Directory UMM :Journals:Journal_of_mathematics:TAC:
Theory and Appli
ations of Categories,
Vol. 6, No. 1, pp. 5{24.
-AUTONOMOUS
CATEGORIES:
ONCE MORE AROUND THE TRACK
To Jim Lambek on the o
asion of his 75th birthday
MICHAEL BARR
ABSTRACT. This represents a new and more
omprehensive approa
h to the autonomous
ategories
onstru
ted in the monograph [Barr, 1979℄. The main tool in
the new approa
h is the Chu
onstru
tion. The main
on
lusion is that the
ategory of
separated extensional Chu obje
ts for
ertain kinds of equational
ategories is equivalent to two usually distin
t sub
ategories of the
ategories of uniform algebras of those
ategories.
1. Introdu
tion
The monograph [Barr, 1979℄ was devoted to the investigation of -autonomous
ategories.
Most of the book was devoted to the dis
overy of -autonomous
ategories as full sub
ategories of seven di erent
ategories of uniform or topologi
al algebras over
on
rete
ategories that were either equational or re
e
tive sub
ategories of equational
ategories.
The base
ategories were:
1. ve
tor spa
es over a dis
rete eld;
2. ve
tor spa
es over the real or
omplex numbers;
3. modules over a ring with a dualizing module;
4. abelian groups;
5. modules over a
o
ommutative Hopf algebra;
6. sup semilatti
es;
7. Bana
h balls.
For de nitions of the ones that are not familiar, see the individual se
tions below.
These
ategories have a number of properties in
ommon as well as some important
di eren
es. First, there are already known partial dualities, often involving topology.
This resear
h has been supported by grants from the NSERC of Canada and the FCAR du Quebe
Re
eived by the editors 1998 November 20 and, in revised form, 1999 August 26.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18D10, 43A40, 46A70, 51A10.
Key words and phrases: duality, topologi
al algebras, Chu
ategories.
Mi
hael Barr 1999. Permission to
opy for private use granted.
5
Theory and Appli
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6
It is these partial dualities that we wish to extend. Se
ond, all are symmetri
losed
monoidal
ategories. All but one are
ategories of models of a
ommutative theory and
get their
losed monoidal stru
ture from that (see 3.7 below). The theory of Bana
h balls
is really di erent from rst six and is treated in detail in [Barr, Kleisli, to appear℄.
What we do here is provide a uniform treatment of the rst six examples. We show that
in ea
h
ase, there is a -autonomous
ategory of uniform spa
e models of the theory. In
most
ases, this is equivalent to the topologi
al spa
e models. The main tool used here is
the so-
alled Chu
onstru
tion as des
ribed in an appendix to the 1979 monograph, [Chu,
1979℄. He des
ribed in detail a very general
onstru
tion of a large
lass of -autonomous
ategories. He starts with any symmetri
monoidal
ategory V and any obje
t ? therein
hosen as dualizing obje
t to produ
e a -autonomous
ategory denoted Chu(V ; ?). The
simpli
ity and generality of this
onstru
tion made it appear at the time unlikely that it
ould have any real interest beyond its original purpose, namely showing that there was
a plenitude of -autonomous
ategories. We des
ribe this
onstru
tion in Se
tion 2.
A preliminary attempt to
arry out this approa
h using the Chu
onstru
tion appeared
in [Barr, 1996℄, limited to only two of the seven example
ategories (ve
tor spa
es over
a dis
rete eld and abelian groups). The arguments there were still very ad ho
and
depended on detailed properties of the two
ategories in question. In this arti
le, we
prove a very general theorem that appeals to very few spe
ial properties of the examples.
In 1987 I dis
overed that models of Jean-Yves Girard's linear logi
were -autonomous
ategories. Within a few years, Vaughan Pratt and his students had found out about the
Chu
onstru
tion and were studying its properties intensively ([Pratt, 1993a,b, 1995,
Gupta, 1994℄). One thing that espe
ially stru
k me was Pratt's elegant, but essentially
obvious, observation that the
ategory of topologi
al spa
es
an be embedded fully into
Chu(Set; 2) (see 2.2). The real signi
an
e|at least to me|of this observation is that
putting a Chu stru
ture on a set
an be viewed as a kind of generalized topology.
A reader who is not familiar with the Chu
onstru
tion is advised at this point to read
Se
tion 2. Thinking of a Chu stru
ture as a generalized topology leads to an interesting
idea whi
h I will illustrate in the
ase of topologi
al abelian groups. If T is an abelian group
(or, for that matter, a set), a topology is given by a
olle
tion of fun
tions from the point
set of T to the Sierpinski spa
e|the spa
e with two points, one open and the other not.
From a
ategori
al point of view, might it not make more sense to repla
e the fun
tions to a
set by group homomorphisms to a standard topologi
al group, thus
reating a de nition
of topologi
al group that was truly intrinsi
to the
ategory of groups. If, for abelian
groups, we take this \standard group" to be the
ir
le group R=Z, the resultant
ategory
is (for separated groups) a
ertain full sub
ategory of Chu(Ab; R=Z)
alled
hu(Ab; R=Z).
This
ategory is not the
ategory of topologi
al abelian groups. Nonetheless the
ategory
of topologi
al abelian groups has an obvious fun
tor into
hu(Ab; R=Z) and this fun
tor
has both a left and a right adjoint, ea
h of whi
h is full and faithful. Thus the
ategory
hu(Ab=; R=Z) is equivalent to two distin
t two full sub
ategories of abelian groups, ea
h
of whi
h is thereby -autonomous. In fa
t, any topologi
al abelian group that
an be
embedded algebrai
ally and topologi
ally into a produ
t of lo
ally
ompa
t groups has
Theory and Appli
ations of Categories, Vol. 6, No. 1
7
both a nest and a
oarsest topology that indu
e the same set of
hara
ters. The two
sub
ategories
onsist of all those that have the nest topology and those that have the
oarsest. These are the images of the left and right adjoint, respe
tively.
1.1. A
knowledgment. I would like to thank Heinri
h Kleisli with whom I had a many
helpful and informative dis
ussions on many aspe
ts of this work during one month visits
to the Universite de Fribourg during the springs of 1997 and 1998. In parti
ular, the
orre
t proof of the existen
e of the Ma
key uniformity (or topology) was worked out
there in the
ontext of the
ategory of balls, see [Barr, Kleisli, 1999℄.
2. The Chu
onstru
tion
There are many referen
es to the Chu
onstru
tion, going ba
k to [Chu, 1979℄, but see
also [Barr, 1991℄, for example. In order to make this paper self-
ontained, we will give
a brief des
ription here. We sti
k to the symmetri
version, although there are also
non-symmetri
variations.
2.1. The
ategory Chu(V ; ?). Suppose that V is a symmetri
losed monoidal
ategory
and ? is a xed obje
t of V . An obje
t of Chu(V ; ?) is a pair (V; V ) of obje
ts of V
together with a homomorphism,
alled a pairing, h ; i: V
V ! ?. A morphism
(f; f ): (V; V ) !(W; W )
onsists of a pair of arrows f : V ! W and f : W ! V in V
that satisfy the symboli
identity hf v; w i = hv; f w i. Diagrammati
ally, this
an be
expressed as the
ommutativity of the diagram
0
0
0
0
0
0
0
V
f
W
W
0
W
?
W
0
0
V
f
0
V
h
;
0
0
V
h
0
0
?
?
i
0
i
;
Using the transposes V ! V Æ ? and V ! V Æ ? of the stru
ture maps, this
ondition
an be expressed as the
ommutativity of either of the squares
0
f
V
V
0
?
Æ?
f
0
Æ?
W
0
W
0
?
Æ?
W
W
f
0
?
Æ?
f
0
-
V
0
?
Æ?
Æ?
V
Theory and Appli
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8
A nal formulation of the
ompatibility
ondition is that
- Hom(
Hom((V; V 0 ); (W; W 0 ))
V; W
?
- Hom(
?
Hom(W 0 ; V 0 )
V
0
)
W ;
?)
is a pullba
k.
The internal hom is gotten by using an internalization of the last formulation. De ne
the obje
t [(V; V 0 ); (W; W 0 )℄ of V as a pullba
k
-
[(V; V 0 ); (W; W 0 )℄
W
0
?
Æ
V
-
0
V
V
Æ
W
?0
W
Æ?
and then de ne
(V; V 0 ) Æ(W; W 0 ) = ([(V; V 0 ); (W; W 0 )℄; V
W 0 )
Sin
e the dual of (V; V 0 ) is (V 0 ; V ), it follows from the interde nability of tensor and
internal hom in a -autonomous
ategory that the tensor produ
t is
(V; V 0 )
(W; W 0 ) = (V
W;
[(V; V 0 ); (W 0; W )℄)
The result is a -autonomous
ategory. See [Barr, 1991℄ for details.
2.2. The
ategory Chu(Set; 2). An obje
t of Chu(Set; 2) is a pair (S; S 0 ) together with
a fun
tion S S 0 ! 2. This is equivalent to a fun
tion S 0 ! 2S . When this fun
tion
is inje
tive we say that (S; S 0 ) is extensional and then S 0 is, up to isomorphism, a set of
subsets of S . Moreover, one easily sees that if (S; S 0 ) and (T ; T 0 ) are both extensional,
then a fun
tion f : S ! T is the rst
omponent of some (f; f 0 ): (S; S 0 ) !(T ; T 0 ) if and
only if U 2 T 0 implies f 1 (U ) 2 S 0 and then f 0 = f 1 is uniquely determined. This
explains Pratt's full embedding of topologi
al spa
es into Chu(Set; 2).
2.3. The
ategory
hu(V ; ?). Suppose V is a
losed symmetri
monoidal
ategory as above
and suppose there is a fa
torization system E =M on V . (See [Barr, 1998℄ for a primer on
fa
torization systems.) In general we suppose that the arrows in E are epimorphisms and
that those in M are
ompatible with the internal hom in the sense that if V ! V 0 belongs
to M, then for any obje
t W , the indu
ed W Æ V ! W Æ V 0 also belongs to M. In
all the examples here, E
onsists of the surje
tions (regular epimorphisms) and M of the
inje
tions (monomorphisms), for whi
h these
onditions are automati
. An obje
t (V; V 0 )
of the Chu
ategory is said to be M-separated, or simply separated, if the transpose V
! V 0 Æ ? is in M and M-extensional, or simply extensional, if the other transpose
0 ! V Æ ? is in M. We denote by Chus(V ; ?), Chue (V ; ?), and Chuse (V ; ?) the
V
Theory and Appli
ations of Categories, Vol. 6, No. 1
9
full sub
ategories of separated, extensional, and separated and extensional, respe
tively,
obje
ts of Chu(V ; ?). Following Pratt, we usually denote the last of these by
hu(V ; ?).
The relevant fa
ts are
1. The full sub
ategory Chus(V ; ?) of separated obje
ts is a re
e
tive sub
ategory of
Chu(V ; ?) with re
e
tor s.
2. The full sub
ategory Chue (V ; ?) of extensional obje
ts is a
ore
e
tive sub
ategory
of Chu(V ; ?) with
ore
e
tor e.
3. If (V; V 0 ) is separated, so is e(V; V 0 ); if (V; V 0 ) is extensional, so is s(V; V 0 ).
4. Therefore
hu(V ; ?) is both a re
e
tive sub
ategory of the extensional
ategory and
a
ore
e
tive sub
ategory of the separated sub
ategory. It is, in parti
ular,
omplete
and
o
omplete.
5. The tensor produ
t of extensional obje
ts is extensional and the internal hom of an
extensional obje
t into a separated obje
t is separated.
6. Therefore by using s(
) as tensor produ
t and r(
ategory
hu(V ; ?) is -autonomous.
Æ
) as internal hom, the
For details, see [Barr, 1998℄.
3. Topologi
al and uniform spa
e ob je
ts
3.1. Topology and duality. In a -autonomous
ategory we have, for any obje
t A, that
Æ A = A Æ > . If we denote > by ?, we see that the duality has the form
A = >
A 7! A
Æ ?. The obje
t ? is
alled the dualizing obje
t and, as we will see, the way
(or at least a way) of
reating a duality is by nding a dualizing obje
t in some
losed
monoidal
ategory.
In order that a
ategory have a duality realized by an internal hom, there has to be a
way of
onstraining the maps so that the dual of a produ
t is a sum. In an additive
ategory, for example, this happens without
onstraint for nite produ
ts, but not normally
for in nite ones. A natural
onstraint is topologi
al. If, for example, the dualizing obje
t
is nite dis
rete, then any
ontinuous map from a produ
t
an depend on only nite many
of the
oordinates.
For example, even for ordinary
Q
Q topologi
al spa
es, for a
ontinuous
fun
tion f : Xi ! 2, f 1 (0) has the form
Y
i2
= J Xi where J is a nite
Q
Q subset of I
1
and Y Q
is a subset
of the nite produ
t i2J Xi . But then f (1) = Z 0 i2= J X0i , where
Z =
Xi
Y . If two elements of the produ
t x = (xi )i2I and x = (xi )i2I are
i2J
elements su
h that xi = x0i for i 2 J then either x 2 Y and x0 2 Y or x 2 Z and x0 2 Z ,
but in either
ase f x = f x0 . Thus f depends Q
only the
oordinates belonging to J , whi
h
means f fa
tors through the nite produ
t i2J Xi . A similar argument works if 2 is
repla
ed by any nite dis
rete set.
