V(Ci,A)
Also let
Then it is easily seen that m.
on
÷ H(Ci,Dj)
A(A,B)
÷ A_#(A,B)
To go the other way,
But
t~(fi)Ti£~(A)
sets B
di
V(C
,A)
on
and so
.
let
d
~(C ) . sup
Similarly,
_V(B,D~)
on
.
(A,B)
(3.10) We say that a map
dominates
the pseudometric
of pseudometrics
A(A,B)
in
÷ K(C ,D~)
Thus it follows
.
Then there is
A'--+ A
just as
Suppose
arising
A(A,B)
sup(di(gi,gi))
on
from a finite number of
is a pseudometric
on
A__#(A,B)
is uniform. is dominating
is clear that this is dual to an embedding.
Proposition.
and
be a p s e u d o m e t r i c
Thus every p s e u d o m e t r i c o n
A__#(A,B) + A(A,B)
~C~ ÷ A
is embedded
and is thus already realized by a finite number of elements of the
and sets from
B ÷ D~
A#(A,B)
fi : C. ÷ A, gi : B ÷ Di, i = l,...,n , sets Ti£#(A i) 1 such that d ~ sup{(di(gifi,gifi),Ti)}, i = 1 ..... n .
D.i
must b e m a j o r i z e d by the
pseudomaps
Thus
function
is uniform.
a finite set o f pseudomaps,say and pseudometrics
, i = 1 ..... n, j = i,...,
is such that any pseudomap _V(C~,A)
(#fi)
, gjcZ(B,D j) , j = 1 ..... m .
induced bv the composite
is uniform when it is induced by elements of A(A,B)
S = SlO...US n , S i =
d = sup{dj(gj,gj)}
A#(A,B) But the uniformity
that the map
Let
(d,S) = sup{(dj(gjfi,gjfi),Ti)}
But this is just the pseudometric
H(C ,D~)
(see 3.7).
{B + D~}
A'~-+ A
provided
A family
represents
is dominating
B
and
~(A')
maps onto
#(A)
{C -+ A}
generates
A
iff
B ÷ B'
B ÷ ~D~
It
iff
is an embedding.
is an embedding.
Then each
map in the square A(A,B)
> A(A',B)
l
1
A(A,B')
> A(A' ,B' )
is an embedding. Proof.
Just repeat the previous
No special property of
argument.
C
or
D
was used
are families
such that
in that argument. Corollary. ~A---~ A
Suppose
{A~---+ A}
is dominating
and
and
B ÷ ~B
{B --+ B~}
Then each map in the square
is an embedding.
A(A,B)---+ ~A(A
,B)
~A(A,B~)---+ ~A(A ,B~) is an embedding. (3.11) Proposition. and
{ E - - + D E}
Proof. (C~,D~)}
Let
represent
We k n o w from represents
{C~---+ A} E .
(3.9) that it.
Then
and
{C~ --~ B}
A(A,A(B,E))
~(B,E)
dominate is embedded
is embedded
A second application
We note that this implies that
of
in
A in
H(C~,D~)
and
B , respectively,
K(C , (C~,D~)) so that
(3.9) yields the result.
A(A,A(B,E))
is embedded
I[[A(C ,A)®A(C~,B)~A__(E,D E), (C , (C~,D~))]
in
.
{A(B,E)--~
30
By s y m m e t r y so is
A(B,A(A,E))
.
This does not imply that they are isomorphic but it
does imply that any inclusion b e t w e e n as a s u b o b j e c t o f
~ ( I A ~ IBI ,IEI )
V(A,A(B,E))
(3.15).
V(B,A(A,E))
under the i s o m o r p h i s m w i t h
m a t i c a l l y lifts to an inclusion b e t w e e n can do now is
and
A(A,A(B,E))
and
(each c o n s i d e r e d
V(IBI~IAI ,IEI))
A(B,A(A,E))
.
auto-
The b e s t w e
W h e n we introduce completeness h y p o t h e s e s in the next chapter,
we will have b e t t e r results• (3.12)
Proposition.
Suppose
A,B,B'6A
and
B --~ B'
is an embedding.
Then there is
a canonical p u l l b a c k V(A,B)--
--+ V(A,B')
I
Proof.
+
+
V(IAI,]B[)--
+ V(IAI,IB' l)
An element of the p u l l b a c k is a p s e u d o m a p
torization
IAI ~
IBI -+
underlies a u n i f o r m
IB'I .
Since
B
is e m b e d d e d in
IAI ~
IBI
) A(A,B')
I [IAI
The d i a g r a m in
V
4
,B]-
-+
[ [AI , B ' ]
w h i c h underlies this is a p u l l b a c k so the actual p u l l b a c k
has the same u n d e r l y i n g V--object as
~(A,B)
it cannot b e c o a r s e r t h a n that i n d u c e d by uniform.
, the map
A --+ B .
A(A,B)
~(A,B)
for w h i c h there is a facB'
There is a canonical p u l l b a c k
C o r o l l a r y i.
Proof.
A - - + B'
with, possibly, A(A,B')
a coarser uniformity.
But that means that the u n i f o r m i t y on the p u l l b a c k is the same as that of a n d so they are isomorphic.
C o r o l l a r y 2.
If
{ B ÷ B m}
is a family such that
B + ~B~
is an embedding,
then
V(A,B) .... --+ KV(A,B )
I
,
4,
4,
Z(Im,[B!) ----+ n v ( I m , I B
I)
is a pullback. We may put these t o g e t h e r to conclude, Corollary
3.
U n d e r the h y p o t h e s e s of C o r o l l a r y 2, there is a canonical pullback,
A(A,B)----+ K A ( A , B m) + • [IA,,B]
(3.13) Proposition.
Let
But
, else the induced map w o u l d not be
+
.... n[ I m , B j
A , B £ A , C 6 ~ . T h e n there is a canonical inclusion
31
A(A,A(C,B)
Proof.
As n o t e d in
c A(C,A(A,B))
(3.11) it is s u f f i c i e n t to show that there is a canonical inclusion
V(A,A(C,B)) Let
{C~ ÷ A}
.
and
{B + D~}
k n o w from 3.9 that the family
dominate
c V(C,A(A,B)) A
and r e p r e s e n t B , respectively.
{A(A,B)--+
(C ,D~)}
represents
~(A,B)
.
T h e n we T h e n from
C o r o l l a r y 2 above there is a c a n o n i c a l p u l l b a c k V(C,A(A,B)) ----~ nZ(C , (C ,D~))
!(Icl ,~(A,B)) ~
~!(lCl , (C ,D~))
and it is thus s u f f i c i e n t to find c a n o n i c a l maps ~,~
and
V(A,A(C,B))---+ ~( ICI , ~(A,B))
the c o m m u t a t i o n of the square.
.
V(A,A(C,B)) --+ V(C, (C ,D~))
for all
The fact that they are canonical implies
The first is the c o m p o s i t e
V(A,A(C,B)) ~
V(C
, (C,D0)) ~ Z(C, (Cw,D~))
and the second the composite
V ( A , A ( C , B ) ) - + Z(A,[ IcI ,B] w h e r e the first map comes from C o r o l l a r y 4 of
~
(IcI ,V(A,B))
(3.5) and the second is the cotensor ad-
junction. Corollary.
A map
(3.14) Example. discrete f i e l d
A ÷ A(C,B)
exchanges to a m a p
C ÷ ~(A,B)
We again c o n s i d e r the c a t e g o r y of t o p o l o g i c a l vector spaces over the K .
For both p o s s i b l e choices o f
C
and
is discrete w h i c h is that o f u n i f o r m c o n v e r g e n c e on all of of all subsets o f
C .
When
C = D =
the later d e v e l o p m e n t t h a t the functor a * - a u t o n o m o u s structure. pact and tor
.
D , the t o p o l o g y on a C .
finite d i m e n s i o n a l spaces, A(-,-)
=
~(A,-)I
commutes w i t h limits.
#(C)
(C,D)
consists
it w i l l follow f r o m
d e s c r i b e d in this section a l r e a d y gives
W h a t w e w i l l do here is to show that w h e n
D = d i s c r e t e spaces, w e do not get a c l o s e d category.
V(A,-)
Thus
Thus to show that
limits, i t is s u f f i c i e n t to show it has the r i g h t topology.
For
C = l i n e a r l y comA
A(A,-)
f i x e d the funccommutes w i t h
But this follows from
(3.9)
It is n o w a n easy a p p l i c a t i o n o f the s p e c i a l a d j o i n t functor t h e o r e m to show that ~(A,-) has an a d j o i n t w h i c h w e will designate
-~A .
Since
A(A,-)
the a d j u n c t i o n is strong w h i c h means t h e r e is a n a t u r a l e q u i v a l e n c e (B,E))
a n d if
A(-,-)
is a V--functor, ~(A®B,E)
= ~(A,
is to b e a c l o s e d c a t e g o r y structure this will h a v e to lift to
a natural equivalence
A(A®B,E) ~ A(A, (B,E)) Let us now suppose that this e q u i v a l e n c e w e r e valid.
T h e n for
E = K , w e get
(A~B)*
32
A(A,B*)
. Let
~(KT,s'K) ~ reflexive,
that
topology on tinuous
A = KT
(TxS)-K
and
B = KS
so that
I A®BI ~
A®B
with
S
~SxTI.
.
T
infinite sets.
Then
(A,B*)
This implies, since e v e r y space is quasi-
L o o k i n g ahead, w e will see in the next chapter the
is c o a r s e r than that of
(= uniform).
and
(A®B)** = K SxT
(A®B)**, i.e. that
(A®B)** ----+ A®B
is con-
Since every map from a linearly c o m p a c t space is closed, this is
an i s o m o r p h i s m and so
A®B ~ K SxT
.
The a l g e b r a i c tensor p r o d u c t o f
c o n s i s t s of those functions
SxT--+ K
fi[K S, g f ~ K T which, w h e n
and
S
T
w h i c h have the form are infinite,
IAI and IBI n (s,t) [ => i~Ifi(s)gi(t) ,
is not all of K S×T .
the a l g e b r a i c tensor p r o d u c t e q u i p p e d w i t h the topology i n d u c e d by
N o w let
E
be
K SxT. The identity
map
IKSI®IKTI
,
IE
t r a n s p o s e s to a m a p IKS I --'+ V( IKTI On the o t h e r hand,
,IE!
the i s o m o r p h i s m KS®K T
c o n s t r u c t e d above o n the h y p o t h e s i s that nonical m a p
> ~
K SxT
was a c l o s e d c a t e g o r y transposes to a ca-
KS___+ A ( K T, KSxT)
w h i c h gives ~ S l _ _ ~ v ( K T ' K SxT) From
.
(3.12) w e get a m a p IKSI --+ ~(KT,E)
Since
~(KT,E)
is e m b e d d e d in
A ( K T, K S×T )
.
the above map
KS
) A ( K T, K SxT)
factors
as a m a p KS---~ A ( K T, E) w h i c h t r a n s p o s e s to K S ® K ~---+ Since A(-,-) (3.15)
K S ® K T ~ K SxT
this implies that
E .
E ~ K SxT
w h i c h is, as n o t e d above,
false.
Thus
does n o t give a n internal hom. The c o u n t e r e x a m p l e suggests its own resolution.
The p r o b l e m arises out of the
n e c e s s i t y of the tensor p r o d u c t of two objects of
C
first that w e stick to the full s u b c a t e g o r y of
c o n s i s t i n g of complete objects. For
A
to lie in
C .
in that case w e w o u l d b e forced to use a c o m p l e t e d tensor product. o r - l e s s obvious, is
KS×T£~
the
E
as required.
Since, as is m o r e -
c o n s t r u c t e d above is the actual tensor product, However,
this leads to another diffficulty.
sure that the dual o f a complete o b j e c t is complete. discontinuous
This suggests at
(or non-uniform)
o t h e r d i f f i c u l t i e s as well.
its c o m p l e t i o n
We c a n n o t be
It c o u l d be c o m p l e t e d
only by
maps w h i c h is o b v i o u s l y u n d e s i r a b l e and w o u l d cause
This p r o b l e m is s o l v e d by u s i n g a m o d i f i e d notion of com-
p l e t e n e s s w h i c h does not get in the way as much w i t h the duality.
C/{APTER ]
C A T ~ G O R Y G.
i. C o m p l e t e n e s s .
(i.i)
We suppose
henceforth
in their u n i f o r m i t y . c l o s u r e of A A~
of
A
belongs
.
to
in that e m b e d d i n g The h y p o t h e s e s UnV
we suppose
C *~--+ A*
in d e c i d i n g
suffice
to
A
whether
C
t h a t if
object
as w e l l as these of A
is e m b e d d e d
and is,
in fact,
to g u a r a n t e e
that
in
D
~D~
are c o m p l e t e , then the
the u n i f o r m A~
completion
is a d m i s s i b l e
so it
.
A~A
is a p r o p e r
(or e q u i v a l e n t l y
Since a completeable
cient,
of
t h a t if
is also c o m p l e t e
of 1.3.10
and t h e n e v i d e n t l y
In a d d i t i o n the i n d u c e d m a p (1.2)
that the o b j e c t s
F r o m this it follows
closed
s u b o b j e c t of a
V(C,T)---+ V(A,T))
is d e n s e l y
embedded
C6~ then
is not injective.
in its completion,
it is suffi-
every d i a g r a m l
-+ B 2
A in w h i c h map
B1
B~--÷ A
sion.
) B2 .
is a dense
Take the i d e n t i t y
is the i d e n t i t y Since
A
embedding
TO see this take
on
A
is dense
in
can b e c o m p l e t e d
B1 = A
, B 2 = A~
for the v e r t i c a l
, the maps A~
A--+
map.
A
~
, they are equal
If there
A~
to a c o m m u t a t i v e
and the m a p b e t w e e n is a r e t r a c t i o n
a n d the i d e n t i t y
and the i n c l u s i o n
of
on
d i a g r a m by a
t h e m the incluA~----+ A
A~
which
agree o n
A--+ A~
A
.
is an isomor-
phism. Later
in these notes w e w i l l h a v e o c c a s i o n
ness g o t t e n b y r e s t r i c t i n g us suppose complete a
that
~
the class of
is a class
g : B2---+ A
(necessarily
m
(see
Proposition. (II.3.7)).
Proof.
We h a v e
elements
of
Ai
Then that
Suppose AA
as these of
is
A
.
The obvious
S6~(A*),
ded
, aeA
is r e f i n e d b y the u n i f o r m
f : B~---+ A
is dense)
AA
such t h a t
is
Let ~ -
, there
is
gm = f .
has a c o n v e r g e n c e
and
in
F(a,d,S)
V
.
We m a y thus
t h i n g to do h e r e
is a b a s i s of u n i f o r m
{F(a,d,S) where
and
is an i s o m o r p h i s m
is the set of p s e u d o m e t r i c s
d
and e v e r y
A
uniformity
~ - complete. ~ IAI
in
If
~
~ - complete
to s h o w that there
.
in
of c o m p l e t e -
is required.
We say that an o b j e c t
since the e m b e d d i n g
do that w e h a v e A
notions
follows.
A
is
I AAI
weaker
for w h i c h a fill-in
embeddings.
: BI- ~ B 2
unique
The m a i n p o i n t of this g e n e r a l i t y (1.3)
BI--+ B 2
of dense
if for e v e r y p s e u d o m a p
to c o n s i d e r
on
T , then a b a s i c I aeA
= {b
is to use
covers of
uniform
F#(a,d,S)
= {b
=
[ aeA}
I d(~a,~b)
~ {b ~eS
(I.2.5).
the To
b y sets c l o s e d cover of
A A is
} I d(~a,~b)
< 1
for all
cover {F#(a,d,S)
AA
identify
, where ~ ½ , for all
] d(~a~b)
~ %}
~eS}
~£S}
.
Th~s
34
which {b
is an i n t e r s e c t i o n
I d(~a,~b)
S %}
of c l o s e d
sets
is the inverse A ~
Moreover (1.4)
F#(a,d,S)
intersection that a m a p
o f all
~
~ - complete
It does
[ 0, %]
(~a,ld)
induce
d
of
A~
f : A---* B
A~A
which
induces
A~
÷ B~
that
~
and an o b j e c t
such a f u n c t i o n
To see that o b s e r v e
the m a p
and is thus a u n i f o r m
sabobjects
pseudomap)
closed. under
T×T
F(a,2d,S)
of dense e m b e d d i n g s
(respectively
~A---* ~B .
T
is r e f i n e d b y
G i v e n a class
and t h e r e f o r e
image of
cover.
, let
contain
a map
~A
A .
denote
the
We w i s h to s h o w
(respectively
pseudomap)
so that w e h a v e a c o m m u t a t i v e
square A ~
~A
~. A ~ f
B----+ ~A-----+ B and w h a t w e n e e d is a fill-in that
f~ -I(~B)
hence
~A
observe E1
.
So let
that
g
is d e n s e l y
extends
fg
El----+__E 2
possesses embedded
.
h
lies in
We let
~A----* A
Since
f
denote
k
(1.7) We n o w suppose mention
(~B)
that
function
for w h i c h
Ce~
satisfy
hypothesis
v) below.
A
~
on
k ~B,
and
f~k
so d o e s
In one example, We s u p p o s e A---+ C
~ ~A
extends
which
~A
f~g
and
which means
to a V - f u n c t o r
is the image of
for a n y
however
~ , then
B
iii)
If
A---* B
belongs
to
~ , then
B*
If
A--+ B
belongs
to
~ , then
B
~
extend f~h
and
which
~ .
V~V
on
A
.
The i n c l u s i o n ~ .
If
The inclu-
A~A
, then
. Unless
there
is
(see
(IV.4)ff.)
this class does not
the f o l l o w i n g
to
object
First
is taken to be the class of dense e m b e d d i n g s
embedding
is - t h e n
both
: E~--*~B
is a given class o f dense embeddings.
belongs
usually
I
[V,A]
A---+ B
We note t h a t s h o u l d
E1
k
and
is c o m p l e t e
is V--enriched.
If
~-complete
is a p s e u d o m a p
A
be a map.
A~
to also d e n o t e b y
Every dense
Every
since
find c o n v e n i e n t
ii)
v)
~ f~ -I(~B)
~ A~
the a d j o i n t n e s s
(II.l.4),
to the contrary,
A--~ C
iv)
there
subset
: E1
: E2
to s h o w
contains
Thus we h a v e proved,
and h e n c e
~
g
h
s~fficient
for it c e r t a i n l y
and
the full s u b c a t e g o r y
u s e d in
.
~
it is c l e a r l y
A~
takes v a l u e s
h a s a l e f t a d j o i n t which w e
by the same a r g u m e n t
i)
to
. In a d d i t i o n
The o b j e c t
~ A
of
extension
2 to t h e dense
sion is V--full a n d faithful
specific
belong
E
everywhere.
(1.5) P r o p o s i t i o n . (1.6)
in
F o r this
subobject
a unique
Restricted
hence a r e equal the image o f
the middle.
is a b-complete
with
C£C
belongs
is e m b e d d e d
to
~ ;
in an o b j e c t of
~
;
is complete; is p r e r e f l e x i v e ;
is q u a s i - r e f l e x i v e . be stable
~-completeness
under the f o r m a t i o n o f closed
for any class
~
satisfying
subobjects
the above
- it
is e q u i v a l e n t
# to
~ -completeness
A + B
~
completion
and A~
tainly implies
B .
for the class is e m b e d d e d
in
The p o s s i b i l i t y
the p o s s i b i l i t y
~# C6C,
of dense e m b e d d i n g s t h e n the c l o s u r e
of e x t e n d i n g
of e x t e n d i n g
of
A----~ C A
a map defined on
it to
B .
in A
with C
C~C.
F o r if
is its u n i f o r m
to all of
A t cer-
35
(1.8) We say that
A
(1.9) ProlDosition.
is Let
A
complete and reflexive. 6A
~-,-complete p r o v i d e d b e ~-complete.
A*
Then
is
6A =
~complete. ({A*)*
is {-complete,
M o r e o v e r there is a canonical b i j e c t i o n
~-*-
~A---+ A , m e a n i n g that
has the same u n d e r l y i n g V--object w i t h a p o s s i b l y finer uniformity.
Proof.
The inclusion
A*---+ ~A*
induces
dense i n c l u s i o n a n d
TeD
morphism.
16A I ~ -> ~**I -
That is,
is complete,
reflexive w h i c h means that
A** =
there is a c a n o n i c a l b i j e c t i 0 n be given.
since
A
6A =
(~A*)*---~ A**
the induced map
Since
AA
A
is
and so
@A--+ A .
.
A*---~ ~A*
~(~A*,T)---+ ~(A*,T)
~-complete,
A**----+ A
N o w let
Since
is a
is an iso-
it is b y h y p o t h e s i s quasi-
exists and is bijective.
BI~
B2
b e l o n g to
6
and
Thus BI-+~A
We get, by c o m p o s i t i o n
is
B 1 ---+
A
B2----+
A
~-complete, A* ----+ B* 2 ~A*
since
B2
B2**
~ 6A
B2
> 6A
is ~-complete.
(~A)* =
B2----+ B 2**
is a s s u m e d p r e r e f l e x i v e m e a n i n g there is a canonical m a p
easy to check that the r e s t r i c t i o n to ~A*
B2
is a s s u m e d complete,
B2
since
÷
(~A*)** =
Since
(~A*) A
~A*
B1
is the given map
is ~-complete,
) ~A*
B1
-> 6A .
.
It is
This shows that
it is e v i d e n t l y quasi-reflexive.
Thus
w h i c h upon d u a l i z i n g gives the canonical map ~A----+ (6A)**
and shows that
~A
it is reflexive. so that (I.i0)
~A
Finally,
Let
C£C
Let
p
, A(C,B)
{B-+ D }
r e p r e s e n t a t i o n of
~A*
As it also is ~-complete, h e n c e q u a s i - r e f l e x i v e
is {-complete, hence so is
is
b e a class of dense inclusions and
({A*)** =
(6A)*
A(C,B)
B .
Then
be
{~(C,B)--~
(see II.3.10, Corollary). F o r any
1 V(E,[ICI ,B] ) (see I I . 3 . 1 2 , C o r o l l a r y 1
it induces isomorphisms
B
p - complete.
p-complete.
b e a D - r e p r e s e n t a t i o n of
V(E,A(C,B))
is a p u l l b a c k
(~A*) A =
~-*-complete.
Proposition.
Then for any Proof.
is
is p r e - r e f l e x i v e .
EeA
(C,D)}
is a D -
the d i a g r a m
~ ~V(E, ( C , D ) )
I , ~V(E,[ ICl ,D~) and apply V(E,-)) . If
V(E2,(C,D ~))---+ V(EI, ( C , D ) )
,
V(E2,[ ICl ,B]) ~ V( Icl ,V(E2,B)) ---- V(ICl,V(EI,B)) ----V(EI,[Ic~B] ) ,
E1
> E2
belongs to
~,
36
!(E2,[ ]C] ,D ]) ~ ! ( [ C ] , V ( E 2 , D
))
Z(]C] ,!(EI,D ))~ ~ ( E l i [ C [ ,D )) since
(C,D)
B
and
are the same w h e t h e r Corollary.
Let
A
D
are all
E = E1
or
p - complete. E = E2
b e reflexive and
,
But then three o f the four vertices
and hence so is the fourth
~-,-complete.
T h e n for any
(the pullback).
D~D
, A(A,D)
is
~-
complete. Proof.
D u a l i z e and use
(1.11)
Theorem.
(A,B)
is
Proof.
Let
(II.3.6). A
and
~-*-complete
b e r e f l e x i v e and
B
Then
~-complete.
A
~-complete.
P r o c e e d e x a c t l y as in the p r o o f of
(1.10)
e x c e p t replace
C
by
A .
The pull-
back is V (E,A (A, B) )
~V(E,A(A,D
l
1 > ~V(E,[ IAI,D])
V_(E,[ IAI ,B] )
with the above c o r o l l a r y p r o v i d i n g the n e c e s s a r y (1.12)
Example.
))
~-completeness of
We r e t u r n o n c e m o r e to the e x a m p l e of v e c t o r spaces.
are the finite d i m e n s i o n a l spaces, t o p o l o g i z e d discretely, ly complete.
The d e f i n i t i o n o f
of a space in
Those of
crete spaces.
D
As well, D
Not e v e r y space is
and
t h e n the spaces are c e r t a i n -
then,
is e v e r y space
~-,-complete.
consists of discrete, and
~-complete.
~-complete.
c o n s i s t i n g o f the elements of the V
~
~-completeness requires that w e b e g i n w i t h a subspace
C
are e v i d e n t l y c o m p l e t e as are those of
c o m p a c t space cannot b e
that space
When
Thus the r e q u i r e d map e x t e n s i o n p r o p e r t y is t r i v i a l l y s a t i s f i e d and ~-comPlete.
tion is q u i t e d i f f e r e n t w h e n ces.
).
~ - n e c e s s a r i l y finite d i m e n s i o n a l - and a dense subspace - n e c e s s a r i l y
the w h o l e thing. every space is
~(A,D
The situa-
o f l i n e a r l y compact spa-
C
w h i c h are products o f d i s -
A n y dense p r o p e r subspace o f a linearly
For example the s u b s p a c e o f
S-fold d i r e c t sum is not
KS ,
~-complete.
S
infinite, If we call
, w e have S.K---+ V---+ K s
w i t h the first m a p b i j e c t i v e and the s e c o n d a dense embedding.
D u a l i z a t i o n gives
S . K - - ~ V*---+KS The first m a p b e i n g the dual of a dense e m b e d d i n g is a b i j e c t i o n and the second is evidently dense.
We k n o w it is dense as soon as w e k n o w that
finite d i m e n s i o n a l linear c o m p a c t subspaces. crete and factor through
S.K , whence
(S.K)*
and
This is p r e s u m a b l y always true b u t c e r t a i n l y is if finite.
V
does not c o n t a i n any in-
F o r then all the V* S
C-+ V
are w i t h
S-K
is countable.
dis-
are e m b e d d e d in the sum space. and
K
are b o t h c o u n t a b l y in-
F o r any infinite d i m e n s i o n a l l i n e a r l y compact space, b e i n g a p o w e r of
uncountable while
C
Thus in that case
V* ~ V
and so
K , is
V** ~ V
. The
former i s o m o r p h i s m is i n d u c e d b y the s t a n d a r d i n n e r p r o d u c t and h e n c e the latter one is
37
induced by the canonical map. V
is reflexive.
If
V
This shows that
is closed in
and hence has continuous functionals. vanish on
V
W , W/V
(2.1)
~-,-complete either, but that
Thus there are non-zero functionals on
V , confirming the hypothesis made in
2. D e f i n i t i o n
is not
is a separated linearly topologized space
and Elementary
Properties
of G
which
.
It will be a standing hypothesis in this chapter that
jects are quasireflexive.
W
(i.i).
~-complete ob-
The demonstrations of this fact in the various examples seem
quite different - save for the cases in which have any common generalization.
(II.2.8) is satisfied - and do not seem to
It is clear that the hypotheses of
(II.2.8) are not al-
ways satisfied. (2.2) We let
~
denote the full subcategory of
flexive,
~-complete and
both
~
and
fact
Un~
D
~-*-complete.
are subcategories of
A
It follows from hypotheses w e h a v e ~ .
Also
G
is complete, essentially by hypothesis
adjoint functor theorem.
A
consisting of objects which are remade that
is complete and cocomplete.
(see
In
(I.3.10)) and cocomplete by the
being reflexive is also complete and cocomplete as is ~A.
The last step follows from the next proposition. (2.3)
Proposition.
Proof. w e have
That
6A
A*---+ G*
The inclusion
belongs to
G
and with
G--+ ~A
has a right adjoint
follows from
G*
A*
Now if
~-complete, we have
Uniqueness follows from the fact that the maps first because
(1.9).
is densely embedded in
~A*
A
G£G
~A *---+ G*
~ and
whence
(~A*)*---+ A**---+ A and
6A . G + A
is a map,
G ~ G**--+(~A*) *.
are bijective,
the second because
A
is
the
~-com-
plete, hence quasi-reflexive. (2.4)
Let
CI,C2£ ~ .
The identity map
(Cl,C~)
~ (Cl,C~)
transposes , by the corollary of II.3.13 to a map
C1 Applying
[-I
' ((C1,C~),C ~) ~ (C2, (C1,C~)*)
we get a map in
•
V
~1 I---+ !(c2, (Cl,C~)*)----+ !(Ic21,1(Cl,C~)*
ICll Let
T(CI,C2)
(2.5) Proof. if
~ ]c21"-"~
I(c1,c~)*
I)
I .
denote the image of that map as an embedded subobject of
Proposition. (CI,C~)
For any
lies in
D
CI,C2£ ~ , T(CI,C 2) is dense in so its dual is in
~ .
(CI,C~)*
By the hypothesis made in
T(CI,C 2) is not dense, its closure is proper and the induced map !(CI,C~)
is not injective.
> !(T(CI,C 2),T)
But the map !(Cl,C~)
> !(T (CI,C2) , T ) ~
(CI,C~)*
i( IC1 I®
Ic21,ITL)
(i.I)
88 can also be factored as
V(Cl,C~)
v(ICll, Ic~l)
....
V(IC II,V(C2,T))---+ V(]C l ),V_(IC2 ), )T)))
v(Ic ll~Ic 21, ITI) , each term of which is injective.
Since the first factor of an injection is an injection,
the result follows. (2.6)
Proposition.
Let
CI, C2eC .
For any
AEA
there is a canonical map
V(CI,A(C2,A))-~ V( ~(CI,C 2) I ,IAI ) Proof.
Let
image of
{A--~ Din} be a D-representation_ of
A .
Since
IT(CI,C 2) [ is a regular
~i ~ [C2[, there is a pullback V_( IT (CI,C 2 ) I,~I )----+ HV_( IT (CI,C2)] , ) O ))
t
1
v( ~Cl~ Ic21, IAI ) ..... nv(~cz;~Ic21, ID) ) Then map V_(CI,A(C2,A)) ~
-~
HV(Cl, (C2,D m))
~v(c1. (D~,C{~---- ~V(D~, (Cl,C?)
--~ nv((Cl,C~)*,D ~) ~ ~V( I(Cz,C~)* (,ID -~
~Z( I~(Cl,C2)l, I
Also map
Z(Cl,#(c2 ,A) ) ---+ Z(~ l I, Z(c2,A) ) .... Z(IcII,Z(I¢21,1 m )) ~ ~(IcI l~ Ic2;,lal ) Since each of these maps is canonical, they give the same map to
]]V_(ICII®IC21,~3 I) and
hence the map required. (2.7)
Pr__~gsition.
Let
C I, C2~C_, A£A .
Then there is a canonical map
V(CI,A(C2,A) )---~ V(T (CI,C 2) ,A) Proof.
Let
{A---~ D }
be a D__-representation of
A .
By (If. 3.12), there is a pullback
V(T (CI,C2),A)----~ RV(y (CI,C2),D)
1
l
V(I T (CI,C2)I ,I AI )--~[[V_(Ir (CI,C 2) [ ,[DJ ) We map
V(CI,A(C2,A)) ~
--- ~v(c l,{D~,c~)
I[V(CI, (C2,D))
~V(D~,{Cl,C~))
KV((C1,C~)*,D )---~ ZV(T(C I,C2),D )
39
while
(2.6) provides
V(CI,A(C2,A))---~ V(IT(CI,C2)I,IAI)
fromwhich
we have the required
map. Corollary.
If, in addition,
A is ~-complete,
there is a canonical map
V(C I,A(C 2,A) )---+ V( (CI,C~)* ,A)) (2.8) Proposition.
Let
CI,C2E~A
.
Then the canonical map
V(CI,A(C2,6A))
~ V(CI,A(C2,A))
is an isomorphism. Proof.
