The purpose of this section is to show that all the basic sites and basic algebraic sites have fibre products. 

Let C be a category. 

Recall that the product of two objects X and Y in C is a triple (X × Y, p, q), where X × Y is an object, p: X × Y ® X and q: X × Y ® Y are morphisms, such that given any object P and any morphisms u: P ® X and v: P ® Y, there exists a unique morphism h: P ® X × Y such that ph = u and qh = v; the morphism h is also denoted by (u, v). If X and Y are two objects over an object S, then the fibre product of X any Y over S is the product of S-objects X and Y in the category C_/S of S-objects, denoted X ×_S Y. 

The fibre product X ×_S Y can also be regarded as an object over Y, via the projection q, denoted X_Y, which is called the result of extending the base object of X from S to Y. If f: X ® X' is an S-morphism, we shall write f_Y for the induced morphism X_Y to X'_Y. 

Example 4.2.1. Fibre products exist in W(X) and w(X) for any topological spaces X. 

Example 4.2.2. In Set fibre products exist. Suppose f: X ® S and g: Y ® S are two maps of sets. Denote by W the subset of the cartesian product X × Y consisting of the pairs (x, y) such that f(x) = g(x). Let p, q be the restrictions of the projections X × Y ® X and X × Y ® Y on W respectively. Then (W, p, q) is clearly a fibre product of X and Y over S. 

Example 4.2.3. In Top fibre products exist. Suppose f: X ® S and g: Y ® S are two continuous maps of topological spaces. Let (X ×_S Y, p, q) be the fibre product of X and Y over S in the category of sets. Give X ×_S Y the smallest topology such that p and q are continuous maps. Then the topological space X ×_S Y together with the continuous maps p and q is a fibre product of X and Y over S in Top. 

Example 4.2.4. In RPot fibre products exist. Suppose f: spot A ® spot C and g: spot B ® spot C are two morphisms of ringed points, where A, B and C are rings. Then spot A Ä_C B is a fibre product of spot A and spot B over spot C. This is because the category RPot is equivalent to the opposite of Ring, and A Ä_C B is a fibre sum of A and B over C in Ring. 

Example 4.2.5. Similarly fibre products exist in the site AVar/k of affine varieties over an algebraically closed field k, since AVar/k is equivalent to the opposite of the category of finitely generated reduced algebras over k. 

Let f: X' ® X, g: Y' ® Y be morphisms over an object S. Let (X' ×_S Y', p', q') and (X ×_S Y, p, q) be the fibre products over S. Then fp': X' ×_S Y' ® X and gq': X' ×_S Y' ® Y induces a morphism (fp', gq')_S: X' ×_S Y' ® X ×_S Y, denoted simply by f ×_S g. (note that if Y = Y' = S', and g = 1_S', then f ×_S 1_S' = f_S'). 

Proposition 4.2.6. Suppose fibre products exist in C. Let X, Y, Z, S' be S-objects. Then 
(a) X ×_S = X; 
(b) X ×_S Y = Y ×_S X; 
(c) (X ×_S Y) ×_S Z = X ×_S (Y ×_S Z); 
(d) Let f: X ® Y be an S-morphism. Then X ×_S S' = X ×_Y (Y ×_S S'), i.e. X_S' = X ×_Y S', and the projection X ×_Y S' ® Y_S' coincides with f_S'. 
(e) (X ×_S Y)_Z = X_Z ×_Z Y_Z. 
(f) Suppose f: X ® Z, g: Y ® Z are two S-morphisms. Then  
(X ×_Z Y) ×_Z S' = (X ×_Z Y)_S' = X_S' ×_ZS' Y_S'. 
(g) Suppose f: X ® Y is a monomorphism, then f_S': X_S' ® Y_S' is a monomorphism. 
(h) A morphism f: X ® S is a monomorphism if and only if the projections p, q: X ×_S X ® X are isomorphisms and p = q. 

Proof. Note that (f) follows from (d) and (e). The other assertions can be verified easily. 

We now consider fibre products in a metric site C. 

