BIBLIOGRAPHY

M. Barr, Duality of vector spaces, Cahiers Topologie Geometrie Differentielle 1l (1976),3-14.

M. Barr, Duality of banach spaces, Cahiers Topologie Geometrie Differentielle 1l (1976), 15-32

M. Barr, Closed categories and topological vector spaces, Cahiers Topologie Geometrie Differentielle 1l (1976), 223-234.

M. Barr, Closed categories and banach spaces, Cahiers Topologie Geometrie Differentielle 1l (1976), 335-342.

M. Barr, A closed category of reflexive topological abelian groups, Cahiers Topologie Geometrie Differentielle 18 (1977), 221-248.

M. Barr, The point of the empty set, Cahiers Topologie Geometrie Differentielle 13 (1973), 357-368.

S. Eilenberg, G.M. Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, 1965), Springer-Verlag, 1966, 421-562.

E. Hewitt, K.A. Ross, Abstract Hamonic Analysis, Vol. I, 1963, Springer-Verlag.

K.H. Hofmann, M. Mislove, A. Stralka, The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Lecture Notes Math. ~. (1974), Springer-Verlag.

J.R. Isbell, Uniform Spaces, Amer. Math. Soc. Surveys no. 12, 1964.

J.L. Kelley,General Topology, Van Nostrand, 1955.

G.M. Kelly, Monomorphisms, epimorphisms and pull-backs, J. Austral. Math. Soc. ~. (1969), 124-142.

F.W. Lawvere, Functional Semantics of Algebraic Theories, Dissertation, Columbia University, 1963.

S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloquium Publications, Vol. XXVII, 1942.

F.E.J. Linton, Some aspects of equational categories, Proc. Conf. Categorical Algebra (La Jolla, 1965), Springer-Verlag, 1966, 84-94.

A. Pietsch, Nuclear Locally Convex Spaces, Springer-Verlag, 1972.

H.H. Schaefer, Topological Vector Spaces, third printing, Springer-Verlag, 1970.

Z.Samadeni, Projectivity, injectivity and duality, Rozprawy Mat.~ (1963).

A. Wiweger, Linear spaces with mixed topology, Studia Math.~ (1961), 47-68.

CONSTRUCTING *-AUTONOMOUS CATEGORIES

Po-Hsiang Chu

CHAPTER I: PRELIMINARIES

We will be dealing with closed symmetric monoidal (autonomous) and

*-autonomous categories as defined in the previous paper. Using the MacLane-

Kelly coherence conditions (see [MacLane,Kelly]), M.F. Szabo has proved the

following useful theorem [to appear] .

Theorem: A diagram commutes in all closed symmetric monoidal categories

iff it commutes in the category of real vector spaces.

This theorem not only points out the notion of closed symmetric

monoidal category is a 'correct' generalization of the category of vector

spaces, but it also provides a very easy method to check if a diagram is

commutative in any closed symmetric monoidal category.

The following is a collection of easy consequences of this theorerr

which we shall use later on:

Corollary 1. Given A,B,C objects in V and map A®B ~ C, then the following diagram commutes:

(A,A)

1 (B,B) ~ (A®B,C)

where the map_ (A,A) ~ (~~B,C) is the composition

(A,A) (~d,f) (A,(B,C)) (A®B,C) Note. A~ (B,C) is the usual transpose of A®B ~ C The map (B,B) ---+ (A®B,C) is obtained in a similar fashion. From now on

we simply denote either composite by f

Corollary 2. Given A,B,C,D,F objects in V and map B®C---+ F, then the

following diagram commutes:

(id, f)®if (A,C)®(D,B)

lid®(id,f)

(A,C)®(D,C,F))

lid®p-1

(A,C)®(D®C,F)

lid®(s,id)

(A,C)®(C®D,F)

lid®p

(A,C)®(C, (D,F))

ls (C, (D,F))®(A,C)

(A, (B,F))®(D,B)

Ir-l®id

(A®B,F)®(D,B)

lcs,id)®id

(B®A,F)®(D,B)

h®id

(B, (A,F))®(D,B)

1M (D,(A,F))

-1 p

1 -1 (A,(D,F)) ~ (A®D,F) (s,id) (D®A,F)

104

PROOF. It is easy to check that the diagram commutes in the category of

real vector spaces.

Remark. The word "coherence" is going to appear frequently throughout this

paper. In particular, if the commutativity of a certain diagram is said to

be implied by coherence, we understand that its commutativity follows easily

from this theorem.

Our second assumption on ~ is that it has pullbacks. Since almost

all interesting examples of closed symmetric monoidal categories have this

property, this restriction is not too drastic.

The following is a collection of examples satisfies our assumption:

(i) The category of vector spaces over a fixed field K;

(ii) The category of Banach spaces;

(iii) The category of compactly generated spaces;

(iv) The category of sets (and functions);

(v) The category of abelian groups;

(vi) The category of lattices.

An example of a closed symmetric monoidal category that does not have

pullbacks is the category of sets and relations.

CHAPTER II: CONSTRUCTION OF ~X AND ITS ENRICHMENT OVER V.

1. The Category ~

Given an arbitrary object X in ~. we shall construct a category ~

as follows:

The objects of ~X consist of triplets (V,V',v) where V,V' are objects

in V and v:V®V' ------+ X is a morphism in ~-

A morphism from (V,V',v) to (W,W',w) is a pair (f,g), where f:V-----+ W

and g:W' -----+V' are morphisms in V such that the square

V®W' id®g > V®V'

lf®id lv

W®W' w X

commutes.

105

If (f,g):(V,V' ,v)----+ (W,W' ,w) and (h,k):(W,W',w)----+ (U,U' ,u)

are morphisms in !x then the following diagram commutes:

V®U' id®k V®W' id®g > V®V'

lf®id 1 f®id W®U' id®k W®W' v

lh®id ~ U®U' u X

This implies the composition of (f,g) and (h,k) is (hof,gok) in !x .

