Suppose C is a category with fibre products. Definition 5.1.1. Suppose X,
S are two objects and i, j are two morphisms from
X to S. The kernel
(or equalizer) of i and
j is an object ker(i,j), together with a map
p: ker(i,j) ®
X such that ip = jp, which has the following universal
property:
Note that p is a monomorphism representing the subfunctor Definition 5.1.2. Suppose X is an object over S with the structure morphism f: X ® S. A morphism s: S ® X such that fs = 1_{S} is called an Ssection of X. (Thus an Ssection of X is an Smorphism s: S ® X.) The set of Ssections of X is denoted by G(X/S). Definition 5.1.3. Suppose X and Y are objects over S. Suppose f: X ® Y is an Smorphism. The morphism (1_{X}, f)_{S}: X ® X ×_{S} Y over S is called the graph morphism of f, denoted G_{f}, which is an Xsection of X ×_{S} Y. If Y = X and f = 1_{X}, then the graph morphism of 1_{X} is called the diagonal morphism of X over S, denoted D_{X/S} (or simply D_{X}, or D_{f}, or D). Remark 5.1.4. Suppose X, Y are two objects over S. The map f ® G_{f} = (1_{X}, f)_{S} is a bijection: The relationship of these concepts are exhibited in the following lemmas. (The proofs are straightforward which will be omitted.) Suppose u: X ® S is a morphism. Consider an Sobject T and an Smorphism t: T ® X ×_{S} X = W. Form the fibre product (X ×_{W} T, i, j) as below: Lemma 5.1.5. Suppose g, h: T ® X are two Smorphisms with t = (g, h)_{S}. Then E = ker (g, h) is the kernel of g and h: Lemma 5.1.6. Suppose T = X, and s: S ® X is an Ssection of u: X ® S with t = (su, 1_{X}). Then E = S and i = j = s canonically: Lemma 5.1.8. Suppose f: Y ® X, g: Z ® X are two Smorphisms with T = Y ×_{S} Z and t = f ×_{S} g. Then E = Y ×_{X} Z canonically. We have j = (p, q)_{S} and i = fp = gq, where p: Y ×_{X} Z ® Y and q: Y ×_{X} Z ® Z are projections: Proposition 5.1.10. (a) Any
regular monomorphism is a mono.
Proof. (a) is obvious and (b) follows from (5.1.6, 5.1.7, 5.1.8); (c) comes from (5.1.5); (d) follows from (c) and (4.2.6d). Proposition 5.1.11. Suppose P is
a property of morphisms satisfying the following conditions:
Proof. Suppose P satisfies the conditions (a) and (b). We have f_{S'}_{ }= f ×_{S} 1_{S'}. Therefore (i) implies (ii) by (a). On the other hand, f ×_{S} g is the composite morphism Hence (ii) implies (i) by (b). Proposition 5.1.12. Suppose P is
a property of morphisms having the following properties:
Proof. Note that f can be factored as where q is the second projection and q = (gf) ×_{Z} 1_{Y}: X ×_{Z} Y d Z ×_{Z} Y = Y. Then q has the property P by (b). Since G_{f} is a special morphism, f has the property P by (a). Proposition 5.1.13. Suppose C
is a site. Suppose P is a property of morphisms having the following
properties:
Proof. Note that f can be factored as where q is the second projection and q = (gf) ×_{Z} 1_{Y} as in (5.1.12). Then q has the property P by (b). By (5.1.7) G_{f} is the result of D_{g} under a base extension (see the diagram below), thus a closed regular monomorphism, since by assumption D_{g} is universally closed. Hence f has the property P by (a).
