Definition 3.1. Suppose (C, GC)
and (D, GD) 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 GC is a frame on C. Suppose F: E ® C is a functor from a category E to C. We define the pullback F-1(GC) of GC along F to be the function sending each X to F-1(GC)(X) = FX-1(GC(F(X))). We say that F induces a frame on E if F-1(GC) is a frame on E and F is a bicontinuous functor from (E, F-1(GC)) to (C, GC).
If B is a subcategory of C such that the inclusion functor I: B ® C induces a framed topology I-1(GC) on B, then we write GC|B for I-1(GC), called the restriction of GC on B, and we say that (B, GC|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, GC) and (B, GC|B) are framed sites then we say that (B, GC|B) (or simply B) is a framed subsite of C.
Suppose (C, GC) is a framed category and B is a full subcategory of C. We say (C, GC) (or GC) 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/B-sieve and f-1(U) is open for any morphism f: Y ® X in B/X. If GC is defined over B, then by (0.3.a) the restriction GC|B on B is a frame on B.
We shall see that there is a natural one-to-one 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 GB
is a frame on B.
Proof. (a) The uniqueness of GC
is obvious. For any X Î C
let GC(X) be the set
of C/B-sieves on X such that f-1(U)
Ç B/Y is an open B-sieve
on Y for any morphism f: Y ® X in
B/X. Then GC|B
= GB. Thus it suffices to
prove that GC is a frame
on C. Let G(B)(X)
be the collection V of B-sieves of X such that fB-1(V)
(see (0.3.b)) is an open B-sieve on
Y for any morphism f: Y ®
X in B/X. Then the poset G(B)(X)
is a limit of the posets GB(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 GC(X) is isomorphic
GC(X) is also a frame.
Now from the universal property of limits it follows that for any morphism
g: Z ® X, g-1:
GC(Z) is a morphism