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 U-set (resp. U-class) and hom_C(X, Y) is a small U-set 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_jku_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_ju_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 non-unit 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. 
Let s_X: X ® J(X), s_Y: Y ® J(Y) and s_S: S ® J(S) be the canonical morphisms of X, Y and S respectively, which are effective morphisms in D by assumption. Applying (2.6) to D and replacing i, V, W by |X|, |Y|, |S|, we obtain the fibre product of X and Y over S in D, which is naturally an effective subobject of J(Z). n 

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. 
Suppose P Î C ' and f: P ® X, g: P ® Y are morphisms over S. Consider an open subset Z of |P| with Z Î C and Z is contained in f^-1(U)) Ç g^-1(V) for some triple i = (U, V, W) as above. Then the restrictions of f and g to Z determines a morphism Z ® T_i; composing with the effective morphism T_i ® T we obtain a morphism t_Z: Z ® T. If Z ' is another choice with respect to another triple j = (U', V', W'), with similar t_{Z '}: Z ' ® T, then the restrictions of t_Z and t_{Z '} to Z Ç Z ' are the same. Since the collection of all such Z forms an open cover of |P|, and C ' is strict, we can glue these morphisms by (1.5) to obtain a morphism t: P ® T. The uniqueness of t can be checked locally. n 

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): 
(a) Suppose (X, O_X) and (Y, O_Y) are ringed spaces over a ringed space (S, O_S) with the structural morphisms f: X ® S and g: Y ® S. Let (Z, p, q) be the fibre product of the spaces X and Y over S in the category of topological spaces. Let O_Z be the tensor product of p^-1(O_X) and q^-1(O_Y) over (fp)^-1(O_S). Then (Z, O_Z) is the fibre product of X and Y over S in RSp. 
(b) According to Chevalley (cf. [Hakim 1972, p.68]) there is a right adjoint functor Spec: RSp ® LSp, such that for any local ringed space X, the canonical morphism X ® Spec X is effective. Applying (2.8) to the everywhere effective sites C = RSp and D = LSp we obtain the fibre products of local ringed spaces. 
(c) The categories of reduced local ringed spaces, local ringed sets, geometric sets (resp. ringed sets) are full coreflective subcategory of LSp (resp. RSp), thus fibre products exist in these categories by (2.8). 

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.