Since the composition is defined explicitly in terms of morphisms in V , the

associativity of maps in !x can now be verified:

If (f,g):(V,V',v)----+ (W,W' ,w)

(h,k):(W,W',w)----+ (U,U',u)

(l,m):(U,U',u)----+ (T,T',t)

are morphisms in !x , then

((l,m)o(h,k))o(f,g) (loh,kom) o(f,g)

((loh)•f, go(kom))

(lo(hof),(gok)om)

(l,m)o(hof,gok)

(l,m)o((h,k)o(f,g))

Moreover, Id(V,V' ,v) = (idV,idV,)

shown that !x is a category. is the obvious identity. Hence we have

2. !x is Enriched over V Definition. If V is a closed monoidal category, then A is enriched over V

if A is equipped with the following:

i) For each A,B in!, an object ~(A,B) in~;

ii) For each A in!, a morphism j(A):I---+ ~(A,A) in V

iii) For each A,B,C in!, a morphism

M'(A,B,C):y(B,C)®Y(A,B)---+ y(A,C) in V

These data are required to satisfy the following axioms:

vc 1. The following diagram commutes: ~(A,B)®~(A,A)

M' --=--->- ~(A,B)

lid~ ~(A,B)®I r

vc 2.

VC 3.

The following diagram commutes:

_:{(A,A)®_:{(B,A) M'

The following diagram commutes:

(V(C,D)®_:{(B,C))®_:{(A,B)

1M'0id

106

_:{(B ,A)

a _:{(C ,D)®(_:{(B,C)®_:{(A,B))

lid®M'

'(B,D)0~~ ;~~)®,(A,C)

_:{(A, D)

Given A= (V,V' ,v),B = (W,W',w) objects in ~X, define _:{(A,B) to be

the object in V such that the following square is a pullback.

_:{(A, B) pl I (V,W)

p2l 1 (W', V') -->- (V®W' ,X)

Here -1

(V,W) -+ (V®W' ,X) (V,W) w (V, (W' , X)) ~ (V®W' ,X) -+

and -1

(W', V') -+ (V®W' ,X) (W', V') v (W',(V,X)) ~(W10V,X)~(V®W' ,X) -+

are the right and bottom maps, respectively. Therefore _:{(A,B) is defined up to

isomorphism in V .

Given A = (V,V' ,v) in~ , the following diagram commutes, by

Corollary 1:

I ---"'i---+ (V, V)

p.b.

(V', V') v

---------+ (V®V' ,X)

Universal property of pullbacks implies that there exists a unique map j(A)

I -+ _:{(A,A) in V such that the diagram

107

I

~ V(A,A) ___.:c::____,_, (V, V) -,,] p:b. ]' (V',V')~ (V®V' ,X)

commutes.

Now suppose A= (V,V' ,v), B = (W,W',w), C = (T,T' ,t) are three objects

in A -x In order to verify iii) it suffices to show the outer square of the diagram.

~(B,C)®~(A,B) pl®pl

) (W,T)®(V,W)

jp2®p2 lM

(T' ,W')®(W' ,V') ~(A, C) £1 ) (V,T)

l$ p21 p.b. F (W',V')®(T',W') M (T', V') v (V®T' ,X)

commutes.

Using the fact that -®- is a bifunctor and

1 1 (W, T)

F (W', V') (V®W' ,X) (T', W') (W®T' ,X)

are pullbacks (hence commute!), we can get the desired result from the commu-

tative diagram in Fig. 1. Note in Fig. 1 that corollary 2 of Szabo's theorem

(Chapter I) implies that (2) commutes; coherence implies that (1) and (3)

commute. Again using the universal property of pullbacks, there exists a

unique morphism M'(A,B,C):~(B,C)®~(A,B)

diagram

------+~(A,C) in V such that the

£1®£1 > (W,T)®(V,W)

jM

(T',W')®(W',V') ~(A,C) 1 (V,T)

p.b. 1~ v (V®T' ,X) (W',V')®(T',W') M ____,_ (T', V')

commutes. Hence i) - iii) are defined.

108

Now we have to show they satisfy the required axioms.

Given A= (V,V',v), B = (W,W' ,w) in~, by construction we have

the pullback diagram:

~(A,B) 1 (V,W)

p21 p.b. lw (W', V') v (V®W',X)

But the coherence of v implies that the diagrams of Fig. 1 commute.

~(B,C)®~(A,B) id®pl • ~(B,C)®(V,W) pl®id >(W,T)18>(V,W)

p2®1dj p2®id j;.,,

(T' ,W')®~(A,B) id®pl (T' ,W' )®(V,W) w®id (W®T' ,X)®(V,W)

1d~2j id®W (2) 1M

(T',W')®(W',V') id®v (T' ,W')®(V®W' ,X) M (V®T' ,X)

s (3)

(W' ,V')®(T' ,W') ___ M ____ _,_ (T', V')

~(A,B)®I r

~(A,B) ~(A,B)®I r

~(A,B)

lpl®id lpl lp2®id lp2

(V,W)®I r (V,W) (W',V')®I r (W', V')

FIGURE 1.

M (V,T)

109

Hence the following diagram

l®id ~(A, B) ®I ------"-"===---------> (V,W)®I

~ ~(A,B)

1 (V,W)

p2®id

lp2 p.b. (W', V' )®I _ ___::or __ _,_ (W', V') v (V®W' ,X)

commutes.

Since the outer square commutes, there exists a unique map ~(A,B)®I ~ ~(A,B) such that (1) and (2) commute. But the map ~(A,B)®I ~ ~(A,B) has this

property as well; therefore it follows from uniqueness that it is the map induced

by pulling back.

Recall that in the construction of j(A) we have the following commutative

diagram:

I

(V, V)

lv ~(V', V') (V®V' ,X)

Then the defining property of M(A,B,A), coherence of V, and the fact that

-0- is a bifunctor imply that the diagram:

~(A,B)®I

s

(W' ,V')®(V' ,V') ~(A,B) pl

l (V,W)

p.b. [w

I®~(A,B) i®pZ > (V', V' )0(W', V') _ _::M:___,_(W', V') v (V®W' ,X)

commutes.

110

Again applying the same argument, we conclude that the map id®j M'

~(A,B)®I > ~(A,B)®~(A,A) ~(A,B) is the map induced by

pulling back.

But this is not sufficient to conclude that VCl. holds, i.e. that the

diagram: M'

~(A,B)®~(A,A) ~(A,B)

lid/ ~(A,B)®I

commutes.

