Definition 3.1. Suppose (C, G_{C})
and (D, G_{D}) are two
framed categories. A functor F: C ®
D is called bicontinuous
if the following conditions are satisfied:
A composite of two bicontinuous functors is bicontinuous. A bicontinuous functor F: C ® D is an embedding (resp. equivalence) of framed categories if F is an embedding (resp. equivalence) of the underlying categories. Suppose G_{C} is a frame on C. Suppose F: E ® C is a functor from a category E to C. We define the pullback F^{1}(G_{C}) of G_{C} along F to be the function sending each X to F^{1}(G_{C})(X) = F_{X}^{1}(G_{C}(F(X))). We say that F induces a frame on E if F^{1}(G_{C}) is a frame on E and F is a bicontinuous functor from (E, F^{1}(G_{C})) to (C, G_{C}). If B is a subcategory of C such that the inclusion functor I: B ® C induces a framed topology I^{1}(G_{C}) on B, then we write G_{C}_{B} for I^{1}(G_{C}), called the restriction of G_{C} on B, and we say that (B, G_{C}_{B}) (or simply B) is a framed subcategory of C; if B is a full subcategory of C then we say that B is a full framed subcategory of C. If both (C, G_{C}) and (B, G_{C}_{B}) are framed sites then we say that (B, G_{C}_{B}) (or simply B) is a framed subsite of C. Remark 3.2. Any full framed subsite of a strict framed site is a strict framed site. Suppose (C, G_{C}) is a framed category and B is a full subcategory of C. We say (C, G_{C}) (or G_{C}) is defined over B if for any X Î C, a sieve U on X is open if and only if U is a C/Bsieve and f^{1}(U) is open for any morphism f: Y ® X in B/X. If G_{C} is defined over B, then by (0.3.a) the restriction G_{C}_{B} on B is a frame on B. We shall see that there is a natural onetoone correspondence between the collection of frames on B and the collection of frames on C defined over B. Theorem 3.3. Suppose C is a category
and B a full subcategory of C. Suppose G_{B}
is a frame on B.
Proof. (a) The uniqueness of G_{C}
is obvious. For any X Î C
let G_{C}(X) be the set
of C/Bsieves on X such that f^{1}(U)
Ç B/Y is an open Bsieve
on Y for any morphism f: Y ® X in
B/X. Then G_{CB}
= G_{B}. Thus it suffices to
prove that G_{C} is a frame
on C. Let G_{(B)}(X)
be the collection V of Bsieves of X such that f_{B}^{1}(V)
(see (0.3.b)) is an open Bsieve on
Y for any morphism f: Y ®
X in B/X. Then the poset G_{(B)}(X)
is a limit of the posets G_{B}(Y)
(indexed by the morphisms f: Y ®
X in B/X). Using this fact one can verify directly
that G_{(B)}(X) is a frame.
Since G_{C}(X) is isomorphic
to G_{(}_{B)}(X),
G_{C}(X) is also a frame.
Now from the universal property of limits it follows that for any morphism
g: Z ® X, g^{1}:
G_{C}(X) ®
G_{C}(Z) is a morphism
of frames.
