Framed Sites 5

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 e_a: ¯a ® A of locales. The subobject ¯a of A (with the inclusion monomorphism e_a) 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 {¯a_i} is an open effective cover of A; then 1_A = Ú {a_i}. Suppose f, g are two morphisms of locales from A to B whose restrictions on each ¯a_i Í A are equal; then for any b Î B we have f^-1(b) Ù a_i = g^-1(b) Ù a_i. Thus f^-1(b) = Ú {f^-1(b) Ù a_i} = Ú {g^-1(b) Ù a_i} = g^-1(b). This shows that f = g. Conversely, suppose for each i there is a morphism f_i: ¯a_i ® B such that the restrictions of f_i and f_j on ¯a_i Ù ¯a_j are equal, i.e., for any b Î B we have f_i^-1(b) Ù a_j = f_j^-1(b) Ù a_i. Let f^-1: B ® A be the map sending each b Î B to Ú {f_i^-1(b)} Î A. Then

f^-1(b) Ù a_j = Ú_i {f_i^-1(b) Ù a_j} = Ú_i {f_j^-1(b) Ù a_i} = f_j^-1(b) Ù (Ú_i {a_i}) = f_j^-1(b). This shows that the composition of f^-1: B ® A with e_a^-1: A ® ¯a_i yields the morphism f_i^-1: B ® ¯a_i. 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 ¯a_i is f_i. This shows that Loc is strict.
(b) Suppose ({A_i}, {¯a_ij}, {u_ij}) is a glueing diagram of Loc. Glueing the sets A_i along the subsets ¯a_ij Í A_i we obtain a set S containing each A_i. Denote by A the collection of subsets U of S such that U Ç A_i is an open sublocale of A_i. Then A is a locale. We may regard each A_i as an open sublocale of A by identifying each a Î A_i with ¯a Í A. This turns A into a glueing colimit of {A_i} in Loc. n

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

Suppose (B, O_B) is another locale over C. A morphism of locales over C from (A, O_A) to (B, O_B) is a pair (f, f^#) of a morphism f: A ® B of locales, and a natural transformation f^#: f_*O_A ® O_B, where f_*O_A = O_Af^-1: B ® C.

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

Denote by Loc(C) the category of locales over C. We have a strict functor O_C: Loc(C) ® C sending each (A, O_A) to O_A(A) and any morphism (f, f^#): (A, O_A) ® (B, O_B) to f^#_B: O_A(A) ® O_B(B).

Theorem 5.2. (a) Loc(C) is a strict effective framed site.
(b) If (E, G_E) is an effective framed site and F: E ® C a strict functor, there is a unique (up to equivalence) strict bicontinuous functor Spec_F: E ® Loc(C) such that F = O_CSpec_F.
(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 O_A are strict.
(b) If X is an object of E the strict functor F: E ® C induces a strict functor O(F)_X: G_E(X) ® C sending each U Î G_E(X) to F(U). We obtain a locale Spec_F(X) = (G_E(X), O(F)_X) over C. Any morphism f: Y ® X in E induces a morphism Spec_F(f) = (G(f), G(f)^#): (G_E(Y), O(F)_Y) ® (G_E(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 Spec_F: E ® Loc(C) such that F = O_CSpec_F.
(c) Suppose C has colimits. Suppose ({A_i, O_Ai}, {¯a_ij}, {u_ij}) is a glueing diagram of Loc(C). Glueing the locales A_i along ¯a_ij Í A_i as in (5.1.b) we obtain a locale A containing each A_i as an open sublocale of A. For each open sublocale ¯a of A let O_A(a) be the colimit of the objects O_Ai(b) for all the open sublocales ¯b Í ¯a with b Î A_i for some i under the restriction morphisms. Then (A, O_A) is a glueing colimit of ({A_i, O_Ai}, {¯a_ij}, {u_ij}).
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 (G_E(X), O(F)_X) over C. As in (b) we can prove that Spec_F: E ® Loc(C) given by X ® (G_E(X), O(F)_X) is a strict functor and F = O_CSpec_F. n

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