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 $F\colon C^{op}\to Set$ 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 (see Proposition 4.83 of Kelly).

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.

References

• 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]

• B. J. Day, Steve Lack, Limits of small functors, JPAA, pdf

• Boris Chorny, William Dwyer, The homotopy theory of small diagrams over large categories, Forum Math. 21 (2009), no. 2, 167–179; preprint

• Paolo Perrone, Walter Tholen, Kan extensions are partial colimits, Applied Categorical Structures, 2022. (arXiv:2101.04531)

The results of Chorny–Dwyer are cited by Rosický in

• Jiří Rosický, Accessible categories and homotopy theory, lecture notes for the Summer School on Contemporary Categorical Methods in Algebra and Topology (2007) pdf

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:

• Jiří Adámek, V. Koubek, and V. Trnková. How large are left exact functors?. Theory and Applications of Categories 8.13 (2001): 377-390. (pdf)