#
nLab

cocomplete category

# Contents

## Definition

A category $C$ is **cocomplete** if it has all small colimits: that is, if every diagram

$F: D \to C$

where $D$ is a small category has a colimit in $C$.

The most natural morphisms between cocomplete categories are the cocontinuous functors.

- Dually, a category with all small limits is a complete category.
- A category $D$ is cocomplete if and only if $D^{op}$ is complete, so the abstract properties of cocompleteness mimic those of completeness.
- If a category has not all small colimits but all
*finite* colimits, then it is a finitely cocomplete category.

## Examples

Many familiar categories of mathematical structures are cocomplete: to name just a few examples, Set, Grp, Ab, Vect and Top are cocomplete.

Last revised on July 30, 2018 at 09:00:39.
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