5. Localeslocale is a frame and
a morphism f: A
® B of locales is a morphism f:
^{-1}B ® A of frames. Denote by
Loc the category of small locales and morphisms of locales, which
is the opposite of the category of small frames.
Suppose a
® A of locales. The subobject ¯a
of A (with the inclusion monomorphism e) is called
an _{a}open sublocale of A. Denote
by W(A) the set of open sublocales of
A. W(A) is a frame isomorphic
to A. If f: B ® A is
a morphism of locales we have f(¯^{-1}a)
= ¯(f(^{-1}a)), thus
f: W(^{-1}A) ®
W(B) is a morphism of frames. We obtain
an effective framed topology W on Loc
sending each locale A to W(A).
Let us consider the effective framed site (Loc, W).
A; then 1 = Ú
{_{A}a}. Suppose _{i}f, g are two morphisms of
locales from A to B whose restrictions on each ¯a
Í _{i}A are equal; then for any b
Î B we have f(^{-1}b)
Ù a = _{i}g(^{-1}b)
Ù a. Thus _{i}f(^{-1}b)
= Ú {f(^{-1}b)
Ù a} = Ú
{_{i}g(^{-1}b) Ù a}
= _{i}g(^{-1}b). This shows that f = g.
Conversely, suppose for each i there is a morphism f:
¯_{i}a ®
_{i}B such that the restrictions of f and _{i}f
on ¯_{j}a Ù
¯_{i}a are equal, i.e.,
for any _{j}b Î B we have f(_{i}^{-1}b)
Ù a = _{j}f(_{j}^{-1}b)
Ù a. Let _{i}f:
^{-1}B ® A be the map sending each
b Î B to Ú
{f_{i}^{-1}(b)} Î
A. Then
f(^{-1}b) Ù
a = Ú_{j}
{_{i}f(_{i}^{-1}b) Ù
a} = Ú_{j}
{_{i}f(_{j}^{-1}b) Ù
a} = _{i}f(_{j}^{-1}b) Ù
(Ú {_{i}a})
= _{i}f(_{j}^{-1}b). f: ^{-1}B ®
A with e: _{a}^{-1}A ®
¯a yields the morphism
_{i}f: _{i}^{-1}B ®
¯a. Using this fact
it is easy to see that _{i}f is a morphism of frames. We
obtain a morphism ^{-1}f: A ® B
of locales whose restriction on each ¯a
is _{i}f. This shows that _{i}Loc is strict.
(b) Suppose ({ A}, {¯_{i}a},
{_{ij}u}) is a glueing diagram of _{ij}Loc. Glueing the
sets A along the subsets ¯_{i}a
Í _{ij}A we obtain a set
_{i}S containing each A. Denote by _{i}A the collection
of subsets U of S such that U Ç
A is an open sublocale of _{i}A. Then
_{i}A is a locale. We may regard each A as an open
sublocale of _{i}A by identifying each a Î
A with ¯a Í
A. This turns _{i}A into a glueing colimit of {A}
in _{i}Loc. n
Suppose A (viewed as a small framed site;
cf. (2.5.3.b)) and a strict functor O:
_{A}A ® C. If a Î
A and U = ¯a we shall
write O(_{A}U) for O(_{A}a),
and O(_{A}A) for O(_{A}1).
_{A}Suppose ( C.
A morphism of locales over C
from (A, O) to (_{A}B, O)
is a pair (_{B}f, f^{#}) of a morphism f: A
® B of locales, and a natural transformation
f^{#}: f_{*}O ®
_{A}O, where _{B}f_{*}O =
_{A}O: _{A}f^{-1}B ®
C.
C and ¯a is an
open sublocale of A. Denote by O|_{A}
the restriction of _{a}O on ¯_{A}a.
Then (¯a, O|_{A})
is a locale over _{a}C, called an open sublocale of (A, O).
For any _{A}b Î A let (e_{a}^{#}):
_{b}O(_{A}b Ù a)
® O(_{A}b) be the
restriction morphism. We obtain a monomorphism (e, _{a}e_{a}^{#}):
(¯a, O|_{A}a)
® (A, O).
_{A}Denote by Loc(C)
® C sending each (A, O)
to _{A}O(_{A}A) and any morphism (f, f^{#}):
(A, O) ® (_{A}B,
O) to _{B}f^{#}: _{B}O(_{A}A)
® O(_{B}B).
F: E ®
C a strict functor, there is a unique (up to equivalence) strict
bicontinuous functor Spec: _{F}E ®
Loc(C) such that F = O_{C}Spec.
_{F}(c) If C has colimits then Loc(C) is complete
and (b) holds for any framed site E and any strict functor F.
(b) If X is an object of E the strict functor F:
E ® C induces a strict functor
O(F): G_{X}(_{E}X)
® C sending each U Î
G(_{E}X) to F().
We obtain a locale USpec(_{F}X) = (G(_{E}X),
O(F)) over _{X}C. Any morphism f:
Y ® X in E induces
a morphism Spec(_{F}f) = (G(f),
G(f)^{#}): (G(_{E}Y),
O(F)) ®
(G_{Y}(_{E}X), O(F))
of locales over _{X}C such that G(f)
= ^{-1}f: G(^{-1}X) ®
G(Y), and for any open sieve U
of X with V = f(^{-1}U), G(f)^{#}:
_{U} ® V
is the restriction morphism of Uf. We obtain a strict bicontinuous
functor Spec: _{F}E ®
Loc(C) such that F = O_{C}Spec.
_{F}(c) Suppose C has colimits. Suppose ({A,
_{i}O}, {¯_{Ai}a},
{_{ij}u}) is a glueing diagram of _{ij}Loc(C).
Glueing the locales A along ¯_{i}a
Í _{ij}A as in (5.1.b)
we obtain a locale _{i}A containing each A as an
open sublocale of _{i}A. For each open sublocale ¯a
of A let O(_{A}a) be the colimit of the objects
O(_{Ai}b) for all the open sublocales ¯b
Í ¯a
with b Î A for
some _{i}i under the restriction morphisms. Then (A, O)
is a glueing colimit of ({_{A}A, _{i}O},
{¯_{Ai}a}, {_{ij}u}).
_{ij}Next suppose E is a framed site and F: E ®
C a strict functor. For any open sieve U on an object X
of E we let O(F)(_{X}U) = F()
(cf. (1.3)). We obtain a locale (GU(_{E}X),
O(F)) over _{X}C. As in (b) we can
prove that Spec: _{F}E ®
Loc(C) given by X ®
(G(_{E}X), O(F))
is a strict functor and _{X}F = O_{C}Spec.
n
_{F} |