BIBLIOGRAPHY M. Barr, Duality of ector spaces, Cahiers Topologie Geometrie Differentielle l (976),3-4. M. Barr, Duality of banach spaces, Cahiers Topologie Geometrie Differentielle l (976), 5-32 M. Barr, Closed categories and topological ector spaces, Cahiers Topologie Geometrie Differentielle l (976), 223-234. M. Barr, Closed categories and banach spaces, Cahiers Topologie Geometrie Differentielle l (976), 335-342. M. Barr, A closed category of reflexie topological abelian groups, Cahiers Topologie Geometrie Differentielle 8 (977), 22-248. M. Barr, The point of the empty set, Cahiers Topologie Geometrie Differentielle 3 (973), 357-368. S. Eilenberg, G.M. Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, 965), Springer-Verlag, 966, 42-562. E. Hewitt, K.A. Ross, Abstract Hamonic Analysis, Vol. I, 963, Springer-Verlag. K.H. Hofmann, M. Misloe, A. Stralka, The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Lecture Notes Math. (974),. Springer-Verlag. J.R. Isbell, Uniform Spaces, Amer. Math. Soc. Sureys no. 2, 964. J.L. Kelley,General Topology, Van Nostrand, 955. G.M. Kelly, Monomorphisms, epimorphisms and pull-backs, J. Austral. Math. Soc..(969), 24-42. F.W. Lawere, Functional Semantics of Algebraic Theories, Dissertation, Columbia Uniersity, 963. S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloquium Publications, Vol. XXVII, 942. F.E.J. Linton, Some aspects of equational categories, Proc. Conf. Categorical Algebra (La Jolla, 965), Springer-Verlag, 966, 84-94. A. Pietsch, Nuclear Locally Conex Spaces, Springer-Verlag, 972. H.H. Schaefer, Topological Vector Spaces, third printing, Springer-Verlag, 970. Z.Samadeni, Projectiity, injectiity and duality, Rozprawy M a t (963).. A. Wiweger, Linear spaces with mixed topology, Studia M a t (96), h. 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 preious paper. Using the MacLane Kelly coherence conditions (see [MacLane,Kelly]), M.F. Szabo has proed the following useful theorem [to appear]. Theorem: A diagram commutes in all closed symmetric monoidal categories iff it commutes in the category of real ector spaces. This theorem not only points out the notion of closed symmetric monoidal category is a 'correct' generalization of the category of ector spaces, but it also proides a ery easy method to check if a diagram is commutatie in any closed symmetric monoidal category. The following is a collection of easy consequences of this theorerr which we shall use later on: Corollary. Gien A,B,C objects in V and map A B C, then the following diagram commutes: (A,A) (B,B) (A B,C) where the map_ (A,A) ( B, is C ) the composition (A,A) ( d,(a,(b,c)) f ) (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 Corollary 2. following diagram commutes: f Gien A,B,C,D,F objects in V and map B C---+ F, then the (A,C) (D,B) lid (id,f) (A,C) (D,C,F)) lid p- (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) (id, f) if (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) M (D,(A,F)) - p - (A,(D,F)) (A D,F) (s,id) (D A,F)
04 PROOF. It is easy to check that the diagram commutes in the category of real ector spaces. Remark. The word "coherence" is going to appear frequently throughout this paper. In particular, if the commutatiity of a certain diagram is said to be implied by coherence, we understand that its commutatiity follows easily from this theorem. Our second assumption on is that it has pullbacks. Since almost all interesting examples of closed symmetric monoidal categories hae this property, this restriction is not too drastic. The following is a collection of examples satisfies our assumption: (i) The category of ector spaces oer a fixed field K; (ii) (iii) (i) () (i) The category of Banach spaces; The category of compactly generated spaces; The category of sets (and functions); The category of abelian groups; The category of lattices. An example of a closed symmetric monoidal category that does not hae pullbacks is the category of sets and relations. CHAPTER II: CONSTRUCTION OF AND X ITS ENRICHMENT OVER V.. The Category as follows: Gien an arbitrary object X in. we shall construct a category The objects of consist X of triplets (V,V',) where V,V' are objects in V and :V V' ------+ X is a morphism in - A morphism from (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 W W' w l X commutes.