Theory and Appli
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10
3.2. Uniformity. Despite the examples above there are reasons for thinking that the te
hni
ally \
orre
t" approa
h to duality is through the use of uniform stru
tures. A very
readable and informative a
ount of uniform spa
es is in [Isbell, 1964℄. However, Isbell uses uniform
overs as his main de nition. Normally I prefer uniform
overs to the
approa
h using entourages, but for our purposes here entourages are more appropriate.
For any equational
ategory, a uniform spa
e obje
t is an obje
t of the
ategory
equipped with a uniformity for whi
h the operations of the theory are uniform. A morphism of uniform spa
e obje
ts is a uniform fun
tion that is also a morphism of models
of the theory. Topologi
al spa
e obje
ts are de ned similarly. Any uniformity leads to
a topologi
al spa
e obje
t in a
anoni
al way and uniform fun
tions be
ome
ontinuous
fun
tions for that
anoni
al topology. But not every topology
omes from a uniformity
and, if it does, it is not ne
essarily from a unique one. For example, the metri
on the
spa
e of integers and on the set of re
ipro
als of integers both give the dis
rete topology, but the asso
iated uniformities are quite di erent. (Metri
spa
es have a
anoni
al
uniformity. See Isbell's book for details.)
If, however, there is an abelian group stru
ture among the operations of an equational
theory, there is a
anoni
al uniformity asso
iated to every topology. Namely, for ea
h
open neighborhood U of the group identity take f(x; y ) j xy 1 2 U g as an entourage.
Moreover, a homomorphism of the algebrai
stru
ture between uniform spa
e obje
ts is
ontinuous if and only if it is uniform. Thus there is no di eren
e, in su
h
ases, between
ategories of uniform and topologi
al spa
e obje
ts. Obviously, the
ategory of topologi
al
spa
e obje
ts is more familiar. However, one of our
ategories, semilatti
es, does not have
an abelian group stru
ture and for that reason, we have
ast our main theorem in terms
of uniform stru
ture. There is another, less important, reason. At one point, in dealing
with topologi
al abelian groups, it be
omes important that the
ir
le group is
omplete
and
ompleteness is a uniform, not topologi
al, notion.
If V is an equational
ategory, we denote by Un(V ), the
ategory of uniform obje
ts
of V . We let j j: Un(V ) ! V to be fun
tor that forgets the uniform stru
ture.
3.3. Small entourage. Let A be a uniform V obje
t. An entourage E A A is
alled a
small entourage if it
ontains no subobje
t of A A that properly
ontains the diagonal
and if any homomorphism f : B ! A of uniform V obje
ts for whi
h (f f ) 1 (E ) is an
entourage of B is uniform.
3.4.. In all the examples we will be
onsidering, there will be a given
lass of uniform
obje
ts D and we will be dealing with the full sub
ategory A of Un(V )
onsisting of those
obje
ts strongly
ogenerated by D, whi
h is to say that those that
an algebrai
ally
and uniformly embedded into a produ
t of obje
ts of D.
3.5. Half-additive
ategories. A
ategory is
alled half-additive if its hom fun
tor fa
tors
through the
ategory of
ommutative monoids. It is well known that in any su
h
ategory
nite sums are also produ
ts (see, for example, [Freyd, S
edrov, 1990℄, 1.59). Of
ourse,
additive
ategories are half-additive. Of the six
ategories
onsidered here, ve are additive
and one, semilatti
es, is not additive, but is still half-additive. In fa
t, a semilatti
e is a
Theory and Appli
ations of Categories, Vol. 6, No. 1
11
ommutative monoid in whi
h every element is idempotent. This monoid stru
ture
an
be equally well viewed as sup or inf.
3.6. The
losed monoidal stru
ture. The
ategories we are dealing with are all symmetri
losed monoidal. With one ex
eption, the
losed stru
ture derives from their being models
of a
ommutative theory.
3.7. Commutative theories. A
ommutative theory is an equational theory whose operations are homomorphisms ([Linton, 1966℄). Thus in any abelian group G, as
ontrasted
with a non-abelian group, the multipli
ation G G ! G is an abelian group homomorphism, as are all the other operations.
3.8.
Theorem.
[Linton℄ The
ategory of models of a
ommutative theory has a
anoni
al
stru
ture of a symmetri
losed monoidal
ategory.
Suppose V is the
ategory of models and U : V ! Set is the underlying set
fun
tor with left adjoint F . If V and W are obje
ts of V , then W Æ V is a subset of V UW
de ned as the simultaneous equalizer, taken over all operations ! of the theory, of
Proof.
- (V n) UW
V UW
(
)n
! UW
R ?
UW
V!
(
V(
)n
)n
Here n is the arity of ! and the top arrow is raising to the nth power. Sin
e the theory is
ommutative, ! is a homomorphism and so the equalizer is a limit of a diagram in V and
hen
e lies in V . In parti
ular, the internal hom of two obje
ts of V
ertainly lies in V . The
free obje
t on one generator is the unit for this internal hom. As for the tensor produ
t,
V
W is
onstru
ted as a quotient of F (UV UW ), similar to the usual
onstru
tion of
the tensor produ
t of two abelian groups.
3.9.
Proposition. If
A
and
B
are obje
ts of an equational
ategory
V
equipped with
uniformities for whi
h their operations are uniform, then the set of uniform morphisms
from
A
!B
enri
hed over
is a subobje
t of
V
j A j Æ jB j
and thus the
ategory of uniform
. It also has tensors and
otensors from
V
V
obje
ts is
.
Let [A; B ℄ denote the set of uniform homomorphisms from A to B . For ea
h nary operation ! , the arrow !B : B n ! B is a uniform homomorphism and hen
e an arrow
[A; B ℄n
= [A; B n ℄ ![A; B ℄ is indu
ed by !B and we de ne this as ! [A; B ℄. This presents
[A; B ℄ as a subobje
t of jAj Æ jB j so that it also satis es all the equations of the theory
and is thus an obje
t of V . The
otensor AV is given by the obje
t V Æ jAj equipped
with the uniformity indu
ed by AUV . The tensor is
onstru
ted using the adjoint fun
tor
theorem with all uniformities on V
jAj as solution set.
Proof.
Theory and Appli
ations of Categories, Vol. 6, No. 1
12
4. The main theorem
We are now ready to state our main theorem.
4.1.
Suppose V is an equational
ategory equipped with a
losed monoidal
stru
ture, D is a
lass of uniform spa
e obje
ts of V and A is the full sub
ategory of the
ategory of uniform spa
e obje
ts of V that is strongly
ogenerated by D. Suppose that ?
is an obje
t of A with the following properties
1. V is half-additive.
2. D is
losed under nite produ
ts.
3. ? has a small entourage.
4. The natural map > ![?; ?℄ is an isomorphism.
5. If D is an obje
t of D, A D is a subobje
t, then the indu
ed arrow [D; ?℄ ![A; ?℄
Theorem.
is surje
tive.
6. For every obje
t D of D, the natural evaluation map D ! ?
7. A is enri
hed over V and has
otensors from V .
[D;
?℄ is inje
tive.
Then, using the regular-epi
/moni
fa
torization system, the
anoni
al fun
tor P : A
!
hu(V ; j?j) de ned by P (A) = (jAj; [A; ?℄) has a right adjoint R and a left adjoint L,
ea
h of whi
h is full and faithful.
4.2.. Before beginning the proof, we make some observations. We will
all a morphism A
! ? a fun
tional on A. In light of
ondition 5,
ondition 6 need be veri ed only for
obje
ts that are algebrai
ally 2-generated (and in the additive
ase, only for those that
are 1-generated) sin
e any separating fun
tional
an be extended to all of D.
In all our examples, ? is
omplete and a
losed subobje
t of an obje
t of D belongs
to D so that it is suÆ
ient to verify
ondition 5 when A belongs to D.
The
on
lusion of the theorem implies that the full images of both R and L are
equivalent to
hu(V ; ?) and hen
e both image
ategories are -autonomous.
The diagonal of A in A A will be denoted . We begin the proof with a lemma.
Q
4.3.
Suppose that A 2 A and ': A ! ? is a uniform fun
tional. Then
Q
Q
there is a nite subset J I su
h that if Ae is the image of A ! 2 A ! 2 A with
'e
the subspa
e uniformity, then ' fa
tors as A ! Ae ! ?.
A
Lemma.
i
I
i
i
I
i
i
J
i
Theory and Appli
ations of Categories, Vol. 6, No. 1
13
Let E ? ? be a small entourage. The de nition Q
of the produ
t uniformity
implies that there is a nite subset J I su
h that if we let B = i2J Ai and C = Qi2=J Ai,
then there is an entourage F B B for whi
h (A A) \ (F (C C )) (' ') 1(E ).
But then (A A) \ (B (C C )) is a subobje
t of A A that is in
luded in (' ') 1 (E ).
This implies that (' ')((A A) \ (B (C C ))) is a subobje
t of ?? lying between
? and E , whi
h is then ?. In parti
ular, if a = (ai) and a0 = (a0i ) are two elements of A
su
h that ai = a0i for all i 2 J , then '(a) = '(a0). Thus, ignoring the uniform stru
ture, we
an fa
tor ' via an algebrai
homomorphism 'e: Ae ! ?. But (Ae Ae) \ F ('e 'e) 1(E )
whi
h means that 'e is uniform in the indu
ed uniformity on Ae.
4.4.
AB A
[B; ?℄ ![A; ?℄
Q
Sin
e there is an embedding
B
i2I Di with Di obje
ts of D , it is suÆ
ient
Q
to do this in the
ase that B = i2I Di . The lemma says Qthat any fun
tional in [A; ?℄
fa
tors as A ! Ae ! ? where, for some nite J I , Ae i2J Di. The latter obje
t is
in D by
ondition 2 and the map extends to it by
ondition 5.
P
4.5. Q
fAi j i 2 I g
A
i2I [Ai ; ?℄
![ i2I Ai; ?℄
By taking A = Qi2I Ai in the lemma, we see that every fun
tional
Q on the
produ
t
Q fa
tors through a nite produ
t. That is, the
anoni
al map
olimJ I [ i2J Ai ; ?℄
![ i2I Ai; ?℄ is an isomorphism, where the
olimit is taken over the nite subsets J I .
On the other hand, half-additivity implies that the
anoni
al map from a nite sum to
nite produ
t is an isomorphism. Putting these together, we
on
lude that
Proof.
Corollary. For any
in
, the indu
ed
is surje
tive.
Proof.
Corollary. For any set
of obje
ts of
, the
anoni
al map
is an isomorphism.
Proof.
X
X
Y
i2 I
i2J
i2J
[Ai ; ?℄ =
olim
[Ai; ?℄ =
olim
[Ai; ?℄ =
olim
J I
J I
J I
=
olim
J I
"
Y
i2J
#
Ai ; ?
=
"
Y
i2I
#
"
X
i2 J
#
Ai ; ?
Ai ; ?
4.6. Proof of the theorem. The right adjoint to P is de ned as follows. If (V; V 0 ) is
an obje
t of
hu(V ; ?), then by de nition of separated V ! V Æ V 0 is moni
. The
underlying fun
tor from the
ategory of uniform obje
ts to V has a left adjoint and hen
e
preserves moni
s so that jV j ![V 0 ; ?℄ is also moni
. Sin
e the latter is a subset of
j?jjV j, we have that jV j j?jV jj. Then we let R(V; V 0 ) denote jV j, equipped with the
uniformity indu
ed as a uniform subspa
e of ?jV j. Also denote by (V; V 0) the uniformity
of R(V; V 0 ). This is the
oarsest uniformity on V for whi
h all the fun
tionals in V 0 are
uniform. A morphism P A !(V; V 0 )
onsists of an arrow f : jAj ! V in V su
h that
for any ' 2 V 0 the
omposite ' f is uniform. This means that the
omposite A ! V
! ?U V is uniform and hen
e that A ! R(V; V 0 ) is. Conversely, if f : A ! R(V; V 0 ) is
given, then we have f : jAj ! V su
h that the
omposite A ! V ! ?UV is uniform and
0
0
0
0
0
Theory and Appli
ations of Categories, Vol. 6, No. 1
14
if we follow it by the
oordinate proje
tion
orresponding to ' 2 V , we get that ' f is
uniform for all ' 2 V , so that there is indu
ed a unique arrow V ![A; ?℄ as required.
This shows that R is right adjoint to P .
Next we
laim that P R is naturally equivalent to the identity. This is equivalent to
showing any fun
tional uniform on R(V; V ) already belongs to V . But any fun
tional
': R(V; V ) ! ? extends by Corollary 4.4 to a fun
tional on ?V . From Corollary 4.5,
there is a nite set of fun
tionals '1 , : : : , 'n 2 V and a fun
tional : ?n ! ? su
h that
0
0
0
0
0
0
0
0
- ?V
A
'
?
?
0
?
?n
where the right hand arrow is the proje
tion on the
oordinates
orresponding to '1 , : : : ,
'n . If the
omponents of are 1 , : : : , n , then this says that ' = 1 '1 + + n 'n ,
whi
h is in V .
For an obje
t A of A it will be
onvenient to denote RP A by A. This is the underlying
V obje
t of A equipped with the weak uniformity for the fun
tionals on A.
De ne a homomorphism f : A ! B to be weakly uniform if the
omposite A ! B
! B is uniform. This is equivalent to the assumption that for every fun
tional ': B
0
! ?, the
omposite A f! B '! ? is a fun
tional on A. It is also equivalent to the
assumption that f : A ! B is uniform. Given A, let fA ! Ai j i 2 I g range over the
isomorphism
lasses of weakly uniform surje
tive homomorphisms out of A. De ne A as
the pullba
k in the diagram
- Q Ai
A
?
?
