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

By F. Borceux, G. Van den Bossche

ISBN-10: 3540127119

ISBN-13: 9783540127116

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If f : A ÷ B is a morphism in C, we must prove that U U o u(f). f o 6A = 6B Ui It suffices to compose each side with ~uA : Ui U Ui o u(f) o BuA = 6B o ~uB ° ui(f) U. 1 = BB =fob o ui(f ) Ui A = f o B~ Ui o ~uA" 38 u~ is a canonical inclusion and any morphism in U has the form u*(f) for some f in C; so the equality u! u* u! u* = uu = u = u! u* shows that in fact u* u~ is the identity on U. So we have two natural transfor- mations id U : i d u = u * u! Bu : u r u* ~ id c, In order to have an adjunction u[ M u*, it remains to show that the compatibili- ty conditions hold : u* * U = idu, 8U * u I = id U I " Let A be some object in C.

If U ~ V are two formal initial segments, the restriction morphism A A(U < V) : (v, v* A)O) ÷ (u, u* A)O) is the one given by proposition 10. a A(U ~< V) = v, v*(~ ) (1) U -- ~ (1). This makes A A into a presheaf. v,v*A Now if f : A ÷ B is some morphism in Sh(~,Ir), the morphism A(f) is defined for any formal initial segment U by A f(H) = (u, u* f)(]). ]T) ~ Pr(H,]T). A remarkable fact, which will be crucial for the sheaf-representation theorems, is that each A A is actually a sheaf. Indeed, let U = v U.

V* u~ u* coincide also on any morphism f : A ÷ B in C, where O ÷ A and O ÷ B are monomorphisms. Finally we have shown that u! u * v! v * ~= u! u * n v~ but this define it f o r m u l a a l s o shows t h a t this v* functor ~ v, V* u: u * , takes its values i n U n V; we t o b e w*, We have just shown that, when O + A is a monomorphism, u o 6uA v = BAu 6A o u(s~) V U = BA o FvA v V(BA u) = BAO Again by [21] - 12 - 2 - 7 and proposition I - I, this implies an equality between natural transformations B U o (B V * u) = B u o ( u * B V) = ~V o (~U, v) = B V o (v* Bu); we choose this natural transfoTmation w~ w* ~ id to be FW.

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

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