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:
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^{^}.
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).
Definition 3.6. A Dedekind
cut of a strict metric site C is a sheaf A
of sets on C such that
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.
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}:
