1. Metric SitesDefinition 1.1. Let C
be a category. A metric pretopology
on C consists of the data
(a) for every object X
C, a topological space (X),
and
(b) for every morphism f: Y --> X in C,
a continuous function (f): (Y)
--> (X),
subject to the conditions (1) (1 )
is the identity of (_{X}X).
(2) ( gf)
= (g)(f)
whenever gf is defined.
Alternatively one can define a metric pretopology
on
Suppose ( We say A morphism A subset Y, and call
an Ueffective subobject of X;
f is called the effective morphism
of .
UMore generally, a subset
is an
f(U)isomorphism of effective subobjects (note that (a) - (d) hold for any active morphism). A metric presite C is effective
(resp. everywhere
effective) if any open subset (resp. any
subset) of any object is effective. Similarly a metric presite C
is called active
(resp. everywhere
active) if any open subset (resp. any subset) of any object
is active. A metric presite is locally
effective if open effective subsets of any object X form
a base for |X|.
U)
is an open effective subset of |Y| (thus the
intersection of two open effective subsets of |X| is effective).
A metric pretopology t on a category C is a metric
topology if (C, t)
is a metric site.
U)
is active).
Suppose ( D, )
are metric presites. An isometry from _{D}C to D is a
functor :
C --> D such that
is isomorphic to _{D}
(i.e., _{C}((X))
is naturally homeomorphic to _{D}(X)
for any X Î _{C}C),
and
sends an open effective morphism in C to an open effective morphism
in D. An isometry
is called an embedding (resp. equivalence) of metric presites if
the functor j is an embedding (resp. equivalence) of the underlying categories.
B a subcategory of C. We define
the induced metric pretopology on B to be the restriction |_{C}
of on _{B}B. We say
(B, |_{C})
(or simply _{B}B) is a subpresite of C if the inclusion functor
B ® C is an isometry of
metric presites from (B, |_{C})
to (_{B}C, ).
_{C}
B, |_{C})
of _{B}C is also a metric site, then we say that
B is a subsite of C.
C/X ® C be the functor
sending each f to Y. Then C/X
is a metric presite with the metric pretopology /_{C}(_{X}f)
= |Y|, and
is an isometry from _{X}C/X to C. If C is a metric
site, then C/X is also a metric site.
X|. Suppose for each
i we have a morphism f: _{i}
® U_{i}Y such that the restrictions of
f and _{i}f to _{j}
Ç U_{i}
are the same. Then there exists a unique morphism
U_{j}f from X to Y such that the restriction of f
to is U_{i}f.
_{i}
Let ( C we mean a pair (X,
x) consisting of an object X Î
C and a point x Î
|X|. A morphism of pointed objects from (Y, y)
to (X, x) is a morphism f: Y ®
X such that f(y) = x.
D a full subcategory of C. A
subset U of an object X Î
C is called D-exact if
the following conditions are satisfied:
(a) For any point x Î
U there is a morphism f: (Y, y) ®
(X, x) of pointed objects of C
such that Y Î
D and |f(Y)| Í
U.
(b) For any x Î
U the category D/(U, x) of pointed objects
f: (Y, y) ® (X,
x) over (X, x) such that Y
Î D and |f(Y)|
Í U is connected.
(c) A subset V of U is open in the subspace
U if for any morphism f: Y ®
X such that Y Î
D and |f(Y)| Í
U, f^{-}(V)
is an open subset of |^{1}Y|.
Suppose
f| is an embedding. Thus f is effective and
|f(Y)| = U is effective.
(b) Suppose open exact subsets of any object X
form a base for |X|. Let U be an open
subset of |X|. We verify the conditions of (1.6.)
for U:
First (1.6.a) follows from the fact that U has an open exact cover {V} and (1.6a)
holds for each open exact subset _{i}V.
