-AUTONOMOUS CATEGORIES:
ONCE MORE AROUND THE TRACK
To Jim Lambek on the oasion of his 75th birthday
MICHAEL BARR
ABSTRACT. This represents a new and more omprehensive approah to the
-autonomous ategories onstruted in the monograph [Barr, 1979℄. The main tool in
thenewapproahistheChuonstrution. Themain onlusionisthattheategoryof
separated extensionalChu objetsfor ertainkinds ofequational ategories is
equiva-lenttotwousuallydistintsubategories oftheategoriesofuniformalgebrasofthose
ategories.
1. Introdution
Themonograph[Barr,1979℄wasdevotedtotheinvestigationof-autonomousategories.
Most of the book was devoted to the disovery of -autonomous ategories as full
sub-ategories of seven dierent ategories of uniform or topologial algebras over onrete
ategories thatwere eitherequationalorreetive subategories ofequationalategories.
The base ategories were:
1. vetor spaes over a disreteeld;
2. vetor spaes over the real or omplex numbers;
3. modules overa ring with a dualizingmodule;
4. abelian groups;
5. modules overa oommutative Hopf algebra;
6. sup semilatties;
7. Banah balls.
For denitions of the ones that are not familiar, see the individual setions below.
These ategories have a number of properties in ommon as well as some important
dierenes. First, there are already known partial dualities, often involving topology.
Thisresearhhasbeensupportedbygrantsfrom theNSERCofCanadaandtheFCARduQuebe
Reeivedbytheeditors1998November20and, inrevisedform,1999August26.
Publishedon1999November30.
1991MathematisSubjetClassiation: 18D10,43A40,46A70,51A10.
Keywordsandphrases: duality,topologialalgebras,Chuategories.
It is these partial dualities that we wish to extend. Seond, all are symmetri losed
monoidal ategories. All but one are ategories of models of a ommutative theory and
get theirlosedmonoidalstruture fromthat(see3.7below). The theoryofBanahballs
is reallydierent from rst six and is treatedin detailin [Barr,Kleisli, toappear℄.
Whatwedohereisprovideauniformtreatmentoftherstsixexamples. Weshowthat
ineah ase, there isa -autonomousategory ofuniform spae models of the theory. In
most ases,this isequivalenttothe topologialspaemodels. The maintoolused here is
the so-alledChuonstrutionasdesribed inanappendix tothe 1979monograph,[Chu,
1979℄. Hedesribed indetaila very generalonstrution ofalarge lassof -autonomous
ategories. He startswith any symmetrimonoidalategory V and any objet ? therein
hosen asdualizing objetto produea -autonomousategory denoted Chu(V;?). The
simpliityand generality of this onstrutionmade it appear at the time unlikely that it
ould have any real interest beyond its originalpurpose, namely showing that there was
a plenitude of -autonomousategories. Wedesribe this onstrution in Setion2.
ApreliminaryattempttoarryoutthisapproahusingtheChuonstrutionappeared
in [Barr, 1996℄, limited to only two of the seven example ategories (vetor spaes over
a disrete 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 artile, we
proveaverygeneraltheorem that appeals toveryfew speialpropertiesof the examples.
In1987IdisoveredthatmodelsofJean-YvesGirard'slinearlogiwere -autonomous
ategories. Withina fewyears, VaughanPratt and hisstudents hadfound out about the
Chu onstrution and were studying its properties intensively ([Pratt, 1993a,b, 1995,
Gupta, 1994℄). One thing that espeially struk me was Pratt's elegant, but essentially
obvious, observation that the ategory of topologial spaes an be embedded fully into
Chu (Set;2) (see 2.2). The real signiane|at least to me|of this observation is that
putting aChu struture ona set an beviewed as akind of generalizedtopology.
A readerwhoisnotfamiliarwith theChuonstrutionisadvisedatthispointtoread
Setion 2. Thinking of a Chu struture as a generalized topology leads to an interesting
ideawhihIwillillustrateintheaseoftopologialabeliangroups. IfT isanabeliangroup
(or, forthat matter,aset),a topologyisgiven bya olletionof funtionsfromthepoint
set of T to the Sierpinskispae|the spae with two points,one open and the other not.
