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
(iii) The induced map FX-1: GD(F(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
of GC along F to be
the function sending each X to F-1(GC)(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,
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
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
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
(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
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
(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)
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
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
(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
(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)]).