nLab
finite category

Finite categories

Definition

A finite category CC 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).

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

(Locally) finite categories may also be called (locally) ω\omega-small; this generalises from ω\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 DD 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.

Revised on February 2, 2014 06:01:01 by Urs Schreiber (82.113.99.28)