In a previous paper [Luo 1995a] we presented a general theory of metric sites emphasizing its geometric applications. A metric site defined in that paper is a pair (C, t) of a category C and a metric functor t from C to the category Top of topological spaces. Replacing Top by the category Loc of locales, we obtain the notion of a framed site. In º0 - º2 we present an intrinsic approach to the theory of framed sites using sieves. Many notions and definitions introduced in [Luo 1995a] for metric sites can be extended to framed sites in a straightforward fashion. A key theorem concerning the Kan extension of a framed topology G is proved in º3, which is then applied to prove that any strict framed site has a completion in º4.
Metric sites and Grothendieck toposes provide two important classes
of framed sites. A metric site (C, t)
determines a spatial framed site (C, O) where O: C
® Loc is the composition of t
with the canonical functor Top ®
Loc. Since Top is also a framed site, formally we can define
a metric site to be an abstract framed site C together with a bicontinuous
functor t from C to the framed site Top.
An advantage of switching to (C, O) is that the theory of
Kan extension for O is very simple. On the other hand the subobject
classifier of a topos E with colimits determines a functor vE:
E ® Loc such that (E,
vE) is a complete canonical
framed site (Theorem 9.3).
Another type of framed sites comes from Loc itself. Any category
C with colimits determines a complete framed site Loc(C),
consisting of the locales (A, OA) over C,
where A is a locale and OA is a strict functor
from A to C. It is a geometric closure of C in the
sense that any strict framed site (C, G)
can be embedded in Loc(C). Thus Loc(C) contains
a completion of (C, G). This geometric
approach to the theory of complete framed sites is covered in º5 -