Contents

category theory

topos theory

Contents

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 $F\colon C^{op}\to Set$ 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 $P C$ 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 $P C$ can be so defined, using small diagrams in $C$ as proxies for small colimits of representable presheaves. Moreover $P C$ is locally small, and there is a Yoneda embedding $C\hookrightarrow P C$.

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

Since small colimits of small colimits are small colimits, $P C$ 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

$P C$ 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 $P C$ is complete.

We also have:

Theorem

If $C$ and $D$ are complete, then a functor $F\colon C\to D$ preserves small limits if and only if the functor $P F\colon P C \to P D$ (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.

The results of Chorny–Dwyer are cited by Rosicky in Accessible categories and homotopy theory, <http://www.math.yorku.ca/~tholen/HB07Rosicky.pdf>