2. Framed Sites
Definition 2.1. A frame
on a category C is a function G which
assigns to each object X of C a frame
G(X) on X such that any morphism
f: Y ® X in C
is a continuous
morphism of framed objects from (Y, G(Y))
to (X, G(X)). A framed
category is a pair (C, G)
of a category C and a topology G on C;
C is called the underlying
category of (C, G). We often write
C for a framed category (C, G),
and X for a framed object (X, G(X)).
We say a framed category (C, G) (or
G) is active
(resp. strict,
resp. strongly
strict, resp. spatial)
if each framed object (X, G(X))
(or G(X)) is so. A framed category (C,
G) is locally
small (resp. small)
if the underlying category C is locally small (resp. small) and
each frame G(X) is small.
Suppose U is an open sieve on an object X. We say U
is exact
if any subsieve V of U such that f^{1}(V)
is open for each f Î U is
open. If U is both active and exact then we say that U is effective.
A monomorphism f: Y ® X
in C is called an open
effective (resp. active)
morphism if the sieve on X generated by f is an open effective
(resp. active) sieve U on X (thus Y = U).
An open cover {U_{i}} of U is called effective
(resp. active,
resp. exact)
if any open sieve U_{i} is so.
A framed category C (or G) is effective
(resp. active,
resp. exact)
if any open sieve of any object is so; thus a framed category is effective
if and only if it is both active and exact. A framed category is locally
effective (resp. locally
active, resp. locally
exact) if the open effective (resp. active, resp. exact) sieves
on any object X form a base for G(X).
A framed category (C, G) is called neat
if any morphism of C is open effective; any neat framed category
is locally effective.
A framed category C (or G) is called essential
if for any morphism f: Y ®
X and any open
effective sieve U of X, f^{1}(U)
is an open effective sieve of Y (thus the intersection of two open
effective sieves of X is effective).
Definition 2.2. A framed
site (resp. framed
topology) is a locally
effective essential framed
category (resp. frame on a category).
Suppose (C, G) is a framed
category and F: C ® D
is a covariant functor (resp. contravariant functor) from C to a
category D. We say F is strict
(resp. strongly
strict) if it is so at
each framed object (X, G(X))
of C; a strict F is also called a cosheaf
(resp. sheaf)
on C with values in D.
Note that a site (C, G) is strict
if and only if the identity functor C ®
C is strict. If U is an open effective sieve on an object
X of C with the inclusion morphism e_{U}:
U ® X the morphism
F(e_{U}): F(U) ®
F(X) (resp. F(e_{U}): F(X)
® F(U)) will be called the restriction
morphism.
Proposition 2.3. (a) Any locally
exact framed category is exact.
(b) Any locally effective
framed category is exact.
(c) A locally effective framed category (C, G)
with pullbacks is essential
(thus is a framed site).
Proof. (a) Suppose C is a locally exact framed category.
Suppose U is an open sieve on an object X of C. Suppose
V is a subsieve of U such that, for any morphism f:
Y ® X in U, f^{1}(V)
is an open sieve on Y. We prove that V is open. Let {U_{i}}
be an exact open cover of U. Put V_{i} = V
Ç U_{i}. If f:
Y ® X is a morphism in U_{i},
then f^{1}(V_{i}) = f^{1}(V
Ç U_{i}) = f^{1}(V)
by (0.1.f), thus f^{1}(V_{i})
is an open sieve on Y by the assumption on V, hence
each V_{i} is open as each U_{i} is exact.
It suffices to prove that V = Ú
{V_{i}}. If f: Y ®
X is a morphism in Ú {V_{i}}
Í U, f^{1}(V)
is open and f^{1}(V) Ê
f^{1}(V_{i}), we see that f^{1}(V)
Ê Ú
{f^{1}(V_{i})} = f^{1}(Ú
{V_{i}}) = 1_{C/Y}, which implies
that f Î V by (0.1.f).
Conversely, suppose f: Y ®
X is a morphism in V, then
1_{C/Y} = f^{1}(Ú
{U_{i}}) = Ú {f^{1}(U_{i})}
= Ú {f^{1}(V Ç
U_{i})} = Ú {f^{1}(V_{i})}
= f^{1}(Ú {V_{i}})
by (0.1.f), which means that f Î
Ú {V_{i}}. This proves
that V = Ú {V_{i}},
so V is open.
(b) Any locally effective framed category is locally exact, thus exact
by (a).
(c) This follows from (0.1.g) and (b).
n
Remark 2.4. Similarly the following assertions
hold for any framed category.
(a) An open active
sieve U of X is exact
if and only if G(U) = G(X)_{U}
(i.e., the framed object U is isomorphic to the open subobject
XU of X; see (1.2.1)).
(b) Any composite of open
effective monomorphisms is effective.
(c) In an exact framed category
any open active morphism is an open effective morphism.
(d) A framed category is effective
(resp. locally effective)
if and only if it is active (resp.
locally active) and any composite
of open active morphism is open active.
(e) Any effective framed category is essential
(thus is a framed site). This is because the
pullback of an open effective morphism is always open active, which is
open effective by (c) in view of (2.3.b).
Remark 2.5. (a) Any strict
exact framed category is strongly
strict.
(b) Any strict locally effective framed category is strongly strict
(by (2.3.b) and (a)).
(c) If C is a strict framed category and F: C
® D is a functor from C to
a category D preserving colimits, then F is strict.
Example 2.5.1. Suppose C is a
category.
(a) The category of framed objects
of C is a natural framed category.
(b) The category of active
framed objects of C is an effective framed category (by (2.4.d)).
(c) The category of topological
objects of C is a strict effective framed category.
Example 2.5.2. (a) (C, w_{C})
is an exact framed category.
(b) If B is a full subcategory of C, then (C,
w_{C/B}) is an exact framed
category by (0.3) and (0.4).
Example 2.5.3. Suppose (A, £)
is a poset (viewed as a category).
(a) (A, w_{A}) is
a strict neat framed category (for
any a Î A the frame w_{A}(a)
consists of the lower subsets of the poset ¯a).
(b) If A is a frame then (A, v_{A})
is a strict effective neat framed site.
Remark 2.6. Suppose (C, G))
is a framed category. An object X of C is called trivial
if G(X) = {1_{C/X}}
is a singleton.
(a) If f: Y ® X
is a morphism and X is a trivial object, then Y is a trivial
object.
(b) If C is a nonempty effective framed site the open sieve
0 of any X Î C is
effective, hence 0 is a trivial object.
(c) Suppose F: C ® D
is a strict functor
from a framed category (C, G) to a category
D. If is a trivial object of C, then F(X)
is an initial object of D. This is because the empty set is an open
cover of X, so F(X) is a colimit of an empty system
of objects of D, thus an initial object of D.
(d) Any trivial object in a strict framed site C is an initial
object (by (c) for the strict functor F = 1_{C}).
(e) Any nonempty strict effective framed site C has a strict
initial object which is trivial (by (a), (b) and (d)).
(f) If C has a trivial object (e.g., C is nonempty effective)
and D has no initial object, then there is no strict functor from
C to D (by (c)).
Example 2.7. Suppose D is a category.
(a) Denote by F the topology on D
with F(X) = {1_{C/X}}
for any X Î C; F
is the smallest framed topology on D; note that by (2.6.d)
(D, F) is not strict unless all the objects
in D are initial.
(b) Suppose D has a strict initial object. For any object X
of D let f(X) be the sieve of
X of all the morphisms from initial objects to X. Denote
by Y the framed topology on D with Y(X)
= {f(X), 1_{C/X}}
for any X Î D; Y(X)
is the smallest strict effective topology on D.
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