We shall fix a universe U which contains the set N of natural numbers. A category C is called small (resp. ordinary) if C is a small Uset (resp. Uclass) and hom_{C}(X, Y) is a small Uset for any X, Y Î C. A metric presite (C, t) is called small (resp. ordinary) if C and t(C) are small (resp. ordinary) categories, and t(C) consists of small topological spaces. In this section we assume all the categories and metric sites are ordinary. A glueing diagram ({X_{i}}, {U_{ij}}, {u_{ij}}) of a metric site C consists of a small set {X_{i}} of objects of C together with, for any i ¹ j, an open effective subset U_{ij} Í X_{i} and an isomorphism of objects u_{ij}: U_{ij} ® U_{ji}, such that (a) u_{ji} = u_{ij}^{1}; (b) u_{ij}(U_{ij} Ç U_{ik}) = U_{ji} Ç U_{jk}; (c) u_{ik} = u_{jk}u_{ij} on U_{ij} Ç U_{ik}. A glueing colimit of a glueing diagram ({X_{i}}, {U_{ij}}, {u_{ij}}) is an object X Í C, together with open effective 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_{j}u_{ij} on U_{ij}. We often simply say that X is the object obtained by glueing {X_{i}} along {U_{ij}} (if U_{ij} are all empty then we say that X is the disjoint union of the X_{i}). Definition 2.1. A complete metric site is a strict metric site C such that any glueing diagram of C has a glueing colimit. Proposition 2.2. Any complete metric site is effective. Proof. Suppose X is an object of a complete metric site C and U an open subset of X. Any open effective cover {U_{i}} of U determines a glueing diagram G({U_{i}}) = ({U_{i}}, {U_{ij}}, {u_{ij}}), where U_{ij} = U_{i} Ç U_{j} and u_{ij}: U_{ij} ® U_{ji} is the canonical isomorphism. Let (Y, {v_{i}}) be a glueing colimit of G({U_{i}}); we may regard {U_{i}} as an open effective cover of Y via effective morphisms v_{i}. Since C is strict, the effective morphisms U_{i} ® X induces an effective morphism f: Y ® X with f(Y) = U, thus U is effective. n Example 2.2.1. (a) The categories of (small) topological spaces, ringed spaces, local ringed spaces are everywhere effective, complete metric sites. (b) The categories of topological manifolds, differential manifolds, complex analytic spaces are effective, strict metric sites (if we ignore the Hausdorff condition then these are also complete metric sites). (c) The categories of schemes, formal schemes, algebraic spaces and algebraic stacks are complete metric sites. (d) The categories of affine schemes and affine varieties are strict metric sites. (e) The category of algebraic varieties is an effective, strict metric site. Example 2.2.2. Suppose X is a local ringed space. We define the geometric radical r(X) of X to be the sheaf of ideas given by r(X)(U) = {s Î O_{X}(U)s_{x} is a nonunit of O_{x} for all x Î U} for every open subset U of X. A local ringed space X is called geometrically reduced (or simply reduced) if r(X) = 0. The category of reduced local ringed spaces is an everywhere effective, complete metric site, which contains the category of reduced schemes as an effective, strict subsite. Example 2.2.3. A ringed (resp. local ringed, resp. geometric) set is a ringed (resp. local ringed, resp. reduced local ringed) space with a discrete underlying space. The category of ringed sets (resp. local ringed sets, resp. geometric sets) is an everywhere effective, complete subsite of the metric site of ringed spaces (resp. local ringed spaces). Note that a geometric set is a scheme. Suppose C ' is a metric site and C a full subsite of C '. Denote by E(C) the full subsite of C ' consisting of objects X such that the open effective subsets U of X with U Î C form a base for X. We have C Í E(C) because C is locally effective. E(C) is called the completion of C in C '. If C ' = E(C) then C is called a base for C '. Definition 2.3. Suppose C is a strict metric site. A completion of C is a complete metric site C ' containing C as a base. Remark 2.4. Suppose C ' is a strict metric site and C a full subsite of C '. Then (a) C is a strict metric site (1.5.1). (b) C is a dense subcategory of E(C). (c) If C ' is complete then the completion E(C) of C in C ' is a completion of C. Proposition 2.5. Suppose C ' is a strict metric site and C a base for C '. Suppose D is a complete metric site. Then any full embedding of metric sites j from C to D can be extended uniquely up to a natural isomorphism to a full embedding from C ' to D. A completion of a strict metric site is unique up to equivalence. Proof. Suppose {U_{i}U_{i} Î C} is an open effective cover of an object X Î C '. Let G({U_{i}}) = ({U_{i}}, {U_{ij}}, {u_{ij}}) be the glueing diagram of C determined by {U_{i}} (see the proof of (2.2)), which may be viewed as a glueing diagram of C. Since j is an embedding, j(G({U_{i}}) = ({j(U_{i})}, {j(U_{ij})}, {j(u_{ij})}) is a glueing diagram of D, which has a glueing colimit j(X) Î D because D is complete. 