nLab cocomplete category



Category theory

Limits and colimits



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. Equivalently, a category CC is cocomplete if it has all small wide pushouts and an initial object.

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


  • 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.


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

The 2-category of cocomplete categories



  • Andrew M. Pitts, On product and change of base for toposes , Cah. Top. Géom. Diff. Cat. XXVI no.1 (1985) pp.43-61.

Last revised on February 17, 2024 at 11:03:09. See the history of this page for a list of all contributions to it.