8. Geometric Locales
Now consider the site Loc(C) of the locales over a locally
small framed site C with the strict functor O_{C}:
Loc(C) ® C. Write GLoc(C)
for Geo(Loc(C)/O_{C}). An object
of GLoc(C) is called a geometric
locale over C; a morphism in GLoc(C) is
called a geometric morphism of geometric
locales over C.
Proposition 8.1. (a) GLoc(C)
is an effective subsite of Loc(C).
(b) A strict functor F: E ®
C from an effective framed site E to C is continuous
if and only if the induced functor Spec_{F}: E ®
Loc(C) has the image in GLoc(C).
(c) If (E, G_{E})
is an effective framed site and F: E ®
C is a strict continuous functor, there is a unique (up to isomorphism)
bicontinuous functor Spec_{F}: E ®
GLoc(C) such that F = O_{C}Spec_{F}.
(d) If C has colimits then GLoc(C) is complete
and (c) holds for any framed site E and any strict continuous functor
F: E ® C.
Proof. (a), (b) and (c) follow from (6.2.c),
(7.2.b) and (5.2.b).
(d) If C has colimits then Loc(C) is complete
by (5.2.c), and we have C(GLoc(C))
= GLoc(C) by (6.7), thus GLoc(C)
is complete by (4.3). The proof for the
second assertion of (d) is similar to that of (5.2.c).
n
Example 8.1.1. (a) If D is a
(locally small) category with a strict initial object viewed
as a site with the trivial strict topology then Loc(D)
= GLoc(D).
(b) Loc = Loc(2) = GLoc(2) where
2 = {0, 1} is the locale of two elements viewd as
a framed site.
(c) Suppose (E, G_{E})
is a framed category. The topology G_{E}
on E may be viewed as a bicontinuous functor from E to Loc.
If E is a framed site this is a special case of (5.2.d)
or (8.1.(d) with C = 2 and F being
the unique strict functor from E to 2.
Suppose C is a strict framed site with colimits. Since C
is strict the identity functor 1_{C}: C ®
C is a strict bicontinuous functor. Applying (8.1.d)
to F = 1_{C} we obtain a bicontinuous functor
Spec: C ® GLoc(C).
If X is an object of C we call Spec(X) the
spectrum of X.
Proposition 8.2. Spec: C
® GLoc(C) is a full embedding
(i.e., Spec(C) is a full subcategory of GLoc(C)
equivalent to C).
Proof. For simplicity we shall write F for Spec.
Then F is an embedding as 1_{C} = O_{C}F.
Thus we only need to prove that F is full. Suppose X and
Y are two objects of C and (f, f^{#}):
F(X) ® F(Y)
is a geometric morphism of geometric locales. Then G((O_{C})F(X))G(f)
= G(f^{#}_{X})G((O_{C})F(Y)).
But G((O_{C})F(X)):
G(X) ®
G(X) and
G((O_{C})F(Y)):
G(Y) ®
G(Y) are identity functors, thus G(f)
= G(f^{#}X). If U
is an open effective sieve of X, then f^{1}(U)
= f^{#}X^{1}(U). It follows that
f^{#}_{U}: f^{1}(U)
= f^{#}X^{1}(U) ®
U is induced by the restriction of f^{#}_{X}
on f^{#}_{X}^{1}(U).
For an arbitrary open sieve U we can show that f^{#}_{U}:
O(f^{1}(U)) ®
O(U) is also induced by f^{#}_{X}
by passing to the colimits. This shows that (f, f^{#})
= F(f^{#}_{X}). Thus F is a
full embedding. n
Definition 8.3. An affine
scheme over C is
a geometric locale over C isomorphic
to the spectrum of some object of C. A scheme
over C is a geometric locale in the completion C(Spec(C))
of Spec(C) in GLoc(C).
Theorem 8.4. Suppose C is a strict
framed site with colimits.
(a) The full subcategory Sch(C) of GLoc(C)
of schemes over C is a complete framed site which is equivalent
to a completion of C.
(b) The full subcategory ASch(C) of GLoc(C)
of affine schemes over C is a reflective subcategory of GLoc(C).
Proof. (a) Since GLoc(C) is complete by (8.1.d),
the assertion follows from (4.3).
