2. Framed Sites 
Definition 2.1.frame on a category C is a function G which assigns to each object X of C a frame G(X) on X such that any morphism f: Y ® X in C is a continuous morphism of framed objects from (Y, G(Y)) to (X, G(X)). A framed category is a pair (C, G) of a category C and a topology G on C; C is called the underlying category of (C, G). We often write C for a framed category (C, G), and X for a framed object (X, G(X)). 

We say a framed category (C, G) (or G) is active (resp. strict, resp. strongly strict, resp. spatial) if each framed object (X, G(X)) (or G(X)) is so. A framed category (C, G) is locally small (resp. small) if the underlying category C is locally small (resp. small) and each frame G(X) is small. 

Suppose U is an open sieve on an object X. We say U is exact if any subsieve V of U such that f-1(V) is open for each f Î U is open. If U is both active and exact then we say that U is effective. A monomorphism f: Y ® X in C is called an open effective (resp. active) morphism if the sieve on X generated by f is an open effective (resp. active) sieve U on X (thus Y = U). An open cover {Ui} of U is called effective (resp. active, resp. exact) if any open sieve Ui is so. 

A framed category C (or G) is effective (resp. active, resp. exact) if any open sieve of any object is so; thus a framed category is effective if and only if it is both active and exact. A framed category is locally effective (resp. locally active, resp. locally exact) if the open effective (resp. active, resp. exact) sieves on any object X form a base for G(X). A framed category (C, G) is called neat if any morphism of C is open effective; any neat framed category is locally effective. 

A framed category C (or G) is called essential if for any morphism f: Y ® X and any open effective sieve U of X, f-1(U) is an open effective sieve of Y (thus the intersection of two open effective sieves of X is effective). 

Definition 2.2.framed site (resp. framed topology) is a locally effective essential framed category (resp. frame on a category). 

Suppose (C, G) is a framed category and F: C ® D is a covariant functor (resp. contravariant functor) from C to a category D. We say F is strict (resp. strongly strict) if it is so at each framed object (X, G(X)) of C; a strict F is also called a cosheaf (resp. sheaf) on C with values in D

Note that a site (C, G) is strict if and only if the identity functor C ® C is strict. If U is an open effective sieve on an object X of C with the inclusion morphism eU: U ® X the morphism F(eU): F(U) ® F(X) (resp. F(eU): F(X) ® F(U)) will be called the restriction morphism

Proposition 2.3. (a) Any locally exact framed category is exact
(b) Any locally effective framed category is exact. 
(c) A locally effective framed category (C, G) with pullbacks is essential (thus is a framed site). 
Proof. (a) Suppose C is a locally exact framed category. Suppose U is an open sieve on an object X of C. Suppose V is a subsieve of U such that, for any morphism f: Y ® X in U, f-1(V) is an open sieve on Y. We prove that V is open. Let {Ui} be an exact open cover of U. Put Vi = V Ç Ui. If f: Y ® X is a morphism in Ui, then f-1(Vi) = f-1(V Ç Ui) = f-1(V) by (0.1.f), thus f-1(Vi) is an open sieve on Y by the assumption on V, hence 
each Vi is open as each Ui is exact. It suffices to prove that V = Ú {Vi}. If f: Y ® X is a morphism in Ú {Vi} Í U, f-1(V) is open and f-1(V) Ê  f-1(Vi), we see that f-1(V) Ê  Ú {f-1(Vi)} = f-1(Ú {Vi}) = 1C/Y, which implies that f Î V by (0.1.f). Conversely, suppose f: Y ® X is a morphism in V, then 

1C/Y = f-1(Ú {Ui}) = Ú {f-1(Ui)} = Ú {f-1(V Ç Ui)} = Ú {f-1(Vi)} = f-1(Ú {Vi}) 
by (0.1.f), which means that f Î Ú {Vi}. This proves that V = Ú {Vi}, so V is open. 
(b) Any locally effective framed category is locally exact, thus exact by (a). 
(c) This follows from (0.1.g) and (b). n 

Remark 2.4. Similarly the following assertions hold for any framed category
(a) An open active sieve U of X is exact if and only if G(U) = G(X)|U (i.e., the framed object U is isomorphic to the open subobject X|U of X; see (1.2.1)). 
(b) Any composite of open effective monomorphisms is effective. 
(c) In an exact framed category any open active morphism is an open effective morphism. 
(d) A framed category is effective (resp. locally effective) if and only if it is active (resp. locally active) and any composite of open active morphism is open active. 
(e) Any effective framed category is essential (thus is a framed site). This is because the pullback of an open effective morphism is always open active, which is open effective by (c) in view of (2.3.b). 

Remark 2.5. (a) Any strict exact framed category is strongly strict
(b) Any strict locally effective framed category is strongly strict (by (2.3.b) and (a)). 
(c) If C is a strict framed category and F: C ® D is a functor from C to a category D preserving colimits, then F is strict

Example 2.5.1. Suppose C is a category. 
(a) The category of framed objects of C is a natural framed category. 
(b) The category of active framed objects of C is an effective framed category (by (2.4.d)). 
(c) The category of topological objects of C is a strict effective framed category. 

Example 2.5.2. (a) (C, wC) is an exact framed category. 
(b) If B is a full subcategory of C, then (C, wC/B) is an exact framed category by (0.3) and (0.4). 

Example 2.5.3. Suppose (A, £) is a poset (viewed as a category). 
(a) (A, wA) is a strict neat framed category (for any a Î A the frame wA(a) consists of the lower subsets of the poset ¯a). 
(b) If A is a frame then (A, vA) is a strict effective neat framed site. 

Remark 2.6. Suppose (C, G)) is a framed category. An object X of C is called trivial if G(X) = {1C/X} is a singleton. 
(a) If f: Y ® X is a morphism and X is a trivial object, then Y is a trivial object. 
(b) If C is a nonempty effective framed site the open sieve 0 of any X Î C is effective, hence 0 is a trivial object. 
(c) Suppose F: C ® D is a strict functor from a framed category (C, G) to a category D. If  is a trivial object of C, then F(X) is an initial object of D. This is because the empty set is an open cover of X, so F(X) is a colimit of an empty system of objects of D, thus an initial object of D
(d) Any trivial object in a strict framed site C is an initial object (by (c) for the strict functor F = 1C). 
(e) Any nonempty strict effective framed site C has a strict initial object which is trivial (by (a), (b) and (d)). 
(f) If C has a trivial object (e.g., C is nonempty effective) and D has no initial object, then there is no strict functor from C to D (by (c)). 

Example 2.7. Suppose D is a category. 
(a) Denote by F the topology on D with F(X) = {1C/X} for any X Î C; F is the smallest framed topology on D; note that by (2.6.d) (D, F) is not strict unless all the objects in D are initial. 
(b) Suppose D has a strict initial object. For any object X of D let f(X) be the sieve of X of all the morphisms from initial objects to X. Denote by Y the framed topology on D with Y(X) = {f(X), 1C/X} for any X Î D; Y(X) is the smallest strict effective topology on D

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