We are still required to show that the following diagrams commute:

~(A,B)®I El®id

~>~v.F' ~(A,B)®I

E2®id > (W',V')®I

plOij /

/

jp20i ]" "'

(V,W)®(V,V) M (V ,W) (W', V' )®(V', V') I®(W', V')

That is, that the induced maps satisfy the same commutative square

(therefore they are same by uniqueness).

But it is trivial once we notice there exist canonical maps

(V,W)®I id®i (V,W)®(V,V) in (1) and I®(W',V') i®id (V',V')®(W',V')

in (2) which break (1) and (2) into two smaller connnutative squares. Hence

VCl. holds.

Applying a similar argument, we conclude VC2. is also true. Next we

are going to verify VC3.

Given A= (V,V',v), B = (W,W'.,w), C = (T,T',t), D = (U,U',u) objects

in~ , then by iii) we have the commutative diagrams of Figure 2.

Coherence of V and property of M(A,B,C) imply that subdiagrams (1) and

(2) of Figure 2 commute; similarly (1') and (2') commute.

Now we apply the same argument as in proving VCl, i.e. the maps id®M' M'

~(C,D)®(~(B,C)®~(A,B)) ~(C,D)®~(A,C) ~(A,D)

M'®id M' --=--+~(A, D)

are the maps induced by pulling back. We only have to show that the composition

a(V(C,D), ~(B,C), V(A,B)) (~(C,D)®~(B,C) )®.Y(A,B - - '.Y_(C,D)®(.Y_(B,C)6)~_(A,B))

M' ~(A,D)

111

}':(C,D)®(}':(B,C)®}':(A,B)) _____ ___..p:.::l:..::®_,_(p~;::l=-:®p:..t::=l.;_) ______ _,_, (T,U)®((W,T)®(V,W))

lp2o(p2op2) ~ (U', T' )®((T' ,W')0(W', V')) }':(C,D)0}':(A,C) ----"-'l=-®--"'-"l'---------~

(1) lidOM

(T,U)0(V,T)

1M lid®s (2) jp2~2 ~ (U', T' )®(T', V') }':(A,D) 1 (V,U) (U' ,T')®((W' ,V')®(T' ,W'))

1 s p21 p.b. lu v ((W', V' )®(T' ,W' ))®(U', T')~(T', V' )®(U' ,T' )----=-M=---~(U', V') _ ____:_ __ -+ (V®U' ,X)

(y(c, D) ®}':(B, C)) ®Y(A,B) -------'('-"p-=l.c.®J?~:..:l::..

112

a (T,U)®((W,T)®(V,W)) ((T,U)®(W,T))®(V,W)

l(pl®pl)®pl

(!(C,D)®!(B,C))®!(A,B)

1 pl®(pl®pl) -----"a'------->- !(C,D)®(!(B,C)®!(A,B))

l(p2®p2)®p2

((U' ,T')®(T' ,W'))®(W' ,V')

ls®id

((T~ ,W')®(U' ,T'))®(W', V')

(W' ,V')®((T' ,W')®(U' ,T'))

a

-1 a

1p2®(p2®p2)

(U' ,T')®((T' ,W')®(W' ,V'))

~id®s (U' ,T')®((W' ,V')®(T' ,W'))

((W' ,V')®(T' ,W'))®(U' ,T')

As for the second part, we observe a simple fact of ! : two

permutations of the tensor product of any three fixed objects are coherently

isomorphic. Therefore it is enough to show the following diagrams commute:

((T,U)®(W,T))®(V,W) a (T,U)®((W,T)®(V,W))

(W,U)®(V,W) (T,U)®(V,T)

~ (V,U)~ ((W' ,V')®(T' ,W'))®(U' ,T') a (W', V' )®( (T', W' )®(U', T'))

lid®M

(T',V')®(U',T') (W',V')®(U',W')

~(U',V')~ This follows trivially from coherence and completes the proof.

113

CHAPTER III: ~ HAS A *-AUTONOMOUS STRUCTURE

1. The Hom-Functor ~(-,-)

Definition. Given any two objects A = (V,V',v) and B = (W,W',w)

in~, define an object ~(A,B) = (~(A,B), V®W', n) in ~X as follows:

First of all recall ~(A,B) is the object in V such that the follow-

ing diagram is a pullback.

~(A,B) pl

' (V,W)

p21 p.b. lw

(W', V') v (V®W' ,X) Since we require ~(A,B) to be an object in ~ , n has to be a

morphism in~. which sends ~(A,B)®(V®W') to X.

It seems there are two (canonical) alternatives for defining n:

(1) Since the above square commutes, let n' be the morphism (along either

route) which sends ~(A, B) to (V®W' ,X) , and define

n:~(A,B)®(V®W')----+ X to be the transpose of n'.

(2) Again since the above square commutes, we have the following

commutative diagram:

~(A,B)®(V®W')

p2®idl

pl®id ' (V,W)®(V®W')

(W' ,V')®(V®W') V®id (V®W' ,X)®(V®W 1 )

Now let ev:(V®W' ,X)®(V®W')-----+ X be the evaluation map, then put

n" ev composed with the above map ~(A,B)®(V®W') --->- (V®W' ,X)®(V®W').

But since V is coherent, it is easy to verify that n is identical

to n', so these two definitions are same.

For the rest of this section we shall prove ~X(-,-) is a bifunctor

which sends ~ op x ~ to ~ . We have to show

i) given any object B = (W,W',w) in~, F

is a contravariant functor;

ii) G = ~(B,-) is a covariant functor;

~x(-,B)

iii) Given A--->- B, C-->- D in~ , then the diagram

114

~X(B,C) ~X(A,C)

1 l ~X(B,D) ~X(A,D)

commutes.

Recall if C; (V,V' ,v) and A; (P,P' ,p) in ~X and (f,g):C ~A

is a morphism in ~X' then the square:

V®P' id®g > V®V'

1f®id 1v

P®P' X

commutes.

In order to show F is contravariant, we must find a map (in ~X)

F(f,g); (f',g'):~(A,B) ----+ ~X(C,B) .

By definition ~(A,B) ; (~(A,B), P®~', n1

) and ~X(C,B)

(~(C,B), V®W', n2

) ; so the choice for g' is clear: g' ; f®id:V®W' ~ P®W'

As for f' , consider the following diagram: ~(A,B)-------L~1-------+ (P, W)

-~P'-'1=-----r> (V, W) ~ (*) p2

j

~(C,B)

p2 p.b.

