cocomplete category



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


  • 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.

Last revised on July 30, 2018 at 09:00:39. See the history of this page for a list of all contributions to it.