On the 2-categories of weak distributive laws

TL;DR

This paper introduces a new 2-category framework for weak mixed distributive laws, generalizing Beck's classical laws by incorporating weak structures and establishing a fully faithful 2-functor under certain conditions.

Contribution

It defines a 2-category of weak mixed distributive laws and constructs a fully faithful 2-functor to a 2-category of 2x2 diagrams, broadening the understanding of distributive laws in 2-categories.

Findings

Weak mixed distributive laws are described as compatible pairs of monads and comonads.

A new 2-category of weak mixed distributive laws is constructed.

A fully faithful 2-functor to K^{2 x 2} is established under certain conditions.

Abstract

A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category K which admits Eilenberg-Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to K^{2 x 2}.