2.1. Normal Sieves
In this chapter we assume that A is an arbitrary
category with a strict initial
object 0. By an initial
map we mean a map with the initial object 0 as domain. 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.
Suppose S and T
are two sets of maps to an object X. We say that S
is disjoint
with T if any map in S
is disjoint with any map in T. Denote
by ØS
the set of maps to X which is disjoint with S.
Denote by J(S)
the sieve on X generated by S.
(i.e. the smallest sieve on X containing S).
Let 1X (resp. 0X
) be the sieve on X generated by the identity map (resp. the initial
map) on X.
Proposition 2.1.1. Suppose S
and T are two set of maps to X.
(a) ØS
is a non-empty sieve which contains the initial map to X; S
Ç ØS
consists of only initial map; Ø(S
È ØS)
= 0X,; ØØ(S
È ØS)
= 1X; S
Í ØØS.
(b) S Í
T implies that ØT
Í ØS
and ØØS
Í ØØT.
(c) ØS
= ØØØS;
T Í
ØØS
iff ØS
Í ØT;
T Í
ØØS
iff ØØT
Í ØØS.
(d) A map t is in
ØØS
iff any non-initial map to X which factors through t
is not disjoint with S; if T
is a sieve then T Í
ØØS
iff any non-initial map in T
is not disjoint with S.
(e) ØS
= ØJ(S)
and ØØS
= ØØJ(S)
(f) (ØØS)
Ç (ØØT)
= ØØ (J(S)
Ç J(T)).
(g) If S and T
are sieves then (ØØS)
Ç (ØØT)
= ØØ (S
Ç T).
Proof. (a), (b) (d) and (e) are obvious; (c) follows from (a) and (b).
(f) We have J(S)
Ç J(T)
Í J(S),
so ØØ (J(S)
Ç J(T))
Í ØØJ(S)
= ØØS
by (e). Similarly ØØ
(J(S)
Ç J(T))
Í ØØJ(T)
= ØØT.
Thus ØØ (J(S)
Ç J(T))
Í (ØØS)
Ç (ØØT).
We prove the other direction using (d). Assume t: Y ®
X is a non-initial map in (ØØS)
Ç (ØØT).
To see that t is in ØØ
(J(S)
Ç J(T)),
it suffices to prove that for any non-initial map s: Z ®
Y, s°t
is not disjoint with J(S)
Ç J(T).
Since t Î
ØØS,
we have s°t
Î ØØS,
so by (d) s°t
is not disjoint with S.
Thus there is a non-initial map u: U ®
Z such that u°s°t
factors through a map in S.
Since t Î
ØØT,
we have u°s°t
Î ØØT,
so by (d) u°s°t
is not disjoint with T.
Thus there is a non-initial map v: V ®
U such that v°u°s°t
factors through a map in T.
It follows that v°u°s°t
Î J(S)
Ç J(T),
which implies that s°t
is not disjoint with J(S)
Ç J(T)
as desired.
(g) follows from (f) as J(S)
= S and J(T)
= T if S
and T are sieves. n
A sieve S of maps to X is
a normal sieve
if S = ØØS.
Clearly 1X and 0X
are normal sieves.
Proposition 2.1.2. (a) ØS
is a normal sieve for any set S of
maps to X .
(b) Any intersection of normal sieves is a normal sieve.
(c) If {Si} is a collection
of sets of maps to X then ØØ(Èi
Si) is the smallest
normal sieve containing each Si.
Proof. (a) follows from (2.1.1.c).
(b) Suppose S is an intersection
of a set of normal sieve {Si}
on X. Then Ç Si
Í Si
implies ØØ(Ç
Si) Í
ØØSi
= Si, thus ØØ
(Ç Si)
Í Ç Si,
which implies the equality ØØ
(Ç Si)
= Ç Si
because the other direction is trivial.
(c) If S is a normal sieve which
contains each Si, then Èi
Si Í
S implies ØØ(Èi
Si) Í
ØØS
= S. n
Consider a map f: Y ®
X. If S is a set of maps to X
we denote by f*(S) the inverse
image of S under f, which consists
of all the maps z: Z ®
Y such that f°z
is in S. If S
is a sieve on X then f*(S)
is a sieve on Y.
Denote by Â(X) the set of normal
sieves on an object X.
Proposition 2.1.3. (a) f*(ØS)
= Øf*(S)
for any sieve S on X.
(b) If S is normal then f*(S)
is normal.
(c) The function f*: Â(X)
® Â(Y)
preserves intersections.
Proof. (a) Suppose z: Z ®
Y is a map in Øf*(S).
If w: W ® Z is a map
such that f°z°w
Î S,
then z°w Î
Øf*(S)
Ç f*(S),
so z°w is an initial
map by (2.1.1.a); thus w in an initial map,
which implies that f°z
Î ØS,
i.e. z Î f*(ØS).
This shows that Øf*(S)
Í f*(ØS).
The other direction is trivial.
(b) If S is normal then S
= ØØS,
by applying (a) twice we obtain f*(S)
= f*(ØØS)=
ØØf*(S),
i.e. f*(S) is normal.
(c) follows from (2.1.2.b) as f* preserves
intersection of sieves. n
Proposition 2.1.4. (a) Â(X)
is a complete boolean algebra with Ù
= Ç.
(b) If f: Y ® X is
a map then f*: Â(X)
® Â(Y)
is a morphism of complete boolean algebras.
Proof. (a) We already know by (2.1.2.b) that Â(X)
is a complete lattice with Ù = Ç.
Suppose S is a normal sieve and {Tk}
is a set of normal sieves on X. We prove the infinite distributive
law
S Ç
(Úk Tk)
= Úk (S
Ç Tk).
We only need to verify the relation Í
because the other direction is always true. Since Úk(S
Ç Tk)
= ØØ(Èk
S Ç
Tk) by (2.1.2.c)
and S Ç
(Úk Tk)
is a sieve, we only need to prove that any map in u: U ®
X in S Ç
(Úk Tk)
is not disjoint with Èk
S Ç
Tk by (2.1.1.d).
As u Î Úk
Tk = ØØ(Èk
Tk) by (2.1.2.c),
u is not disjoint with a Tk
for some k by (2.1.1.d), and u Î
S implies that u is not disjoint
with S Ç
Tk as required.
The normal sieve ØS
is a complement of any normal sieve S
in Â(X) by (2.1.1.a).
Thus Â(X) is
a complete boolean algebra.
(b) f* preserves arbitrary intersection (2.1.3.c)
and complement by (a), thus also preserves arbitrary join. n
It follows from (2.1.4) that Â
is a functor from A to the (meta)category of
boolean locales, called the boolean
functor on A.
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