V --> 

 

 

Example 2.6.1.1. (a) The framed topologies discussed in (2.5.2.1) are subnormal. 
(b) All the metric topologies given in [Luo 1995a, Example (2.2.1) - (2.2.3)] may be viewed naturally as subnormal framed topologies. 

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 non-initial 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. 
Conversely, assume the condition is satisfied. Suppose v: V --> (X) is a non-isomorphic open embedding of locales, which is a join of open effective sublocales (v_i): (V_i) --> (X) by (2.5.2.b). Then {v_i} is not unipotent by assumption. So we can find a non-initial map t: T --> X which is disjoint with each v_i. By (2.5.4.a) (t) is disjoint with the join v of (v_i). This shows that  is subnormal by definition (2.6.1).  

Proposition 2.6.3. (a) A framed topology  is subnormal iff every open sieve is normal. 
(b) The open divisor D() of a subnormal framed topology  is subnormal. 

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 non-initial 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. 
Conversely assume that any open sieve is normal. Suppose v: V --> (X) is a non-isomorphic open embedding of locales. The open sieve U determined by V is a proper normal sieve on X. Thus U is not unipotent. So we can find a non-initial map t: T --> X which is disjoint with U. This means in particular that t is disjoint with an open effective cover of V. Thus (t): (T) --> (X) is disjoint with v by (2.5.4.a). 
(b) Suppose  is subnormal and u: U --> X is an open effective mono, then u generates an open effective sieve, which is normal by (a). Thus 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. 
(b) A subnormal framed topology  is equivalent to the subnormal framed topology _D(). 

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 D-sieve on X may be viewed naturally as an open sublocale of (X). Since any D-mono 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 D-sieve is generated by a set of D-monos 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). 
(b) Suppose  is a subnormal framed topology. The open divisor D() of  is subnormal by (2.6.3.b), and open sieves coincide with D()-sieves. We obtain a mapping from (X) to _D()(X), which is a natural isomorphism.  

Corollary 2.6.5. (a) A subnormal framed topology  is spatial iff for any non-initial object X the locale (X) has a point. 
(b) If A is locally atomic then any subnormal framed topology on A is spatial. 

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. 
(b) If A is an extensive category the framed topology _E generated by the extensive divisor E of direct monos is called the extensive topology. 
(c) If A is an analytic category the framed topology _A generated by the analytic divisor A of analytic monos is called the analytic 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. 
(a) A divisor is called subanalytic if it consists of analytic monos. 
(b) A framed (or metric) topology on A is called subanalytic if it is generated by a subanalytic divisor. 

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)). 
(b) An analytic category is called strict if its analytic topology is strict. 

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. 
(b) The analytic topologies of the categories of Hausdorff spaces, affine schemes, or Stone spaces are strict.