Proposition 4.2.7. Suppose i: X ® S and j: Y ® S are two morphisms in a site C. Suppose U Í |X|, V Í |Y|, and W Í |S| are effective subsets of |X|, |Y| and |S| respectively such that i(U) Í W and j(V) Í W. Suppose the fibre product X ×_S Y of X and Y over S exists. Let E = p^-1(U) Ç q^-1(V). Suppose E is effective. The subobject E together with the morphism p|_E and q|_E is the fibre product U ×_W V of 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, thus |h(P)| Í p^-1(U) Ç q^-1(V) = E. The morphism 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. Now taking W = |S|, one has the isomorphism E = U ×_S V. The last assertion is obvious. 

Proposition 4.2.8. Suppose I: D ® C is a covariant functor of categories and J: C ® D is a right adjoint of I. Suppose X, Y are S-objects with the structure morphisms i: X ® S and j: Y ® S in C. Suppose (Z, p, q) is the fibre product of X and Y over S in C. Then (J(Z), J(p), J(q)) is the fibre product of J(X) and J(Y) over J(S) in D, i.e, J preserves fibre products. 

Proof. Let W be an object of D over J(S). Then 

Theorem 4.2.9. Let C be an everywhere effective site and D an embedded subsite of C. Suppose fibre products exist in C. Then fibre products exist in D. 

Proof. Let J: C ® D be the right adjoint of the inclusion functor D ® C. Suppose X and Y are objects over S in D with the structural morphisms i: X ® S and j: Y ® S. Let (Z, p, q) be the fibre product of X and Y over S in C. 
Let s_X: X ® J(X), s_Y: Y ® J(Y) and s_S: S ® J(S) be the units of X, Y and S in D respectively, which are effective morphisms in D (4.1.3). Applying (4.2.7) to D and replacing U, V, W by X, Y, S, we see that the fibre product of X and Y over S in D exists which is naturally a subobject of J(Z). 

If D satisfies the conditions of (4.1.5), we can describe the fibre product of X and Y over S in D more explicitly: 

Theorem 4.2.10. Let D be a coreflective everywhere effective subsite of an everywhere effective site C satisfying the conditions of (4.1.5). Suppose X and Y are objects over S in D with the structural morphisms i: X ® S and j: Y ® S. Let (Z, p, q) be the fibre product of X and Y over S in C. Let J(Z) be a left associate object of Z in D and let r: J(Z) ® Z be the counit. Let W = {z Î |J(Z)| ½ (pr)_z and (qr)_z are in D}. Let p', q' be the restrictions of pj and qj on the subobject W of J(Z) respectively. Then (W, p', q') is the fibre product of X and Y over S in D. 

Proof. Note that p' and q' are morphisms in D by (4.1.5.b). Suppose P is an S-object of D and u: P ® X, v: P ® Y are morphisms of D over S. Since (Z, p, q) is the fibre product of X, Y over S in C, we obtain a morphism h = (u, v)_S: P ® Z. Since P is an object of D, by the universal property of J(Z) there exists a unique morphism h': P ® J(Z) such that rh' = h. But u, v are morphisms of D, and |h'(P)| Í W, thus we may regard h' as a morphism from P to W in D, which has the property that p'h' = u and q'h' = v. This proves that (W, p', q') is the fibre product of X and Y over S in D. 

Proposition 4.2.11. (a) Fibre products exist in the site RSp of ringed spaces. 
(b) Fibre products exist in the site LSp of local ringed spaces. 
(c) Fibre products exist in the site GSp of geometric spaces.  
(d) The functors Spec: RSp ® LSp and red: LSp ® GSp preserve fibre products. 

Proof. (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) (these are sheaves over Z). For each z = (x, y) Î Z we have O_Z,z = O_X,_x Ä_OS,_s O_Y,_y where s = p(x) = q(y). The canonical maps O_X,x ® O_Z,z determines a morphism p#: O_X ® O_Z, thus we obtain a projection morphism (p, p#): (Z, O_Z) ® (X, O_X). Similarly we define the projection morphism (q, q#): (Z, O_Z) ® (Y, O_Y). Using the universal property of tensor products it is easy to see that ((Z, O_Z), (p, p#), (q, q#)) is the fibre product of X and Y over S. 
(b) LSp is an embedded subsite of RSp, hence fibre products exist in LSp by (4.2.9) or (4.2.10).  
(c) GSp is an embedded subsite of LSp. 
(d) follows from (4.2.8). 

