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 Cauchycomplete 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 {U_{i}} 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 {V_{j}} be an open effective cover of Z such
that each h(V_{j}) is contained in some U_{i}.
Since B is a sheaf on C, and Z Î
C, we can find an open subset V_{k} such that the
restrictions of fh and gh to V_{k}
are different. Suppose h(V_{k}) is contained U_{k}.
Then the restrictions of f and g to U_{k}
are different.
For any standard metric site C denote by G^{~}(C), F^{~}(C) and E^{~}(C) the Cauchycompletion, finite Cauchycompletion, and effective completion of C in C^{~} respectively, then they are the Cauchycompletion, finitely Cauchycompletion, and effective completion of C respectively by (2.1.10), because C^{~} is Cauchycomplete, therefore finitely Cauchycomplete and effective. Thus we obtain Theorem 2.3.5. Any strict metric site C has a Cauchycompletion (resp. finite Cauchycompletion, resp. effective completion). Any Cauchycompletion (resp. finite Cauchycompletion, 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.
