Eilenberg-Moore Bicategories for Opmonoidal Pseudomonads

TL;DR

This paper investigates how monads interact with monoidal structures in a 2-category setting, providing conditions for lifting structures and extending results to braided and symmetric cases, introducing the Gray-tensor product of pseudomonads.

Contribution

It introduces the Gray-tensor product of pseudomonads and establishes conditions for lifting monoidal structures to Eilenberg-Moore pseudoalgebras in a 2-categorical context.

Findings

Conditions for lifting monoidal structures to pseudoalgebras

Extension of results to braids, syllapses, and symmetries

Introduction of Gray-tensor product of pseudomonads

Abstract

We analyse compatibility between monads and monoidal structures in the two-dimensional setting. We describe sufficient conditions for monoidal structures to lift to the Eilenberg-Moore pseudoalgebras. We then extend these results to braids, syllapses and symmetries. To achieve these results we define the Gray-tensor product of pseudomonads, and examine its interaction with the Eilenberg-Moore construction.