1. Metric Sites
Definition 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) (1X)
is the identity of (X).
(2) (gf)
= (g)(f)
whenever gf is defined.
Alternatively one can define a metric pretopology
on C to be a functor from C to the metacategory
of topological spaces (cf. [Mac Lane
1971, p.9]). The set (C)
of topological spaces together with all the continuous
functions (f) is a category,
thus t may be viewed as a functor
from C to the category (C).
Definition 1.2. A metric
presite is a pair (C, )
consisting of a category C and a metric pretopology
on C; C is called the underlying category of (C, ).
Suppose (C, )
is a metric presite. We often simply write C for (C, ).
If X is an object and f: Y
--> X a morphism, we shall adopt the following notation:
|X| (or simply X) for the underlying space (X)
(a subset U of |X| is often called a subset
of X);
|f| (or simply f) for the underlying map
t(f);
|f(X)| for the subset |f|(|X|)
of |Y|;
f(U) for |f|(U) if U 
|Y|, and f-1(V) for |f|-1(V)
if V |X|.
We say f: Y --> X is surjective,
injective, homeomorphism,
bicontinuous, open,
closed, etc., if the
underlying map |f|: |Y| --> |X| is so. An object
X is called an empty object
if the underlying space |X| of X is empty.
A morphism f: Y --> X is called active
if any morphism g: Z --> X with |g(Z)|
|f(Y)| factors through f uniquely.
An effective
morphism is an active morphism f: Y --> X
such that the continuous map |f|: |Y|
-->|X| is an embedding (i.e., f is bicontinuous).
A subset U of |X| is called effective
if there is an effective morphism f: Y --> X such
that |f(Y)| = U. Since Y is uniquely
determined by U up to isomorphism, we often write U
for Y, and call U
an effective subobject of X;
f is called the effective morphism
of U.
More generally, a subset U of |X| is called
active if there is a morphism f:
Y --> X such
that |f(Y)|
U, and any morphism g: Z --> X
with |g(Z)|
U factors through f uniquely. The morphism
f is an active morphism, uniquely determined by U (but in
general |f(Y)| 
U), called the active morphism
of U. Any effective subset of |X| is active.
Remark 1.3.
An effective morphism plays the role of a local isomorphism in categorical
geometry. It has all the properties one would expect for a local isomorphism:
(a) An effective morphism is uniquely determined
by its image.
(b) An effective morphism is a monomorphism.
(c) An isomorphism is effective.
(d) A surjective effective morphism is an isomorphism.
(e) A composition of two effective morphisms is
effective.
(f) Suppose f: Y ®
X is an effective morphism. Then for any subset U
|Y|, U is
effective if and only if f(U) is effective,
in which case the induced morphism U --> f(U)
is an
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|.
Definition 1.4.
A metric site is a locally effective
metric presite satisfying the following condition:
If f: Y --> X
is a morphism in C and U an open effective subset of |X|,
then f-1(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.
Example 1.4.1.
(a) The empty metric presite is a metric site.
(b) Any effective (resp. everywhere effective) metric
presite is a metric site, called an effective (resp.
everywhere effective) metric site.
(c) Since the empty subset 
of any object X Î
C belongs to every base of |X|,
is effective if C is locally effective. Thus
any non-empty locally effective metric presite has at least
one empty object. This also implies that any initial object of a locally
effective metric presite (if exists) must be an empty
object.
(d) Any locally effective metric presite with fibre products
is a metric site (this follows from (1.7b), and the
fact that, if f: Y --> X is a morphism and U
an open active subset of |X|, then f-1(U)
is active).
Suppose (C, C)
and (D, D)
are metric presites. An isometry from C to D is a
functor  :
C --> D such that D
is isomorphic to C
(i.e., D((X))
is naturally homeomorphic to C(X)
for any X Î 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.
Example 1.4.2.
Suppose (C, C)
is a metric presite, B a subcategory of C. We define
the induced metric pretopology on B to be the restriction C|B
of on B. We say
(B, C|B)
(or simply B) is a subpresite of C if the inclusion functor
B ® C is an isometry of
metric presites from (B, C|B)
to (C, C).
Example 1.4.3.
Suppose (C, C)
is a metric site. If a subpresite (B, C|B)
of C is also a metric site, then we say that
B is a subsite of C.
Example 1.4.4.
Suppose (C, ) is
a metric presite. For any object X Î
C denote by C/X the slice category
of objects f: Y ® X
over X. Let  X:
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 X
is an isometry from C/X to C. If C is a metric
site, then C/X is also a metric site.
Definition 1.5.
A strict
metric site is a metric site C in which the following
glueing lemma for morphisms holds:
Suppose X, Y are objects and {Ui}
is an open effective cover of |X|. Suppose for each
i we have a morphism fi: Ui
® Y such that the restrictions of
fi and fj to Ui
Ç Uj
are the same. Then there exists a unique morphism
f from X to Y such that the restriction of f
to Ui is fi.
Example 1.5.1.
