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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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.

V! U* V~ u! U~ u , u* v , v* = u , u* = v, v* u, u* • U U v , v (~A) = a v , v T. v * ( ~ ) : v , v* A + u , = ~U v~ v • : u, u* A ÷ v,. V* A. A The following equalities hold u! u* v! v* = u! W* V ~ = u! , U* = v, w, U* = LL, U* and so for any object A in C we have natural morphisms V, v * ( ~ ) : v , v* A ÷ u , u* A 8U v* : ul u* A-~ v, v* A. v! A But we a l s o have v~ V ~ u! U* = v! w~ U ~ = u! U* • By taking their right adjoints we get u, u* v , v* = u, w* v* = u, u*. So f o r any A in C we have n a t u r a l morphisms 0 v,v v!

We are unable to explain these differences; we simply point out that they vanish when the morphism 0 ÷ A is always a monomorphism in S h ( ~ , ~ ) : in particular this is the case when ~ff"has, at each leve~ a single generic constant or none at all. The notion of formal initial segment is essentially a tool which makes possible the proof of the characterization theorem in chapter 4, so we may freely adapt it to make the proofs work : this is what we do. However, in chapter 6, the notion of formal initial segment itself turns out to be interesting, but then the theory ~C considered there is the theory of modules on a ring R and thus ~ has a single constant : therefore the differences between theorem 5 and definition 6 vanish.

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