nLab
finite category
Finite categories
Context
Category theory
category theory

Concepts
Universal constructions
Theorems
Extensions
Applications
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).

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

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

Limits and colimits
One is often interested in whether an arbitrary category $D$ has limit s and colimit s 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 January 18, 2018 at 07:05:18.
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