nLab large site

Idea

Often sites are required to be small categories (small sites). If not, if one has a large site, there are complications.

Many of the good properties of sheaves depend on such smallness. To begin with, even the category of sheaves may have to be extra-large, but there are other issues, such as:

• the existence of sheafification,
• the existence of colimits in the category of sheaves,
• cartesian closedness of the category of sheaves, and
• the category of sheaves being a topos.

However, for many purposes it is desirable to consider the notion of sheaves on large sites.

Small (dense) sub-sites

In some cases, sheaves on a large site can be identified with sheaves on some small full sub-site, for instance a dense sub-site. A large site with a small dense sub-site is called an essentially small site.

For example, if $C$ is a Grothendieck topos with its canonical coverage, then every sheaf on $C$ is representable, so $C\simeq Sh(C)$; thus $Sh(C)$ is equivalent to the category of sheaves on some small site (a defining site for $C$ itself).

Small subsites of large sites

On the other hand, one sometimes wants to consider sheaves on large categories such as Top or Diff, which are certainly not Grothendieck toposes. One idea for dealing with this is to consider full subcategories of these large categories on objects whose size is bounded by some large (in the non-technical sense) cardinal number $\kappa$. In an extreme case, $\kappa$ could be an inaccessible cardinal.

Although the resulting subsite is often not a dense subsite, for sufficiently large $\kappa$ the difference may be imperceptible when using constructions typically employed in practice, such as limits and colimits of diagrams of size less than $\kappa$, subobjects and quotient objects, etc.

For an example, see the definition of the small site of schemes in the Stacks Project.

Further discussion can be found in Jardine 2007 p. 2, where this issue arises in the study of simplicial presheaves and the model structure on simplicial presheaves.

Brief discussion of this idea may be found here.

Related concepts

• WISC

References

• J. F. Jardine: Fields Lectures: Simplicial presheaves, notes (2007) [pdf,pdf, webpage]