3. Dedekind Cuts  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 {Ui} of |X|, A(X) is the limit (resp. colimit) of the system A(Uij) for all i, j, where Uij = Ui Ç Uj, via the morphisms induced by the inclusions among Uij.  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 ® homE(Z, A(X)) (resp. homE(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 {Ui} of |X|, together with fi Î A(Ui), such that the restrictions of fi and fj to Ui Ç Uj 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 AX 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 AX+ the associated sheaf of AX. Then X ® AX+(X) defines a sheaf A+ of sets on C, called the associated sheaf of A. The canonical maps AX(X) ® AX+(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 ({Ai}, {Uij}, {uij}) be a glueing diagram of C~. We glue the sheaves Ai along uij as presheaves, and take the associated sheaf A. Then A together with the associated morphisms Ai ® A is a glueing colimit of ({Ai}, {Uij}, {uij}) 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 {Ui} of |A| such that each Ui is representable ({Ui} 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 tD of t^ on D(C); tD is also the extension of t on D(C).  Theorem 3.7. (D(C), tD) 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 {Ui} is an open representable cover of |A|. Then U = È (Ui Ç U) is a small set, and U has an open representable cover (as each Ui 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 ({Ai}, {Uij}, {uij}) be a glueing diagram of D(C). We glue the sheaves Ai along uij as presheaves, and take the associated sheaf A. Then A is a Dedekind cut because |A| = È |Ai| is a small set, and |A| has an open representable cover as each |Ai| has such. Thus A together with the associated morphisms Ai ® A is a glueing colimit of ({Ai}, {Uij}, {uij}) 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 Ringop 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 Ringop:  (a) The category D(Ringop) of Dedekind cuts is a completion of Ringop. 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: Ringop ® 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(Ringop) in LSp. Therefore by (2.4.c) the category Sch of schemes is equivalent to a completion of Ringop. 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]).    [Next Section][Content][References][Notations][Home] 