2.4. Analytic Sieves
Recall that a frame is a complete lattice with infinite distributive law, and a morphism of frames is a function preserving arbitrary joins and finite meets (cf. [Johnstone 1982] or [Borceux 1994 Vol III]). The opposite of the category of frames is the category of locales. Any complete boolean algebra is a boolean locale. We know from (2.1.4) that the set of normal sieves (X) on an object is a boolean locales. 

We now show that any divisor D determines a functor from A to the category of locales. Recall that by (2.1.2.c) if S is a set of maps to X then S is the smallest normal sieve containing S, which is the normal sieve generated by S

Definition 2.4.1. A normal sieve U on an object X is called a D-sieve if it is generated by a set of D-maps. 

Denote by D(X) (or simply (X) if no confusion will thereby arise) the set of D-sieves on X. 

Proposition 2.4.2. (a) D(X) is a subframe of the frame (X) of normal sieves on XD(X) is an initial frame iff X is initial. 
(b) Suppose f: Y --> X is a map. If U is a D-sieve on X, then f*(U) is a D-sieve on Y

Proof. (a) We prove that D(X) is closed under infinite  and finite  in (X). If {Ui} is a collection of D-sieves on X, the smallest normal sieve containing each Ui is the D-sieve (i Ui) by (2.1.2.c). Thus (X) is a complete lattice whose  operation coincides with that of (X). The second assertion follows from the first as it is true for (X). 
Suppose U and V are two D-sieves. We prove that the normal sieve U  V is a D-sieve. Since U and V are generated by D-maps, by the infinite distributive law in (X), U  V is the join of all the normal sieves {t {s} for all the D-maps t in U and s in V. Thus it suffices to prove that {t}  {s} is a D-sieve. Denote by S the set of D-maps to X which can be factored through t and s. Applying (2.3.1.c) twice one can show that S dominates sie(s)  sie(t). But the later dominates {t}  {s} by (2.1.1.f). Thus {t}  {s}S is a D-sieve. 
(b) If t: T --> X is a D-map then f*({t}){f*(t)} by (2.1.3.a); so f*({t}) is dominated by f*(t). By (2.3.1.c) f*(t) is dominated by the D-maps in it. Thus f*({t}) is dominated by the D-maps in it. This shows that f*({t}) is a D-sieve. The general case follows from the facts that any D-sieve is the join of such D-sieves {t} with t in D and that f*: (X) --> (X) preserves infinite join by (2.1.4). 

We conclude from (2.4.2) that FD is a functor from A to the category of locales, called the functor generated by D

Proposition 2.4.3. If E is a subdivisor of D, then for any object X the frame E(X) is a subframe of D(X). 

Proof. Clearly each E-sieve is also a D-sieve. 

Remark 2.4.4. Consider the divisor O of all the maps. An O-sieve on an object X is precisely a normal sieve on X. Thus O(X) coincides with the complete boolean algebra (X) of X

Definition 2.4.5. (a) A divisor D is called spatial if the locale D(X) for any object X is spatial. 
(b) We say A is locally atomic if the divisor O of all the maps is spatial (i.e. each complete boolean algebra (X)O(X) is atomic). 

Proposition 2.4.6. (a) A divisor D is spatial iff for any non-initial object X the locale D(X) has a point. 
(b) A subdivisor of a spatial divisor is spatial. 
(c) If A is locally atomic then any divisor D on A is spatial. 

Proof. Assume the condition is satisfied. Suppose S and T are two D -sieves on an object X, and S is not contained in T. Then we can find a non-initial D-map y: Y --> X in S but not in T. Since T is a D-sieve, y is not dominated by T. Thus we can find a non-initial map z: Z --> Y such that y° z is disjoint with any map in T. Since Z is non-initial by assumption D(Z) has a point P (as a morphism 2 --> D(Z)). The image of P under yz is a point contained in S but not in T. This shows that the D-sieves on X are separated by points, so D(X) is spatial. The other direction is trivial. 
(b) follows from (2.4.3) as a sublocale of a spatial locale is spatial. 
(c) follows from (b). 

Example (a) All the categories in [Luo 1997a, Example 3 - 6] are locally atomic. 
(b) All the categories in [Luo 1995a, Example (2.2.1) - (2.2.3)] are locally atomic. 

Suppose A is an analytic category. Consider the stable divisor A of analytic monos on A. An A-cover (resp. A-sieve) is called an analytic cover (resp. analytic sieve). As a special case of (2.3.8) and (2.4.2) we obtain the following 

Proposition 2.4.7. (a) Analytic covers form a unipotent Grothendieck topology on an analytic category A
(b) The analytic divisor A generates a functor A from A to the category of locales. 

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