2.5. Framed Topologies   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, which means that (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.  Next we prove (2.3.1.c) for D(). 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.  (c) Suppose {ui: Ui --> X} is an open effective cover on X. Suppose for each i one has a open effective cover {uij: Uij --> Ui}, then (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].     [Next Section][Content][References][Notations][Home]