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).

Sheaves on a large site

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 way to deal 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. The idea is that for sheaves and in particular for any homotopy theory of sheaves the choice of these cardinality bounds is “inessential.”

See for instance p. 2 of Jardine, Fields Lectuere: Simplicial presheaves where this issue arises in the study of simplicial presheaves and the model structure on simplicial presheaves.

Can any of you size-issue experts help to clarify this?

Mike: I wish. I added some stuff, but I still don’t really understand this business. In particular I don’t really know what is meant by “inessential.” It certainly seems unlikely that you would get equivalent homotopy theories, but it does seem likely that you would get similar behavior no matter where you draw the line. And if all you care about is, say, having a good category of sheaves in which you can embed any particular space or manifold you happen to care about, then that may be good enough. But I don’t really know what the goal is of considering such large sites.