9. Canonical Sites 

We say that a framed site (C, G) is canonical if G = vC, i.e., G(X) is the set of subobjects of X Î C
Remark 9.1. (a) Any canonical framed site is effective by (2.4.d). 
(b) Any frame viewed as a category is a neat canonical framed site. 

Remark 9.2. (a) The category Loc has colimts. Denote by J: Loc ® Setop the functor sending each locale to its underlying set. Then J lifts colimits (cf. [Borceux 1994, Vol III, Section 1.4]). 
(b) Suppose G: C ® Loc is a framed topology on a category C such that the composition JG: Cop ® Set is representable. Since any representable functor preserving limits, JG preserves limits in Cop. It follows that JG: C ® Setop preserves colimits. Combining with (a) we conclude that JG: C ® Loc preserve colimits. 

Theorem 9.3. (a) Any elementary topos such that the poset vC(X) is complete for any X Î C is a canonical framed site and G: C ® Loc preserves colimits. 
(b) Any elementary topos with colimits is a complete framed site. 
(c) Any Grothendieck topos is a complete canonical framed site. 
(d) Any elementary topos such that vC(X) is finite for any X Î C is a strict canonical framed site. 

Proof. (a) The poset vC(X) of subobjects of each object X of an elementary topos is a Heyting algebra, and the pullback along any morphism f: Y ® X induces a morphism f-1: vC(X) ® vC(Y) which has both a left adjoint and a right adjoint [Borceux 1994, Vol III, Prop. (6.2.3)]. Thus (C, vC) is an effective framed site if each vC(X) is complete. The second assertion follows from (9.2.b). 
(b) Suppose {Ui} is an open cover of an object X. Let Y be the coproduct of {Ui}and p: Y ® X the canonical morphism. Suppose p = uv where v is an epimorphism and u a monomorphism. Since each monomorphism Ui ® X factors through u, and the joint of {Ui} is 1X, we see that u is an isomorphism, so p = v is an epimorphism. The epimorphism p: Y ® X is the coequalizer of its kernel pair by [Mac Lane and Moerdijk 1993, p. 197]. This implies that X is a colimit of {Ui Ç Uj} via restrictions, so C is strict. 
Next we prove that C is complete. Suppose ({Xi}, {Uij}, {uij}) is a glueing diagram of C. Suppose (X, {vi}) is the colimit of ({Xi}, {Uij}, {uij}) in C, where each vi: Xi ® X is a morphism in C. The joint of the images of all the vi is X by (a) because X is a colimit of Xi. Thus it suffices to prove that each vi: Xi ® X is a monomorphism. We imitate the proof of [Mac Lane and Moerdijk 1993, p.211, Corollary 4]. For a fixed vi let PXi be the power object of Xi with the monomorphism t: Xi ® PXi. Since PXi is injective and ej: Uij = Uji ® Xj is a 
monomorphism for each j, the morphism tej: Uij ® Xi ® PXi extends to Xj, to give a morphism gj: Xj ® PXi. Since X is a colimit, there is a unique h: X ® PXi such that t = hvi and gj = hvj. Since t = hvi is a monomorphism, so is vi. This proves the assertion. 
(c) is a special case of (b). 
(d) The proof that C is strict is similar to (b) because any finite Heyting algebra is complete, and any elementary topos has finite colimits. n

Example 9.3.1. (a) The category FSet of finite sets is a canonical framed site by (9.3.c). 
(b) Set is a complete canonical framed site and is a completion of FSet