The inverse of the canonical
map is the composite
Z(CI,A(C2,A))
~ !((CI,C~)*,A)
--~ !(A*, (Cl,C~)) ~ !(~A*, (Cl,C~)) !(CI,A(~A*,C~))
÷ Z(CI,A(C2,~A))
where the next to last map is the V - m o r p h i s m (2.9) Proposition.
Let
Cc~,
GeG, A 6 ~
.
underlying
the map of II.3.13.
Then the canonical
V(C,A(G~A))
map
.7 V(C,A(G,A))
is an isomorphism. Proof.
Let
{C
) G}
be a C--representation of V(C,A(G,6A))
G .
~ HV(C,A(C
1
Then there is a pullback ,~A))
1
V(ICI,V(G,6A)) . . .
) HV(ICI,V(C
.
,~A)) ¢0
We map V(C,A(G,A)) as established
in
(2.8).
, HV(C,A(C
,A))
~ nV(C,A(C
,~A))
Also we have
V(C,A(G,A))
~ V(ICI ,V(G,A))
~
V(ICI,V(G,6A))
This gives the required map. (2.10) Proposition.
Let
C£C, D e D A(G,
and
(C,D))
G~G
.
Then the canonical
map of
~ A(C,A(G,D))
is an isomorphism. Proof.
As o b s e r v e d in
(II.3.11)
it is sufficient
to show that
V(G, (C,D))---+ V (C,A (G,D)) is an isomorphism.
The inverse of the canonical
map is given by
V(C,A(G,D))---+ V(C,A(D*,G*))---+V_((C,D)*,G*) the second map coming from the corollary (2.11) Proposition.
Let
A,B~A, G e G .
to
~ V(G, (C,D))
(2.7).
Then there is a canonical
map
(II.3.13)
40
A (A,A (G,B)) Proof.
As o b s e r v e d
in
(II.3.11)
~ A (G,A (A,B))
it is sufficient
V (A,A (G, B) ) Let
{C----~ A}
and
{B---~ D }
to have a canonical
map
~ V (G,A (A, B))
be a C - d o m i n a t i o n
and D-representation,
respectively.
Then there is a pullback V(G,A(A,B))
> KV(G, (C ,D ))
V(IGI ,V (A, B) )---~ KV([GI,V(C We have a canonical
map
V(A,A(G,B)) by
(2.10).
,D )) .
~ EV(C
,A(G,D~)) ~ EV(G, (C ,D~))
As well we have V(A,A(G,B))----~ V(A,[I GI, B] ) ~ V(IGI,V(A,B))
induced by the embedding Corollary
i.
Let
A(G,B)----+ [ IGI,B]
AeA, G, H 6 G .
Then there is a canonical
A(G,A(H,A)) Corollary
2.
Let
A£~A, C£~, G ~ G .
isomorphism
~ A (H,A(G,A))
Then the canonical
V (G,A (C,6A))
map
~ V (G,A (C,A))
is an isomorphism. (2.12) Proposition.
Let
Ae~A
, G, H£G .
Then the natural map
V (H,A (G, 6A) )
~ V (H,A (G,A))
is an isomorphism. Proof.
Replace
C
by
H
everywhere
in the proof of
V(H,A(C,~A)) is e s t a b l i s h e d Corollary. Proof.
in corollary
A map
Apply
(2.13) Proposition.
Let
~ V(H,A(C~,A))
is equivalent
to a map
H--~ A(G,A)
to the above. A,B,E£~A
.
Then there is a canonical map
V (A,A (B,E)) Proof.
The necessary
2 above.
H---+ A(G,~A)
Hom(I,-)
(2.9).
V(6A,6A(~B,6E))
We have V (A,A (B,E))
~ V (A,A (6B, E) )
V (6B,A_ (A,E))--+ V_(6B,A,(6 A, E) ) V(~B,A(~A,6E))---~ V(~A,A(6B,6E)) V(~A, ~A (~B, 6E) )
.
isomorphism
41
Here the first a n d t h i r d a r r o w are i n d u c e d b y b a c k adjunctions,
the second and fourth
from 2.11, the fifth f r o m 2.12 and the last i s o m o r p h i s m is the one given b y the adjunction. Corollary. Proof.
U n d e r the same h y p o t h e s e s a m a p
A - - + A(B,E)
gives a map
6A--~ ~A (6B,6E) .
A p p l y Hom(I,-) to the above.
3.
(3.1)
Let
= 6A(G,H)
G,HcG .
.
The Closed M o n o i d a l Structure on
Since b y
Note that since
[G(G,H) I ~ V(G,H)
.
(i.ii), A(G,H)
is ~-complete,
is b i j e c t i v e for all
6A---+ A
In p a r t i c u l a r e v e r y e l e m e n t of
G (G,H)
G
6A(G,H)~G . Ae~A,
We define G(G,H)
it follows that
is a u n i f o r m pseudomap. This
explains the p r i n c i p a l a d v a n t a g e of u s i n g ~-completeness i n s t e a d of completeness.
No
n o n - u n i f o r m maps n e e d b e a d d e d to h o m o b j e c t to make it ~-complete. (3.2) Proposition. Proof.
By
Let
G,H~G .
T h e n there is an e v a l u a t i o n map
G - ~ A(A(G,H),H)
.
(2.11) there is a canonical map A(A(G,H),A(G,H))
~ A(G,A(A(G,H),H))
.
The r e q u i r e d m a p is, o f course, the image of the identity. (3.3) Proposition.
Let
G,H,K6G .
T h e n c o m p o s i t i o n of m o r p h i s m s determines a canoni-
cal map A(G,H) Proof.
F r o m the d i s c u s s i o n following
into w h i c h b o t h from
~
A (G,A (A (H, K), K) )
and
A(A(H,i),A(G,K)) (II.3.11) it follows that there is a single A (A (H, K) ,A (G,K))
m a y be embedded.
A£A
Moreover,
(II.3. 12 ) it follows that
is a pullback.
V (A (G,H) ,A (A (H,K) ,A(G, K) ) )
V (A (G,H) ,A)
V(V(G, H) ,V(A(H,K) ,A(G,K) ) )
V(V(G,H), ]AI)
N o w the p r e c e d i n g p r o p o s i t i o n p r o v i d e s a map
n
~ A_(A (H,~) ,K)
to w h i c h we may apply the functor A(G,H) w h i c h is an e l e m e n t o f
A(G,-)
to get
~ A(G,A(A(H,K),K))---~ A ,
V(A(G,H),A)
.
From
(II.3.4), w e h a v e a map
V(G, H)----+ V ( A ( H , K ) , A ( G , K ) ) w h i c h expresses the fact that
A(-,K)
is a V--functor.
This is a canonical element o f
V_(V(G,H) ,V_(A(G,H) ,A(G, K) ) ) a n d b o t h b e i n g canonical give the same element o f element of and
V(A(G,H),A(A(G,H),A(G,K)))
Hom(I,-)
commutes w i t h p u l l b a c k s ,
.
V (V (G,H), IAI)
Thus we get an
Since the elements we b e g a n w i t h were maps
this e l e m e n t too is a map
42
A (G, H)---~ (3.4) Proposition.
Proof.
Let
G,H,K£G
.
A (A (G, H) ,A(G,K) )
Then c o m p o s i t i o n gives maps
G(G,H)
>
G(G(H,K),G(G,K))
G(H,K)
~
G (G (G, H) ,G(G,K) )
The first comes by a p p l y i n g the c o r o l l a r y o f
comes f r o m t r a n s p o s i n g
G(G,H)---~ G(G(H,K),G(G,K))---+
G(H,K)---+ A ( G ( G , H ) , G ( G , K ) ) (3.5)
Theorem
Proof.
Since
G---+ G(H,K*)
G
(2.11) to the above.
and a p p l y i n g
~
(G(H,K),G(G,K))
The s e c o n d
to get
once more.
is ,-autonomous.
I£C, I £ G .
If
, t h e n we h a v e
G e G , G* ~ G~
(G,I)£G
G(H,K*)~
and h e n c e
G(K,H*)
and
G* = G(G,I)
K~
If
G(G,H*) ~ G(H,G*)
.
T h e n b y I°4.4 we h a v e all the data required.
4.
(4.1)
StuJmary of the Hypotheses.
The h y p o t h e s e s u s e d in this c o n s t r u c t i o n are rather complicated.
In a d d i t i o n
they are s c a t t e r e d all o v e r the p r e c e d i n g three chapters, b e i n g i n t r o d u c e d as needed. Thus it seems useful to collect in one place a summary of these h y p o t h e s e s and a reference to a fuller e x p o s i t i o n o f each. under which
(3.4) is proved.
It is u n d e r s t o o d that these are the a s s u m p t i o n s
In some cases more r e s t r i c t i v e forms of these h y p o t h e s e s
are s t a t e d that are s a t i s f i e d in some of the examples.
This will be m e n t i o n e d in the
examples. (4.2)
The h y p o t h e s e s are (i)
V
is an a u t o n o m o u s
(i.e. closed, s y m m e t r i c , monoidal)
category
(see
(I.l.l), (I.i.2)) . (ii)
V
is a semi-variety;
that is a full s u b c a t e g o r y of a variety closed u n d e r
projective limits and c o n t a i n i n g all the free algebras (I.i.8)) (iii)
a n d that the h y p o t h e s i s o f
The s u b c a t e g o r i e s
C
and
D
a pre-*-autonomous situation representation (C,D) (iv)
Every (see
(v)
(see
C~C
C*--~
is not injective; C
(see
(I.3)) have the structure o f
(see
(II.3.7))
(III.l.4) and
and every c l o s e d p r o p e r s u b o b j e c t
C
has a
D--
•
(III.l.7)) is p r e r e f l e x i v e
equivalently
C*---+
A---~ C Aw
and every o b j e c t of
D
, the map
injective iff
(see(iII.l.l)).
E v e r y o b j e c t in
C~C
(II.2.9)).
F o r every A*
(I.i.4) -
(II.l.2)); m o r e o v e r the u n i f o r m i t y on the objects
~-complete o b j e c t (II.2.7) also
Un V
(see 1.4.6.) and that e v e r y
is a c o n v e r g e n c e u n i f o r m i t y
dense in (vi)
(see
of
(see
(I.3.10) is satisfied.
is complete.
A
is
CHAPTER
Before Top M o d R
getting
of t o p o l o g i c a l
In fact, A
, let
into e x a m p l e s
u(M)
A
= {a+M
neighborhoods f i n e m e n t of
let
of
R-modules
is e q u i v a l e n t
be a topological
I a£A}
.
.
module.
The c o l l e c t i o n
To see that,
there
is a
c~A
with
N e x t I c l a i m that s u b t r a c t i o n traction special
of
~(N) x u(N)
then
In fact if
f-l(~(M))
To go the other way, logical
spaces d e s c r i b e d
1.8, p.17),
that the u n i f o r m the o r i g i n a l a uniform
topology
topology.
group
group and let
is a u n i f o r m
cover
~
u
M
u
VI,V 2 6 ~ 0
in
T h e n for all
~
V~v
u
cated c o m p l e t e l y
for t o p o l o g i c a l in the e a r l i e r
clear that other choices follows we t a k e
KS
Let
~
K
and
of
K = ~
D
gized w i t h l o c a l l y
(1.3) for
for
.
M
b~a+M
.
, the image under sub-
topological and
or
is u n i f o r m
modules
is a
induces a
a neighborhood
of
0
in
from u n i f o r m
their
spaces
products
continuity.
It is clear
from a topological
of a n e i g h b o r h o o d
(Let
of
By the u n i f o r m i t y
M
there
is a
contain
0
group is
u
.
cover
.
with
in
So let
of a d d i t i o n
U£u
st(0,w__)
is in some set in
to topo-
([ Isbell ] ,
G
there
V I + V 2 c U.
for some star
In p a r t i c u l a r
for
I a£A}
or
on compact
convex D °p
V e c t o r Spaces.
and
K = •
D
However, will result
, always of
for special
of finite
consisting
choices
(explained below). K
of
theories.
K
, it is In w h a t
topology.
of finite
and c o u n t a b l e
is just the h o m into
field has b e e n expli-
in d i f f e r e n t
w i t h the usual
Top V
sum t o p o l o g y > C
spaces over a d i s c r e t e
Chapters.
C
be the c a t e g o r y
direct
and c o u n t a b l e sums
The d u a l i t y
, topologized
S.K
powers
topolo-
C °p---~ D
by uniform
as con-
sets.
In fact a c o m p a c t s£S
N-N c M
we m u s t s h o w that every u n i f o r m
implies
, V+M
vector
b e the s u b c a t e g o r y
well as its inverse vergence
is a star re-
.
The theory
(1.2)
in
and so
1. (i.i)
u(N)
N-N c M
group constructed
{a+M refines
implies
b e a u n i f o r m cover.
of
is in some set in
b-aE
functor
by cover b y t r a n s l a t e s
aEV
, a+M
v.)
if
map between
of the o p e r a t i o n s
of
0
Then
Since t h a t f u n c t o r p r e s e r v e s
such t h a t
is a n e i g h b o r h o o d
of
ranges o v e r all
scalar m u l t i p l i c a t i o n
just use the s t a n d a r d
refinement
M
, I claim
.
is continuous,
of a u n i f o r m
be a uniform
There
That
To go the other way,
is r e f i n e d
as
.
in 1.2.
the u n i f o r m i t y
u(M)
so t h a t
In fact,
.
f : A'----+ A
= u(f-l(M))
of u n i f o r m R-modules.
is a n e i g h b o r h o o d
N-N c M
, bEc+N
case of the fact t h a t a c o n t i n u o u s
u n i f o r m one. A
M
I a~c+N}
is uniform. ~(M)
Un M o d R
If
b£~*(a,N)
aec+N
refines
to
of all
For if
suppose b£~c+N
w h i c h means
EXAMPLES.
we need to k n o w that for any ring R, the c a t e g o r y
0 , is a uniformity.
u(M)
ZV~
the p r o j e c t i o n
s e t in
KS
o n the
s
has c o m p a c t coordinate
projection
is c o n t a i n e d
o n every coordinate. in the c l o s e d d i s c
Thus A(rs)
44 of radius
rs
a continuous
Hence
every c o m p a c t
linear m a p
Ks ~ ~ K
set is a subset of
K A(r s£S b y an e l e m e n t
is r e p r e s e n t e d
It is e v i d e n t
)
of
(Xs)
S-K
that
b y the
formula (x s) (Ys)
= ZXsY s •
Then the n e i g h b o r h o o d
{(x s) I (xs)nA(r s) < I} = {(x s) can b e e a s i l y d e s c r i b e d i/r s (or all of
K
as follows.
where
lYsi < r s
we h a v e
is a s e q u e n c e an e l e m e n t
ZysZ s = r < 1 .
Let
S-K
Xs6A°(i/rs )
~ilsXsYsl
< Zilsl
(ys)~KA(rs) value
Is = rsZs/r IXsl =
= 1 .
implies rs
denote
.
We h a v e
=
consi-
Zllsl = 1 . In other w o r d s o (i/r s) in the sum. Then
< 1 .
YsZs
S-K
A
On the o t h e r h a n d
Zllsl
=
the o p e n d i s c of radius
the subset of
and
l~YsZsl
such that
rlZsl/rslZsi
< i}
suppose
T h e n choose
(Zs)£S-K for e a c h s~S
is real and positive.
~IrsZsi/r
r/r s < i/r s .
= ZYsZs/r
This
Then
= i. If n o w
shows
that
*
~ KS
(KS) *
.
(1.3)
To go the o t h e r way,
only question vergence.
Proof.
Every
compact
Z
The p r o d u c t
on
K
on topological contradicts
set in
is a c o m p a c t
s = 1,2,3,...,n,... the s e m i n o r m on
(1.4)
first o b s e r v e
that a l g e b r a i c a l l y topology
on
KS
(S.K)
.
The
is that of p o i n t w i s e
con-
We b e g i n with,
Suppose
seminorm
we
is the topology.
Proposition.
there
S.K , p =
is a p o i n t
(pn)
vector
is in a finite d i m e n s i o n a l
Pn(X)
S.K
and
for
subspace.
countably
m a n y s, say
(x (n)) 6 Z w i t h x n # 0 . Let p = (ps) be s (n) = nix/x n I • T h e n since Pn is a c o n t i n u o u s
is a c o n t i n u o u s
spaces)
a n d suppose
s e m i n o r m on
p ( x (n)) a n s
.
Thus
S-K p
(this is s t a n d a r d is u n b o u n d e d
on
Z
result which
its compactness.
Now a compact
set in
S'K
subspace.
is a c o m p a c t
in a space
as the set of all e l e m e n t s
mensional
S'K
subset of
s u c h that
In fact it is c o n t a i n e d SO
A°(i/rs ) c K
F{A°(i/rs ) I seS}
such t h a t
of a b s o l u t e
x s = Zs/X s , we h a v e
~IXsrsl
c i r c l e d h u l l of the images of the
for w h i c h
Ys~K
Let
= 0) and
s (IsXs)
sting of all e l e m e n t s o FA (i/rs) is the c o n v e x if
r
I
of
A compact
S0-K S
set in
set in some
for some
necessary S0-K
finite
to e x p r e s s
is c o n t a i n e d
finite d i m e n s i o n a l subset
SO c S
a basis
.
subspaee.
Just take
for this finite di-
in a set of the f o r m
A~ = {~{~sS I s~s 0} I ~l~sl ~ ~} where
1
~(s)
< ~/I
is a fixed p o s i t i v e , ~llsl
< I
real number.
N o w if
~
is a linear
functional
and
implies
l~(~ss) l = IZ~s~(S) l < Zl~sl~/Z _< so t h a t the basic
neighborhood
of
0
{~I~(A~) < ~} ~ {~l~(s) < c/~ , s~s 0} , which
shows
that
(S.K)
h a s the t o p o l o g y of p o i n t w i s e
convergence
and h e n c e
is
KS
.
45
(1.5) and
The h y p o t h e s e s D
of III.4.2,
and thus are still
denote
the space of c o n t i n u o u s
p a c t subsets
of
Proposition.
C
parts
satisfied.
(i) and
maps
C---~ D
topologized
Let
w e let,
on the c h o i c e of for
C~C
, D{D
by uniform convergence
C
,(C,D) on com-
.
Let
C = KS
and
D = P.K
.
Then
(C,D) ~ Proof.
(ii) do not d e p e n d
For the third part,
ti-li : p.K---+ ~
(SxP)-K
b e the n o r m d e f i n e d
.
by
ll(Xp)il = % Ixpl . which
is d e f i n e d
be a continuous s
coordinate
since there are o n l y linear map.
(all other
For
finitely many non-zero
seS
, let
If
t h a t for at least one
peP
in
coordinate
in all o t h e r s
s
topology
also converge. converge
and
0
to the e l e m e n t But exactly
finitely many
cannot be a Cauchy
which
This shows
p
f
denote
sp s, an
.
Thus the
of o r d i n a r y
p
x
of finite infinite
converges
s
sums of
series
in the
f(Xs)
must
this net can
£ > 0 , it is the case that
(The p r o o f
is t h e same,
exceptions.
coordi-
x £K can be c h o s e n s the e l e m e n t w i t h xs
sums of
net
the
and every
P(Xs)<£
the finite p a r t i a l
sums
net otherwise.) f
s finitely many
involves
are non-zero. peP
.
Thus
Now
f
: K---~ P . K is d e t e r m i n e d b y s in all o n l y finitely m a n y fsp ~ 0 .
that a l g e b r a i c a l l y (KS,p.K)
tained
let
the net of finite
(x s)
as in c o n v e r g e n c e
Thus only finitely many fs(1)
x =
o n l y if for every s e m i n o r m
w i t h at m o s t
peP,
x Let f : KS----+ P - K P the r e s t r i c t i o n of f to the
f ~ 0 for i n f i n i t e l y m a n y s , fsp(Xs) = 1 . If XseKS denotes
so
product
denote
s 0) and for
coordinates
n a t e of that restriction.
f
As for the topology,
we have
in a set of the f o r m
KA(rs)
neighborhood
of
or
f : K S.
0
in
P-K
~
(SxP)-K
seen a l r e a d y where
rs
is the set F~°(tp)
.
that every c o m p a c t is a n o n - n e g a t i v e where
t
set in
KS
real number.
is a p o s i t i v e
is conA basic
real number
P ~
If
w i t h elements
of
K
> P-K .
has components
f
sp
the
f
sp
: K---+ K
can be identified
W e h a v e that f(HA(rs) ) c FA°(tp)
iff (*)
~ fsp r s / t p s,p
T h e r e a s o n is t h a t and
(~fsprs s
I peP)
m u s t be able to b e w r i t t e n
~I~pI which (,)
is e a s i l y
seen to b e e q u i v a l e n t
is e q u i v a l e n t
< 1 .
to the c o n d i t i o n
o
£ FA
(lptp)
with
IpeK
< 1 ,
to (fsp)
as
(tp/r s)
(*)
Similarly
the c o n d i t i o n
46
o
which in
is an o p e n
(SxP)-K
.
set in
(S×P)-K
.
To go the other way,
T h a t this is o p e n in the f u n c t i o n
space
let
FA
(Usp)
topology
be an o p e n set
is an easy c o n s e q u e n c e
of the following. (1.6)
Lemma.
reals.
Let
f
be a function
T h e n there are f u n c t i o n s
from
g
: ~---+ R
f(i,j) whenever Proof.
(i,j)
is in the d o m a i n
By l e t t i n g
f : ~ x~---+ R +
.
f(i,j)
e.g.,
of
- or a subset t h e r e o f
, h
: ~--+ ~
- to the p o s i t i v e
such t h a t
~ g(i)h(j)
f .
wherever
f
was hitherto
undefined,
we can suppose
Let g(n)
T h e n if,
= 1
~ x~
= h(n)
= inf{l,f(i,j)
I i ~ n, j ~ n}
.
i -< j, we h a v e 1 >- g(i)
Note:
It is the failure
the r e s t r i c t i o n (1.7) index
set
S×P
-> h(j)
f(i,j)
-> g(i)h(j)
of this lemma to h o l d
to finite
N o w then,
f(i,j)
and c o u n t a b l e
g i v e n an o p e n s e t where
e a c h of
w e can find two s e q u e n c e s
S
FA°(Usp) and
(rs)
and
for u n c o u n t a b l e
index sets t h a t
forces
index sets.
P
(S×P)-K
is e i t h e r
(tp)
Usp
in
, Usp
is a f u n c t i o n
finite or countable.
on an
B y the lemma
such that
-> r stp
This means that FA°(Usp) and the l a t t e r
is the set o f d o u b l y
D FA°(rstp)
indexed
f(~A(i/tp)) and h e n c e (1.8)
is a n e i g h b o r h o o d
This decribes
of
0
sequences
f =
(fsp)
such t h a t
~ FA°(rs )
in the f u n c t i o n
space.
the b i f u n c t o r C °p × D - - ~ D
which
t o g e t h e r w i t h the d u a l i t y b e t w e e n
C
and
D
describes
a pre-,-autonomous
situ-
ation. That every object the f u n c t i o n
space
(C,D)
(1.9)
Proposition.
(with
T = K)
Proof.
~
has a D--representation
is, b y d e f i n i t i o n ,
The h y p o t h e s e s
and hence
the c o n c l u s i o n
and the u n i f o r m i t y
on
uniformity. of II.2.9
are s a t i s f i e d
.
factors
can b e r e p e a t e d
is a t o p o l o g i c a l theorem.
is e v i d e n t
a convergence
The a r g u m e n t we used to s h o w t h a t any m a p
tely m a n y
Banach
in
vector
space
to s h o w t h a t
K
KS
so that the i n j e c t i v i t y
The third hypothesis
is evident.
~ K
is zero e x c e p t
is cosmall. of
K
Evidently follows
for fini-
any space in
f r o m the H a h n -
47
(i.i0) means of
If
A---~ B
is a p r o p e r
t h a t there is a n o n - z e r o
K
allows
an e x t e n s i o n
It is e v i d e n t see [ S c h a e f e r ]
t h a t objects
tails
for
R
~
or
.
b~A
, A+Kb/A ~ K
is zero on
Thus
A
.
B*---~ A*
which
The i n j e c t i v i t y
is not injective.
For the c o m p l e t e n e s s
of
D
,
out with
sequences
above pairing.
(ri)
= 0
.
of finite
sums
The r e q u i r e d
Next I claim that
C
.
l i m i n+2 r. = 0
of
A sequence
de-
D
Pn
which means
s =
Again
lim in+2ri=0
is said t o b e slowly
large
i . (s i)
Clearly
for e v e r y
, the d o t p r o d u c t
b e the s p a c e o f r a p i d l y d e c r e a s i n g
= ~ inlri ]
convergence
if
spaces.
is said to b e rapidly
K
(s i)
increasing
and
C
on compact
is the dual of
For
vector
it is t h e n a l s o true t h a t
We let Pn(r)
by u n i f o r m
D
.
slowly
convergent.
in
of elements Since
= 0
b y the s e m i n o r m s
topologized
of topological
for all s u f f i c i e n t l y
(ri) a n d e v e r y
are dense
C = D = the c a t e g o r y
of a c a t e g o r y
A sequence
is a b s o l u t e l y
are satisfied.
f r o m the above.
, Isil S i n
r =
sequences,topologized
we know that
.
for o u r c o n s t r u c t i o n
n , lim inr
n
rapidly decreasing
sequences
which B
and
, e q u i p p e d w i t h the usual t o p o l o g y .
interesting •
growing if for some
r-s = ~ ris i
A
are complete.
l s e e n to imply t h a % ÷ ~ ~ inlril
is e a s i l y
nite
K
inferred
decreasing if for all
growing
in
can also b e w o r k e d
H e r e is a n o t h e r
stands
A+Kb---~ K
required
o f copies of
can b e r e a d i l y
(i.ii)
in
, II.6.2.
The t h e o r y (or products)
map
embedding
of this to all of
T h u s the h y p o t h e s e s
K
closed
D
.
the space o f slowly sets in
D
First observe
is the s e m i n o r m
t h a t for some
m
above and
under the
that the fir =
l.n+2 r i
, i > m
(ri)6D
,
< 66/ 2
or t h a t
I ~ i nr. I < £ • Thus r is a p p r o x i m a t e d b y the finite s e q u e n c e w h i c h agrees m+l 1 w i t h the first m terms of r and is 0 thereafter. T h e n a functional on D is det e r m i n e d b y its v a l u e quence
(si)
can choose
on the finite
b y the f o r m u l a
a sequence
i(1) ,. ..,i(j)
i(1)
ri I < i(2)
sequences
which means
~ Z ris i. If, < i(3)
it is d e t e r m i n e d
in fact,s i
< ...
such t h a t
b y a se-
is n o t s l o w l y growing, > ij .
si(j)
we
For h a v i n g
s. ~ i j+l is not s a t i s f i e d for all s u f f i c i e n t l y l large i so there are a r b i t r a r i l y large i for w h i c h it fails. Let i(j+l) be the -i first one of these larger t h a n i(j) T h e n let r. = s• when i = i(j) and 0 .-n-~ 1 otherwise. Then ~ r.s. diverges while r. < 1 for all i > i(n+l) implies ii 1 inri < i -I---~ 0 as i--+ ~ This contradiction establishes that (si)£C chosen
(i.12)
There
we know that
is a c e r t a i n
called nuclear spaces. lowed
characterization
by
and
T.
Y. K o m u r a
of a c a r t e s i a n
class of spaces
of n u c l e a r
p o w e r of
D
is d e t e r m i n e d
.
D
so t h a t is a c o m p l e t e
by a sequence
w i t h the b o u n d e d
convergence
D
(in p a r t i c u l a r
IV.6.1;
II.7.1,
IV.9.6..
convergence
corollary
metric
C
II.81;
Next I claim that
on b o u n d e d
sets,
conjectured a space
especially
of p s e u d o - n o r m s
space,
topology,
isolated
i.e.
has the t o p o l o g y
Hom(C,D)
is i s o m o r p h i c
to
is n u c l e a r ii.i.i
and e s t a b l i s h e d
for details.
or F-space. DF-space
of b o u n d e d 2; IV.5.6
.
In fact if
The topo-
s e e n to be Its dual
and b o t h
convergence
on
C
C
,
and
D). See
and its first corollary;
, e q u i p p e d w i t h the t o p o l o g y D
the fol '
iff it is a s u b s p a c e
and it is e a s i l y
nuclear
corollary
for our p u r p o s e
by Grothendieck
a Fr4chet
is a c o m p l e t e
II.7.2.,
for s t u d y b y G r o t h e n d i e c k
is r a t h e r o p a q u e b u t
Namely,
See [ P i e t s c h ]
complete
are r e f l e x i v e
spaces
is m o r e useful.
l o g y on
[Schaefer]
first
T h e usual d e f i n i t i o n
f : C --+ D
of u n i f o r m is c o n t i n -
48
uous,
t h e n it is c o n t i n u o u s
the c o m p o s i t e
C---+ D---+ ~
f o l l o w e d by each c o o r d i n a t e , the second m a p p r o j e c t i o n
projection.
So let
f. 3
on the jth coordinate.
be Then
f. is a linear functional on C and hence r e p r e s e n t e d b y a s e q u e n c e (r..) which 3 13 for each j is r a p i d l y d e c r e a s i n g . In o r d e r that for each (si)EC , the values s.r.. E D i 13 c r e a s i n g in sufficient
.
double
j .
Although
sequence
versely,
into a sequence.
logy,
it follows
hoods
of 0
Chapter
this m u s t be done (i n ) .
for all
Thus
Clearly
(rij)
(s i)
in
IV,
C
and
j < k(i,j)
vector
space of
D
especially
.
in
it is s u f f i c i e n t
k(i,j)
An
I Pm(tj)
f
b e the r e a r r a n g i n g
t h a t the polars sequence
by
seminorms
w h i c h are,
of b o u n d e d
< in
by (r..) 13 ~ inj TM Irij I < £
iff
of fundamental
to
i---~ ~ .
C
a Since
Con-
or as
j ÷~.
As for the topo-
sequence .
of n e i g h b o r -
(See [ Schaefer]
of II.7.1.)
form a f u n d a m e n t a l
takes .
function. k---+ ~ .
sets in
with the corollary
Isil
represented
< ~}
m i n e d b y these
defined
a m a p iff
to r e s t r i c t
< i2n'2nlr3 ij I ~ 0 as .n.n 2n i ] r.. < k r k + 0 as 13 Hom(C,D) is i s o m o r p h i c to D
5.2 in c o n j u n c t i o n C
it is c l e a r l y
represents
so
form a fundamental
r a p i d l y de-
we use the usual m e t h o d of r e a r r a n g i n g
So let
from reflexivity
is the sets
sets in
that the latter be a s e q u e n c e
-< (i+j) 2 , knlrk I < (i+j)2nlrij]
i < k(i,j)
Thus the u n d e r l y i n g
{(tj)
in p r i n c i p l e
to c o n s i d e r
k = k(i,j)
means
and s u f f i c i e n t
the test s e q u e n c e s .n.m n,m, w e have lim i 3 r . = 0 . 13 To s h o w the i s o m o r p h i s m w i t h D
for all m = n
it is n e c e s s a r y
W h a t this
s y s t e m of b o u n d e d
the set into the n e i g h b o r h o o d
Thus the t o p o l o g y
as seen above,
equivalent
on
Hom(C,D)
is deter-
to the o n e s on D
under
the isomorphism. (1.13)
Let
C
b e the full s u b c a t e g o r y
of
and
C
and
K
(=products) --+ D
(K,K)
with f~ = 0
f(a~) .
= 1 .
Thus
sists, As for
of
C
ces is dense. which means sing sequence
required
to
a
.
are satisfied.
shows
a =
, ~ ~ 0
r e s u l t of
f
~ IF = 0
0
f(a) A ~-
example
a = = 0
.
But
~
linear
is r e p r e s e n t e d
is n o n - z e r o
Thus
there
has f
fac-
f~ ~ 0
III.4.2.
(see i.i0)
.
