Algebra in a Localic Topos with Applications to Ring Theory by F. Borceux, G. Van den Bossche

By F. Borceux, G. Van den Bossche

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Reflects monomorphisms. (F 8) if 0 ÷ M is a monomorphism in U, u! preserves monomorphisms with codo- main M. As u I is full and faithful so is u. (cfr. [2]] - ]6 - 8 - 9). Also u I reflects monomorphisms because it is faithful. Now consider a monomorphism m : N ÷ M in U with 0 ÷ M a monomorphism in U. Consider the image of m in C N By (F 3 ) , I is is invariant adjoint; I still m ~ M (up t o an isomorphism) i n U; t h u s t h e whole d i a g r a m i s i n U and u n d e r u* ( c f r . [21] - ]6 - 5 - 4 ) .

Define U = v U. to be the full image of u, saturated iEI l u~ is the inclusion of U in C az~d u* is the corestriction Thus u = u~ u*. UIf A is any object in C, we have a cone (BA l : ui(A) ÷ A)i61 on the diagram defining u(A); indeed if U i ~ Uj, the pullback we constructed in order to define U. ^ U. reduces to the commutative triangle 1 j U ~u. (A) u i(A) 3 ,, uj (A) A. So we obtain a unique factorization U u* BA : uT (A) = u(A) ÷ A 36 U. 1 of the cone (BA)iCI through the limit-cone (s i : ui(A) + u(A))iC I.

V But for any g in (a F hv, a F hv) the cor~nutativity of the diagram a F hv g ~ a F hv Ba F h v ga F h v u L u a F hu gtu ....... , a F hu shows that glu corresponds universally to g under the bijection (a F hu, a F hv) ~ (a F h u, a F hu). Lenmm 8. Given u ~ v in H and f : a F h u ÷ a F h v in Sh(Iq,~F), there is some operation ~ 6 01 (U) such that f factors into the following composite : u ~aFh v aF hu ~ aFhu r ~ aFh v. By the adj~iction u! q u*, f factors through 8ua F hv into a morphism a : a F h u ÷ a F h u, By ler~aa 7, a comes from some operation a 6 01(u).

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