Suppose _{i}x Î
U. We have to prove that the category C/(U, x)
is connected. Let V be an open exact neighborhood
of x contained in U. Then C/(V, x) is a connected
subcategory of C/(U, x) (1.6b). It suffices
to prove that any object of C/(U, x) is connected
to an object of C/(V, x). Suppose f: (Y,
y) ® (X,
x) is an object of C/(U, x). Then
f^{-}(^{1}V)
is an open subset of Y containing y. Applying (1.6.a)
to an open exact subset of f^{-}(^{1}V)
containing y we can find a morphism g: (Z, z) ®
(Y, y) such that g(Z) Í
f(^{-}^{1}V).
Since fg: (Z, z) ®
(X, x) is an object of C/(V, x), we
see that the object f Î
C/(U, x) is connected to an object
fg Î C/(V,
x) by g. Suppose V is a subset of U such that
(1.6.c) holds for V. Take an open exact cover
{V} of _{i}U. Then (1.6.c)
holds for the subset V Ç
V of _{i}V for each _{i}i. Thus each
V Ç V
is open in _{i}V. Hence _{i}V is
open in U. n
D. A pretopology
on _{C}C is called an extension of
on _{D}C if
= _{D}|_{C}
and |_{D}X| is D-exact for any X
C.
always exists: for any _{C}X Î
C consider the triples (Y, y,
f), where Y Î
D, y Î
|Y|, and f: Y ®
X is a morphism. Write (Y, y, f) ~ (Z,
z, g) if there is another triple (W,
w, h) with morphisms p: W ®
Y, q: W ®
Z, such that fp = h = gq,
p(w) = y and q(w) = z. Denote
by (_{C}X)
the set of equivalence classes of these
triples under the equivalence relation generated by ~. Any morphism u: X ®
S in C induces a map (_{C}u): (_{C}X)
® (_{C}S)
sending each (Y, y, f) Î (_{C}Z)
to (Y, y, uf) Î (_{C}S).
If X Î D,
then (_{C}X)
may be identified with the set (_{D}X).
A subset U of (_{C}X)
is open if for any Y Î
D and f: Y ®
X, f^{-}(^{1}U)
is an open subset of |Y|. These open subsets turns (_{C}X)
into a topological spaces. If X Î
D, then (X)
is naturally homeomorphic to the space _{C}(_{D}X).
Let
be the pretopology on _{C}C defined by X ® (_{C}X)
for any X Î
C. Then we have
=
_{D}|_{C}.
It is easy to verify that _{D}(_{C}X)
is D-exact for any X Î
C. Thus
is an extension of _{C}
on _{D}C.
D a full subcategory of C.
Suppose t is
an extension of _{C}
= _{D}|_{C}.
Then (_{D}C, )
is exact if and only if (_{C}D, )
is exact.
_{D}Proof. First suppose ( C, )
is exact. Let _{C}U be an open subset of
an object X of C. We show that U is C-exact
by verifying the conditions of (1.6).
(a) Since | X| is D-exact, for any point
x Î U
there is a morphism f: (Y, y) ®
(X, x) such that Y Î
D (1.6a). Since D is exact, we can find a D-exact
open neighborhood V of y contained in
f^{-}(^{1}U).
Let g: (Z, z) ® (Y,
y) be a morphism with Z Î
D and |g(Z)| Í
V (1.6.a). Then fg:
(Z, z) ® (X, x)
is a morphism such that |fg(Z)| Í
U. This proves (1.6.a) for
U.
(b) Suppose x Î
U. Consider two pointed objects f: (Y, y) ®
(X, x) and g: (Z, z) ®
(X, x) over (X, x) in C/(U,
x). We have to prove that f and g are connected in
C/(U, x). We have seen in (a)
that any pointed object of C/(U, x) is connected to
a pointed object of D/(U, x).
Thus we may assume Y, X
D. Then f and g are connected in D/(|X|,
x) because |X| is D-exact. Using
the fact that D is exact we can easily verify that f and
g are connected in D/(U, x).
