Download Categorical Algebra and its Applications: Proceedings of a by J. Adámek, J. Rosický, V. Trnková (auth.), Francis Borceux PDF

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Addition, -ORS 41 The key is to say, concept for f necessary to be for F a descent to be full morphism, is and that faithful, of that a weak retract. 2 DEFINITION. 3 if A morphism g' is, (the when following and only every PROOF. The if suppose and LR(P,h)'OLR(P,g' g'°h = M' the Since in If w e are closed in Ban P@RM such is ~ N is said is in is to be a weak a retraction Mod R weak such in that retracts. a weak a regular retract monomorphism in Mod R in , P@g end, (P@g)(Xn) iy • i(Xn) i(y) is ~ isometric Mod R , we an ~ (P@RM) " inclusion , where into its (P@g)" is also be .

6 Corollary. 7 Proposition. Two maps to a separated object that agree on a dense subobject are equal. Suppose j is a topology on a category C. Then for any object C, R C is an equivalence relation on C. P r o o f . Since the intersection of dense subobjects is dense, one easily shows t h a t the relation on Hom(A, C) of agreeing on a dense subobject of C is an equivalence relation and is the relation of factoring t h r o u g h R C . 8 We will call a topology j on C e f f e c t i v e if for each object C of C, R C is an effective equivalence relation on C.

Mentioned earlier theorem. 3, for diagonal defined map. S any commutative n S = $ R i=l biproduct The m o d u l e a multiplication be is a r e t r a c t some As d i s c u s s e d under f:R ~ S S of the M o r i t a unit, MOdR ® this equivalence , where recalling context EXAMPLES. Theorem). follows by the 0 n e ~ result We to e s t a b l i s h . (Morita n @ R , for s o m e i=l are e q u i v a l e n t . PROOF. 4, namely by = (rltl,rlt2+r2tl+rlr2 .... r l t n + r n t l + r l r n ) , 54 also has R as a Mod R - r e t r a c t Therefore, r ~ (r,0,...