nLab finite category

Definition

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

This means that a finite category consists of

• a finite set of objects;
• a finite set of morphisms.

(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.

In dependent type theory

In dependent type theory, there are multiple notions of category. Because the type of objects in a category is required to be a finite set, all finite categories are strict categories, and the only finite univalent categories are the finite gaunt categories.

Related concepts

• finite set

• finite graph

• homotopy type with finite homotopy groups

• finite homotopy type

• finite (∞,1)-category

• locally finite type