nLab cocomplete category

Definition

A category $C$ is cocomplete if it has all small colimits: that is, if every small diagram $F: D \to C$ where $D$ is a small category has a colimit in $C$. Equivalently, a category $C$ is cocomplete if it has all small wide pushouts and an initial object.

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

Properties

• A category is cocomplete if and only if it has small coproducts and reflexive coequalisers.

• Dually, a category with all small limits is a complete category.

• A category $D$ is cocomplete if and only if $D^{op}$ 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.

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

The 2-category of cocomplete categories

Properties

• The 2-category of cocomplete categories is 2-monadic over $Cat$, and complete and cocomplete as a 2-category. (See this MathOverflow answer by Mike Shulman.)
• The copower $C \cdot A$ where $C$ is a category and $A$ is a cocomplete category, is given by $[C^{op}, A]$. The power $C \pitchfork A$ is given by $[C, A]$ (see Pitts 1985).

Related concepts

• complete category

• bicomplete category

• M-complete category

References

• J.-M. Maranda, Some remarks on limits in categories. Canadian Mathematical Bulletin 5 2 (1962) 133-146 [doi:10.4153/CMB-1962-015-0]
• Ernest Manes, A triple miscellany: Some aspects of the theory of algebra over a triple, PhD thesis, Wesleyan University, 1967. [pdf]
• Andrew M. Pitts, On product and change of base for toposes , Cah. Top. Géom. Diff. Cat. XXVI no.1 (1985) pp.43-61.