nLab
cocomplete category

Contents

Context

Category theory

Limits and colimits

Contents

Definition

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

F:DC F: D \to C

where DD is a small category has a colimit in CC.

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 DD is cocomplete if and only if D opD^{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.

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

Last revised on May 26, 2020 at 12:31:05. See the history of this page for a list of all contributions to it.