nLab site

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 $(C,J)$ is a category $C$ equipped with a coverage $J$.

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

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

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

Some classes of sites have their special names

Definition

A site is called

• a small site, large site, essentially small site if the underlying category is a small category, large category, essentially small category, respectively;

• a cartesian site if the underlying category is finitely complete (which the Elephant calls a cartesian category);

• a standard site if it is a cartesian site equipped with a subcanonical coverage.

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 $PSh(S)$.

Definition

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

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 $(\mathcal{C}, J)$ of $\mathcal{E}$ such that $J$ 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

• Every frame is canonically a site, where $U$ is covered by $\{U_i\}$ precisely if it is their join.

  A subclass of examples is the category of open subsets of a topological space.

  This are examples of posites/(0,1)-site.

• Various categories come with canonical structures of sites on them:

  • For every category $C$ there is its canonical coverage.

  • On every regular category there is its regular coverage.

  • On every coherent category there is its coherent coverage.

  • Generalizing the previous two examples, on an ∞-ary regular category there is a $\kappa$-canonical coverage.

  If the category in question is the syntactic category of a theory, the corresponding site is the syntactic site.

• For every site there is the corresponding double negation topology that forces the sheaf topos to a Boolean topos.

Other classes of sites are listed in the following.

• big site

• dense sub-site

• large site, essentially small site

• concrete site

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site

• atomic site

Specific sites

• Sites for big toposes defining notions of geometry are:

  • The sites that define the geometry called differential geometry are CartSp${}_{smooth}$, SmoothMfd, etc, equipped with the open cover coverage. Or more generally smooth loci etc.

  • The sites that induce topological geometry? are small versions of Top equipped with the open cover coverage.

  • The sites that induce the higher geometry modeled on Euclidean topology are the large site of paracompact manifolds and its dense sub-site CartSp${}_{top}$.

• The sites that define the geometry called algebraic geometry are site structures on categories of formal duals of commutative rings or commutative associative algebras

  • fpqc-site$\to$ fppf-site $\to$ syntomic site $\to$ étale site $\to$ Nisnevich site $\to$ Zariski site

    crystalline site

Related concepts

• posite

• site

  • internal site
  • ∞-ary site

• 2-site, (2,1)-site

• (∞,1)-site

  • model site, simplicial site

• morphism of sites, covering lifting property

References

In

• Peter Johnstone, Sketches of an Elephant

sites are discussed in section C2.1.