.
is satis-
H e n c e all
The category
The s u b s p a c e
is the
(a~)£HA~ .
for w h i c h
one of its c o n s e q u e n c e s
coordinate
a~ = 0 w h e n e v e r
c a n n o t b e de-
(V) of
are satisfied.
sequences.
In fact is an a~A~_-
in the r e m a i n i n g
f(a)
If now
of those
with
finite.
at such an e l e m e n t
that
if n o t there w o u l d b e a n o n - z e r o
and as s o o n as single
and
there
T. and Y. K o m u r a of e x a c t l y
w e can examine
but
(a~)
The condition
as in the p r e c e d i n g
of slowly g r o w i n g
In fact,
f~ = 0
this grows w i t h o u t bound,
limit a r g u m e n t
(-,-):c°PxD
It is also cosmall.
Unless
T h e value of
for our m a i n c o n s t r u c t i o n
spaces,
consists
~£D
•
o f t e n let
on the finite p r o d u c t
the same a r g u m e n t
~-complete
infinitely
Since
II.2.8
b y t h e above m e n t i o n e d
T h e space
is injective.
h o m functor
sums
the index sets b e i n g
in f i n i t e l y m a n y c o o r d i n a t e s
, the same
the p r o j e c t i o n
the h y p o t h e s e s
(C~,D),
f~ = flA~
sums of copies
are finite d i r e c t
the i n t e r n a l
D ) = H~,
R
are finite d i r e c t
objects
w i t h o n l y f i n i t e l y m a n y exceptions. a
the c o n d i t i o n s
fied b y e x a c t l y
a~
whose
We d e f i n e
let
topology,
coordinates.
f~ = 0
.
that
If this h a p p e n s
for all
tors t h r o u g h Hence
implies
in the p r o d u c t
= 0
D
(~C~,~
is a functional,
number o f n o n - z e r o
f~(a~)
and and
The e l e m e n t s w i t h
converge,
fined.
K
= K
theorem
f : ~A~---+ K
whose objects
the full s u b c a t e g o r y
of c o p i e s of
so t h a t
The H a h n - B a n a c h if
D
A
con-
the n u c l e a r
spaces.
fairly c o n c r e t e l y F
of finite
functional
on
here.
sequenC/c£ (F)
by a rapidly decreais an
a6F
with
49
~(a) ~ 0 .
For
A£A , a map
elements of
A .
Here
a. l
nate and
0
elsewhere.
sequence
Pal,Pa2,...
f : F---+ A
(pa i)
say that a sequence (Pai) (ai)
C
of elements of
A
is a r a p i d l y d e c r e a s i n g sequence and the sum
E c.a.l l
p
on
A , the
= E ciPa i
m u s t b e a r a p i d l y d e c r e a s i n g s e q u e n c e of real numbers.
(ai)
converge.
Since
of
b y the formula
= E P(ciai)
Now
is r a p i d l y d e c r e a s i n g if the sequence
is r a p i d l y d e c r e a s i n g for all seminorms
numbers,
al,a2,..,
1 in the ith coordi-
C o n t i n u i t y r e q u i r e s that for every s e m i n o r m b e e x t e n d a b l e to all of pf(c) = pf((ci))
w h i c h m e a n s that
is d e t e r m i n e d b y a sequence
is the image of the s e q u e n c e w i t h a
p .
(ci)
T h e n we require that w h e n e v e r
a slowly i n c r e a s i n g sequence of real
(ciai)
is also r a p i d l y decreasing, this
can b e r e p l a c e d b y the simpler h y p o t h e s i s that every r a p i d l y d e c r e a s i n g series converge.
I m u s t emphasize that this is o n l y a c o n s e q u e n c e of ~ - c o m p l e t e n e s s .
It is
t e m p t i n g to c o n j e c t u r e that it is e q u i v a l e n t b u t there is no real evidence for that. If the c o n j e c t u r e w e r e valid,
it w o u l d follow t h a t
A
is
~ - * - c o m p a c t p r o v i d e d the
sum of rapidly c o n v e r g i n g series of c o n t i n u o u s f u n c t i o n a l s w e r e always continuous. 2. D u a l i z i n g M o d u l e s . (2.1)
This example is in r e s p o n s e to a q u e s t i o n of Robert Raphael as to w h e t h e r the
theory of v e c t o r spaces over a d i s c r e t e field had any natural g e n e r a l i z a t i o n to modules over o t h e r c o m m u t a t i v e rings. k e l y that the d u a l i z i n g o b j e c t
T
If there is such a generalization,
its e n d o m o r p h i s m ring - m u s t be the g i v e n ring. quire that
T
m u t a t i v e ring.
it seems li-
will h a v e to be injective and its dual - w h i c h is Technical c o n s i d e r a t i o n s s e e m to re-
be finitely g e n e r a t e d and a cogenerator. We say that an R - m o d u l e
T
A c c o r d i n g l y let
R
is a dualizing module p r o v i d e d
finitely g e n e r a t e d injective c o g e n e r a t o r and the canonical m a p
b e a comT
is a
R---~HomR(T,T)
is an
isomorphism. We leave till 2.10 the q u e s k i o n of the existence of a d u a l i z i n g module. is a field,
it is clear that
R
itself is the unique d u a l i z i n g module.
If
R
The theory in
t h a t case reduces to that o f vector spaces c o n s i d e r e d earlier. (2.2)
N o w w e let
structure.
V
be the c a t e g o r y of all R - m o d u l e s with its usual m o n o i d a l closed
The c a t e g o r y
Un V
take all m o d u l e s of the form modules of the f o r m
S.T
. S
(2.3)
and
D*
= RS
D~D , D X S-T
w i t h the p r o d u c t t o p o l o g y and for
Hom(D,T) ~ Hom(S-T,T) ~ R S .
e q u i p p e d w i t h the p r o d u c t topology. D
Rs
and
RF
any
we
T---+ S.T
T---~ S-T
We w o u l d like to define
Instead, define ~ R
factors through
h a v e the p r o d u c t t o p o l o g y
F-T
D
to b e Hom(D,T)
is c o n t i n u o u s for a l l
d u a l i z e to the p r o j e c t i o n s
t o p o l o g y is at least as fine as the p r o d u c t topology. finitely generated,
~
But w e must show that this is indepen-
as a d i r e c t sum. , e q u i p p e d w i t h the coarsest t o p o l o g y such that D
When
For
all d i s c r e t e
such as that it be finite or countable.
d e n t of the r e p r e s e n t a t i o n of
The c o o r d i n a t e injections
D
V a r i a t i o n s o n the t h e o r y can b e o b t a i n e d b y p u t t i n g car-
d i n a l i t y r e s t r i c t i o n s on If
is simply that of t o p o l o g i c a l R-modules. RS
RS---+ R
O n the other hand,
since
for a finite subset
(the latter b e i n g discrete)
T---~ D.
so that this T
is
F c S .
the p r o j e c -
50
tion D*
RS----~ R F
is c o n t i n u o u s
is the p r o d u c t
topology
(2.4)
If
module
in its kernel.
kernels
C ~ RS ~ C
of m a p s
the d i r e c t
and h e n c e
RS----+ R ~
on
, w e see t h a t any c o n t i n u o u s The p r o d u c t
RS----+ R F
limit of
topology
where
Hom(RF,T)
limit of the finite
sums
cribes
the duality between
C
(2.5)
Since
R .
F
is a finite
is just and
D
is a cogenerator,
p o l o g i z e d b y o p e n submodules.
Thus the t o p o l o g y
on
m u s t have an o p e n sub-
subset of
S
base consisting .
Thus
~ Hom(R,T) F ~ F.Hom(R,T)
S-T
.
Thus we d e f i n e
C
of the
Hom(RS,T) ~ F'T
= S-T
is
and the
.
This des-
.
every
Then
R S---+ T
has a n e i g h b o r h o o d
~ Hom(F-R,T)
direct
T
so is
RS
A
AEUn~
has a D_-representation
consists
of these
"linearly
iff it is to-
topologized"
mo-
dules. Proposition. Proof.
T
Let
A 0 = ker~ U D B0
be a n e m b e d d i n g
is o p e n in
where
B0
a = a0+b 0 ~ A Thus
is i n j e c t i v e w i t h r e s p e c t
A---~ B
A
and
, A 0 = ANU
where
is an o p e n submodule.
where
a0eA0,
a = a0+b 0 £ A 0
b0~B 0
as well.
to the class of e m b e d d i n g s q : A --+ T U
a map.
is a n e i g h b o r h o o d
I c l a i m that
, then
Since
in
T
of
0
A 0 = AN(A0+B0)
b 0 = a-a 0 £ A
while
~
.
is d i s c r e t e in .
B
.
Then
In fact if
b£B 0 c U so b ~ A N U
= A 0.
Thus w e have A/A0-- -+ B / ( A 0 + B 0)
# is an i n j e c t i o n --+ T.
Then extension
injective (2.6)
and b o t h m o d u l e s
in
V
to
o f II.2.9
f r o m the e a s i l y p r o v e d
For a continuous
for some finite
P-T
factors
map
subset
are n o w satisfied.
through
subsets
S
.
~ lim and
c°P×D--~ D
factors,
Since
for some
The c o s m a l l n e s s
finite
(F×G) .T ~
G c p
(S×P)-T
P respectively.
.
where
and the d i s c r e t e
the h y p o t h e s e s
of III.4.2.
W e c a n now v e r i f y
In fact the r e q u i r e d
isomorphisms,
(RS,(RP,Q.T)) are i m m e d i a t e w h e n
S,P
argument
Since b o t h
as above.
in
C
formity o n
are p o w e r s (C,D)
all subset of Since III.4.2 T
(iv)
T
is
and
Q
~
is a c o n v e r g e n c e
~
follows
remark,
through
generated
a m a p to
the limits
are taken o v e r
(C,D)
to b e Hom(C,D)
topology.
The first two are clear while the such as
and the general
sides are discrete, in
T
(RQ,(RP,s.T))
are finite
of objects
of
Thus H o m ( R S , p -T) ~ lim H o m
Thus w e d e f i n e
of an R - m o d u l e
jects
that
: A/A 0
is v e r y s i m p l y the hom-
is f i n i t e l y
(2.7)
is easy.
~
group is u n i f o r m l y
by the p r e v i o u s
R F ~ F-R
w i t h the usual s t r u c t u r e
third
induces
from the h y p o t h e s i s
m a p on a t o p o l o g i c a l
functor
R S---+ P - T
G-T ~ T G
of
The
F c S
(F.R,T G) ~ l i m Hom(R,T) FxG the finite
A 0 = k e r 9, ~
n o w follows
fact that a c o n t i n u o u s
t o g e t h e r w i t h 1.2.12.
functor. RF
B/(A0+B0)----+ T
Since
.
The hypotheses
continuous
are discrete.
case
follows
no t o p o l o g i c a l
by a limit
question
arises.
and hence have a D--representation.
uniformity,
~(C)
consisting
of
C
alone
Ob-
The uni(or of
C ). the h y p o t h e s e s is satisfied.
is a cogenerator.
of II.2.9
are satisfied,
The fifth h y p o t h e s i s
In fact if
A ---~ B
every o b j e c t
is p r e r e f l e x i v e
is an e a s y c o n s e q u e n c e
is a p r o p e r
embedding
and
so
of the fact t h a t
x(B,
x{A
, let B 0
51
b e an open submodule of
B
such t h a t
x+B 0
does not m e e t
m e a n s the latter is a n o n - z e r o open submodule. B/A+B 0
admits a n o n - z e r o map to the c o g e n e r a t o r
but
on
0
A
.
But then
x~A+B 0 w h i c h
Thus the non-zero d i s c r e t e module T .
This is a m a p n o n - z e r o on
B
A , as required.
That the o b j e c t s in
C
and
D
are complete is evident and h e n c e all the hypo-
theses are s a t i s f i e d and the c o n s t r u c t i o n works. (2.8)
If
R
is noetherian,
the full s u b c a t e g o r y of finitely g e n e r a t e d R-modules is
a s u b - , - a u t o n o m o u s category. M
and
N
Hom(M,N)
In fact, T
are finitely generated, is a submodule of
is finitely generated by h y p o t h e s i s and if
choose a s u r j e c t i o n
Hom(F.R,N) ~ F-N
.
F-R---+ M
with
F
finite. T h e n
In that case it b e c o m e s natural to ask
w h e t h e r this is a compact category in the sense of Kelly,
n e c e s s a r y that
A c t u a l l y the q u e s t i o n cannot b e p r o p e r l y formulated w i t h o u t
R ~ T .
G r a n t i n g that, there are natural maps
w h i c h compose to give
M*~M---+ (N,N)
.
is a field and h e n c e w h e n
R
is not semisimple,
R
M*@M--+ R
This t r a n s p o s e s to
really a s k e d is w h e t h e r this m a p is an isomorphism. R
M = N = R
(M,N) is ca-
M*~ N .
that hypothesis.
F i r s t off, b y l e t t i n g
that is w h e t h e r
n o n i c a l l y i s o m o r p h i c to
is s e m i s i m p l e
exact w h i l e
M . M~-
M*~N---+ (M,N)
In fact if
is zero
If
K
(Tor commutes w i t h (M ,-)
is left
is a field and
R =
is a d u a l i z i n g m o d u l e and that w h e n
m a d e into an R - m o d u l e via the o b v i o u s a r g u m e n t a t i o n K*~K---+ (K,K)
and what is
(yon N e u m a n n regular s e l f - i n j e c t i v e
T h e n S t e v e Schamuel's elegant o b s e r v a t i o n that
is not settles the question. R
R---+ (N,N)
(i.e. a finite p r o d u c t of fields).
there is a n o n - f l a t m o d u l e
x ] / ( x 2) , then it is easily seen that
and
It is easy to see that it is w h e n
rings are fields), and h e n c e a finitely g e n e r a t e d n o n - f l a t module li~m), call it
w e see that it is
R---+ K ,
K
is
t h e n the canonical map
(although they are isomorphic).
(2.10) Now w e turn to the q u e s t i o n of the existence of a d u a l i z i n g module.
The only
result I k n o w is the following. Proposition.
Suppose
K
is a c o m m u t a t i v e ring w i t h a d u a l i z i n g module
a finitely g e n e r a t e d K - p r o j e c t i v e K-algebra.
Q
and
T h e n for any c o n s t a n t l y rank
g e n e r a t e d R - p r o j e c t i v e R - m o d u l e P , T = HomK(P,Q)
1
is a d u a l i z i n g module for
R
is
finitely R .
Before b e g i n n i n g the p r o o f let me o b s e r v e that this requires b e g i n n i n g w i t h a ring that has a d u a l i z i n g module.
Of course
K
m i g h t b e a field.
The result of this
p r o p o s i t i o n a p p l i e d in t h a t case is t h a t any finite d i m e n s i o n a l c o m m u t a t i v e K - a l g e b r a has a d u a l i z i n g m o d u l e and m a y w e l l have more than one. (R,K) ~ R
iff
R
P r o o f of the p r o p o s i t i o n . For if
and w i t h
M---+ N
It can b e shown that
Hom
K
is self injective. The p r o o f t h a t the
T
so d e f i n e d is injective is standard.
is an i n j e c t i o n of R-modules, we have
H°mR(i'H°mK(P'Q))
HomR(i,HOmK(P,Q))
H o m K (P@RN ,Q)
HOmE (P~RM, Q )
P R-projective
P@RM
> P @ RN
is still an i n j e c t i o n and since
Q
is K - i n -
52
jective it follows that the map is a surjection. (R,Q)) ~ HomK(M,Q) that
HomR(M,T)
and the fact that
~ 0
ted as an R-module,
so that
T
Q
c o g e n e r a t e s R-modules.
there is a s u r j e c t i o n
F.R---~ p
this m a p has a left inverse.
Similarly
G
is a retract of
and thus as a K-module, P
tract of
QF×G
.
Since
Q
R
from
M ~ 0 , HOmR(M,HomK it follows
Since
P
is finitely genera-
and since
P
is p r o j e c t i v e
is a retract of (FXG)-K
G.K
whence
for some finite set
T = HomK(P, Q)
is finitely g e n e r a t e d as a K - m o d u l e so is
it is finitely generated as an R-module. finitely additive.
Similarly,
is a c o g e n e r a t o r in K-modules,
The functor
M
~
T
.
is a reAfortiori,
HomK(HOmK(M,Q),Q))
is
The natural m a p M ---+ H o m K ( H o ~ (M, Q) ,Q)
is an i s o m o r p h i s m w h e n
M = K , b y hypothesis,
hence is w h e n
w e l l w h e n it is f i n i t e l y g e n e r a t e d and projective. c o n s i d e r e d as a K-module.
M
is finite free and as
This is, in particular,
true of
P,
Now
H O m R ( T , T ) = HomR(HOmK(P, Q) ,HOmK(P,Q) H O m K (P@RHOmK (P, Q),Q) H o m R ( P , H O m K (HOmK(P, Q) ,Q) HomR(P,P) But
P
is locally isomorphic to
R
and hence the natural m a p
e v e r y w h e r e locally an i s o m o r p h i s m and hence is an i s o m o r p h i s m
R--~ HomR(P,D )
is
(its k e r n e l and c o k e r n e l
are e v e r y w h e r e locally zero). 3. B a n a c h Spaces. (3.1)
Let
V
b e the c a t e g o r y of b a n a c h spaces and c o n t i n u o u s linear maps of n o r m Nl.
It
is k n o w n that this is a semivariety.
V
its unit ball
uV
of all functions lasl
.
The u n d e r l y i n g functor assigns to each spare 1 S the b a n a c h space £ (S)
The left a d j o i n t assigns to each set
a : S---+ K
(where
K
is the real or c o m p l e x field)
converges, w i t h n o r m d e f i n e d b y
Itai] =
~iasi
such that
A n a l g e b r a for the theory de-
s£S t e r m i n e d b y this a d j o i n t pair is a set closed under o p e r a t i o n s
(x i)
(li)
Z
is any finite or countable sequence of scalars for w h i c h
I
> ~ l.x.1 1
ili] S 1 .
where
The op-
erations m u s t s a t i s f y equations w h i c h look like double s u m m a t i o n identities and are fairly obvious.
E x a m p l e s of a l g e b r a s for this theory w h i c h are not
b a n a c h spaces are the o p e n
interval
(unit b a l l s of)
(-i,i) as w e l l as the q u o t i e n t space [ - i , i ] / (-i,i).
This latter space is more a c c u r a t e l y the c o e q u a l i z e r of the inclusion m a p and the zero map.
It has three elements i,-i and the third w h i c h m i g h t b e d e n o h e d
sents the w h o l e i n t e r i o r of the ball. take the value (3.2)
1
Proposition.
or
-i , takes the value Let
V
let
W
Since
uV
0 .
b e a b a n a c h space and
and invariant under the above theory. Proof.
V
B c uV
b e t o p o l o g i c a l l y closed
T h e n there is b a n a c h space
is closed in V , so is
b e the linear subspace of
0 , and repre-
Any o p e r a t i o n not r e q u i r e d b y an i d e n t i t y to
B .
Since
generated by
B .
V
W
such that B ~ W.
is complete, Since
B
so is
B . Now
is invariant under
53
the o p e r a t i o n s
it is c o n v e x and c i r c l e d
and hence d e t e r m i n e s
a norm
p
on
W
b y the
formula p(w) If
w£B
, p(w)
~ 1 , while
all
( > 0 .
Now
W
is c o m p l e t e
Since
B
a IB
.
This is c l o s e d in
if
p(w)
is closed
finer t h a n t h a t induced b y
V
since
formity (3.3)
follows,
in
V
b y 1.2.5,
uniform.
.
First
let
I
cause the v a l u e
be
B
is•
B
is.
I---+ j
preserves
lute value ~ 1 . (-i,i)
(3.4)
structure
and banach
D
V
There
C
space w i t h n o r m
euclidean
norm determined
in
and by
IB
V
of 1.3.10
choices
banach
U = max(p(al),...,P(an))
.
IB
on
IB of
in is
6B
in the
p uni-
It is not clear
object which
is not
w i t h its usual uniformity•
Let
This is a p r e u n i f o r m
value
object be-
iff all the genuine
varies)
are 1 (resp.
< 1
(-i,i)
variables
-i).
A map
b y a scalar of abso-
are uniform.
Thus Hom(I,J)
Un V
A
for
C
spaces.
.
It is s u f f i c i e n t
consist and
Let
A
to des-
of all o b j e c t s w i t h a DD
.
The first
is to take
b e a finite d i m e n s i o n a l
be a linear basis.
Let
li II
denote
the
, i.e.
Ullal+...+inanll
R n , w e have
p
1 and -i are isolated w h i l e
interval.
and let
al,...,a n
N o w let
in this norm.
is e v e n t u a l l y
. is satisfied.
[-i,i]
in w h i c h
genuinely
al,...,a n
d u c t in
of
of a p r e u n i f o r m
to d e s c r i b e
D
are two n a t u r a l
p
of
for
.
and
to b e finite d i m e n s i o n a l
is closed
sequence
iff it is m u l t i p l i c a t i o n
it is n o t n e c e s s a r y
cribe the full s u b c a t e g o r i e s representation.
to
(l-£)w£B
are b y t r a n s l a t e s
o n l y those of a b s o l u t e
w h i c h does n o t b e l o n g
or
covers
interval
the o p e r a t i o n
the a l g e b r a
Fortunately,
B
Thus
is the value of an o p e r a t i o n
on w h i c h
Of these,
.
Thus the c o m p l e t e n e s s
f r o m the c l o s e d
-i)
£ > 0
induced by
is an e x a m p l e
the closed
inherited
(i.e. all v a r i a b l e s
Here
for all
The u n i f o r m i t y
The uniform
w i t h the u n i f o r m i t y
1 (resp.
wEB
.
is since e v e r y C a u c h y
f r o m its c o m p l e t e n e s s
are.
be the same interval
h a s the u n i f o r m i t y
is
IB
F r o m this w e see t h a t the h y p o t h e s i s
w h a t the u n i f o r m o b j e c t s
J
this implies
since
and these are c l o s e d
I wclB}
~ 1 , w6(l+6)B
iff every set V
= inf {I
=
(a~+...+a~) ½ .
First observe
[(Lll[ ..... [lnl)-([l ..... i) llll+...+il n
that f r o m the o r d i n a r y
inner pro-
~ [Illll ..... lln])[[ [[(i..... 1) I[ ~ ,~
2)h (Illl2+...+lln I
.
Then I p ( l l a l + . . . + I n a n)
~
l l l l P ( a l ) + . . . + l l n l P ( a n) U(llll+...+llnl)
•
2)%
~[n(illl2+..+llnl = ~ so t h a t pact
p
is b o u n d e d
in the n o r m
in a finite d i m e n s i o n a l
11 II
euclidean
takes on a m i n i m u m v a l u e there,
Conversely,
space
say l/M,
Lillal+...+Inanli
and
p
the
(n-l)
, sphere
never v a n i s h e s
and it is clear t h a t
ilali = 1
on it.
[iall K Mp(a)
Hence
is comp
for all a£A.
54
Thus each of the norms defines (3.5)
F r o m this it follows
cessarily
the same t o p o l o g y
every
a m a p in the category)
A**
w i t h a under
and its second dual. (3.6)
Proposition.
Proof.
F o r any
a£A
If(a) I ~ p(a)
Let
p**
p** = p
.
p*(f)
Then
T h i s shows
spaces.
denote
identification
~ p*(f)
.
We may,
at a c e r t a i n l y
A
such t h a t
p**(a) that
~ f(a)
A ~ A**
observations spaces,
this means
extension
to the w h o l e
and e s t a b l i s h e s A
nach
space
.
If
not a b a n a c h
is c o n t i n u o u s
is e q u i v a l e n t
"MT"
with
p*(f)
~ i,
b~A
suppose p(a) = i .
f(a)
.
f
= 1 . The Hahnto a functio-
This implies
that
which
for finite d i m e n s i o n a l
is the unit b a l l of a b a n a c h into u n i t
uniformity
F i r s t off,
a product
does not
is a t o p o l o g i c a l restricted
that a d d i t i o n
n o t the case
of finite d i m e n vector
The first
on the p r o d u c t
on the unit ball, in
A
it makes
A
.
described
induced b y the c o a r s e s t
which
see [ S e m a d e n i ]
of d i s c r e t e
A
con-
As long as this is done
no d i f f e r e n c e
Thus w e take
ba-
locally
locally
is chosen.
, [Wiweger]
The
) spaces
It is clear that the first a p p r o a c h
development.
of p r o d u c t s
J
is to use the
of finite d i m e n s i o n a l
is continuous.
(for m i x e d topology,
In
has an
and scalar m u l t i p l i c a t i o n
for the o b j e c t
this uniformity.
space.
to the u n i t ball,
such t h a t e v e r y m a p to an a r b i t r a r y
to the c a t e g o r y
in the s p i r i t of our p r e v i o u s are subspaces
A
space,
uniformity
space
in the same w a y for e v e r y o b j e c t of
for all
is to use the u n i f o r m i t y
v e x space w h i c h
is a c a t e g o r y
f
t h a t we can e x t e n d
uniformity,
This is a s s u r d l y
i n d u c e d b y the p r o d u c t The second
loss of generality,
N o t e that the p r o d u c t
space of such a nature
continuous.
spaces.
ces w h i c h
p*
induced b y a family of e m b e d d i n g s
spaces.
that the induced
o n the w h o l e
which
by
induced b y the norm.
convex topology
result
A*
of a finite d i m e n s i o n a l induced by
the d u a l i t y
is a set
There are two w a y s of e x t e n d i n g
uniformity
We norm
.
a functional
guarantees
m a y be m a d e here.
although
particular
in 3.3.
(but not ne-
= 1 = p(a)
banach
w i t h the u n i f o r m i t y
Several
are u n i f o r m l y
.
~ i} ~ p(a)
If(b) l~ p(b)
A n o b j e c t of the c a t e g o r y
sional b a n a c h
A
If(a) I ~ p*(f)p(a)
without
admits
] , II.3.2.)
balls of finite d i m e n s i o n a l
(3.8)
A
so t h a t
space a n d has the n o r m and the u n i f o r m i t y
coincide
~ i}
the n o r m on
= sup { If(a) l lP*(f)
a ~ 0
([Schaefer
on all of
~ 1 .
(3.7)
let
generated
theorem
nal d e f i n e d
the usual
on
is c o n t i n u o u s
be the set of them.
= sup { If(a) l lp(a)
, If(a/p(a))l
To go the o t h e r way, The s u b s p a c e
A*
A---+ R
and h e n c e p**(a)
Banach
(or uniformity)
functional
and we let
p*(f) We i d e n t i f y
linear
is m o r e
to b e the c a t e g o r y
of spa-
spaces w i t h the n o r m and the t o p o l o g y
induced. It s h o u l d b e noted t h a t if logically)
unit balls
map of
to a b a n a c h
A
Since
B
is e m b e d d e d
(3.9)
As internal
A
and
B
they are isomorphic. space,
homfunctor
For the t o p o l o g y
is c o n t i n u o u s
in a p r o d u c t
(algebraically on
A
and topo-
is such t h a t a n y
as soon as it is on the unit h a l l of
of b a n a c h
c ° P × D ---+ D
have isomorphic
spaces,
w e take
the same is true of (C,D)
B
A
.
to b e all linear m a p s b e -
.
55
t w e e n the spaces spectively,
C
and
D
.
d e f i n e the n o r m
If
p
and
q
are the n o r m f u n c t i o n s on
(q/p) (f) = sup {qf(c)
i p(c)
~ i}
.
C
and
D re-
This c o u l d also be
described by sup {qf(c/p(c)) except when nomous
C = 0 .
situation.
I c ~ 0}
U s i n g this it is e a s y to see t h a t
C
and
D
form a pre * - a u t o -
The o n l y p a r t of I I I . 4 . 2 w h i c h m u s t be d e m o n s t r a t e d is that e v e r y
o b j e c t is p r e r e f l e x i v e
(the ~ - c o m p } e t e n e s s here
comes h a r d e r to v e r i f y as
C
and D
is a v a c u o u s h y p o t h e s i s ) .
grow larger
o b j e c t b u t its u n i f o r m i t y b e c o m e s
This be-
for the dual has the same u n d e r l y i n g
finer m e a n i n g
it m i g h t a d m i t m o r e
functionals.
Ac-
c o r d i n g l y we t u r n to the next example. (3.10)
We let
through
D
d e n o t e the c a t e g o r y of b a n a c h
spaces.
u n c h a n g e d w i t h f i n i t e d i m e n s i o n a l b a n a c h s p a c e s r e p l a c e d by b a n a c h space.
we can c o n s i d e r determined
A
to be the c a t e g o r y of those
b y m a p s to b a n a c h
spaces.
MT
n o r m b u t in t h a t t o p o l o g y ) .
Semademi
m a y be d e s c r i b e d as follows. pologized by pointwise f o r m u l a as in is a proof.
(3.5).
D(D
(simple, weak)
([Semademi]),
, D*
D*
I uD
takes which
which preserves
nations automatically
the
uD
of these
that
C °p ~ D
.
in the
The d u a l i t y
is the set of l i n e a r f u n c t i o n a l s on
to
D*
KD
.
I = u]<
is compact.
finitary
induces a map
~
(not, of course,
D to-
c o n v e r g e n c e and w i t h the n o r m g i v e n by the same
It is t o p o l o g i z e d as a s u b s p a c e of
in the u n i t ball of
uD---~ I
is c o m p a c t
The fact t h a t the u n i t ball of
g i z e d as a s u b s p a c e of map
For
shows
Thus
s p a c e s w h o s e t o p o l o g y and n o r m are
A m o n g t h e m are the full s u b c a t e g o r y
spaces w h i c h are l o c a l l y c o n v e x and w h o s e u n i t b a l l
a map
The d i s c u s s i o n of 3.8 goes
If and
uD
is standard.
Here
is the unit ball of
D,
so that unit ball is t o p o l o -
It is in fact a c l o s e d s u b s p a c e for any
operations
D---+ ~
is c o m p a c t
.
essentially convex linear combi-
The p r e s e r v a t i o n of a f i n i t a r y o p e r a t i o n
i n v o l v e s o n l y a f i n i t e n u m b e r of c o o r d i n a t e s and the set of m a p s p r e s e r v i n g a g i v e n ope r a t i o n is e a s i l y seen to b e closed. of c o n t i n u o u s l i n e a r m a p s u n i t ball of
C
(3.11)
Proposition.
uD*
Proof.
.
E a c h such m a p
and the least u p p e r b o u n d
p o l o g y is r e q u i r e d on
and
C---+ K
On the other hand,
C* Let
.
and
DeD
C(C
.
C
{9~D* £ > 0 .
where
uD
i llgll ~ 1
So let
and
DeD m
uC*
in norm.
{9£D* is a n e i g h b o r h o o d of and
0 x6D
D
.
i [1911 ~ 1
and
X
be a c o m p a c t subset.
A zero
is
ig(x) I < e
N o w t h e r e is a f i n i t e s u b s e t
is the u n i t b a l l of
the l a t t e r set
No a d d i t i o n a l to-
we r e q u i r e the following.
T h e n the t o p o l o g y on the u n i t b a l l s
in the c o m p a c t c o n v e r g e n c e t o p o l o g y
where
be the set
t h a t is c l e a r since it is the t o p o l o g y of u n i f o r m c o n v e r g e n c e on the
u n i t b a l l and its scalar m u l t i p l e s . neighborhood
C*
is b o u n d e d a b o v e on the c o m p a c t
is that of u n i f o r m c o n v e r g e n c e o n c o m p a c t sets b o u n d e d For
, let
is the n o r m of the map.
Before continuing,
CeC
if
for all
x I .... ,x n
xcX}
such that
X c U
(xi+(e/2)uD)
The set and
19(xi) i < {/2,
i=l ..... n}
in the p o i n t w i s e c o n v e r g e n c e topology. , x = xi+(£/2)y
for some
y
with
But if
ilytl ~ 1 .
