small presheaf


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


Let CC be a category which is locally small, but possibly large. A presheaf F:C opSetF\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 CC is itself small, then every presheaf is small.

Categories of small presheaves

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

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

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


PCP C is complete if and only if for every small diagram in CC, 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.


If CC is either complete or small, then PCP C is complete.

We also have:


If CC and DD are complete, then a functor F:CDF\colon C\to D preserves small limits if and only if the functor PF:PCPDP 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 Φ\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,

Revised on April 11, 2016 15:08:59 by Zoran Škoda (