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 G(u): G(U) ® G(X) is called open effective if the following conditions are satisfied:  (a) G(u) is an open embedding of locales.  (b) If t: T ® X is a map in A such that G(t) factors through G(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 G(u) or G(U) is an open effective sublocale of G(X).  Definition 2.5.2. A framed topology on A is a functor G from A to the category of locales, satisfying the following conditions:  (a) An object X is initial iff G(X) is initial.  (b) Any open sublocale of G(X) is a join of open effective sublocales of G(X).  A framed topology G is spatial if for any object X the locale G(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 G is a framed topology on A. If {Ui} is a set of open effective subobjects of X such that G(X) is the join of {G(Ui)}, then we say that {Ui} (resp. {G(Ui)}) is an open effective cover on X (resp. G(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 G(t) is disjoint with G(u) (i.e. G(t)-1(G(U) = 0).  (b) If u is a normal mono then t is dominated by u iff G(t) factors through G(u).  Proof. (a) If s: S ® X is a map factors through both u and t then G(s): G(S) ® G(X) factors through G(t)-1(G(U)), so s must be initial if G(t)-1(G(U)) = 0 by (2.5.2.a). Conversely if the open sublocale G(t)-1(G(U) ¹ 0 then it is a join of non-initial open effective sublocales {vi: Vi ® T} by (2.5.2.b), and each t°vi 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 G(t) factors through G(u) by (2.5.1.b) as u is effective. n  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 G(t) is disjoint with the join of G(ui).  (b) t is dominated by {ui: Ui ® X} if G(t) factors through the join of G(Ui).  Proof. (a) Assume t is disjoint with each ui. Then G(t)-1(G(Ui)) = 0 for each Ui by (2.5.3.a). Thus  G(t)-1(Ú(G(Ui)) = Ú(G(t)-1(G(Ui)) = 0, which means that G(t) is disjoint with the join of G(ui). The other direction is obvious.  (b) Suppose G(t) factors through the join of G(ui). Consider a map s: S ® T such that t°s is disjoint with each ui. Then G(t°s) is disjoint with the join of G(ui) by (a). But G(t°s) also factors through the join of G(ui) because G(t) is so. This means that G(t°s) is the initial locale. Hence S is initial by (2.5.2.a). This shows that t is dominated by {ui}. n  Denote by D(G) the class of open effective monos for a framed topology G.  Proposition 2.5.5. D(G) 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(G) is closed under composition. Suppose u: U ® X and v: V ® U are two open effective monos. Then G(u°v) = G(u)°G(v) as a composite of open embeddings of locales is an open embedding. If t: T ® X is a map in A such that G(t) factors through G(u)°G(v), then t factors through u uniquely in a map s: T ® U because u is open effective and G(t) factors through G(u). It follows that s factors through v in a map r: T ® V as v is open effective and G(s) factors through G(v). This shows that t factors through u°v. Thus u°v is open effective. Next we prove (2.3.1.c) for D(G). Suppose f: Y ® X is a map and u: U ® X is an open effective map. By (2.5.2.b) the open sublocale G(f)-1(G(U)) of G(Y) is the join of a set of open effective sublocales G(Vi), where each vi: Vi ® Y is an open effective subobject of Y. Then f°vi factors through u for each vi. Suppose t: T ® Y is a map to Y such that f°t factors through u. Then G(t) factors through the open sublocale G(f)-1(G(U)) of G(Y). Applying (2.5.4.b) we see that t is dominated by {vi}. n The monic divisor D(G) is called the open divisor of G.  Proposition 2.5.6. Any open effective cover is a unipotent cover.  Proof. Suppose {Ui} is an open effective cover on X. Then G(X) is the join of {G(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. n  Proposition 2.5.7. The collection T(G) of open effective covers is a unipotent Grothendieck topology on A.  Proof. We verify the three conditions of a Grothendieck topology for T(G) (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 G(f)-1(G(Ui) is a join of open effective sublocales {G(vij): G(Vij) ® G(Y)}. Since G(Y) is the join of {G(f)-1(G(ui)}, it is the join of {G(vij)}. Thus {vij: Vij ® Y} is an open effective cover on Y such that for each ij, f°vij 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 G(X) is the join of {G(uij)}. Thus the collection {ui°uij: Uij ® X} is an open effective cover on X.  These together with (2.5.6) show that T(G) is a unipotent Grothendieck topology on A. n  Remark 2.5.8. If A has pullbacks then one can show that the open divisor D(G) of any framed topology G is a stable divisor (see [L2]).  Suppose G 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 ® G(X) of locales (i.e., a map s is in U iff G(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 G(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]
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