Distributive Laws for Relative Monads

Abstract

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category \(\mathcal {K}\). In order to do that, we introduce the 2-category of relative monads in a 2-category \(\mathcal {K}\) with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in \(\mathcal {K}\) defined by Street. Using this perspective, we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg–Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories.