Distributive laws for relative monads

TL;DR

This paper introduces the concept of relative distributive laws between relative monads and monads within any 2-category, extending classical monad theory and establishing foundational theorems and examples.

Contribution

It defines relative distributive laws in 2-categories, relates them to existing monad frameworks, and proves Beck-type theorems for this new concept.

Findings

Defined relative distributive laws in 2-categories

Proved two Beck-type theorems for these laws

Provided examples in the 2-category of locally small categories

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. Thanks to this view 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.