In a representable presite monomorphisms are always injetive and surjective morphisms are stable under base extension. Formally these important settheoretic properties can be derived from a very simple condition which is satisfied virtually by all the presites arising in geometry: Definition 1.3.1.
We say a presite C is separable
if the following condition is satisfied:
(We may take W to be the fibre product X ×_{S} Y if it exists.) Remark 1.3.2. If X ×_{S}
Y exists in C then the condition (1.3.1) can be rephrased
as follows:
We shall see that all the presites discussed in (1.1) are separable. Example 1.3.3. A presite (C, t) is separable if and only if its underlying presite (C, t*) of sets is separable. Example 1.3.4. Any representable presite is separable. Thus Set, Top, RSp, RSet, RPot, PVar/k and AVar/k are separable. Example 1.3.5. Any presite (C, t) with fibre products such that t: C ® Top preserves fibre products is separable. Thus the presites w(X) and W(X) for any topological space X are separable. Proposition 1.3.6. Any monomorphism in a separable presite C is injective. (Thus the metric functor t: C ® Top preserves monomorphisms.) Proof. Suppose a morphism f: Y ® X is not injective. There are two distinct points x and y of Y such that f(x) = f(y). Since C is separable, we can find an object W, a point z of W and two morphisms p: W ® Y and q: W ® Y such that fp = fq, p(z) = x, and q(z) = y, which implies that p ¹ q. Thus f is not a monomorphism. Example 1.3.7. Since active morphisms are monomorphisms, an active morphism in a separable presite is injective by (1.3.6). It follows from (1.2.5.b) that a composite of two active morphisms in a separable presite is active. Proposition 1.3.8. Suppose f: X
® S, g: Y ®
S are two objects over an object S in a separable presite. Suppose the
fibre products (X ×_{S} Y, p,
q) of X and Y over S exists. We have
Proof. (a) can be checked directly using (1.3.2); (b) and (c) follow from (a). 1.3.9 We say a category D is separable if it is separable as a presite of spots. Thus a category D is separable if and only if for any two objects X and Y over an object S, there exist an Sobject W and two Smorphisms W ® X and W ® Y. Clearly a presite C is separable if and only if the category C_{*} of pointed objects of C is separable. Example 1.3.10. (a) A category with
an initial object is separable.
Example 1.3.11. The category Gpot of geometric points is separable. To see this suppose F, G and H are fields such that spot F and spot G are geometric points over spot H. Then F and G are field extensions of H. We can find a large field extension E of H such that F and G are embedded in E as subextensions. Then spot E is over spot F and spot G. Proposition 1.3.12. Suppose D is a category and D' a full coreflective subcategory of D. Then D is separable if and only if D' is separable. Proof. Suppose D is separable and f: X ®
S and g: Y ® S
are two morphisms in D'. Since C is separable, there exists
an Sobject W Î C
and two Smorphisms p: W ®
X and q: W ® Y.
Denote by r: J(W) ®
W the left associated object (cf. (4.1)) of W in D'.
Then the Sobject J(W) and the Smorphisms
pr and qr are in D' since D' is a full subcategory,
which are what we are looking for. This proves that D' is separable.
Proposition 1.3.13. Suppose D is a generic subpresite of a presite C. Then C is separable if and only if D is separable. Proof. We know that the presite C is separable if and only if the category C_{*} is separable. Since D is a full coreflective subcategory of C_{*}, C_{*} is separable if and only if D is separable by (1.3.12). Example 1.3.14. GPot is a generic subpresite of all the nonrepresentable basic presites LSp, GSp, LSet, GSet, LPot and the basic algebraic presites Sch, GSch, ASch, GSch and GASch. Since GPot is separable (1.3.11), these presites are separable by (1.3.13). Remark 1.3.15. Suppose D is
a separable generic subpresite of a presite C. Then the underlying
functor t*: C ®
Set of C is completely determined by the subcategory D.
This is because t* is the Kan extension of the
restriction t*_{D}: D
® Set, which is the functor sending
each object of D to a singleton.
