nLab completely distributive category

Definition

For locally small category $\mathcal{K}$ denote by $\mathcal{PK}$ the full subcategory of $[\mathcal{K}^{op},Set]$ on small colimits of representables. The category $\mathcal{K}$ is cocomplete iff the Yoneda embedding $Y \colon \mathcal{K}\to\mathcal{PK}$ has a left adjoint $colim \colon \mathcal{PK}\to\mathcal{K}$. If $colim$ has a further left adjoint $W \colon \mathcal{K}\to\mathcal{PK}$ and $\mathcal{K}$ is moreover complete, then it is called completely distributive.

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

• Completely distributive categories are an example of continuous algebras for a lax-idempotent 2-monad. (But the condition of being completely distributive seems to be stronger since it also requires completeness.)

References

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