6. Geometric Subsites
Suppose (C, G) is a framed site. If
U is a sieve on an object X of C we denote by U*
the joint of open subsieves U_{i} of U in G(X)
(i.e. U* = Ú_{i}
U_{i}).
Proposition 6.1. Suppose U and
V are two sieves on X.
(a) Suppose V is open. Then V Í
U* if and only if V has an open cover {V_{i}}
with each V_{i} Í U.
(b) (U Ç V)* = U*
Ç V*.
(c) If f: Y ® X
is a morphism then f^{1}(U*) Í
(f^{1}(U))*.
(d) If f: Y ® X
is an open effective morphism then f^{1}(U*) = (f^{1}(U))*.
(e) If C is neat then U Í
U*.
(
Proof. (a) If V Í U*
then V = V Ç U*
= V Ç (Ú_{i}
U_{i}) = Ú_{i}
V Ç U_{i}. This
shows that {V_{i} = V Ç
U_{i}  U_{i} is any open subsieve of U}
is an open cover of V. The other direction is trivial.
(b) Clearly we have (U Ç V)*
Í U* Ç
V*. Since U* Ç V*
Í U*, U* Ç
V* has an open cover {U_{i}  iÎ
I} with each U_{i} Í
U by (a). Similarly U* Ç
V* has an open cover {V_{j}  j Î
J} with each V_{j} Í
V. Then {U_{i} Ç
U_{j}  i Î I
and j Î J} is an open cover
of U* Ç V* and each U_{i}
Ç V_{j} Í
U Ç V. It follows that
U* Ç V* Í
(U Ç V)* by (a). The other
direction is trivial.
(c) Suppose {U_{i}} is an open cover of U* with
each U_{i} Í U,
then f^{1}(U_{i}) is an open cover of f^{1}(U*)
and f^{1}(U_{i}) Í
f^{1}(U). Thus f^{1}(U*)
Í (f^{1}(U))*
by (a).
(d) Suppose Y = V for an open effective sieve
V on X. Then f^{1}(U*) = V
Ç U* = V* Ç
U* = (V Ç U)* =
(f^{1}(U))*.
(e) If C is neat any sieve U on X is the union
of its open effective subsieves. By definition U* is the joint of
open sieves on U, so U Í
U*.
Suppose (C, G_{C})
and (D, G_{D}) are two
framed sites. We consider a fixed functor F: C ®
D. Suppose X is an object of C. If W is any
sieve on F(X) we write F_{X}*(W) for
(F_{X}^{1}(W))*. We obtain an orderpreserving
map F_{X}*: w_{D}(F(X))
® G_{C}(X).
Proposition 6.2. (a) If U and V
are sieves on F(X), then F_{X}*(U Ç
V) = F_{X}*(U) Ç
F_{X}*(V).
(b) For any morphism f: Y ®
X in C and any sieve W on F(X) we have
f^{1}(F_{X}*(W)) Í
F_{Y}*(F(f)^{1}(W)).
(c) For any open effective morphism f: Y ®
X in C and any sieve W on F(X) we have
f^{1}(F_{X}*(W)) = F_{Y}*(F(f)^{1}(W)).
(d) Suppose F: C ®
D is a strict functor. Suppose X is an object of C
and W an active sieve on F(X). Then F_{X}*(W)
Í F_{X}^{1}(W).
If C is neat then F_{X}*(W) = F_{X}^{1}(W).
Proof. (a) We have F_{X}*(U Ç
V) = (F_{X}^{1}(U Ç
V))* = (F_{X}^{1}(U) Ç
F_{X}^{1}(V))* = (F_{X}^{1}(U))*
Ç (F_{X}^{1}(V))*
= F_{X}*(U) Ç F_{X}*(V)
by (6.1.b).
(b) We have f^{1}(F_{X}*(W))
Í (f^{1}(F_{X}^{1}(W))*
= (F_{Y}^{1}(F(f)^{1}(W))*
= F_{Y}*(F(f)^{1}(W))
by (6.1.c).
(c) We have f^{1}(F_{X}*(W))
= (f^{1}(F_{X}^{1}(W))*
= (F_{Y}^{1}(F(f)]^{1}(W))*
= F_{Y}*(F(f)^{1}(W))
by (6.1.d).
(d) Suppose {U_{i}} is an open cover of F_{X}*(W)
such that each U_{i} is an open subsieve of F_{X}^{1}(W).
Suppose f: Y ® X is
a morphism in F_{X}*(W). Then U = f^{1}(È
{U_{i}}) is the unoin of an open cover of Y. Since
F is strict, F(Y) is a colimit of F(U)
(i.e., F(Y) = F(U); cf. (1.3)).
For each g Î U we have F(fg)
Í W, thus F(fg)
can be factored uniquely through the inclusion morphism W ®
F(X). This implies that the morphism F(f):
F(Y) ® F(X)
can be factored uniquely through the inclusion W ®
F(X), i.e., f Î F_{X}^{1}(W).
This shows that F_{X}*(W) Í
F_{X}^{1}(W). If C is neat then F_{X}^{1}(W)
Í (F_{X}^{1}(W))*
= F_{X}*(W) by (6.1.e), thus F_{X}*(W)
= F_{X}^{1}(W).
