Let C be a category and a covariant functor from C to a category of topological spaces. We are interested in the geometry of C imposed by . In practice the functor always comes from the underlying topology of the objects, therefore should be regarded as an invariant of C. We introduce the concept of a metric site, which is a pair (C, ) satisfying certain categoricalgeometric properties. A metric site has enough local isomorphisms (called effective morphisms), so that the open effective covers generate a Grothendieck topology (in the sense of [SGA4]). Virtually all the major categories arising in modern geometry are metric sites, and many results obtained by Grothendieck in [EGAI] for schemes can be reformulated and proved for the objects of any metric site. Therefore we believe it is worth to develop a general theory of metric sites, which will provide a categorical foundation for modern geometry (see [Luo 1998]). In a metric site we are able to glue objects directly
in categorical terms (instead of "glueing charts"
in differential geometry, or "glueing sheaves" in algebraic geometry).
We study the phenomenon associated with the glueing
procedures in º2, introducing the concepts of
strict and complete metric sites. As in the theory of metric spaces, afundamental
fact in this direction is the following theorem concerning the completion
of a metric site (see º3):
We mention another approach to the theory of metric sites.
Recall that a frame is a complete lattice with infinite
distributive law, and a morphism of frames is a map of lattices
preserving finite meets and arbitrary joints (see [Johnstone
1982]). We define a framed site to be a category together
with a function which assigns to each object a frame of sieves in a continuous
way. For instance, every elementary topos is naturally a framed site (assigning
to each object t he frame of its subobjects). Using the language of framed
sites we can give an intrinsic definition for a sober metric site (see
[Luo 1995b]).
