nLab
small category

Contents

Context

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 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 CC is an essentially surjective functor from a set (as a discrete category) to CC. 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.

Smallness in the context of universes

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

  • a UU-small category is a category internal to USetU Set.

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

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

and that a category CC essentially UU-small if it is locally UU-small and admits an essentially surjective functor from a discrete UU-small category.

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

Last revised on January 19, 2020 at 13:41:48. See the history of this page for a list of all contributions to it.