By Michael Barr
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Additional info for Exact Categories and Categories of Sheaves
3}. Let C = ~n' and we have an embedding of ~n >~(cOP S) which, since the cardinality of each covering of the topology is 1, embeds X as objects of finite rank. 1) are -n satisfied. 44 Chapter 1. Statements of result. 1) Theorem. 2) III. Th9 Embeddinq Every locally presentable into a functor Theorem. category has a full exact category. Every topos has a full exact embedding into a functor category. 3) Theorem. Every small regular into a functor category. 4) Theorem. Every small, full exact embedding category has a full exact embedding finitely complete into objects regular category has a of finite rank of a functor cate- gory.
5) eO f. = u is true )X. Factor ~X. of this kind of generator. 6) Proof. (G,f) If b) (G,f) is ~ c) (G,f) is definition is equi- to the same m a p being an is )X t . Then all G ~ F, f is ~for )>. ) for all G ~ F if and o n l y if f i s ~ ~ ;. > for all G ~ F if and only if f is follows easily >. -X ~ X One w a y is trivial. XVtt ' ~ from /L i (G,X) b) To distinguish simply call them generators. Let f: X a) a) This of coproducts) whose could have these could be called a set of regular generators.
Of I has an upper b o u n d with ~ [U1]) although the cardinal numbers u s e d to satisfy some of the d e f i n i t i o n s m i g h t become number. as an outline theory monomorphism. generators >X, (X, colim Di) ~ycolim(X,Di). 3) Definition. Sometimes, and a m a p G IX' w h i c h is not an isomorphism )X' w h i c h does not factor through X is said to be locally p r e s e n t a b l e if it has a r b i t r a r y coproducts (denoted II ) and a set of generators each one of w h i c h has rank. 4) P;oposition.
Exact Categories and Categories of Sheaves by Michael Barr