Limits of small functors ∗

For a small category K enriched over a suitable monoidal category V , the free completion of K under colimits is the presheaf category [K ,V ]. If K is large, its free completion under colimits is the V -category PK of small presheaves on K , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on PK . A fundamental construction in category theory is the category of presheaves [K op,Set] on a small category K . Among many other important properties, it is the free completion of K under colimits. If the category K is large, then the full presheaf category [K op,Set] is not the free completion of K under colimits; indeed it is not even a legitimate category, insofar as its hom-sets are not in general small. In some contexts it is more appropriate to consider not all the presheaves on K , but only the small ones: a presheaf F : K op → Set is said to be small if it is the left Kan extension of some presheaf whose domain is small. This is equivalent to F being the left Kan extension of its restriction to some small full subcategory of its domain, or equally to its being a small colimit of representables. The natural transformations between two small presheaves on K do form a small set, and so the totality of small presheaves on K forms a genuine category PK with small hom-sets. Furthermore, PK is in fact the free completion of K under colimits. Of course if K is small, then every presheaf on K is small, and so PK is just [K op,Set], but in general this is not the case. Although PK is the free completion of K under colimits, it does not have all the good properties of [K op,Set] for small K . For example it is not necessarily complete or cartesian closed. In this paper we study, among other things, when PK does have such good properties. In fact we work not just with ordinary categories, but with categories enriched over a suitable monoidal category V . Once again, if K is small then [K op,V ] is the free completion of K under colimits, but for large K this is no longer the case; the illegitimacy of [K op,V ] in that case is more drastic: it is not even a V -category. The free completion of K under colimits is the V -category PK of small presheaves on K , where once again a presheaf is small if it is the left Kan extension of some presheaf with small domain; and once again the two reformulations of this notion can be made. ∗Both authors gratefully acknowledge the support of the Australian Research Council.