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.
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.)
Last revised on November 27, 2022 at 05:50:55. See the history of this page for a list of all contributions to it.