5. Locales
Recall that a locale is a frame and a morphism  f: A ® B of locales is a morphism f-1: B ® A of frames. Denote by Loc the category of small locales and morphisms of locales, which is the opposite of the category of small frames. 

Suppose A is a locale. For any element a Î A the principal lower subset ¯a of A with the induced partial order is a locale, and the function sending each b Î A to b Ù a Î ¯a is a surjective morphism of frames, thus defines a monomorphism ea: ¯a ® A of locales. The subobject ¯a of A (with the inclusion monomorphism ea) is called an open sublocale of A. Denote by W(A) the set of open sublocales of A. W(A) is a frame isomorphic to A. If f: B ® A is a morphism of locales we have f-1(¯a) = ¯(f-1(a)), thus f-1: W(A) ® W(B) is a morphism of frames. We obtain an effective framed topology W on Loc sending each locale A to W(A). Let us consider the effective framed site (Loc, W). 

Proposition 5.1. (Loc, W) is a complete framed site. 

Proof. (a) We first prove that (Loc, W) is strict. Suppose A is a locale and {¯ai} is an open effective cover of A; then 1A = Ú {ai}. Suppose f, g are two morphisms of locales from A to B whose restrictions on each ¯ai Í A are equal; then for any b Î B we have f-1(b) Ù ai = g-1(b) Ù ai. Thus f-1(b) = Ú {f-1(b) Ù ai} = Ú {g-1(b) Ù ai} = g-1(b). This shows that f = g. Conversely, suppose for each i there is a morphism fi: ¯ai ® B such that the restrictions of fi and fj on ¯ai Ù ¯aj are equal, i.e., for any b Î B we have fi-1(b) Ù aj = fj-1(b) Ù ai. Let f-1: B ® A be the map sending each b Î B to Ú {fi-1(b)} Î A. Then 

f-1(b) Ù aj = Úi {fi-1(b) Ù aj} = Úi {fj-1(b) Ù ai} = fj-1(b) Ù (Úi {ai}) = fj-1(b). 
This shows that the composition of f-1: B ® A with ea-1: A ® ¯ai yields the morphism fi-1: B ® ¯ai. Using this fact it is easy to see that f-1 is a morphism of frames. We obtain a morphism f: A ® B of locales whose restriction on each ¯ai is fi. This shows that Loc is strict. 
(b) Suppose ({Ai}, {¯aij}, {uij}) is a glueing diagram of Loc. Glueing the sets Ai along the subsets ¯aij Í Ai we obtain a set S containing each Ai. Denote by A the collection of subsets U of S such that U Ç Ai is an open sublocale of Ai. Then A is a locale. We may regard each Ai as an open sublocale of A by identifying each a Î Ai with ¯a Í A. This turns A into a glueing colimit of {Ai} in Loc. n 

Suppose C is a locally small category. A locale over C is a pair (A, OA) of a locale A (viewed as a small framed site; cf. (2.5.3.b)) and a strict functor OA: A ® C. If a Î A and U = ¯a we shall write OA(U) for OA(a), and OA(A) for OA(1A). 

Suppose (B, OB) is another locale over C. A morphism of locales over C from (A, OA) to (B, OB) is a pair (f, f#) of a morphism f: A ® B of locales, and a natural transformation f#: f*OA ® OB, where f*OA = OAf-1: B ® C

Example 5.1.1. Suppose (A, OA) is a locale over C and ¯a is an open sublocale of A. Denote by OA|a the restriction of OA on ¯a. Then (¯a, OA|a) is a locale over C, called an open sublocale of (A, OA). For any b Î A let (ea#)b: OA(b Ù a) ® OA(b) be the restriction morphism. We obtain a monomorphism (ea, ea#): (¯a, OA|a) ® (A, OA). 

Denote by Loc(C) the category of locales over C. We have a strict functor OC: Loc(C) ® C sending each (A, OA) to OA(A) and any morphism (f, f#): (A, OA) ® (B, OB) to f#B: OA(A) ® OB(B). 

Theorem 5.2. (a) Loc(C) is a strict effective framed site. 
(b) If (E, GE) is an effective framed site and F: E ® C a strict functor, there is a unique (up to equivalence) strict bicontinuous functor SpecF: E ® Loc(C) such that F = OCSpecF
(c) If C has colimits then Loc(C) is complete and (b) holds for any framed site E and any strict functor F

Proof. (a) Loc(C) is an effective framed site with open sublocales over C defined in (5.1.1) as open subobjects. That Loc(C) is strict follows easily from the fact that Loc and each OA are strict. 
(b) If X is an object of E the strict functor F: E ® C induces a strict functor O(F)X: GE(X) ® C sending each U Î GE(X) to F(U). We obtain a locale SpecF(X) = (GE(X), O(F)X) over C. Any morphism f: Y ® X in E induces a morphism SpecF(f) = (G(f), G(f)#): (GE(Y), O(F)Y) ® (GE(X), O(F)X) of locales over C such that G(f)-1 = f-1: G(X) ® G(Y), and for any open sieve U of X with V = f-1(U), G(f)#U: V  ® U  is the restriction morphism of f. We obtain a strict bicontinuous functor SpecF: E ® Loc(C) such that F = OCSpecF
(c) Suppose C has colimits. Suppose ({Ai, OAi}, {¯aij}, {uij}) is a glueing diagram of Loc(C). Glueing the locales Ai along ¯aij Í Ai as in (5.1.b) we obtain a locale A containing each Ai as an open sublocale of A. For each open sublocale ¯a of A let OA(a) be the colimit of the objects OAi(b) for all the open sublocales ¯b Í ¯a with b Î Ai for some i under the restriction morphisms. Then (A, OA) is a glueing colimit of ({Ai, OAi}, {¯aij}, {uij}). 
Next suppose E is a framed site and F: E ® C a strict functor. For any open sieve U on an object X of E we let O(F)X(U) = F(U) (cf. (1.3)). We obtain a locale (GE(X), O(F)X) over C. As in (b) we can prove that SpecF: E ® Loc(C) given by X ® (GE(X), O(F)X) is a strict functor and F = OCSpecF. n 

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