Recall that a frame
is a complete lattice with the infinite distributive law. A morphism
of frames is a map preserving finite meets and arbitrary joints
(thus preserving order and 1, 0). A subset B of a
frame A is called a base
of A if any element of A is a joint of some elements of B.
Suppose (X, A) is a framed object. Any sieve U Î A is called an open sieve on (X, A). An open cover of an open sieve U is a set {U_{i}} of open subsieves of U such that U = Ú {U_{i}}; if U is the unit sieve on X then we say that {U_{i}} is an open cover of X (or {U_{i}} covers X). A framed object (X, A) (or the frame A) is called active if any open sieve on X is active. Definition 1.2. If (X, A) and (Y, B) are two framed objects, a continuous morphism from (X, A) to (Y, B) is a morphism f: X ® Y such that f^{1}(B) Í A and the induced map f^{1}: B ® A is a morphism of frames. Framed objects of C with continuous morphisms form a category. Example 1.2.1. Suppose U is an open active sieve on a framed object (X, A) with the inclusion morphism e_{U}: U ® X. The set A_{U} of open sieves contained in U is naturally a frame on U. Thus we obtain a framed object X_{U} = (U, A_{U}) together with a monomorphism e_{U}: X_{U} ® (X, A) in the category of framed objects, called an open subobject of (X, A). Remark 1.3. Let F be a covariant (resp. contravariant) functor from a category C to a category D. Suppose U is a sieve on an object X of C. Denote by F/U the composite of the natural functor C/U ® C (resp. (C/U)^{op} ® C^{op}) with F: C ® D (resp. F: C^{op} ® D). If Z Î D is a colimit (resp. limit) of F/U: C/U ® D (resp. (C/U)^{op} ® D) we shall write F(U) for Z. Note that F(U) is determined up to isomorphism, and F(U) = F(U) if U is active. We say F is strict at a sieve U of X if F(X) together with F(f): F(Y) ® F(X) (resp. F(X) ® F(Y)) for all the morphisms f: Y ® X in C/U is a colimit (resp. limit) of F/U (thus F(U) = F(X)). Note that if F is strict at U then F is strict at any larger sieve V on X containing U. Suppose (X, A) is a framed object of C. We say F is strict at (X, A) if it is strict at the union U = È {U_{i}} of any open cover {U_{i}} of X. A framed object (X, A) is called strict (resp. strongly strict) if the identity functor 1_{C}: C ® C is strict at (X, A) (resp. at any open subobject of (X, A)). An active strongly strict framed object is called a topological object of C. Example 1.3.1. The category Top
of topological spaces is the category of topological objects of the category
Set of small sets.
