nLab cocomplete category

Definition

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

$$F: D \to C$$

where $D$ is a small category has a colimit in $C$. Equivalently, a category $C$ is cocomplete if it has all small wide pushouts and an initial object.

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

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.

For a small category $C$, the presheaf category $[C^{op},Set]$ is cocomplete, and the Yoneda embedding exhibits it as the free cocompletion of $C$.

Related concepts

• complete category

• bicomplete category

• M-complete category