nLab
small category

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 is called large.

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. CC is essentially 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 condition is non-evil.

So let USetU\Set be the category of UU-small sets. Then

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.

Revised on February 7, 2014 06:34:05 by Urs Schreiber (89.204.153.37)