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: 
(i) F sends an open effective morphism in C to an open effective morphism in D. 
(ii) For any X Î C we have F_X^-1(G_D(F(X)) Í G_C(X). 
(iii) The induced map F_X^-1: G_D(F(X)) ® G_C(X) is an isomorphism of frames. 

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/B-sieve 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 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 G_B is a frame on B. 
(a) There is a unique frame G_C on C defined over B extending G_B (i.e., G_C|_B = G_B). 
(b) G_B is exact if and only if G_C is so. 
(c) G_B is spatial if and only if G_C is so. 
(d) If B is dense in C then (B, G_B) is a framed subcategory of (C, G_C). 

Proof. (a) The uniqueness of G_C is obvious. For any X Î C let G_C(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 G_C|B = G_B. Thus it suffices to prove that G_C is a frame on C. Let G_(B)(X) be the collection V of B-sieves of X such that f_B^-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 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. 
(b) First assume G_B is exact. Suppose U Ê V are two sieves on an object X Î C, and U is open in (C, G_C). Suppose for any morphism f: Y ® X in U, f^-1(V) is an open sieve on Y. We prove that V is an open sieve on X. By (0.4.c) V is a C/B-sieve of X. To see that V is open it suffices to prove that V ' = f^-1(V) Ç B/Y is open for any morphism f: Y ® X in B/X. But U ' = f^-1(U) Ç B/Y is open and V ' Í U '. Since U ' is an open exact sieve on Y, to see that V ' is an open sieve of Y, it suffices to prove that, for any g: Z ® Y in U ', g_B^-1(V ') is an open sieve on Z in B. But g(Z) Í f^-1(U) implies that fg Î U, thus fg^-1(V) is open in (C, G_C), hence fg^-1(V) Ç B/Z = g_B^-1(V ') is open. This proves that V is open, thus (C, G_C) is exact. 
Conversely, suppose (C, G_C) is exact. Suppose V Í U are two B-sieves on an object X Î B and U is open. Suppose for any f: Y ® X in U, f_B^-1(V) is open. We prove that V is an open B-sieve. Let V ' and U ' be the C/B-sieve of X determined by V and U, respectively. It suffices to prove that V ' is an open sieve on X in C, which would implies that V = V ' Ç B/X is open in B. Since U ' is exact, it suffices to prove that, for any 
morphism g: Z ® X in U ', g^-1(V ') is open. Since g^-1(V ') is a C/B-sieve (0.4.b), it suffices to prove that, for any h: W ® Z in B/Z, h^-1(g^-1(V ')) Ç B/W = (gh)_B^-1(V) is open, but this follows from the assumption on V because gh Î U. 
(c) The category of spatial frames is a reflective subcategory of the category of frames. Thus the inclusion functor preserves limits. Since each G_C(X) is the limit of the system of frames G_B(Y) indexed by the morphisms Y ® X in B/X, G_C is spatial if and only if G_B is so. 
(d) For (3.1.i) it suffices to show that any open effective morphism in B is open active in C. This follows easily from the assumption that B is a dense subcategory of C (cf. [Luo 1995a, (1.11)]). n