- Q A
i
A
If f : A ! B is weakly uniform, it fa
tors A !
! A B and the rst arrow is weakly
uniform sin
e every uniform fun
tional on A extends to a uniform fun
tional on B . Sin
e,
up to isomorphism, A !
! A is among the A !! Ai , it follows that f : A ! B is uniform.
Sin
e the identity is a weakly uniform surje
tion, the lower arrow in the square above is
a subspa
e in
lusion and hen
e so is the upper arrow. That implies that the lower arrow
in the diagram of fun
tionals
0
0
0
Q Ai; ?℄
- [A; ?℄
Q A?i; ?℄
- [ A;??℄
[
[
P
P
The left hand arrow is equivalent to [A ; ?℄ ! [A ; ?℄ (Corol-
Theory and Appli
ations of Categories, Vol. 6, No. 1
15
is a surje
tion.
i
i
lary 4.5), whi
h we have just seen is an isomorphism. Thus the right hand arrow is a
surje
tion, while it is evidently an inje
tion. This shows that A has the same fun
tionals as A. If A were a stri
tly ner uniformity than that of A on the same underlying
V stru
ture that had the same set of fun
tionals as A, then the identity A ! A would
be weakly uniform and then A ! A would be uniform, a
ontradi
tion. Thus if we
de ne L(V; V ) = R(V; V ) we know at least that P L
= Id so that L is full and faithful. If (f; f ): (V; V ) ! P A is a Chu morphism, then f : V ! jAj is a homomorphism
su
h that for ea
h uniform fun
tional ': A ! ? the
omposite ' f 2 V . Thus R(V; V )
! A is weakly uniform and hen
e L(V; V ) = R(V; V ) ! A is uniform. Conversely,
if f : L(V; V ) ! A is uniform, then for any fun
tional ': A ! ?, the
omposite ' f is
uniform on L(V; V ) and hen
e belongs to V so that we have (V; V ) ! P A.
b
0
0
0
b
b
0
0
0
0
0
0
0
0
0
weak uniformity weak
Ma
key uniformity Ma
key
We will say that an obje
t A with A = A has the
(or
) and that one for whi
h A = A has the
(or
). The latter name is taken from the theory of lo
ally
onvex topologi
al ve
tor
spa
es where a Ma
key topology is
hara
terized by the property of having the nest
topology with a given set of fun
tionals.
topology
topology
4.7. Ex
eptions. In verifying the hypotheses of Theorem 4.1, one notes that ea
h example
satis es simpler hypotheses. And ea
h simpler hypothesis is satis ed by most of the
examples. Most of the
ategories are additive (ex
eption: semilatti
es) and then we
an
use topologies instead of uniformities. In most
ases, the dualizing obje
t is dis
rete
(ex
eptions: abelian groups and real or
omplex ve
tor spa
es) and the existen
e of a
small entourage (or neighborhood of 0 in the additive
ase) is automati
. In most
ases,
the theory is
ommutative and the
losed monoidal stru
ture
omes from that (ex
eption:
modules over a Hopf algebra) so that the enri
hment of the uniform
ategory over the
base is automati
. Thus most of the examples are ex
eptional in some way (ex
eptions:
ve
tor spa
es over a eld and modules with a dualizing module), so that we may
on
lude
that they are all ex
eptional.
5. Ve
tor spa
es: the
ase of a dis
rete eld
The simplest example of the theory is that of ve
tor spa
es over a dis
rete eld. Let K be
a xed dis
rete eld. We let V be the
ategory of K -ve
tor spa
es with the usual
losed
monoidal stru
ture and let D be the dis
rete spa
es. Sin
e the
ategory is additive, we
an work with topologies rather than uniformities. We take the dualizing obje
t ? as the
eld K with the dis
rete topology.
The
onditions of Theorem 4.1 are all evident and so we
on
lude that the full sub
ategories of the
ategory of topologi
al K -ve
tor spa
es
onsisting of weakly topologized
spa
e and of Ma
key spa
es are -autonomous.
We note that in nite dimensional dis
rete spa
es do not have the weak uniformity. In
fa
t the weak uniformity asso
iated to the
hu spa
e (V; V Æ K ) is V with the uniform
Theory and Appli
ations of Categories, Vol. 6, No. 1
16
topology in whi
h the open subspa
es are the
o nite dimensional ones. Sin
e the 0
subspa
e is not
o nite dimensional, the spa
e is not dis
rete. On the other hand, the
map to the dis
rete V is weakly uniform and so the Ma
key spa
e asso
iated is dis
rete.
6. Dualizing modules
The
ase of a ve
tor spa
e over a dis
rete eld has one generalization, suggested by R.
Raphael (private
ommuni
ation). Let R be a
ommutative ring. Say that an R-module ?
is a dualizing module if it is a nitely generated inje
tive
ogenerator and the
anoni
al
map R ! HomR (?; ?) is an isomorphism. Let A be the
ategory of topologi
al (=
uniform) R-modules that are strongly
ogenerated by the dis
rete ones. Then taking
the small neighborhood to be f0g and D the
lass of dis
rete modules, the
onditions of
Theorem 4.1 are satis ed and we draw the same
on
lusion.
6.1. Existen
e of dualizing modules. Not every ring has a dualizing module. For example,
no nitely generated abelian group is inje
tive as a Z-module, so Z la
ks a dualizing
module. On the other hand, If R is a nite dimensional
ommutative algebra over a
eld K , then HomK (R; K ) is a dualizing module for K . It follows that any artinian
ommutative ring has a dualizing module:
6.2. Proposition. Suppose K is a
ommutative ring with a dualizing module D and R
is a
ommutative K -algebra nitely generated and proje
tive as a K -module. Then for
any nitely generated R-proje
tive P of
onstant rank one, the R-module HomK (P; D) is
a dualizing module for R.
It is standard that su
h a module is inje
tive. In fa
t, for an inje
tive homomorphism f : M ! N , we have that
Proof.
HomR (f; HomK (P; D))
= HomK (P
R
f; D
)
whi
h is surje
tive sin
e P is R-
at. Sin
e P is nitely generated proje
tive as an Rmodule, it is retra
t of a nite sum of
opies of R. Similarly, R is a retra
t of a nite sum
of
opies of K , when
e P is a retra
t of a nite sum of
opies of K . Then Hom(P; D) is
a retra
t of a nite sum of
opies of D and is thus nitely generated as a K -module, a
fortiori as an R-module. Next we note that a
onstantly rank one proje
tive P has endomorphism ring R. In fa
t, the
anoni
al R ! HomR (P; P ) lo
alizes to the isomorphism
RQ ! HomRQ (PQ ; PQ ) = HomRQ (RQ ; RQ ) whi
h is an isomorphism, at ea
h prime ideal
Q and hen
e is an isomorphism. Then we have that
HomR (HomK (P; D); HomK (P; D))
= HomK (P
= HomR(
= HomR(
P;
sin
e
D
R HomK (
)
P; D ; D
)
HomK (HomK (P; D); D))
P; P
)
=R
is a dualizing module for K and P is a nitely generated
K
module.
Theory and Appli
ations of Categories, Vol. 6, No. 1
17
Whether any non-artinian
ommutative ring has a dualizing module is an open question. For example, the produ
t of
ountably many elds does not appear to have a
dualizing module. The obvious
hoi
e would be the produ
t ring itself and, although it
is inje
tive, it is not a
ogenerator sin
e the quotient of the ring mod the ideal whi
h is
the dire
t sum is a module that is annihilated by every minimal idempotent so that the
quotient module has no non-zero homomorphism into the ring.
7. Ve
tor spa
es:
ase of the real or
omplex eld
We will treat the
ase of the
omplex eld. The real
ase is similar. We take for D the
lass
of Bana
h spa
es and the base eld C as dualizing obje
t. The D-
ogenerated obje
ts are
just the spa
es whose topology is given by seminorms. These are just the lo
ally
onvex
spa
es (see [Kelly, Namioka, 1963℄, 2.6.4). The
onditions of 4.1 follow immediately from
the Hahn-Bana
h theorem and we
on
lude that the
ategory
hu(V ; ?) is equivalent to
both full sub
ategories of weakly topologized and Ma
key spa
es and that both
ategories
are -autonomous. In parti
ular, the existen
e of the Ma
key topology follows quite easily
from this point of view.
We
an also give a relatively easy proof of the fa
t that the Ma
key topology is
onvergen
e on weakly
ompa
t,
onvex,
ir
led subsets of the dual. In fa
t, let A be a
lo
ally
onvex spa
e and A denote the weak dual. If f : A ! D is a weakly
ontinuous
map, then we have an indu
ed map, evidently
ontinuous in the weak topology, f : D
! A and one sees immediately that the weakly
ontinuous seminorm indu
ed on A
jj jj
by the
omposite A ! D
! R is simply the sup on f (C ), where C is the unit
ball of D , whi
h is
ompa
t in the weak topology. On the other hand, if C A is
ompa
t,
onvex, and
ir
led, let B be the linear subspa
e of A generated by C made
into a Bana
h spa
e with C as unit ball. With the topology indu
ed by that of C , so
that a morphism out of B is
ontinuous if and only if its restri
tion to C is, B be
omes
an obje
t of the
ategory C as des
ribed in [Barr, 1979℄, IV.3.10. This
ategory
onsists
of the mixed topology spa
es whose unit balls are
ompa
t. The dis
ussion in IV.3.16 of
the same referen
e then implies that every fun
tional on B is represented by an element
of B . This means that the indu
ed A ! B is weakly
ontinuous. But B is a Bana
h
spa
e whose norm is the absolute sup on C , as is the indu
ed seminorm on A.
f
8. Bana
h balls
A Bana
h ball is the unit ball of a Bana
h spa
e. The
on
lusions, but not the hypotheses
of Theorem 4.1 are valid in this
ase too. However, the proof is di erent and will appear
elsewhere [Barr, Kleisli, 1999℄. The proof given here of the existen
e of the right adjoint
and the Ma
key topology was rst found in this
ontext.
Theory and Appli
ations of Categories, Vol. 6, No. 1
18
9. Abelian groups
The
ategory of abelian groups is an example of the theory. For D we take the
lass of
lo
ally
ompa
t groups. The dualizing obje
t is, as usual, the
ir
le group, R=Z. Sin
e
the
ategory is additive, we
an deal with topologies instead of uniformities. A small
neighborhood of 0 is an open neighborhood of 0 that
ontains no non-zero subgroup
and for whi
h a homomorphism to the
ir
le is
ontinuous if and only if the inverse image
of that neighborhood is
ontinuous.
9.1. Proposition. The image U R=Z of the interval ( 1=3; 1=3) R is a small
neighborhood of 0.
Suppose x 6= 0 in U . Suppose, say, that x is in the image of a point in (0; 1=3).
Then the rst one of x, 2x, 4x, : : : , that is larger than 1=3 will be less than 2=3 1=3,
whi
h shows that U
ontains no non-zero subgroup.
It is
lear that the set of all 2 nU; n = 0; 1; 2; : : : is a neighborhood base at 0. Suppose
that f : A ! R=Z is a homomorphism su
h that V = f 1 (U ) is open in A. Let V0 = V
and indu
tively
hoose an open neighborhood Vn of 0 so that Vn Vn Vn 1 . Then one
easily sees by indu
tion that Vn f 1 (2 n U ).
The remaining
onditions of Theorem 4.1 are almost trivial, given Pontrjagin duality.
The only thing of note is that if D 2 D and A D then any
ontinuous homomorphism
': A ! R=Z
an rst be extended to the
losure of A, sin
e the
ir
le is
ompa
t and
hen
e
omplete in the uniformity. A
losed subgroup of a lo
ally
ompa
t group is lo
ally
ompa
t and the duality theory of lo
ally
ompa
t groups gives the extension to all of D.
9.2. Proposition. Lo
ally
ompa
t groups are Ma
key groups.
Proof. Sin
e all the groups in A are embedded in a produ
t of lo
ally
ompa
t groups,
it suÆ
es to know that a weakly
ontinuous map between lo
ally
ompa
t topologi
al
groups is
ontinuous. This is found in [Gli
ksberg, 1962℄.
Proof.
9.3. Other
hoi
es for D. One thing to note is that it is possible to
hoose a di erent
ategory A. The result
an be a di erent notion of Ma
key group. For instan
e, you
ould
hoose for A the subspa
es of
ompa
t spa
es. In that
ase weakly
ontinuous
oin
ides with
ontinuous and weak and Ma
key topologies
oin
ide. Another possibility
is to use
ompa
t and dis
rete spa
es. It is easy to see that the real line
annot be
embedded into a produ
t of
ompa
t and dis
rete spa
es. There are no non-zero maps
to a dis
rete spa
e, so it would have to embedded into a produ
t of
ompa
t spa
es. But
the real line is
omplete, so the only way it
ould be embedded into a produ
t of
ompa
t
spa
es would be if it were
ompa
t.
In the original monograph,
ountable sums of
opies of R were permitted in D. But
the sum of
ountably many
opies of R is not lo
ally
ompa
t. Here we show that we also
get a model of the theory by allowing D to
onsist of
ountable sums of lo
ally
ompa
t
groups. The only issue here is the inje
tivity of the
ir
le. So suppose A D, where
D = D1 D2 is a
ountable sum of lo
ally
ompa
t groups. As above, we
an suppose
Theory and Appli
ations of Categories, Vol. 6, No. 1
19
that A is
losed in D. Let Fn (D) = D1 Dn and Fn (A) = A \ Fn (D). Every element
of A is in some nite summand, so that, algebrai
ally at least, A =
olim Fn (A). Whether
it is topologi
ally is not important, sin
e we will show that every
ontinuous
hara
ter
on the
olimit extends to a
ontinuous
hara
ter on D. What does matter is that, by
de nition of the topology on the
ountable sum, D =
olim Fn (D) both algebrai
ally and
topologi
ally. The square
Fn 1(A)
Fn(A)
?-
- ?
