Definition 2.1.1. A metric site (or simply site) is a separable, locally effective presite (C, t). The functor t is called a metric topology on C. Suppose (C, t) and (C', t) are two metric sites. An isometry of sites from (C, t) to (C', t') is an isometry of presites j: C ® C' such that j(f) is effective for any effective morphism f in C; j is called an isometric embedding if the functor j is faithful, full, and C is equivalent to its image in C'. Suppose (C, t) is a metric site, B a subcategory of C. If the subpresite (B, t_{B}) is a metric site and any open effective morphism f: X ® Y in (B, t_{B}) is an effective morphism in C, we say (B, t_{B}) (or simply B) a subsite of (C, t). If B is a subsite of C the inclusion functor I: B ® C is an isometry of sites. Definition 2.1.2. A metric
site C is strict (or subcanonical)
if the following glueing lemma for morphisms
holds:
Definition 2.1.3.
A Cauchycomplete metric site is a
strict, effective metric site C such that the following glueing
lemma for objects holds:
We say C is a finitely Cauchycomplete metric site if (2.1.3) holds for any finite family {X_{i}} of objects. Example 2.1.4. If C is a strict site and B a full subsite of C, then B is a strict site. Example 2.1.5. All the basic presites (resp. basic algebraic presites) are separable and locally effective, hence are metric sites, which will be called the basic sites (resp. basic algebraic sites) from now on. The glueing lemma for morphisms (2.1.2) holds for all these sites. Thus they are strict. Example 2.1.6. The glueing lemma for objects (2.1.3) holds for all the basic sites and the sites Sch, GSch, PVar/k, hence they are Cauchycomplete metric sites. 2.1.7 Suppose C' is a strict site
and C a full subsite of C'. We introduce three subsites of
C':
2.1.8 Clearly C is a subsite of G(C), F(C) and E(C). If C' = G(C) (resp. C' = F(C), resp. C' = E(C)) then we say that C is a Cauchybase (resp. finite Cauchybase, resp. effective base) of C'. Definition 2.1.9. Suppose C is a strict metric site. A Cauchycompletion (resp. finite Cauchycompletion, resp. effective completion) of C is a Cauchycomplete (resp. finite Cauchycomplete, resp. standard and effective) metric site C' containing C as a Cauchybase (resp. finite Cauchybase, resp. effective base). Remark 2.1.10. Suppose C' is a Cauchycomplete (finitely Cauchycomplete, resp. effective) metric site and C a full subsite of C'. Then G(C) (resp. F(C), resp. E(C)) is a Cauchycompletion (resp. finite Cauchycompletion, resp. effective completion) of C. Example 2.1.11. (a) Sch is the
Cauchycompletion of ASch in LSp.
Proposition 2.1.12. Suppose C' is a strict metric site and C is a Cauchybase (resp. finite Cauchybase, resp. effective base) for C'. Suppose D is a Cauchycomplete (resp. finitely Cauchycomplete, resp. effective) metric site. Then any full isometric embedding of metric sites j from C to D can be extended (noncanonically) to a full isometric embedding from C' to D. A Cauchycompletion (resp. finite Cauchycompletion, resp. effective completion) of a strict metric site is unique up to equivalence. Proof. We treat the case of Cauchycomplete sites. The other
cases are similar. Suppose {U_{i}U_{i}
Î C} is an effective open cover
of an object X Î C'. Put
U_{ij} = U_{i} Ç
U_{j}. Then we can glue {j(U_{i})}
along {j(U_{ij})} to obtain
an object j(X) Î
D. Suppose f: Y ® X
is a morphism in C' and {V_{j}V_{j}
Î C} is an effective open cover
of Y such that f(V_{j}) Í
U_{i} for some i. Let f_{j}: V_{j}
® U_{i} be the restriction
of f to V_{j}; then f_{j} is
a morphism in C. Glueing these j(f_{j})
we obtain a morphism j(f): j(Y)
® j(X).
We thus obtain an extension of j on C'
which is an isometric embedding. The last assertion follows from the first
one.
