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Topos Theory

topos theory



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In higher category theory




A site is a presentation of a sheaf topos as a structure freely generated under colimits from a category, subject to the relation that certain covering colimits are preserved.

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.



A site (C,J)(C,J) is a category CC equipped with a coverage JJ.

For \mathcal{E} a topos equipped with an equivalence of categories

Sh(C,J) \mathcal{E} \simeq Sh(C,J)

to the sheaf topos over a site, one says that (C,J)(C,J) is a site of definition for \mathcal{E}.

Some classes of sites have their special names


A site is called

The term standard site appears in (Johnstone, example A2.1.11).


Often a site is required to be a small category. But also large sites play a role.


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 PSh(S)PSh(S).


For (C,J)(C,J) a site, we write Sh J(C)Sh_J(C) for the category of sheaves on CC with respect to the coverage JJ.


Morita equivalent sites

Many inequivalent sites may have equivalent sheaf toposes.


Every sheaf topos has a standard site of definition.

This appears as (Johnstone, theorem C2.2.8 (iii)).


By this corollary at classifying topos this means that every sheaf topos is the classifying topos for some theory of local algebras.

Subcanonical sites


For \mathcal{E} a sheaf topos, the essentially small sites of definition (𝒞,J)(\mathcal{C}, J) of \mathcal{E} such that JJ is a subcanonical coverage are precisely the full subcategories on generating families of objects equipped with the coverages induced from the canonical coverage of \mathcal{E}.

This appears as (Johnstone, prop. C2.2.16).


Classes of sites

Other classes of sites are listed in the following.

Specific sites



sites are discussed in section C2.1.

Last revised on August 4, 2016 at 07:27:42. See the history of this page for a list of all contributions to it.