Suppose A is a category with a strict initial object 0. Consider a functor G from A to the category of locales.
Definition 2.5.1. A mono u:
U --> X in A and its image 
(u): (U)
--> (X) is called open
effective if the following conditions are satisfied:
(a) (u) is an
open embedding of locales.
(b) If t: T --> X is a map in A
such that (t) factors
through (u) then
t factors through u (uniquely).
If u is open effective then u or U is called an open
effective subobject of X, and (u)
or (U) is an open
effective sublocale of (X).
Definition 2.5.2. A framed
topology on A is a functor
from A to the category of locales, satisfying
the following conditions:
(a) An object X is initial iff (X)
is initial.
(b) Any open sublocale of (X)
is a join of open effective sublocales of (X).
A framed topology
is spatial
if for any object X the locale (X)
is spatial.
Example 2.5.2.1. (a) The identity
functor on the category of locales is a framed topology.
(b) The functor on the category of topological spaces sending each
topological space to the locale of its open sets is a spatial framed topology.
Suppose is a framed
topology on A. If {Ui} is
a set of open effective subobjects of X such that (X)
is the join of {(Ui)},
then we say that {Ui} (resp. {(Ui)})
is an open
effective cover on X (resp. (X)).
Proposition 2.5.3. Suppose u:
U --> X is an open effective mono and t: T --> X is a
map.
(a) t is disjoint with u iff (t)
is disjoint with (u)
(i.e. (t)-1((U)
= 0).
(b) If u is a normal mono then t is dominated by u
iff (t) factors
through (u).
Proof. (a) If s: S --> X is a map factors through
both u and t then (s): (S)
--> (X) factors
through (t)-1((U)),
so s must be initial if (t)-1((U))
= 0 by (2.5.2.a). Conversely if the open sublocale (t)-1((U) 
0 then it is a join of non-initial open effective sublocales {vi:
Vi --> T} by (2.5.2.b), and each tvi
factors through both u and t. Thus t is not disjoint
with u.
(b) If u is normal then t is dominated by u iff
t factors through u, which is equivalent to that (t)
factors through (u)
by (2.5.1.b) as u is effective.
Proposition 2.5.4. Suppose {ui:
Ui --> X} is a set of open effective monos and t:
T --> X is a map.
(a) t is disjoint with each ui iff (t)
is disjoint with the join of (ui).
(b) t is dominated by {ui: Ui
--> X} if (t)
factors through the join of (Ui).
Proof. (a) Assume t is disjoint with each ui.
Then (t)-1((Ui))
= 0 for each Ui by (2.5.3.a).
Thus
(t)-1(
((Ui))
= ((t)-1((Ui))
= 0,(t)
is disjoint with the join of (ui).
The other direction is obvious.
(b) Suppose
(t)
factors through the join of (ui).
Consider a map s: S --> T such that ts is disjoint
with each ui. Then (ts)
is disjoint with the join of (ui)
by (a). But (ts)
also factors through the join of (ui)
because (t) is so.
This means that (ts)
is the initial locale. Hence S is initial by (2.5.2.a).
This shows that t is dominated by {ui}.
Denote by D()
the class of open effective monos for a framed topology .
Proposition 2.5.5. D()
is a submonic divisor.
Proof. Clearly the isomorphisms are open effective. The initial
maps are open effective as a consequence of (2.5.2.a).
We prove that D()
is closed under composition. Suppose u: U --> X and v:
V --> U are two open effective monos. Then (uv)
= (u)(v)
as a composite of open embeddings of locales is an open embedding. If t:
T --> X is a map in A such that (t)
factors through (u)(v),
then t factors through u uniquely in a map s: T
--> U because u is open effective and (t)
factors through (u).
It follows that s factors through v in a map r:
T --> V as v is open effective and (s)
factors through (v).
This shows that t factors through uv. Thus uv is open
effective.
).
Suppose f: Y --> X is a map and u: U --> X
is an open effective map. By (2.5.2.b) the open sublocale (f)-1((U))
of (Y) is the join
of a set of open effective sublocales (Vi),
where each vi: Vi --> Y is an open
effective subobject of Y. Then fvi factors through
u for each vi. Suppose t: T --> Y
is a map to Y such that ft factors through u. Then (t)
factors through the open sublocale (f)-1((U))
of (Y). Applying
(2.5.4.b) we see that t is dominated by {vi}.
The monic divisor D()
is called the open
divisor of .
Proposition 2.5.6. Any open effective cover is a unipotent cover.
Proof. Suppose {Ui} is an open effective cover
on X. Then (X)
is the join of {(Ui)}.
Applying (2.5.4.b) we see that the identity map X
® X is dominated by {ui:
Ui ® X}. Thus {ui}
is a unipotent cover on X.
Proposition 2.5.7. The collection T()
of open effective covers is a unipotent Grothendieck topology on A.
Proof. We verify the three conditions of a Grothendieck topology
for T() (cf. (2.3.4)).
(a) Clearly any isomorphism is an open cover and the empty set is an
open cover only on 0 .
(b) Suppose {ui: Ui --> X} is an
open cover on X and f: Y --> X is any map. Then each (f)-1((Ui)
is a join of open effective sublocales {(vij): (Vij)
® (Y)}.
Since (Y) is the
join of {(f)-1((ui)},
it is the join of {(vij)}.
Thus {vij: Vij --> Y} is an open effective
cover on Y such that for each ij, fvij
factors through ui.
(X)
is the join of {(uij)}.
Thus the collection {uiuij: Uij
--> X} is an open effective cover on X.
These together with (2.5.6) show that T(
)
is a unipotent Grothendieck topology on A.
Remark 2.5.8. If A
has pullbacks then one can show that the open divisor D()
of any framed topology
is a stable divisor (see [L2]).
Suppose is a framed
topology on A. A sieve U on an object
X is called open if it 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); an open sieve is effective
if it is generated by an open effective mono. The set of open sieves on
X is a locale isomorphic to (X)
naturally. Thus a framed topology can be defined intrinsically as a function
which assigns to each object a locale of sieves (cf. [Luo
1995b].