05 If (f,g):(v,v',)----+ (W,W',w) and (h,k):(w,w',w)----+ (U,U',u) are morphisms in!x then the following diagram commutes: id k id g V U' V W' lf id f id W U' lh id U U' id k W W' u > V V' 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 associatiity of maps in!x can now be erified: If (f,g):(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)) Moreoer, Id(V,V',) = (idv,idv,) is the obious identity. Hence we hae shown that!x is a category. 2.!x is Enriched oer V Definition. If V is a closed monoidal category, then A is enriched oer V if A is equipped with the following: i) For each A,B in!, an object ( A, ib ) n ; ii) For each A in!, a morphism j(a):i---+ ( A, in A 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: c. The following diagram commutes: M' ( A, B ) --=--->- ( A, A ) ( A, B ) l i d ( A, B ) I r
06 c 2. The following diagram commutes: _:{(A,A) _:{(B,A) M' _:{(B,A) VC 3. The following diagram commutes: (V(C,D) _:{(B,C)) _:{(A,B) a _:{(C,D) (_:{(B,C) _:{(A,B)) M'0id lid M' ' ( B, D ) 0 ; ), ( A, C ) _:{(A, D) Gien A= (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) l pl I (V,W) (W', V') -->- (V W',X) Here (V,W) -+ (V W',X) and (W', V') -+ (V W',X) (V,W) (W', V') - w -+ (V, (W', X)) (V W',X) - -+ (W',(V,X)) ( 0WV, X ),X)( V W ' are the right and bottom maps, respectiely. isomorphism in V. Therefore _:{(A,B) is defined up to Gien A = (V,V',) i n, the following diagram commutes, by Corollary : I ---"'i---+ (V, V) (V', V') p.b. ---------+ (V V',X) Uniersal property of pullbacks implies that there exists a unique map j(a) I -+ _:{(A,A) in V such that the diagram
07 I V(A,A).:c::,_, (V, V) -,,] p:b. ]' commutes. in A -x diagram. Now suppose ( V ', (V V' V ',X)) A= (V,V',), B = (W,W',w), C = (T,T',t) are three objects In order to erify iii) it suffices to show the outer square of the pl pl ( B, C ) ( A, B ) ) (W,T) (V,W) j (T',W') (W',V') ( C) A, lm ) (V,T) l$ p.b. F (W',V') (T',W') M (T', V') (V T',X) commutes. Using the fact that - - is a bifunctor and (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 commutatie diagram in Fig.. Note in Fig. that corollary 2 of Szabo's theorem (Chapter I) implies that (2) commutes; coherence implies that () and (3) commute. Again using the uniersal property of pullbacks, there exists a unique morphism M ' ( A, B, C ) : ( B -,- C -)- --( + A in, ( B A V ) such, C ) that the diagram > (W,T) (V,W) (T',W') (W',V') ( A, C ) (V,T) (W',V') (T',W'),_ M (T', V') commutes. Hence i) - iii) are defined. jm p.b. (V T',X)
08 Now we hae to show they satisfy the required axioms. Gien A= (V,V',), B = (W,W',w) i n by construction, we hae the pullback diagram: ( A, B ) (V,W) p.b. lw (W', V') (V W',X) But the coherence of implies that the diagrams of Fig. commute. id pl ( B, C ) ( A, B ) pl id ( B, C ) ( V, W ) >(W,T)8>(V,W) M (V,T) dj id j;.,, id pl (T', W ' ) ( A, B ) d 2 j id W (2) (T',W' ) (V,W) w id (W T',X) (V,W) M (T',W') (W',V') id (T',W') (V W',X) M (V T',X) s (3) (W',V') (T',W') M,_ (T', V') r ( A, B ) I r ( A, B ) ( A, B ) I ( A, B ) lpl id lpl l id l (V,W) I r (V,W) (W',V') I r (W', V') FIGURE.
09 commutes. Hence the following diagram B) I ------"-"===---------> l id (V,W) I ( A, id (V,W) ( A, B ) l (W', V' ) I ::or,_ (W', V') p.b. Since the outer square commutes, there exists a unique map (V W',X) such that () and (2) commute. But the map ( A, B ) I ( A, has B ) this property as well; by pulling back. diagram: ( A, B ) I ( A, B ) therefore it follows from uniqueness that it is the map induced Recall that in the construction of j(a) we hae the following commutatie I (V, V) l ( 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 pl (W',V') (V',V') ( A, B ) l (V,W) p.b. [w commutes. I ( A i pz, B > )(V', V' )0(W', V') ::M:,_(W', V') (V W',X)