9 Then
b e l o n g s to
56
1~(x) I S (3.12)
i~(xi) I + £/2
The
result
I ~(Y) t < 6/2 + E/2 = 6
of t h i s
continuous
on
uD*
tion.
the
topology
For
immediately is a l s o
C ---~ D
a continuous
linear
norm.
with
and
which
(~elley]
, Chapter
Proposition. A
of b a n a c h
the unit ball
neighborhood
N
defined
AeA
linear
functions
This means
norm, a
of norm
(l-c)ilail
.
Thus
follows C ---+ D
some scalar multiple
The previous
gives
of
us a g a i n
works
5, p.223)
implies are
and
a6A
be
an e l e m e n t and
space
0
such that
in
(a+M)
as w e l l
of
of n o r m
II~LI ~ 1
0
The only
> i.
are comthing
.
there
It f o l l o w s
whose
is a
.
Thus
.
Then there
so is t h e u n i t b a l l
for a s u b s p a c e
a{uA
N uA = ~
is a n e i g h b o r h o o d
fact that
spaces
reflexive.
is c l o s e d
and
The
the d o m a i n
the s a m e h e r e .
D
implies
set o f a l l b i l i n e a r
o f the u n i t b a l l s .
and
> 1
as t h e
well when
C
~(a)
paragraph
C--+ D
to see t h a t t h a t m a p p r e s e r v e s
identified
on the p r o d u c t
A
closure
in a p r o -
of such a product.
is a c o n v e x that
does
circled
(a+½M) N ( u A + ½ M )
not contain
a
=
.
by
with
p(b)
an extension
the natural
~A*
c a n be
of a b a n a c h
on the one dimensional
(3.15)
norm.
It is r o u t i n e
holds
A
theorem,
is func-
function
s p a c e of ~ o n t i n u o u s
is c l o s e d
, defined
Banach
a fact which
linear
that
linear
scalar
This property
of
on
a continuous
former.
D*---+ C*
by the
p(b) is a s e m i n o r m
map
and dividing
uA
N = uA+½M of
in
such that
spaces.
M
so t h a t The gauge
Let
on
duct
t h a t of t h e norm,
Thus
adjointness
7, t h e o r e m
the unit ball
Thus
induced
is t r u e of a c o n t i n u o u s
D*--+ C*
(C,(C',D))
product
is t o s h o w t h a t o b j e c t s
~
.
the
to t h e
It f o l l o w s
are continuous
hom/cartesian
Since
D
sup-on-the-unit-ball
(C',(C,D))
functional
than
(C,D) ---+ (D*, C*)
left
Proof.
is a m a p same
belongs
bounded.
to be t h e
Pact
(3.14)
in
a map
(C,D)
t h e usual
C×C'---~ D
the usual
The
~
D*---+ C * .
is a m a p
Both
maps
.
is c o a r s e r
, induces
We n o w d e f i n e
that there
C
C---+ D
D*
in n o r m a n d h e n c e
it is a m a p
equipped
on
if
on
from the representation
continuous
(3.13)
is t h a t
and hence
and thus
= inf { ~ I b £ 1 N }
Jlbti
~
for all
subspace
to a l l of
b
and
generated A
p(a)
> i.
~(a)
= p(a)
by a
for w h i c h
that
no m a t t e r
how
map
A-
is n o r m p r e s e r v i n g .
~ 1
such that
+ A **
A*
,
~(b)
is t o p o l o g i z e d ,
l~(a/(l-E)llalL)
For
I > 1
~ p(b)
The
given
has,
by the Hahn-
K llbll .
as l o n g as
which
functional
any
means
it b e a r s
the
~ > 0 , there that
sup is
i~(a) I >
the sup { i~(a) I i ii~II = 17 Z llall
while
the other
nal on
A*
(3.16)
This
topology ous, image ber
on
, has
direction
norm equal
implies, C**
it f o l l o w s is d e n s e . ~i,...,~ n
is a u t o m a t i c . to
C---+ C * *
Suppose of
f£C**
functionals
that
a , considered
as a f u n c t i o -
ilall.
in p a r t i c u l a r ,
that
is t h a t o f s i m p l e that
It f o l l o w s
A --~ A**
convergence
is c o n t i n u o u s . on {g
on
and
an
Let
C*
~£C*
in t h a t
A neighborhood C
is a n i n j e c t i o n .
~ > 0
I i (g-f(~i) I < £}
.
Since
topology.
of
f as
•
each
We w i l l
is d e t e r m i n e d
C£~
.
The
is c o n t i n u -
show that the b y a f i n i t e num-
57
Now
ker
~i
is a s u b s p a c e
codimension. mensional
Let
subspace
t i o n of
f
to
of c o d i m e n s i o n
A = C/ker of
A*
C*
(in fact for a n y
(3.7),
m a y a l s o be c h o s e n
to h a v e n o r m C**
of
and t h a t of
uC
is
morphism
uC**
on u n i t balls
C**
< 1
of n o r m
Then
lies
.
< 1 C
b y an
by
If
in the n e i g h b o r h o o d Then
Since
is compact,
C**
.
Since
The r e s t r i c -
is a p r e i m a g e
of
f
of
described
its image
of that image.
uC
is c o m p a c t
a
above
ILall = lJflA*Li~iifi1< 1 and so
is an i s o m o r p h i s m
C---+ C **
cEC
has finite
to a finite di-
~l,...,~n).
.
[Ifl] < 1 . uC
ker ~n
is isomorphic
a£A
is in the closure
is
and h e n c e
ker ~in...n A*
they are g e n e r a t e d
N o w suppose
every e l e m e n t o f
so t h a t
represented
e l e m e n t of
£ > 0).
1
ker ~n
, (in fact,
is, b y
t h e n the c o r r e s p o n d i n g
~IN...N
c
is c l o s e d b u t
Thus the image
t h a t m a p is homeo-
(see the d i s c u s s i o n
in
(3.8)) . (3.17)
Proposition.
Let
A
61~II < 1
h a s an e x t e n s i o n
Proof.
Since
extension
mension
1
nal
~
B/A 0
with
in
of n o r m
B
A
.
T h e n any functional
m a p s are u n i f o r m l y
continuous, in w h i c h
.
Thus we m a y
suppose
A/A 0
is e m b e d d e d
in
ilall > ~(a)
case t h a t
1 .
> 1
A
B/A 0
~ = 0)
We k n o w
and
~ : A---+ K
with
< 1 .
and c o n t i n u o u s
in the trivial
= 1 , whence
on
of
In that case (except
~(a)
B
is c o m p l e t e
to the c l o s u r e
A 0 = ker ~ .
that
K
be e m b e d d e d
to
closed and
A/A 0
generated
.
The r e q u i r e d
has an
case
so is
is a space of di-
by an e l e m e n t
from 3.14 t h a t there
LI~Ii ~ 1
~
is a linear
functional
a
such
functio-
is
b ----+ ~ (b)/~ (a) (3.18)
Proposition.
Proof.
We have
of m a p s
Every
A£A
is q u a s i r e f l e x i v e .
shown t h a t e v e r y
CE~
is a functional
with
]Lfll < 1 .
whose
n o r m is still
t i o n a l on
~C *
if m o r e t h a n c o u n t a b l y
llflii+'''+lifntl > alf#11 .
119iI] = 1
and
nate and
0
A*---+ HC
Let
We have
f
Let # 0)
{q
:C~
f
f
.
Then
let
A}
= f# LC~
we could
•
If
(~)e~C
9i
f: A * - - - + ~ f#
to a func-
~11f i] > i]f#1[ (in
find a finite
have
is a family
Suppose
has an e x t e n s i o n
~ = SlflU+'''+llfnll - llf#11 and
fi(gi ) > IIfill - £/n elsewhere.
.
< 1
many
Suppose
is an embedding.
T h e n we k n o w t h a t
particular such
is reflexive.
C ~C such that the induced
9icCi
set
~=l,...,n
such t h a t
in the ith coordi-
f(9 ) = fl(~l)+...+fn(~n) > [iflll+llf211+...+LI fnll -
= lif#11 while C
II(~)II
= 1 , a contradiction.
and of c o u r s e
converges = Z~g
(in norm,
(c)
(3.19) jective
D£D
, both
and p r e s e r v e s C
and
We should stated.
f eC
is r e p e s e n t e d
by an e l e m e n t
a~ = g(c ) , ZIIa~IL < 1 to an
a{A
.
T h e next to last e q u a l i t y
D
and
.
If
follows
9cA*
so t h a t
c ~
~ae
, f(9)=f#((gg
))
from the fact that
and continuous.
Now with
proof that
Then with
in the topology)
afortiori
= 9(Zg c ) = ~(a)
is a d d i t i v e
N o w each
ZIic~II converges.
D
norm
constitute
now v e r i f y
The c o n d i t i o n
D**
(see 3.15)
are b a n a c h
a pre-*-autonomous
III.4.2(v) .
is s a t i s f i e d
spaces,
the m a p b e t w e e n
and is thus an isomorphism.
t h e m is bi-
This f i n i s h e s
the
situation.
Unfortunately
for n o r m - p r e s e r v i n g
it is not quite true in the form embeddings
(use the same argu-
58
ment
as in i.i0).
the topology
of
necessarily this,
out
is to let
Then
be c o n t i n u o u s
we b e g i n
(3.19)
The way (CI,C2)
and
T ( C I , C 2)
the map
have
constructed
can be proved
ad h o c
of
IClI@IC21
and
to t h e C o r o l l a r y
the norm
of III.27
will
to b e
norm
non-increasing.
To see
with
Proposition.
A,B6A
For any
, the
natural
embedding
A(A,B) ---+ A ( B * , A * ) is n o r m p r e s e r v i n g . Proof.
If
f : A ---+ B
, IIfll = s u p {ilf(a) il lllal] = i}
Suppose
llfll = 1
llf(a) ll > 1-6/2. 18f(a) i > 1-6.
.
Then There
But
for a n y
6 > 0 , there
is a l i n e a r
functional
implies
this
is t r u e
for all
ilf*ll _< llfll = 1
6 > 0
and
> I-E
, it f o l l o w s
(3.20)
The natural
norm
suppose
.
.
.
V(B*,A*)
that
1 and show
that ~
~
: (Cl,C~)*---+
has norm
-< 1 f#
still
has
norm
1 and
~
has
.
A
A(C2,A)
is the m a p
induced
by
the C o r o l l a r y
The map
: Ci---+ A(C2,A)---+ the
same
norm
A(A*,C~)
as
~* : A * - ~ (Cl,C ~) NOW
if
cI6C 1 , d6A*
, c2£C 2
are
such
that
llClLi = ildll = lIy211 = 1 , t h e n
IIf# (Cl) II _< 1 which
and and
norm.
Now we
We m u s t
ilall = 1 liS]i = 1
embedding
f : Cl~-~ has
that
that
that
V(A,B)---+ preserves
such such
lIBil = 1 ,
IIf*ll : 1 Corollary.
B
on
llf*Z(a) II > -6
IIf*ll >_ 1 Since
aEA
B
that llf*Sil
Since
is a n
then i]ai[ = i,
which
.
implies
that llf#(Cl) (~)II <_ 1
and hence
that If#(Cl ) (e) (C 2) I < 1
.
to III.2.7.
sg
But
and
f # ( C I) Ca) (C 2) = ~(a) (c I) (c 2)
1 , the
since
d,CI,C 2
are arbitrary
elements
of n o r m
inequality
I~(c~) successively
(C I) (C2) I -< 1
implies
IIf(~) (Cl)ll
~ 1
I1~(~)11 ~ 1 I1~11 ~ i which nore
establishes that
what
condition
The objects
call t h a t
the
w e need.
Since
in
D
are banach
"underlying
Thus t h e r e q u i s i t e s
set"
for o u r
spaces
is t h e u n i t
construction
4. M o d u l e s (4.1)
Let
H
is a v e c t o r
be
a cocommutative
space
no o t h e r
use was made
of III.4.2(v),
we may
ig-
here.
over
K
, equipped
a Hopf
algebra
with
: K---+ H
and
are complete.
for s p a c e s
As for
in
C
.
This
C
, this
, re-
is c o m p a c t .
are present.
over
hopf
and hence ball
over
K-linear
; e
: H ® H ---+ H
Algebra. a field
K
means
that
H
maps
: H---+ K
; 6 : H---+ H6~H
1 : H---+ H such
that
c
and
6
determine
a cocommutative
in the c a t e g o r y
of cocommutative
(4.2)
known
6(g)
The best = gQg
is. T h e
and
second
best
The operations +£®i
and
l(g)
are
I(~)
of groups
(4.3)
If
tive
H
dules
is s i m p l y
(4.4)
We a d o p t
let
for
Let
M
H(M,N)
the
denote denote
.
algebra
Note
that
homomorphisms
Most
algebras
of the m a n y
belong
in f a c t
we u n d e r s t a n d is g i v e n
homomorphism.
Sweedler's
is a m o d u l e , H
algebra
£6L
algebra
a module
.
is t h e e n v e l o p i n g
multiplication
in p r a c t i c e , If
(4.5)
g(G
example
and o f lie
whose
is t h e g r o u p for
the u n i q u e
is a h o p f
algebra
although
= g-i
known
= -£
theory
example
coalgebra
and
notation
case
We
algebra
object
for w h i c h features to the
p .
let
H for
let
PM
the c a t e g o r y
the abelian
: H@M---~ M of H-modules
group
Hom(M,N)
G
, this
formula
o f a lie 6(£)
which theory
heH
denote
~(g)
G
algebra
L
= 0 , 6(£)
of h o p f
= i,
iff
are common
a module
A morphism denote
Here
is c o m m u t a t i v e
Le
.
= i~£
to t h e
algebras.
for t h e a s s o c i a -
f : M---+ N
the category
of H-mo-
of H-modules.
,
6h = E h ( 1 ) ® h ( 2 ) h o n t h e s u m m a t i o n is u s u a l l y
index
of a g r o u p
a group
of a g r o u p .
b y an H - m o d u l e
by
and write
~ G]
the a l g e b r a
omitted.
the a c t i o n
of
H
on
and morphisms
described
equipped
the H-action
with
(hf) (m) = E h ( 1 ) f ( ~ h ( 2 ) m ) In t h e
~,~,i
coalgebras.
amounts
to
M
above.
. If M,N£H,
defined
by
60
(xf) (m) = x f ( x - l m ) , which
is m o r e - o r - l e s s
standard
while
(4.6)
Proposition.
H-modules
iff
Let
M,N
for all
h£H
in
H
, mEM
.
To m o t i v a t e
In t h a t case,
the
the proof formula
is e v i d e n t l y
x
-i
is
that case
above becomes,
for a l l
for a l l
x£G
arbitrary,
sible.
.
The proof
lution
-i
f(m)
is g o t t e n b y
is a g r o u p .
,
of t h i s
combinatorial
el = 6
into diagrammatic
argument
commutative , 61 = @
.
would
diagrams. If w e
language.
seem nearly Since
replace
h
1 by
impos-
is an i n v o lh
the
=
(Elh) f(m)
= Elh(1) f(h(2)m)
can be written [email protected]~M.I~H~M.6~M
where
we use
mutative
the c a n o n i c a l
isomorphism
= f. E S M
to i d e n t i f y
K~M
with
M
.
Then we have
a com-
The polygon
marked
diagram HSM
H~H~M
H®(*)
x~G
G
= f(xm)
juxtaposing
structure
(*)
The
where
H = ~ G]
above becomes (6h) f(m)
This
is a h o m o m o r p h i s m
= f(x-lm)
is a t r a n s l a t i o n
that a purely
of the c o a l g e b r a
formula
f : M---+ N
this means
The proof below
mentioning
.
= xf(x-lm)
xf(m)
It is w o r t h
"bracket"
to x
and since
map
the
= ~ h(1) f ( l h ( 2 ) m )
we c o n s i d e r
equivalent
- f(xm) , x ~ L
A K-linear
f(m) which
, it b e c o m e s
,
(6h) f(m) Proof.
L
for a lie a l g e b r a
(xf) (m) = xf(m)
of
xEG
left-lower
sequence
is e x a c t l y
that,
HSF
H~H~f
~
H~N
-w H ~ H ~ N
of this diagram (*)
being
the
~N
~ HSN
is the u p p e r - r i g h t formula
labelled
N
.
of next.
above.
6~
H@f
H~uM
H~N
7/
8~M I H~uHef
H~H~uM
> H~N
H~C~N~ H~H~uH~M H ~ I(9/I(9.MI
H~M
H~(*)
H~ f
H~uH~H~UM H~H~ M H®HSf
Then we have a c o m m u t a t i v e d i a g r a m
H~M~
H~HOM
[ [
I
6~M
H~60M
d®H®M
H®HSM
H@M
) H@H@M--
I
H®f H~N ~ N
H~H~H~M
I I
H®H~M
HSHSz M
H@~ M
6~M
H~ISH®M
•+ H ~ H ~ H ~ M
H@I~M
H~H@M H@H~f
H~H®f
+ H~H~N--
H@I@N
H~ N HSN
H~H~N
I
£~N
K®N
>
H~N
~N
*
N
Finally w e have a c o m m u t a t i v e d i a g r a m H®ZM H(~H(gM-
/ H~M
K~N-I~M.
H@f > H@M
K~laM
> K~M
~ H~N
K~f
~ K~N
r/~N
> H~N
> H~N
62
Putting
these
diagrams
is, o f c o u r s e ,
together,
the condition
we
see f
that
that
(*)
implies
that
b e an H - h o m o m o r p h i s m .
f ' Z M = ~N "H@f T o go the o t h e r
"
This
way
the
equations f(hm)
= hf(m)
,
{h = E h ( l ) (ih(2)) imply
that ~ h ( l ) f (ih(2) m)
: E h ( l ) (ih(2))f(m) =
(4.7)
If
an o b v i o u s K ---+ M
H
is a h o p f
way
an H - m o d u l e
is d e t e r m i n e d
in t h e c a t e g o r y
over
K
structure
on
algebra
uniquely by
the element
m
(~h) f ( m )
K
.
an e l e m e n t must
If meM
satisfy,
In o r d e r heH
K
module,
that
determines
in
a K-linear
map
the map be a morphism
,
.
a map K
is d e t e r m i n e d
is a n o t h e r
M .
for all
(eh)m = h m In p a r t i c u l a r ,
: H ~
, the a u g m e n t a t i o n
uniquely
by
a K-linear
+ H(M,N) map
f : M---+ N
such
that
(~h) f = h f which
means,
in v i e w
of the H - a c t i o n
defined
(6h) (fm) i.e.
that
f
Proposition.
any
M,N
in
H
Thus
Define
MSN
~
to b e t h e K - m o d u l e
the algebra
a symmetric,
is c o c o m m u t a t i v e
associative
tensor.
shown,
Hom(K,H(M,N))
M~ N K
h(mSn) Since
we h a v e
,
,
Hom(M,N) (4.8)
(4.4),
: Eh(1)f(ih(2)m)
is an H - h o m o m o r p h i s m . For
in
with
H-action
: ~h (i) m ~ h (2) n
counit
by
"
and coassociative, The
given
it is e a s y
to see t h a t
this
gives
identity
h = ~ h ( l ) (ch(2)) gives
rise
to a n i s o m o r p h i s m M
so t h a t (4.9)
this
~
makes
Proposition.
H
into
For any
a symmetric
M,N,P
in
Hom(M~N,P) Proof.
Since
isomorphism,
as a K - m o d u l e ,
M~N
~
= MOKN
H
M~K
monoidal , there
category. is a n a t u r a l
isomorphism,
+ Hom(M,H(N,P)) and
H(M,N)
= HomK(M,N),
there
is a c a n o n i c a l
63
HomK(M~N,P )
~ HomK(M,H(N,P))
.
Thus the only thing to check is that a morphism on one side corresponds the other. all
If
f : M®N --+ P
h6H , m E M , nEN
is a K - l i n e a r
m~p it is a m o r p h i s m
to a morphism on
iff it satisfies,
for
, (6h) f(m®n)
= ~ h(l ) f(lh(2 ) (m~n))
(i)
= E h(1)f(lh(2)(1)m~lh(2 ) (2)n)
The corresponding
map
~
: M---+ H(N,P),
defined by
~(m) (n) : f(m@n) must satisfy for all
hEH
, mEM (~h)(fm) = ~ h(1)~(lh(2)m)
which means,
for all
n6N
, ({h) (fro) (n) = ~ h(1)~(lh(2)m) (n) (Eh) f(m~n)
= ~ h(l ) (1)f(lh(2)m) (lh(l) (2)n)
(ii)
= E h(l ) (1)f(lh(2)m~lh(1) (2)n)
To see that these are equal, of
H
on
M,N,P,
let
respectively
note the interchange
ZM ' ~N ' ~P
(e.g.
isomorphism.
denote the maps describing
~M : H~M---+ M)
Also let
From the cocommutativity
the action
o : H®H
> H@H
and coassociativity,
deit
follows that the d i a g r a m H
~H H~
H~H
commutes.
This implies that for
'÷
H~
~ H~H~H
II®H®H--
hEH
h(1)®h(2) (1)®h(2) (2) = ~ h(1) (1)®h(2)~h(1) (2) By applying
H®I@I
, tensoring with
m®n
and applying
an interchange,
we conclude that
~h(1)®lh(2) (l)~m~lh(2) (2)~n = ~h(1) (1)®lh(2)~m®lh(1) (2)®n • Now apply
~p.H~f.H®~M~ N
(4.10) Theorem.
H
and the required
is an autonomous
Note that the set underlying
identity of
is
HomK(M,N)
distinct
have led in the current
, not
from maps.
chapter III it may have appeared that the pseudomaps Although
(ii) results.
category. ~(M,N)
have an example which features pseudomaps
mary category.
(i) and
instance to considering
.
Thus we
In the development
in
should have been taken as the pri-
that point of view would have simplified
but whose maps are just K-linear.
Hom(M,N)
the language,
the category whose objects
it would
are H-modules
64
(4.11) Until n o w we h a v e not used the fact that sults are v a l i d field a l t h o u g h (4.12)
for a r b i t r a r y it w o u l d
It is e v i d e n t
h o p f algebras.
likely be s u f f i c i e n t
that
H
is a v a r i e t y
should be n o t e d that the theory of G]-modules
in c o m m u t a t i v e
the c a n o n i c a l
one.
The p s e u d o m a p s
This
M--~ N
iff
H
G
that
is a field.
Thus the p r e c e d i n g
follows w i l l
require
K
have a dualizing
as w e l l as c l o s e d m o n o i d a l
is n o t in general
is)
is r e f l e c t e d - i.e.
K What
in the e x i s t e n c e of
commutative
re-
be a
module. It
(the theory of
of p s e u d o m a p s
H(M,N)
K
category.
and even if it is, the closed
the e l e m e n t s
that
structure distinct
- are the K - l i n e a r
is not
from maps.
maps while
the m a p s are the H - h o m o m o r p h i s m s . Since
the c a t e g o r y
gical H - m o d u l e s .
W e let
logical H - m o d u l e s
which
is
K
H
is abelian,
D are,
as K - v e c t o r
e q u i p p e d w i t h the H s t r u c t u r e
subcategory derlying
of o b j e c t s w h i c h h a v e
any such o b j e c t
entations. the dual
space of the v e c t o r
spaces,
• .
M•A
K-vector
, the space
space u n d e r l y i n g
is e q u i v a l e n t C
compact.
Evidently
.
A
denote
the v e c t o r
M(IM],K)
The same
underlying
of topo-
The d u a l i z i n g
object the
space un-
space w i t h enough d i s c r e t e underlying
M
to topolo-
b e the c a t e g o r y
As usual w e let
a D--representation.
IMI and hence
and
linearly
by
Thus the linear t r a n s f o r m a t i o n
is an i s o m o r p h i s m
Un H
H-modules
induced
is a t o p o l o g i c a l
We also k n o w that if
M* = A(M,A)
the c a t e g o r y
b e the d i s c r e t e
repre-
is the same as
is e v i d e n t l y
the c a n o n i c a l
true of
map
+ H (M*,K)
this m a p is an isomorphism.
This
shows t h a t every o b j e c t
is q u a s i - r e f l e x i v e . To s h o w that o b j e c t s that every t o p o l o g i c a l spaces,
in
~
K-vector
has a r e p r e s e n t a t i o n
have
a D--representation,
by discrete
spaces.
underlying
an o b j e c t of
sufficient
to find a family of pseudomaps,
maps.
For that,
structures
The p r e - * - a u t o n o m o u s
ity r e q u i r e d ning parts spaces
sums of copies structure
discretely
underlying
C*
of
already belongs linear,
to d i s c r e t e K
is evident.
insures
that
are immediate. and
i.e.
.
spaces w h i c h
The m o d u l e
A*
Thus the g e n e r a l
Recall,
applies
in s h o w i n g
A
.
space
For it is H-linear,
can b e g i v e n H - m o d u l e for all m a p s of
in
(4.5).
when
f
the fifth,
K-linear
here.
to
of c o n t i n u o u s
described
to o b s e r v e of o p e n sub-
is the v e c t o r
this h o l d s
is c o n t i n u o u s
are the c o n t i n u o u s theory
0
but not necessarily
In p a r t i c u l a r
hf
basis at
If such a space
and g i v e n the H - s t r u c t u r e
of the H a c t i o n
of III.4.2
respectively.
, t h e n that o b j e c t
y o u can take the maps
as d i r e c t
is t o p o l o g i z e d
Un H
it is s u f f i c i e n t
space w h i c h has a n e i g h b o r h o o d
C£C
.
C ---+ D
The c o n t i n u -
is.
The remai-
that the v e c t o r
functionals
on
C
and
A,
65
5.
(5.1)
In t h i s
as o b s e r v e d
section
we
Topological
l e t V be
at t h e b e g i n n i n g
of t o p o l o g i c a l
abelian
L e t A be
the c a t e g o r y
of t h i s
chapter,
Groups.
of abelian
identify
groups.
Un V w i t h
group.
By a seminorm
on A is m e a n t
p(0)
ii)
iii) iv) v) vi) (5.2)
An
;
p(a-a')
< p(a)
Trivial
consequences
p(a)
~< 0
p(a)
= p(-a)
Ip(a)
invariant ±)
+ p(a')
= d(a+a",
iii)
d(a-a')
= d(a-a",
~(p)
on
in
) ;
(ii)
.
for all
d
a,a',a" e
such A
that
.
are ;
a'-a")
= d(0,
a seminorm
) ;
;
a'+a"),
consequences
d(-a,-a')
a, a' E A.
on A is a p s e u d o m e t r i c
= d(-a,-a')
be
(ii)
- p(a') I ~< p ( a - a ' )
d(a,a')
p
in
+ p(a')
pseudometric
d(a,a')
(ii),
for all
(take a ' = 0
~< p(a)
Trivial
Let
,
are
(take a=a'
ii)
(To p r o v e (5.3)
= 0
p(a+a')
.
a-a') A
.
= d(a-a',0)
Define
: A x A
= d(a,a').
a function
,
by (a,a') for
= p(a-a')
a , a ' , a " E A, e(p) (a,a')
= p(a-a') ~< p ( a - a " )
= e so t h a t ~(p)
is a p s e u d o m e t r i e .
+ p(a"-a')
(p) (a,a")
+ e(p) (a",a')
Also,
e(p) (a,a')
= p(a-a') = p (a+a"-a"-a') = ~ a+a"-(a"+a')]
= e(p) (a+a', and we
see t h a t ~(p) Let
d
be
can and will, the category
a function
: A--~
that i)
Then
We
T o p V,
groups.
an a b e l i a n
p Such
Abelian
is an i n v a r i a n t
an invariant
a'+a")
pseudometric.
seminorm B(d)
on
A
and define
: A----+~
by Bd(a)
= d(a,0)
.
a function
66
Then f,d(0) = d(0,O) ~d(a-a')
= 0
: d(a-a',
0)
: d(a-a'+a',
O+a')
: d(a,a') < d(a,0)
+ d(0,a')
: d(a,O)
+ d(a',0)
=
so t h a t (5.4)
8d
8d(a)
+
8d(a')
is a s e m i n o r m .
Proposition.
The
correspondences p ---+ ~ (p) d ---+ B (d)
determine on
A
a
correspondence
1-1
between
seminorms
A
on
and invariant
pseudometrics
.
Proof.
We have ~Bd(a,a')
- 6d(a-a') : d(a-a',
0)
= d(a-a'+a',
a')
: d(a,a') Also, 8~p(a)
Thus
~
(5.5)
and
8
are i n v e r s e
Proposition.
following
Let
=
p(a-0)
=
p(a)
isomorphisms.
A
be a t o p o l o g i c a l
p
is c o n t i n u o u s
ii)
p
is u n i f o r m l y
~(p)
is c o n t i n u o u s
~(p)
is u n i f o r m l y
The
If
p
is c o n t i n u o u s
M
the
f i r s t map,
is c o n t i n u o u s
components
ous. of
Finally, 0
for
a seminorm.
Then
the
;
(resp.
(resp.
of the suppose
.
uniformly
continuous)
uniformly
first map that
p
are
continuous,
continuous),
a' E
a+M
the
identity
is c o n t i n u o u s .
and For
0
composite
implies
p(a)
< s .
- p(a') I <_ p ( a - a ' )
< s .
composite
is.
If
is the c o m p o s i t e
and
C > 0
implies IP(a)
is the
so the
p = ~(p)
such that
a @ A,
~(p)
A ~E+FR
s u b t r a c t i o n , is u n i f o r m l y
a E M Then
p
;
continuous
A x A ~
~(p)
and
; continuous
iv)
in w h i c h
group
are equivalent: i)
iii)
Proof.
= ~p(a,O)
it is u n i f o r m l y
, choose
continu-
a neighborhood
87
(5.6)
Let
A
so is
-M
and
h o o d of
0
be
a topological
M0 = M n
contained
(-M)
in
M
abelian .
.
The
group
set
We m a y
and
M0
M
be
a neighborhood
is a s y m m e t r i c
inductively
choose
of
M.
(i.e. M 0 = - M 0)
Then
neighbor-
a sequence
M I , M 2 , M 3 , .... of s y m m e t r i c
neighborhoods
of
0
such M
for a l l
n
+M
~
is a f i n i t e
set of strictly I(~)
the c o r r e s p o n d i n g
If
1
is a p o s i t i v e
Proposition.
dyadic
rational
dyadic
rational,
For
If
~+v >
the c o n c l u s i o n
1 ,
define = A : M0
means
there
are
finite
is b y
the s m a l l e s t
a double
integer
sets
~ n
;
~
8 •
;
case,
v >
v
If
,
~=i
and
v=O
or v i c e
versa,
i
I(B)
and =
T h u s we b e g i n
8
such
that
v
f i r s t on t h e
cardinality
of
supposing
that
by
N 8
and
second
on
.
we h a v e
union
U
e
: Z{2-ili the
by
1
and
~
l(a U 9) : Z { 2 - 1 1 i
since
A
suppose
of integers :
of
M(~+v)
is e v i d e n t .
i
M(1)
~} D
C
~ n s : ~ In t h a t
i >
rationals
induction;
in
if
Thus we may
i(~)
The proof
,
+ M(v)
D > This
let
~}
a subset
: ~{MiJi E
conclusion
is a l s o e v i d e n t .
E
integers,
number.
M(1)
any dyadic
the
positive
M(1)
M(~) Proof.
Mn-i
= Z{2-ili
M (i(~)) (5.5)
C
n
n > 0 If
be
that
8}
e ~} + E { 2 - i l i
E
8}
Thus
is d i s j o i n t .
