As such, sites generalise topological spaces and locales, which present localic sheaf toposes. More precisely, sites generalise and categorify posites, which present localic toposes but also present locales themselves in a decategorified manner.
In technical terms, a site is a small category equipped with a coverage or Grothendieck topology. The category of sheaves over a site is a sheaf topos and the site is a site of definition for this topos.
For a topos equipped with an equivalence of categories
to the sheaf topos over a site, one says that is a site of definition for .
Some classes of sites have their special names
A site is called
Every coverage induces a Grothendieck topology. Often sites are defined to be categories equipped with a Grothendieck topology. Some constructions and properties are more elegantly handled with coverages, some with Grothendieck topologies.
Notice that there are many equivalent ways to define a Grothendieck topology, for instance in terms of a system of local isomorphisms, or in terms of a system of dense monomorphisms in the category of presheaves .
Many inequivalent sites may have equivalent sheaf toposes.
This appears as (Johnstone, theorem C2.2.8 (iii)).
For a sheaf topos, the essentially small sites of definition of such that is a subcanonical coverage are precisely the full subcategories on generating families of objects equipped with the coverages induced from the canonical coverage of .
This appears as (Johnstone, prop. C2.2.16).
Various categories come with canonical structures of sites on them:
For every category there is its canonical coverage.
Generalizing the previous two examples, on an κ-ary regular category there is a -canonical coverage.
Other classes of sites are listed in the following.
sites are discussed in section C2.1.