?
- Fn(?D)
Fn 1(D)
is a pullba
k by de nition. There is no reason for it to be a pushout, but if we denote
the pushout by Pn , it is trivial diagram
hase to see that Pn ) ! Fn (D) is inje
tive. The
group Fn (D) is lo
ally
ompa
t and so, therefore, is the
losed subgroup Fn (A) and so is
the
losure Pn . Thus, taking Pontrjagin duals, all the arrows in the diagram below are
surje
tive and the square is a pullba
k:
Fn(D)
AAHHH
AA HHHHH
AA RRPn --HjHjFn 1(D)
AA
AAU
? -- ?
U
A
Fn(A)
Fn 1(A)
The surje
tivity of the arrow Fn (D) !
! Pn implies that ea
h square of
Fn+1 (D)
Fn (D)
Fn 1 (D)
--
--
--
--
? -- ? -- ? -- ? -- ?
Fn+1(A)
Fn(A)
Fn 1(A)
is a weak pullba
k. From this, it is a simple argument to see that the indu
ed arrow
lim Fn (D) !
! lim Fn(A) is surje
tive.
10. Modules over a
o
ommutative Hopf algebra
Modules over a Hopf algebra are not models of a
ommutative theory, unless the algebra
should also be
ommutative and, even so, the
losed stru
ture does not
ome from there.
Theory and Appli
ations of Categories, Vol. 6, No. 1
20
Thus we will have to verify dire
tly that the
ategory of the topologi
al algebras is enri
hed
over the
ategory of dis
rete algebras for the theory.
There are two important spe
ial
ases and we begin with brief des
riptions of them.
10.1. Group representations. Let G be a group and K be a eld. A K -representation of
G is a homomorphism of G into the group of automorphisms of some ve
tor spa
e over
K . Equivalently, it is the a
tion of G on a K -ve
tor spa
e. A third equivalen
e is with
a module over the group algebra K [G℄. The
ategory of K -representations of G is thus
an equational
ategory, that of the K [G℄-modules, but the theory is not
ommutative
unless G should be
ommutative and even in that
ase, we do not want that
losed
monoidal stru
ture. The one we want has as tensor produ
t of modules M and N the
K -tensor produ
t M
N = M
K N . The G-a
tion is the so-
alled diagonal a
tion
x(m
n) = xm
xn, x 2 G, extended linearly. The internal hom takes for M Æ N the
set of K -linear maps with a
tion given by (xf )m = x(f (x 1 m)) for x 2 G. This gives a
symmetri
losed monoidal stru
ture for whi
h the unit obje
t is K with trivial G a
tion,
meaning every element of G a
ts as the identity on K .
If M and N are topologi
al ve
tor spa
es with
ontinuous a
tion of G (whi
h is assumed dis
rete, at least here), then it is easily seen that the set of
ontinuous linear
transformations M ! N is a G-representation with the same de nition of a
tion and we
denote it by [M; N ℄ as before. Thus the
ategory of topologi
al (= uniform) G-modules is
enri
hed over the
ategory of G-modules. The
otensor is also easy. De ne AV as jAj Æ V
topologized as a subspa
e of AUV .
10.2. Lie algebras. Let K be a eld and g be a K -Lie algebra. A K -representation of g
is a Lie algebra homomorphism of g into the Lie algebra of endomorphisms of a K -ve
tor
spa
e V . In other words, for x 2 g and v 2 V , there is de ned a K -linear produ
t xv
in su
h a way that [x; y ℄v = x(yv ) y (xv ). If V and W are two su
h a
tions, there is
an a
tion on V
W = V
K W given by x(v
w) = xv
w + v
xw. The spa
e
V Æ W of K -linear transformations has an a
tion given by (xf )(v ) = x(f v ) f (xv ). If
g a
ts
ontinuously on topologi
al ve
tor spa
es V and W , then xf is
ontinuous when
f is so that the
ategory of topologi
al representations is enri
hed over the
ategory of
representations. The
otensor works in the same way as with the groups.
10.3. Modules over a
o
ommutative Hopf algebra. These two notions above
ome together in the notion of a module over a
o
ommutative Hopf algebra. Let K be a eld. A
over K is a K -algebra given by a multipli
ation : H
H
! H (all tensor produ
ts in this se
tion are over K ), a unit : K ! H , a
omultipli
ation
Æ : H ! H
H , a
ounit : H ! K and an involution : H ! H su
h that
o
ommutative Hopf algebra
HA{1. (H; ; ) is an asso
iative, unitary algebra;
HA{2. (H; Æ; ) is a
oasso
iative,
ounitary,
o
ommutative
oalgebra;
HA{3. Æ and are algebra homomorphisms; equivalently,
morphisms;
and
are
oalgebra homo-
Theory and Appli
ations of Categories, Vol. 6, No. 1
HA{4.
21
makes (H; ; ) into a group obje
t in the
ategory of
o
ommutative
oalgebras.
This last
ondition is equivalent to the
ommutativity of
H
Æ
Æ
?
H
-H
H
1
H
?
H
The leading examples of Hopf algebras are group algebras and the enveloping algebras
of Lie algebras. If G is a group, the group algebra K [G℄ is a Hopf algebra with Æ (x) = x
x,
(x) = 1 and (x) = x 1 , all for x 2 G and extended linearly. In the
ase of a Lie algebra
g, the de nitions are Æ (x) = 1
x + x
1, (x) = 0 and (x) = x, all for x 2 g.
10.4. The general
ase. Let H be a
o
ommutative Hopf algebra. By an H -module we
simply mean a module over the algebra part of H . If M and N are modules, we de ne
M
N to be the tensor produ
t over K with H a
tion given by the
omposite
H
M
N
Æ
1
1! H
H
M
N ! H
M
H
N ! M
N
The se
ond arrow is the symmetry isomorphism of the tensor and the third is simply
the two a
tions. We de ne M Æ N to be the set of K -linear arrows with the a
tion
H
(M Æ N ) ! M Æ N the transpose of the arrow H
M
(M Æ N ) ! N given by
H
M
(M Æ N )
1
1! H
H
M
(M Æ N )
1
1
1! H
H
M
(M Æ N )
Æ
! H
M
(M Æ N ) ! H
N ! N
The third arrow is the a
tion of H on M , the fourth is evaluation and the fth is the
a
tion of H on N .
That this gives an autonomous
ategory
an be shown by a long diagram
hase. The
tensor unit is the eld with the a
tion xa = (x)a.
We have to show that the
ategory of topologi
al modules is enri
hed over the
ategory
of modules. We
an des
ribe the enri
hed stru
ture as
onsisting of the
ontinuous linear
maps with the module stru
ture given as before, that is by
onjugation. The
ontinuity
of the module stru
ture guarantees that the a
tion preserves
ontinuity. From then on
the argument is the same. The dualizing obje
t is the dis
rete eld K whi
h has a small
neighborhood and the rest of the argument is the same. The
otensor is just as in the
ase of group representations.
The
lass D
onsists of the dis
rete obje
ts. The dualizing obje
t is the tensor unit.
Sin
e the internal hom is just that of the ve
tor spa
es, the
onditions of Theorem 4.1 are
easy.
Theory and Appli
ations of Categories, Vol. 6, No. 1
22
11. Semilatti
es
By semilatti
e we will mean inf semilatti
e, whi
h is a partially ordered set in whi
h
every nite set of elements has an inf. It is obviously suÆ
ient that there be a top element
and that every pair of elements have an inf. The
ategory is obviously equivalent to that
of sup semilatti
es, sin
e you
an turn the one upside down to get the other. The
ategory
is equational having a single
onstant, 1 (the top element) and a single binary operation ^
whi
h is unitary (with respe
t to 1),
ommutative, asso
iative and idempotent. (In fa
t,
sup semilatti
es have exa
tly the same des
ription|it all depends on how you interpret
the operations.)
Semilatti
es do not form an additive
ategory, but they are half-additive sin
e they
are
ommutative monoids. The dualizing obje
t is the 2 element
hain with the dis
rete
uniformity, whi
h evidently has a small entourage. Sin
e it is the tensor unit,
ondition 4
of Theorem 4.1 is satis ed. For D, we take the
lass of dis
rete latti
es. We need show
only
onditions 5 and 6.
Suppose we have an in
lusion L1 L2 of dis
rete semilatti
es and f : L1 ! ? is a
semilatti
e homomorphism. We will show that if x 2 L2 L1 , then f
an be extended to
the semilatti
e generated by L1 and x. This semilatti
e is L1 [ (L1 ^ x). We rst de ne
f x = 1 unless there are elements a, b 2 L1 su
h that f a = 1, f b = 0 and a ^ x b in whi
h
ase we de ne f x = 0. Then we de ne f (a ^ x) = f a ^ f x for any a 2 L1 . The only thing
we have to worry about is if a ^ x 2 L1 for some a 2 L1 . If f a = 0, then f (a ^ x) f a
so that f (a ^ x) = 0 = f a ^ f x. If f a = 1, then either f (a ^ x) = 1 or taking b = a ^ x
we satisfy the
ondition for de ning f x = 0 and then 0 = f (a ^ x) = f a ^ f x as required.
The rest of the argument is a routine appli
ation of Zorn's lemma. This
ompletes the
proof of 5. Now 6 follows immediately sin
e given any two elements of a dis
rete latti
e,
they generate a sublatti
e of at most 4 four elements and it is easy to nd a separating
fun
tional on that sublatti
e.
12. The
ategory of
Æ -obje
ts
We will very brie
y explain why the
ategories of Æ -obje
ts
onsidered in [Barr, 1979℄
is also -autonomous. Of
ourse, it is likely more interesting that the larger
ategories
onstru
ted here are -autonomous, but in the interests of re
overing all the results from
the monograph, we in
lude it.
An obje
t T is
alled -
omplete if it is inje
tive with respe
t to dense subobje
ts of
ompa
t obje
ts. That is, if in any diagram
C0
?
T
-C
Theory and Appli
ations of Categories, Vol. 6, No. 1
23
with C
ompa
t and C0 a dense subobje
t,
an be
ompleted by an arrow C ! T . The
obje
t T is -
omplete if T is -
omplete. An obje
t is
alled a Æ -obje
t if is -
omplete,
-
omplete and re
exive.
12.1.
Theorem. The full sub
ategories of
Æ -obje
ts
are
-autonomous sub
ategories the
ategories of Ma
key obje
ts.
The proof uses one property that we will not verify. Namely that all
ompa
t
obje
ts are Æ -obje
ts. The duals of the
ompa
t obje
ts are
omplete (in most
ases
dis
rete). For an obje
t T , we de ne T as the interse
tion of all the -
omplete subobje
ts
of the
ompletion of T . The
ru
ial
laim is that if T is -
omplete, so is (T ) . In fa
t,
the adjun
tion arrow T ! T gives an arrow (T ) ! T
= T . Now
onsider a
diagram
Proof.
C0
?
-C
-
(T )
T
Sin
e T is -
omplete, there is an arrow C ! T that makes the square
ommute. This
gives T ! C and, sin
e C is
omplete, T ! C , and then C
= C !(T ) , as
required. We now invoke Theorem 2.3 of [Barr, 1996℄ to
on
lude that the Æ -obje
ts form
a -autonomous
ategory.
Referen
es
M. Barr (1991),
M. Barr (1979),
-Autonomous Categories
. Le
ture Notes in Mathemati
s
-Autonomous
ategories and linear logi
Computer S
ien
e 1, 159{178.
.
752
. Mathemati
al Stru
tures in
M. Barr (1996), -Autonomous
ategories, revisited. J. Pure Applied Algebra, 111, 1{20.
M. Barr (1998), The separated
Categories, 4.6, 137{147.
extensional Chu
ategory
. Theory and Appli
ations of
M. Barr and H. Kleisli (1999), Topologi
al balls. To appear in Cahiers de Topologie et
Geometrie Differentielle Categorique, 40.
P.-H. Chu (1979),
Constru
ting
P.J. Freyd, A. S
edrov (1990),
I. Gli
ksberg (1962),
V. Gupta (1994),
University.
. Appendix to [Barr, 1979℄.
-autonomous
ategories
Categories, Allegories
Uniform boundedness for groups
. North-Holland.
. Canadian J. Math. 14, 269-276.
Chu Spa
es: A Model of Con
urren
y.
Ph.D. Thesis, Stanford
Theory and Appli
ations of Categories, Vol. 6, No. 1
24
J.L. Kelley, I. Namioka (and
oauthors) (1963),
Nostrand.
Linear Topologi
al Spa
es.
Van
F.E.J. Linton (1966), Autonomous equational
ategories. J. Math. Me
h. 15, 637{642.
V.R. Pratt (1993a), Linear Logi
for Generalized Quantum Me
hani
s. In Pro
eedings
Workshop on Physi
s and Computation, Dallas, IEEE Computer So
iety.
V.R. Pratt (1993b), The Se
ond Cal
ulus
MFCS'93, Gda
nsk, Poland, 142{155.
of Binary Relations.
In Pro
eedings of
V.R. Pratt (1995), The Stone Gamut: A Coordinatization of Mathemati
s. In Pro
eedings
of the
onferen
e Logi
in Computer S
ien
e IEEE Computer So
iety.
Dept. of Math. and Stats.