~(~ u
8)
: l(e)
+
(8)
Similarly M(D+V)
= M (I(~) =
M
+ I(8))
(X(~
u
8))
= z{Mili
e
~ u
: E{Mil
i e
= M (I(~)) =
Next we
suppose
that
~ N
8 # 0 :
and has
~, n
B'
has
fewer
the s a m e n u m b e r
and
of elements
;
than but
+
the
v
e ~
8
e} + E { M i l i E
6}
+ M (I(B))
M(v)
t h a t the
~(~')
elements
M(~)
8}
conclusion :
is v a l i d w h e n e v e r
~(8')
and
that
least integer
it is a l s o v a l i d w h e n of
~, n
8'
is s m a l l e r
a' n than
8'
68
that Now
of
e N
let
i
B • be
the s m a l l e s t
element
of
a N
~ .
1 2 a n d the
only
possibility
consistent
Then
if
i = 1 ,
1 2
with ~ + ~
<
1
is 1 2 in w h i c h
case
the
result
follows
from M1 + M1
Now
suppose
i > 1
.
Then
if
they
have
which
in at l e a s t
i-i ~ the
B , same
is s m a l l e ~
M0
.
Since i - i
it is n o t
C
o n e o f them.
a' n
~,
number
than
~n
Suppose, =
~ - {i} U
B'
=
~-
has
fewer
of
a n
B
say, t h a t
~'
i-i ~
~
.
In t h a t
case,
let
{i - i}
{i}. elements
of e l e m e n t s
that
~
but
S •
the
than least
In e i t h e r
d n
B
element
case,
while of
our
if
a' A
inductive
i-i • B'
is
B , i-i
hypothesls
implies M (I(~'))
+ M (ICB'))
C
M (ICa')
+ I(B'))
Evidently I(~')
=
l(e)
- 2 -i + 2 - i + l
I(B')
=
I(B)
- 2 -i
so t h a t
Finally, M (l(e))
If
=
Z{Mj]j
6 a} + X { M j ] j
=
Z{M.Ij ]
• a-{i}
} + M i + X{Mj]j
•
X{Mj[j
• a-{i}
} + M i _ 1 + ~{Mjl j e
=
Z{Mjlj
• ~'} + Z { M j [ j •
=
S (l(a'))
c Corollary.
+ M (I(8))
~-{i}
B}
} + M. x
B'}
S'}
+ S (i(8'))
M (~(a') ~ < ~
•
6
+ I(~'))
are
dyadic
=
M (~(a)
rationals,
+ I(8))
then
M(~) C M(~) Proof.
For
the difference
of t w o d y a d i c
M(~) • M(~) (5.6)
Proposition.
is g i v e n Proof.
by Let
The
a family A
be
topology
(resp.
of continuous
a topological
rationals
+ M(~-~)
uniformity)
seminorms group
{M(1) ] I
and
(resp. M
a dyadic
is o n e
so t h a t
C M(~) of a n y t o p o l o g i c a l uniform
invariant
a neighborhood. rational
}
There
abelian
group
pseudometrics). is t h e n
a family
69
o f s!nmnetric
neighborhoods
of
0
such
that
M(1) M(I) and whenever
I ~ ~
+ M(~)
C M(I+~)
, M(1)
Define
C M
C M(~)
a function p
: A
~
by p(a) as
1
runs
(5.7)
over
The
function
(i)
p(0)
= 0
;
(ii)
p(a)
= p(-a)
(iv) (v)
(ii)
(i) Since
p(a-a')
-< 1
is e v i d e n t each
M. 1
dyadic p
has
the
+ p(a')
if
if a n d o n l y
if
following
properties.
;
;
implies
a E M
f r o m the
fact
is s y m m e t r i c
. for e v e r y
0 E M(1)
so is e a c h p(a)
if a n d o n l y
E M(1)}
rationals.
;
= p(a)
p is c o n t i n u o u s p(a)
= inf{IIa
the positive
Proposition.
(iii)
Proof.
all
M(1)
positive
dyadic
rational
1
Thus
_< 1
a e
M(I)
-a E -M(1) if a n d o n l y
if - a E M(I)
if a n d o n l y
if p (-a)
Since
the
dyadic
rationals
are
dense,
p(a) (iii)
It is s u f f i c i e n t ,
now,
to s h o w
p(a Now
let
e > 0
Since
the
I and
I'
dyadic such
be
< 1
this
is o n l y
possible
= p(-a)
that
+ a')
-<
p(a)
+ p(a')
are
dense
o n the
line,
there
that p(a)
p(a')
< I < p(a)
< I'
< p(a')
that a E M(1)
and a' E M(I') so t h a t
.
given.
rationals
+ e/2
and
It f o l l o w s
if
+ £/2
are positive
dyadic
rationals
70
a + a'
C
M(1)
C
M(Z
+ M
(i')
+ l'
so t h a t p(a+a')
Since
s > 0
is a r b i t r a r y ,
this
Let
a C A
Then
a + M(1)
and
s > 0
I + I'
<
p(a)
is p o s s i b l e
p(a+a') (iv)
<-
be
<
p(a)
given.
+ p(a')
only
+ c
if
+ p(a')
Let
I
be
a dyadic
rational
such
that
)~ < 6 . is a n e i g h b o r h o o d
of
0
and
a'
E
a + M(1)
if ,
we h a v e , a'
- a
E
M(1)
or p(a'-a)
<- I < a .
But p(a)
: p(a-a'+a')
<- p ( a - a ' )
+ p(a')
gives p(a)
- p(a')
-<
p(a-a')
Similarly, p(a')
- p(a)
-< p ( a ' - a )
= p(a-a')
so t h a t Ip(a') (v)
This
(5.8)
is b y
This
borhood
M
essentially of
- p(a) I
0
completes
, there
the
(5.9)
At
abelian
continuity
of
this point we
groups.
p(a'-a)
Let
{A
p
~
}
be
p
<
For we have
such
p-l([ 0,1])
guarantees
require w
the a r g u m e n t .
is a s e m i n o r m 0
while
S
definition.
that
a digression a family
shown
that
for a n y n e i g h -
that
C
M
p-l([ 0,i])
is a n e i g h b o r h o o d
o n the
in the
of a b e l i a n
sums
groups.
category
of 0.
of t o p o l o g i c a l
Then
A : ~A denotes, finite p =
as u s u a l ,
number
(pw)
be
the subgroup
of n o n - z e r o the s e m i n o r m
of
terms. on
A
HA If
consisting
for e a c h
defined
It is t r i v i a l
is a s e m i n o r m
on
only A
,
a let
to see t h a t p is a s e m i n o r m th w summand, then
on
A
and that
if
u
: A
w
> A
is the
of the
supposing
topology over
Pw
with
E Pwaw
PW NOW
,
sequences
by
P(aw)
inclusion
w
of all
each
Ai
=
is a t o p o l o g i c a l
for the s e t of s e m i n o r m s
the s e t of s e m i n o r m s
on
A
w
PU abelian
p =
(pw)
We
call
g r o u p we e n d o w
so d e f i n e d this
as e a c h
the d i r e c t
A -wP
with runs
sum topology
the w e a k independently on
A
, a
71
term which
we w i l l
Before seminorms
p
assertion
justify
later.
stating
the n e x t p r o p o s i t i o n ,
and
on
that
q
for a l l
A
,
p
£ > 0
refines
the
q
cover
{M C A l a , a ' refines
it w i l l b e
of s e m i n o r m s
q
(5.10)
there
~
is an
continuous Proof.
Let
Supposing
p /
on
B
b + M
where
M
and
B
discrete ,
p.f
let
p
be
if
p.f
groups.
so is .
of
a seminorm
is a s e m i n o r m
on
A
p.f
.
on
B
of
O
.
if a n d o n l y
groups f
and
the
N
be
such
that
f : IAI
on
A
seminorm
neighborhood
Let
for a n y
of
semi-
. ~IBI b e
is c o n t i n u o u s
seminorm
while
that sq
iff
a
for e v e r y
. property
b
is e v i -
is o f the
a neighborhood
of
0
form s u c h thai
, < i} C f - l ( s )
We have + M)
if f(a)
if a n d o n l y
refines
< l} c N
a E f-l(b
e
b +M
if £ (a) - b
is t r u e
a
to the
the p r o p e r t y p
Then
Every
0
{ a l p (f (a)) is a n e i g h b o r h o o d
with that
topological
{b'Ip(b')
and
is e q u i v a l e n t
< s }
is a c o n t i n u o u s
b C B
is a n e i g h b o r h o o d let
such
be
is c o n t i n u o u s ,
To go the o t h e r w a y ,
if
This
< ~ }
--)q(a-a')
p E p
and
of t h e u n d e r l y i n g seminorm
N + N C M
and A
dent.
Now
.
of two
by
is a s e t o f s e m i n o r m s
s > 0
Proposition.
homomorphism
which
p k q
to s a y t h a t
the cover by
A basis
Then
A
E M =~p(a-a')
{M C Ala, a, E M
norm
if
of
convenient
M
p (f (a) - b)
<
if
=
E A
~
is s u c h
1
-
p
(f
(a)
-
1
b)
>
0
that Ipf(a)
- pf(a') I
<
s
,
we have p (f(a')
- b)
-<
p (](a')
- f(a))
<
6 + 1 - e = 1
+ p (f(a)
- b)
so t h a t f (a')
E
b + M
Thus f-l(b is an o p e n (5.11) topology
set
in
A
Proposition.
+ M)
and hence
f
is c o n t i n u o u s .
Let
be
a family
{A
w
}
of topological
on A
=
EA
oJ
abelian
groups.
Then
the
72
described
above
is the
finest
for w h i c h u
each
inclusion
: A ---+A w
w
is c o n t i n u o u s . Proof. and
It is e v i d e n t
p =
hence
(pw)
each
that
if for e a c h
, then p~ = P-u
uw
is c o n t i n u o u s
is.
-
Conversely,
for e a c h
w
.
~
Thus
'
suppose
Let
p =
q
is a c o n t i n u o u s
:
such
is a n y
(pw).
a
Pw
for e a c h
Then
p
,
seminorm
p-u
seminorm
on
A
is c o n t i n u o u s
on
A
such
and
that q.u
p~
for
(a w) ~ A
we have :
Euw(a )
=
q (Z u
_<
Z qum(a )
so t h a t q(a)
and consequently defined
by
Corollary. Proof.
The
If
for all
p
refines
all s u c h
B
~
p
q
.
defined
with
ness
direct
sum
(5.12)
in the u s u a l w a y . on
f
A
,
qfu
is the
group
Then
= q[w
and henceforth
abelian
group
A, A*
ways.
Here we
proposition.
and
f
is a c o n t i n u o u s
: A
>B
fu
w
= f
If
Thus
f
on
A
q
is a s e m i n o r m and hence
w
is a c o n t i n u o u s
Let
{A
is
in this
the g r o u p
take }
the
be
section,
we
let
of c o n t i n u o u s
topology
a family
T
qf
on
B
, qf
is
is a s e m i n o r m
homomorphism.
For
is a n a t u r a l
of topological
each
~
there
A * w isomorphism.
dualizes
universal
The
on
unique-
abelian
~ (HA w) *
is a p r o j e c t i o n
property
of the s u m
----+ A
gives
Z A * ---+ ( ~ A ) * is e v i d e n t l y
canonical
and natural.
Now
~/Z
.
For
a topological
A----+ T
topologized
convergence.
to
mapping
=
homomorphisms
of compact
A * ---+ ( ~ A ) * w W
which
homomorphism
: ~ A ----+B w
: ~A
The
in t h e t o p o l o g y
sum.
map
and that map
which
is c o n t i n u o u s
categorical
is a s e m i n o r m
sum topology.
varying
Proof.
q
is c l e a r .
Here
canonical
p(a')
In p a r t i c u l a r ,
topology
is a t o p o l o g i c a l
the direct
of
E pw(a)
=
, let
a seminorm A
=
.
f be
(a))
suppose
groups.
Then
there
is a
in
73
f is a c o n t i n u o u s
homomorphism.
:HA
> T
w
Let -
be
the m a p w h i c h
is t h e
identity
)]]A
on the
:
~
coordinate
and
0
o n all o t h e r s .
Let
=:.u w
I first
claim
that only
the n e i g h b o r h o o d
contains
of
0
no n o n - z e r o
finitely in
[ w represented
T
subgroup
of
eventually, by the archimedean
Similarly
for a
M
f-l(M)
.
Now
t C (1
, 0)
is to b e
many
T
.
Thus
M
= ~i'
is a n e i g h b o r h o o d "''' m
n
M
~
= A
of
0
doubling
the i m a g e
a neighborhood
of
D
in
from
interval/
of the r e a l s ,
[ -i (M) where
the
[ Repeated
property
.]
are d i f f e r e n t by
~A
: -i (M)
and except
for
B
=
nB
where I = I0, B
!
A
Since
B
is a s u b g r o u p
w = w I, ,
that
...,
otherwise
mn .
and f (B) C
it f o l l o w s
: (B) = 0
.
M
In p a r t i c u l a r
since
u (A) w
C B
;
w
#
a =
(a)
for
it f o l l o w s
that [~
Then
=
0
w I, . . . ,w n
if ~ A
,
the difference • --r
a' belongs
to
B
SO t h a t
= f (a')
a --
l{umawl
= 0
.
fa
~ : ~i'
Hence =
[(Zua w
)
=
Zfu
=
Z
=
Z:
~a
=
L:
~T
[
a
a
so t h a t :
M
be M
will
(0,41-)
the i n t e r v a l / i k4
subgroup
cannot
~3
}
n
finitely
that
many
w
,
i~
27
lie in
HM~
M
D
t E
let
Evidently
This means
In p a r t i c u l a r
w
of a
For
1 ~.
p u t it i n t o
in
.
,~ J
of a non-zero
0
0
1 k- ~
say
74
This
shows Now
t h a t the
canonical
let
be
Mi
map
is at
the n e i g h b o r h o o d
least of
(_9_-I, for
i -> 2
.
Then
the
family
of all
an a l g e b r a i c
0
in
T
isomorphism.
represented
by
the i n t e r v a l
2 -i)
Mi
is a n e i g h b o r h o o d
base
at
0
.
In p a r t i c u -
lar M2 (as d e f i n e d
above).
Now
If
{f as
X
ranges
compact
when
over X
compact
sets
and
over
0
all
to
X
which
contain
0
for all
x E X
, and
.
jx for all s u c h
f
integer
i
:
jx +
consists
of
> 2 .
X U
Since
{0}
is
,
, ,
(2 i-2 - j)0
C
2i-2x
j ,
(X) ~ M i , t h e r e
is s a m e
j < 2 i-2
M such
f (jx) ~ which
integers
C M2 : M
f (jx) e If
A*
If
0 < j < 2 i-2
then
in
~ {sls(xu{0)) cMil
f (2i-2x) since
at
is a n d
can restrict
then,
base
(X) C Mi} i
{:Is(x) c M )1 we
M
=
a neighborhood
is a c o n t r a d i c t i o n .
In fact,
j
that
M
may
taken
as
2 raised
to the p o w e r
4 +[log 2 Irl] where
is the a b s o l u t e l y
least
residue
of
x
(mod i).
Thus {f If (X) C M i } Since under
2i-2x the
is c o m p a c t
addition
map of
when
X
D
is
{f If (2i-2x)
--
that power
of
C M}
it is the
image
A
--
to
A
of t h e
2i-2nd
the sets of the
power
of
X
form
{flf(x) c M) as
X
Now
let
least
ranges p
over
be
residue,
the c o m p a c t
the s e m i n o r m modulo
i.
on
sets T
in
A
which
form neighborhood assigns
to
x
base
, 4 times
at
0
in
A
its a b s o l u t e l y
least
Then x E M
{--~ p(x)
< 1
Thus f (X) C M if a n d o n l y
if
p f
< i
on all of
~(s) then
f
belongs
:
X
. If w e
if
~
sup {p~(x) Ixe x)
to
{gIg(x) c M} if a n d o n l y
let
denote
the
seminorm
defined
by
75
~(f) Thus of
the
seminorms
seminorms Now
determine,
<
X
as
1
ranges
X C A
is
compact,
~ X
is
sets
of
A
, a basis
compact
,
in
A
Evidently
w
X
that
<s or,
equivalently,
Now
if
is
Is
<x> c M}
refined
by
D
is
also
sists
of
compact ~
where
A Y
and X
is
<s
If
<x #) c M}
X #i
Y : x u ¥
compact
to
X D X# : ~ so
the
A*
of
returning
A = ~A if
over
refines
~
{0}
.
u
(-x)
Thus
a basis
of
seminorms
of
A = HA
con-
compact X = ~
X
0 E
,
X
,
and X = -X I claim
that
under
these
hypotheses
: ( (~ x) ^ ) (5.13)
At
Lemma.
Let
of
is
x.
this
p o i n t , we r e q u i r e ,
n > 1
and
x_, ±
...,
x
E
T
be
such
that
the
absolutely
least
residue
n
positive, px i < 1
,
j = l,
...,
n-i
,
i=l and px n < 1 Then
the
absolutely
least
residue
.
of
g xi i:l is p o s i t i v e .
Moreover,
n Px i i=l Proof.
We will
hypotheses
are
confuse
xi
and
its
=
p
absolutely
least
that 0
-<
Xl
<
0
<-
X2
<
1 4 1
residue.
When
n =
2 ,
the
76
and,
of c o u r s e , 1 xI + x 2 <
This
implies
residue
that
is its o w n
xI + x2
is p o s i t i v e
and moreover
absolutely
p ( x 1 + x 2)
For
larger
Then
~e
n
, we may
hypotheses
suppose
least
residue
and hence
that that
that =
4 ( x 1 + x 2)
=
4x 1 + 4x 2
=
p ( x I)
inductively
are s a t i s f i e d
+ p ( x 2)
Vat
the assertion
is v a l i d
for
n-i
.
by n-i and
xn
i=l
Hence
the a b s o l u t e l y
least
residue
of
p
i n~ i=l
=
p
is p o s i t i v e
X x. i=l
and
+ px n
n-1 =
~ PX i i=l
=
~
+ px n
Px i
i=l
(5.14)
Now we
A
that
such
return
to
~e
proof
of 5.12.
We
X
=
H~ X
0
E
X
X
=
-X
suppose
t h a t is a c o m p a c t
;
and
Let f be
such
: A
~T
that ~(f)
T h e n is t h e r e
some
x E X
such
>_
p [ (x) Now
f
is a f i n i t e
1
that >-
i
sum f (x)
=
Z fu~x
and -<
p f (x)
=
p(Z f U
~ n)
_<
Z p f u~X
-<
Z sup
=
A Z X
{p f u n Ix e X } c0 w ~ L0
(fu)
s~set
of
77
Thus
{siS(s) T o go the o t h e r
way,
l}
<
D
{sl Z
X^
(fU a)
< I}
.
suppose A
k (f u w) Then
there
are
x
E X
such
w
that
Zp[ux (of c o u r s e
X
a n d the
arguments
would
weren't.)
Now
are
complicate since
lutely
least
residue
nitely
many
~
Now we may
X
X
sum by
somewhat
is
[u
lemma,
so t h e s e
sups
but
the s a m e
principle
X
is s y m m e t r i c ,
x
finite.
induction,
this means
either
a is p o s i t i v e . Suppose
are
we
actually would
work
can suppose
Since
the
attained.
[u
= 0
s e t of i n d i c e s
even that
The if t h e y
the abso-
for all b u t
involved
is
fi-
~l,...,~n.
that
E {p [ u By o u r
>- 1
w
compact,
and hence
of each
, the
suppose,
-> 1
n
Iw = ~I . . . . .
~n-i }
<
1
that p fu~
x
->
n
1
n
or that p(E
In the
first
case,
there.
In t h e
where.
In e i t h e r
let
second
=
Z {p f u
->
1
x E X
case
case
{fu x x
have
let
x E X
Iw = al . . . . .
an})
la = ~ i . . . . .
an }
0
x
in e v e r y
have
xwi
coordinate
in t h e
~. 1
but
th
th
the
n
coordinate
and
and x
0
L0 n
else-
while f (x)
>
1
~(f )
>
1
so t h a t
This
shows
that
{~l~(s) < l} Thus
topologies
the
on
E A
:
and
{slz ~ (Su a) < 1}
(~A.)
are
identical
.
and these
groups
are
iso-
morphic. (5.15)
Proposition.
Proof.
Let
{A }
be
A direct
sum of complete
a family
of
complete A
and which
A'
be
the
same
the product
topological
group.
abelian
of any In
group
family
fact
for
~
,
M
but
groups
is a n e i g h b o r h o o d
)
and
topologized
of open
+ M
is c o m p l e t e .
~A
sets
a neighborhood H(a
where
=
groups
= of
by
is open.
of
(a)
(a a) + H M 0
in
A
the b o x First
topology I claim
contains
a set
-- the o n e
that of
the
A'
in
is a
form
w Moreover
if
N
is a n e i g h -
78
borhood
of
0
such
that N
- N
to
TIN
-
HN
LJ
Next
I claim
it
is
0
in
that
A'
sufficient, A'
closed
is
since
consisting
of
neighborhoods
B
be
the
subgroup
B
is
a closed
is
sets
of
0
(of
A')
subgroup.
In
closed the
For
A
a
~ 0
w neighborhood
for of
infinitely 0
such
w elements
M
be
w
an
by
show
an o b v i o u s
additive
that
is
But
are
of
there course
the
HM
those
of
the
of
1.2.5,
base
at
the
M m a y b e t a k e n as to c l o s e d in A . Now let
is
w
analogue
a neighborhood
direct
sum.
I claim
that
if
many
(a w)
g
For
all
to .
B
,
these
w
all
other
let
M
to
be
a symmetric
that a
Let
TIM tO
.
,
to
, whence
a then
C
to
in A
whose
M
fact,
complete,
in
C
LO
complete. A
w
arbitrary
{
w
neighborhood
M
of
w
0
for
M
KM
(ato)
+
coordinates
and
let
w
Then
is
a neighborhood
of
(a w)
(bto) then
C
have
so
that
+
aw
#
0
a
~
M
~
M
-a
M
w
to
,
then 0
so
(aw)
when
we
and
M
If
~
a
+
to
M
that b
#
to
0
Thus (bto) This
is d i s j o i n t
from
In p a r t i c u l a r The
B
B
.
is
topology
Hence
B
also on
(a) + M to complement of
the
for
each
to '
Pw
can be
is
described
of
0
be
A
such
on
to
in
=
a seminorm M
is a n e i g h b o r h o o d a seminorm
B
is o p e n
in
A'
and
B
is
closed.
complete.
B
p ( a w) where
~
means
B
where
as
sup A
B
N
M
w
w
(Note
.
generated
by
the
seminorms
that
the
sup
is
finite.)
For
EM
w is a n e i g h b o r h o o d
that
Mto C
one
(Pwaw)
on
:
the
{atolpato < l}
of
zero
in
A
to
, let
pto _
if
79
Then (a)
E
to
if a n d o n l y
]]M
pwato for all
w
, if a n d o n l y
<
1
if sup pmaw
Thus
the
topology
determined
by
completeness cient
on
EPw
B
is c o a r s e r
where
each
of t h e d i r e c t
to s h o w
that
sum
the basic
in
B
.
than
p~
<
that
means
from
another
on
A w )"
application
=
sum
(which,
Now we may
of
1.2.5.
recall, infer
is
the
It is the
suffi-
_< i}
{ (a w) I~ P w a ~
Suppose ~
M
that Z pwaw
Choose
of the direct
is a s e m i n o r m
(a) w which
1
neighborhood M
is c l o s e d
to
if
e > 0
> 1
so t h a t Z pwaw
Since
(a) LO
E
Z A
number
of i n d i c e s
03
,
this
>
i
s u m is a c t u a l l y
involved.
Then
for
+
e
finite.
i = i,
Let
...,
n
,
eI
'
define
...,
~
n
be
seminorms
the q~
finite by
1 qw i
nP~.l/~
Let qto
for i n d i c e s Moreover,
w ¢ el,
.. ., Wn.
Then
=
q~
is a c o n t i n u o u s
if (b) w
is s u c h
0
evidently
e
B
that sup qwb
<
1 ,
we h a v e q~ibwi
=
nPwibw'l
~
<
1
or < Pwibwi for
i - 1 .....
n
.
~/n
Then we have,
Z p
(a
+ b
)
-
D
~i a i
~,i.
i=l
>
a
i=l >
1
tO 1
O] 1
+
s
-
~ P~ibwi i=l s/n i=l
seminorm
on
A~.
80
=
1
=
1
£
+
-
E
If
N
=
{blsu p q b
< i}
,
we have that ( ( a M) + N ) and
M
(5.16)
is closed. Proposition.
This completes
A
Let
an e l e m e n t
Let
M~
X (a)
A
=
E A
lies in a sum of finitely many factors.
be a compact subset of • X
M
Let
Then every compact set in Proof.
~
the proof.
A .
Let
~
be an index such that there is
with
be a neighborhood
of
0
with
a~ ~ M~; and let
PC
be a s e m i n o r m on
A~
S~
D
such that {a • A~Ip~(a)
< i}
whence p~ (a~) Now assuming this to be possible
->
1
for at least countably many coordinates, =
say
~i' ~2 . . . . .
then we can find such seminorms P~l' P~2 . . . . Let
q
be the s e m i n o r m on
A
w h i c h has the seminorm qi
in the
i th
coordinate
i = 1,2,...
an e l e m e n t
and
0
=
lP~ i
elsewhere.
(a .) el
By hypothesis
6
there is for each
X
such that P~ia~i I
>
1
->
i
so that qia~i I so that q(a i )
=
~qj a~j j
-> qi a~i i >But then
is a continuous
unbounded
i
real valued map on the compact set
which
81
is impossible. (5.17)
Proposition.
Let
A
and
B
b e t o p o l o g i c a l a b e l i a n groups.
T h e n the canon-
ical b i j e c t i o n A @ B
) A x B
is an isomorphism. Proof.
This map exists in any p o i n t e d c a t e g o r y and is, of course, a b i j e c t i o n of the
u n d e r l y i n g a b e l i a n groups.
If
p
and
q
are seminorms on
I(a,b) Ipa + qb < 1 I
D
I (a'b) Ipa < 1} 2
The left h a n d side is a b a s i c n e i g h b o r h o o d of side is the i n t e r s e c t i o n of two such in C o r o l l a r y i.
0
(3 in
A x B .
A
and
B ,respectively,
I(a,b)[qb < 12 } " A • B
w h i l e the right h a n d
Thus the m a p is also open.
The canonical map (A • B)*
, A* x B*
is an isomorphism. Proof.
B y the c a n o n i c a l map is m e a n t the one w h o s e
A*
coordinate,
for example,
is
g o t t e n b y d u a l i z i n g the i n j e c t i o n A
~ A ~ B
At any rate, we h a v e (A • B)
(A x B)
B*
*
~
C o r o l l a r y 2.
If
AI,
..., A
A
*
X
B*
are t o p o l o g i c a l a b e l i a n groups, the canonical map
n
(Z A i)
~ HA i
is an isomorphism. (5.18)
Proposition.
Let
{A } ca
be a family of t o p o l o g i c a l a b e l i a n groups.
Then
the canonical map (~ A )
~ KA w
is an isomorphism. Proof.
The fact that the canonical m a p is a c o n t i n u o u s b i j e c t i o n is trivial.
b a s i c n e i g h b o r h o o d of
0
in
(EA)
ca
is N(X,M)
= {[ I[ (X) C M}
where X is c o m p a c t and
M
c o n t a i n e d in a finite s u m of X
C
A
A
N(X,M)
A
ca in
0
T .
It follows from 5.16 that
, say
• ... @ A 1
Thus
C
is a n e i g h b o r h o o d of
ca n
=
A(cal' "''' can)
is the i n v e r s e image of the set of the same name in A(ca 1 ,
....
can )
X
is
82
under
the
d u a l o f the
canonical
inclusion
A ( m l, From
the
commutativity
...,
) Z A
w n)
(Z Am)
>
]I A m
1 ...,
A(m I , together it
with
follows
arrow
the p r e v i o u s l y
that
the
(5.19)
w ) n
~
image
follows
A
x...x A mI
fact that
of t h a t that
mn
set
in
inverse
the
lower
Z A
image
arrow
is an i s o m o r p h i s m ,
is o p e n .
m is t h e
image
Since of
the
N(X,M)
upper under
map.
Before
and discrete
it
1
established
inverse
is a b i j e c t i o n ,
the c a n o n i c a l
w
of
continuing,
groups.
it is n e c e s s a r y
If
D
is a d i s c r e t e D
is t o p o l o g i z e d Proposition.
by The
the p r o d u c t group
=
to d i s c u s s
Pontijagin
group,
group
the
duality
for c o m p a c t
(D,T)
topology,
i.e.
of h o m o m o r p h i s m s
of
as a s u b s p a c e D
~ T
of
T
D
is c l o s e d
. in
TD
and hence
compact. Proof.
If
x,y C D
, the m a p T
defined
D
) T
by £ ~
~- f (x)
+
- f (x + y)
[ (y)
factors TD where
the
first map
a n d the s e c o n d
> T x T x T
is p r o j e c t e d
onto
are
continuous
and
so the
I
>
is c l o s e d .
The
(5.20)
C
inverse
of
0 =
intersection is a c o m p a c t
of t h e s e group,
we
over
the
discrete
of topological Theorem.
If
to
x,y
topology.
We
all
y))
x,y E D
is t h e n
closed.
use here
(C,T) one h i g h l y
non-trivial
fact
f r o m the t h e o r y
groups. C
is c o m p a c t ,
then
for any
~ : C that
x + y
, f (x +
x E C
,
x ~ 0
, there
is a c o n t i n u o u s
homomorphism
such
and
let
C with
corresponding
tI + t2 - t3
{f If (x) + f (y)
If
~ T
factors
takes ( t l , t 2 , t 3)
Both
the
$(x)
~ 0
Proof.
See [ Hewitt-Ross] , 22.17 on p.345.
(5.21)
This
implies
T
.
that
for a n y
compact
group
C
, the
canonical
map
83
C is i n j e c t i v e . known
fact
The
that
The
analogous
T
fact
is an i n j e c t i v e
continuity,
) C**
for d i s c r e t e cogenerator
for a c o m p a c t
C
immediately
fact that every requires,
from
map
in
of c o u r s e ,
(5.22)
Proposition.
Proof.
The
the p o i n t w i s e C*
to a m a p contains
~ :
0 < ~ < e . n l E K
.
= 0.
rR ----+ T
In t h e
such
~
if
K
topology
The
analogous
n
let
and
;
T*
~
~(27 ) : 0 ~ x ~
is c h o s e n
there
T
for the s e c o n d ,
number
case,
is d e n s e
Otherwise,
As that
positive
latter
Then
Thus
category
from the well
of a b e l i a n
groups.
~ C** convergence
is c o n t i n u o u s .
is o b v i o u s .
a smallest
numbers.
in t h e
readily
of t h e
fact
latter
and
for d i s c r e t e
the
groups
no p r o o f .
~* first
follows
, of
Cfollows
groups
so
nl
The kernel
[R
s > 0
Then
and n~
_< x <
.
(n+l)~
is s e p a r a t e d , Choose
1
<
( n+
n I
-
i
is e q u i v a l e n t a map either
small positive
there
is a
~ ~- K
, Inl - x I < s
K = m
T
of such
arbitrary
~ E K
-<
K
contains
T
is a s m a l l e s t
~ : T --÷
or e l s e
that
since
a map
~
such
and
@ = 0
with
and whence
that
i)~
Then 0 Since
i ~ K
and
n X E K
this
~
<
is o n l y p o s s i b l e
1 if
n I = i
or
1 n Now
choose
e > 0
so t h a t
~,~) Choose
an i n t e g e r
k > i
such
that 2-kll
so t h a t
for
j >- k
let
a
residue
of
be
absolutely
~(2-k-ix)
or
a > 0 ).
the
absolutely
least
the
residue
But
if
least of
least
can only be a e
residue
~(2-k-Jl)
of
E
~
~,~)
residue either
i
of ~a
~(2-k-ll)
or
~a + _ i
n2ka
=
2-J-kl
2-k-Jl
.
