The Eilenberg-Moore category and a Beck-type theorem for a Morita context

TL;DR

This paper introduces a new framework connecting pairs of monads via Morita contexts, constructs their Eilenberg-Moore categories, and establishes conditions under which these categories are equivalent, enriching the theory of monad interactions.

Contribution

It defines Morita contexts for pairs of monads, constructs their Eilenberg-Moore categories, and proves a Beck-type theorem relating these contexts to double adjunctions.

Findings

Morita contexts induce equivalences between categories of algebras in many cases.

A construction of the Eilenberg-Moore category for a Morita context is provided.

Conditions for the comparison functor to be an equivalence are established.

Abstract

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\em double adjunction} to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the {\em moritability} of a pair of functors with a common domain) are derived.