cocomplete

In category theory, a category is cocomplete if it has all small colimits. That means for every small indexing category J and every functor D: J → C, the colimit of D exists in C. Equivalently, C has all small coproducts, coequalizers, pushouts, and, more generally, all small colimits.

Common cocomplete categories include Set, the category of abelian groups Ab, and the category Cat of small categories. In addition, presheaf categories such as Set^C^op are cocomplete for any small category C. If D is cocomplete, then the functor category [C, D] is cocomplete, with colimits computed pointwise; in particular, presheaf categories Set^C^op are cocomplete for any small C.

The notion of cocompleteness is central to the concept of cocompletion. For a small category C, the presheaf category [C^op, Set] is the free cocompletion of C, together with the Yoneda embedding y: C → [C^op, Set]. This embedding sends an object to its representable presheaf. The universal property states that for any cocomplete category D and any functor F: C → D, there exists a unique cocontinuous functor F̂: [C^op, Set] → D such that F̂ ∘ y ≅ F; thus F̂ is obtained by left Kan extension of F along y.

Cocompleteness is dual to completeness: C is cocomplete iff C^op is complete. Not all categories are cocomplete; for example, FinSet has only finite colimits and is not cocomplete because it lacks certain small colimits such as infinite coproducts.