(W' ,V')-~v __ _,

/'d,g)

(W' ,PI)

(1)

(V®W' ,X)

~d) (P®W' , X)

We know the outer square commutes, therefore it suffices to show (1)

and (2) are commutative.

For (1), we prove it by looking at the following commutative diagram:

(P,W) (f. id) (V,W)

/, (P, (W' ,X))~ (P®W' ,X) (f®id, id) ~ (V®W',X) (V,(W',X))

115

As for (2) we have a similar diagramatical proof:

(W' ,P) -----'-(=-id:::.z..,, g"')'-----+> (W' , V' )

lp lv (P®W' ,X) (f®id,id) > (V®W' ,X)

lcs,id) lcs,id)

(W', (P,X)) (W'®P,X) (id®f,id) (W'®V,X) (W', (V,X))

But in this case the commutativity of the outer square is due to

the fact that (f,g) is a morphism which sends C to A (hence the

diagram (*) above commutes) .

This implies that there is a unique map f' :~(A,B) ---+ ~(C,B)

induced by pullback such that the diagram

l (W' ,P')

commutes.

1 (P, W)

/.id)

(V,W) 1

p.b. l· . (W', V') (V®W' ,X)

"' ~ (f®id,id) I (id,g) ~ -------"--------+ (P®W' ,X)

Therefore the following diagram commutes:

~(A,B) 2 7(W', P) (P®W' ,X)

l(id,g) l(fOid,id)

~(C, B) 2

(W', V') v (V®W' ,X)

This implies that:

~(A,B)®(V®W') id®g' ----==-=---+> ~(A,B)®(P®W')

~(C,B)®(V®W') X

commutes.

116

Therefore (f',g') has the property required of a morphism in

~· It is trivial to see that F(idA) = idF(A)

F preserves composition.

Now we have to show

Suppose A= (P,P' ,p), C = (V,V',v) and E = (U,U' ,u) are three

objects in~, moreover (f,g):E----+ C and (h,k):C----+ A then we

want to show that

(h' ,k') !x (A, B) -------''-'----''-=---"'------+ !x ( C, B)

((hof)',(gok)') ~ / (f',g') ~(E,B)

commutes.

By definition: ~(A,B) (~(A,B), P®W', nl)

~(C,B) (~(C,B), V®W', n2)

~(E,B) (~(E,B), U®W', n3)

Now we consider the following commutative diagram:

V(A,B) 1

~ V(C,B) l (V,W) - ~ (f,iy

p2 y(F ---":'-=~'---b--+: l' (W' ,U') ------'il=-----+(U®W' ,X)

p2

~d,g) - (f®id,~ (W',V') v (V®W',X)

~ (h®~ (W' ,P') -------------~-------------+(P®W',X)

117

But the following diagrams also commute:

(W',P') (id k) (W',V')

(id,g•k) l ~ cw• ,u•)

(P®W' ,X) (h®id,id) (V®W' ,X)

((hof)®id,id) j ~ ~ (f®id,id)

(U®W' ,X)

(hof,id~Pj,W) (h,id/ (V,W) ~ (f,id)

(U,W)

This implies that the map induced by pullback is identical to f'oh',

and clearly k'og' = (h®id)o(f®id) = ((hof)®id) = (gok)' . Hence F is a

contravariant functor.

As for G, we have a similar series of diagrammatical proofs: Suppose

B = (W,W' ,w), A= (P,P' ,p), C = (V,V' ,v) are objects in~ with (f,g):

A----+ C a morphism in~. We need G(f,g) = (f' ,g'):G(A)----+ G(C).

By definition

G(A) = ~(B,A)

and

G(C) = ~(B,C)

Hence the choice of

g' = id®g:W®V 1 -----+ W®P'

is clear. And the following commutative diagram shows the existence and

uniqueness of f':

118

~(B,A) pl

(W ,P)

~(B,C) pl

(W, V) ~ p2 lp2 p.b. l· p

w ------'-'--------;- (W®V I 'X)

~d)

(W®P 1 ,X)

Again the preservation of the identity is clear.

Now if A= (P,P 1 ,p), C = (V,V 1 ,v), E = (U,U 1 ,u) are objects in

~X and (f,g):A----+ C, (h,k):C----+ Dare morphisms, then the

commutative diagrams of Figure 3 imply G preserves composition.

To prove (iii):

Suppose A= (V,V 1 v), B = (W,W 1 ,w), C = (P,P 1 ,p), D = (U,U' ,u) are

objects in ~X and (f,g):A----+ B, (h,k):C----+ D are maps in ~X

then the following diagrams commute:

~(B,C) 1

(W,P)

~ (W,U)

~ V(B,D)-----1'-"'-2 --->-

pl - lp2 p.b. lu P w

(U' ,W' )----"------>- (W®U' ,X)

~,id) (id®k,~ (P',W') w (W®P 1 ,X)

V(B,D) - ~

p2 Y'T (U', V1 )

hd,g)

1

1

(W, U)

~id (V,U)

p.b. l u a (V®U' ,X)

(g®id,i~ w (U I' W' )---------'~------>- (W®U' ,X)

119

(W, P) _ _,(=i.=cd ''-"£_,_) --+ (W, V) (W0P I, X) (id®g, id) ) (W®V' , X)

(id,h•f) \ ~) (Jd0(gok),id)\ ~,id) (W0U 1 ,X) (W, U)

(P' ,W') (g,id) > (V 1 ,W')

(g•k,id)j ~ (U' ,W')

~(B,A) 1 (W,P)

~ (i:Y ~(B,C)

1 (W, V) •

~ pl (idy '{_(B,E) > (W, U)

I

p2 p2

)'' p.b. ii v p

(U 1 , W1 ) w

(W0U 1 ,X)

/'·''' (idOk,i~ (V' ,W 1 ) w (W®V' ,X)

~) (>d®g,i~ (PI, W1 ) w (W0P I ,X)

FIGURE 3.