Then
the a b s o l u t e l y
(depending
and similarly
is =
-
can only be
is
implies
i
#(2-kl)
a - ~1 < _ 1
(0, ¼),
2-3a That
a
, ~(2-3X)
Now
<
a ~ .
least
on w h e t h e r
in the o t h e r
Similarly,
case.
a < 0 Thus
the a b s o l u t e l y
84
(x) This x
continues
x
to be true w h e n
are c l e a r l y d e n s e in
T
.
=
x 6
R .
2ka
:
~(I)
is an integer.
=
~ (2k2-kl)
1
12kal
is a s m a l l e r p o s i t i v e
m u l t i p l i c a t i o n by
n
or by
=
2ka
If 12kal
so that
n2kax
Moreover, 0
so that
and such
Hence ~(x)
for all
n2kax
is a l i n e a r c o m b i n a t i o n of such e l e m e n t s
-n
.
>
1
,
Hence
e l e m e n t in the kernel.
This
~
is e i t h e r
shows t h a t the c a n o n i c a l map ~ (T, T)
is s u r j e c t i v e .
T h a t it is i n j e c t i v e
(5.23)
It is clear
and
are also r e f l e x i v e .
T
Proposition. Proof.
Let
from
(5.12)
is c l e a r and h e n c e
and
(5.18)
of c o p i e s of
A c o m p a c t g r o u p is i s o m o r p h i c to a c l o s e d s u b g r o u p of a p o w e r of C
be
compact.
Since
C
T
.
The n a t u r a l m a p
C is i n j e c t i v e .
it is an i s o m o r p h i s m .
that sums a n d p r o d u c t s
is compact,
~ TC it is h o m e o m o r p h i c ,
hence
isomorphic,
to its
image. (5.24)
Proposition.
d i r e c t s u m m a n d of Proof.
C
Let
T
be e m b e d d e d
c o m p a c t group
C
.
Then
T
is a
.
It is c l e a r l y s u f f i c i e n t to show that the map C
i n d u c e d b y the e m b e d d i n g ,
T*
subgroup
D C
a c t e r on
T /D .
D
in the
> T
is s u r j e c t i v e .
is the image. This m e a n s
Since
Suppose, T
to the contrary,
is a c o g e n e r a t o r ,
there is a n o n - z e r o c h a r a c t e r on
there T
t h a t the p r o p e r is a n o n - z e r o
char.
w h i c h v a n i s h e s on
or t h a t T
is not injective.
B u t we h a v e
~D*
a commutative
diagram
T
'C
1
1
T in w h i c h
~D
~C
the top and right h a n d m a p are i n j e c t i o n s
and the left an i s o m o r p h i s m from
w h i c h it f o l l o w s t h a t the b o t t o m m a p is an i n j e c t i o n (5.25)
Proposition.
Proof.
The p r o o f is standard.
The group
T
as well.
is i n j e c t i v e in the c a t e g o r y of c o m p a c t groups.
If C1
~ C2
85
is an i n j e c t i o n
and
¢ : C1
) T
is a map, C1
> C2
1
[
T This
is the
compact
group
) C
modulo
T x C2
f o r m the p u s h o u t
the
compact,
hence
closed
subgroup,
consist-
i n g o f all { (-#(x), In o t h e r w o r d s , is a s u b g r o u p
the p u s h o u t
of
C
(5.26)
Proposition.
Proof.
If
nite).
Then
C
is the s a m e
a n d the p r e c e d i n g Every
is c o m p a c t ,
compact embed
0 • is e x a c t compact
with one
to
and
Tn/c
abelian
maps
work
pact,
out
it
is
(5.27)
that
the
If
Then
is, b y
D
since
composite
0
~ C
Since
It
Every
follows
map must
discrete
on
D
D
be
from
( n
in b o t h
diagram
(5.12) is
0,
surjective.
, it
follows
needn't
be
fi-
categories
with
exact
of
rows,
and
(5.18)
that
it is an e a s y
Since
C
C**
and
the m i d d l e
diagram
chase
are
com-
the v e r t i c a l
left hand
it is a l w a y s
d u a l o f the
the dual
map
true
natural
(in a n y
closed
category)
map
of t h a t m a p
is i n j e c t i o n .
Thus
~ D** /D
•* with
exact
' D• *
>
is an i s o m o r p h i s m .
there
is an e x a c t
Now
sequence
70
rows, D** / D
~ 0
1
+ D •* maps
canonical
~D
that
~D''~D diagram
The
Since
+ 0
reflexive.
~D
the
this map
~D
is
is c o m p a c t .
an i s o m o r p h i s m .
a commutative 0
This
Tn
~ 0
composite
group
of that map with
0
the
argument.
T
n ** n ** ) ---+(T /C) ---+ 0
the bottom
hand
is a c o g e n e r a t o r ,
Then we have
in w h i c h
is i n j e c t i v e
~ Tn/C
' Tn
---+(T
the
Thus
~ 0
) C
D
T
T
, say
0
is d i s c r e t e ,
is the i d e n t i t y since
T
a commutative
left
the p r e c e d i n g ,
the
of
~ Tn/c
D
that
completes
groups.
is r e f l e x i v e .
groups we have
•*
abelian
an isomorphism.
Proposition.
Proof.
proposition
in a p o w e r
injections.
is an i s o m o r p h i s m .
as of the u n d e r l y i n g
group
C
I X @ CI }
~ C -----+ T n
is c o m p a c t .
and discrete
the vertical
x )
are
~ D**** injections,
~ (D ** /D) ** the m i d d l e
) 0
an i s o m o r p h i s m
and hence
so is
one.
shows
that
the d u a l i t y
is v a l i d b e t w e e n
the
compact
a n d the d i s c r e t e
groups
88
(5.28)
Proposition.
topology Proof.
The
of uniform
dual
o f the g r o u p
convergence
on
compact
~
of r e a l n u m b e r s ,
sets,
equipped
with
the
is
If :
is a map,
~
> T
let :
By continuity,
0(I)
= 0
inf
and
{xlx
~ = 0
> 0
&
~(x)
only when
~(i/l)
:
: 0}
~ : 0
~
.
. Otherwise
let
~ T
by @(i/~) (x) : x / A We h a v e
an e x a c t
sequence 0
and
@(~
) = 0
,
@
~ ~
induces
> ~
~ ~/~
from 5.22
that
is an i n j e c t i o n .
~
Thus
If we d e f i n e ,
for all
~ 0
~ T
is m u l t i p l i c a t i o n
The o n l y m a p s
the i n v e r s e .
~ T
a map : T
We k n o w
(mod i)
T --÷
=
by
T
±
an i n t e g e r .
which
are
Since
injections
ker are
@ = ker @
the
identity
, and
}(1/~)
~ E ~(I)
: ~R
~ T
by ~(~) (x) then
the a b o v e
shows
that
every
~
=
~x
is of
(mod l)
the
form
~(i)
for
al C i~
which
is c l e a r l y
unique. A basis natural
for the c o m p a c t
number.
t h a t the b a s i c (5.29)
We We
leads
Evidently
~(I)
neighborhoods
of
are now ready consider
in
of t h e
choice
of
are
just
the u s u a l
D
consist
i)
iii) The three
IlI<
n
i/4n
a so
ones.
and
C
finally,
a n d as m a n y of
G
.
We
for let
D
.
Each
C
consist
.
A power
form
of all g r o u p s
C
is c o m p a c t
n = 0
, n
m
is
finite,
in
(ii)
choices
A compact ~
iff
n ],
x
of the
mn
x
~m
form
•
n. [R
•
m.T
to
ii)
of
(_ 14 ' i )
subcategory
, of duality
D subject
into
I-n,
theory. for the
C and
of the i n t e r v a l s
set
the
A
consists
that
0
to a p p l y
~
takes
six passibilities
to a d i f f e r e n t
of all g r o u p s
sets
group
in a p r o d u c t
of d i s c r e t e
or
and two
c a n be
can of course be
and
D
is f i n i t e , m
discrete; or
in
(iii)
embedded,
embedded groups.
n
is at m o s t
is at m o s t
the s i x p o s s i b i l i t i e s .
as m e n t i o n e d
in a p o w e r Thus
give
of
countable;
countable.
~
.
above, A power
for all possibile
in a p o w e r
choices
of
T
can be
of of
C
,
embedded
every
group
87
in
C
can be e m b e d d e d
is made,
we let
of groups (5.30)
in
A
D
in a p r o d u c t
, as usual,
of groups
consist
in
D
.
of all groups
Whatever
choice
of
t h a t can be e m b e d d e d
C
and
D
in a p r o d u c t
.
W e define
the functor (-,-)
: C °p
x D
) D
by letting (C x m
be the d i r e c t
sum of nine
i) ii)
(c, p . ~ )
= 0
(C, q.T)
= q.C*
(~m,
= 0
( [R
( m,
q.T)
(~n,
ix) We w i s h
p.~)
maps
) D topologized
C
discretely;
;
=
(m x n ) . •
=
(mxq).~
=
;
;
;
;
(n x q ) . T that these
of a m a d for same
@ q.T)
;
to e s t a b l i s h
qo.T
D @ p.~
as follows.
= (m x p ) . ~
(Z~ n, q.T)
the image
,
;
(77n, D) = n . D
viii)
it lies in
D)
TM, n.~)
v) vi) vii)
(iii)
terms
77 n
is the group of continuous
iii)
iv)
in
(C,D)
x
C ~
are the c o r r e c t
q.T
finite
lies
subset
Hom(C,q.T)
in a c o m p a c t
qo C q
~
underlying
.
F o r example,
groups.
subgroup
of
T
.
By
(5.16)
Hence
l i m ) H o m (C,qo.T) lira+Horn (C, T
qo
lira)Horn (C,T)
)
qo
lim qo.HOm(C,T) q. Hom(C,T) Here
Hem
(-, -)
refers
are g e n e r a t e d b y c o m p a c t any map sum. of
from either
Analogously 0
~RTM
(and,
or
in 77 n
subsets,
of these
t h a t contains
group c o n t a i n e d power
to the a b e l i a n
if fact,
no n o n - z e r o
We c o n s i d e r
in its k e r n e l
the p r o d u c t
is
Let
the cases
C E C
shows
and
on c o m p a c t
already
When
D
must contain
convergence
i)
in
of
Proposition.
We have
the groups
subgroup.
0, i} n
in the k e r n e l
Hom(A,B).
Proof.
dually)
{-i,
D E D
established
.
rRTM
and
respectively. factor
all h a v e
f
Then
(compact
.
Thus
case,
(C,D)
a finite
a neighborhood
a continuous
of all b u t
map
from a
abelian
Hom(C,D)
group
topologized
convergence).
t h a t the a b e l i a n
group
underlying
(C,D)
is
Hom(C,D)
separately. C
is c o m p a c t
{ ::
and
c--+D
is o p e n in the c o m p a c t
D
discrete,
[ :(c)
: 0 ]
convergence
M
any sub-
finitely many.
the discrete
is
77n
Thus
through
is a map to such a group,
t h a t in each
subsets
Both groups
s u m m u s t also
lies
f-l(M)
underlying (5.31)
and
f
to the above
by uniform
1 ]m
to a d i r e c t
If
T h e n a dual a r g u m e n t (A,B)
[-i,
groups
group v a l u e d hom.
=
0
topology
so that the t o p o l o g y
is
88
discrete. ii)
There is nothing
to prove.
*
iii)
The group hood
M
q'C of
~ q ,
M
is discrete.
O
= (-1/4,
will do. Again,
v)
Thus
has a neighbor-
For example,
if for all
1/4), M
iv)
On the other hand, q-T
w i t h o u t any small subgroups.
=
F(M )
Hom(C,q-T)
there is nothing
is discrete
in the compact open topology.
to prove.
We must show that the abstract (n x m)- ~
isomorphism
~ (~n, m.~)
is a homeomorphism. Here
n
and
m
one is easier.
may be finite or countable. We first show it is open.
We consider
A neighborhood
the latter case as the other of
0
on the left is of the
form F(-rB , r~) where
{r B}
so d e t e r m i n e d
is a doubly indexed sequence
ii)
real numbers.
consists of all doubly indexed sequences
of real numbers i)
of positive
such that Only
[~
finitely many
E I [~8/r~BI
Given such a sequence,
~ 0 ,
and
< 1
let s
2 ~ sup {i, i/rsyIB,y
=
~ a}
and t Then for
=
2 -~ inf {i, rBy I B,y N ~}
e ~ ~ , we have s
~
2~
and t while
~ k B
~
2-Br8
implies >-
s
2 /r8
and -S t8
~
2
and in either case ts/s ~
~
2-~-8r 8
On the right h a n d side the set {/I / ( H [-s~,s~]) is a n e i g h b o r h o o d
of
O .
C
Every f
=
(l ~8)
F(-ts,t ~) }
The n e i g h b o r h o o d
89 in that set must in p a r t i c u l a r have the property f
8[-sa,
sa ]
C
that
(-tB, ts)
or
If as sa] 1[ aS l
<
ts/s a
t8
<
2-a-B r 8
-<
so that
IZ f C~B/rc~8l <- ZIf as/rasl
<-
To go the other way,
This shows that the map is open.
1
every compact set in
is
[An
contained in one of the form
H [-s a, s] while sets of the form F(-t B, t B) are a basis
m-~
for the open sets in
.
s,
{fl'(~[
Thus a basic open set in
s l)
C
F(-t8,
( n,
m'[A)
is
ts)}
If f then
f
belongs
=
{f S}
to that set iff [
I[ [ ~ 8
S~ I /
<
Its[
1
Now let
rc~ ~
2-~-B
=
tB/s a
Then supposing {f 8}
6
F(-r 8, r B)
C
(n x m)" •
,
we have,
I I lS~Bsl / Ihl
~ l l l s ~ BsJtSI
B < Thus t h e map i s
continuous
The failure of such an argument reason for the restriction topology on
(C,D)
in several places. correct w i t h o u t
The remaining arguments
=
1
for larger than countable
to finite and countable
is not a convergence
exponents.
exponents
figures
the main results of [Barr,
to at most countable
is the
Without this,
topology w h i c h hypothesis
Thus I do not know w h e t h e r
restriction
~Z2 -~-B
and h e n c e a h o m e o m o r p h i s m .
the
crucially 1977]
are
exponents.
cases are h a n d l e d b y trivial m o d i f i c a t i o n
of one of the above
and are left to the reader.
(5.32)
Proposition.
For any
C E C
Proof.
Just examine the nine cases individually.
(C, D)
,
D E D ~
,
(D , C*) Each of them is trivial in view of
the definitions. The error is in the proof of 7.9 in which the caveat
of 5.3 is ignored.
g0
(5.33)
Proposition.
For
any
Cl,
(C I, In p a r t i c u l a r ,
there
C2 E C
,
(C2,D))~
D ~ D_ , ( C 2,
is a i-i c o r r e s p o n d e n c e
(C1,D))
between
maps
) (C2, D)
C1 and maps C 2 -Proof. above and
There and
C2
are,
the are
so t h a t we
in p r i n c i p a l ,
fact that some compact
need only
27 c a s e s
are
zero,
and
D
discrete,
show
they have $
is a m a p
of a c o m p a c t
a finite
number
of w h i c h
is an o p e n
has
open kernel.
subgroup
which
that map
is c o n t i n u o u s .
contains
the k e r n e l
using
the s y m m e t r y
collapsing.
If
a n d (C 2, (CI,D))
elements.
Now
C1
are d i s c r e t e
a map
(C2, D)
set and hence
The
2
has
a finite
image.
It thus
determines
-----+ D
intersection
Each
of
[
of the
.
element Since
of the
induced
) ('C I iI,
C2
kernel
(C I, (C2,D))
same
is the k e r n e l
whence
However,
is a c o n s i d e r a b l e
maps C
each
both
the
.
to c o n s i d e r .
there
: C 1 --+
to d i s c r e t e
of continuous
(C 1 ' D)
D)
of
that
finitely
many open
subgroups
map
,
C 2 determines is open,
each
a map
ICI]
such map
, D
whose
is c o n t i n u o u s
so
we have (Cl, D)
C2 It is e v i d e n t l y
exactly
the
variable
and
coefficients
consider
the
cases
same
in the o t h e r
in the s e c o n d
come
C1 since
(C2,D) what
is an
choice
D
in
D
leave
the s a m e
nothing
c a n be (5.34)
to p r o v e .
reduced This
to o n e
shows
it is s u f f i c i e n t T
as the
is the c a s e
as c o e f f i c i e n t s ,
that
cosmallness
discrete
in the
first
it is s u f f i c i e n t
or
PR
or
T
.
If
is d i s c r e t e
no
to C] p
(CI,D)
to the
:
0
C 1 = ~.
The
cases
reader
to s h o w
Cl = 77
that
using
or
C 2 : Z[
(5.32),
every
case
considered.
III.4.2(iii)
will
D
On the other hand,
(c I, D))
It is l e f t
to v e r i f y
exponents
, so a l s o
C 2 compact,
already
Since
~ (C2, D)
r R - v e c t o r space.
of
( c 2, Exactly
out
C. c o m p a c t or ~ or ~ and l C 2 = 59 , t h e r e a r e no n o n - z e r o m a p s
is c o m p a c t ,
matter
direction.
is s a t i s f i e d .
the h y p o t h e s e s appear
of
as a w a y
II.2.9.
station
We n o w We
t u r n to
turn
a n d the
(iv),
for w h i c h
to the i n j e c t i v i t y
third hypothesis
of
is e v i d e n t .
Suppose A is an e m b e d d i n g . that
every
I
: A
Since > T
B
can be can b e
* B
embedded
extended
to
in a p r o d u c t B
> T
of
in the
D
it s u f f i c e s case
B = ~D
w
to s h o w The
set
91
f-if_ is o p e n
in
A
and hence
contains
a set
of the N
A where
M
is o p e n
indices
Wl,
....
in
B
Wn
such
.
The
B0
is a s u b g r o u p
and that set
contains
topology
on
D
B0
so is
=
A/A 0
denote
B
is s u c h
K {Bwle
that
~ e I .....
A0 = A n B0 .
there
is a f i n i t e
set of
en }
Since
no s u b g r o u p s , .{ (A 0 )
Let
form
M
that
M Since
1
the a b s t r a c t
quotient
=
0
group
topologized
(here only')
as a s u b g r o u p
of B/B 0
:
Dwl
x...x
Den
The homomorphism
F induced
by
$
the i n v e r s e
image
image
in
B
in
.
Then
B
is s t i l l of
: A/A 0
continuous.
(-1/4,
of an o p e n
1/4)
set M
In
7
has
f
is c o n t i n u o u s
a unique
products, a closed case A and
subgroup
there
continuous
.
A0
is an
Let
finite
sums
splits
sequence
that
of
D ----+ A
topological
retraction
on
Since
is c o n t i n u o u s
to b e
iff
the i n v e r s e
neighborhoods
Since
D
to
D E D
D
.
~-subspaces A
.
Then
T
of
0
is c o m p l e t e ,
is c l o s e d
t h a t if
We of
A0
under
and
consider A
.
finite
A C D
is
first
In fact,
is a s u b s p a c e
the
since
of
D
S i n c e any ~ - l i n e a r m a p o n a f i n i t e 0 v e c t o r s p a c e is c o n t i n u o u s , t h i s m a p is
subspace
of
supposing
m-~
.
m = ~ on
0 D
) A -----+ A / A 0
is i s o m o r p h i c
chosen
A/A 0 .
to s h o w
is c o n t i n u o u s
~ A0
T
n
A / A 0.
of
to
are b a s i c
an e x t e n s i o n
subgroup
(I am t a c i t l y
the
A/A 0
induced
sets
the s u m of all
dimensional
0 means
(A/A0)
has
retraction
finite
to p r o v e ) ,
which
be
to an a b i t r a r y
is n o t h i n g the
A0
can be
D
topology
A ----+ T
M
such
1/4)
is t h e d i v i s i b l e
on e v e r y
But
it is s u f f i c i e n t
map
a homomorphism
since
to the c l o s u r e
Thus
R-linear
space
of the
.
every
D = mJR
closed,
dimensional
limit
extension
B/B 0 E D
that
is
in t h e
fact,
is o p e n .
in B / B 0
7 -l(-1/4, a n d so
~ T
to a s u b g r o u p
Since
m-~ is
is the d i r e c t
; if
m
. The
restriction
finite,
there
to
A
~ 0 of
D
.
Let
A
be
a subspace
1 of
D
containing
A/A 0
such
that D
algebraically. while
A 0 and
the a b o v e
The map A 1 being
isomorphism
~
A0 • A 1
D ---+ A 0 • A 1 subspaces
implies
is t o p o l o g i c a l
is c o n t i n u o u s the
as w e l l .
by
continuity Now
the s a m e
argument
in the o p p o s i t e
A/A 0 C A 1
so t h a t
as a b o v e
direction
so
92
A
A For
every
hence
finite
subspace
discrete.
is t h e r e f o r e
Thus
every
relatively
The
duality
of
is
surjective
C
V C
and
D
A1
,
subset
open
in
• A/A 0
0 V
~
of
A/A 0
A/A 0
A/A 0
implies
Next
the
suppose
result
+ n-T
D = m-~
Consider
the
to
a map
kernel
of each
: B ----+ T
such
the
) m ~
vertical
that
map
evidently,
Finally,
:
g ( n "77 ) = 0
D1
discrete.
Then
that
m.
~
a map,
the
D1
and .
pullback "
A
" ~
so
map
a map
A ----+ T
has,
by
the
is
the
above,
same
as
a
an extension
) T
induces
the
desired
map
D
T
.
@ m- ~
@ n.T
Let
+ n.T
is
=
uniquely
A N
m- ~{ @ n . T
determined,
f0
The
map
: A
to
m. ~
+ n.T
being
the
identity
component
of
D.)
f
vanishes
on
A0
and hence
-
induces
A/A 0
continuous get
is
a subgroup
extension
to
of
D1
a map
h
> T
then
to
: D
gLA
a map > T
:
A
~ T
a map
f Since
}T
: A0
and g
: A/A 0 , that
> T is
: D1 ~
discrete T
.
and
Composed
a map h
: D
~ T
that hIA
=
f
- gli
or f Thus
V
A C D
restriction
an e x t e n s i o n
such
such have
if
has
we
and we
n-T
Such
(m + n)
f is
in e v e r y
is
let
A0 (Note
@
n'~ .
, and
= with
is
f (n~ ) = 0
g for w h i c h ,
open
discrete
and
I
A
f
is
subspace
.
(rn + n)
! map
relatively A/A 0
vector
) A for
B
where
no
that
follows
D = m-~
is
Thus
D and
contains
g + h
is
the
desired
=
gIA
+ hIA
extension.
=
(g + h) IA
T with
is
injective
the
projection
f
has D
a ~ D]
93
(5. 35)
To take care of I II.4.2(v),
s u b g r o u p such t h a t the class of
C
.
A*
C * ---+ Suppose
we
c o n s i d e r the case of
is injective.
F
C E D
If
A
g e n e r a t e d by
is a p r o p e r x
and
A
and
A C C
a
n
F x~
is a f i n i t e l y g e n e r a t e d a b e l i a n g r o u p and
.
C E C
is done b y s u c c e s s i v e l y e n l a r g i n g
first t h a t C
where
This
.
closed subgroup Then
B/A
let
n
is finite.
x E C - A
and
In that case,
B
be the s u b g r o u p
is cyclic and has a n o n - z e r o map to
T
.
The
composite B can, by the p r e v i o u s The
m
and
n
fRTM ---+ T m
where
~ T
s e c t i o n be e x t e n d e d to
C
.
Thus
C
---+ A
is not injective.
case C
with
~ B/A
finite
Rewrite
F0
x
T
m
is e a s i l y r e d u c e d to the p r e v i o u s
one b y p u l l i n g b a c k along
t h a t as
is finite.
torus and is,
F x~R n
:
Then
in fact,
F 0 ® Tm
is a c o m p a c t s u b g r o u p of a finite d i m e n s i o n a l
the m o s t g e n e r a l such.
For the dual of a c o m p a c t s u b g r o u p of a
torus is the q u o t i e n t of a f i n i t e l y g e n e r a t e d
free group.
This
is the d i r e c t sum of a
free group and finite group and its dual is the s u m of a torus and a finite group. N o w in the m o s t g e n e r a l case, :
C It follows
from the d u a l i t y
arbitrary. topology of
A
If
A
that
C1 x P R n x 77 k
C1
is not dense in
that there are
is n o t dense
can be e m b e d d e d in a p o w e r C
finite sets
in the image
, Jt follows m0 C m,
CO
of
m T0x~0
n O C n,
C
CO
n
<
and
A0
(5.36)
~
k0 C k
readily
it now follows that the t h e o r y applies. scribed here extends here to show that. 9.6 and 9 . 8 i.
Every
in
n x~
0 x
such that the image A 0
T
with
It is the
no ~k 0 x . fR
T h a t is,
22k 0
from 5.15.
from the p r e v i o u s
A l l the h y p o t h e s e s
case. satisfied,
case that the d u a l i t y t h e o r y de-
a l t h o u g h not s u f f i c i e n t m a c h i n e r y
facts are these:
(See [Hewitt-Ross],
is d e v e l o p e d
89, e s p e c i a l l y
.)
locally compact group with
only at zero) finite ;
that of P o n t r j a z i n The m i s s i n g
C1
The r e s u l t now follows
follows
may be
k0
m) F0 • T C
is a n o n - d e n s e subgroup. The last h y p o t h e s i s
m
x~Z
is the p r o d u c t of the image of
CO
where
in
m0 The group
Tm
from the d e f i n i t i o n of the p r o d u c t
is of the
form
a norm
D ~ n-rR @ m-T
(i.e. a s e m i n o r m t a k i n g on the value where
D
is d i s c r e t e
and
n, m
are
zero
94
2.
Every
C x £Rn
locally
zzm
x
For
if
it is c l e a r compact group
p
C
~
group
generated
is c o m p a c t
and
that
group
and
compact
where
~
~
are
and
~
is a s u b g r o u p a seminorm,
and
the
by
a compact
n
are
m,
full
set
subcategories
are d u a l
under
of a p r o d u c t
is of the
form
finite.
the
of
D
duality.
of g r o u p s
in
and
C
Moreover For
~O
if
described every L
above,
locally
is s u c h
a
let L
:
{x • Lip(x).
=
L/L
: 0}
P h/p
P Then L and
L/p E ~
that they Let
M
empty
all
be
.
Since
lie
in
interior.
a n d is h e n c e
open. The s e t
~
.
.
Let
L
subgroup An open set X C L0
to b e o p e n b y
a seminorm
compact
be
0
.
generated
shows
(5.37) To s e e
L*
that
let
has
A
n ~
by
X
is a c o m p a c t
X + M
also has
space
be
it f o l l o w s
a compact
set with
non-empty
is l o c a l l y
set.
non-
interior
compact
so
1 1 (- ~, ~)}
0
topology.
the w a y ,
that every
dual has
group
of
integers
topologized
x~
x~
the
of
~ L
the c o m p a c t / o p e n
be
is a s u b g r o u p
and
X + M
compact
T[f (X) C
2
which
by
group
~-complete,
the m a p
It is n o t true, this,
} hence
Hence
L This
L
compact
Then
in a l o c a l l y
.
on
is c o m p l e t e ,
a locally
of
L0
{f: is f o r c e d
~{L/plp
locally
neighborhood
The
L0
E
G
a compact
C
every
T 0
4
The maps
8
x
.
.
A ----+ T
the
compact/open
as a s u b g r o u p
topology. of
.
are
found among
the m a p s
}7 - - +
T
,
, i.e.
the e l e m e n t s
of
T
.
2n~
for all s u f f i c i e n t l y
order
is a p o w e r
of
2 .
A
consists
large In
n
A 2, 4,
converges
to
0
, so t h a t X
is c o m p a c t .
.
the
That
s > 0
they
are
the
of
T
which
elements
of
annihilate T
whose
8,
16,
...,
2 n,
...
the s e t =
{0,
2, 4,
such
8, 16 . . . . .
2n .... }
: A
that
, there
f= is a
~ T
that [ (X)
If
is,
elements
sequence
[
I claim
the
Suppose
is a h o m o m o r p h i s m
Then
of all
0
.
For
k
such
if not,
1 1 (- ~, ~) 4
C let
s
be
<
1 2
that 1 4
--
k <
2
£
the a b s o l u t e l y
least
residue
of
f (2)
95
and then Using
t h a t the the
f (2 k) ~
the b i n a r y
group
(-1/4,
zero homomorphism ~7 2
1/4)
Thus
representation
of
is o p e n
of e l e m e n t s
of
f (2) : 0
n
,
we
in the
2 power A*
=
from which
see
that
compact
order, lim --+
open
topology
topologized
2n
Then A**
which
is the
2-adie
completion
= of
* lim ~ ~---2n
~
.
=
f (2 k)
f (n) = 0
lira ~[ <----2n
= 0
as w e l l . on
A*.
discretely.
for a l l
k
.
This means Thus In f a c t
A*
is
98
6.
(6.1)
By an i n f s e m i l a t t i c e
of e l e m e n t s pair y
has
an inf.
of e l e m e n t s
by
xy
in w h i c h
we
see t h a t element
f : LI----+L 2
If
f
let
and
g
a partially
If the e m p t y
is i d e m p o t e n t .
L
which denote
u : L1
preserves
1
obviously
a unit
is an o b j e c t pointwise,
of
L
that the theory
of
Thus
parts
(6.3)
take
for
crete
d e d in a p r o d u c t
we
We
let
the
let
C*
one e a s i l y
C
of
III.4.2
we
take
the
of
the u n i f o r m i t y
sees,
by
and
discrete
I)
If
by
arguments,
, the
x
finite
and
monoid
a morphism
inf,
i.e.
a monoid
by
is a h o m o m o r p h i s m
Thus
the s e t o f m a p s
the i n t e r n a l
and hence
that
hom
L
and L1
) L2
is c o m p u t e d
is an a u t o n o m o u s
satisfied. consisting
compact
ones
(or,
we
Since
that
ultimately, with
with
D E D
which
D*
be
of 2 e l e m e n t
is c l o s e d ,
If
ones).
C E C
uniformity
the
is c o m p a c t ,
dis-
can b e e m b e d -
0 < 1 .
the discrete
let I
~(D,I)
of all the u n i f o r m l y
semilattices
{0,i}
C --+ I
I D.
i n f of
as a c o m m u t a t i v e
fg d e f i n e d
above.
lattice
maps .
induced
the u s u a l x E D
the
two e l e m e n t
L(Icl,
are
and the
set
every
semi-lattices.
x E L1
since
of U n L
all
the set of continuous
is d i s c r e t e
defined
finite
.
for all
is c o m m u t a t i v e
of topologically be
and preserves
It is e v i d e n t ,
L
a n d ii)
For
structure
with
.
any
s e t as w e l l
are s e m i l a t t i c e s
of these
f(x) g(x)
= 1
1
the s a m e L2
so is t h e m a p =
u(x)
the s u b c a t e g o r y
I = T
be
lattice
equipped
D
i) D
semilattices.
(6.4)
that
is d e n o t e d
and
the c a t e g o r y
f o r the m u l t i p l i c a t i o n
L(LI,L2)
variety. We
such
s e t in w h i c h
t h a t the e m p t y
is e x a c t l y L1
are h o m o m o r p h i s m s ,
~ L2
inf
If
fg(x) The map
ordered
it is s u f f i c i e n t
an i n f s e m i l a t t i c e
is a f u n c t i o n We
is m e a n t
Obviously
an inf.
every
homomorphism. (6.2)
have
Semilattices
lattice, so is
hence
ID
,
and
L(D,I), and
compact.
If
discrete
semi-
function : D ---+ T
defined
by {~
~(y)
, if
x
y
= , otherwise
is r e a d i l y lattice
seen
to p r e s e r v e
and hence
the c a n o n i c a l
every
inf.
object
of
From C
this,
it is i m m e d i a t e
has enough
that every
representations
into
T
and
that
maps D
) D**
C
~ C**
are i n j e c t i o n s . (6.5)
T o see
tice.
