By Chazelle B., Goodman J.E., Pollack R. (eds.)

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The n o t a t i o n ~k or if t care with To nomorphism. ~ XA If all are m o n o m o r p h i s m s . • For each : /\ XeA f \/ TET 0he n e w ' T,~ • is an i n t e r s e c t i o n T Im gTX and, On the o t h e r then by when all that ~ P~gi (Cj) , of hand, : fiPi ~/Im iel gi= I . 8. that for all applies from w h i c h prea re- to the pre- theorem it is c l e a r the lemma Im x' : i T case X ~X x T' = keA the . First are m o n o m o r p h i s m s . l we can now p r o v e to s h o w then f 1 : is C*i ; then p r o d u c t s we can reduce X i such With ( ~T ) they induce T If e v e r y to the f a m i l i e s it is easily deduced epimorphism.

The by all EC U,V through c. ] E cU , V on the 9. Let E(U) = c(U) : C. If V ~ U and and the m o r p h i s m s , > P(CAV*CNV) : Ec(U) restriction > EC~v(V) map (unique EU, V : E(U) such > E(V) is (C e ~(U)) One sees that and f u r t h e r m o r e on as follows. C) induces such that of = ( U i n V ) i e I e {(V) nuknv P (~)gc ' . One sees that does not d e p e n d h" = j,kel ~ PU j n U k , U j on equalizers unique ) E(U) > P(C nv) induced ) E from two steps should the case in a C% regular 3. The result is now a m o r p h i s m P result to a sheaf is that, colimits terminate actually of p r e s h e a v e s factors uniquely in an exact category and a p r o j e c t i v e a sufficient (MR 26 ~ 1887), tegories step : Ec(U) p.

I as above; cm. m. = c I assume furthermore (i e $) for exists. all i. 3 c C inducing a morphism Im m = i is c of m, injections ker Applying ' through and \/ Im ie~ , with be epimorphism, c. 4. 3. Then all con- 4,5. 2. i = so the result. Proof. i. Proof. 5 Csl = \/ ie~ epimorphism. must contain all c Im m. s Also, = i (c i) ie~ Im(mi,mjD(f)). 43 If c o n v e r s e l y tible ker d family, factors hence through c Lemma codomain A. 5. Then in p a r t i c u l a r , contains these, through ker Let all then (dmi)ie $ (ci)ie ~ = is a c o c o m p a - (cmi)ie ~ , so that d be a m o n o m o r p h i s m of c S ker d • f,g : A m e Equ(f,g) ) B if and only s ~A(g-lf) Equ(f,g) = and m if (where Im(m,m) AA :A = g-lf ) A xA A ¢ ; is the diagonal).

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