Definition 6.3.1. A simple
site is a strict site C having the following properties:
Example 6.3.2. Set, Top, GSp, GSet, GASch, PVar/k and AVar/k are simple sites. Proposition 6.3.3. Suppose C
is a simple site with finite limits.
Proof. (a) Suppose g: S ®
X is a section of a morphism f: X ®
S. Since g is bicontinuous, to prove that g is effective,
it suffices to prove that it is active. Suppose h: Z ®
X is a morphism with h(Z) Í
g(S). We have to prove that h factors through g.
Consider the composition gfh: Z ®
X ® S ®
X. It suffices to prove that g(fh) = h. For
any z Î Z write h(z)
= z' and fh(z) = z". Then f_{z'}g_{z"}
= 1_{z"}, which implies that g_{z"} is a
section. Since C is simple, g_{z"} must be an isomorphism
(6.3.1.b), therefore we have g_{z"}f_{z'}
= 1_{z'}, hence (g(fh))_{z}
= g_{z"}f_{z'}h_{z} = h_{z}
for any z Î Z. It follows
that g(fh) = h (6.3.1.b). This
shows that g is effective.
Corollary 6.3.4. Suppose C
is an everywhere effective simple site with finite limits. Then any
special morphism s: Z ® W is uniquely
determined by the image s(Z) of Z in W. In particular, we have
A category D is called simple if it is a simple site as a site of spots. Thus D is simple if and only if any section in D is an isomorphism. (If D has fibre products then this is equivalent to the assertion that any regular monomorphism is an isomorphism.) If D is a simple category, the site D/Set of Dsets is an everywhere effective simple site. Applying (6.3.3.c) we see that any regular monomorphism in D/Set is effective. Example 6.3.5. If C is a simple site, then Spot(C) is a simple category. Example 6.3.6. The most important simple category is the category GPot = Field^{op} of geometric points. Suppose C is a strict metric site. By a simple model of C we mean a full, simple, isometric coreflective subsite C' of C. Proposition 6.3.7.
Suppose C is a strict metric site with finite limits. Suppose
C' is a simple model of C. Then
Proof. (a) follows from (4.2.9). (b) and (c) hold for the simple site C', therefore also hold for C because J: C ® C' is an isometry preserving finite limits by (a). By the same reason (d) holds. Example 6.3.8. GSp is a simple model of LSp, GSet is a simple model of LSet, and GPot is a simple model of LPot. We now study the universally injective morphisms in a metric site C with finite limits. Recall that any monomorphism in a metric site is universally injective. In a simple site we shall show that the converse is also true. First we make some observations. Suppose f: X ® Y is a universally injective morphism. Then the projection q: X ×_{Y} X ® X as a result of base extension is injective. Since the diagonal morphism D_{f}: X ® X ×_{Y} X is always injective (bicontinuous) and qD_{f} = 1_{X}, we see that D_{f} and q both are bijective. Thus we have: Proposition 6.3.9. Suppose f: X ® Y is a universally injective morphism. Then the projection q: X ×_{Y} X ® X and the diagonal morphism D_{f}: X ® X ×_{Y} X are both bijective. Suppose f: X ® Y is a morphism. For any object T, there is a map f_{T}: X(T) ® Y(T) of points with value in T (see (1.1.21)) determined by f. Then f is a monomorphism if and only if f_{T} is injective for any T Î C. Suppose C is a simple site. Then Spot(C) is a simple generic subpresite of C. Any object X determines a Spot(C)set f(X) (1.1.28)) consisting of spots of X. Proposition 6.3.10.
Suppose C is a simple site. Suppose f:
X ® Y is
a morphism. Then the following conditions are equivalent:
Proof. The equivalences of (i)  (vi) follow from (6.3.1). Since C is separable, any monomorphism is universally injective, thus (i) implies (vii). That (vii) implies (viii) follows from (6.3.9). We now prove that (viii) implies (iv). Note that if D_{f}: X ® X ×_{Y} X is bijective, then D_{f}_{(f)}: f(X) ® f(X) ×_{f(Y)} f(X) is bijective. Since Spot(C) is simple, the section D_{f}_{(f)} is an isomorphism, hence (iv) holds. Remark 6.3.11. Suppose D is a generic subsite of a Cauchycomplete site C. Then D/Set is naturally an isogenous coreflective subsite of C. Thus a morphism f is universally injective if and only if the morphism f(f) is universally injective in D/Set. Corollary 6.3.12.
Suppose C is a Cauchy complete site and D
a simple, generic subsite of C. The following conditions
are equivalent for a morphism f: X ®
Y in C:
Proof. Note that D/Set is a simple site and Spot(D/Set) is equivalent to D. The assertions then follow from (6.3.10) by (6.3.11). Example 6.3.13. A morphism spot F ® spot G in GPot = Field^{o} is a monomorphism if and only if the field extension F of G is a radical extension. We say a morphism f: X ® Y of local ringed spaces is radical if it is injective, and the induced homomorphism k_{f}(x) ® k_{x} of residue fields for any point x Î X is radical. Example 6.3.14.
Since GSp is a geometric site and Spot(GSp) = GPot,
applying (6.3.10) we see that the following are equivalent
for any morphism f in GSp:
Example 6.3.15.
Since GPot is a generic subsite of LSp and is simple, the
following are equivalent for a morphism f in LSp by (6.3.12):
