# 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 $C$.

# Remarks

• 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.

Revised on January 28, 2014 11:41:05 by Adeel Khan (77.9.213.189)