M
Gill University
805 Sherbrooke St. W
Montreal, QC H3A 2K6
Email:
barrmath.m
gill.
a
This arti
le may be a
essed via WWW at http://www.ta
.mta.
a/ta
/ or by
anonymous ftp at ftp://ftp
ations of Categories,
Vol. 6, No. 1, pp. 5{24.
-AUTONOMOUS
CATEGORIES:
ONCE MORE AROUND THE TRACK
To Jim Lambek on the o
asion of his 75th birthday
MICHAEL BARR
ABSTRACT. This represents a new and more
omprehensive approa
h to the autonomous
ategories
onstru
ted in the monograph [Barr, 1979℄. The main tool in
the new approa
h is the Chu
onstru
tion. The main
on
lusion is that the
ategory of
separated extensional Chu obje
ts for
ertain kinds of equational
ategories is equivalent to two usually distin
t sub
ategories of the
ategories of uniform algebras of those
ategories.
1. Introdu
tion
The monograph [Barr, 1979℄ was devoted to the investigation of -autonomous
ategories.
Most of the book was devoted to the dis
overy of -autonomous
ategories as full sub
ategories of seven di erent
ategories of uniform or topologi
al algebras over
on
rete
ategories that were either equational or re
e
tive sub
ategories of equational
ategories.
The base
ategories were:
1. ve
tor spa
es over a dis
rete eld;
2. ve
tor spa
es over the real or
omplex numbers;
3. modules over a ring with a dualizing module;
4. abelian groups;
5. modules over a
o
ommutative Hopf algebra;
6. sup semilatti
es;
7. Bana
h balls.
For de nitions of the ones that are not familiar, see the individual se
tions below.
These
ategories have a number of properties in
ommon as well as some important
di eren
es. First, there are already known partial dualities, often involving topology.
This resear
h has been supported by grants from the NSERC of Canada and the FCAR du Quebe
Re
eived by the editors 1998 November 20 and, in revised form, 1999 August 26.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18D10, 43A40, 46A70, 51A10.
Key words and phrases: duality, topologi
al algebras, Chu
ategories.
Mi
hael Barr 1999. Permission to
opy for private use granted.
5
Theory and Appli
ations of Categories, Vol. 6, No. 1
6
It is these partial dualities that we wish to extend. Se
ond, all are symmetri
losed
monoidal
ategories. All but one are
ategories of models of a
ommutative theory and
get their
losed monoidal stru
ture from that (see 3.7 below). The theory of Bana
h balls
is really di erent from rst six and is treated in detail in [Barr, Kleisli, to appear℄.
What we do here is provide a uniform treatment of the rst six examples. We show that
in ea
h
ase, there is a -autonomous
ategory of uniform spa
e models of the theory. In
most
ases, this is equivalent to the topologi
al spa
e models. The main tool used here is
the so-
alled Chu
onstru
tion as des
ribed in an appendix to the 1979 monograph, [Chu,
1979℄. He des
ribed in detail a very general
onstru
tion of a large
lass of -autonomous
ategories. He starts with any symmetri
monoidal
ategory V and any obje
t ? therein
hosen as dualizing obje
t to produ
e a -autonomous
ategory denoted Chu(V ; ?). The
simpli
ity and generality of this
onstru
tion made it appear at the time unlikely that it
ould have any real interest beyond its original purpose, namely showing that there was
a plenitude of -autonomous
ategories. We des
ribe this
onstru
tion in Se
tion 2.
A preliminary attempt to
arry out this approa
h using the Chu
onstru
tion appeared
in [Barr, 1996℄, limited to only two of the seven example
ategories (ve
tor spa
es over
a dis
rete eld and abelian groups). The arguments there were still very ad ho
and
depended on detailed properties of the two
ategories in question. In this arti
le, we
prove a very general theorem that appeals to very few spe
ial properties of the examples.
In 1987 I dis
overed that models of Jean-Yves Girard's linear logi
were -autonomous
ategories. Within a few years, Vaughan Pratt and his students had found out about the
Chu
onstru
tion and were studying its properties intensively ([Pratt, 1993a,b, 1995,
Gupta, 1994℄). One thing that espe
ially stru
k me was Pratt's elegant, but essentially
obvious, observation that the
ategory of topologi
al spa
es
an be embedded fully into
Chu(Set; 2) (see 2.2). The real signi
an
e|at least to me|of this observation is that
putting a Chu stru
ture on a set
an be viewed as a kind of generalized topology.
A reader who is not familiar with the Chu
onstru
tion is advised at this point to read
Se
tion 2. Thinking of a Chu stru
ture as a generalized topology leads to an interesting
idea whi
h I will illustrate in the
ase of topologi
al abelian groups. If T is an abelian group
(or, for that matter, a set), a topology is given by a
olle
tion of fun
tions from the point
set of T to the Sierpinski spa
e|the spa
e with two points, one open and the other not.
From a
ategori
al point of view, might it not make more sense to repla
e the fun
tions to a
set by group homomorphisms to a standard topologi
al group, thus
reating a de nition
of topologi
al group that was truly intrinsi
to the
ategory of groups. If, for abelian
groups, we take this \standard group" to be the
ir
le group R=Z, the resultant
ategory
is (for separated groups) a
ertain full sub
ategory of Chu(Ab; R=Z)
alled
hu(Ab; R=Z).
This
ategory is not the
ategory of topologi
al abelian groups. Nonetheless the
ategory
of topologi
al abelian groups has an obvious fun
tor into
hu(Ab; R=Z) and this fun
tor
has both a left and a right adjoint, ea
h of whi
h is full and faithful. Thus the
ategory
hu(Ab=; R=Z) is equivalent to two distin
t two full sub
ategories of abelian groups, ea
h
of whi
h is thereby -autonomous. In fa
t, any topologi
al abelian group that
an be
embedded algebrai
ally and topologi
ally into a produ
t of lo
ally
ompa
t groups has
Theory and Appli
ations of Categories, Vol. 6, No. 1
7
both a nest and a
oarsest topology that indu
e the same set of
hara
ters. The two
sub
ategories
onsist of all those that have the nest topology and those that have the
oarsest. These are the images of the left and right adjoint, respe
tively.
1.1. A
knowledgment. I would like to thank Heinri
h Kleisli with whom I had a many
helpful and informative dis
ussions on many aspe
ts of this work during one month visits
to the Universite de Fribourg during the springs of 1997 and 1998. In parti
ular, the
orre
t proof of the existen
e of the Ma
key uniformity (or topology) was worked out
there in the
ontext of the
ategory of balls, see [Barr, Kleisli, 1999℄.
2. The Chu
onstru
tion
There are many referen
es to the Chu
onstru
tion, going ba
k to [Chu, 1979℄, but see
also [Barr, 1991℄, for example. In order to make this paper self-
ontained, we will give
a brief des
ription here. We sti
k to the symmetri
version, although there are also
non-symmetri
variations.
2.1. The
ategory Chu(V ; ?). Suppose that V is a symmetri
losed monoidal
ategory
and ? is a xed obje
t of V . An obje
t of Chu(V ; ?) is a pair (V; V ) of obje
ts of V
together with a homomorphism,
alled a pairing, h ; i: V
V ! ?. A morphism
(f; f ): (V; V ) !(W; W )
onsists of a pair of arrows f : V ! W and f : W ! V in V
that satisfy the symboli
identity hf v; w i = hv; f w i. Diagrammati
ally, this
an be
expressed as the
ommutativity of the diagram
0
0
0
0
0
0
0
V
f
W
W
0
W
?
W
0
0
V
f
0
V
h
;
0
0
V
h
0
0
?
?
i
0
i
;
Using the transposes V ! V Æ ? and V ! V Æ ? of the stru
ture maps, this
ondition
an be expressed as the
ommutativity of either of the squares
0
f
V
V
0
?
Æ?
f
0
Æ?
W
0
W
0
?
Æ?
W
W
f
0
?
Æ?
f
0
-
V
0
?
Æ?
Æ?
V
Theory and Appli
ations of Categories, Vol. 6, No. 1
8
A nal formulation of the
ompatibility
ondition is that
- Hom(
Hom((V; V 0 ); (W; W 0 ))
V; W
?
- Hom(
?
Hom(W 0 ; V 0 )
V
0
)
W ;
?)
is a pullba
k.
The internal hom is gotten by using an internalization of the last formulation. De ne
the obje
t [(V; V 0 ); (W; W 0 )℄ of V as a pullba
k
-
[(V; V 0 ); (W; W 0 )℄
W
0
?
Æ
V
-
0
V
V
Æ
W
?0
W
Æ?
and then de ne
(V; V 0 ) Æ(W; W 0 ) = ([(V; V 0 ); (W; W 0 )℄; V
W 0 )
Sin
e the dual of (V; V 0 ) is (V 0 ; V ), it follows from the interde nability of tensor and
internal hom in a -autonomous
ategory that the tensor produ
t is
(V; V 0 )
(W; W 0 ) = (V
W;
[(V; V 0 ); (W 0; W )℄)
The result is a -autonomous
ategory. See [Barr, 1991℄ for details.
2.2. The
ategory Chu(Set; 2). An obje
t of Chu(Set; 2) is a pair (S; S 0 ) together with
a fun
tion S S 0 ! 2. This is equivalent to a fun
tion S 0 ! 2S . When this fun
tion
is inje
tive we say that (S; S 0 ) is extensional and then S 0 is, up to isomorphism, a set of
subsets of S . Moreover, one easily sees that if (S; S 0 ) and (T ; T 0 ) are both extensional,
then a fun
tion f : S ! T is the rst
omponent of some (f; f 0 ): (S; S 0 ) !(T ; T 0 ) if and
only if U 2 T 0 implies f 1 (U ) 2 S 0 and then f 0 = f 1 is uniquely determined. This
explains Pratt's full embedding of topologi
al spa
es into Chu(Set; 2).
2.3. The
ategory
hu(V ; ?). Suppose V is a
losed symmetri
monoidal
ategory as above
and suppose there is a fa
torization system E =M on V . (See [Barr, 1998℄ for a primer on
fa
torization systems.) In general we suppose that the arrows in E are epimorphisms and
that those in M are
ompatible with the internal hom in the sense that if V ! V 0 belongs
to M, then for any obje
t W , the indu
ed W Æ V ! W Æ V 0 also belongs to M. In
all the examples here, E
onsists of the surje
tions (regular epimorphisms) and M of the
inje
tions (monomorphisms), for whi
h these
onditions are automati
. An obje
t (V; V 0 )
of the Chu
ategory is said to be M-separated, or simply separated, if the transpose V
! V 0 Æ ? is in M and M-extensional, or simply extensional, if the other transpose
0 ! V Æ ? is in M. We denote by Chus(V ; ?), Chue (V ; ?), and Chuse (V ; ?) the
V
Theory and Appli
ations of Categories, Vol. 6, No. 1
9
full sub
ategories of separated, extensional, and separated and extensional, respe
tively,
obje
ts of Chu(V ; ?). Following Pratt, we usually denote the last of these by
hu(V ; ?).
The relevant fa
ts are
1. The full sub
ategory Chus(V ; ?) of separated obje
ts is a re
e
tive sub
ategory of
Chu(V ; ?) with re
e
tor s.
2. The full sub
ategory Chue (V ; ?) of extensional obje
ts is a
ore
e
tive sub
ategory
of Chu(V ; ?) with
ore
e
tor e.
3. If (V; V 0 ) is separated, so is e(V; V 0 ); if (V; V 0 ) is extensional, so is s(V; V 0 ).
4. Therefore
hu(V ; ?) is both a re
e
tive sub
ategory of the extensional
ategory and
a
ore
e
tive sub
ategory of the separated sub
ategory. It is, in parti
ular,
omplete
and
o
omplete.
5. The tensor produ
t of extensional obje
ts is extensional and the internal hom of an
extensional obje
t into a separated obje
t is separated.
6. Therefore by using s(
) as tensor produ
t and r(
ategory
hu(V ; ?) is -autonomous.
Æ
) as internal hom, the
For details, see [Barr, 1998℄.
3. Topologi
al and uniform spa
e ob je
ts
3.1. Topology and duality. In a -autonomous
ategory we have, for any obje
t A, that
Æ A = A Æ > . If we denote > by ?, we see that the duality has the form
A = >
A 7! A
Æ ?. The obje
t ? is
alled the dualizing obje
t and, as we will see, the way
(or at least a way) of
reating a duality is by nding a dualizing obje
t in some
losed
monoidal
ategory.
In order that a
ategory have a duality realized by an internal hom, there has to be a
way of
onstraining the maps so that the dual of a produ
t is a sum. In an additive
ategory, for example, this happens without
onstraint for nite produ
ts, but not normally
for in nite ones. A natural
onstraint is topologi
al. If, for example, the dualizing obje
t
is nite dis
rete, then any
ontinuous map from a produ
t
an depend on only nite many
of the
oordinates.
For example, even for ordinary
Q
Q topologi
al spa
es, for a
ontinuous
fun
tion f : Xi ! 2, f 1 (0) has the form
Y
i2
= J Xi where J is a nite
Q
Q subset of I
1
and Y Q
is a subset
of the nite produ
t i2J Xi . But then f (1) = Z 0 i2= J X0i , where
Z =
Xi
Y . If two elements of the produ
t x = (xi )i2I and x = (xi )i2I are
i2J
elements su
h that xi = x0i for i 2 J then either x 2 Y and x0 2 Y or x 2 Z and x0 2 Z ,
but in either
ase f x = f x0 . Thus f depends Q
only the
oordinates belonging to J , whi
h
means f fa
tors through the nite produ
t i2J Xi . A similar argument works if 2 is
repla
ed by any nite dis
rete set.