Fromaategorialpointofview,mightitnotmakemoresensetoreplaethefuntionstoa
set by group homomorphisms to a standard topologialgroup, thus reating adenition
of topologial group that was truly intrinsi to the ategory of groups. If, for abelian
groups,we take this \standardgroup"to bethe irle groupR=Z, the resultant ategory
is(forseparatedgroups)aertainfullsubategoryofChu (Ab;R=Z)alledhu(Ab;R=Z).
This ategoryisnot the ategoryoftopologialabelian groups. Nonetheless the ategory
of topologialabelian groupshas anobvious funtor intohu (Ab;R=Z) and this funtor
has both a left and a right adjoint, eah of whih is full and faithful. Thus the ategory
hu (Ab=;R=Z) isequivalenttotwodistinttwofullsubategoriesof abeliangroups,eah
both a nest and a oarsest topology that indue the same set of haraters. The two
subategories onsist of all those that have the nest topology and those that have the
oarsest. These are the imagesof the leftand right adjoint,respetively.
1.1. Aknowledgment. I would like to thank Heinrih Kleisli with whom I had a many
helpfuland informativedisussionsonmanyaspets ofthis work duringonemonthvisits
to the Universite de Fribourg during the springs of 1997 and 1998. In partiular, the
orret proof of the existene of the Makey uniformity (or topology) was worked out
there inthe ontext of the ategory of balls, see [Barr, Kleisli,1999℄.
2. The Chu onstrution
There are many referenes to the Chu onstrution, going bak to [Chu, 1979℄, but see
also [Barr, 1991℄, for example. In order to make this paper self-ontained, we will give
a brief desription here. We stik 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 objet of V. An objet of Chu (V;?) is a pair (V;V 0
) of objets of V
together with a homomorphism, alled a pairing, h ; i:V V 0
!?. A morphism
(f;f 0
):(V;V 0
) !(W;W 0
) onsists of a pair of arrows f:V !W and f 0
:W 0
!V 0
in V
that satisfy the symboli identity hfv;w 0
i = hv;f 0
w 0
i. Diagrammatially, this an be
expressed asthe ommutativity of the diagram
W W 0
?
-h ; i V W
0
V V 0
-V f 0
? f W
0
? h ; i
Using the transposes V !V 0
Æ? and V 0
!V Æ? of the struture maps, this
ondi-tion an beexpressed asthe ommutativity of eitherof the squares
V 0
Æ? W
0
Æ?
-f 0
Æ?
V W
-f
? ?
W Æ? V Æ?
-f Æ? W
0
V 0 -f
0
A nal formulation of the ompatibilityondition is that
Hom(W 0
;V 0
) Hom(V W
0
;?)
-Hom((V;V 0
);(W;W 0
)) Hom(V;W)
-? ?
is apullbak.
The internalhom isgottenby usinganinternalizationof the lastformulation. Dene
the objet[(V;V 0
);(W;W 0
)℄ of V as apullbak
W 0
ÆV 0
V W 0
Æ?
-[(V;V 0
);(W;W 0
)℄ V ÆW
-? ?
and then dene
(V;V 0
) Æ(W;W 0
)=([(V;V 0
);(W;W 0
)℄;V W 0
)
Sine the dual of (V;V 0
) is (V 0
;V), it follows from the interdenability of tensor and
internal hom ina -autonomous ategorythat the tensorprodut 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 objet of Chu (Set;2) is a pair (S;S 0
) together with
a funtion SS 0
!2. This is equivalent to a funtion S 0
!2 S
. When this funtion
is injetive 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 funtion 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 embeddingof topologial spaesintoChu (Set;2).