0 Again applying the same argument, we conclude that the map id j M' ( A, is B ) the map induced by ( A, B ) I > ( A, B ) ( A, A ) pulling back. diagram: But this is not sufficient to conclude that VCl. holds, i.e. that the M' ( A lid/, B ) ( A, A ) ( A, B ) ( A, B ) I commutes. We are still required to show that the following diagrams commute: El id ( A, B ) I ploij "' (V,W) (V,V) / / M E2 id >. F ' ( A, B ) I j0i > (W',V') I (V,W) (W', V' ) (V', V') I (W', V') ]" That is, that the induced maps satisfy the same commutatie square (therefore they are same by uniqueness). But it is triial once we notice there exist canonical maps (V,W) I id i (V,W) (V,V) in () and I (W',V') i id (V',V') (W',V') in (2) which break () and (2) into two smaller connnutatie squares. VCl. holds. Hence Applying a similar argument, we conclude VC2. is also true. Next we are going to erify VC3. Gien A= (V,V',), B = (W,W'.,w), C = (T,T',t), D = (U,U',u) objects i n, then by iii) we hae the commutatie diagrams of Figure 2. Coherence of V and property of M(A,B,C) imply that subdiagrams () and (2) of Figure 2 commute; similarly (') and (2') commute. Now we apply the same argument as in proing VCl, i.e. the maps id M' M' ( C, D ) ( ( B, C ) ( A (, C B, ) D ) ( A, C )( A, D ) M' id are the maps induced by pulling back. M' - - = - - + ( A, D) We only hae to show that the composition a(v(c,d), ( B, V(A,B)) C ), ( ( C, D ) ).Y(A,B ( B, -C ) - '. Y _ ( C, D ) (. Y _ ( B, C ) 6 ) _ ( A, B ) ) M' ( A, D )
}':(C,D) (}':(B,C) }':(A,B)) _.. p :. : : l :.. : : _, _ ( p ; : : l = - : p :.. t : : = l. ; _ ) _,_, () lo(o) (T,U) ((W,T) (V,W)) lidom (U', T' ) ((T',W')0(W', V')) }':(C,D)0}':(A,C) - - - - " - ' l = - - - " ' -(T,U)0(V,T) " l ' - - - - - - - - lid s (2) j p 2 2 (U',T') ((W',V') (T',W')) (U', T' ) (T', V') }':(A,D) M (V,U) s p.b. lu ((W', V' ) (T',W' )) (U', T ' ) V' () (U' T ',T',) - - - - = - M = V') - -_ - :_ ( U ', -+ (V U',X) (y(c, D) }':(B, C)) Y(A,B) j (0) - - - - - - - ' ( ' - " p - = l. c. J? :.. : l : :.. < ) - ' -,_, - " ((T, ' - p U) (W, " ' - l T)) (V, W) O p 2 i d ((U',T')0(T',W')) (W',V') }':(B,D) }':(A,B) pl pl ' (W,U) (V,W) ('), W ' ) : : ) ) o ( W '. J : : ((T' :::: (U' ls (W',V') ((T',W')0(U',T')) : A, (V,U) D ) p.b. ls lu M (,V') (U' W ',W') --"-'c._--+ (U', V') --'------T (V U',X) FIGURE 2. is also a map induced by pullback and it satisfies the same commutatie square as the map: (}':(C,D)0}':(B,C)) }':(A,B) diagram: M' id }':(B,D) }':(A,B) M' - - = - - }':(A,D) The first part follows easily from the following commutatie
2 ((T,U) (W,T)) (V,W) l(pl pl) pl a (T,U) ((W,T) (V,W)) pl (pl pl) (!(C,D)!(B,C))!(A,B) -----"a'------->-!(c,d) (!(B,C)!(A,B)) l( ) ( ) ((U',T') (T',W')) (W',V') ls id ( (,W') (U' T,T')) (W', V') a (U',T') ((T',W') (W',V')) i d s (U',T') ((W',V') (T',W')) (W',V') ((T',W') (U',T')) - a ((W',V') (T',W')) (U',T') As for the second part, we obsere 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: a ((T,U) (W,T)) (V,W) (T,U) ((W,T) (V,W)) (W,U) (V,W) ((W',V') (T',W')) (U',T') a (T,U) (V,T) ( V, U ) (W', V' ) ( (T', W' ) (U', T')) lid M (T',V') (U',T') (W',V') (U',W') ( U ', V ' ) This follows triially from coherence and completes the proof.
3 CHAPTER III: HAS A *-AUTONOMOUS STRUCTURE. The Hom-Functor ( -, - ) Definition. Gien any two objects A = (V,V',) and B = (W,W',w) i n define, an object ( A, = B () ( A, V W', B ), n) in as X follows: First of all recall ( A, is B the ) object in V such that the following diagram is a pullback. ( A, B ) (W', V') pl p.b. ' (V,W) lw (V W',X) Since we require ( A, to B be ) an object in, n has to be a morphism i n which. sends ( A, B ) to ( V X. W ' ) It seems there are two (canonical) alternaties for defining n: () Since the aboe square commutes, let n' be the morphism (along either route) which sends ( B) A, to (V W',X), and define n : ( A, B ) ( V X to W be ' ) the - -transpose - - + of n'. (2) Again since the aboe square commutes, we hae the following commutatie diagram: pl id ( A, B ) ( V W ' ) ' (V,W) (V W') idl (W',V') (V W') V id (V W',X) (V W ) Now let e:(v W',X) (V W')-----+ X be the ealuation map, then put n" e composed with the aboe map ( A, B ) --->- ( V (V W' W ' ),X) (V W'). But since V is coherent, it is easy to erify that n to n', so these two definitions are same. is identical For the rest of this section we shall proe X ( -, is - a ) bifunctor which sends op x to. We hae to show i) gien any object B = (W,W',w) i n F, x ( -, B ) is a contraariant functor; ii) G = ( B, is - a ) coariant functor; iii) Gien A--->- B, C-->- D i n, then the diagram
