A category is cocomplete if it has all small colimits: that is, if every diagram
where is a small category has a colimit in .
The most natural morphisms between cocomplete categories are the cocontinuous functors.
- Dually, a category with all small limits is a complete category.
- A category is cocomplete if and only if 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.
Revised on January 24, 2017 15:33:58
by Max New?