2.3. Theategoryhu(V;?). SupposeVisalosedsymmetrimonoidalategoryasabove
and suppose thereis afatorizationsystem E=MonV. (See [Barr,1998℄ fora primeron
fatorization systems.) In generalwesuppose thatthe arrows inE are epimorphismsand
thatthoseinMareompatiblewiththeinternalhominthesensethatifV !V 0
belongs
to M, then for any objet W, the indued W ÆV !W ÆV 0
also belongs to M. In
allthe examples here,E onsists of the surjetions(regularepimorphisms)and Mof the
injetions(monomorphisms),forwhihtheseonditions areautomati. Anobjet(V;V 0
)
of the Chu ategoryis said to beM-separated, orsimplyseparated, if the transpose V
!V 0
Æ? is in M and M-extensional, or simply extensional, if the other transpose
V 0
full subategories of separated, extensional, and separated and extensional, respetively,
objets of Chu (V;?). FollowingPratt, we usually denote the lastof these by hu (V;?).
The relevant fatsare
1. The full subategory Chu
s
(V;?)of separated objetsis a reetive subategory of
Chu (V;?)with reetor s.
2. Thefullsubategory Chu
e
(V;?)ofextensionalobjetsisaoreetivesubategory
of Chu (V;?)with oreetor e.
3. If(V;V 0
)is separated, so ise(V;V 0
);if (V;V 0
)is extensional, so iss(V;V 0
).
4. Therefore hu (V;?)isbothareetivesubategoryofthe extensionalategoryand
aoreetivesubategoryoftheseparatedsubategory. Itis,inpartiular,omplete
and oomplete.
5. The tensorprodut ofextensionalobjets isextensionalandthe internalhomof an
extensional objetintoa separated objet is separated.
6. Therefore by usings( )as tensorprodut and r( Æ ) as internalhom, the
ategory hu (V;?) is-autonomous.
For details,see [Barr,1998℄.
3. Topologial and uniform spae objets
3.1. Topologyand duality. In a -autonomous ategory we have, for any objet A, that
A
=
> ÆA
=
A Æ>
. If we denote >
by ?, we see that the duality has the form
A 7!A Æ?. The objet ? is alled the dualizing objet and, as we willsee, the way
(or at least a way) of reating a duality is by nding a dualizing objet in some losed
monoidalategory.
In order thata ategoryhave adualityrealizedby aninternalhom, there has tobe a
way of onstrainingthe maps sothat the dual of aprodut is asum. Inan additive
at-egory, forexample, this happens without onstraint for niteproduts, but not normally
for inniteones. A naturalonstraintis topologial. If,forexample, thedualizingobjet
isnitedisrete, thenanyontinuousmapfromaprodutandependononlynitemany
of the oordinates. For example, even for ordinary topologial spaes, for a ontinuous
funtion f: Q
X
i
!2, f 1
(0) has the form Y Q
i2J= X
i
where J is a nite subset of I
and Y is asubset of the nite produt Q
i2J X
i
. But then f 1
(1)=Z Q
i2J= X
i
, where
Z = Q
i2J X
i
Y. If two elements of the produt x = (x
i )
i2I
and x 0
= (x 0
i )
i2I are
elements suh that x
i =x
0
i
for i 2J then either x2Y and x 0
2Y orx 2Z and x 0
2Z,
but in either ase fx=fx 0
. Thusf depends onlythe oordinates belongingto J, whih
means f fators through the nite produt Q
i2J X
i
3.2. Uniformity. Despitethe examplesabovetherearereasonsforthinkingthatthe
teh-nially \orret" approah to duality is through the use of uniform strutures. A very
readable and informative aount of uniform spaes is in [Isbell, 1964℄. However,
Is-bell uses uniform overs as his main denition. Normally I prefer uniform overs to the
approah using entourages, but for our purposes here entourages are more appropriate.
For any equational ategory, a uniform spae objet is an objet of the ategory
equipped with a uniformity for whih the operations of the theory are uniform. A
mor-phism of uniform spae objets is a uniform funtion that is also a morphism of models
of the theory. Topologial spae objets are dened similarly. Any uniformity leads to
a topologial spae objet in a anonial way and uniform funtions beome ontinuous
funtions for that anonial topology. But not every topology omes from a uniformity
and, if it does, it is not neessarily from a unique one. For example, the metri on the
spae of integers and on the set of reiproals of integers both give the disrete
topol-ogy, but the assoiated uniformitiesare quite dierent. (Metri spaes have a anonial
uniformity. See Isbell's book for details.)
