Could not include topos theory - contents
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.