By F. Borceux, G. Van den Bossche
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Extra resources for Algebra in a Localic Topos with Applications to Ring Theory
Then for any sub- object R ~ F h v in Pr(]1, ~),g-1(R) = F h u and thus for any subobject S ~ in Sh(lq, 3Y), (ag) -I (S) = a F h u.
Cfr.  - 16 - S - 4). (M) ~ M is a monomorphism in Sh(u¢,]ru¢). It rmm~ins to prove that u! u* f % f. diagram in Pr( I~, IT) : u' u, u*A ............ M where the horizontal mappings arise from the adjunctions. If v ~ u , the v-com- portent of this diagrmn is A(v) f(v) r id M(v) , A(v] If(v) ~ M(v) id which is a pullback. M(v) is the smallest subobject of u~M(v) - (the set of constants) - and is thus contained 24 in A(v). u*A + A u! M id So the upper one is an isomor- The lower horizontal morphism is an isomorphism.
So o is left adjoint to r. Now @' is defined by a filtered colimit. But in Sh( I~, TF) and Sh(H, IT) filtered colimits commute with finite limits (proposition I - 3). So 0' commutes with finite limits. But the associated sheaf functor also ¢or~autes with finite limits. Thus @ is exact, i 49 Proposition ]4. L e t ~ be a frame, ~ a classical theory and H the frame of formal initial segments in S h ( ~ , ~ ) . The restriction functor F : Sh(H,~) + S h ( ~ , ~ ) has a right inverse A which is continuous and faithful.
Algebra in a Localic Topos with Applications to Ring Theory by F. Borceux, G. Van den Bossche