4 commutes. X ( B, C ) X ( A, C ) l X ( B, D ) X ( A, D ) Recall if C; (V,V',) and A; (P,P',p) in and X (f,g):c is a morphism in X then ' the square: A V P' id g > V V' f id commutes. P P' X In order to show F is contraariant, we must find a map (in X ) F(f,g); ( f ', g ' ) :----+ ( A, B X )( C., B ) By definition ( A,; B () ( A, P B ), n' ), and X ( C, B ) ( ( C, V W', B ), n ) ; so the choice for g' is clear: g' ; f id:v W' P W' 2 As for f', consider the following diagram: ( A, B ) - -- - --- (P, -L W) - - + ( C,- B ) P = '- -'- -(V, - r W) > (*) p.b. () j (W',PI) (W' /'d,g), V ' ) - _, (V W',X) d ) (P W', X) We know the outer square commutes, therefore it suffices to show () and (2) are commutatie. For (), we proe it by looking at the following commutatie diagram: (P,W) /, (P, (W', X ) ) (f. id) (V,W) (P W',X) (f id, id) (V W',X) (V,(W',X))
5 As for (2) we hae a similar diagramatical proof: (W',P) -----'-(=-id:::.z..,, g"')'-----+> (W', V' ) lp (P W',X) (f id,id) l > (V W',X) (W', (P,X)) (W' P,X) lcs,id) (id f,id) lcs,id) (W' V,X) (W', (V,X)) But in this case the commutatiity of the outer square is due to the fact that (f,g) is a morphism which sends C to A (hence the diagram (*) aboe commutes). This implies that there is a unique map f' : ( A ---+, B ) ( C, B ) induced by pullback such that the diagram (P, W) l (W',P') (W', V') p.b. l /.id). (V,W) (V W',X) "' (f id,id) I (id,g) -------"--------+ (P W',X) commutes. Therefore the following diagram commutes: 2 ( A, B ) 7(W', P) (P W',X) l(id,g) l(foid,id) ( B) C, 2 (W', V') (V W',X) This implies that: id g' ( A, B ) ----==-=---+> ( V W ' ) ( A, B ) ( P W ' ) commutes. ( C, B ) ( V W ' ) X
6 F Therefore (f',g') has the property required of a morphism in It is triial to see that F(idA) = idf(a) preseres composition. Now we hae to show Suppose A= (P,P',p), C = (V,V',) and E = (U,U',u) are three objects i n moreoer, (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') commutes. ( E, B ) By definition: ( A, B () ( A, P W', B ), nl) ( C, B () ( C, V W', B ), n2) ( E, B () ( E, U W', B ), n3) Now we consider the following commutatie diagram: V(A,B) V(C,B) l (V,W) - y(f l' (f,iy - - - " : ' - = ' - - - b - - + : (W',U') ------'il=-----+(u W',X) d, g -)( f i d, (W',V') (V W',X) ( h (W',P') - - - - - - - - - - - - - - - - - - - - - -
7 But the following diagrams also commute: (id,g k) l (W',P') (id k) (W',V') cw,u ) (P W',X) (h id,id) (V W',X) ((hof) id,id) j (f id,id) (U W',X) ( h o f, i d P j (h,id/, W ) (V,W) (U,W) (f,id) 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 contraariant functor. As for G, we hae a similar series of diagrammatical proofs: Suppose B = (W,W',w), A= (P,P',p), C = (V,V',) are objects i n with A----+ C a morphism i n We. need By definition (f,g): G(f,g) = (f',g'):g(a)----+ G(C). and G(A) = ( B, A ) G(C) = ( B, C ) Hence the choice of g' = id g:w V -----+ W P' is clear. And the following commutatie diagram shows the existence and uniqueness of f':
8 ( B, A ) ( B, C ) l p.b. l Again the preseration of the identity is clear. Now if A= (P,P,p), pl (W,P) pl (W, V) p w ------'-'--------;- (W V I 'X) d ) (W P,X) C = (V,V,), E = (U,U,u) are objects in X and (f,g):a----+ C, (h,k):c----+ Dare morphisms, then the commutatie diagrams of Figure 3 imply To proe (iii): G preseres composition. Suppose A= (V,V ), B = (W,W,w), C = (P,P,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 ) V(B,D)-----'-"'- 2 --->- (W,U) pl - l p.b. lu P (W,P) w (U',W' )----"------>- (W U',X), i d ) ( i d k, (P',W') w (W P,X) V(B,D) - Y'T p.b. l u (V,U) (W, U) i d a (U', V ) hd,g) (V U',X) ( g i d, i w (U I' W' ) - - - - - - - - - (W U' ',X) - - - - - - > -
9 (W, P),(=i.=cd''-" _,_)--+ (W, V) (W0P I, X) (id g, id) ) (W V', X) (id,h f) \ (Jd0(gok),id)\ ), i d ) (W, U) (W0U,X) (P',W') (g,id) > (V,W') (g k,id)j (U',W') ( B, A ) (i:y ( B, C ) (W, V) pl (idy '{_(B,E) > (W, U) I (W,P) )'' p.b. ii p (U, W ) w (W0U,X) /' ''' ( i d O k, i (V',W ) w (W V',X) ) ( > d g, i (PI, W ) w (W0P I,X) FIGURE 3.
