A small presheaf on a category is a presheaf which is determined by a small amount of data. If is itself small, then every presheaf on is small, but this is no longer true when is large. In many cases, when is large, it is the small presheaves which seem to be more important and useful.
Let be a category which is locally small, but possibly large. A presheaf 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 (see Proposition 4.83 of Kelly).
Of course, if is itself small, then every presheaf is small.
We write for the category of small presheaves on . Observe that although the category of all presheaves on cannot be defined without the assumption of a universe, the category can be so defined, using small diagrams in as proxies for small colimits of representable presheaves. Moreover is locally small, and there is a Yoneda embedding .
Of course, if is small, then is the usual category of all presheaves on .
Since small colimits of small colimits are small colimits, is cocomplete. In fact, it is easily seen to be the free cocompletion of , even when is not small. It is not, in general, complete, but we can characterize when it is (cf. Day–Lack).
is complete if and only if for every small diagram in , 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 is either complete or small, then is complete.
We also have:
If and are complete, then a functor preserves small limits if and only if the functor (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 (which, for some purposes, we might want to assume to be “saturated”). See the paper by Day and Lack.
Max Kelly, Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]
Brian J. Day, Stephen Lack, Limits of small functors, Journal of Pure and Applied Algebra 210:3 (2007), 651-663. doi, arXiv, pdf.
Boris Chorny, William Dwyer, The homotopy theory of small diagrams over large categories, Forum Math. 21:2 (2009), 167–179. doi, arXiv
Paolo Perrone, Walter Tholen, Kan extensions are partial colimits, Applied Categorical Structures 30:4 (2022), 685-753. (arXiv:2101.04531, doi)
The results of Chorny–Dwyer are cited by Rosický in
See also
Georg Biedermann, Boris Chorny, Duality and small functors, Algebraic & Geometric Topology 15 (2015) 2609–2657. doi
Boris Chorny, David White, A variant of a Dwyer-Kan theorem for model categories (v1: Homotopy theory of homotopy presheaves), Algebraic & Geometric Topology 24:4, 2185–2208 arXiv:1805.05378
On when finitely continuous presheaves are small:
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