Theory and Appli
ations of Categories, Vol. 6, No. 1
10
3.2. Uniformity. Despite the examples above there are reasons for thinking that the te
hni
ally \
orre
t" approa
h to duality is through the use of uniform stru
tures. A very
readable and informative a
ount of uniform spa
es is in [Isbell, 1964℄. However, Isbell uses uniform
overs as his main de nition. Normally I prefer uniform
overs to the
approa
h using entourages, but for our purposes here entourages are more appropriate.
For any equational
ategory, a uniform spa
e obje
t is an obje
t of the
ategory
equipped with a uniformity for whi
h the operations of the theory are uniform. A morphism of uniform spa
e obje
ts is a uniform fun
tion that is also a morphism of models
of the theory. Topologi
al spa
e obje
ts are de ned similarly. Any uniformity leads to
a topologi
al spa
e obje
t in a
anoni
al way and uniform fun
tions be
ome
ontinuous
fun
tions for that
anoni
al topology. But not every topology
omes from a uniformity
and, if it does, it is not ne
essarily from a unique one. For example, the metri
on the
spa
e of integers and on the set of re
ipro
als of integers both give the dis
rete topology, but the asso
iated uniformities are quite di erent. (Metri
spa
es have a
anoni
al
uniformity. See Isbell's book for details.)
If, however, there is an abelian group stru
ture among the operations of an equational
theory, there is a
anoni
al uniformity asso
iated to every topology. Namely, for ea
h
open neighborhood U of the group identity take f(x; y ) j xy 1 2 U g as an entourage.
Moreover, a homomorphism of the algebrai
stru
ture between uniform spa
e obje
ts is
ontinuous if and only if it is uniform. Thus there is no di eren
e, in su
h
ases, between
ategories of uniform and topologi
al spa
e obje
ts. Obviously, the
ategory of topologi
al
spa
e obje
ts is more familiar. However, one of our
ategories, semilatti
es, does not have
an abelian group stru
ture and for that reason, we have
ast our main theorem in terms
of uniform stru
ture. There is another, less important, reason. At one point, in dealing
with topologi
al abelian groups, it be
omes important that the
ir
le group is
omplete
and
ompleteness is a uniform, not topologi
al, notion.
If V is an equational
ategory, we denote by Un(V ), the
ategory of uniform obje
ts
of V . We let j j: Un(V ) ! V to be fun
tor that forgets the uniform stru
ture.
3.3. Small entourage. Let A be a uniform V obje
t. An entourage E A A is
alled a
small entourage if it
ontains no subobje
t of A A that properly
ontains the diagonal
and if any homomorphism f : B ! A of uniform V obje
ts for whi
h (f f ) 1 (E ) is an
entourage of B is uniform.
3.4.. In all the examples we will be
onsidering, there will be a given
lass of uniform
obje
ts D and we will be dealing with the full sub
ategory A of Un(V )
onsisting of those
obje
ts strongly
ogenerated by D, whi
h is to say that those that
an algebrai
ally
and uniformly embedded into a produ
t of obje
ts of D.
3.5. Half-additive
ategories. A
ategory is
alled half-additive if its hom fun
tor fa
tors
through the
ategory of
ommutative monoids. It is well known that in any su
h
ategory
nite sums are also produ
ts (see, for example, [Freyd, S
edrov, 1990℄, 1.59). Of
ourse,
additive
ategories are half-additive. Of the six
ategories
onsidered here, ve are additive
and one, semilatti
es, is not additive, but is still half-additive. In fa
t, a semilatti
e is a
Theory and Appli
ations of Categories, Vol. 6, No. 1
11
ommutative monoid in whi
h every element is idempotent. This monoid stru
ture
an
be equally well viewed as sup or inf.
3.6. The
losed monoidal stru
ture. The
ategories we are dealing with are all symmetri
losed monoidal. With one ex
eption, the
losed stru
ture derives from their being models
of a
ommutative theory.
3.7. Commutative theories. A
ommutative theory is an equational theory whose operations are homomorphisms ([Linton, 1966℄). Thus in any abelian group G, as
ontrasted
with a non-abelian group, the multipli
ation G G ! G is an abelian group homomorphism, as are all the other operations.
3.8.
Theorem.
[Linton℄ The
ategory of models of a
ommutative theory has a
anoni
al
stru
ture of a symmetri
losed monoidal
ategory.
Suppose V is the
ategory of models and U : V ! Set is the underlying set
fun
tor with left adjoint F . If V and W are obje
ts of V , then W Æ V is a subset of V UW
de ned as the simultaneous equalizer, taken over all operations ! of the theory, of
Proof.
- (V n) UW
V UW
(
)n
! UW
R ?
UW
V!
(
V(
)n
)n
Here n is the arity of ! and the top arrow is raising to the nth power. Sin
e the theory is
ommutative, ! is a homomorphism and so the equalizer is a limit of a diagram in V and
hen
e lies in V . In parti
ular, the internal hom of two obje
ts of V
ertainly lies in V . The
free obje
t on one generator is the unit for this internal hom. As for the tensor produ
t,
V
W is
onstru
ted as a quotient of F (UV UW ), similar to the usual
onstru
tion of
the tensor produ
t of two abelian groups.
3.9.
Proposition. If
A
and
B
are obje
ts of an equational
ategory
V
equipped with
uniformities for whi
h their operations are uniform, then the set of uniform morphisms
from
A
!B
enri
hed over
is a subobje
t of
V
j A j Æ jB j
and thus the
ategory of uniform
. It also has tensors and
otensors from
V
V
obje
ts is
.
Let [A; B ℄ denote the set of uniform homomorphisms from A to B . For ea
h nary operation ! , the arrow !B : B n ! B is a uniform homomorphism and hen
e an arrow
[A; B ℄n
= [A; B n ℄ ![A; B ℄ is indu
ed by !B and we de ne this as ! [A; B ℄. This presents
[A; B ℄ as a subobje
t of jAj Æ jB j so that it also satis es all the equations of the theory
and is thus an obje
t of V . The
otensor AV is given by the obje
t V Æ jAj equipped
with the uniformity indu
ed by AUV . The tensor is
onstru
ted using the adjoint fun
tor
theorem with all uniformities on V
jAj as solution set.
Proof.
Theory and Appli
ations of Categories, Vol. 6, No. 1
12
4. The main theorem
We are now ready to state our main theorem.
4.1.
Suppose V is an equational
ategory equipped with a
losed monoidal
stru
ture, D is a
lass of uniform spa
e obje
ts of V and A is the full sub
ategory of the
ategory of uniform spa
e obje
ts of V that is strongly
ogenerated by D. Suppose that ?
is an obje
t of A with the following properties
1. V is half-additive.
2. D is
losed under nite produ
ts.
3. ? has a small entourage.
4. The natural map > ![?; ?℄ is an isomorphism.
5. If D is an obje
t of D, A D is a subobje
t, then the indu
ed arrow [D; ?℄ ![A; ?℄
Theorem.
is surje
tive.
6. For every obje
t D of D, the natural evaluation map D ! ?
7. A is enri
hed over V and has
otensors from V .
[D;
?℄ is inje
tive.
Then, using the regular-epi
/moni
fa
torization system, the
anoni
al fun
tor P : A
!
hu(V ; j?j) de ned by P (A) = (jAj; [A; ?℄) has a right adjoint R and a left adjoint L,
ea
h of whi
h is full and faithful.
4.2.. Before beginning the proof, we make some observations. We will
all a morphism A
! ? a fun
tional on A. In light of
ondition 5,
ondition 6 need be veri ed only for
obje
ts that are algebrai
ally 2-generated (and in the additive
ase, only for those that
are 1-generated) sin
e any separating fun
tional
an be extended to all of D.
In all our examples, ? is
omplete and a
losed subobje
t of an obje
t of D belongs
to D so that it is suÆ
ient to verify
ondition 5 when A belongs to D.
The
on
lusion of the theorem implies that the full images of both R and L are
equivalent to
hu(V ; ?) and hen
e both image
ategories are -autonomous.
The diagonal of A in A A will be denoted . We begin the proof with a lemma.
Q
4.3.
Suppose that A 2 A and ': A ! ? is a uniform fun
tional. Then
Q
Q
there is a nite subset J I su
h that if Ae is the image of A ! 2 A ! 2 A with
'e
the subspa
e uniformity, then ' fa
tors as A ! Ae ! ?.
A
Lemma.
i
I
i
i
I
i
i
J
i
Theory and Appli
ations of Categories, Vol. 6, No. 1
13
Let E ? ? be a small entourage. The de nition Q
of the produ
t uniformity
implies that there is a nite subset J I su
h that if we let B = i2J Ai and C = Qi2=J Ai,
then there is an entourage F B B for whi
h (A A) \ (F (C C )) (' ') 1(E ).
But then (A A) \ (B (C C )) is a subobje
t of A A that is in
luded in (' ') 1 (E ).
This implies that (' ')((A A) \ (B (C C ))) is a subobje
t of ?? lying between
? and E , whi
h is then ?. In parti
ular, if a = (ai) and a0 = (a0i ) are two elements of A
su
h that ai = a0i for all i 2 J , then '(a) = '(a0). Thus, ignoring the uniform stru
ture, we
an fa
tor ' via an algebrai
homomorphism 'e: Ae ! ?. But (Ae Ae) \ F ('e 'e) 1(E )
whi
h means that 'e is uniform in the indu
ed uniformity on Ae.
4.4.
AB A
[B; ?℄ ![A; ?℄
Q
Sin
e there is an embedding
B
i2I Di with Di obje
ts of D , it is suÆ
ient
Q
to do this in the
ase that B = i2I Di . The lemma says Qthat any fun
tional in [A; ?℄
fa
tors as A ! Ae ! ? where, for some nite J I , Ae i2J Di. The latter obje
t is
in D by
ondition 2 and the map extends to it by
ondition 5.
P
4.5. Q
fAi j i 2 I g
A
i2I [Ai ; ?℄
![ i2I Ai; ?℄
By taking A = Qi2I Ai in the lemma, we see that every fun
tional
Q on the
produ
t
Q fa
tors through a nite produ
t. That is, the
anoni
al map
olimJ I [ i2J Ai ; ?℄
![ i2I Ai; ?℄ is an isomorphism, where the
olimit is taken over the nite subsets J I .
On the other hand, half-additivity implies that the
anoni
al map from a nite sum to
nite produ
t is an isomorphism. Putting these together, we
on
lude that
Proof.
Corollary. For any
in
, the indu
ed
is surje
tive.
Proof.
Corollary. For any set
of obje
ts of
, the
anoni
al map
is an isomorphism.
Proof.
X
X
Y
i2 I
i2J
i2J
[Ai ; ?℄ =
olim
[Ai; ?℄ =
olim
[Ai; ?℄ =
olim
J I
J I
J I
=
olim
J I
"
Y
i2J
#
Ai ; ?
=
"
Y
i2I
#
"
X
i2 J
#
Ai ; ?
Ai ; ?
4.6. Proof of the theorem. The right adjoint to P is de ned as follows. If (V; V 0 ) is
an obje
t of
hu(V ; ?), then by de nition of separated V ! V Æ V 0 is moni
. The
underlying fun
tor from the
ategory of uniform obje
ts to V has a left adjoint and hen
e
preserves moni
s so that jV j ![V 0 ; ?℄ is also moni
. Sin
e the latter is a subset of
j?jjV j, we have that jV j j?jV jj. Then we let R(V; V 0 ) denote jV j, equipped with the
uniformity indu
ed as a uniform subspa
e of ?jV j. Also denote by (V; V 0) the uniformity
of R(V; V 0 ). This is the
oarsest uniformity on V for whi
h all the fun
tionals in V 0 are
uniform. A morphism P A !(V; V 0 )
onsists of an arrow f : jAj ! V in V su
h that
for any ' 2 V 0 the
omposite ' f is uniform. This means that the
omposite A ! V
! ?U V is uniform and hen
e that A ! R(V; V 0 ) is. Conversely, if f : A ! R(V; V 0 ) is
given, then we have f : jAj ! V su
h that the
omposite A ! V ! ?UV is uniform and
0
0
0
0
0
Theory and Appli
ations of Categories, Vol. 6, No. 1
14
if we follow it by the
oordinate proje
tion
orresponding to ' 2 V , we get that ' f is
uniform for all ' 2 V , so that there is indu
ed a unique arrow V ![A; ?℄ as required.
This shows that R is right adjoint to P .
Next we
laim that P R is naturally equivalent to the identity. This is equivalent to
showing any fun
tional uniform on R(V; V ) already belongs to V . But any fun
tional
': R(V; V ) ! ? extends by Corollary 4.4 to a fun
tional on ?V . From Corollary 4.5,
there is a nite set of fun
tionals '1 , : : : , 'n 2 V and a fun
tional : ?n ! ? su
h that
0
0
0
0
0
0
0
0
- ?V
A
'
?
?
0
?
?n
where the right hand arrow is the proje
tion on the
oordinates
orresponding to '1 , : : : ,
'n . If the
omponents of are 1 , : : : , n , then this says that ' = 1 '1 + + n 'n ,
whi
h is in V .
For an obje
t A of A it will be
onvenient to denote RP A by A. This is the underlying
V obje
t of A equipped with the weak uniformity for the fun
tionals on A.
De ne a homomorphism f : A ! B to be weakly uniform if the
omposite A ! B
! B is uniform. This is equivalent to the assumption that for every fun
tional ': B
0
! ?, the
omposite A f! B '! ? is a fun
tional on A. It is also equivalent to the
assumption that f : A ! B is uniform. Given A, let fA ! Ai j i 2 I g range over the
isomorphism
lasses of weakly uniform surje
tive homomorphisms out of A. De ne A as
the pullba
k in the diagram
- Q Ai
A
?
