3. Framed Subsites  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:  (i) F sends an open effective morphism in C to an open effective morphism in D.  (ii) For any X Î C we have FX-1(GD(F(X)) Í GC(X).  (iii) The induced map FX-1: GD(F(X)) ® GC(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 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.  Remark 3.2. Any full framed subsite of a strict framed site is a strict framed site.  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.  (a) There is a unique frame GC on C defined over B extending GB (i.e., GC|B = GB).  (b) GB is exact if and only if GC is so.  (c) GB is spatial if and only if GC is so.  (d) If B is dense in C then (B, GB) is a framed subcategory of (C, GC).  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 to G(B)(X), 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(X) ® GC(Z) is a morphism of frames.  (b) First assume GB is exact. Suppose U Ê V are two sieves on an object X Î C, and U is open in (C, GC). 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 ', gB-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, GC), hence fg-1(V) Ç B/Z = gB-1(V ') is open. This proves that V is open, thus (C, GC) is exact.  Conversely, suppose (C, GC) 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, fB-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 GC(X) is the limit of the system of frames GB(Y) indexed by the morphisms Y ® X in B/X, GC is spatial if and only if GB 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    [Next Section][Content][References][Notations][Home]