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 X; D(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 2.4.6.1. (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.