R. Street. Unpublished manuscript. 1979.  Home/Search   Document Not in Database   Summary   Related Articles

....(which follows from fully faithfulness of I) B(I; I Gamma) B(I; X Delta 9I ) Gamma) MB(W I; 9I ( Gamma) MB(Y I; 9I ( Gamma) MI Delta 9I ) Gamma) Gamma: Thus B(I; 1) B Gamma ME is an equivalence. Since both E and now ME are locally small it follows from [7] (see also [2] that E is small as required. If C and D are total then a functor F : C Gamma D preserves all colimits if and only if it has a right adjoint. If, moreover, F is Kan then preservation of all colimits is equivalent to invertibility of the canonical natural transformation XD 9F ....

....Again, the reader is advised to think of X as a general counterpart of the supremum arrow for a complete ordered set. Now replace D in the immediately preceding discussion by MD; where D is an arbitrary locally small category. According to our definition of total category and again invoking [7] (or [2] MD is total if and only if D is small. But we do have MYD assuming only that D is locally small. If F is both Kan and a left adjoint then a canonical isomorphism as in the right hand diagram is produced by a modification of the calculations which establish that the canonical arrow in ....

R. Street. Unpublished manuscript. 1979.

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