nLab small category

Definition

A category is small if it has a small set of objects and a small set of morphisms.

In other words, a small category is an internal category in the category Set.

A category which is not small may be called large, especially when it is not essentially small (see below).

Properties

Small categories are free of some of the subtleties that apply to large categories.

A category is said to be essentially small (or, rarely, svelte) if it is equivalent to a small category. Assuming the axiom of choice, this is the same as saying that it has a small skeleton, or equivalently that it is locally small and has a small number of isomorphism classes of objects.

A small category structure on a locally small category $C$ is an essentially surjective functor from a set (as a discrete category) to $C$. A category is essentially small iff it is locally small and has a small category structure; unlike the previous paragraph, this result does not require the axiom of choice.

Characterisations

The following are equivalent for a locally small category $B$ (see the linked MathOverflow answer).

1. $B$ is essentially small.
2. The category of presheaves $[B^{op}, Set]$ is locally small.
3. For every presheaf $q : B^{op} \to Set$ and copresheaf $p : B \to Set$ the coend $\int^{y \in B} py \times qy$ is small.
4. Every presheaf on $B$ is small.
5. For every functor $F : B \to C$ with locally small codomain, and for every object $c \in C$, the presheaf $C(F{-}, c) : B^{op} \to Set$ is small.

These different characterisations are useful, because they give a way to capture the notion of size in formal category theory. For instance, characterisation (2) is axiomatised in the formalism of small objects in Yoneda structures; whereas characterisation (5) is axiomatised in the formalism of petit objects in KZ-doctrines (see DL23).

Smallness in the context of universes

If Grothendieck universes are being used, then for $U$ a fixed Grothendieck universe, a category $C$ is $U$-small if its collection of objects and collection of morphisms are both elements of $U$. Thus,

• a $U$-small category is a category internal to $U Set$.

This of course is a material formulation. We may call $C$ structurally $U$-small if there is a bijection from its set of morphisms to an element of $U$ (the same for the set of objects follows). This gives an up-to-isomorphism version of $U$-smallness (see universe in a topos for an alternative structural formulation). Such structural $U$-smallness may be substituted in the discussion below.

Let $U\Set$ be the category of $U$-small sets. Similar considerations lead us to say

• a $U$-category (a locally U-small)-category is a category enriched over $U Set$,

and that a category $C$ essentially $U$-small if it is locally $U$-small and admits an essentially surjective functor from a discrete $U$-small category.

A category is $U$-moderate if its set of objects and set of morphisms are both subsets of $U$. However, some categories (such as the category of $U$-moderate categories!) are larger yet.

Related concepts

• small category, locally small category, complete small category

• essentially small (∞,1)-category

• small site

• large category

References

• Ross Street and Robert Walters, Yoneda structures on 2-categories, Journal of Algebra, Vol. 50, No. 2, 1978, pp. 350-379. [doi:10.1016/0021-8693(78)90160-6]

• Peter Freyd and Ross Street, On the Size of Categories, Theory and Applications of Categories, Vol. 1, No. 9, 1995, pp. 174-181. [TAC]

• Ivan Di Liberti, Fosco Loregian, Accessibility and Presentability in 2-Categories, Journal of Pure and Applied Algebra 227 1 (2023) [arXiv:1804.08710, doi:10.1016/j.jpaa.2022.107155]

• Seerp Roald Koudenburg, Formal category theory in augmented virtual double categories, Theory and Applications of Categories 41.10 (2024): 288-413.

• Seerp Roald Koudenburg, answer to “Is every petite category essentially small?”: MathOverflow