20!(A,C) h (V,P) ( i d, V(A,D) --= =------> (V,U) -l p.b. [u. p (U, V ) ) ("\I,X) (k id,id)'\ (P, V ) (V P,X)!(B,C) (7 (W,P) V(A,C) - - - - - - - - (i:; - - - (V,P) - - - -!(A,D) ( V, U ) ]'' p b ]". (U, V ) (,X) V U i d () k i d, i (P, V ) ;_-----------+ (V P, X) ( f (P,W ) - - - - - - - - - - - - - - - - - - - - w (W P -,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)
2 Next consider Figure 5. Since the center square of the first diagram is a pullback, f'oh' is the unique map ( B, C ) -(-A-, - that D + ) makes the diagram commute. Next consider the lower diagram of Figure 5. fact that the following diagram commutes: Using this and the V U' id k )'""' id k W U' V P' l' id W P' ( B, C ) ' V(B,D) (W, U) - f' (f,id)""" V(A,D) ---'p"-'l"--->-> (V, U) -l lu (W,P) ( i (U', V') --'--->(V U',X) i d, g f ) i d, i d (U',W') w (\J U',X) d ) ( i d k, (P',W') - - - - - - - - - - - - - - - - - - - w - - - - - - - - - - - - - - - - - - - - (P', V') ) p.b. l' (V,P) : -+(V P', X) (W,P) (f/ ' -+ (P',W') - - - - - - - - - - - - - - - - w ( f i. (W P',X) FIGURE 5.
22 we obtain the desired result that the diagram - - - - - (A, - D) > - X commutes. 2. The Functor* In this section we shall define a functor and examine its relationship with X ( -., - ) Definition. Gien any object A; (V,V',) i n define *(A) * : -o p - - - - X to be the object (V',V,os) where s:v' V V V X ' is a map in V. Suppose B; (W,W',w) is another object in X and a morphism in (f,g) :A----+ B X, then define *(f,g) ; (g,f) :*(B)-----+ *(A). This definition is justified since the commutatiity of the diagram: V W id g I V V' jf id j 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 aboe formula on morphisms we can easily conclude that * a functor. is Moreoer * has an inerse (contraariant), since *a* ida -x The following are some properties of *: ProEOSition. Gien A (V,V',) ; ( A, = B )( * ( B ), * ( A ) ) ' B ; (W,W',w') in X ' then PROOF. By definition *(A) = (V',V,os) and *(B) (W',W,wos). Consider the commutatie diagram of Figure 6. commute. Notice that the coherence of V implies squares (), (2), (3), (4) It also implies that the diagram (V Wj' ::' < ' ' ", x l (s,id) (V W',X)
23 conunutes. The fact that ( *,*(A)) ( B ) :el > (V,W) j 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 inole ( A, induces B ) a unique map q : ( * ( B ), * ( A ), such ) B -) - - + that the diagram of Figure 6 conunutes. induced by the pullback square: This implies qop is the map ( A, B ) (V,W) j p.b. l (W', V') (V W',X) ( A, B ) (W', V') 7 ( * ( B (, * ( A ) ) (V,W) / pl ( A, B ) > (V,W) ]'' p.b. wos j (W', V') (V W',X) (W', V') /< ( s, (W' V,X) (V,W) w ( s (V W',X) FIGURE 6.
24 But by the remark aboe i d ( A also, B ) has this property. Hence qop i d ( A., B ) Now switch ( A, and B ) ( * ( B ), in * ( the A ) preious ) diagram, and apply the same argument to conclude that poq i d ( * ( B )., This * ( A ) ) completes the proof. Corollar:l Let A,B be two objects in!2.x, then ( A, * ( B )( *(A)) ) B, PROOF. For any object c in!2.x, *(''(C)) Corollary. Let A (V, V',), B then f2.x(a,b) f2.x (*(B),*(A)). (W,W' c.,w), be two objects in!2.x, PROOF. By definition *(A) (V',V,os), *(B) (W', W,wos) implies that f2.x(*(b),*(a)) But recall that f2.x(a,b) ( ( ), * ( A ), W ' V, n ) ( ( A, B ), V W ', n 2 which ) moreoer we hae isomorphism p : ( A -->-, B ) ( <,*(A)) ( B ) and q : (,*(A)) * ( B )-->- ( A, B ) such that i d ( A, qop B ), i d (,*(A)) < ( B ) poq We also hae s(v,w') :V W' -->- 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 commutatie diagram: But we see this by considering the following ( A, B ) ( * ( <(A)) B ), ( A, B ) j j j (V W',X) (s,id) (W' V,X) and complete the proof by taking the transpose. (s, id) Corollary. Let A,B be two objects in f2.x, then (V W',X) PROOF. Proposition If C is an object in f2.x, then C *(*C)) 2. Let A,B,C be three objects in!2.x, then ( A, f 2 *(C))). x ( B,- ( C,! 2 _ X ( B, * ( A ) ) ). PROOF. definition Let A (V, V',), B *(A) (V',V;os) and < (C) Now put Be Ba f2.x(b,*(c))!2_x(b, *(A)) (W,W',F), c (U', U, uos). (U,U',u). Then by ( ( B, * ( C ), W U, n ) and ( ( B, * ( A ), W V, n 2 ) squares pullbacks: Recall that ( B, * ( and C ) ) ( B, * make ( A ) the ) following
25 pl pl ( B, * ( C ) ) > (W,U') ( B, * ( A ) ) > (W, V') (,,!":' Wos (U,W') (W U,X) (V,W') (,, l:, W 0 S (W V,X) Now consider Figure 7. Since (U,-) and (V,-) hae left adjoints, they presere pullbacks, still commutatie. hence the outer and inner squares are But () 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,- / (V (W U), X).::s::,_ (U (W V),X) (U, (W V, X)) ( A, B c ) P :. : l ' - - - - -l - - - - - 7 ( C, B pl a ) ( U, B a ) ( i d, p l ( U, ( W, V ' ) ) j (id, ) () j (id, V) (U, (V,W')) /. (id,-w) (U, (W V,X)) j,- (W V, U') u (U (W V),X),-/ l (V, (*(C))) B, (V, (U, W')) '' p.b. (id,-w) - (id,u) (V, (W, U' )) (V, (W U, X)) (V (W U),X) FIGURE 7.
