In this section we assume D is a strict metric site with colimits. We present an axiomatic approach to the theory of local ringed sites, replacing Ring^op by D. If D is a T₀-site this approach also provides a general method to construct a Cauchy-completion for D, using the notion of D-spaces (see 3.2.16). This method also works for an arbitrary strict T₀-site (which may not have colimits); see (3.2.18). 

Definition 3.2.1. A site over D (or simply D-site) is a pair (C, O) consisting of a metric site C and a cosheaf O: C ® D on C with values in D sending any empty object of C to an empty object of D; C and O are called the underlying site and the structure cosheaf of (C, O) respectively. 

Suppose (C, O) is a D-site. For any object X Î C and any open subset V of |O(X)| we define an open subset |X|_V of |X|:  
|X|_V = {x Î |X| ½ there is an effective open neighborhood i: U ® |X| of x such that |O(i)O(U)| Í V}. 

Definition 3.2.2. A local site over D (or local D-site, or simply local site) is a D-site (C, O) satisfying the following conditions for any X Î C: 
(a) O(X) is empty if and only if X is empty. 
(b) For any open cover {V_i} of |O(X)|, {|X|_Vi} is an open cover of |X|. 
(c) Suppose f: Y ® X is a morphism in C. Then for any open subset V of |O(X)|, f(Y) Í |X|_V if and only if |O(f)(O(Y))| Í V.  

Remark 3.2.3. If B is a basis for D, then in (3.2.2) we may assume that V_i and V are in B. Moreover, if the space of any object of D is quasi-compact, we may assume that {V_i} is a finite cover. 

Example 3.2.4. Suppose (C, O) is a D-site and B a subsite of a C. Then (B, O|_B) is a D-site, denoted simply by B. If C is a local site, then so is B.  

Example 3.2.5. D is a local D-site with the identity functor D ® D as the structure cosheaf. 

Example 3.2.6. A ringed site (C, O) is a site over Ring^op with the structure cosheaf O: C ® Ring^op. It is easy to see that (C, O) is a local ringed site if and only if it is a local site over Ring^op. (The equivalence of (3.2.2.a) and (3.1.3.a) can be established by an inductive argument based on (3.2.3).)  

Next we define D-spaces and local D-spaces, generalizing the concepts of ringed spaces and local ringed spaces respectively. 

Definition 3.2.7. A space over D (or simply D-space) is a pair (X, P) consisting of a topological space X and a cosheaf P: W(X) ® D on X such that P(Æ) ® D is an empty object.  

Thus a D-space is a D-site whose underlying site consists of open sets of a topological space. Suppose (X, P) and (Y, Q) are two D-spaces. A morphism from (Y, Q) to (X, P) is a pair (|f|, f) consisting of a continuous map |f|: Y ® X, and a morphism f (i.e., natural transformation) from Qf^-1: W(X) ® D to P: W(X) ® D, where f^-1 = W(X) ® W(Y) is the functor sending each open set U of X to f^-1(U) Í Y. We obtain a category of D-spaces, denoted D-Sp.  

Remark 3.2.8. D-Sp is a metric site with the obvious metric topology. As in the case of ringed spaces, D-Sp is strict and Cauchy-complete (here we need the assumption that D has colimits). It is a D-site with the structure cosheaf sending each D-space (X, P) to P(X).  

Definition 3.2.9. A local space over D (or local D-space, or simply local space) is a D-space (X, P) which is a local D-site. A morphism of local spaces over D is a morphism (|f|, f): (Y, Q) ® (X, P) of D-spaces such that for any open subsets V Í |P(X)| and U Í |Y|, f(U) Í |X|_V if and only if the morphism Q(U) ® P(X) induced by f: Q(Y) ® P(X) sends |O(U)| into V. 

Remark 3.2.10. Denote by D-LSp the category of local D-spaces. Then D-LSp is a Cauchy-complete site. It is a local D-site. If D = Ring^op we obtain the site of local ringed spaces. 

Suppose (C, O) is a D-site. Since D has colimits, as in (3.1) we can define the Kan extensions (C^{^}, O^{^}) and (C^~, O^~) which are again D-sites. We have the following generalization of (3.1.18) which can be proved in a similar way: 

Proposition 3.2.11. Suppose (C, O) is a local D-site. Then (C^{^}, O^{^}) and (C^~, O^~) are local D-sites. 

3.2.12 Suppose (C, O) is a D-site. Any object X determines a space b(X) = (X, O|_W(X)) over D where O|_W(X) is the restriction of the structure sheaf O: C^{^} ® D sending each open subset U of X to O(U). We obtain a functor b: (C, O) ® D-Sp which is an isometry of metric sites. Note that the structure cosheaf O: C ® D is isomorphic to the pullback of the structure cosheaf on D-Sp along b. Also (C, O) is a local D-site if and only if its image b(C) in D-Sp is in D-LSp.  

