Completely and totally distributive categories I

In 1978, Street and Walters defined a locally small category K to be totally cocomplete if its Yoneda functor Y has a left adjoint X. Such a K is totally distributive if X has a left adjoint W. Small powers of the category of small sets are totally distributive, as are certain sheaf categories. A locally small category K is small cocomplete if it is a P-algebra, where P is the small-colimit completion monad on . In 2007, Day and Lack showed that P lifts to ℛ-algebras, where ℛ is the small-limit completion monad on . It follows that there is a distributive law and we say that K is completely distributive if K is a Pℛ-algebra, meaning that K is small cocomplete, small complete, and preserves small limits. Totally distributive implies completely distributive. We show that there is a further supply of totally distributive categories provided by categories of interpolative bimodules between small taxons as introduced by Koslowski in 1997.