?
- Q A
i
A
If f : A ! B is weakly uniform, it fa
tors A !
! A B and the rst arrow is weakly
uniform sin
e every uniform fun
tional on A extends to a uniform fun
tional on B . Sin
e,
up to isomorphism, A !
! A is among the A !! Ai , it follows that f : A ! B is uniform.
Sin
e the identity is a weakly uniform surje
tion, the lower arrow in the square above is
a subspa
e in
lusion and hen
e so is the upper arrow. That implies that the lower arrow
in the diagram of fun
tionals
0
0
0
Q Ai; ?℄
- [A; ?℄
Q A?i; ?℄
- [ A;??℄
[
[
P
P
The left hand arrow is equivalent to [A ; ?℄ ! [A ; ?℄ (Corol-
Theory and Appli
ations of Categories, Vol. 6, No. 1
15
is a surje
tion.
i
i
lary 4.5), whi
h we have just seen is an isomorphism. Thus the right hand arrow is a
surje
tion, while it is evidently an inje
tion. This shows that A has the same fun
tionals as A. If A were a stri
tly ner uniformity than that of A on the same underlying
V stru
ture that had the same set of fun
tionals as A, then the identity A ! A would
be weakly uniform and then A ! A would be uniform, a
ontradi
tion. Thus if we
de ne L(V; V ) = R(V; V ) we know at least that P L
= Id so that L is full and faithful. If (f; f ): (V; V ) ! P A is a Chu morphism, then f : V ! jAj is a homomorphism
su
h that for ea
h uniform fun
tional ': A ! ? the
omposite ' f 2 V . Thus R(V; V )
! A is weakly uniform and hen
e L(V; V ) = R(V; V ) ! A is uniform. Conversely,
if f : L(V; V ) ! A is uniform, then for any fun
tional ': A ! ?, the
omposite ' f is
uniform on L(V; V ) and hen
e belongs to V so that we have (V; V ) ! P A.
b
0
0
0
b
b
0
0
0
0
0
0
0
0
0
weak uniformity weak
Ma
key uniformity Ma
key
We will say that an obje
t A with A = A has the
(or
) and that one for whi
h A = A has the
(or
). The latter name is taken from the theory of lo
ally
onvex topologi
al ve
tor
spa
es where a Ma
key topology is
hara
terized by the property of having the nest
topology with a given set of fun
tionals.
topology
topology
4.7. Ex
eptions. In verifying the hypotheses of Theorem 4.1, one notes that ea
h example
satis es simpler hypotheses. And ea
h simpler hypothesis is satis ed by most of the
examples. Most of the
ategories are additive (ex
eption: semilatti
es) and then we
an
use topologies instead of uniformities. In most
ases, the dualizing obje
t is dis
rete
(ex
eptions: abelian groups and real or
omplex ve
tor spa
es) and the existen
e of a
small entourage (or neighborhood of 0 in the additive
ase) is automati
. In most
ases,
the theory is
ommutative and the
losed monoidal stru
ture
omes from that (ex
eption:
modules over a Hopf algebra) so that the enri
hment of the uniform
ategory over the
base is automati
. Thus most of the examples are ex
eptional in some way (ex
eptions:
ve
tor spa
es over a eld and modules with a dualizing module), so that we may
on
lude
that they are all ex
eptional.
5. Ve
tor spa
es: the
ase of a dis
rete eld
The simplest example of the theory is that of ve
tor spa
es over a dis
rete eld. Let K be
a xed dis
rete eld. We let V be the
ategory of K -ve
tor spa
es with the usual
losed
monoidal stru
ture and let D be the dis
rete spa
es. Sin
e the
ategory is additive, we
an work with topologies rather than uniformities. We take the dualizing obje
t ? as the
eld K with the dis
rete topology.
The
onditions of Theorem 4.1 are all evident and so we
on
lude that the full sub
ategories of the
ategory of topologi
al K -ve
tor spa
es
onsisting of weakly topologized
spa
e and of Ma
key spa
es are -autonomous.
We note that in nite dimensional dis
rete spa
es do not have the weak uniformity. In
fa
t the weak uniformity asso
iated to the
hu spa
e (V; V Æ K ) is V with the uniform
Theory and Appli
ations of Categories, Vol. 6, No. 1
16
topology in whi
h the open subspa
es are the
o nite dimensional ones. Sin
e the 0
subspa
e is not
o nite dimensional, the spa
e is not dis
rete. On the other hand, the
map to the dis
rete V is weakly uniform and so the Ma
key spa
e asso
iated is dis
rete.
6. Dualizing modules
The
ase of a ve
tor spa
e over a dis
rete eld has one generalization, suggested by R.
Raphael (private
ommuni
ation). Let R be a
ommutative ring. Say that an R-module ?
is a dualizing module if it is a nitely generated inje
tive
ogenerator and the
anoni
al
map R ! HomR (?; ?) is an isomorphism. Let A be the
ategory of topologi
al (=
uniform) R-modules that are strongly
ogenerated by the dis
rete ones. Then taking
the small neighborhood to be f0g and D the
lass of dis
rete modules, the
onditions of
Theorem 4.1 are satis ed and we draw the same
on
lusion.
6.1. Existen
e of dualizing modules. Not every ring has a dualizing module. For example,
no nitely generated abelian group is inje
tive as a Z-module, so Z la
ks a dualizing
module. On the other hand, If R is a nite dimensional
ommutative algebra over a
eld K , then HomK (R; K ) is a dualizing module for K . It follows that any artinian
ommutative ring has a dualizing module:
6.2. Proposition. Suppose K is a
ommutative ring with a dualizing module D and R
is a
ommutative K -algebra nitely generated and proje
tive as a K -module. Then for
any nitely generated R-proje
tive P of
onstant rank one, the R-module HomK (P; D) is
a dualizing module for R.
It is standard that su
h a module is inje
tive. In fa
t, for an inje
tive homomorphism f : M ! N , we have that
Proof.
HomR (f; HomK (P; D))
= HomK (P
R
f; D
)
whi
h is surje
tive sin
e P is R-
at. Sin
e P is nitely generated proje
tive as an Rmodule, it is retra
t of a nite sum of
opies of R. Similarly, R is a retra
t of a nite sum
of
opies of K , when
e P is a retra
t of a nite sum of
opies of K . Then Hom(P; D) is
a retra
t of a nite sum of
opies of D and is thus nitely generated as a K -module, a
fortiori as an R-module. Next we note that a
onstantly rank one proje
tive P has endomorphism ring R. In fa
t, the
anoni
al R ! HomR (P; P ) lo
alizes to the isomorphism
RQ ! HomRQ (PQ ; PQ ) = HomRQ (RQ ; RQ ) whi
h is an isomorphism, at ea
h prime ideal
Q and hen
e is an isomorphism. Then we have that
HomR (HomK (P; D); HomK (P; D))
= HomK (P
= HomR(
= HomR(
P;
sin
e
D
R HomK (
)
P; D ; D
)
HomK (HomK (P; D); D))
P; P
)
=R
is a dualizing module for K and P is a nitely generated
K
module.
Theory and Appli
ations of Categories, Vol. 6, No. 1
17
Whether any non-artinian
ommutative ring has a dualizing module is an open question. For example, the produ
t of
ountably many elds does not appear to have a
dualizing module. The obvious
hoi
e would be the produ
t ring itself and, although it
is inje
tive, it is not a
ogenerator sin
e the quotient of the ring mod the ideal whi
h is
the dire
t sum is a module that is annihilated by every minimal idempotent so that the
quotient module has no non-zero homomorphism into the ring.
7. Ve
tor spa
es:
ase of the real or
omplex eld
We will treat the
ase of the
omplex eld. The real
ase is similar. We take for D the
lass
of Bana
h spa
es and the base eld C as dualizing obje
t. The D-
ogenerated obje
ts are
just the spa
es whose topology is given by seminorms. These are just the lo
ally
onvex
spa
es (see [Kelly, Namioka, 1963℄, 2.6.4). The
onditions of 4.1 follow immediately from
the Hahn-Bana
h theorem and we
on
lude that the
ategory
hu(V ; ?) is equivalent to
both full sub
ategories of weakly topologized and Ma
key spa
es and that both
ategories
are -autonomous. In parti
ular, the existen
e of the Ma
key topology follows quite easily
from this point of view.
We
an also give a relatively easy proof of the fa
t that the Ma
key topology is
onvergen
e on weakly
ompa
t,
onvex,
ir
led subsets of the dual. In fa
t, let A be a
lo
ally
onvex spa
e and A denote the weak dual. If f : A ! D is a weakly
ontinuous
map, then we have an indu
ed map, evidently
ontinuous in the weak topology, f : D
! A and one sees immediately that the weakly
ontinuous seminorm indu
ed on A
jj jj
by the
omposite A ! D
! R is simply the sup on f (C ), where C is the unit
ball of D , whi
h is
ompa
t in the weak topology. On the other hand, if C A is
ompa
t,
onvex, and
ir
led, let B be the linear subspa
e of A generated by C made
into a Bana
h spa
e with C as unit ball. With the topology indu
ed by that of C , so
that a morphism out of B is
ontinuous if and only if its restri
tion to C is, B be
omes
an obje
t of the
ategory C as des
ribed in [Barr, 1979℄, IV.3.10. This
ategory
onsists
of the mixed topology spa
es whose unit balls are
ompa
t. The dis
ussion in IV.3.16 of
the same referen
e then implies that every fun
tional on B is represented by an element
of B . This means that the indu
ed A ! B is weakly
ontinuous. But B is a Bana
h
spa
e whose norm is the absolute sup on C , as is the indu
ed seminorm on A.
f
8. Bana
h balls
A Bana
h ball is the unit ball of a Bana
h spa
e. The
on
lusions, but not the hypotheses
of Theorem 4.1 are valid in this
ase too. However, the proof is di erent and will appear
elsewhere [Barr, Kleisli, 1999℄. The proof given here of the existen
e of the right adjoint
and the Ma
key topology was rst found in this
ontext.
Theory and Appli
ations of Categories, Vol. 6, No. 1
18
9. Abelian groups
The
ategory of abelian groups is an example of the theory. For D we take the
lass of
lo
ally
ompa
t groups. The dualizing obje
t is, as usual, the
ir
le group, R=Z. Sin
e
the
ategory is additive, we
an deal with topologies instead of uniformities. A small
neighborhood of 0 is an open neighborhood of 0 that
ontains no non-zero subgroup
and for whi
h a homomorphism to the
ir
le is
ontinuous if and only if the inverse image
of that neighborhood is
ontinuous.
9.1. Proposition. The image U R=Z of the interval ( 1=3; 1=3) R is a small
neighborhood of 0.
Suppose x 6= 0 in U . Suppose, say, that x is in the image of a point in (0; 1=3).
Then the rst one of x, 2x, 4x, : : : , that is larger than 1=3 will be less than 2=3 1=3,
whi
h shows that U
ontains no non-zero subgroup.
It is
lear that the set of all 2 nU; n = 0; 1; 2; : : : is a neighborhood base at 0. Suppose
that f : A ! R=Z is a homomorphism su
h that V = f 1 (U ) is open in A. Let V0 = V
and indu
tively
hoose an open neighborhood Vn of 0 so that Vn Vn Vn 1 . Then one
easily sees by indu
tion that Vn f 1 (2 n U ).
The remaining
onditions of Theorem 4.1 are almost trivial, given Pontrjagin duality.
The only thing of note is that if D 2 D and A D then any
ontinuous homomorphism
': A ! R=Z
an rst be extended to the
losure of A, sin
e the
ir
le is
ompa
t and
hen
e
omplete in the uniformity. A
losed subgroup of a lo
ally
ompa
t group is lo
ally
ompa
t and the duality theory of lo
ally
ompa
t groups gives the extension to all of D.
9.2. Proposition. Lo
ally
ompa
t groups are Ma
key groups.
Proof. Sin
e all the groups in A are embedded in a produ
t of lo
ally
ompa
t groups,
it suÆ
es to know that a weakly
ontinuous map between lo
ally
ompa
t topologi
al
groups is
ontinuous. This is found in [Gli
ksberg, 1962℄.
Proof.
9.3. Other
hoi
es for D. One thing to note is that it is possible to
hoose a di erent
ategory A. The result
an be a di erent notion of Ma
key group. For instan
e, you
ould
hoose for A the subspa
es of
ompa
t spa
es. In that
ase weakly
ontinuous
oin
ides with
ontinuous and weak and Ma
key topologies
oin
ide. Another possibility
is to use
ompa
t and dis
rete spa
es. It is easy to see that the real line
annot be
embedded into a produ
t of
ompa
t and dis
rete spa
es. There are no non-zero maps
to a dis
rete spa
e, so it would have to embedded into a produ
t of
ompa
t spa
es. But
the real line is
omplete, so the only way it
ould be embedded into a produ
t of
ompa
t
spa
es would be if it were
ompa
t.