Since D is a local D-site (3.2.5), we obtain a canonical functor b: D d D-LSp by (3.2.12). 

Proposition 3.2.13. Suppose X, Y are two objects of D and |X| is a T₀-space. Then any morphism from b(Y) to b(X) in D-LSp is induced by a morphism from Y to X in D. 

Proof. Suppose (|f|, f): b(Y) ® b(X) is a morphism of local spaces. Since b(X) = (X, O|_W(X)), b(Y) = (Y, O|_W(Y)) and O is the identity functor of D, we have O(Y) = Y and O(X) = X. We prove that (|f|, f) is induced by the morphism f_X: Y = O(Y) = O(f^-1(X)) ® O(X) = X. Since f is uniquely determined by f_X, it suffices to prove that |f_X| coincides with |f|. Assume there is a point y Î |Y| such that f_X(y) / f(y). Since |X| is a T₀-space, we can find an open neighborhood U of f_X(y) such that f(y) Ï U, or an open neighborhood V of f(y) such that f_X(y) Ï V. In the first case there is an effective open neighborhood W of y in |O(Y)| = |Y| such that f_X(W) Í U. Then |f(W)| Í X_U = U by (3.2.9), so f(y) Î U which is a contradiction. Similarly we can prove that the second case is impossible. This shows that f(y) = f_X(y). Thus |f_X| coincides with |f|. 

Theorem 3.2.14. Suppose D is a T₀-site (i.e., the space of any object of D is a T₀-space). Then the canonical morphism b: D ® D-LSp is an isometric embedding. 

Proof. Clearly b: D ® D-LSp is faithful. It is full by (3.2.13).  

Definition 3.2.15. Suppose D is a T₀-site. An affine D-scheme is a local D-space which is isomorphic to b(X) for an object X e D. A D-scheme is a local D-space (X, P) in which any point x has an open neighborhood U such that the D-subspace (U, P|_U) of X is an affine D-scheme.  

Denote by D-Sch and D-ASch the sites of D-schemes and affine D-schemes respectively.  

Theorem 3.2.16. Assume D is a T₀-site. Then 
(a) D is equivalent to D-ASch. 
(b) D-Sch is a Cauchy-completion of D.  

Proof. (a) follows from (3.2.14). (b) follows from (a) and (2.1.10) as D-LSp is Cauchy-complete and D-Sch is the completion of D-ASch in D-LSp. 

Example 3.2.17. If D = Ring^op we obtain the ordinary schemes and affine schemes. 

Remark 3.2.18. If C is any standard T₀-site we first embed C isometrically into a standard T₀-site D with colimits. Define an affine C-scheme to be an affine D-scheme which is isomorphic to b(X) for an object X e C. Define a C-scheme to be a D-scheme (X, P) in which any point x has an open neighborhood U such that the D-subspace (U, P|_U) of X is an affine C-scheme. Then the site C-ASch of affine C-schemes is equivalent to C and the site C-Sch of C-schemes is a Cauchy-completion of C. 

Example 3.2.19. If C is the opposite of the category k-Alg_f of finitely generated reduced algebras over an algebraically closed field k, and D is the opposite of the category k-Alg of k-algebras with the Zariski topology, then a C-scheme (resp. affine C-scheme) in the sense of (3.2.18) is precisely a prevariety (resp. affine variety) over k in the sense of (1.1.19). Thus the site PVAr/k of prevarieties is a Cauchy-completion of the T₀-site k-Alg_f^op.  

A topological space is called sober if every closed irreducible set has a unique generic point. A site D is called sober if the space of any object of D is sober.  

Proposition 3.2.20. Suppose D is a sober site. A D-site (C, O) is a local site if and only if for any X Î C there is a continuous map |X| ® |O(X)| satisfying the following conditions: 
(a) For any morphism f: Y ® X in C, the following diagram of maps of topological spaces commutes:  

Proof. First assume (C, O) is a local site. Consider an object X Î C. For any point x Î |X| let W be the union of open subsets V of |O(X)| such that x is not contained in |X|_V. Then |O(X)| - W is an irreducible closed set. Since |O(X)| is sober, |O(X)| - W has a unique generic point, denoted by t(x). We obtain a map t: |X| ® |O(X)| sending each x Î |X| to t(x). This is a continuous map satisfying the conditions (a) and (b). 
Conversely, suppose the conditions are satisfied. One can easily verify the conditions of (3.2.2) are satisfied for (C, O). Thus (C, O) is a local site. 

Remark 3.2.21. Suppose (C, O) is a D-site. For any X Î C let W^o(X) be the subsite of C consisting of effective open subobjects of X. We say X is a local object of C if the D-subsite W^o(X) of C is a local site. Suppose X, Z are two local objects. A morphism f: Z ® X is called local if (3.2.2.b) holds for any object Y of W^o(Z). Denote by Loc(C) the subsite of local objects of C with local morphisms. It is the largest local D-subsite of C. Clearly we have Loc(D-Sp) = D-LSp.