26 Tgis implies that there exists a unique map ( A,---+ B c )(U,Ba) such that the diagram of Figure 7 still commutes. Now using the fact that (U,Ba) j, (W V, U') (U (W V), X) is a pullback, there exists a unique map ( A, B c ) ( C, B a ) A similar argument (Figure 8) shows the existence of a map p : ( C, B a ) ( A, B c ) Applying the same argument as in the preious proposition, we conclude that poq = i d ( A, and B c qop ) = i d ( C, B a ) Corollary. If A, B, C are objects i n, then PROOF. Apply the same argument as in preious corollary. Corollary. Let A, B, C be objects in X, then PROOF. ( A, X ( = B, x C ()* )( * ( 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. Gien 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 -x X X (*,*) X ( -, - ) A (id,*) X X X X
27 ( Ba) C,---------- t=-------------+(w V, U ) l ( i d, 2 ) p.b. ( A, B c P ) (V,V(B,*(C))), (id,p ), (V, (W, U )) /, [(id,u) (V,(U,W )) --'("'i,_,d_,_, W=._,-)'------+- ( ' (W U' X) ) I" (W U,V ) - - - - - - - - - - - - + - - o s lp- (V (W U),X) (U,Ba) (id,p) ' r7 (U, (V,W )) p.b.l (id,w) - (U' (W' I)) ( id' ) (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).
28 4. The Dualising Object and the Unit fr 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',) be an object in!x, then we hae the following commutatie diagram Y._(A, T) (V, X) j j' (I, V') (V I, X) l' V' l' j (id,i) (V, (I, X)) (V,X) But V' lid (V,X) lid V' (V,X) is triially a pullback in ':!_, which implies that we hae 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)
29 On the other hand, A T * T * A A. This completes the proof. Theorem. Let A,B,C be objects in then PROOF. ( A B, Ac x )( A, (. B, C ) ) Ax(A B,C) = Ax<Ax(A,B *) *,c) * * Ax<c Ax(A,B )) * * Ax<c Ax(B,A )) Proposition. Let PROOF. Remark. A ( A, A x ( B, C ) ). be an object in then Ax<T,A) Ax<T *,A) A. * *,T) ( A A. () There is an obious embedding functor from the comma category (y,x) to sending V----+ X to V I ----+ X: context (y,x) has a *-autonomous structure. hence in this (2) It is easy to erify also satisfies our first assumption, i.e. the MacLane-Kelly coherence conditions. CHAPTER IV: APPLICATIONS. Functor Categories In this chapter, we shall apply the theory deeloped thus far to the double enelope of a symmetric monoidal category C. Before defining the double enelope, let us recall some elementary results of the functor categories. Gien categories X and Y we hae the functor category W =. We know that if X is complete, then so is. in the case =, the category of sets W also has a closed symmetric monoidal structure. The tensor is the cartesian product while the internal GF is defined as the functor whose alue at D is the set of nature transformations F(-) x Hom(D,-)---+ G(-).
30 2. The Double Enelope. Definition. Gien a symmetric monoidal category with a faithful functor - :s/->- we, denote the double enelope of C by E ( ) The. objects of E ( are ) all triplets (F,G;t) where F and G are functors from C 0 to, t is a natural transformation from F x G to - -. 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') - - - = i d F(C) x g x G(C'), F' (C) X G' (C') ::S: -* I C C I I commutes for eery object (C,C') of C 0 x C 0 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 eery (C,C') in x Q_o. F(C)XG" (C') lidxg' F(C)XG' (C') lidxg F(C)XG(C I) This implies that (f,g):(f,g;t) f : c ; X ;. : : i :. F. I : d (c) :.. XG" ( c I ) lidxg' fxid --==-=----+F' (C)XG' (C') f'xid F" (C)XG" (C') u -------------------=t---------------------+ C C' (f',g'): (F',G' ;s) (f",g"): (F",G";u) (F',G';s) (F", G"; u) (F"',G"';) are maps in E ( ), then ( f", G") o ( ( f', g' ) o ( f, g) ) (f", g") o (f of, gog ) ( f 0 ( f I of) ' (gog I ) o g ) ((f of )of,go(g og")) (f of,g' og") o(f,g) ( ( f ' g") 0 ( f I 'g I ) ) 0 ( f' g) Moreoer, gien (F,g;t) then (idf,idg) is the obious choice for identity.
