Definition 4.1. A metric presite C
is separable if the following condition
All the metric sites given in (1.5.2), (2.2.1) - (2.2.3), (3.11) are separable. A separable metric presite C has the important properties that any monomorphism is injective, and surjective morphisms are stable under base extension:
Proof. Suppose a morphism f: Y ® X is not injective. There are two distinct points y and y' of |Y| such that f(y) = f(y'). Since C is separable, we can find a pointed object (W, w) and two morphisms p: (W, w) ® (Y, y) and q: (W, w) ® (Y, y') such that fp = fq. Since p ¹ q, the morphism f is not a monomorphism.
Proposition 4.3. Suppose f: X
® S, g: Y ®
S are two objects over an object S in a separable metric
presite. Suppose the fibre product X ×S
Y of X and Y over S exists, with the projections
p: X ×S Y ®
X and q: X ×S Y ®
Y. We have
Proof. First suppose (D, tD)
is separable. Suppose f: (X, x) ®
(S, s) and g: (Y, y) ®
(S, s) are two morphisms of pointed objects of C.
Since |X|, |Y| are D-exact, there are two morphisms
f': (X', x') ® (X,
x) and g': (Y', y') ®
(Y, y) of pointed objects with X', Y' Î
D by (1.6.a). We obtain two morphisms
u = f f': (X', x') ®
(S, s) and v = gg': (Y', y')
® (S, s). Since |S|
is D-exact, u is connected to v in D/(|S|,
s) by (1.6.b). In the notation of
(1.9) this means that the triple (X',
x', u) is equivalent to (Y', y', v).
Since D is separable, it is easy to see that (X', x',
u) is equivalent to (Y', y', v) if and only
if (X', x', u) ~ (Y', y', v).
Thus we can find two morphisms p: (Z, z) ®
(X', x') and q: (Z, z) ®
(Y', y') with Z Î
D such that up = vq. The compositions f'p:
(Z, z) ® (X, x) and g'q:
(Z, z) ® (Y, y)
are what we are looking for. This shows that (C, tC)
Remark 4.5. It follows from (4.4) that if C is an ordinary metric (resp. strict metric) site, then C^ (resp. C~ or D(C)) is separable if and only if C is separable.
Finally we introduce the important notion of a separated morphism in a metric site. Suppose C is a metric site with fibre products and a terminate object Z. Let f: X ® Y be a morphism in C. The diagonal morphism of f is the unique morphism Df: X ® X ×Y X whose composition with both projection maps p, q: X ×Y X ® X is the identity map of X.
Definition 4.6. A morphism f: X ® Y in C is called separated (or X is separated over Y) if the diagonal morphism Df: X ® X ×Y X is universally closed (i.e., Df is a closed morphism which remains closed under any base extension). An object X is called separated if it is separated over the final object Z.
Remark 4.7. In categorical geometry a separated object plays the role of a Hausdorff space in topology. Indeed, if the pretopology t of a metric site C preserves fibre products and |Z| is a one-point space (e.g., C is the metric site of topological spaces, or ringed spaces), then one can prove that an object X is separated if and only if its underlying space is Hausdorff.
Remark 4.8. A monomorphism in a category is regular if it occurs as an equalizer of a pair of morphisms with the same domain and codomain. Many natural metric sites (with fibre products) have the good property that any regular monomorphism is universally bicontinuous (e.g., the metric sites of local ringed spaces or schemes). Since the diagonal morphism Df for any morphism f is a regular monomorphism, it follows that a morphism f: X ® Y in these sites is separated if and only if |Df(X)| is a closed subset of |X ×Y X|.
Following the notation and method of [EGAI, Ch. I, §3] we can prove the following general properties for separated morphisms:
Proposition 4.9. (cf. [Hartshorne
1977, p.99]). (a) Any monomorphism is separated.
Corollary 4.10. (a) Let f: X
® S be a separated morphism. Suppose
U Í |X| and V Í
|S| are effective subsets and f(U) Í
V. Then U is separated over V.
A metric site C is called separated if it has fibre products and a terminate object such that any object (or equivalently by (4.10.b), any morphism) in C is separated. An ordinary, strict, separated metric site C is called separately complete if any separated Dedekind cut in D(C) is representable.
Theorem 4.12. Any ordinary, strict, separated metric site C has a separated completion, which is uniquely determined by C up to equivalence. The collection of separated Dedekind cuts forms a separated completion of C. Fibre products exist in any separated completion of C. n
Example 4.12.1. Fibre products exist in the metric sites of complex analytic spaces, separated prevarieties and separated schemes because each of these metric sites is a separated completion of a strict, separated metric site.