(c) Suppose V is a subset of U such that
for any morphism f: Y ®
X with |f(Y)| Í
U, |f^{ -}(^{1}V)|
is an open subset of |Y|. Suppose g: Z ®
X is any morphism with Z Î
D. For any h: W ®
Z with W Î
D and |h(W)| Í
g^{-}(U),
the subset (^{1}gh)^{-}(^{1}V)
= h^{-}(^{1}g^{-}(V))
is open because |^{1}gh(W)| Í
U. Since g^{-}(^{1}U)
is D-exact, this implies that g^{-}(^{1}V)
is an open subset of g^{-}(^{1}U),
thus an open subset of |Z|. Since |X| is D-exact, this means
that V is open in |X|, thus also open
in U. Conversely, suppose (C, t) is exact.
Suppose _{C}U is an open subset of an object X
Î D. We show
that U is D-exact by verifying the conditions of (1.6).
(a') Since U is C-exact, for any point
x Î U
there is a morphism f: (Y, y) ®
(X, x) with f(Y) Í
U by (1.6.a). Since |Y| is D-exact,
we can find a morphism g: (Z, z) ®
(Y, y) with Z Î
D (1.6.a). Then fg: (Z, z)
® (X, x) is a morphism of
pointed objects of D such that |fg(Z)|
Í U. This
proves (1.6.a) for U.
(b') We prove that for any x Î
U, the category D/(U, x) is connected. Consider
two pointed objects f: (Y, y)
® (X, x) and g: (Z,
z) ® (X, x) over
(X, x) in D/(U, x). Since U is
C-exact, f and g are connected in C/(U,
x). Using the fact that
is an extension of _{C}
we can show that _{D}f and g are also connected
in D/(U, x).
(c') Suppose V is a subset of U such that
for any morphism f: Y ®
X with |f(Y)| Í
U and Y Î
D, f^{-}(V)|
is an open subset of |^{1}Y|. Suppose g: Z ®
X is any morphism with g(Z) Í
U. For any morphism h: Y ®
Z with Y Î
D, we have |gh(Y)| Í
U. Thus (gh)^{-}(^{1}V)
= h^{-}(^{1}g^{-}(^{1}V))
is open. Since |Z| is D-exact, this implies that g^{-}(V)
is open. Since ^{1}U is C-exact, this means
that V is open in U. n
D a full subcategory of C.
Suppose
is an extension of _{C}
= _{D}|_{C}.
Suppose (_{D}D, )
is exact and (_{D}C, )
is active. Then (_{C}C, )
is an effective metric site, and (_{C}D, )
is a subsite of (C, )
if _{C}D is dense in C.
D a full dense subcategory of
C. Suppose the glueing lemma (1.5) holds for
the morphisms from X to Y with X Î
D. Then (C, )
is a strict metric site.
_{C}
X|.
Suppose f, g Î
hom(_{C}X, Y) and f ¹
g. Since D is dense
in C, We can find Z Î
D and h: Z ®
X such that fh ¹ gh.
Since {h^{-}(^{1}U)}
forms an open cover of _{i}Z Î
D, by the glueing lemma we can find an open effective subset
V of |Z| contained in some h^{-}(^{1}U)
such that the restrictions of _{i}fh and gh to V are different
(because the collection of all such V forms an open
effective cover of |Z|). It follows that
the restrictions of f and g to U
are different. Now suppose for any _{i}i we have
a morphism f: _{i} ®
U_{i}Y such that the restrictions of f
and _{i}f to _{j} Ç
U_{i} are the same. For any object U_{j}Z Î
D and any morphism h: Z ®
X, let
{ V}be an effective open cover of |_{j}Z|
such that h(V) Í
_{j}U for some _{i}U. Denote by _{i}h
the restriction of _{i}h to .
Then V_{i}f and _{i}h_{i}f
agree on _{j}h_{j} Ç
V_{i}. By the glueing lemma for morphisms
from V_{j}Z to Y these f determine
a morphism _{i}h_{i}h': Z ®
Y. We obtain a map hom(_{C}Z,
X) ® hom(_{C}Z,
Y) given by h ®
h' for each Z Î
D. Since D is dense in C, the collection
of all these maps determines a morphism f Î
hom(_{C}X, Y) whose restriction to
each is U_{i}f.
The uniqueness of _{i}f is obvious. This proves that (C, )
is strict. n
_{C} |