(b) SpecO_{C}: GLoc(C) ®
Spec(C) is the left adjoint of the inclusion functor Spec(C)
® GLoc(C). Since ASch(C)
is equivalent to Spec(C), it is a reflective subcategory
of GLoc(C). n
Remark 8.5. If C is a locally small
category (resp. framed site) with colimits, then one can show that Loc(C)
(resp. GLoc(C)) have colimits.
Suppose C, D, and E are framed sites and F:
C ® E, G: D
® E are two bicontinuous functors.
Suppose (X, Y) is a pair with X Î
C and Y Î D such
that F(X) = G(Y). Since F and G
are bicontinuous, G_{C}(X),
G_{E}(F(X)) = G_{E}(G(Y)),
and G_{D}(Y) are all isomorphic,
so we may identify G_{C}(X)
with G_{D}(Y). We say
(X, Y) is compatible
if X and Y has a common open effective cover {U_{i}}
Í G_{C}(X)
= G_{D}(Y).
Let C ×_{E} D be the collection of
all such compatible pairs (X, Y). Suppose (X', Y')
is another compatible pair. We define a morphism from (X', Y')
to (X, Y) to be a pair (f, g) with f:
X' ® X and g: Y'
® Y such that F(f)
= G(g). This turns C ×_{E} D
into a category, which is naturally a framed site such that a morphism
(f, g): (X, Y) ®
(X', Y') in C ×_{E} D
is open effective if and only if both f: X' ®
X and g: Y' ® Y
are open effective. The natural functors p_{1}: C
×_{E} D ® C
and p_{2}: C ×_{E} D
® D are then bicontinuous functors.
It is easy to see that C ×_{E} D together
with p_{1} and p_{2} is the fibre product
of C and D over E in the metacategory of framed sites
and bicontinuous functors. Thus we can talk about base extension for framed
sites. If E = Loc with p_{1} = G_{C}
and p_{2} = G_{D }then
C ×_{Loc} D is the product of C
and D, denoted simply by C × D.
Remark 8.6. (a) C ×_{E}
D is effective if and only if both C and D are so.
(b) If any one of C and D is effective, then the underlying
category of C ×_{E} D is the fibre product
of C and D over E as categories because then any pair
(X, Y) with F(X) = G(Y) is compatible.
(c) Faithful (resp. full) bicontinuous functors are stable under base
extension (i.e., if F is faithful or full then so is p_{2}:
C ×_{E} D ®
D).
(d) If C is a full effective base of E, then C
×_{E} D is naturally a full base of D.
Definition 8.7. Suppose (M, O_{M})
is a strict effective framed site (viewed as a framed site over Loc
via O_{M}).
(a) If C is a category we write Loc_{M}(C)
for the effective framed site M ×_{Loc }Loc(C).
An object of Loc_{M}(C) is called an Mspace
of C.
(b) If C is a framed site with colimits we write GLoc_{M}(C)
for the effective framed site M ×_{Loc} GLoc(C).
An object of GLoc_{M}(C) is called a geometric Mspace
of C.
Example 8.7.1. (a) Let M be a
strict effective framed site. If C is a category and t:
C ® M is a functor inducing
a framed topology G_{t} on C
such that (C, G_{t}) is an essential
locally effective framed site, then we say that (C, t)
is a Mmetric site. Note that
any framed site is naturally an Locmetric site.
(b) A Topmetric site is simply called a metric
site (see [Luo 1995a]) .
(c) Denote by STop the complete framed subsite of Top
consisting of sober topological spaces. A metric site (C, t)
is called sober if t(C)
® STop.
(d) Suppose C is a strict Mmetric site. We say C
is Mcomplete if any glueing
diagram S of C such that t(S)
has a glueing colimit in M has a glueing colimit in C. Any
strict Msite C has an Mcompletion which is a complete
Mmetric site containing C as a base (the proof is similar
to that of (4.4)).
Example 8.7.2. Consider the opposite
Ring^{op} of the category Ring of small commutative
rings with 1. Ring^{op} is a locally small category
with colimits, which is a strict effective framed site with the topology
sending each commutative ring R to the frame of radical ideals of
R. We have
(a) Loc_{Top}(Ring^{op}) = Top
× Loc(Ring^{op}) is the complete metric site
of ringed spaces.
(b) GLoc_{Top}(Ring^{op}) = Top
× GLoc(Ring^{op}) is the complete metric site
of local ringed spaces.
(c) SchS_{Top}(Ring^{op}) = STop
× Sch(Ring^{op}) is the complete metric sites
of schemes.
[Next Section][Content][References][Notations][Home]
