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Topos Theory

topos theory

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Idea

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.

Definition

Definition

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

Definition

A site is called

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

Remark

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

Remark

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).

Definition

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.

Properties

Morita equivalent sites

Many inequivalent sites may have equivalent sheaf toposes.

Proposition

Every sheaf topos has a standard site of definition.

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

Remark

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

Subcanonical sites

Proposition

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).

Examples

Classes of sites

Other classes of sites are listed in the following.

Specific sites

References

In

sites are discussed in section C2.1.

Revised on January 5, 2014 04:46:37 by heisenbug? (84.148.59.172)