8. Geometric Locales 
Now consider the site Loc(C) of the locales over a locally small framed site C with the strict functor OC: Loc(C) ® C. Write GLoc(C) for Geo(Loc(C)/OC). An object of GLoc(C) is called a geometric locale over C; a morphism in GLoc(C) is called a geometric morphism of geometric locales over C

Proposition 8.1. (a) GLoc(C) is an effective subsite of Loc(C). 
(b) A strict functor F: E ® C from an effective framed site E to C is continuous if and only if the induced functor SpecF: E ® Loc(C) has the image in GLoc(C). 
(c) If (E, GE) is an effective framed site and F: E ® C is a strict continuous functor, there is a unique (up to isomorphism) bicontinuous functor SpecF: E ® GLoc(C) such that F = OCSpecF
(d) If C has colimits then GLoc(C) is complete and (c) holds for any framed site E and any strict continuous functor F: E ® C

Proof. (a), (b) and (c) follow from (6.2.c), (7.2.b) and (5.2.b). 
(d) If C has colimits then Loc(C) is complete by (5.2.c), and we have C(GLoc(C)) = GLoc(C) by (6.7), thus GLoc(C) is complete by (4.3). The proof for the second assertion of (d) is similar to that of (5.2.c). n 

Example 8.1.1. (a) If D is a (locally small) category with a strict initial object viewed 
as a site with the trivial strict topology then Loc(D) = GLoc(D). 
(b) Loc = Loc(2) = GLoc(2) where 2 = {0, 1} is the locale of two elements viewd as a framed site. 
(c) Suppose (E, GE) is a framed category. The topology GE on E may be viewed as a bicontinuous functor from E to Loc. If E is a framed site this is a special case of (5.2.d) or (8.1.(d) with C = 2 and F being the unique strict functor from E to 2

Suppose C is a strict framed site with colimits. Since C is strict the identity functor 1C: C ® C is a strict bicontinuous functor. Applying (8.1.d) to F = 1C we obtain a bicontinuous functor Spec: C ® GLoc(C). If X is an object of C we call Spec(X) the spectrum of X

Proposition 8.2. Spec: C ® GLoc(C) is a full embedding (i.e., Spec(C) is a full subcategory of GLoc(C) equivalent to C). 

Proof. For simplicity we shall write F for Spec. Then F is an embedding as 1C = OCF. Thus we only need to prove that F is full. Suppose X and Y are two objects of C and (f, f#): F(X) ® F(Y) is a geometric morphism of geometric locales. Then G((OC)F(X))G(f) = G(f#X)G((OC)F(Y)). But G((OC)F(X)): G(X) ® G(X) and 
G((OC)F(Y)): G(Y) ® G(Y) are identity functors, thus G(f) = G(f#X). If U is an open effective sieve of X, then f-1(U) = f#X-1(U). It follows that f#U: f-1(U) = f#X-1(U) ® U is induced by the restriction of f#X on f#X-1(U). For an arbitrary open sieve U we can show that f#U: O(f-1(U)) ® O(U) is also induced by f#X by passing to the colimits. This shows that (f, f#) = F(f#X). Thus F is a full embedding. n 
Definition 8.3. An affine scheme over C is a geometric locale over C isomorphic to the spectrum of some object of C. A scheme over C is a geometric locale in the completion C(Spec(C)) of Spec(C) in GLoc(C). 

Theorem 8.4. Suppose C is a strict framed site with colimits. 
(a) The full subcategory Sch(C) of GLoc(C) of schemes over C is a complete framed site which is equivalent to a completion of C
(b) The full subcategory ASch(C) of GLoc(C) of affine schemes over C is a reflective subcategory of GLoc(C). 