Corollary 4.2.12. (a) The forgetful functor RSp d Top preserves fibre products.  
(b) If f: X ® S is bicontinuous, then f_Y = q: X ×_S Y ® Y is bicontinuous. 
(c) Suppose z = (x, y) Î Z = X ×_S Y. Then O_Z,z = O_X,x Ä_OS,s O_Y,y.  
(d) Suppose x Î X and y Î Y and f_x^#: O_S,f(x) ® O_X,x is surjective with the kernel I. Put z = (x, y) Î Z = X ×_S Y. Then q_z#: O_Y,y ® O_Z,z is surjective with the kernel generated by g#(I).  

Proof. These assertions are obvious by the explicit construction of X ×_S Y given in (4.2.11). Note that (d) follows from (c). 

Theorem 4.2.13. Suppose C' is a metric site with fibre products. Suppose C is a full, subsite of C' such that for any objects X, Y over S in C, the fibre product X ×_S Y in C' is an object of C. Then the Cauchy-completion E(C) of C in C' also has this property. Fibre products exist in E(C).  

Proof. Suppose X, Y and S are objects in E(C), and f: X ® S, g: Y ® S are two morphisms. Let (X ×_S Y, p, q) be the fibre product of X and Y over S in C'. For any z in |X ×_S Y| let W be an open neighborhood of fp(z) in S such that W Î C. Since f^-1(W) and g^-1(W) are open subsets of X and Y respectively, we can find sufficiently small open neighborhoods U Í f^-1(W) of p(z) and V Í g^-1(W) of p(z) such that U and V are objects of C. Then the fibre product U ×_W V of U and V over W is an object of C by assumption, which is an open subobject of X ×_S Y containing z (4.2.7). This proves that X ×_S Y Î E(C), which means that X ×_S Y is a fibre product of X and Y over S in E(C).  

Theorem 4.2.14. If C is a standard site with fibre products (resp. products), then any Cauchy-completion C' of C has fibre products (resp. products).  

Proof. We give two proofs for fibre products. (The case of products is similar.) Suppose C has fibre products. 
(1) We may realize C' as the Cauchy-completion E(C) of C in C^~. Since fibre products exist in C^~ and the conditions of (4.2.13) are satisfied, it follows that C' has fibre products.  
(2) (cf. [Ha, Theorem 3.3]) Suppose X and Y are two S-objects in C'. For any triple i = (x, y, s), where x Î |X| and y Î |Y| are over s Î |S|, take open neighborhoods U_x, V_y and W_s of x, y and s respectively such that U_x, V_y and W_s are objects of C, such that U_x and V_y are above W_s. Let T_i = U_x ×_Ws V_y be the fibre product in C with the projections p_i and q_i. If j = (x', y', s') is another similar triple as i, with T_j = U_x' ×_Ws' V_y', we define T_ij = p_i^-1(U_x Ç U_x') Ç q_i^-1(U_y Ç U_y'). By (4.2.7) both T_ij and T_ji are fibre products of U_xÇ U_x' and V_y Ç V_y' over W_s Ç W_s'. 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 in the sense of (2.1.3). Glueing these T_i along the isomorphisms u_ij by (2.1.3) we obtain an object X ×_S Y. Glueing the projections from the pieces T_i we obtain the projection morphisms p and q of X ×_S Y. We prove that X ×_S Y is the fibre product. 
Suppose Z Î C' and f: Z ® X, g: Z ® Y are S-morphisms. For each z Î |Z| let i = (f(z), g(z), s) be the triple determined by z as above. Let Z_z be an effective open neighborhood of z contained in f^-1(U_f(z)) Ç g^-1(V_g(z)) such that Z_z Î C. Then the restrictions of f and g to Z_z determines a morphism t_z: Z_z ® T_i. If z' Î |Z| is another point with similar j = (f(z'), g(z'), s'), Z_z' and t_z': Z_z' ® T_i', then the restrictions of t_z and t_z' to Z_z Ç Z_z' are the same if we identify T_i with T_j through u_ij, so by (2.1.2) we can glue these morphisms to obtain a morphism t: Z ® X ×_S Y. The uniqueness of t can be checked locally. 