Then
they are
isomorphisms,
we p r o c e e d
as
follows.
Let
L
be
a finite
lat-
given : L ----+ T
let = x tice
x = inf{y E Llg(y ) = 1 } . . L °P
It is i m m e d i a t e .
maps preserve
(Of c o u r s e only
infs.
that
x -< y
a finite rf
Since
L
is
implies
semilattice
finite y -< x
and
~
so t h a t
is a l a t t i c e .
preserves L*
finite
is s i m p l y
However
remember
inf, the
lat-
that
97
f : L 0 ---+L I is an inf p r e s e r v i n g
morphism,
the i n d u c e d f*: L~P----+ L~ p
also p r e s e r v e s course
infs.
The c o r r e s p o n d i n g
and is a c t u a l l y
clear.
An a r b i t r a r y
discrete
finite,
L* = lim L
of
semilattice L
L
f *op : LI----+ L 0
function
the left a d j o i n t
=
f:)
Thus
is the d i r e c t
li T L
equipped with
preserves
sups of
in this case the d u a l i t y
is
limit of finite ones.
If
,
the i n v e r s e
limit topology.
If
: L ---+ 2 is u n i f o r m means
(or e v e n continuous)
it contains
projection inverse
a set of the form
~
image of i is o p e n
L* N M
where
of a s u b s e t of a finite p r o d u c t
image of
is that
the i n v e r s e
0
depends
a factorization
o n l y on finite m a n y
indices,
This
image u n d e r
say
el'
"''' em
Thus there is
that when
In p a r t i c u l a r
extends
~ .
L0>---+ L 1
there
Now we have
whence
In the process,
so is
of r e p r e s e n t a t i o n s
is an injection,
on finite
L1
> L0
lattices
is a
is a map
shown EL
is a surjection,
2
The i d e n t i f i c a t i o n
: L * x...x ~i which
C ~L*
of
in particular,
surjection.
L
x...x L The same is true of the ~i ~n d i f f e r e n t finite set of indices). The r e s u l t
(with a p o s s i b l y
L 0 C L*al x . . . x L*~ m
implies,
in
is the i n v e r s e
L
L ---+ L 0 where
M
L*
----+ T
~m
that
= EL
--+ L
,
L ----+ L T
we h a v e s e e n t h a t
is c o s m a l l
and that w h e n
L = +--lim L
,
then
L*
lim L * +---
=
B u t then L and with each (6.6) (C,D)
L
reflexive,
This establishes to c o n s i s t
equipped with
=
so is
lim L
L
.
the d u a l i t y b e t w e e n
of the s u b l a t t i c e
the d i s c r e t e
of
C
(ICI,D)
uniformity.
and
D
.
consisting
As
for the hom,
we take
of the u n i f o r m morphisms,
The m a p
(C,D) ---+ (D*,C*) is the e v i d e n t
one and as for its b e i n g
uniform,
there
is n o t h i n g
to prove.
T h e com-
posite (C,D) is the i d e n t i t y
~ (D*,C*)
~ (C**,D**)
so the first is injective.
first it is i n j e c t i v e
as w e l l
Since
~
(C,D)
the s e c o n d
so t h a t e a c h is an isomorphism. (I,D) ~ D
is an instance
of the
The equivalences
98
Horn (I, (C,D)) are clear.
In view of
(topologically)
Hom(C,D)
, 1.4.6(iii)
(e l, (C2,D)) Since b o t h sides are
~
(C,D) --~ (D*,C*)
~
(C 2, (CI,D))
discrete,
Horn (CI, (c2,m))
follows from
it is s u f f i c i e n t to show that
--~ Hom (C2, (Cl,m))
If f : C I ----+ (C2,D) then the image of
f
e q u i v a l e n c e relation that
f
factors
Xl,
..., x n
is compact and discrete, hence finite. E
on
C1
Cl---+ Cl/E ~ Let
That is, there is an
w i t h only finitely many equivalence classes such
(C2,D)
be a set of r e p r e s e n t a t i v e s mod f(x , 1
. E
.
Each
xj
determines a function
) : C 2 ----+ D
w h i c h s i m i l a r l y factors C2~ where
E
1
C2/Ei ~
D Since
is an e q u i v a l e n c e relation with only finitely m a n y classes. C2 / N E i ----+ ZC2/E i
is injective,
there are only finitely many classes mod
(N El).
Thus
f
determines
an e l e m e n t of (Cl/E
, (C2/ n E i ,D))
(C2/ N E 1 ,(CI/E , D)) ---+ (C 2 , (C1,D)) the exchange p o s s i b l e there b e c a u s e all three are d i s c r e t e and category.
t
L
is an autonomous
This determines a map (C I, (C2,E)) ---+ (C2, (CI,E))
w h i c h is e v i d e n t l y an involution.
Thus we h a v e a pre - * - automonous situation.
The d i s c r e t e u n i f o r m i t y of
(C,D)
is evidently that of global u n i f o r m
c o n v e r g e n c e and is a c o n v e r g e n c e uniformity.
Thus the first three parts of III.4.2
are satisfied. (6.7)
We now consider III.4.2(v).
lattice. image
A0
Since
C
Let
C E C
and
A C C
is p r o f i n i t e there is a finite q u o t i e n t
remains proper,
else
A
w o u l d b e dense.
Since
be a p r o p e r closed subC of C such that the ,0 C O ---+,A0 is s u r j e c t i v e
and not an i s o m o r p h i s m it cannot be i n j e c t i v e w h e n c e neither is
C ---+ A*
(6.8)
The first is already
We now show that the h y p o t h e s e s of II.2°9 are satisfied.
shown and the third is evident.
For the second,
it suffices to show that every dia-
gram of the form A C
) nD
w
T
can b e completed. such that
As in 6.5 above there is a finite set of indices
depends only on those indices.
Wl'
"''' ~n
We can thus factor the d i a g r a m as
99
A
C
~ zD
D Ix...x
A 0 C__+
D
= B0
n + T Now
A0
and
B0
are d i s c r e t e .
If
is n o t
B 0 ----+ A 0
surjective
let
C
be
its
image. We h a v e
the
composite A0
is i n j e c t i v e ,
whence
the
proper
The
condition
last
in
C*
A0
first
and by hypothesis
the
~
factor
A0
is.
is
contradicts
is i m m e d i a t e
B0
compact
so
that
the p r e v i o u s
and hence
we have
C
is
closed
paragraph. another
model
of
theory.
(6.9)
The
model
of a
category
category
maps
fact models follows.
if
that
X
semilattiees
complete
a set,
: X ----+ Y
for
A C X
that
is n o t
is the
inf preserving theory.
assigns
N
complete
homomorphisms
constructed lattice.
category
triple
whose
TX
=
~ =
in the w a y
provides
described
It is u n d e r s t o o d objects
functions.)
The
are
complete
It is a c l o s e d
(T , ~
, ~)
that
a
here. the
lattices
category,
in
can be described
as
let
is a map,
T f
2X
is t h e
right
adjoint
This means
to 2f : 2 y ---+ 2x
,
Tf(A) unit
and
is in f a c t a c o m p l e t e
of commutative
For
f
are
semilattices
category
semilattice
of c o m p l e t e
and whose
and
of complete
*-autonomous
(A c o m p l e t e
The
~
B0
But
and this
of III.4.2
B0
is t h e
: X ---+ T X
to a f a m i l y
of s u b s e t s
{yl/l(y)
singleton
map
C A } .
and
~X
: 2 2 x ---+ 2 x
their
intersection.
Thus
(x ,Y) consists by
the
of all
infs
in
inf preserving Y
.
homomorphisms
map
category. readily
X ----+ 2 Hence
seen
*-autonomous a compact
into a complete
lattice
(X , 2)
is a l i m i t p r e s e r v i n g X*
and
to be o r d e r structure
category:
made
In p a r t i c u l a r , X* =
Every
X ---+ Y
X
have
inverting.
Thus
are o b v i o u s .
a *-autonomous
set-valued
isomorphic
X* ~ X °p
It h a s b e e n category
functor
underlying
such
sets
I ; T
X
considered
as a
a n d the i s o m o r p h i s m
from which
incorrectly that
on
the d u a l i t y
conjectured and such
that
that
is
a n d the this
the
is
natural
map X* ® Y - - ~ induced
by
exchanging
Y
and
X X*
is an i s o m o r p h i s m .
Here
X
® X
(X ,Y)
in ®
X --~
> I
I ---+
(Y
,Y)
is e v a l u a t i o n
and
I ---+ (X,X)
t h e u n i t map.
100
Since
(X,Y)
either
of the n a t u r a l
=
(X ® Y )
, compactness
maps
(whose
is e q u i v a l e n t
constructions
X
® Y
an i s o m o r p h i s m .
(x°P,Y °p)
were
dual
isomo~hic
~derlying
In the
sets.
~
and,
this would
in p a r t i c u l a r ,
I have
to
(X ,Y )
c a s e of l a t t i c e s ,
lattices
category)
I = T )
---+ (X ® Y)
(X,Y) being
(in a * - a u t o n o m o u s
are e a s y -- g i v e n
verified,
that
using
imply
(X,Y)
an
that
and
(X,Y)
(xOP,
HP67 progra~le
~d
yOp)
have
calculator,
that when
x:xOp
/i\
\i/
and
I\
/\1\ \i/ then there The
are 94
computation
integers would
inf preserving
is c a r r i e d
ordered
X ---+ Y
out by modeling
by divisibility
s e e m to be p o s s i b l e To have
maps
such
for any
compactness,
that
finite
and only
Y
(as w e l l
the
88 s u c h m a p s as
yOp)
X °p ~ X ---+ y O p
as a s e t o f p o s i t i v e
inf of two numbers
is t h e i r
gcd
.
This
lattice.
it is n e c e s s a r y
-- a n d a l m o s t
surely
sufficient
-- to h a v e
a
trace map tr such that
the
(X,I) is the i d e n t i t y . the
Here
first and second
lattices triple)
of complete has
e d to a n y
: (X,X)
---+ (X, (X,X)) the
first map
copies
of
X
subcategory.
---+ (X, (X,X)) is i n d u c e d The
atomic boolean
such a trace map but
larger
.
its
by
the u n i t a n d the s e c o n d
(which
form suggests
Namely,
(X'tr') ~ (X,I)
full subcategory
algebras
tr be defined
~ X* ® X ---+ I ,
composite
if the
: (X,X)
o f the
category
is t h e K l e i s l i very much
lattice
interchanges of complete
category
for the
t h a t it c a n n o t b e e x t e n d -
X = 2A
, then
let
----+ 2
by
tr(f)
=
In o t h e r w o r d s ,
trace
for a n y s u b s e t
A0 < A
is c o m p l e m e n t e d
and atomic
i,
if
O,
otherwise
is r e p r e s e n t e d g(A)
= A
,
that omits seems
a @ f (A - {a})
by
g(A-{a})
crucial
.
the m a p
at l e a s t
for a l l a @ A
= a
g : X ---+ X ,
g ( A 0)
two elements
here.
defined
by
= of
A
.
The
fact
that
X
BIBLIOGRAPHY
M. Barr, Duality of vector spaces, Cahiers Topologie G~om~trie Diff~rentielle 17 (1976),3-14. M. Barr, Duality of banach spaces, Cahiers Topologie G~om~trie Diff~rentielle 17 (1976), 15-32 M. Barr, Closed categories and topological vector spaces, G~om~trie Diff~rentielle 17 (1976), 223-234.
Cahiers Topologie
M. Barr, Closed categories and banach spaces, Cahiers Topologie G~om~trie Diff~rentielle 17 (1976), 335-342. M. Barr, A closed category of reflexive topological abelian groups, Topologie G~om~trie Diff~rentielle 18 (1977), 221-248.
Cahiers
M. Barr, The point of the empty set, Cahiers Topologie G~om~trie Diff~rentielle 13 (1973), 357-368. 8. Eilenberg, G.M. Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, 1965), Springer-Verlag, 1966, 421-562. E. Hewitt, K.A. Ross, Abstract Hamonic Analysis, Vol. I, 1963, Springer-Verlag. K.H. Hofmann, M. Mislove, A. Stralka, The Pontryagin Duality of Compact O-Dimensional Semilattices and its Applications, Lecture Notes Math. 396, (1974), Springer-Verlag. J.R. Isbell, Uniform Spaces, Amer. Math. Soc. Surveys no. 12, 1964. J.L. Kelley,General Topology, Van Nostrand, 1955. G.M. Kelly, Monomorphisms, epimorphisms and pull-backs, J. Austral. Math. Soc. 9, (1969), 124-142. F.W. Lawvere, Functional Semantics of Algebraic Theories, Dissertation, Columbia University, 1963. S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloquium Publications, Vol. XXVII, 1942. F.E.J. Linton, Some aspects of equational categories, Proc. Conf. Categorical Algebra (La Jolla, 1965), Springer-Verlag, 1966, 84-94. A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag, 1972. H.H. Schaefer, Topological Vector Spaces,
third printing, Springer-Verlag, 1970.
Z. Samadeni, Projectivity, injectivity and duality, Rozprawy Mat. 35 (1963). A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47-68.
CONSTRUCTING
*-AUTONOMOUS CATEGORIES
Po-Hsiang Chu
CHAPTER I:
PRELIMINARIES
We will be dealing with closed symmetric monoidal *-autonomous
Kelly coherence conditions
(see [MacLane,Kelly]),
following useful theorem
[to appear].
Theorem:
(autonomous)
categories as defined in the previous paper.
and
Using the MacLane-
M.F. Szabo has proved the
A diagram commutes in all closed symmetric monoidal categories
iff it commutes in the category of real vector spaces. This theorem not only points out the notion of closed symmetric monoidal category is a 'correct' generalization
of the category of vector
spaces, but it also provides a very easy method to check if a diagram is commutative in any closed symmetric monoidal category. The following is a collection of easy consequences
of this theoren
which we shall use later on: Corollary i.
Given A,B,C objects in
V
and map
f
A®B
> C, then the
following diagram commutes: i
I
~ (A,A)
(B,B) where the map~ (A,A) (A,A) Note. The map
÷ (A~B,C)
(id'fJ>(A,(B,C)) x A
?
) (B,C)
(B,B)
,
P
is the composition >(A®B,C)
is the usual transpose of
A®B
f
~ C .
÷ (A®B,C) is obtained in a similar fashion.
we simply denote either composite by Corollary 2.
(A®B,C)
From now on
f .
Given A,B,C,D,F objects in
V and map B®C
> F, then the
following diagram commutes: (A,C)®(D,B)
(id,f)®if
lid®(id, f) (A,C)®(D,C,F))
Ip-l®id (A®B,F)®(D,B)
Iid®p-i
I (s,id)®id
(A,C)®(D®C,F)
(B®A,F)®(D,B)
lid®(s,id) (A,C)®(C®D, F)
Ip®id (B, (A,F))®(D,B)
iid~p
iM
(A,C)®(C, (D,F))
(D, (A, F))
(C,(D,F))®(A,C)
(A,(D,F))
> (A,(B,F))®(D,B)
P
p-i
, (A®D,F)
'ts'id)'(D®A,F)
104
PROOF.
It is easy to check that the diagram commutes in the category of
real vector spaces.
Remark. The word "coherence" is going to appear frequently throughout this paper.
In particular,
if the commutativity of a certain diagram is said to
be implied by coherence, we understand that its commutativity follows easily from this theorem. Our second assumption on V is that it has pullbacks.
Since almost
all interesting examples of closed symmetric monoidal categories have this property,
this restriction is not too drastic.
The following is a collection of examples satisfies our assumption: (i)
The category of vector spaces over a fixed field K;
(ii)
The category of Banach spaces;
(iii)
The category of compactly generated spaces;
(iv)
The category of sets (and functions);
(v)
The category of abelian groups;
(vi)
The category of lattices. An example of a closed symmetric monoidal category that does not have
pullbacks is the category of sets and relations.
CHAPTER II:
CONSTRUCTION OF ~ X AND ITS ENRICHMENT OVER V.
i.
The Category ~
Given an arbitrary object X in V, we shall construct a category_A X as follows: The objects of ~ X consist of triplets in V and v:VeV'
> X is a morphism in V.
A morphism from (V,V',v) and g:W'
commutes.
(V,V',v) where V,V' are objects
to (W,W',w) is a pair (f,g), where f:V
~V' are morphisms in V such that the square
V®W'
id®g
W®W '
w
~ V®V'
>
X
~W
105
If (f,g):(V,V',v)
~ (W,W',w) and (h,k):(W,W',w)
(U,U',u)
are morphisms in Ax then the following diagram commutes: V®U'
W®U'
id®k
id®$
> V®W'
id®k
> V®V'
+ W®W'
lh®id
v
~
~
.
u
U®U'
> X
This implies the composition of (f,g) and (h,k) is (hof,gok) in AX . Since the composition is defined explicitly in terms of morphisms in V , the associativity of maps in AX
can now be verified:
If (f,g):(V,V',v)
(W,W',w)
(h,k):(W,W',w)
÷ (U,U',u)
(l,m):(U,U',u)
(T,T',t)
are morphisms in AX , then ((l,m) o(h,k))o(f,g)
= (loh,kom) o(f,g) = ((loh)of, go(kom) = (lo(hof),(gok)om) = (l,m) o (hof, gok) = (l,m) o((h,k)o(f,g))
Moreover,
Id(V,V',v)
= (idv,idv,)
.
is the obvious identity.
Hence we have
shown that Ax is a category.
2.
__~ is ~hriched over X
Definition.
If V is a closed monoidal category,
then A is enriched over
if A is equipped with the following:
i) For each A,B in A, an object V(A,B)
in V;
ii) For each A in A, a morphism j(A):I
> V(A,A) in V ;
iii) For each A,B,C in A, a morphism M'(A,B,C):~(B,C)®V(A,B)
~ X(A,C) in ~ .
These data are required to satisfy the following axioms: VC i.
The following diagram commutes: M'
V(A,B)®V(A,A)
V(A,B) ®I ~
r
~ V(A,B)
106
VC 2.
The following diagram commutes: M'
V(A,A)®V(B,A)
~(B,A)
j @ i d f I®V(B,A) I VC 3.
The following diagram commutes: (V(C,D)®V(B,C))®V(A,B)
a
) V(C,D)®(V(B,C)®V(A,B))
iM'eid
[id®M'
V(B,D)®V(A,B)
V(C,D)®X(A,C)
~ X ( A , D ) / M ' Given
A = (V,V',v),B = (W,W',w)
the object in
V
objects in A X , define V(A,B) to be
such that the following square is a pullback. V(A,B)
pl
> (V,W)
(W',V')
~ (V~W',X)
Here
-i (v,w)
~ (ww',x)
:
(v,w)
w
~ (v,(w',x))
P
and
> (v®w',x)
-i (w',v')
~ (ww',x)
=
(w',v')
v
~ (w',(v,x))
are the right and bottom maps, respectively. isomorphism in Given
P
~(w'~v,x)
s
~(V®W',X)
Therefore V(A,B) is defined up to
V .
A = (V,V',v)
in _~ , the following diagram commutes, by
Corollary i: I
(v',v')
i
> (V,V)
, (v®v',x)
Universal property of pullbacks implies that there exists a unique map j(A) I ----~V(A,A)
in
V
such that the diagram
107
-
V(A,A)
~
~(V,V)
~
(v',v')
v + (v®v',x)
commutes. Now suppose in ~ X "
A = (V,V',v),
In order to verify
B = (W,W',w), C = (T,T',t)
are three objects
iii) it suffices to show the outer square of the
diagram. _V(B, C) ®V(A, B)
pl~pl
) (W,T)®(V,W)
p2®p2
IM
(T',W')®(W',V')
V(A,C)
~ (V,T)
p1
I
It
(W',V')®(T',W')
M
,
(T',V')
9
, (V®T',X)
commutes. Using the fact that -~_V(A,B)
pl
(W',V')
is a bifunctor and > (V,W)
V(B,C)
> (V®W',X)
(T',W')
are pullbacks (hence commute!), tative diagram in Fig. 1.
~ (W,T)
, (W®T',X)
we can get the desired result from the commu-
Note in Fig. 1 that corollary 2 of Szabo's theorem
(Chapter I) implies that (2) commutes; commute.
pl
coherence implies that (i) and (3)
Again using the universal property of pullbacks, there exists a
unique morphism
M'(A,B,C):V(B,C)®V(A,B)
÷ V(A,C)
in
V such that the
diagram V(B,C)®V(A,B)
pl®pl
(T',W')®(W',V')
(W',V')®(T',W') commutes.
Hence
V(A,C)
M
, (T',V')
i) - iii) are defined.
> (W,T)®(V,W)
pl
9
, (V,T)
, (V®T',X)
108
Now we have to show they satisfy the required axioms. Given
A = (V,V',v),
B = (W,W',w)
in _~ , by construction we have
the pullback diagram: V(A,B)
pl
p21
p.b.
(w',v') But the coherence of V(B,C)OX(A,B)
V
id~pl
> (V,W) I~
~ (vaw',x)
implies that the diagrams of Fig. i commute. V(B,C)®(V,W)
plaid
~(W,T)e(V,W)
p2aid
p2®id
(T' ,W')®V(A,B)
id~pl
,
(T',W')a(V,W)
ltaid waid
,
(WaT',X)a(V,W)/t
id~p2
s
(T' ,W')a(W' ,V')
id®~
, (T',W')®(VaW',X)
M
, (VAT'
(3)
(w' ,v')a(T' ,W')
M
V(A,B)®I
r
~ V(A,B)
(V,W) ®I
r
÷ (V,W)
(T',V')
V(A,B) ®I
(W' ,V')®I FIGURE i.
r
> V(A,B)
r
~ (W' ,V')
)
M
~ (V,T)
109
Hence the following diagram pl®id
V(A,B)®I
(V,W)®I
V(A, B) ,2®id
lr 1°
(v,w)
pl
I
p2
p.b.
(V®W', X)
(W' ,V' )
(W',V')®l commutes.
rI
Since the outer square commutes, such that (1)
and (2)
property as well;
commute.
there exists a unique map V(A,B)®I
But the map
V(A,B)®I
r
,
V(A, B)
~ V(A,B) has this
therefore it follows from uniqueness that it is the map induced
by pulling back. Recall that in the construction of
j(A)
we have the following commutative
diagram:
I
~
~ ~~
~ -~V ( A~, A )'
"
w
~ v ( v ' , v ' ) Then the defining property of M(A,B,A), -~-
~ (v,v)
~
> (v®v' ,x)
coherence of V_ , and the fact that
is a bifunctor imply that the diagram:
V(A, B) ®V(A,A)
Ip2
(W' ,V')®(V' ,V')
V(A,B)
Is I®V(A,B) i®p2>(V',V')®(W',V') commutes.
pl®pl
~ (V,W)®(V,V)
pl
ip pb M
,(U' ,V')
v
> (V,W)
Io , (V®W' ,X)
110
Again applying _V(A,B)®I
id®j
the same argument, M'
+ V(A,B)®_V(A,A)
we conclude + V(A,B)
that the map
is the map induced by
pulling back. But this is not sufficient
to conclude
that VCI. holds,
i.e.
that the
diagram: M'
V(A,B) ®V(A,A)
> V(A, B)
V(A,B)®I commutes. We are still required V(A,B)®I
pl®id
]
+ (V,W)@I .-
/ t
M
(V,W)e(V,V)
to show that the following
I
V(A,B)®I
diagrams
p2®id
> (W',V')®I
L
r
+ (V,W)
commute:
I S
(W',V')®(V',V')
I®(W',V')
]
11 / j
j/~/
S
1
I M
(V',V')e(W',V') That is, that the induced maps (therefore
(V,W)®I
satisfy
,(W',V')
the same commutative
square
they are same by uniqueness).
But it is trivial once we notice there exist canonical maps id®i ieid ~ (V,W)®(V,V) in (i) and I®(W',V') ) (V',V')®(W',V')
in (2) which break
(i) and (2) into two smaller
conmlutative
squares.
Hence
VCI. holds. Applying
a similar
argument,
we conclude
VC2.
is also
true.
Next we
are going to verify VC3. Given in _ ~
A = (V,V',v),
, then by iii) we have Coherence
(2) of Figure
B = (W,W',w),
of V and property
2 commute;
Now we apply
C = (T,T',t),
the commutative
diagrams
of M(A,B,C)
similarly
(i') and
imply
that subdiagrams
the same argument as in proving VCI, id®M' + !(C,D)®V(A,C) M'®id
(V(C,D)®V(B,C))®V(A,B)
by pulling back. a(V(C,D),
V(B,C),
+ V(B,D)®V(A,B) We only have
*_V(C,D)®V(A,C)
M'
> V(A,D)
to show that the composition
, V(C, D) ® (V_(B, C) @V (A, B) ) M'
(i) and
i.e. the maps M' ~ V(A,D)
_V(A,B))
(V(C, D) ®V (B, C) ) ®V(A, B) id®M'
objects
2.
(2') commute.
V(C,D)®(V(B,C)®V(A,B))
are the maps induced
D = (U,U',u)
of Figure
+
V(A,D)
111 pl®(pl~pl)
V(C, D)® (V(B, C) ®V(A, B) )
> (T,U)®((W,T)®(V,W))
(i)
(U',T')®((T' ,W')®(W',V')) id®s
idoM
V(C, D) ®_V(A, C)
l
plopl
> (T,U)®(V,T)
(2)
(U' ,T')®((W' ,V' )®(r' ,W'))
(U', T')®(T' ,V' )
Is ((W',V')®(T',W'))®(U',T')M®id~(T',V')®(U',T ')
M
V(A,D)
pl
pl
p.b.
>(u',v')
(pl~pl)®pl
(~(C,D)®V(B,C))®~(A,B)
, (V,U) ]~ >(VoU',X)
> ((T,U)®(W,T))®(V,W) L
(p2~p2)~p2~®id ((U',T')®(T',W'))®(W',V')
s®id
(i ') V(B, D)®V(A,B)
(2')
((T',W')®(U',T'))®(W',V')
IM®i d pl~pl
~ (W,U)®(V,W)
1
p2®
M
(U' ,W')®(W' ,V')
Ls (W',V')O((T',W')®(U',T')) id®M~(w',v')®(U';W')
V(A,D)
M
pl
plpb , (U',V')
~ (V,U)
~r
I , (V®U',X)
FIGURE 2. is also a map induced by pullback and it satisfies the same commutative square as the map: (V(C,D)®V(B,C))®V(A,B)
__
M'oid > V(B,D)®V(A,B )
M T
> V(A,D)
The first part follows easily from the following commutative diagram:
112 (T,U)®((W,T)®(V,W))
((T,U)®(W,T))e(V,W)
pl®(pl@pl)
(pl®pl)@pl
X(C,D)®(X(B,C)@V(A,B))
(X(C,D)®~(B,C))®V(A,B)
p2®(p2@p2)
(p2@p2)@p2
(U',T')®((T',W')®(W',V'))
((U' ,T')®(T' ,W'))®(W' ,V' )
t
l
id®s
s®id
(U',T')®((W',V')®(T',W'))
((T'.,W')e(U' ,T'))®(W' ,V')
-1
a
(W' ,V')®((T',W')e(U',T'))
((W',V')®(T',W'))®(U',T')
As for the second part, we observe a simple fact of
V : two
permutations of the tensor product of any three fixed objects are coherently isomorphic.
Therefore it is enough to show the following diagrams commute: ((T,U)®(W,T))®(V,W)
a
1
1
(W,U)®(V,W)
((W',V')®(T',W'))®(U',T') iM®id (T' ,V')®(U' ,T')
> (T,U)®((W,T)®(V,W))
(T,U)®(V,T)
a
> (W',V')®((T',W')®(U',T')) lid®M (W' ,V')®(U' ,W')
This follows trivially from coherence and completes the proof.
113
CHAPTER III:
1.
STRUCTURE
The H o m - F u n c t o r ~ ( - , - )
Definition. in ~
~_ --x HAS A *-AUTONOMOUS
Given any two objects
= (V,V',v)
A
and B = (W,W',w)
, define an object _~(A,B)
= (V(A,B), VOW', n)
First of all recall V(A,B)
is the object in
in ~ X as follows: such that the follow-
ing diagram is a pullback. V(A,B)
pl
~ (V,W)
(w',v')
, (v®w',x)
Since we require _~(A,B)
to be an object i n _ ~
morphism in V, which sends V(A,B)®(V®W')
to X.
It seems there are two (canonical) (i)
Since the above square commutes, route) which sends V(A,B) n:~(A,B)®(V®W')
(2)
alternatives
for defining
let n' be the morphism
to (V®W',X)
n:
(along either
, and define
> X to be the transpose of n'.
Again since the above square commutes, we have the following commutative
diagram: pl®id
V(A,B)®(V®W')
> (V,W)®(V®W')
l
~®id
p2®id I (W',V')®(V®W') Now let
~id
ev:(V®W',X)®(V®W')
But since
V
is coherent,
n', so these two definitions
) (V®W',X)®(V®W')
> X be the evaluation map, then put
n" = ev composed with the above map
to
, n has to be a
V(A,B)®(V®W')
_~op
×
_~
to
n
A X(-,-)
is a bifunctor
_~ .
We have to show i)
given any object
B = (W,W',w)
is a contravariant ii) iii)
G =_~(B,-) Given A
in _ ~ , F = A_X(-,B)
functor;
is a covariant ~ B,
C
is identical
are same.
For the rest of this section we shall prove which sends
~ (V®W',X)®(V®W').
it is easy to verify that
funetor;
~ D in A X , then the diagram
114
A_x(B,C)
~ Ax(A,C)
t
[
A_x(B,D)
, Ax(A,D)
commutes. Recall if
C = (V,V',v) and
A = (P,P',p)
in A X and (f,g):C
is a morphism in AX, then the square: V®P '
id®g
P®P '
P
~ V®V'
~ X
commutes. In order to show
F
is contravariant,
we must find a map (in A X)
F(f,g) = (f',g'):_Ax(A,B) By definition
.
_~(A,B) = (V(A,B), P®W', n I) and Ax(C,B) =
(V(C,B), V®W', n 2) ; As for
~ Ax(C,B)
so the choice for
g'
is clear:
g' = f®id:V®W'
f', consider the following diagram: pl V(A B)
~(C,B)
pl
(*) p2
p21
~
I ~
p.b.
(W',V,)-- ~
(P,W)
(V,W)(~
i
+ P®W'
f,id)
(i)
> (V®W',X)
d,g)
id)
(W' ,P' )
~ ( P ~ W '
,X)
We know the outer square commutes,
therefore it suffices to show (i)
and (2) are commutative. For (i), we prove it by looking at the following commutative diagram:
(P,(W',X))
p
(P,W)
(f,id)
> (V,W)
~ (P®W',X)
(f®id,
id)
~ (V®W',X) +---P
(V,(W',X))
115
As for (2) we have a similar diagramatical proof: (w',e)
(W',(P,X))
P
(id,$)
÷ (W'eP,X)
+ (W',V')
(id®f,id)
But in this case the commutativity the fact that diagram
(*)
(f,g)
, (W'®V,X) ~
P
(W',(V,X))
of the outer square is due to
is a morphism which sends
C
to
A
(hence the
above commutes).
This implies that there is a unique map
f':V(A,B)
+ V(C,B)
induced by pullback such that the diagram pl
V(A,B)
+ (P,W) id)
pl
~(C,B)
, (v,w)
1°
p.b. (W',V')
°
(V®W',X) K (f®id,id)
iN,g) (W',P')
~'.
(P®W',X)
commutes. Therefore the following diagram commutes: V(A,B)
p2
I
~(W',P) -
f'
> (P®W',X)
(id,g)
V(C,B)
p2
(feid,id)
~(W',V')
(V®W',X)
This implies that: V(A,B)®(VOW, )
I
id®g'
1.
f'Qid
V(C,B)®(V®W') commutes.
