1. Framed Objects 

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
Definition 1.1. Suppose C is a category. A frame on an object X Î C is a frame A consisting of sieves on X with Ù = Ç. Specifically, A is a subset of wC(X) with the following properties: 
(i) The unit sieve 1C/X is in A
(ii) If U, V Î A, then U Ç V Î A
(iii) A has arbitrary joints. 
(iv) The infinite distribute law V Ç (Ú {Ui}) = Ú {V Ç Ui} holds in A
framed object of C is a pair (X, A) of an object X and a frame A on X

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 {Ui} of open subsieves of U such that U = Ú {Ui}; if U is the unit sieve on X then we say that {Ui} is an open cover of X (or {Ui} 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 eU: 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 eU: 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 ® Cop) with F: C ® D (resp. F: Cop ® 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 {Ui} of any open cover {Ui} of X. A framed object (X, A) is called strict (resp. strongly strict) if the identity functor 1C: 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. 

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