120

1 !(A,C) h 11 (V,P)

~ (id,~ V(A,D) --=1=------> (V,U)

p2 -lp2 p.b. [u. p

(U 1 , V 1 )

~) ("\I ,X)

(k®id,id)'\

(P 1 , V1 ) (V®P 1 ,X)

!(B,C) 1 (W ,P) (7 (V,P) V(A,C) --------~1----------

~ (i:;v !(A,D) ~(V,U)

p2 p2 ]'' p~b. ]"

(U 1 , V1 )~(V®U 1 ,X)

~id) (k®id,i~ (P 1 , V 1 ) ________ v;_-----------+ (V®P 1 , X)

~ (f~ (P 1 ,W 1 ) --------------------~w~------------------~ (W®P 1 ,X)

FIGURE 4.

This implies that the first diagram in Figure 4 commutes which implies, in

turn, that the second one does.

Applying the same argument, the map from !(B,C) to !(A,D) induced

by pullback is the same as h"of" hence the following diagram commutes:

f"

fl !(B,D) ----=----+ !(A,D)

121

Next consider Figure 5. Since the center square of the first diagram

is a pullback, f'oh' is the unique map ~(B,C)----+ ~(A,D) that

makes the diagram commute.

Next consider the lower diagram of Figure 5. Using this and the

fact that the following diagram commutes:

V®U' id®k V®P'

)'""' id®k

l'•id W®U' W®P'

~(B,C) 1 (W,P)

(i~ '~ 1

V(B,D) - f' ~

V(A,D)

(W, U) (f,id)"""

---'p"-'l"--->-> (V, U)

p2 p2 -lp2 v

lu (U', V') --'--->(V®U' ,X)

~id,g) ~f®id,id~ (U' ,W') w (\J®U' ,X)

~d) (id®k,~ (P' ,W') -------------------w~----------------------+(W®P',X)

1 (W ,P) (f/

(V,P) 1

p.b. l' ' (P', V')

v ____ _:_ ___ -+(V®P' , X)

(f®i~. (W®P' ,X)

~) (P',W') ----------------~w ________________ -+

FIGURE 5.

122

we obtain the desired result that the diagram

------>-~X (A, D)

commutes.

2. The Functor*·

In this section we shall define a functor *:~op -----~~X

and examine its relationship with ~X(-,-) .

Definition. Given any object A; (V,V' ,v) in~ define *(A) to be the

object (V',V,vos) where s:V'®V ~ V®V'~ X is a map in V.

Suppose B; (W,W' ,w) is another object in ~X and (f,g) :A----+ B

a morphism in ~X , then define *(f,g) ; (g,f) :*(B)-----+ *(A). This

definition is justified since the commutativity of the diagram:

V®W 1 id®g I V®V'

jf®id jv

W®W' w X

implies that the diagram

W'®V id®f

W'®W

jg®id jwos

V'®V X V 0 S

commutes.

From the above formula on morphisms we can easily conclude that * is

a functor.

Moreover * has an inverse (contravariant), since *a* idA -x

The following are some properties of *:

ProEOSition 1. Given A ; (V,V',v) '

B ; (W,W' ,w') in ~X ' then

~(A,B) = ~(*(B),*(A)) PROOF. By definition *(A) = (V' ,V,vos) and *(B) (W' ,W,wos).

Consider the commutative diagram of Figure 6.

Notice that the coherence of V implies squares (1), (2), (3), (4)

commute. It also implies that the diagram

(V®Wj' ::'

123

conunutes.

The fact that

~(*(B) ,*(A)) :el > (V ,W)

jp2 p.b. ]·~" (W', V')

V 0 S (W'®V,X)

is a pullback square implies that there exists a unique p:~(A,B)---+ ~(*(B),*(A))

such that the diagram of Figure 6 still conunutes. Similarly the pullback

square involve ~(A,B) induces a unique map q:~(*(B),*(A))---+ ~(A,B) such

that the diagram of Figure 6 conunutes. This implies qop is the map

induced by the pullback square:

~(A,B) 1

(V,W)

jp2 p.b. l· v (W', V') (V®W' ,X)

~(A,B) 1 (V,W)

1 7

~(*(B(,*(A)) (V,W)

pl /

~(A,B) > (V ,W)

p2 p2 ]'' p.b. j· wos w v

(W', V') (V®W' ,X)

/< (s,~ (W', V') (W'®V,X)

(s~ (W', V') (V®W' ,X)

FIGURE 6.

124

But by the remark above id~(A,B) also has this property. Hence

qop id~(A,B) . Now switch ~(A,B) and ~(*(B),*(A)) in the previous diagram, and

apply the same argument to conclude that poq ~ id~(*(B),*(A)) .

completes the proof.

This

Corollar:l· Let A,B be two objects in !2.x , then ~(A,*(B)) ~(B, *(A)) PROOF. For any object c in !2.x , *(''(C)) ~ c. Corollary. Let A ~ (V, V' ,v), B ~ (W,W' ,w), be two objects in !2.x , then f2.x(A,B) ~ f2.x (*(B),*(A)).

PROOF. By definition *(A) ~ (V' ,V,vos), *(B) (W', W ,wos) which

implies that f2.x(*(B),*(A)) ~ (~(~),*(A)),W'®V,n1 )

But recall that f2.x(A,B) ~ (~(A,B),V®W',n2 ) moreover we have isomorphism p:~(A,B) -->- ~(1- ~(A,B)

such that id~(A,B) ~ qop , id~(1- W'®V and S(W' ,V):W'®V-->- V®W' such that s(V,W')os(W',V) ~ i~'®V

and s(W',V)os(V,W') ~ idV®W'.

Hence it is sufficient to check that the pair (p,s(W',V)) is indeed

an isomorphism in f2.x

commutative diagram:

But we see this by considering the following

~(A,B) ~(*(B), 1

125

~(B,*(C)) pl

> (W,U') ~(B,*(A)) pl

> (W, V')

(,, Wos !":'

(,, lv:, (U,W') (W®U,X) (V,W')

W 0 S (W®V,X)

Now consider Figure 7. Since (U,-) and (V,-) have left

adjoints, they preserve pullbacks, hence the outer and inner squares are

still commutative.