Any full subsite of a strict metric site is a strict metric site.
Example 1.5.2.
The metric presite w(X)
of subspaces of a topological space X is an
everywhere effective, strict metric site. The metric presite W(X)
of open subsets of X is an effective, strict
subsite of w(X).
All the morphisms in w(X)
and W(X) are effective.
Let (C, C)
be a metric presite. By a pointed object of 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.
Definition 1.6.
Suppose (C, C)
is a metric presite and 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-1(V)
is an open subset of |Y|.
Suppose C is a metric presite. A subset U
of an object X Î
C is called exact if it is
a C-exact subset of |X|. A metric presite C
is called exact (resp. everywhere
exact) if any open subset (resp. any subset)
of any object X Î
C is exact.
Proposition 1.7.
Suppose C is a metric presite.
(a) A subset U of an object X is effective
if and only if U is active and exact (thus any
active, exact metric presite is an effective metric site).
(b) C is exact if the open exact subsets of any
object X form a base for |X| (thus any
locally effective metric presite is exact).
Proof. (a) One direction is obvious. Suppose U
is an active subset of X and f: Y ®
X is the active morphism of U. Suppose U
is exact. Then (1.6.a) means that |f(Y)|
= U, (1.6b) implies that |f| is injective,
and (1.6c) indicates that any subset V of U is open if f-1(V)
is open, so |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 {Vi} and (1.6a)
holds for each open exact subset Vi.
Suppose 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
{Vi} of U. Then (1.6.c)
holds for the subset V Ç
Vi of Vi for each i. Thus each
V Ç Vi
is open in Vi. Hence V is
open in U. n
Definition 1.8.
Suppose C is a category and D a full subcategory of C.
Suppose D
is a pretopology on D. A pretopology C
on C is called an extension of D
on C if D
= C|D
and |X| is D-exact for any X
C.
Remark 1.9.
It is easy to see that C
is unique up to a natural isomorphism, if it exists.
We now show that such an extension C
always exists: for any 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 C(X)
is naturally homeomorphic to the space D(X).
Let C
be the pretopology on C defined by X ® C(X)
for any X Î
C. Then we have D
=
C|D.
It is easy to verify that C(X)
is D-exact for any X Î
C. Thus C
is an extension of D
on C.
Theorem 1.10.
Suppose (C, C)
is a metric presite and D a full subcategory of C.
Suppose tC is
an extension of D
= C|D.
Then (C, C)
is exact if and only if (D, D)
is exact.
Proof. First suppose (D, D)
is exact. We prove that the extension (C, C)
is exact. Let 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-1(U),
the subset (gh)-1(V)
= h-1(g-1(V))
is open because |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, tC) is exact.
Suppose 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 C
is an extension of D
we can show that 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-1(V)|
is an open subset of |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-1(V)
is open. Since U is C-exact, this means
that V is open in U. n
Corollary 1.11.
Suppose (C, C)
is a metric presite and D a full subcategory of C.
Suppose C
is an extension of D
= C|D.
Suppose (D, D)
is exact and (C, C)
is active. Then (C, C)
is an effective metric site, and (D, )
is a subsite of (C, C)
if D is dense in C.
Proof. The first assertion follows from (1.10)
and (1.7.a). Now suppose D
is dense in C. Suppose f: Y ®
X is an open effective morphism in D; we prove that f
is effective in C. Since f is bicontinuous,
it suffices to prove that it is active in C. Suppose g: Z
® X is a morphism
in C such that |g(Z)| Í
|f(Y)|. Then for any morphism h: W ®
Z with W ” D, gh: W
® X has the image |gh(W)|
Í |f(Y)|.
Since f is effective in D, gh factors through f
uniquely. Since D is dense in C, this implies
that h factors through f uniquely, thus f is
active in C. This shows that D is a subsite
of C. n
Theorem 1.12.
Suppose (C, C)
is a metric presite and 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, C)
is a strict metric site.
Proof. We prove that C is strict by verifying
the condition of (1.5) for any objects X,
Y Î C
and an open effective cover {Ui} of |X|.
Suppose f, g Î
homC(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(Ui)}
forms an open cover of Z Î
D, by the glueing lemma we can find an open effective subset
V of |Z| contained in some h-1(Ui)
such that the restrictions of 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 Ui
are different. Now suppose for any i we have
a morphism fi: Ui ®
Y such that the restrictions of fi
and fj to Ui Ç
Uj are the same. For any object Z Î
D and any morphism h: Z ®
X, let
{Vj}be an effective open cover of |Z|
such that h(Vj) Í
Ui for some Ui. Denote by hi
the restriction of h to Vi.
Then fihi and fjhj
agree on Vi Ç
Vj. By the glueing lemma for morphisms
from Z to Y these fihi determine
a morphism h': Z ®
Y. We obtain a map homC(Z,
X) ® homC(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
homC(X, Y) whose restriction to
each Ui is fi.
The uniqueness of f is obvious. This proves that (C, C)
is strict. n
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