nLab completely distributive category

Definition

A locally small category $\mathcal{K}$ is completely distributive if it has small colimits, small limits, and the colimit functor $colim \colon \mathcal{PK} \to \mathcal{K}$ is continuous.

More abstractly, there is a pseudodistributive law between the free cocompletion and the free completion, for which the pseudoalgebras are the completely distributive categories.

Examples

• Every totally distributive category, in particular every presheaf category $[C^{op},Set]$ for small $C$ is completely distributive.

Related pages

• totally distributive category

• completely distributive lattice

References

• Francisco Marmolejo, Bob Rosebrugh, and Richard Wood, Completely and totally distributive categories I, JPAA 216 no. 8-9 (2012). PDF