Proof. (a) Since GLoc(C) is complete by (8.1.d), the assertion follows from (4.3). 
(b) SpecOC: GLoc(C) ® Spec(C) is the left adjoint of the inclusion functor Spec(C) ® GLoc(C). Since ASch(C) is equivalent to Spec(C), it is a reflective subcategory of GLoc(C). n 

Remark 8.5. If C is a locally small category (resp. framed site) with colimits, then one can show that Loc(C) (resp. GLoc(C)) have colimits. 

Suppose C, D, and E are framed sites and F: C ® E, G: D ® E are two bicontinuous functors. Suppose (X, Y) is a pair with X Î C and Y Î D such that F(X) = G(Y). Since F and G are bicontinuous, GC(X), GE(F(X)) = GE(G(Y)), and GD(Y) are all isomorphic, so we may identify GC(X) with GD(Y). We say (X, Y) is compatible if X and Y has a common open effective cover {Ui} Í GC(X) = GD(Y). 

Let C ×E D be the collection of all such compatible pairs (X, Y). Suppose (X', Y') is another compatible pair. We define a morphism from (X', Y') to (X, Y) to be a pair (f, g) with f: X' ® X and g: Y' ® Y such that F(f) = G(g). This turns C ×E D into a category, which is naturally a framed site such that a morphism (f, g): (X, Y) ® (X', Y') in C ×E D is open effective if and only if both f: X' ® X and g: Y' ® Y are open effective. The natural functors p1: C ×E D ® C and p2: C ×E D ® D are then bicontinuous functors. It is easy to see that C ×E D together with p1 and p2 is the fibre product of C and D over E in the metacategory of framed sites and bicontinuous functors. Thus we can talk about base extension for framed sites. If E = Loc with p1 = GC and p2 = GD then C ×Loc D is the product of C and D, denoted simply by C × D

Remark 8.6. (a) C ×E D is effective if and only if both C and D are so. 
(b) If any one of C and D is effective, then the underlying category of C ×E D is the fibre product of C and D over E as categories because then any pair (X, Y) with F(X) = G(Y) is compatible. 
(c) Faithful (resp. full) bicontinuous functors are stable under base extension (i.e., if F is faithful or full then so is p2: C ×E D ® D). 
(d) If C is a full effective base of E, then C ×E D is naturally a full base of D

Definition 8.7. Suppose (M, OM) is a strict effective framed site (viewed as a framed site over Loc via OM). 
(a) If C is a category we write LocM(C) for the effective framed site M ×Loc Loc(C). An object of LocM(C) is called an M-space of C
(b) If C is a framed site with colimits we write GLocM(C) for the effective framed site M ×Loc GLoc(C). An object of GLocM(C) is called a geometric M-space of C

Example 8.7.1. (a) Let M be a strict effective framed site. If C is a category and t: C ® M is a functor inducing a framed topology Gt on C such that (C, Gt) is an essential locally effective framed site, then we say that (C, t) is a M-metric site. Note that any framed site is naturally an Loc-metric site. 
(b) A Top-metric site is simply called a metric site (see [Luo 1995a]) . 
(c) Denote by STop the complete framed subsite of Top consisting of sober topological spaces. A metric site (C, t) is called sober if t(C) ® STop
(d) Suppose C is a strict M-metric site. We say C is M-complete if any glueing diagram S of C such that t(S) has a glueing colimit in M has a glueing colimit in C. Any strict M-site C has an M-completion which is a complete M-metric site containing C as a base (the proof is similar to that of (4.4)). 

Example 8.7.2. Consider the opposite Ringop of the category Ring of small commutative rings with 1. Ringop is a locally small category with colimits, which is a strict effective framed site with the topology sending each commutative ring R to the frame of radical ideals of R. We have 
(a) LocTop(Ringop) =  Top × Loc(Ringop) is the complete metric site of ringed spaces. 
(b) GLocTop(Ringop) = Top × GLoc(Ringop) is the complete metric site of local ringed spaces. 
 (c) SchSTop(Ringop) = STop × Sch(Ringop) is the complete metric sites of schemes. 

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