Definition 2.6.1. A framed topology on A is called subnormal if for any object X and any nonisomorphic open embedding v: V > (X) of locales there is a noninitial map t: T > X such that (t): (T) ® (X) is disjoint with v. Example 2.6.1.1. (a) The framed topologies
discussed in (2.5.2.1) are subnormal.
Proposition 2.6.2. A framed topology is subnormal iff any unipotent cover consisting of open effective subobjects is an open effective cover. Proof. First suppose
is subnormal. Given a unipotent cover {u_{i}: U_{i}
> X} consisting of open effective monos. Let v: V > (X)
be the join of {(U_{i})}.
We have to prove that {u_{i}: U_{i} > X}
is an open effective cover, i.e. V = (X).
We prove it by contradiction. Assume that V is a proper sublocale
of (X). Then by
(2.6.1) there is a noninitial map t: T
> X such that (t): (T)
> (X) is disjoint
with v. Then t is disjoint with each u_{i}
by (2.5.4.a). But this is impossible as
by assumption {u_{i}} is a unipotent cover. Thus V = (X),
i.e. {u_{i}} is an open effective cover on X.
Proposition 2.6.3. (a) A framed topology
is subnormal iff every open sieve is normal.
Proof. (a) First suppose
is subnormal. Consider an open sieve U on an object X, which
is the pullback of an open embedding v: V > (X)
of locales (i.e., a map s is in U iff (s)
factors through v). To see that U is normal we have to prove
that if t: T > X is a map dominated by U, then t
is in U, i.e. (t)
factors through v. Assume that this is not the case. Then (t)^{1}(V)
is a proper open sublocale of (T).
Since is normal we can
find a noninitial map s: S > T such that (s)
is disjoint with (t)^{1}(V).
Thus (t_{°}s)
is disjoint with V. Hence ts is disjoint with U. But
this contradicts the fact that t is dominated by U. This
shows that U is normal.
We now show that a subnormal framed topology is completely determined by the subnormal divisor of open effective monos. Proposition 2.6.4. (a) Suppose D
is a subnormal divisor on A. Then the functor _{D}
generated by D is a subnormal framed topology on A.
Proof. (a) We already know from (2.4.2)
that _{D}
is a functor from A to the category of locales and (X)
= 0 iff X is initial; each Dsieve on X
may be viewed naturally as an open sublocale of (X).
Since any Dmono u: U > X is normal, a map
t: T > X factors through u iff it is dominated by
u, and the later is equivalent to that (t)
factors through (u).
Also one can show easily that (u): (U)
> (X) is an
open embedding of locales. Hence u and (u)
is open effective for .
By (2.4.1) any Dsieve is
generated by a set of Dmonos U = {u_{i}:
U_{i} > X}, thus any open sublocale of (X)
is a join of open effective sublocales. This shows that
is a framed topology, which is subnormal by (2.6.3.a).
Corollary 2.6.5. (a) A subnormal framed
topology is spatial
iff for any noninitial object X the locale (X)
has a point.
Proof. These follow from (2.4.6) in view of (2.6.4). Definition 2.6.6. (a) If A
has pullbacks the framed topology _{N}
generated by the normal divisor N of normal
monos is called the normal
topology.
Clearly the normal topology is the finest subnormal framed topology on a category A with pullbacks. If A is a coflat disjunctable analytic category (e.g. a topos) then any normal mono is analytic, thus the normal and analytic topologies are the same in this case. On the other hand, if A is an analytic category in which any strong mono is an intersection of direct monos (e.g. the category of Stone spaces), then analytic category reduces to extensive topology. Some special cases of extensive topologies have been studied by several authors (see, for instance, Barr and Pare [1980] and Diers [1986]). We will study the main properties of these canonical topologies in Chapter 3. Definition 2.6.7. Suppose A
is an analytic category.
Definition 2.6.8. (a) We say a framed
topology is strict
if the Grothendieck topology T()
defined by open effective covers is subcanonical (i.e., any representable
presheaf of sets is a sheaf) (see (2.5.7)).
In practice most of the natural framed topologies are subanalytic. The general rule is that if a natural analytic category is not strict (i.e. its analytic topology is not strict), then it carries another natural strict subanalytic framed topology which is more useful than the analytic topology. Example 2.6.8.1. (a) The analytic
topologies of the categories of topological spaces, locales, or coherent
spaces are not strict, yet each of these categories carries a natural strict
subanalytic topology defined by the inclusion functor to the category of
locales.