3 Before proing the main theorem of this chapter, let us inco C 0 X C 0 estigate the functor c a t e g o r eand s s- There are C0 X co two obious embeddings of s into s- namely h e r e ( F ; ) F x I and co I is the unit in S- i.e. r(f) ; I x F for eery F in s. and r, co s-, and I sends eery object into the singl5ton Hence we can regard objects in S as (the terminal object) in C 0 X CO objects in s- ia either embedding. Now we can proe. C 0 X CO Proposition. E ( is ) enriched oer V ; S- PROOF. By preious remark V is a closed symmetric monoidal category with pullbacks, moreoer it is coherent. Now gien A; (G,F;t) and B ; ( G, F ' in ; s E )( we ) hae to 0 define ( A, an B 0 b object in ( ; x ) Suppose (C,C') is an object of S x C, then V(A,B) is the functor whose alue at (C,C') is defined by requiring that the diagram V(A,B)(C,C') U(G). ( (C,C') G ' ) be a pullback. (r(f'),r(f))(c,c') (G X F', - -I)(C,C') Note. (-,-) denotes the internal hom-functor of map ( ( G ), ( G '--r ) ) ( C (G, x C F', ' ) - -I)(C,C'), As for the we simply obsere 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' - - 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 ealuation. This concludes the proof. Theorem. E ( is ) a subcategory of a *-autonomous category!; moreoer A is enriched oer V. PROOF. Put X; - - then follow the construction in Chapter III. 3. Miscellaneous Results. In this section, we are assuming V has all the properties as gien in Chapter I and we shall proe that there is a functor F maps V to - C A T CAT ( -is the category of all categories which are enriched oer ). The functor F on objects of V is obious: gien X in. then put F(X) ; - -
32 Now we hae to show gien a map f:x - - - -S- - in. this induces a V-functor T( = F(f)) from The notion of a II, Section 6. to!s V-functor can be found in [Eilenberg & Kelly] Chapter In this case we hae 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, to C ) ( T ( T(C)) B ), in V such that the following axioms are satisfied: () The following diagram commutes: T ( B,-----"-------+ B ) ( T,T(B)) ( B ) j (2) The following diagram commutes: I :_: M' ----+ (B, D) ( C, D ) ( B, C ) T T T ( T ( C ), T ( D ) ) -----''-'----------+ ( T ( B ), T ( (T C (B) ) ), T (D) ) M' 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 then T(B) is the composition V V' T(B) = (V,V',fo). To show (ii): is obious; gien B = (V,V',) f X s i.e. Suppose B = (V,V',) C = (W,W',w) objects in!x then T(B) (V,V',fo), T(C) = (W,W',fow) and the following diagram commutes in ( B, C ) Y(TFT(C)) :d (V,W) p.b. ) t (V,W) (W', V') /,:') - - - - - - - - f _ o,s) - - - - ( V W ' ( i d, f (V W',X)
33 Since the inner square is a pullback, there exists a (unique) map T(B,C) from ( B, to C ) ( T ( B ), T ( C ) ). To show () commutes let B = (V,V',) in Then and the following diagrams cornrnu te: T(B) = (V,V',fo) : p.b. (V', V') ----'"---+(V V',X) p.b.. fo (V', V' )---=-..:...--->- (V V',S) Y<B, B) --------...L.:::'-------------+ (V' V) '' Y(T(B)[::B)) p.b. ( V fo (V', V' ) ----"--.: ---->- ( W9V', X) ( i d (V', V') -----------------+ (V V',X) I V (B, B) --------"-= '------------+ (V, V) - T y F» (r Y(T(B) (V',V') fo (V', V) ----=--'------+(V V', S) ( i d, f (W!lV',X)
34 T Hence the composition I., -+ ( B, B ) - - ( = T -( T(B)) -B- ) +, and map I _.,, _ ( are T ( B both ), T induced ( B ) ) by pulling back. the uniqueness property they are "equal". Thus by To show (2) commutes, let B = (V,V,), C = (W,W,w), D = (U,U,u) be three objects in. Then T(B) = (V,V,fo), T(C) = (W,W,fow), T(D) = (U,U,fou) and the following four diagrams commute: ( C, D ) ( B, C ) - - - - r P - - -(W,U) (V,W) - " (U,W ) (W,V ) ( B,--.<:.=----+(V,U) D ) p.b. (W',V ) (U,W ) (U I '' ) -----''-------->- (V U,X) y(t(c), T (Dj::::(B), T pl pl ( ) \---"-=L-"'--------+> Ur, (W, W (U',W ) (W',V ) ( T,T(D))-P<=-= ( B ) =----+> (V,U) p.b. (W',V ) (U',W') ---"M'----+-(U I '') - - = - f _ o ',S) - - ( V U ' ( C, D ) (U',W ) A ( T ( T (D)) C ), l pl '(W, U) l':" fow (U,W ) (W U,S) /o w (W,U) ij ( i d, (W U',X)
