By F. Borceux, G. Van den Bossche
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Extra info for Algebra in a Localic Topos with Applications to Ring Theory
The following triangular equality arises from the adjunction v* -4 v, * v* V ~ V * Now v! v* i s l e f t v . v* A t o u . u V* V, V ~ U a d j o i n t t o v , v*; thus v , v (aA) i s the unique morphism from A such t h a t , , U V ~V oV! v v,v ~A o @A = u, u * A But the f o l l o w i n g e q u a l i t i e s U (~). hold : Bv o v I v*(~ U ) u, u* A ' v , v* A o ~V = aU v, v* A v, v* A = ~ U o v . v* A = a BV naturality o v, v , V (aA) v . v* A U V V ° aA o @A v, v* A naturality V U V aA ° ~A ° @A naturality = u.
V* u! u* we obtain, by taking the right adjoints of both sides, v. v* u. u* ~ u. u* v. v*. Now we define w. = v, v* IL, u* W! ~ u, U* V, V* w! which is a f~mctor from U N V to C. By the adjunction just mentioned, we have natural isomorphisms (u! u* v! v* A, B) ~ (A, v, v* u. u* B) with A and B objects in C. In particular for A in C and M in U N V one has natural isomorphisms (w* A, M~ ~ (u! u* v! v* A, w, N0 (A, v, v* u. u* w, M) (A, w, M) which show that w. is right adjoint to w*. This concludes the proof of the fact that W = U a V is an initial segment of C.
22 M(u ^ v); so f A V = fv and finally u. u* f = f. Thus u. is full. We now turn our attention to the definition of u!. M will be the sheaf associated to sone presheaf u'M that we define now. For any v in 14 = I M(v) if v-<-u 0 0 (v) if V ~ u (u'M) (v) where 0 0 (v) denotes the zero-operations (the constants) of the theory ]~(v). If w ~< v in 14, we have a restriction morphism (u'M) (w < v) which is M(w < v) if v < u and the composite 00(w ~< v) 00(v) if V ~ u. > " 00(w) (u'M) (w) (Recall that O0(w ) is the initial object in Sets~r(w)).
Algebra in a Localic Topos with Applications to Ring Theory by F. Borceux, G. Van den Bossche