Kleisli enriched

AbstractFor a monad S on a category K whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V-(bi)modules (=profunctors, distributors), for some quantale V. The assignment is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative and unital) quantale V the prototypical example of a modular monad, namely the presheaf monad on V-Cat, the category of (small) V-categories. We discuss in particular the question whether the Hausdorff monad on V-Cat is modular