# *- Autonomous Categories by Michael Barr, Po-Hsiang Chu., By Michael Barr, Po-Hsiang Chu.,

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Example text

Then the canonical map of ~ A(C,A(G,D)) is an isomorphism. Proof. 11) it is sufficient to show that V(G, (C,D))---+ V (C,A (G,D)) is an isomorphism. 11) Proposition. Let A,B~A, G e G . 7). 13) 40 A (A,A (G,B)) Proof. 11) ~ A (G,A (A,B)) it is sufficient V (A,A (G, B) ) Let {C----~ A} and {B---~ D } to have a canonical map ~ V (G,A (A, B)) be a C - d o m i n a t i o n and D-representation, respectively. 10). ,D )) . ~ EV(C ,A(G,D~)) ~ EV(G, (C ,D~)) As well we have V(A,A(G,B))----~ V(A,[I GI, B] ) ~ V(IGI,V(A,B)) induced by the embedding Corollary i.

B e a D - r e p r e s e n t a t i o n of V(E,A(C,B)) is a p u l l b a c k (~A*) A = ~-*-complete. Proposition. Then for any Proof. is is p r e - r e f l e x i v e . EeA (C,D)} is a D - the d i a g r a m ~ ~V(E, ( C , D ) ) I , ~V(E,[ ICl ,D~) and apply V(E,-)) . (E2,[ ]C] ,D ]) ~ ! (EI,D ))~ ~ ( E l i [ C [ ,D )) since (C,D) B and are the same w h e t h e r Corollary. Let A D are all E = E1 or p - complete. E = E2 b e reflexive and , But then three o f the four vertices and hence so is the fourth ~-,-complete.

Now a compact set in S'K subspace. is a c o m p a c t in a space as the set of all e l e m e n t s mensional S'K subset of s u c h that In fact it is c o n t a i n e d SO A°(i/rs ) c K F{A°(i/rs ) I seS} such t h a t of a b s o l u t e x s = Zs/X s , we h a v e ~IXsrsl c i r c l e d h u l l of the images of the for w h i c h Ys~K Let = 0) and s (IsXs) sting of all e l e m e n t s o FA (i/rs) is the c o n v e x if r I of A compact S0-K S set in set in some for some necessary S0-K finite to e x p r e s s is c o n t a i n e d finite d i m e n s i o n a l subset SO c S a basis .