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. Since D is strict, we can glue these j(f_{j}) to obtain a morphism j(f): j(Y) ® j(X). We obtain an extension of j on C ' which is an embedding. The last assertion follows from the first one. n Lemma 2.6. Suppose i: X ® S and j: Y ® S are two morphisms in a metric presite C. Suppose U Í X, V Í Y, and W Í S are effective subsets such that i(U) Í W and j(V) Í W. Suppose the fibre product (Z, p, q) of X and Y over S exists. Put E = p^{1}(U) Ç q^{1}(V) and suppose E is effective. Then the subobject E of Z together with the restrictions p_{E} and q_{E} is the fibre product U ×_{W} V of the effective subobjects U and V over W, which is isomorphic to U ×_{S} V. If U and V are open (resp. closed) subobjects of X and Y respectively, then E is an open (resp. closed) subobject of Z. Proof. Let P be an object over W and let u: P ® U and v: P ® V be morphisms over W. Composing u, v with the effective morphisms: U ® X, V ® Y, we obtain two morphisms u': P ® X and v': P ® Y over S, which determines a morphism h = (u', v')_{S}: P ® X ×_{S} Y, with ph(P) = u(P) Í U and qh(P) = v(P) Í V, so h(P) Í p^{1}(U) Ç q^{1}(V) = E. Thus h induces a morphism from P to the effective subobject E having the required properties. On the other hand, p_{E} and q_{E} are clearly morphisms over W. Thus (E, p_{E}, q_{E}) is the fibre product of U and V over W. Now taking W = S, we obtain an isomorphism E ® U ×_{S} V. The last assertion is obvious. n Lemma 2.7. Suppose C ' is a metric site with fibre products. Suppose C is a full subsite of C ' with fibre products such that the inclusion functor C ® C ' preserves fibre products. Then the completion E(C) of C in C ' has fibre products and the inclusion functor C ® E(C) preserves fibre products. Proof. Suppose X, Y and S are objects in E(C), and f: X ® S, g: Y ® S are two morphisms. Let Z = X ×_{S} Y be the fibre product of X and Y over S in C '. It suffices to prove that Z Î E(C). The collection of objects U ×_{W} V, where U Í X, V Í Y and W Í S are open effective subsets with f(U) Í W and g(V) Í W, forms an open effective cover of Z by (2.6), so Z is in E(C). n Lemma 2.8. Suppose C is an everywhere effective metric site with fibre products, and D an everywhere effective subsite of C. Suppose the inclusion functor I: D ® C has a right adjoint J: C ® D (i.e., D is a coreflective subcategory of C) such that, for any object X Î D, the canonical morphism s: X ® J(X) is an effective morphism in D (e.g., D is a full coreflective subcategory of C). Then fibre products exist in D. Proof. Suppose i: X ® S
and j: Y ® S are objects
over S in D. Let Z = X ×_{S}
Y be the fibre product of X and Y over S in
C. Since J is a right adjoint functor, it preserves fibre
products, thus J(Z) is a fibre product of J(X)
and J(Y) over J(S) in D.
Theorem 2.9. If C is a strict metric site with fibre products, then any completion C' of C has fibre products. Proof. Suppose X and Y are two objects over S
in C'. For any triple i = (U, V, W)
of open subsets of X, Y and S respectively such
that U, V and W are in C
and U, V are above W, let T_{i} = U
×_{W} V be the fibre product
in C with the projections p_{i} and q_{i}.
If j = (U', V', W') is another similar triple
as i, with T_{j} = U' ×_{W'}
V', we define T_{ij} = p_{i}^{1}(U
Ç U') Ç
q_{i}^{1}(U Ç
U'). By (2.6) both T_{ij}
and T_{ji} are fibre products of U Ç
U' and V Ç V'
over W Ç W'. Thus
there is a unique isomorphism u_{ij}: T_{ij}
® T_{ji} compatible
with all the projections. Furthermore for each i, j, k
these isomorphisms are compatible with each other. Thus we obtain a glueing
diagram ({T_{i}}, {T_{ij}}, {u_{ij}})
of C ', which has a glueing colimit T as C ' is complete.
Glueing the projections from the pieces T_{i} we obtain
the projections p: T ® X
and q: T ® Y. We prove
that T is the fibre product of X and Y over i.
Theorem (2.9) provides a unified proof for the existence of fibre products in the complete metric sites of prevarieties, schemes, reduced schemes and formal schemes, because each of these metric sites is a completion of a strict metric site with fibre products. Remark 2.10. Denote by RSp and
LSp the complete metric sites of (small) ringed spaces and local
ringed spaces respectively. Then fibre products also exist in these complete
metric sites, but the proofs are quite different from that of (2.9):
Remark 2.11. The site Sch of schemes is the completion of the subsite ASch of affine schemes in LSp. Applying (2.7) to C ' = LSp and C = ASch we obtain another definition of fibre product of schemes. Remark 2.12. A morphism f: Y
® X of local ringed spaces is called
an immersion if f is bicontinuous, and for any y Î
Y, the induced map f_{y}^{#}: O_{X,f}_{(y)}
® O_{Y,y} of stalks is surjective.
Immersions are bicontinuous monomorphisms, hence universally bicontinuous
because they are stable under base extension. Note that a morphism f:
X ® Y of reduced local ringed
spaces is an immersion if and only if it is effective in the category of
reduced local ringed spaces.