35 - /td V(B,C) ------"-' =-------->- (V,W) ( T ( B ), T P( C ) ) >(V,W) - p.b. fow W r - (WI 'VI ) ::f: o_: -----+(V W I ' S) ( i d,t) "" (W,V ) ---------'-------+ (V W,X) ( C, D ) ------------------>- ( B, C ) (W,U) (V,W) (T (B), T (C))..:P.::: p--"-= '-------+> (W, U) (V, W) M 7 (C), T T ( D ) ) M (T(B),T(D)) (V,U) (V,U) p.b. fou - (W,V') (U,W ) (, V) U ( V, S ) U u /- ( i d (W,V ) (U,W ) M (U,V ) (V U,X)
36 V(C,D) V(B,C) l pl (W,U) (V,W) - - tm ( B, D ) pl pl ' (V,U) (W',V') (U',W') M (U', V') r ( T ( B ), T ( D (V,U) ) ) t (U' V') p.b. fo ij V;u (V U',S) u (ida (V U',X) This implies that the diagrams 3boe commute, which implies that the composition ( C, D ) C) () - T ( B J ',. ( _ T _.( T(D)), C ), (B), ( T T(C)) is the map induced by pulling back. This also implies that the composition M' T ( C, D () B, _ C _o_:_ ),_ ( B, - D ) --( T + ( T(D)) B ), M' - --( T + ( B ), T ( D ) ) is the map induced by pulling back. commutes. Hence by the uniqueness property, they are "equal", therefore (2) 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 preseres composition. and (ii). then All we hae to check is that the composition is presered in (i) It is easy to show (i) is presered. For if B (V,V',) in, (F(g)oF(f))(B) F(g) (F(f) (B)) F(g)(V,V',fo) (V,V',go(fo)) (V, V', (go f) o) F(gof) (B). To show (ii) is presered: Let B = (V,V',), C = (W,W',w) in X then ' F(f)(B) (V,V',fo),F(f)(C) = (W,W',fow),(F(g)oF(f))(B) = F(gof)(B) (V,V',(gof)o),F(gof)(C) = (F(g)oF(f))(C) = (W,W',(gof)ow) and the diagrams (*), (**) and (***) commute
(*) 37 V(B,C) --------"-' :::.. _. (V,W) - y(f(f)(b) ff)(c)) -:-" :c=.:- T (W', V') - f _ o " - - - S) - - + ( V W ' / (W',V') ---------"-'---------+ (V W',X) ( i d (**) ( F (B) ( f,f(f) ) (C)) (V,W) 7 V ( F ( g o f ) ( B ), F ( g o f ) ( C ) ) ( V, W ) - l p.b. f:w ) (g:f') (W' V') -----'-"'---"-L----+>(V W',K) /- ' ( i d, g o (W',V') fo. ""' (V W',S) Note. F(gof) (-) (F(g) of(f)) (-). V(B,C) l (V,W) - ) V ( F ( g o f ) ( B ), F ( g o f ) ( C ) ) ( V, W ) (***) - l - w (W', V') (W', V') (go f) o > (V W',K) ( i (V W',X) But (*) and (**) imply the diagram of Figure 9 connnutes. This implies that both F(gof) in (***) and the composition ( B,---=F--"(f::.L) C ) ( F (B) ( f,f(f) ) (C)) --=-F->-<(g'-'-)- ( F ( (B) g o,f(gof) f ) (C)) are induced by pulling back. Hence it follows they are equal.
38 ( B, C ) f ) y V(F(f)(B),F(f)(C)) l (V,W) ) y V(F(gof)(B),F(gof)(C)) (V,W) fow - l ------ w (W', V'),("'g'--o-"'f-"-)_o...:.,_, (V W, K) (V,W) fo ( i (W', V') (V W',S) /. ( i d, f (W', V') (V W',X) FIGURE 9. BIBLIOGRAPHY. M. BARR, Duality of ector spaces, Cahiers Topologie Geometrie Differentielle, XVII- (976), 3-4. 2. M. BARR, Duality of Banach spaces, Ibid., 5-32. 3. M. BARR, Closed categories and topological ector 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 reflexie topological abelian groups, Ibid. XVIII-3, 22-248. 6. M. BARR, *-autonomous categories, This olume. 7. S. ElLENBERG,.G.M. KELLY, Closed categories, Proc. Conf. Categorical Alg. (La Jolla, 965), Springer (966), 42-562. 8. M.E. SZABO, Commutatiity in closed categories. To appear.
Index of Definitions Admissible (uniform object) Autonomous (category) 3 Basis (for a pseudometric) Completable 9 Conergence uniformity Cos mall 24 Dominating 29 Double enelope 30 Dualizing module 49 Embedding 8 Entourage 8 Linearly compact 8 Linearly totally bounded \ complete 33 Nuclear 47 Pre-reflexie 23 Pre-*-autonomous 5 Pre-uniform structure 8 Product uniformity 0 Pseudomap 5 Pseudometric 7 8 20 7 28 9 Quasi-reflexie 23 Quasi-ariety 3 Refine (of seminorms) 7 Reflexie 23 Represent 28 Semi-norm 65 Semi-ariety 3 Separated uniform 6 *-autonomous 3 Uniform conergence Uniform coer 6 Uniform object 0 Uniform space 6 on <!> 28 Variety 3 V-enriched pre-*- autonomous 7 V-enriched *-autonomous 7 34 35 ; ; - c o m p l e t e ; ; - * - c o m p l e t e
Index of Notation ( -, - ) 2 II 33 <-,-> 5 )I:!. 33-6 r 33 <-> 9 s 34 (-)- 9 8 37 UnV 0 T 37 [-,-] :!. 43 T 3 :!. 0 44 (-)* 3 u 52 (-) (-) 8 h() h(2) 59! 2 9 (-) " 74 A 9 :!x 04 - ( -, - ) 25 E ( ) 30 28