In the original monograph,
ountable sums of
opies of R were permitted in D. But
the sum of
ountably many
opies of R is not lo
ally
ompa
t. Here we show that we also
get a model of the theory by allowing D to
onsist of
ountable sums of lo
ally
ompa
t
groups. The only issue here is the inje
tivity of the
ir
le. So suppose A D, where
D = D1 D2 is a
ountable sum of lo
ally
ompa
t groups. As above, we
an suppose
Theory and Appli
ations of Categories, Vol. 6, No. 1
19
that A is
losed in D. Let Fn (D) = D1 Dn and Fn (A) = A \ Fn (D). Every element
of A is in some nite summand, so that, algebrai
ally at least, A =
olim Fn (A). Whether
it is topologi
ally is not important, sin
e we will show that every
ontinuous
hara
ter
on the
olimit extends to a
ontinuous
hara
ter on D. What does matter is that, by
de nition of the topology on the
ountable sum, D =
olim Fn (D) both algebrai
ally and
topologi
ally. The square
Fn 1(A)
Fn(A)
?-
- ?
?
- Fn(?D)
Fn 1(D)
is a pullba
k by de nition. There is no reason for it to be a pushout, but if we denote
the pushout by Pn , it is trivial diagram
hase to see that Pn ) ! Fn (D) is inje
tive. The
group Fn (D) is lo
ally
ompa
t and so, therefore, is the
losed subgroup Fn (A) and so is
the
losure Pn . Thus, taking Pontrjagin duals, all the arrows in the diagram below are
surje
tive and the square is a pullba
k:
Fn(D)
AAHHH
AA HHHHH
AA RRPn --HjHjFn 1(D)
AA
AAU
? -- ?
U
A
Fn(A)
Fn 1(A)
The surje
tivity of the arrow Fn (D) !
! Pn implies that ea
h square of
Fn+1 (D)
Fn (D)
Fn 1 (D)
--
--
--
--
? -- ? -- ? -- ? -- ?
Fn+1(A)
Fn(A)
Fn 1(A)
is a weak pullba
k. From this, it is a simple argument to see that the indu
ed arrow
lim Fn (D) !
! lim Fn(A) is surje
tive.
10. Modules over a
o
ommutative Hopf algebra
Modules over a Hopf algebra are not models of a
ommutative theory, unless the algebra
should also be
ommutative and, even so, the
losed stru
ture does not
ome from there.
Theory and Appli
ations of Categories, Vol. 6, No. 1
20
Thus we will have to verify dire
tly that the
ategory of the topologi
al algebras is enri
hed
over the
ategory of dis
rete algebras for the theory.
There are two important spe
ial
ases and we begin with brief des
riptions of them.
10.1. Group representations. Let G be a group and K be a eld. A K -representation of
G is a homomorphism of G into the group of automorphisms of some ve
tor spa
e over
K . Equivalently, it is the a
tion of G on a K -ve
tor spa
e. A third equivalen
e is with
a module over the group algebra K [G℄. The
ategory of K -representations of G is thus
an equational
ategory, that of the K [G℄-modules, but the theory is not
ommutative
unless G should be
ommutative and even in that
ase, we do not want that
losed
monoidal stru
ture. The one we want has as tensor produ
t of modules M and N the
K -tensor produ
t M
N = M
K N . The G-a
tion is the so-
alled diagonal a
tion
x(m
n) = xm
xn, x 2 G, extended linearly. The internal hom takes for M Æ N the
set of K -linear maps with a
tion given by (xf )m = x(f (x 1 m)) for x 2 G. This gives a
symmetri
losed monoidal stru
ture for whi
h the unit obje
t is K with trivial G a
tion,
meaning every element of G a
ts as the identity on K .
If M and N are topologi
al ve
tor spa
es with
ontinuous a
tion of G (whi
h is assumed dis
rete, at least here), then it is easily seen that the set of
ontinuous linear
transformations M ! N is a G-representation with the same de nition of a
tion and we
denote it by [M; N ℄ as before. Thus the
ategory of topologi
al (= uniform) G-modules is
enri
hed over the
ategory of G-modules. The
otensor is also easy. De ne AV as jAj Æ V
topologized as a subspa
e of AUV .
10.2. Lie algebras. Let K be a eld and g be a K -Lie algebra. A K -representation of g
is a Lie algebra homomorphism of g into the Lie algebra of endomorphisms of a K -ve
tor
spa
e V . In other words, for x 2 g and v 2 V , there is de ned a K -linear produ
t xv
in su
h a way that [x; y ℄v = x(yv ) y (xv ). If V and W are two su
h a
tions, there is
an a
tion on V
W = V
K W given by x(v
w) = xv
w + v
xw. The spa
e
V Æ W of K -linear transformations has an a
tion given by (xf )(v ) = x(f v ) f (xv ). If
g a
ts
ontinuously on topologi
al ve
tor spa
es V and W , then xf is
ontinuous when
f is so that the
ategory of topologi
al representations is enri
hed over the
ategory of
representations. The
otensor works in the same way as with the groups.
10.3. Modules over a
o
ommutative Hopf algebra. These two notions above
ome together in the notion of a module over a
o
ommutative Hopf algebra. Let K be a eld. A
over K is a K -algebra given by a multipli
ation : H
H
! H (all tensor produ
ts in this se
tion are over K ), a unit : K ! H , a
omultipli
ation
Æ : H ! H
H , a
ounit : H ! K and an involution : H ! H su
h that
o
ommutative Hopf algebra
HA{1. (H; ; ) is an asso
iative, unitary algebra;
HA{2. (H; Æ; ) is a
oasso
iative,
ounitary,
o
ommutative
oalgebra;
HA{3. Æ and are algebra homomorphisms; equivalently,
morphisms;
and
are
oalgebra homo-
Theory and Appli
ations of Categories, Vol. 6, No. 1
HA{4.
21
makes (H; ; ) into a group obje
t in the
ategory of
o
ommutative
oalgebras.
This last
ondition is equivalent to the
ommutativity of
H
Æ
Æ
?
H
-H
H
1
H
?
H
The leading examples of Hopf algebras are group algebras and the enveloping algebras
of Lie algebras. If G is a group, the group algebra K [G℄ is a Hopf algebra with Æ (x) = x
x,
(x) = 1 and (x) = x 1 , all for x 2 G and extended linearly. In the
ase of a Lie algebra
g, the de nitions are Æ (x) = 1
x + x
1, (x) = 0 and (x) = x, all for x 2 g.
10.4. The general
ase. Let H be a
o
ommutative Hopf algebra. By an H -module we
simply mean a module over the algebra part of H . If M and N are modules, we de ne
M
N to be the tensor produ
t over K with H a
tion given by the
omposite
H
M
N
Æ
1
1! H
H
M
N ! H
M
H
N ! M
N
The se
ond arrow is the symmetry isomorphism of the tensor and the third is simply
the two a
tions. We de ne M Æ N to be the set of K -linear arrows with the a
tion
H
(M Æ N ) ! M Æ N the transpose of the arrow H
M
(M Æ N ) ! N given by
H
M
(M Æ N )
1
1! H
H
M
(M Æ N )
1
1
1! H
H
M
(M Æ N )
Æ
! H
M
(M Æ N ) ! H
N ! N
The third arrow is the a
tion of H on M , the fourth is evaluation and the fth is the
a
tion of H on N .
That this gives an autonomous
ategory
an be shown by a long diagram
hase. The
tensor unit is the eld with the a
tion xa = (x)a.
We have to show that the
ategory of topologi
al modules is enri
hed over the
ategory
of modules. We
an des
ribe the enri
hed stru
ture as
onsisting of the
ontinuous linear
maps with the module stru
ture given as before, that is by
onjugation. The
ontinuity
of the module stru
ture guarantees that the a
tion preserves
ontinuity. From then on
the argument is the same. The dualizing obje
t is the dis
rete eld K whi
h has a small
neighborhood and the rest of the argument is the same. The
otensor is just as in the
ase of group representations.
The
lass D
onsists of the dis
rete obje
ts. The dualizing obje
t is the tensor unit.
Sin
e the internal hom is just that of the ve
tor spa
es, the
onditions of Theorem 4.1 are
easy.
Theory and Appli
ations of Categories, Vol. 6, No. 1
22
11. Semilatti
es
By semilatti
e we will mean inf semilatti
e, whi
h is a partially ordered set in whi
h
every nite set of elements has an inf. It is obviously suÆ
ient that there be a top element
and that every pair of elements have an inf. The
ategory is obviously equivalent to that
of sup semilatti
es, sin
e you
an turn the one upside down to get the other. The
ategory
is equational having a single
onstant, 1 (the top element) and a single binary operation ^
whi
h is unitary (with respe
t to 1),
ommutative, asso
iative and idempotent. (In fa
t,
sup semilatti
es have exa
tly the same des
ription|it all depends on how you interpret
the operations.)
Semilatti
es do not form an additive
ategory, but they are half-additive sin
e they
are
ommutative monoids. The dualizing obje
t is the 2 element
hain with the dis
rete
uniformity, whi
h evidently has a small entourage. Sin
e it is the tensor unit,
ondition 4
of Theorem 4.1 is satis ed. For D, we take the
lass of dis
rete latti
es. We need show
only
onditions 5 and 6.
Suppose we have an in
lusion L1 L2 of dis
rete semilatti
es and f : L1 ! ? is a
semilatti
e homomorphism. We will show that if x 2 L2 L1 , then f
an be extended to
the semilatti
e generated by L1 and x. This semilatti
e is L1 [ (L1 ^ x). We rst de ne
f x = 1 unless there are elements a, b 2 L1 su
h that f a = 1, f b = 0 and a ^ x b in whi
h
ase we de ne f x = 0. Then we de ne f (a ^ x) = f a ^ f x for any a 2 L1 . The only thing
we have to worry about is if a ^ x 2 L1 for some a 2 L1 . If f a = 0, then f (a ^ x) f a
so that f (a ^ x) = 0 = f a ^ f x. If f a = 1, then either f (a ^ x) = 1 or taking b = a ^ x
we satisfy the
ondition for de ning f x = 0 and then 0 = f (a ^ x) = f a ^ f x as required.
The rest of the argument is a routine appli
ation of Zorn's lemma. This
ompletes the
proof of 5. Now 6 follows immediately sin
e given any two elements of a dis
rete latti
e,
they generate a sublatti
e of at most 4 four elements and it is easy to nd a separating
fun
tional on that sublatti
e.
12. The
ategory of
Æ -obje
ts
We will very brie
y explain why the
ategories of Æ -obje
ts
onsidered in [Barr, 1979℄
is also -autonomous. Of
ourse, it is likely more interesting that the larger
ategories
onstru
ted here are -autonomous, but in the interests of re
overing all the results from
the monograph, we in
lude it.
An obje
t T is
alled -
omplete if it is inje
tive with respe
t to dense subobje
ts of
ompa
t obje
ts. That is, if in any diagram
C0
?
T
-C
Theory and Appli
ations of Categories, Vol. 6, No. 1
23
with C
ompa
t and C0 a dense subobje
t,
an be
ompleted by an arrow C ! T . The
obje
t T is -
omplete if T is -
omplete. An obje
t is
alled a Æ -obje
t if is -
omplete,
-
omplete and re
exive.
12.1.
Theorem. The full sub
ategories of
Æ -obje
ts
are
-autonomous sub
ategories the
ategories of Ma
key obje
ts.
The proof uses one property that we will not verify. Namely that all
ompa
t
obje
ts are Æ -obje
ts. The duals of the
ompa
t obje
ts are
omplete (in most
ases
dis
rete). For an obje
t T , we de ne T as the interse
tion of all the -
omplete subobje
ts
of the
ompletion of T . The
ru
ial
laim is that if T is -
omplete, so is (T ) . In fa
t,
the adjun
tion arrow T ! T gives an arrow (T ) ! T
= T . Now
onsider a
diagram
Proof.
C0
?
-C
-
(T )
T
Sin
e T is -
omplete, there is an arrow C ! T that makes the square
ommute. This
gives T ! C and, sin
e C is
omplete, T ! C , and then C
= C !(T ) , as
required. We now invoke Theorem 2.3 of [Barr, 1996℄ to
on
lude that the Æ -obje
ts form
a -autonomous
ategory.
Referen
es
M. Barr (1991),
M. Barr (1979),
-Autonomous Categories
. Le
ture Notes in Mathemati
s
-Autonomous
ategories and linear logi
Computer S
ien
e 1, 159{178.
.
752
. Mathemati
al Stru
tures in
M. Barr (1996), -Autonomous
ategories, revisited. J. Pure Applied Algebra, 111, 1{20.
M. Barr (1998), The separated
Categories, 4.6, 137{147.
extensional Chu
ategory
. Theory and Appli
ations of
M. Barr and H. Kleisli (1999), Topologi
al balls. To appear in Cahiers de Topologie et
Geometrie Differentielle Categorique, 40.
P.-H. Chu (1979),
Constru
ting
P.J. Freyd, A. S
edrov (1990),
I. Gli
ksberg (1962),
V. Gupta (1994),
University.
. Appendix to [Barr, 1979℄.
-autonomous
ategories
Categories, Allegories
Uniform boundedness for groups
. North-Holland.
. Canadian J. Math. 14, 269-276.
Chu Spa
es: A Model of Con
urren
y.
Ph.D. Thesis, Stanford
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J.L. Kelley, I. Namioka (and
oauthors) (1963),
Nostrand.
Linear Topologi
al Spa
es.
Van
F.E.J. Linton (1966), Autonomous equational
ategories. J. Math. Me
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V.R. Pratt (1993a), Linear Logi
for Generalized Quantum Me
hani
s. In Pro
eedings
Workshop on Physi
s and Computation, Dallas, IEEE Computer So
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ond Cal
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nsk, Poland, 142{155.
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onferen
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in Computer S
ien
e IEEE Computer So
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Dept. of Math. and Stats.
M
Gill University
805 Sherbrooke St. W
Montreal, QC H3A 2K6
Email:
barrmath.m
gill.
a
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le may be a
essed via WWW at http://www.ta
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a/ta
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