Definition 6.3. We say an object X
of C is geometric over D
(for F) if F_{X}* maps an open cover of F(X)
to an open cover of X. We say that X is strongly
geometric over D if
any open subobject of X is geometric over D. A morphism f:
Y ® X in C is a geometric
morphism over D if f^{1}(F_{X}*(W))
= F_{Y}*(F(f)^{1}(W))
for any open effective sieve W of F(X). We say that
f is strongly geometric over
D if for any open effective sieve U of X
with V = f^{1}(U), the restriction f_{V}:
V ® U is geometric.
Proposition 6.4. (a) Any open subobject
of a strongly geometric object over D is strongly geometric.
(b) Any open effective morphism in C is strongly geometric over
D.
(c) A composition of strongly geometric morphisms of strongly geometric
objects over D is a strongly geometric morphism.
Proof. (a) and (c) follow directly from the definition (6.3),
and (b) is the content of (6.2.c).
It follows from (6.4) that the collection of strongly
geometric objects with the strongly geometric morphisms over D forms
a subsite Geo(C/F) of C, called the geometric
subsite of C over D.
Lemma 6.5. Suppose U is
a geometric open subobject of an object X of C. Suppose U
Í F_{X}^{1}(W)
for an open sieve W on F(X) and {W_{i}}
is an open cover of W. Then {U Ç
F_{X}*(W_{i})} is an open cover of U.
Proof. Applying (6.2.c) to the open effective
morphism f: U ® X
we get U Ç F_{X}*(W_{i})
= f^{1}(F_{X}*(W_{i})) =
F_{U}*(F(f)^{1}(W_{i})).
Since {W_{i}} is an open cover of F(X), {F(f)^{1}(W_{i})}
is an open cover of F(U). But U is geometric
over C, so {F_{U}*(F(f)^{1}(W_{i}))}
is an open cover of U. This implies that {U Ç
F_{X}*(W_{i})} is an open cover of U.
Proposition 6.6. (a) Suppose X
is strongly geometric over D. The map F_{X}*: G_{C}(F(X))
® G_{D}(X)
is a morphism of frames (i.e., G(F_{X}):
G_{D}(X) ®
G_{C}(F(X)) is
a morphism of locales).
(b) Suppose X and Y are strongly geometric objects over
D and f: Y ® X
is a geometric morphism over D. Then f^{1}(F_{X}*(W))
= F_{Y}*(F(f)^{1}(W))
for any open sieve W of F(X) (i.e., G(F_{X})G(f)
= G(F(f))G(F_{Y})).
Proof. (a) We only need to show that F_{X}* preserves
joints in view of (6.1.b). Suppose W is an open
sieve on F(X) and {W_{j}  j Î
J} is an open cover of W. Let {U_{i}  i
Î I} be an open effective cover
of F_{X}*(W) with U_{i} Í
F_{X}^{1}(W). Since X is strongly
geometric, each U_{i} is geometric, thus by (6.5)
{U_{i} Ç F_{X}*(W_{j})
 j Î J} is an open cover
of U_{i}. This shows that {F_{X}*(W_{j})
 j Î J} is an open cover
of F_{X}*(W).
(b) Since X and Y are strongly geometric objects over
D, F_{X}* and F_{Y}* are morphisms
of frames by (a). Thus f^{1}F_{X}* and F_{Y}*F(f)^{1}
are two morphisms of frames from G(F(X))
to G(Y). Since f is geometric
over D, these two maps agree at any open effective sieve on F(X),
and therefore also agree at any open sieve on F(X) because
open effective sieves on F(X) form a base for G(F(X)).
Theorem 6.7. Suppose X is an object
of C and {U_{i}  i Î
I} an open effective cover of X.
(a) If each U_{i} is geometric (resp. strongly
geometric) over D, then X is geometric (resp. strongly geometric)
over D.
(b) Suppose X and Y are strongly geometric objects over
D. Suppose f: Y ® X
is a morphism such that the restriction f_{Vi}:
V_{i} ® U_{i}
of f on each V_{i} = f^{1}(U_{i})
is strongly geometric. Then f is strongly geometric over D.
Proof. (a) First suppose each U_{i} is
geometric over D. Suppose {W_{j}  j Î
J} is an open cover of F(X). Then by (6.5)
{U_{i} Ç F_{X}*(W_{j})
 j Î J} is an open cover
of U_{i}. Thus {F_{X}*(W_{j})
 j Î J} is an open cover
of X, i.e., X is geometric over D.
Next suppose each U_{i} is strongly geometric.
Then for any open subobject U of X there exist an
open effective cover of U consisting of geometric objects
over C. Applying the above result we see that U is
geometric. Thus X is strongly geometric over D.
(b) Under the assumption we have G(F_{Ui})G(f_{Vi})
= G(F(f_{Vi}))G(F_{Vi})
for each i by (6.6.b). Since G(F_{Ui})
and G(F(f_{Vi}))
are morphisms of locales by (6.6.a), G(F_{Ui})G(f_{Vi})
and G(F(f_{Vi}))G(F_{Vi})
are morphisms of locales. This implies that the restrictions of the morphisms
of locales G(F_{X})G(f)
and G(F(f))G(F_{Y})
on each V_{i} are equal. Since {V_{i}} is
an open effective cover of Y, we have G(F_{X})G(f)
= G(F(f))G(F_{Y})
as Loc is strict by (5.1). This proves
that f is geometric. For any open effective sieve U of X
with V = f^{1}(U), the induced morphism f_{V}:
V ® U satisfies
the same condition, thus f_{V} is geometric, hence
f is strongly geometric over D.
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