+ _V(A,B)®(P®W' )
n2 ~ X
116
Therefore
(f',g')
has the property required of a morphism in
It is trivial to see that F
F(id A) = idF(A) . Now we have to show
preserves composition. Suppose
A = (P,P',p),
C = (V,V',v)
objects in _~ , moreover (f,g):E
~C
and and
E = (U,U',u) (h,k):C
~A
are three then we
want to show that Ax(A,B ) ((h°f)''(g°k)')
(h',k')
~
> A_x(C,B) f
(f''g')
_~(E,B) commutes. By definition:
_~(A,B) = (V(A,B), P®W', nl) _~(C,B) = (V(C,B), V®W', n 2) _Ax(E,B) = (V(E,B), U®W', n3)
Now we consider the following commutative diagram: pl
V(A,B)
(P,W)
pl
V(C,B)
~ (V,W) (f, i d a /
_V(E,C)
pl ~ (U,W) 1
p2
p2
p.b.
[~
1
~(U®W',X)
(W' ,U' ) (id,g)
(W' ,V' )
(W',P')
~(f®id,id)~ v
~ (V®W',X)
~(P®W',X)
117
But the following diagrams also commute: (W' ,P')
(id~k)
~ (W' ,V')
(W' ,U' ) (P®W',X) (h®id,id)
((h°f)®id'id) I
> (V®W',X)
~~~(f~id,id)
(UoW',X)
(h,id)
(P,W)
(v,w)
(hof,id)
(u,w) This implies that the map induced by pullback is identical to and clearly
k'og' = (h®id) o(f®id) = ((hof)®id) = (gok)'
Hence
f'oh', F
is a
contravariant functor. As for B = (W,W',w), A
> C
G,
we have a similar series of diagrammatical proofs:
A = (P,P',p),
a morphism i n _ ~
C = (V,V',v) are objects i n _ ~ with . We need
G(f,g) = (f',g'):G(A)
Suppose
(f,g): > G(C).
By definition G(A) = _~(B,A) = (V(B,A) , W®P', n I) and
G(C) = _~(B,C) = (_V(B,C) , W®V', n 2) Hence the choice of g' = id®g:W®V' is clear.
> W®P'
And the following commutative diagram shows the existence and
uniqueness of
f':
118
pl
V(B,A)
pl
V(B,C) p2
> (w,e)
(W,V) ( ~ i d , f )
p.b.
]p2
) (W®V',X) id) ~ > (W®P',X)
(V' ,W')
(P',W')
Again the preservation of the identity is clear. Now if AX
and
A = (P,P',p),
(f,g):A
> C,
C = (V,V',v), (h,k):C
E = (U,U',u) are objects in
) D are morphisms, then the
commutative diagrams of Figure 3 imply
G
preserves composition.
To prove (iii): Suppose
A = (V,V'v),
objects in
~X
B = (W,W',w),
and (f,g):A
C = (P,P',p),
) B,
(h,k):C
D = (U,U',u) ~ D
are maps in
then the following diagrams commute: pl
V(B C)
V(B,D)
I
p2
(w,P)
p2
, (w,u) I~
p .b.
# .
(W®U',X)
(U' ,W' )
(id®k,id)~ > (W®P',X)
(P',w')
pl
V(B,D) V(A,D)
lp2
p2
(U',V') Sd,g) (U',W')
> (W, U)
~,idl pl ~ (V,U) p.b.
lfi ~ (V®U',X)
(g®id,id)~ > (W®U',X)
are ~X '
119
(W,P)
(id,f)
(W®P',X)(ideg,id) ~ (W®V',X)
+ (W,V)
(id,h°f)1
(id®(gok),id)I
~k,id)
(w®u',x)
(w,u)
(P',W')(g,id)___+ (V',W') (g°k'id)I (u',w')
V(B
pl
,A)
v_(B,c)
> (W,P) (id,/ ÷ (w,v)~
pl !(B,E) -- pl i
~2
lp2
p.b.
, (w,u)
I~
(U',W')~ +(W®U',X) ,id) (id®k,id)~ + (W®V',X)
(V',w')
(id®g,id)~ (W®P',X)
(P',W') FIGURE 3.
120
pl
~(A,C~
> (V,P)
J
(id,h) pl
V(A,D)
lp2
~2
(V,U)
p.b. 9
(U' ,V')
(k®idi ~ ~X) > (V®P',X)
(P',V')
pl
V(B,C)
V(A,C)
(w,p)
pl
(V,P)
\
V(A,D) pl ,(V,U) p2 lp2 p.b. fi ~(V®U' ,X)
(u' ,v')
(k®id,
id)~ (V®P ', X)
(P' ,V' )
(W®P',X)
(P' ,W' ) FIGURE 4.
This implies that the first diagram in Figure 4 commutes which implies, in turn, that the second one does. Applying the same argument, the map from by pullback is the same as
V(B,D)
V(A,D)
induced
h"of" , hence the following diagram commutes:
lh
V(B,C)
V(B,C) to
f"
f,
lh
÷V(A, C)
>_V(A,D)
121 Next consider Figure 5. is a pullback,
f'oh'
Since the center square of the first diagram is the unique map
V(B,C)
> V(A,D)
that
makes the diagram commute. Next consider the lower diagram of Figure 5.
Using this and the
fact that the following diagram commutes: id®k
V®U'
> V®P '
If®id
[f®id id®k
W®U'
-.%
~ W®P '
pl
V(B,C)
pl
V(B,D),
--
-~I
lp2
V(A,D) p2
• [
(U',V')
~id, (U',W') (kid)
,
g)
pl
~ (W,U)
1
,(V,U) (f' id)k/ >(v®u' ,X)
(f®id,id~, ~(W®U' ,X) (id®k,~ ~(W®P', X)
(P' ,w' )
pl
V(B,C)
V(A,C) p2
(W,P)
[p2
pl p.b.
(W,P)
,
(V,P)
1°
(V®P ',X)
(P' ,V')
( f ® i ~ (W®P',X)
(P',W') FIGURE 5.
122
we obtain
the desired
result
that the diagram
Ax(B,C)
~Ax(A,C)
Ax(B ,D)
~Ax(A,D )
commutes.
2.
The Functor
*.
In this section we shall define and examine
its relationship
Definition.
Given any object
object
(V',V,vos)
where
with
a morphism
in
definition
is justified
AX
, then define since
VoW'
in A X define
~ vev'
v
object
*(f,g)
+ X in
id®g
w
~X
*(A)
to be the
is a map in ~X
and
V .
(f,g):A
= (g,f):*(B)
the commutativity
W®W ' implies
s
is another
,:_~op
.
A = (V,V',v)
s:V'®V
B = (W,W',w)
Suppose
AX(-, -)
a functor
+ *(A).
> B This
of the diagram:
> V®V'
>
X
that the diagram id®f
W'®V
+ W'®W
I gOid V'®V
Iw°s > X
vos
commutes. From the above
formula on morphisms
we can easily conclude
that
*
a functor. Moreover
*
has an inverse
The following Proposition V(A,B)
i. Given
~ V(*(B),*(A))
PROOF.
are some properties A = (V,V',v)
the commutative
Notice commute.
,
of
since
*o* = idAx_
*:
B = (W,W',w')
in
~X
, then
.
By definition
Consider
(contravariant),
*(A) = (V',V,vos) diagram of Figure
that the coherence
It also implies
of
V
and
*(B) = (W',W,wos).
6.
implies
squares
that the diagram
(V®W', X)
(V®W',X)
(s, id)
~ (W' ®V, X)
(I),
(2),
(3),
(4)
is
123
commutes. The fact that V(* (B) ,*(A))
pl
> (V,W)
Vos
(w',v')
, (w'~v,x)
is a pullback square implies that there exists a unique such that the diagram of Figure 6 still commutes. square involve
V(A,B)
induces a unique map
that the diagram of Figure 6 commutes.
p:V(A,B)
~ V(*(B),*(A))
Similarly the pullback
q:V(*(B),*(A))
This implies
~ V(A,B) such
qop is the map
induced by the pullback square: V(A, B) Ip2
pl
(V,W)
p'b"
(w' ,v' )
If~ (v®w' ,x)
p1
V(A,B)
pl
_V(* (B(,*(A))
V(A,B) p2
~2
(v,w)
pl
, (v,w)
, (v,w)
Wos
p.b.
(W',V')
(V®W',X)
~os
(W' ,V' )
2/
(W'®V,X)
(V®W',X)
(w' ,v' ) FIGURE 6.
124
But by the remark above qop = idv(A,B)
V(A,B)
and
the same argument
completes
Let
For any object
Corollary.
Let
A,B
PROOF.
C
By definition
diagram,
and
This
that > W'®V
in
that
Ax(A,B)
and
,
V(A,*(B))
= V(B,*(A))
= C.
be two objects
in
*(B) = (W',W,wos)
it is sufficient in
= (V(A,B),V®W',n2) and
idv(,(B),,(A))
S(W',V):W'®V
_AX ,
which
~ VoW'
;
we
q:V(*(B),*(A))
= poq
~ V(A,B)
We also have s(V,W'):V®W'
.
such that
s(V,W')os(W',V)
=
(p,s(W',V))
to check that the pair
P
~ V(*(B),*(A))
(s,id)
A,B
is indeed
the following
Ax(A,*(B)) If
C
is an object
Proposition
2. Let A,B,C
in
in
= Ax(B,*(A)) ~X
, then
V(A,_Ax(B,*(C))) Let
definition
*(A) = (V',V,vos)
A = (V,V',v),
Now put
Recall
(V®E (V®W',X)
m
in
C ~X
= *(*C) ' then
V(C,Ax(B,*(A))).
B = (W,W',w), and
•
C = (U,U',u).
that
Ax(B,*(A)) V(B,*(C))
Then by
*(C) = (U',U,uos).
Bc = Ax(B,*(C)) = (V(B,*(C)),W®U,n I) Ba =
pullbacks:
(s,id),
_V(A,B)
A X , then
be three objects
PROOF.
,
the transpose.
be two objects
PROOF.
q
, (W'®V,X)
the proof by taking Let
idw,®v
.
But we see this by considering
_AX .
moreover
diagram:
(V®W',X) and complete
' then
*(*(C))
+ V(*(B),*(A))
= qop
V(A,B)
Corollary.
~X
*(A) = (V',V,vos),
p:V(A,B)
idv(A,B)
an isomorphism commutative
'
B = (W,W',w),
s(W',V) os(V,W ') = idv®w, Hence
in
~X
AX(*(B),*(A)) = (V(~B),*(A)),W'®V,n I) .
that
have isomorphism
squares
in the previous
poq = idv(,(B),,(A) ) .
(*(B),*(A))-
But recall
and
that
be two objects
A = (V,V',v),
A--x(A'B) ~ ~ X
such
Hence
the proof.
PROOF.
implies
V(*(B),*(A))
to conclude
Corollary.
then
also has this property.
.
Now switch apply
idv(A,B)
and
= (V(B,*(A)),W®V,n 2) and
V(B,*(A))
make
the following
125
V(B,*(C))
pl
~ (W,U')
V(B,*(A))
pl
> (W,V')
]P2 (u,w')
(v,w')
, (weu,x)
]v~s wos
> (w~v, x)
Now consider Figure 7. Since (U,-) and (V,-) have left adjoints, they preserve pullbacks, hence the outer and inner squares are still commutative. But (i) is a pullback, hence the following diagrams commute: (v, (u,w'))
pos
> (u,(v,w'))
(v, (w®u, x))
l
(u, (wov, x))
s
~
~ (w(w~u),x)
(u, (w,v'))
I
p-1
(w (weu), x)
(w~u,v)
(u® (w®v) ,x)
\/
~ (ue(w~v),x)
(u, (wev,x))
/
pl
V(A,Bc) pl
V(C Ba)
> (U,Ba)(id~pI~(u,(w,v')) (id,p2)
,2
(I)
(U, (V,W')) (id,~)
(W®V,U')
(V,V(B,* (C)))
w.uv)
(id,9)
(u, (w~v,x))
, (u®(w®v),x)
(v,(u,w'))
(S~
°
(id,~)
(V, (W,U'))
(id,~)
(v,(w®u,x))
FIGURE 7.
P
-i (w(w~u),x)
126
This implies
that there exists
that the diagram of
Figure
7
a unique map still
V(C, Ba)
pl
a pullback,
there
A similar p:V(C,Ba)
argument
If
map
8) shows
q
> V(C,Ba)
the existence
as in the previous
and
.
of a map
A, B, C are objects
PROOF.
Apply
the same argument
Corollary.
Let A, B, C be objects _Ax(A,Ax(B,C)) _~(A,Ax(B,C))
proposition,
qop = idv(c,Ba) in _~
Ax(A,Ax(B,*(C)))
PROOF.
V(A,Bc)
.
the same argument
that poq = idv(A,Bc )
Corollary.
such
(u®(w®v),x) a unique
(Figure
> V(A,Bc)
Applying conclude
exists
~ (U,Ba)
Now using the fact that
(U,Ba)
(WOV,U') is
V(A,Bc)
commutes.
.
, then
~Ax(C,_~(B*(A)))
as in previous in
we
•
corollary.
A X , then
~Ax(*(C),Ax(B,*(A)))
.
~ AX(*(*(A)),Ax(B,*(*(C)))) ~ AX(*(C),Ax(B,*(A)))
Remark.
These propositions
the foundation
3.
The Functor
Note:
Henceforth
Definition.
and corollaries
of our construction,
the duality
lay
-ewe write,
Given
It is clear
concerning
as we shall see later on.
that
A,B
for an object
A
objects
' then define
-®-
in
~X
is a bifunctor,
of
~X
since
'
A* instead
of
*(A).
A®B = Ax(A,B*)*
-®-
.
is the composition AX(-,-)
~X × ~X
(*'*)
+ &
×
AX
(id,*)
~ _~ ×
~X
> ~X
127
p1
_V(C,Ba)
>(WoV, U'
/ pl
V(A Bc)
(V,V(B,*(C))) (id,pl) (V,(W,U')) (id,p2) 1
~2
p2
(V,(U,W'))
(id,~)
/ (W®U,V')
(U,Ba)
(id,
(id,~)
p.b.
(v,(w®u,x)) p-i (v®(w®u),x)
/pos
(s,id)~
, (U, (V,W'))
(u,(w,v')) ~.~-v-rr-=v+~(u,(w®v,x))
tlO,v)
P
-i
FIGURE 8. Proposition.
Let A,B be objects in _~ , then A®B m B®A.
PROOF. Proposition.
PROOF.
A®B = Ax(A,B*)* m Ax(B,A*)* = B®A. Let A,B,C
be objects in _AX , then
(A®B) C ~ A®(B~C) (A®B) ®C = Ax(A, B*)*®C = Ax(Ax(A, B*)*, C*)* Ax(C,Ax(A, B*) )* Ax(C,_~(B,A*) )* Ax(A,_Ax(B,C*) )* Ax(Ax(B, C*)*, A*)* Ax(A,Ax(B,C*)**)* = A®Ax(B, C*)* = A®(B~C).
>(u®(w®v),x)
128
4.
The Dualisin$ Let
is the cannonical Claim.
T
Object
and the Unit
T = (X,l,r) isomorphism
in
is the dualising
Let
following
A = (V,V',v)
commutative
in
~X
r:X®l
' such that
X
V. object,
_~(A,T) PROOF.
for Tensor.
be the object
i.e.
for any object
A
in
_AX.
m A
be an object
_~
in
, then we have the
diagram
V(A,T)_
pl
~ (V,X)
pl
Ir
(1,v')
~
, (v®I,X)
[
1p (V,(I,X))
V'
(id,i)
(V,X) But V'
~ (V,X)
V' is trivially (unique)
~
a pullback
morphism
unique map
g:V'
Corollary.
T
PROOF.
Suppose
in
V,
Apply
such that
~ V(A,T) is the identity is
implies
V'.
f:V(A,T)
A
which
(V,X)
for
an object
•
in
AX ,
m A x ( A ,T) ~A =A
fog = idv,
-®-
T ®A = __~(T ,A )
that we have an induced the same argument
then
and
to get a
gof = id.(A,T)~
.
129
On the other hand, Theorem.
Let
AoT
~ T ®A ~ A.
This completes
A,B,C be objects in
PROOF.
_~,
the proof.
then
_~(A®B,C)
-~ A x ( A , ~ ( B , C ) )
Ax(A®B,C)
= Ax(Ax(A,B
.
) ,C)
Ax(C
,Ax(A,B ))
-~ Ax(C
,Ax(B,A ))
_~(A,Ax(B,C)) . Proposition.
Let
A
be an object in
_~
then
Ax(T ,A) ~- A. PROOF.
Ax(T
,A) ~ _ ~ ( A
,T) ~ A.
Remark. (i)
There is an obvious embedding
(V,X)
to
context (2)
_AX (V,X)
sending
V
~ X
has a *-autonomous
It is easy to verify
the MacLane-Kelly
_AX
V®I
> X:
hence in this
structure.
also satisfies our first assumption,
APPLICATIONS
Functor Categories In this chapter, we shall apply the theory developed
far to the double envelope of a symmetric monoidal Before defining the double envelope, elementary
.
We know that if
X
and
Y
X is complete,
X = S , the category of sets structure.
category
thus
C.
let us recall some
results of the functor categories. Given categories
W = X~
i.e.
coherence conditions.
CHAPTER IV:
i.
functor from the comma category to
W
we have the functor category then so is
W,
in the case
also has a closed symmetric monoidal
The tensor is the cartesian product while the internal
is defined as the functor whose value at transformations
F(-) × IIom(D,-)
D
--+ G(-).
is the set of nature
GF
130
2.
The Double Envelope.
Definition.
Given a symmetric
I-I:C ° objects
> S , we denote of
E(C)
from ~o
to
morphism
from
f
are all triplets
S,t
is a natural
(F,G;t)
is a natural
transformation
monoidal
to
G'
with a faithful
(F,G;t)
(F',G';s)
to
in
from G
~
where
F
F
from
E(~)
to
by and
~ F(C)
G
and
g
The
are functors
to
is a pair
F'
functor
E(C).
F x G
I-~-I.
(f,g)
A
where
is a natural
such that the following
idxg
F(C) x G'(C')
of
transformation
transformation from
category
the double envelope
diagram
x G(C')
fxid
F'(C) commutes
x G'(C')
s
for every object
Proposition. PROOF.
E(C)
(C,C')
in
E(C) x CO .
of
C° x C° .
is a category.
Suppose
(f,g):
(F,G;t)
(f',g'):
cO
~Ic~c' I
, then the following
F(C)XG"(C')
fXid
> (F',G';s)
(F',G';s) diagram
commutes
) F'(C)XG"(C')
idXg'
are maps
> (F",G";u)
(C,C')
for every
f'Xid
) F"(C)XG"(C')
idXg'
F(C)XG'(C')
fXid
in
U
>F'(C)XG'(C')
idXg F(C)XG(C') This implies
are maps
in
that
E(C)
t
) C~C'
(f,g) : (F,G;t)
) (F' ,G' ;s)
(f',g') : (F' ,G' ;s)
~ (F", G" ;u)
(f",g"):(F",G";u)
> (F'",G'";v)
, then
(f",G")o((f',g')o(f,g))
= (f",g")o(f'of,gog') = (f" o (f' o f), (gog') og") = ((f' of') of,go (g' og")) = (f"of', g' og") o (f, g) ((f", g") o(f' ,g')) o (f, g).
Moreover,
given
(F,g;t)
then
(idF,id G) is the obvious
choice
for identity.
131
Before p r o v i n g the m a i n t h e o r e m of this chapter O
v e s t i g a t e the functor categories
~o
co
two obvious embeddings of
S
and
into
S
--
where
O
S~ --
let us in~
× ~
.
T h e r e are
, namely
~
--
and r, O
~(F) = F x I
and r(F) = I x F for every F in S~ , and C° -I is the unit in S-i.e. I sends every object into the singleton -C° (the terminal object) in S. H e n c e w e can regard objects in S-- as C° × CO objects in S= -v i a either embedding. N o w we can prove. -O O C × C Proposition. E(C) is e n r i c h e d over V = S--- .
PROOF.
By previous r e m a r k
V
is a closed symmetric m o n o i d a l
c a t e g o r y w i t h pullbacks, m o r e o v e r it is coherent. N o w given
A = (G,F;t)
V(A,B)
an object in
and
B = (G',F';s) O
define
--
of
object
C °
S--
x
C°
--
, then
V(A,B)
in E(C) w e have to
O
V(= S~C
× ~ ) .
Suppose
(C,C')
is an
is the functor w h o s e value at (C,C')
is defined by requiring that the d i a g r a m V(A,B)(C,C')
pl
~ (~(G),%(G')(C,C')
(r(F'),r(F))(C,C')
~ (G x F',
I-~-I)(C,C')
be a pullback. o Note.
(-,-)
denotes the internal h o m - f u n c t o r of
map (~(G),~(G'))(C,C') that in
V,
p r o p e r t y of
G x F' V
> (G x F',
is isomorphic to
S~
cO x
As for the
I-~-I)(C,C'),
w e simply o b s e r v e
~(G) × r(F').
Then the adjoint
constructs such a map (in the same fashion as in Chapter
II, Section 2.)
A similar argument constructs
(r(F'),r(F))(C,C')
) (g x F,
map
I-®-I)(C,C').
N o w the enrichment follows i m m e d i a t e l y f r o m the result in Chapter II, since this is h o w p u l l b a c k s are defined in the functor category, i.e. b y p o i n t - w i s e evaluation. Theorem. A
E(~)
is e n r i c h e d over
PROOF. 3.
Put
This concludes the proof.
is a s u b c a t e g o r y of a * - a u t o n o m o u s category
A; m o r e o v e r
V.
X =
I-®-I
, then follow the c o n s t r u c t i o n in Chapter III.
M i s c e l l a n e o u s Results. In this section, we are a s s u m i n g
V
has all the p r o p e r t i e s as given
in Chapter I and we shall prove that there is a functor - C A T ( V - CAT
F maps
V
to
is the c a t e g o r y of all categories w h i c h are e n r i c h e d over
X)The functor put
F(X) = ~ .
F
on objects of
V
is obvious:
given
X
in
V,
then
132
Now we have to show given a map a
V-functor T( = F(f)) The notion of a
II, Section 6. a function
(ii)
for each
(i)
V
_~
to
> S
in
V, this induces
AS.
V-functor can be found in [Eilenberg & Kelly] Chapter
In this case we have to show:
(i)
in
from
f:X
T
maps objects of
B,C
in
_~
to objects of
AX, a morphism
T(B,C) maps
AS.
V(B,C) to
V(T(B),T(C))
such that the following axioms are satisfied: The following diagram commutes: T
V(B,B)
, V(T(B) ,T(B))
I
(2)
The following diagram cormnutes: M v
V(C,D)®V(B,C) .
.
.
.
_V(B,D)
I,l t
V(T(C),T(D))®V(T(B),T(C))
, _V(T(B),T(D))
Note. In both categories we denote the enriched object by
V(-,-)
,
it
is clear from the context which one we are referring to. The function
T
on objects of
_Ax, then T(B) is the composition
_~ is obvious;
V®V'
v
given
B =
f
, S
i.e.
objects in
AX,
then
~ X
(V,V',v)
T(B) = (V,V',fov). To show (ii): Suppose
B = (V,V',v)
C = (W,W',w)
and the following diagram commutes
T(B) = (V,V',fov), T(C) = (W,W',fow) pl
V(B,C)
V(T(B) ,T(C))
, (v,w)
pl
p.b.
p2
(W' ,V' )
fov
, (V,W)
I
f~w
~(V~W',S)
( i d , f ~
(W',V')
(V~W',X)
in
133
Since the inner square is a pullback, there exists a (unique) map T(B,C)
from
V(B,C)
to V(T(B),T(C)).
TO show (i) commutes let B = (V,V',v) and the following diagrams commute:
in
_AX.
Then
T(B) = (V,V',fov)
i3 V(T(B), T(B)) p2
P'I"
(v',v')
(V' ,V')
~(wv',x)
--,
pl
V(T(B),T(B))
lp2 (V',V')
pl
(V,V)
÷(V,V)
p.b. fov
> (v~v' ,x)
(id,f~~ (V',V')
(V®V' ,X)
(v' ,v)
(v' ,v')
> (V,V)
[~
V(B,B)
p2
P
v
,(wv' ,s)
+ (wv',x)
for , (V®V' ,S)
134
Hence the composition map
I
J
> V(T(B),T(B))
I
J
> V(B,B)
T
> V(T(B),T(B))
are both induced by pulling back.
and
Thus by
the uniqueness property they are "equal". To show
(2)
commutes, let
B = (V,V',v),
C = (W,W',w),
D = (U,U',u)
be three objects in _AX . Then T(B) = (V,V',fov), T(C) = (W,W',fow), T(D) = (U,U',fou) and the following four diagrams commute: V(C,D) ®V(B, C)
pl~pl
(U' ,W')®(W' ,V')
V(B,D)
Is
~ (W,U)®(V,W)
pl
p2[
(w',v')®(u',w')
M
p.b.
[fi
~ (u',v')
V(T(C),T(D))®V_(T(B),T(C))
(U' ,W')®(W' ,V')
~ (wu',x)
pl®pl
> (W,U)®(V,W)
V(T(B) ,T(D))
l
pl
, (V,U)
p.b.
(W' ,V')®(U' ,W')
M
,(U' , V ' )
fov
~(V®U',S)
pl
V(C,D) V(T(C),T(D)) p2
,(V,U)
p21
(U',W')
(w,u) pl
~(W,U)
I
f~u
fow
~(W®U',S) ( i d , ~
(U',W')
> (W®U',X)
135
pl
V(B,C)
(V,W)
~a
\ V(T (B) ,T(C))
~2
pl
p2
(w' ,V')
,(v,w)
I
f~w
p .b.
fov
~(V®W',S)
(id,f)~ (V®W',X)
(w' ,V')
(w,u)~(v,w)
V(C,D) ®V(B, C)
V(T (C) ,T (D)) ®V(T (B), T (C))
~2®p2
Y
pl~l
(w,u)~(v,w)
p2~2 ~'~V(T(B),T(D))
pl> (V,U)
(v,u)
p21 pb Ifou (w' ,v')~(u' ,w')
(w' ,v')~(u' ,w')
M
+(u' ,v')
, (U' ,V')
fov
~(wu' ,s)
v
(V®U',X)
136
pl®p1
V(C,D) ®V(B,C)
(w,u)®(v,w)
,
h ~
~"~V(B,D)
plepl
~ (V,U)
J
V(T(B),T(D))
pl + (V,U)
p2ep2
~ U',V') M
f°~(V®U'
/~d (u', v')
(w',v')®(u',w')
9
,S)
(id'f ) ~ , (v®u',x)
This implies that the diagrams above commute, which implies that the composition ~(C,D)®(V(B,C))
T®~
M'
> V(T(C),T(D))®V(T(B),T(C))
--+ V(T(B),T(D))
is the map induced by pulling back. This also implies that the composition V(C,D)® V(B,C)
M' '
~
T
V(B,D)
+ V(T(B),T(D))
is the map induced by pulling back. Hence by the uniqueness property, they are "equal", therefore (2) commutes. Now we are left to show that if maps in
V, then
F(g) oF(f)
f:X
F(gof), i.e.
> S F
and
g:S
>K
are
preserves composition.
All we have to cheek is that the composition is preserved in (i) and (ii). It is easy to show (i) is preserved.
For if
B = (V,V',v) in
_~ ,
then (F(g) oF(f))(B) = F(g)(F(f)(B)) = F(g)(V,V',fov) = (V,V',go(fov)) = (V,V',(gof) ov) = F(gof)(B). To show (ii) is preserved: Let B = (V,V',v),
C = (W,W',w)
F(f)(B) = (V,V',fov),F(f)(C) = (W,W',fow),(F(g)oF(f))(B)
in
~X' then
= F(gof)(B)
= (V,V',(gof) ov) ,F(gof)(C) = (F(g) oF(f))(C) = (W,W',(gof) ow) and the diagrams (*), (**) and (***) commute
137
p1
V(B ,c)
+ (v,w)
~(F(f)(B),F(f)(C))
(*)
[p2
~2
(W',V')
pl
~P'b" fov
(v,w)
I
f~w
~(V®W'.S) (id,f)~
(V®W',X)
(W',V')
V(F(f)(B),F(f)(C))
~
pl
÷(v,w)
~ X(F(gof)(B),F(gof) (c)) pl >(v,w)
(**)
f ow
(W',V')
(g°f) f~v
(V®W ',K) (id,gof)~
(w' v') Note.
>(V®W',S)
F(gof) (-)
=
(F(g) oF(f)) (-).
pl
X(s,c)
+ (V,W)
V(F(gof)(B),F(gof)(C))
p2 (W',V')
p.b. ($of)ov
,(V,W)
(gof)ow ~ , (WW',K)
(V®W',X)
(W',V')
But (*) and (**) imply the diagram of Figure 9 connnutes. This implies that both F(gof) in (***) and the composition V(B,C)
F(f)
V(F(f)(B),F(f)(C))
are induced by pulling back.
F(g)
V(F(gof)(B),F(gof)(C))
Hence it follows they are equal.
138
pl
V(B,C)
(v,w) pl
V(F(f) (B) ,F(f) (C))
~
~ (v,w)
) V(F~of)(B),F(gof)(C))
p2
Y
J
pl ~ (V,W)
[p2
I
fow
(gof)ov ,(V®W',K)
(W' ,V')
fov
(W',V') ..~
(i
(V®W' ,S) (id,f)~
(w',v')
~ (ww' ,x)
FIGURE
9.
BIBLIOGRAPHY i.
M. BARR, Duality of vector spaces, Cahiers Topologie G~om~trie Diff~rentielle, XVII-I (1976), 3-14.
2.
M. BARR, Duality of Banach spaces, Ibid., 15-32.
3.
M. BARR, Closed categories and topological vector spaces, Ibid. XVII-3, 223-234.
4.
M. BARR, Closed categories and Banach spaces, Ibid. XVII-4, 335-342.
5.
M. BARR, A closed category of reflexive topological abelian groups, Ibid. XVIII-3, 221-248.
6.
M. BARR, *-autonomous categories,
7.
S. EILENBERG, G.M. KELLY, Closed categories, Proc. Conf. Categorical
8.
M.E. SZABO, Commutativity in closed categories.
This volume.
Alg. (La Jolla, 1965), Springer (1966), 421-562. To appear.
Index of Definitions
Admissible Autonomous Basis
(uniform object) (category) 13
9
(for a pseudometric)
Completable 9 Convergence uniformity Cosmall 24
8
7
Semi-norm 65 Semi-variety 3 Separated uniform *-autonomous 13 Uniform Uniform Uniform Uniform
8 8
Linearly compact 18 Linearly totally bounded complete Nuclear
Refine (of seminorms) Reflexive 23 Represent 28
33
47
Pre-reflexive 23 Pre-*-autonomous 15 Pre-uniform structure Product uniformity i0 Pseudomap 5 Pseudometric 7
71
28
Dominating 29 Double envelope 130 Dualizing module 49 Embedding Entourage
Quasi-reflexive 23 Quasi-variety 3
20
convergence cover 6 object I0 space 6
6
on
28
Variety 3 V-enriched pre-*- autonomous V-enriched *-autonomous 17 ~-complete 34 ~-*-complete 35
17
Index of Notation
_v(-,-)
2
<--~-->
5
I-I
6
<-->
9
34
(_)~
9
37
UnV
i0
T
[-,-]
ii
A
33
()A
33
r
33
37 43 o
T
13
A
44
(-)
13
u
52
(-). (-)
18
h(1)®h(2) A
59
C~D
19
(-)
74
A
19
_Ax
104
A(-,-)
25
E(C)
130
28
