2.3 Cauchy-completions 

Suppose (C, t) is a metric site. Denote by t^: C^ ® Top the Kan extensions of t on C^. We obtain a presite (C^, t^). Denote by (C~, t~) the subpresite of C^ with the induced metric functor t~ = t^|C~. We shall see that (C^, t^) and (C~, t~) are metric sites. 

Remark 2.3.1. If C is strict, then C Í C~ and t~ is the Kan extension of t on C~

Proposition 2.3.2. (C^, t^) and (C~, t~) are everywhere active. 

Proof. If U is a subset of the space |A| of a presheaf A Î C^. Then U determines a subfunctor U of A given by U(Z) = {f Î A(Z) ½ |f(Z)| Í U} for each Z Î C. Clearly the canonical morphism h: U ® A is active in C^. Thus C^ is everywhere active. If A Î C~ is a sheaf on C the subfunctor U of A is also a sheaf on C. Then h: U ® A is a active morphism in C~. Thus C~ is everywhere active. 

Theorem 2.3.3. (C^, t^) and (C~, t~) are effective metric sites. 

Proof. Since C is exact and separable, C^ is exact and separable by (1.4.13) and (1.4.17). Also C^ is active (2.3.2). Thus C is effective, hence a metric site. Next we assume C is strict. Then C~ is effective by the proof of (2.3.2). Since fibre products exist C~ and the inclusion functor C~ ® C^ preserves fibre products (2.2.7 or 2.2.8), C~ is separable by (1.3.2). Thus C~ is a metric site. (If C is strict then this also follows from (1.4.13) and (1.4.17) in view of (2.3.1).) 

Theorem 2.3.4. Suppose C is a strict metric site. Then C~ is a Cauchy-complete metric site containing C as a subsite. 

Proof. We prove that C is strict by verifying the definition (2.1.2) of strict metric sites for two sheaves A, B Î C~ and an open cover {Ui} of |A|. Suppose f, g Î hom (A, B) and f ¹ g. We can find Z Î C and h Î A(Z) = hom (Z, A) such that fh ¹ gh. Let {Vj} be an open effective cover of |Z| such that each h(Vj) is contained in some Ui. Since B is a sheaf on C, and Z Î C, we can find an open subset Vk such that the restrictions of fh and gh to Vk are different. Suppose h(Vk) is contained Uk. Then the restrictions of f and g to Uk are different. 
Now suppose for any i we have an fi Î hom (Ui, B) such that the restrictions of fi and fj to Ui Ç Uj are the same. For any object Z Î C and h Î hom (Z, A), let {Vj} be an open effective cover of |Z| as above. Denote by hj the restriction of h to Vj. Since B is a sheaf, the compositions fihj for all h(Vj) Í Ui determine a g' Î hom (Z, B). We obtain a map hom (Z, A) ® hom (Z, B) given by g ® g' for each Z Î C. The collection of all these maps forms a morphism f Î hom (A, B) whose restriction to each Ui is fi. This proves that C~ is strict. 
Next we prove that C~ is Cauchy-complete. Let {Ai} be a family of sheaves on C. For each i ¹ j, suppose given an open subset Uij Í |Ai|. Suppose also given for each i ¹ j an isomorphism of sheaves uij: Uij ® Uji such that for each i, j, k these isomorphisms are compatible with each other in the sense of the definition of Cauchy-complete metric sites (2.1.3). We glue these Ai along uij as presheaves, and take the associated sheaf A. Then A together with the associated morphisms Ai ® A satisfies the conditions of (2.1.3). This proves that C~ is Cauchy-complete. 

For any standard metric site C denote by G~(C), F~(C) and E~(C) the Cauchy-completion, finite Cauchy-completion, and effective completion of C in C~ respectively, then they are the Cauchy-completion, finitely Cauchy-completion, and effective completion of C respectively by (2.1.10), because C~ is Cauchy-complete, therefore finitely Cauchy-complete and effective. Thus we obtain 

Theorem 2.3.5. Any strict metric site C has a Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion). Any Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion) of C is canonically equivalent to G~(C) (resp. F~(C), resp. E~(C)) via the functor X d hom (~, X) for any X Î C'. 

Example 2.3.6. Consider the site C = W(X) of open sets of a nonempty topological space X. Denote by Sh(X) = C~ the site of sheaves of sets on X
(a) Any morphism in Sh(X) is a local isomorphism and X is the final object of Sh(X). 
(b) Sh(X) is the Cauchy-completion of W(X). 
(c) For any sheaf A on X the space |A| of A is usually called the espace étalé of A, denoted Spé(A). 
(d) The metric topology Spé: Sh(X) ® Top on Sh(X) induces an equivalence from Sh(X) to the subcategory T of Top/X consisting of local homeomorphisms to X. Thus T is also a Cauchy-completion of W(X). Note that each Spé(A) is a local space over Top/X in the sense of (3.2). 

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