But (1) is a pullback, hence the following diagrams commute:

(V, (U, W' )) _ _._p_o"'s _ __,, (U, (V, W' ) ) (W®U, V) ---"''----+ (V® (W®U) , X)

j. j. j, (V, (W®U,X)) (U, (W®V, X)) (U,(W,V')) (U®(W®V),X)

j,-1 ~/ (V® (W®U) , X) _ _______.::s::_______,_ (U®(W®V) ,X) (U, (W®V, X))

~(A,Bc) _____________ P~:.:l'----------7wou, v• l

pl (U,Ba)(id,pl~(U,(W,V')) ~(C,Ba)

j (id, p2) (1) j (id, V) p2 (U, (V,W'))

(id,-w) (U, (W®V,X))

p2 /. j,-1 v (W®V, U')

u (U®(W®V),X)

,-/ l (V ,~(B, *(C))) (V, (U, W'))

''1 p.b. (id,-w) (id,u)

-1 (V, (W, U' )) (V, (W®U, X)) (V®(W®U) ,X)

FIGURE 7.

126

Tgis implies that there exists a unique map ~(A,Bc) ---+ (U,Ba) such

that the diagram of Figure 7 still commutes. Now using the fact that

1 (U,Ba)

j·, (W®V, U') (U®(W®V), X)

is a pullback, there exists a unique map ~(A,Bc) ~ ~(C,Ba)

A similar argument (Figure 8) shows the existence of a map

p:~(C,Ba) ~ ~(A,Bc)

Applying the same argument as in the previous proposition, we

conclude that poq = id~(A,Bc) and qop = id~(C,Ba)

Corollary. If A, B, C are objects in~ , then

PROOF. Apply the same argument as in previous corollary.

Corollary. Let A, B, C be objects in ~X , then

PROOF. ~(A,~X(B,C)) = ~x(*(*(A)),~X(B,*(*(C)))) = ~x(*(C),~X(B,*(A)))

Remark. These propositions and corollaries concerning the duality lay

the foundation of our construction, as we shall see later on.

3. The Functor -®-

Note: Henceforth we write, for an object A of ~X, A* instead of *(A).

Definition. Given A,B objects in ~X , then define A®B = ~X(A,B*)* •

It is clear that -0- is a bifunctor, since -®- is the composition

(*,*) ~ A~

(id,*) ~X

~X(-,-) A X ~X X ~X ~X -x

127

1 ~(C, Ba) ----------____t=-------------+(W®V, U 1 )

/, ~(A,Bc) __ P~1~~, (V,V(B,*(C))) (id,p1),

(id,~2) l p.b. (V, (W, U1 )) [(id,u) p2 p2 (V,(U,W 1 )) --'("'i,_,d_,_, W=._,-)'------+-• ( v ' (W®U' X) )

I" (W®U,V 1 ) ------~------+-~-1 ~os

(U,Ba) (id,p1) '

r7 (U, (V,W 1 ))

p.b.l (id,W) -1

lp-1

(V®(W®U),X)

(U' (W' vI)) ( id' v) (U' (W®V' X)) ____ __j"-------------+(U® (W®V) 'X)

FIGURE 8.

Proposition. Let A,B be objects in ~ , then

A®B : B®A.

PROOF.

Proposition. Let A,B,C be objects in ~ , then

PROOF.

(A®B) C : A®(B®C)

(A®B)®C = !x(A,B*)*®C

= !x(!X(A,B*)*,C*)*

: !x(C,!X(A,B*))*

= ~(C,~(B,A*))* : !x(A,~(B,C*))*

: !x(!x(B,C*)*,A*)*

: !x(A,!X(B,C*)**)*

= A®!x(B,C*)*

= A®(B®C).

128

4. The Dualising Object and the Unit fvr Tensor.

Let T = (X,I,r) be the object in !x , such that r:X®I----+ X

is the cannonical isomorphism in V.

Claim. T is the dualising object, i.e. for any object A in !x·

!x(A, T) =A *

PROOF. Let A = (V,V',v) be an object in !x , then we have the

following commutative diagram

Y._(A, T) 1 (V, X)

p2j j' (I, V') v (V®I, X)

l' l' V' (V, (I, X))

j (id,i) (V,X)

But

V' v (V,X)

lid lid V' v (V,X)

is trivially a pullback in ':!_, which implies that we have an induced

(unique) morphism f:Y._(A,T)----+ V'. Apply the same argument to get a

unique map g:V' ----+ Y._(A,T) such that fog= idV' and gof

* Corollary. T is the identity for -®-PROOF. Suppose A is an object in !x , then

* * * * T ®A = !x(T ,A )

* = !x(A, T) ** =A

= A

idY._(A,T)

129

* On the other hand, A®T * T ®A~ A. This completes the proof. Theorem. Let A,B,C be objects in ~· then

PROOF.

Proposition. Let

PROOF.

Remark.

~(A®B,c) ~ Ax(A,~(B,C)) .

* * Ax(A®B,C) = Ax

130

2. The Double Envelope.

Definition. Given a symmetric monoidal category with a faithful functor

1-1 :s/->- ~, we denote the double envelope of C by E(~). The objects of E(~) are all triplets (F,G;t) where F and G are functors

from C0 to ~,t is a natural transformation from F x G to 1-®-1. A morphism from (F,G;t) to (F',G';s) in E(~) is a pair (f,g) where

f is a natural transformation from F to F' and g is a natural

transformation from G' to G such that the following diagram

F(C) X G' (C') ---=id~x~g~~, F(C) x G(C')

F' (C) X G' (C') _ ____::S:___-* I C®C I I

commutes for every object (C,C') of C0 x C0

Proposition. E(~) is a category.

PROOF. Suppose (f,g): (F,G;t)-----+ (F',G';s)

(f',g'): (F',G';s)-----+ (F",G";u) are maps

in E(~) , then the following diagram commutes for every (C,C') in

~ x Q_o.

F(C)XG" (C')

lidXg'

F(C)XG' (C')

lidXg

F(C)XG(C I)

__ f:c;X;.::i:..:d:..__~ F I (c) XG" ( c I )

lidXg'

fXid --==-=----+F' (C)XG' (C')

f'Xid F" (C)XG" (C')

u

~ -------------------=t---------------------+ C®C'

This implies that (f,g):(F,G;t) (F',G';s)

(F", G"; u)

(F"' ,G"';v)

(f' ,g'): (F' ,G' ;s)

(f" ,g"): (F" ,G";u)

are maps in E (~) , then ( f", G") o ( ( f' , g' ) o ( f, g) ) (f", g") o (f 1 of, gog 1 )

( f 11 0 ( f I of) ' (gog I ) o g11 )

((f 1 of 1 )of,go(g 1 og"))

(f 11 of 1 ,g' og") o(f,g)

( ( f 11 ' g") 0 ( f I 'g I ) ) 0 ( f' g) •

Moreover, given (F,g;t) then (idF,idG) is the obvious choice for identity.

