0. Sieves
Suppose C is a category and X is an object of C.
Recall that a set S of maps
to an object X is called a sieve
on X if S contains
any map to X that factors through a map in S.
The largest sieve on X is denoted by 1_{C/X},
called the unit
sieve on X, which consists of all the maps to X.
A sieve on X not equal to 1_{C/X} is
called a proper
sieve on X. If V Í
U are two sieves of X we say that V is a subsieve
of U.
A sieve U on X is called active
if it is generated by a monomorphism e: Y ®
X; since Y is uniquely determined by U up to isomorphism,
we often write U for Y (and e_{U} for
e), viewed as a subobject of X (with the inclusion morphism
e_{U}). Sometimes we simply identify a subobject Y
of X with the active sieve on X generated by it. If f:
Y ® X is a map we simply write
f(Y) for the sieve on X generated by f.
We shall use the following notation:
(i) C/X for the slice category of all the morphisms with
codomain X.
(ii) If f: Y ® X
is a morphism we write C/f: C/Y ®
C/X for the induced functor,
(iii) w_{C}(X) is
the poset of sieves on X.
(iv) v_{C}(X) is the
poset of subobjects of X.
(v) C/U for the slice category of all the morphisms in
a sieve U of X.
Remark 0.1. Suppose f: Y
® X is a morphism and W is
a sieve on Y. Suppose U and V are two sieves on X.
(a) f^{1}(U) is a sieve on Y and f(W)
is a sieve on X.
(b) The assertions that f Î
U, f(1_{C/Y}) Í
U and f^{1}(U) = 1_{C/Y}
are equivalent.
(c) W Í f^{1}(U)
if and only if f(W) Í U.
(d) A morphism g: Z ®
Y is in f^{1}(U) if and only if fg(Z)
Í U (by (b) and (c)).
(e) If {U_{i}} is a set of sieves on X then Ç
{U_{i}} and È {U_{i}}
are two sieves on X. We have f^{1}(Ç
{U_{i}}) = Ç {f^{1}(U_{i})}
and f^{1}(È {U_{i}})
= È {f^{1}(U_{i})}.
(f) If f Î U then f^{1}(U
Ç V ) = 1_{C/Y}
Ç f^{1}(V ) =
f^{1}(V ) (by (b) and (e)).
(g) Suppose C has pullbacks. If f: Y ®
X is a morphism and U is an active
sieve on X, then f^{1}(U) is an active
sieve of Y generated by the monomorphism Y ×_{X}
U ® Y.
Remark 0.2. If D is a category
and F: C ® D is a functor
we write F_{X}: C/X ®
D/F(X) for the induced functor.
(a) If W is a sieve on F(X) then F_{X}^{1}(W)
is a sieve on X.
(b) If {W_{i}} is a set of sieves on F(X)
then F_{X}^{1}(Ç
{W_{i}}) = Ç {F_{X}^{1}(W_{i})}
and F_{X}^{1}(È
{W_{i}}) = È {F_{X}^{1}(W_{i})}.
Next we introduce a relative version of the notion of a sieve, which
will be useful in the study of extensions of framed topologies in º3.
If A is a full subcategory of C and X Î
C we write A/X for the full subcategory of C/X
of all the morphisms f: Y ®
X with Y Î A. Suppose
A Ê B are two full subcategories
of C. By an A/Bsieve
on an object X Î C we mean
a subset U Í A/X
such that a morphism f: Y ® X
in A/X is in U if and only if
for any morphism g: Z ®
Y in B/Y, fg is in U. If B =
A then we simply say that U is a Bsieve
on X. Denote by w_{A/B}(X)
(resp. w_{B}(X)) the set
of A/Bsieves (resp. Bsieves) on X.
Remark 0.3. (a) There is a natural onetoone
correspondence between w_{C/B}(X)
and w_{B}(X) preserving
orders, sending each C/Bsieve U on X to the
Bsieve U Ç B/X
on X.
(b) If f: Y ® X
is a morphism and U a Bsieve on X then f_{B}^{1}(U)
= {u: Z ® Y  Z
Î B and fu Î
U} is a Bsieve on Y.
Remark 0.4. (a) w_{C/B}(X)
is closed under arbitrary intersections.
(b) If f: Y ® X
is a morphism and U is a C/Bsieve on X, then
f^{1}(U) is a C/Bsieve on Y.
(c) Suppose U Ê V are
two sieves on X, and U is a C/Bsieve. Suppose
for any f: Y ® X in
U, f^{1}(V) is a C/Bsieve
on Y, then V is a C/Bsieve.
Example 0.4.1. Suppose B is
a small category and C = B^{^} is the category of
contravariant functors from B to the category Set of small
sets, then a Csieve U on an object X in C
is a C/Bsieve if and only if U is active
in C (i.e. U is determined by a subfunctor of X).
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