Contents

category theory

foundations

# Contents

## 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.

## References

Last revised on July 6, 2024 at 09:44:06. See the history of this page for a list of all contributions to it.