In this section we assume C is an ordinary metric site and E an ordinary category. 

Definition 3.1. A presheaf (resp. copresheaf) on C with values in E is a contravariant (resp. covariant) functor A from C to E. We say A is a sheaf (resp. cosheaf) if for any object X of C, and any open effective cover {U_i} of |X|, A(X) is the limit (resp. colimit) of the system A(U_ij) for all i, j, where U_ij = U_i Ç U_j, via the morphisms induced by the inclusions among U_ij. 

Remark 3.2. If E = Set is the category of (small) sets, then we say that A is a presheaf (resp. copresheaf) of sets on C. A presheaf (resp. copresheaf) A on C with values in any ordinary category E is a sheaf (resp. cosheaf) if and only if for any Z Î E, the presheaf of sets on C given by X ® hom_E(Z, A(X)) (resp. hom_E(A(X), Z)) is a sheaf. 

Remark 3.3. A presheaf A of sets on C is a sheaf if and only if the following condition is satisfied: 
If X is an object, then to give an element f of A(X), it is equivalent to give an open effective cover {U_i} of |X|, together with f_i Î A(U_i), such that the restrictions of f_i and f_j to U_i Ç U_j are the same. 

Example 3.3.1. A metric site C is strict if and only if the identity functor C ® C is a cosheaf (this follows from (3.2) and (3.3)). 

Denote by C^{^} and C^~ the categories of presheaves and sheaves of sets on C respectively. We identify C with a subcategory of C^{^} via Yoneda embedding; if C is strict then C Í C^~. 

Remark 3.4. A presheaf A of sets on C induces a presheaf A_X on the space |X| of any object X, generated by A(U) for open effective sets U of |X| (cf. [Grothendieck and Dieudonne EGAI, Ch. 0, (3.2.1)]). Denote by A_X⁺ the associated sheaf of A_X. Then X ® A_X⁺(X) defines a sheaf A⁺ of sets on C, called the associated sheaf of A. The canonical maps A_X(X) ® A_X⁺(X) for every X Î C determine a canonical morphism A ® A⁺, which has the universal property that any morphism from A to a sheaf B of sets on C factors through A ® A⁺ uniquely. The functor (+): C^{^} ® C^~ is a left adjoint of the inclusion functor C^~ ® C^{^}. Thus C^~ is a full reflective subcategory of C^{^}. Since limits exist in C^{^}, limits exist in C^~ and the inclusion C^~ ® C^{^} preserves limits. It is easy to see that (+) preserves fibre products and final objects. 

Since C is a full subcategory of C^{^}, applying (1.9) we obtain an extension of t on C^{^}, denoted by t^{^}. 

Theorem 3.5. (a) (C^{^}, t^) is an effective metric site; C is a subsite of C^{^}. 
(b) If C is strict then C^~ is a strict, effective subsite of C^{^}. 
(c) If C is a small strict metric site then C^~ is an ordinary metric site which is complete. 

Proof. (a) Suppose A Î C^{^} is a presheaf. Any subset U of |A| = t^(A) determines a subfunctor U of A defined by X ® {f Î A(X)| |f(X)| Í U} for each X Î C. Clearly the inclusion U ® A is active in C^{^}, thus C^{^} is everywhere active. Since C is exact, C^{^} is an effective metric site by the first assertion of (1.11). Since C is a full dense subcategory of C^{^}, it is a subsite of C^{^} by the second assertion of (1.11). 
(b) Suppose C is strict. Then it is a full subcategory of C^~. The extension of t on C^~ is the restriction of t^ on C^~. If A Î C^~ is a sheaf, and U an open subset of |A|, then U is a sheaf on C. This means that U is effective in C^~, hence C^~ is an effective subsite of C^{^}. It is strict by (1.12) because C is dense in C^~. 
(c) Suppose C is a small strict metric site. Then C^{^} and C^~ are ordinary effective sites. We prove that C^~ is complete. Let ({A_i}, {U_ij}, {u_ij}) be a glueing diagram of C^~. We glue the sheaves A_i along u_ij as presheaves, and take the associated sheaf A. Then A together with the associated morphisms A_i ® A is a glueing colimit of ({A_i}, {U_ij}, {u_ij}) in C^~. This proves that C^~ is complete. 

