After recalling a few fundamental notions about V-Cat, V-Rel and V-Dist, where V is a quantale, we will present two new characterizations of the submonads of the presheaf monad: one in terms of a special class of V-distributors; and another as those monads which are fully (BC)*, lax idempotent and satisfy certain fully faithfulness conditions. By (BC)* we mean a new Beck-Chevalley type condition which gives us an interaction between V-Cat and V-Dist, analogous to the interaction between Set and Rel given by the usual Beck-Chevalley condition on Set. The algebras for these submonads are the V-categories satisfying a condition involving their multiplication. In the special case of the formal ball monad, the algebras can also be seen as the V-categories with a particular class of weighted colimits.

(This talk is based on joint work with Maria Manuel Clementino.)