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 Dsieve if it is generated by a set of Dmaps. Denote by _{D}(X) (or simply (X) if no confusion will thereby arise) the set of Dsieves on X. Proposition 2.4.2. (a) _{D}(X)
is a subframe of the frame (X)
of normal sieves on X; _{D}(X)
is an initial frame iff X is initial.
Proof. (a) We prove that _{D}(X)
is closed under infinite and
finite in (X).
If {U_{i}} is a collection
of Dsieves on X, the smallest normal sieve containing
each U_{i} is the Dsieve (_{i}
U_{i}) 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).
We conclude from (2.4.2) that F_{D} 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 Esieve is also a Dsieve. Remark 2.4.4. Consider the divisor O of all the maps. An Osieve 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.
Proposition 2.4.6. (a) A divisor D
is spatial iff for any noninitial object X the locale _{D}(X)
has a point.
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
noninitial Dmap y: Y > X in S
but not in T. Since T
is a Dsieve, y is not dominated by T.
Thus we can find a noninitial map z: Z > Y such that y_{°}
z is disjoint with any map in T.
Since Z is noninitial 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 Dsieves
on X are separated by points, so _{D}(X)
is spatial. The other direction is trivial.
Example 2.4.6.1. (a) All the categories
in [Luo 1997a, Example 3  6] are locally atomic.
Suppose A is an analytic category. Consider the stable divisor A of analytic monos on A. An Acover (resp. Asieve) 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.