Definition 3.6. A Dedekind cut of a strict metric site C  is a sheaf A of sets on C such that 
(a) |A| is a small topological space (note that if C is small then |A| is always small). 
(b) There is an open cover {U_i} of |A| such that each U_i is representable ({U_i} is called an open representable cover of |A|). 

The collection of Dedekind cuts of C is an ordinary full subcategory of C^~, denoted by D(C), which is an ordinary metric presite with the induced pretopology t_D of t^ on D(C); t_D is also the extension of t on D(C). 

Theorem 3.7. (D(C), t_D) is a complete metric site for any strict metric site C; C is a base of D(C). 

Proof. (a) We prove that D(C) is effective. Suppose A is a Dedekind cut. Any open subset U of |A| is effective in C^~ by (3.5.b). To see that U is effective in D(C), it suffices to prove that the sheaf U is a Dedekind cut. Suppose {U_i} is an open representable cover of |A|. Then U = È (U_i Ç U) is a small set, and U has an open representable cover (as each U_i Ç U has such). This shows that U is a Dedekind cut. 
(b) D(C) is strict because it is a full subsite of the strict metric site C^~ (1.5.1). 
(c) We prove that D(C) is complete. Let ({A_i}, {U_ij}, {u_ij}) be a glueing diagram of D(C). We glue the sheaves A_i along u_ij as presheaves, and take the associated sheaf A. Then A is a Dedekind cut because |A| = È |A_i| is a small set, and |A| has an open representable cover as each |A_i| has such. Thus A together with the associated morphisms A_i ® A is a glueing colimit of ({A_i}, {U_ij}, {u_ij}) in D(C), which shows that D(C) is complete. 
Since C is dense in D(C), it is a subsite of D(C) by (1.11). It is a base of D(C) by the definition of a Dedekind cut. n 

Now our main theorem follows from (3.7), (2.5) and (2.9) immediately: 

Theorem 3.8. Any strict metric site C has a completion, which is uniquely determined by C up to equivalence. The collection D(C) of Dedekind cuts forms a completion of C, and C is complete if and only if any Dedekind cut of C is representable. If fibre products exist in C, then fibre products exist in any completion of C. n 

Definition 3.9. A metric topos is an ordinary, strict metric site which is equivalent to the metric site (C^~, t) of sheaves on some small strict metric site C. 

Remark 3.10. Any metric topos is complete since C^~ is so; the metric site C^~ of sheaves on any small strict metric site C is a metric topos, and D(C) is the completion of C in C^~. Since fibre products exist in C^~, (2.7) implies that fibre products exist in D(C) if C has fibre products. This yields another proof for (2.9) in the case that C is small. 

Example 3.10.1. Consider the strict metric site W(X) of the open subsets of a (small) topological space X. It is easy to see that any sheaf of sets on W(X) is a Dedekind cut of W(X). Thus the topos Sh(X) = (W(X))^~ of sheaves on X is a completion of W(X), i.e. D(W(X)) = W(X)^~. 

Remark 3.11. The opposite Ring^op of the category Ring of (small) commutative rings (with unit) is an ordinary strict metric site with the pretopology sending each ring A to the space Spec A of prime ideals of A with the Zariski topology. There are essentially two methods to construct a completion of Ring^op: 
(a) The category D(Ring^op) of Dedekind cuts is a completion of Ring^op. This is the functorial approach in algebraic geometry. 
(b) The category LSp of (small) local ringed space is a complete metric site and there is an embedding j: Ring^op ® LSp sending each ring A to the affine scheme (Spec A, O). A scheme is a local ringed space with an open affine cover, thus the site Sch of schemes is the completion of the subsite j(Ring^op) in LSp. Therefore by (2.4.c) the category Sch of schemes is equivalent to a completion of Ring^op. This is the geometric approach in algebraic geometry. One can show that this geometric approach also works for any strict metric site with colimits. The key step is to define a local site over any strict metric site with colimits, generalizing the notion of a local ringed spaces (see [Luo 1995b]).