# nLab finite category

Finite categories

category theory

# Finite categories

## Definition

A finite category $C$ is a category internal to the category FinSet of finite sets.

This means that a finite category consists of

(Notice that the latter implies the former, since for every object there is the identity morphism on that object). Finite categories form the 2-category, FinCat.

Similarly, a locally finite category is a category enriched over $Fin Set$, that is a category whose hom-sets are all finite.

(Locally) finite categories may also be called (locally) $\omega$-small; this generalises from $\omega$ (the set of natural numbers) to (other) inaccessible cardinals (or, equivalently, Grothendieck universes).

## Limits and colimits

One is often interested in whether an arbitrary category $D$ has limits and colimits indexed by finite categories. A category with all finite limits is called finitely complete or left exact (or lex for short). A category with all finite colimits is called finitely cocomplete or right exact.

Last revised on February 15, 2021 at 05:43:11. See the history of this page for a list of all contributions to it.