nLab
small presheaf

Idea

A small presheaf on a category $C$ is a presheaf which is determined by a small amount of data. If $C$ is itself small, then every presheaf on $C$ is small, but this is no longer true when $C$ is large. In many cases, when $C$ is large, it is the small presheaves which seem to be more important and useful.

Definition

Let $C$ be a category which is locally small, but possibly large. A presheaf $F:C^{\mathrm{op}}\to \mathrm{Set}$ is small if it is the left Kan extension of some functor whose domain is a small category, or equivalently if it is a small colimit of representable functors.

Of course, if $C$ is itself small, then every presheaf is small.

Categories of small presheaves

We write $PC$ for the category of small presheaves on $C$. Observe that although the category of all presheaves on $C$ cannot be defined without the assumption of a universe, the category $PC$ can be so defined, using small diagrams in $C$ as proxies for small colimits of representable presheaves. Moreover $PC$ is locally small, and there is a Yoneda embedding $C\hookrightarrow PC$.

Of course, if $C$ is small, then $PC$ is the usual category of all presheaves on $C$.

Since small colimits of small colimits are small colimits, $PC$ is cocomplete. In fact, it is easily seen to be the free cocompletion of $C$, even when $C$ is not small. It is not, in general, complete, but we can characterize when it is (cf. Day–Lack).

Theorem

$PC$ is complete if and only if for every small diagram in $C$, the category of cones over that diagram has a small weakly terminal set, i.e. there is a small set of cones such that every cone factors through one in that set.

Corollary

If $C$ is either complete or small, then $PC$ is complete.

We also have:

Theorem

If $C$ and $D$ are complete, then a functor $F:C\to D$ preserves small limits if and only if the functor $PF:PC\to PD$ (induced by left Kan extension) also preserves small limits.

These results can all be generalized to enriched categories, and also relativized to limits in some class $\Phi$ (which, for some purposes, we might want to assume to be “saturated”). See the paper by Day and Lack.

References

• Day and Lack, “Limits of small functors”, JPAA, http://www.maths.usyd.edu.au/u/stevel/papers/small.pdf

• Some work of Chorny and Dwyer: http://wwwmaths.anu.edu.au/~chorny/personal/research/ and also earlier minipreprint version http://www.math.uwo.ca/~bchorny2/research/smalldiag.ps the results of Chorny–Dwyer is cited by Rosicky in Accessible categories and homotopy theory http://www.math.yorku.ca/~tholen/HB07Rosicky.pdf