Before proving the main theorem co

vestigate the functor categor~es ~

two obvious embeddings of s~ into

131

of this chapter, C0 X C0

and s-C0 X co s- namely

r(F) ; I x F for every F in

let us in-

There are

~ and r, co

s- , and ~here ~(F) ; F x I and co

I is the unit in S-

(the terminal object) in C0 X CO

objects in s-

i.e.

s. I sends every object into the singl5ton

Hence we can regard objects in S~ as via either embedding. Now we can prove.

C0 X CO is enriched over V ; S-Proposition. E(~)

PROOF. By previous remark V is a closed symmetric monoidal

category with pullbacks, moreover it is coherent.

Now given A; (G,F;t) and B0

; (G~,F';s) in E(~) we have to

define ~(A,Bb an0object in ~(; ~ x ~) • Suppose (C,C') is an

object of S~ x C , then V(A,B) is the functor whose value at (C,C') is defined by requiring that the diagram

V(A,B)(C,C') 1 U(G) .~(G') (C,C')

p2

(r(F'),r(F))(C,C') (G X F', 1-®-I)(C,C')

be a pullback.

Note. (-,-) denotes the internal hom-functor of As for the

map (~(G),~(G'))(C,C') --r (G x F', 1-®-I)(C,C'), we simply observe

that in ~. G x F' is isomorphic to ~(G) x r(F'). Then the adjoint

property of V constructs such a map (in the same fashion as in Chapter

II, Section 2.) A similar argument constructs map

(r(F'),r(F))(C,C') ( G X F' 1-®- I ) ( c 'c I ) • Now the enrichment follows immediately from the result in Chapter

II, since this is how pullbacks are defined in the functor category, i.e.

by point-wise evaluation. This concludes the proof.

Theorem. E(~) is a subcategory of a *-autonomous category !; moreover A is enriched over V.

PROOF. Put X; 1-®-1 then follow the construction in Chapter III.

3. Miscellaneous Results.

In this section, we are assuming V has all the properties as given

in Chapter I and we shall prove that there is a functor F maps V to

v - CAT(~- CAT is the category of all categories which are enriched over

~).

The functor F on objects of V is obvious: given X in ~. then

put F(X) ; ~--

132

Now we have to show given a map f:X ------~

a V-functor T( = F(f)) from ~ to !s· S in ~. this induces

The notion of a V-functor can be found in [Eilenberg & Kelly] Chapter

II, Section 6. In this case we have to show:

(i) a function T maps objects of ~ to objects of !s· (ii) for each B,C in !x· a morphism T(B,C) maps ~(B,C) to ~(T(B), T(C)) in V such that the following axioms are satisfied:

(1) The following diagram commutes:

~(B,B) T

-----"-------+ ~(T(B) ,T(B))

j

I

(2) The following diagram commutes:

~(C,D)®~(B,C) M' ___ ___:_: ___ ----+ ~ (B, D)

T®T T

~(T(C),T(D))®~(T(B),T(C)) M'

-----''-'----------+ ~ (T (B) , T (D) )

Note. In both categories we denote the enriched object by ~(-,-) , it

is clear from the context which one we are referring to.

The function T on objects of ~ is obvious; given B = (V,V',v)

~· then T(B) is the composition V®V' v

X f s i.e.

T(B) = (V,V',fov).

To show (ii):

Suppose B = (V,V',v) C = (W,W',w) objects in !x• then T(B) (V,V',fov), T(C) = (W,W',fow) and the following diagram commutes

~(B,C)

p2

(W', V')

1 (V,W)

Y(TFT(C)) 1 v:d (V ,W) 1

/,:') p.b. )t~

--------~f_o~v----~(V®W' ,S)

(id,f~ (V®W' ,X)

in

133

Since the inner square is a pullback, there exists a (unique) map

T(B,C) from ~(B,C) to ~(T(B),T(C)).

To show (1) commutes let B = (V,V',v) in ~· Then T(B) = (V,V',fov) and the following diagrams cornrnu te:

:~

I

p.b. p.b.

__.

(V', V') ----'"---+(V®V' ,X) fov (V', V' )---=-..:......--->-

1 Y

134

T Hence the composition I _ ___.,___-+ ~(B,B)--=---+ ~(T(B), T(B)) and

map I _ __.,____,_~(T(B),T(B)) are both induced by pulling back. Thus by

the uniqueness property they are "equal".

To show (2) commutes, let B = (V,V 1 ,v), C = (W,W 1 ,w), D = (U,U 1 ,u)

be three objects in ~. Then T(B) = (V,V 1 ,fov), T(C) = (W,W 1 ,fow),

T(D) = (U,U 1 ,fou) and the following four diagrams commute:

~(C,D)®~(B,C) ----~P~1®r~1~----~• (W,U)®(V,W)

(U 1 ,W 1 )®(W 1 ,V 1 ) 1 ~(B,D) --. (W, Ur, W ~(T(B) ,T(D))-P (V,U) (U' ,W 1 )®(W' ,V 1 )

p.b.

~

(W' ,V 1 )®(U' ,W') ---"M'----+-(U I 'v') --=-f_ov'--~(V®U' ,S)

~(C,D) 1 (W,U)

~ A ~(T(C), T (D))

pl '(W, U)

p2 p2l l':" ij

(U 1 ,W 1 ) fow

(W®U 1 ,S)

/o w (id,~ (U' ,W 1 ) (W®U' ,X)

135

V(B,C) ------"-'1=-------->- (V,W)

- ~ /td ~(T(B),T(C)) P1 >(V,W)

-p.b. fow W p2 r -

(WI 'VI ) __::f:__o_:__v -----+(V®W I ' S)

(id~ v ,t) ""

(W 1 ,V 1 ) ---------'-------+ (V®W 1 ,X)

~(C,D)®~(B,C) ------------------>- (W,U)®(V,W)

~ 71 ~(T (C), T (D))®~ (T (B) , T (C)) _ _..:P.:::1 ®p--"-=1'-------+> (W, U) ®(V, W)

