Let be a category with a pretopology (i.e. a site) and an object of . As an analogy with sheaves on a topological space , which are defined on the site of open sets of , we can try to define sheaves on , using the elements of covering families of from . This is called the little site of , in contrast to the big site of which is the slice category with its induced topology.
A little site may sometimes be called a small site, but it's probably best to save that name for a site which is a small category.
David Roberts: The following is experimental, use at own risk, although I’m sure it has been thought about before.
Consider the subcategory of with objects such that is a member of some covering family . Given two such objects , , and covering families , that contain them, there is a covering family which is the pullback (or at least a weak pullback) of and in . There is then some element of such that there is a square
so is ‘a bit like’ the category of opens of a space (it’s probably cofiltered, but I haven’t checked that there are weak equalisers).
Now the morphisms of are those triangles
such that is an element of a covering family of , so the arrows and really are morphisms of . Then we say a covering family of is a collection of triangles that, when we forget the maps to , form a covering family of in . This is at the very least a coverage, and so we have a site.
To be continued…