If,however, thereisanabelian groupstrutureamongtheoperationsofanequational
theory, there is a anonial uniformity assoiated to every topology. Namely, for eah
open neighborhood U of the group identity take f(x;y) j xy 1
2 Ug as an entourage.
Moreover, a homomorphism of the algebraistruture between uniform spae objets is
ontinuous ifand onlyif itisuniform. Thusthereisno dierene, insuh ases, between
ategoriesofuniformandtopologialspaeobjets. Obviously,theategoryoftopologial
spaeobjetsismorefamiliar. However, oneofourategories, semilatties,doesnot have
an abelian group struture and for that reason, we have ast our main theorem interms
of uniform struture. There is another, less important, reason. At one point, in dealing
with topologial abelian groups, it beomes important that the irle group is omplete
and ompleteness isa uniform, not topologial,notion.
IfV isanequationalategory,wedenoteby Un(V),theategoryof uniform objets
of V. Welet j j:Un(V) !V tobe funtor that forgets the uniformstruture.
3.3. Small entourage. LetA beauniform V objet. AnentourageE AA isalled a
smallentourage ifitontainsnosubobjetofAAthatproperlyontainsthediagonal
and if any homomorphism f:B !A of uniform V objets for whih (f f) 1
(E) is an
entourage ofB isuniform.
3.4.. In all the examples we will be onsidering, there will be a given lass of uniform
objetsD andwewillbedealingwith thefullsubategoryAof Un(V)onsistingofthose
objets strongly ogenerated by D, whih is to say that those that an algebraially
and uniformlyembedded into aprodut of objetsof D.
3.5. Half-additiveategories. Aategoryisalledhalf-additiveifitshomfuntorfators
through theategoryofommutativemonoids. Itiswellknown thatinanysuhategory
nite sums are also produts (see, for example, [Freyd, Sedrov, 1990℄, 1.59). Of ourse,
ommutative monoid in whih every element is idempotent. This monoid struture an
be equallywellviewed assup or inf.
3.6. The losedmonoidalstruture. Theategoriesweare dealingwithareallsymmetri
losedmonoidal. Withoneexeption,thelosedstruturederivesfromtheirbeingmodels
of a ommutative theory.
3.7. Commutativetheories. A ommutative theory isanequational theorywhose
op-erationsarehomomorphisms([Linton,1966℄). ThusinanyabeliangroupG,asontrasted
with a non-abelian group, the multipliation GG !Gis anabelian group
homomor-phism, asare allthe other operations.
3.8. Theorem. [Linton℄ Theategoryof modelsof aommutative theoryhasa anonial
struture of a symmetri losed monoidal ategory.
Proof. Suppose V is the ategory of models and U:V !Set is the underlying set
funtorwithleftadjointF. IfV andW areobjetsofV,thenW ÆV isasubsetofV UW
dened asthe simultaneous equalizer, taken overall operations! of the theory, of
V UW
(V n
) (UW)
n
-V !
R
V (UW)
n ?
! (UW)
n
Here n isthe arityof ! andthe top arrowisraisingtothe nthpower. Sinethe theory is
ommutative,! is ahomomorphismandsothe equalizeris alimitofadiagraminV and
heneliesinV. Inpartiular,the internalhomof twoobjetsofV ertainlyliesinV. The
free objet onone generator is the unit for this internalhom. As for the tensor produt,
V W isonstrutedasa quotientof F(UV UW), similartothe usualonstrutionof
the tensor produt of two abelian groups.
3.9. Proposition. If A and B are objets of an equational ategory V equipped with
uniformities for whih their operations are uniform, then the set of uniform morphisms
from A !B is a subobjet of jAj ÆjBj and thus the ategory of uniform V objets is
enrihed over V. It also has tensors and otensors from V.
Proof. Let [A;B℄denote the set of uniform homomorphisms fromA to B. For eah
n-aryoperation!,thearrow!B:B n
!B isauniformhomomorphismand heneanarrow
[A;B℄ n
=[A;B n
℄ 