M

M

p2®p2 (T(B),T(D)) ~ (V,U) (V,U)

p2 p.b. fou

-(W 1 ,V')®(U 1 ,W 1 ) ~(U1 ,V 1 ) ~(V®U1 ,S) u

/- (id~ (W 1 ,V 1 )®(U 1 ,W 1 ) M (U 1 ,V 1 ) v (V®U 1 ,X)

136

V(C,D)®V(B,C) l®pl (W,U)®(V,W) - - ~ tM

~ ~(B,D) pl®pl ' (V ,U)

~ r ~(T(B),T(D)) ~ (V,U)

p2®p2 p2 tp2

p.b. V;u

fov u (U' V') (V®U' ,S)

~· (idA (W',V')®(U',W') M (U', V')

ij (V®U' ,X)

This implies that the diagrams 3bove commute, which implies

that the composition

~(C,D)®(~(B, C) )-T~®J',.____., ~(T(C), T(D)) ®~(T (B), T(C)) M' -~--+ ~(T(B),T(D))

is the map induced by pulling back.

This also implies that the composition

~(C,D)® ~(B,C) M'

_ _o_:_ _ _,_ ~(B,D) T

-~--+ ~(T(B), T(D))

is the map induced by pulling back.

Hence by the uniqueness property, they are "equal", therefore (2)

commutes.

Now we are left to show that if f:X----+ S and g:S----+ K are

maps in ~. then F(g)oF(f) F(gof), i.e. F preserves composition.

All we have to check is that the composition is preserved in (i)

and (ii).

It is easy to show (i) is preserved. For if B (V,V' ,v) in ~ ,

then

(F(g)oF(f))(B) F(g) (F(f) (B)) F(g)(V,V',fov)

(V,V',go(fov))

(V, V', (go f) ov)

F(gof) (B).

To show (ii) is preserved: Let B = (V,V' ,v), C = (W,W' ,w) in ~X' then

F(f)(B) (V,V',fov),F(f)(C) = (W,W',fow),(F(g)oF(f))(B) = F(gof)(B)

(V,V',(gof)ov) ,F(gof)(C) = (F(g)oF(f))(C) = (W,W' ,(gof)ow)

and the diagrams (*), (**) and (***) commute

137

V(B,C) --------"-'1:::.._ ______ _. (V ,W)

(*)

-~ v y(F(f)(B) ff)(C)) -:-"· :c=.:-· T~·

-~f_ov"-----+(V®W' • S)

p2

(W', V')

(id~ (W' ,V') ---------"-'---------+ (V®W' ,X)

/ 1

~(F(f) (B) ,F(f) (C)) (V,W)

(**)

~ 7 V(F(gof)(B),F(gof)(C))~(V,W) - lp2 p.b. 1~) f:w

(g:f') (W' V') -----'-"'---"-L----+>(V®W' ,K)

/- ' (id,go~ fov . ""' (W',V') (V®W',S)

p2

Note. F(gof) (-) (F(g) oF(f)) (-).

V(B,C) l (V,W)

- ~) ~ V(F(gof)(B),F(gof)(C))~(V,W)

(***) p2 - lp2 ~- 1~ w (W', V') (go f) ov > (V®W' ,K)

~ v (i~ (W', V') (V®W' ,X)

But (*) and (**) imply the diagram of Figure 9 connnutes.

This implies that both F(gof) in (***) and the composition

~(B,C) ---=F--"(f::.L) __ ~(F(f) (B) ,F(f) (C)) --=-F->-

138

1 ~(B,C) (V,W)

~f) y V(F(f)(B),F(f)(C)) l (V,W)

~) y V(F(gof)(B),F(gof)(C)) ~ (V,W)

p2 - lp2 ------ 1 ~w _ __,("'g'--o-"'f-"-)_o...:.v___,_, (V®W 1 , K)

p2

(W', V')

fow

(i~ (V®W' ,S) (W', V')

~ fov /. (id,f~

(W', V') (V®W' ,X)

FIGURE 9.

BIBLIOGRAPHY

1. M. BARR, Duality of vector spaces, Cahiers Topologie Geometrie

Differentielle, XVII-1 (1976), 3-14.

2. M. BARR, Duality of Banach spaces, Ibid., 15-32.

3. M. BARR, Closed categories and topological vector spaces, Ibid.

XVII-3, 223-234.

4. M. BARR, Closed categories and Banach spaces, Ibid. XVII-4, 335-342.

5. M. BARR, A closed category of reflexive topological abelian groups,

Ibid. XVIII-3, 221-248.

6. M. BARR, *-autonomous categories, This volume.

7. S. ElLENBERG, .G.M. KELLY, Closed categories, Proc. Conf. Categorical

Alg. (La Jolla, 1965), Springer (1966), 421-562.

8. M.E. SZABO, Commutativity in closed categories. To appear.

Index of Definitions

Admissible (uniform object) Autonomous (category) 13

Basis (for a pseudometric)

Completable 9 Convergence uniformity Cos mall 24

Dominating 29 Double envelope 130 Dualizing module 49

Embedding 8 Entourage 8

Linearly compact 18 Linearly totally bounded

\1 complete 33

Nuclear 47

Pre-reflexive 23 Pre-*-autonomous 15 Pre-uniform structure 8 Product uniformity 10 Pseudomap 5 Pseudometric 7

8

7

28

20

9 Quasi-reflexive 23 Quasi-variety 3

Refine (of seminorms) 71 Reflexive 23 Represent 28

Semi-norm 65 Semi-variety 3 Separated uniform 6 *-autonomous 13

Uniform convergence Uniform cover 6 Uniform object 10 Uniform space 6

Variety 3

on 28

V-enriched pre-*- autonomous 17 V-enriched *-autonomous 17

~;;-complete 34 ~;;-*-complete 35

Index of Notation

~(-,-) 2 II 33

5 )I:!. 33

1-1 6 r 33 9 s 34

-(-) 9 8 37 UnV 10 T 37

[-,-] 11 1:!. 43

T 13 1:!. 0

44

* (-) 13 u 52 (-)·(-) 18 h(1) ®h(2) 59

" ~·!2 19 (-) 74 A 19 :!x 104 -~(-,-) 25 E(~) 130

~ 28