In a representable presite monomorphisms are always injetive and surjective morphisms are stable under base extension. Formally these important set-theoretic properties can be derived from a very simple condition which is satisfied virtually by all the presites arising in geometry:
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
We shall see that all the presites discussed in (1.1) are separable.
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 S-object W and two S-morphisms 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 1.3.11. 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.
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 S-object W Î C
and two S-morphisms 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 S-object J(W) and the S-morphisms
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.
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 non-representable 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.