[[Category theory MOC]]
# Completeness and cocompleteness

A [[category]] $\cat C$ is said to be #m/def/cat

- **finitely (co)complete** iff for every [[finite category]] $\cat J$,
  $\cat C$ has all $\cat J$-[[Limits and colimits|(co)limits]].
- **(co)complete** iff for every [[small category]] $\cat J$, 
  $\cat C$ has all $\cat J$-[[Limits and colimits|(co)limits]].

## Properties

1. [[Finite (co)completeness criterion]]

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