Definition 2.1.1. A metric site (or simply site) is a separable, locally effective presite (C, t). The functor t is called a metric topology on C. 

Suppose (C, t) and (C', t) are two metric sites. An isometry of sites from (C, t) to (C', t') is an isometry of presites j: C ® C' such that j(f) is effective for any effective morphism f in C; j is called an isometric embedding if the functor j is faithful, full, and C is equivalent to its image in C'. 

Suppose (C, t) is a metric site, B a subcategory of C. If the subpresite (B, t|_B) is a metric site and any open effective morphism f: X ® Y in (B, t|_B) is an effective morphism in C, we say (B, t|_B) (or simply B) a subsite of (C, t). If B is a subsite of C the inclusion functor I: B ® C is an isometry of sites. 

Definition 2.1.2. A metric site C is strict (or subcanonical) if the following glueing lemma for morphisms holds: 
Suppose X, Y are objects and {U_i} is an open effective cover of X. Suppose for each i we have a morphism f_i: U_i ® Y such that the restrictions of f_i and f_j to U_i Ç U_j are the same. Then there exists a unique morphism f from X to Y such that the restriction of f to U_i is f_i. 

Definition 2.1.3. A Cauchy-complete metric site is a strict, effective metric site C such that the following glueing lemma for objects holds: 
Let {X_i} be a family of objects. For each i ¹ j, suppose given an open subset U_ij Í |X_i|. Suppose also given for each i ¹ j an isomorphism of objects u_ij: U_ij ® U_ji such that 
(1) for each i, j, u_ji = u_ji^-1, and 
(2) for each i, j, k, u_ij(U_ij Ç U_ik) = U_ji Ç U_jk, and |u_ik| = |u_jku_ij| on U_ij Ç U_ik. Then there is an object X, together with effective open morphisms v_i: X_i ® X for each i, such that {|v_i(X_i)|} is an open cover of |X|, with v_i(U_ij) = |v_i(X_i)| Ç |v_j(X_j)|, and |v_i| = |v_ju_ij| on U_ij (if U_ij and u_ij are all empty then we say that X is the disjoint union of the X_i). 

We say C is a finitely Cauchy-complete metric site if (2.1.3) holds for any finite family {X_i} of objects. 

Example 2.1.4. If C is a strict site and B a full subsite of C, then B is a strict site. 

Example 2.1.5. All the basic presites (resp. basic algebraic presites) are separable and locally effective, hence are metric sites, which will be called the basic sites (resp. basic algebraic sites) from now on. The glueing lemma for morphisms (2.1.2) holds for all these sites. Thus they are strict. 

Example 2.1.6. The glueing lemma for objects (2.1.3) holds for all the basic sites and the sites Sch, GSch, PVar/k, hence they are Cauchy-complete metric sites. 

2.1.7 Suppose C' is a strict site and C a full subsite of C'. We introduce three subsites of C': 
(a) The Cauchy-completion G(C) of C in C' consists of objects X Î C' such that the effective open subsets U of |X| with U Î C form a base for the underlying space |X| of X. 
(b) The finite Cauchy-completion F(C) of C in C' consists of objects X Î C' such that |X| has a finite open effective cover {U_i} with U_i Î C. 
(c) The effective completion E(C) of C in C' consists of all the effective open subobjects of the objects of C. 

2.1.8 Clearly C is a subsite of G(C), F(C) and E(C). If C' = G(C) (resp. C' = F(C), resp. C' = E(C)) then we say that C is a Cauchy-base (resp. finite Cauchy-base, resp. effective base) of C'. 

Definition 2.1.9. Suppose C is a strict metric site. A Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion) of C is a Cauchy-complete (resp. finite Cauchy-complete, resp. standard and effective) metric site C' containing C as a Cauchy-base (resp. finite Cauchy-base, resp. effective base). 

Remark 2.1.10. Suppose C' is a Cauchy-complete (finitely Cauchy-complete, resp. effective) metric site and C a full subsite of C'. Then G(C) (resp. F(C), resp. E(C)) is a Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion) of C. 

Example 2.1.11. (a) Sch is the Cauchy-completion of ASch in LSp. 
(b) PVar/k is the Cauchy-completion of AVar/k in GSp/k. 
(c) If X is a topological space and B an open basis of X which is closed under intersection, then B is an effective base of W(X). 

Proposition 2.1.12. Suppose C' is a strict metric site and C  is a Cauchy-base (resp. finite Cauchy-base, resp. effective base) for C'. Suppose D is a Cauchy-complete (resp. finitely Cauchy-complete, resp. effective) metric site. Then any full isometric embedding of metric sites j from C to D can be extended (non-canonically) to a full isometric embedding from C' to D. A Cauchy-completion (resp. finite Cauchy-completion, resp. effective completion) of a strict metric site is unique up to equivalence. 

Proof. We treat the case of Cauchy-complete sites. The other cases are similar. Suppose {U_i|U_i Î C} is an effective open cover of an object X Î C'. Put U_ij = U_i Ç U_j. Then we can glue {j(U_i)} along {j(U_ij)} to obtain an object j(X) Î D. Suppose f: Y ® X is a morphism in C' and {V_j|V_j Î C} is an effective open cover of |Y| such that f(V_j) Í U_i for some i. Let f_j: V_j ® U_i be the restriction of f to V_j; then f_j is a morphism in C. Glueing these j(f_j) we obtain a morphism j(f): j(Y) ® j(X). We thus obtain an extension of j on C' which is an isometric embedding. The last assertion follows from the first one.