Theorem 4.2.15. (a) Suppose X = Spec A, Y = Spec B are affine schemes over an affine scheme S = Spec C. Then Spec A Ä_C B is the fibre product of X and Y over S in the category of local ringed spaces.  
(b) Fibre products exist in the site of schemes. 
(c) Suppose X, Y are schemes over a scheme S. Then the fibre product X ×_S Y in the site of local ringed spaces is a scheme.  

Proof. (a) Denote by Ring^op the opposite of the category of rings (commutative with unit). Then the functor J: Ring^op ® LSp given by A ® Spec A for any ring A is a right adjoint of the functor LSp ® Ring^op given by X ® G(X) for any local ringed space X. Thus the assertion follows from (4.2.8). 
(b) follows from (4.2.14), and (c) from (4.2.13) in view of (a). 

Similarly we can prove the following theorem concerning prevarieties over an algebraically closed fields k (using (4.2.5)). 

Theorem 4.2.16. Suppose X and Y are affine k-varieties (resp. k-prevarieties) over an affine k-variety (resp. k-prevariety) S. Then the fibre product of X and Y over S in the site of geometric spaces over k is an affine k-variety (resp. k-prevariety). 

Remark 4.2.17. (4.2.15.c) was proved in [EGA I, Ch. I, (3.2.1)] by a different method, since the existence of fibre products of local ringed space was not established in [EGA].  

Proposition 4.2.18. Suppose D is a generic subsite of a presite C.  
(a) Suppose (X, i) and (Y, j) are S-objects in C. Suppose X ×_S Y is a fibre product of X and Y over S in C.  
Suppose U Í |X|, V Í |Y| and W Í |S| such that i(U) Í W, j(V) Í W. Then f(i^-1(U) Ç j^-1(V)) is a fibre product of f(U) and f(V) over f(W) in D/Set (see (1.1.28)).  
(b) Suppose fibre products exist in C. Then fibre products exist in D/Set. The functor f_D: C ® D/Set preserves fibre products. 
If C is an everywhere effective site with fibre products then Dis(C) has fibre products.  

Proof. (a) The proof of (a) is similar to the proof of (4.2.7). 
(b) Suppose X and Y are S-objects in D/Set. For each triple (x, y, s) where x Î |X|, y Î |Y|, s Î |S| such that x and y are over s, the D-set f(x ×_s y) is a fibre product of x and y over s. The union of such f(x ×_s y) for all triples (x, y, s) is the fibre product of X and Y over S in D/Set.  
The last assertion follows from (b) since in this case Dis(C) is a generic subsite of C. 

In a standard site if {U_i} is a disjoint open cover of an object X, then X is a sum of the subobjects U_i. If {X_i} is any collection of objects in a complete site C, the disjoint union of the X_i exists. This implies that in a complete     site arbitrary sum exist. 

Remark 4.2.19. Suppose D is a generic subsite of a complete site C. Then any D-set may be viewed as a discrete object of C as a disjoint union of spots in D. Thus D/Set is an isogenous coreflective subsite of C. (This is the reason behind the facts that RSet, LSet, and GSet are coreflective subsite of RSp, LSp and GSp respectively.) 

Remark 4.2.20. If C is a Cauchy-complete site and D a generic subsite of C then (4.2.18a & b) follows easily from the fact that D/Set is naturally an embedded subsite of C (4.2.19). We see that D/Set has fibre products if C does by (4.2.9) and the coreflector J = f: C ® D/Set preserves fibre products.  

Example 4.2.21. Fibre products exist in RSet, LSet and GSet. The functors f: RSp ® RSet, f: LSp ® LSet, and f_k: LSp ® GSet preserve fibre products. Note that for LSet this has been proved in (3.4.4).  

Example 4.2.22. If D is a category has fibre products, then D/Set is a Cauchy-complete site with fibre products defined pointwise, as in the case of RSet. On the other hand, even fibre products doest not exist in D, D/Set may still have fibre products, as in the case of D = GPot and D/Set = GSet.