The formal theory of multimonoidal monads

TL;DR

This paper extends Street's formal theory of monads to multimonoidal monads within symmetric strict monoidal 2-categories, introducing a new framework for understanding complex monad structures with multiple monoidal and opmonoidal components.

Contribution

It develops a comprehensive theory of $(p,q)$-oidal monads in symmetric strict monoidal 2-categories, generalizing previous work and providing new insights into their structure and Eilenberg-Moore objects.

Findings

Defines $(p+q)$-oidal objects with compatible structures

Shows how $(p,q)$-oidal monads induce $(p+q)$-oidal structures on Eilenberg-Moore objects

Provides a new proof of recent results in the case of $ ext{Cat}$

Abstract

Certain aspects of Street's formal theory of monads in 2-categories are extended to multimonoidal monads in symmetric strict monoidal 2-categories. Namely, any symmetric strict monoidal 2-category $\mathcal M$ admits a symmetric strict monoidal 2-category of pseudomonoids, monoidal 1-cells and monoidal 2-cells in $\mathcal M$. Dually, there is a symmetric strict monoidal 2-category of pseudomonoids, opmonoidal 1-cells and opmonoidal 2-cells in $\mathcal M$. Extending a construction due to Aguiar and Mahajan for $\mathcal M=\mathsf{Cat}$, we may apply the first construction $p$-times and the second one $q$-times (in any order). It yields a 2-category $\mathcal M_{pq}$. A 0-cell therein is an object $A$ of $\mathcal M$ together with $p+q$ compatible pseudomonoid structures; it is termed a $(p+q)$-oidal object in $\mathcal M$. A monad in $\mathcal M_{pq}$ is called a $(p,q)$-oidal monad in $\mathcal M$; it is a monad $t$ on $A$ in $\mathcal M$ together with $p$ monoidal, and $q$ opmonoidal structures in a compatible way. If $\mathcal M$ has monoidal Eilenberg-Moore construction, and certain (Linton type) stable coequalizers exist, then a $(p+q)$-oidal structure on the Eilenberg-Moore object $A^t$ of a $(p,q)$-oidal monad $(A,t)$ is shown to arise via a symmetric strict monoidal double functor to Ehresmann's double category $\mathbb S\mathsf{qr} (\mathcal M)$ of squares in $\mathcal M$, from the double category of monads in $\mathbb S\mathsf{qr} (\mathcal M)$ in the sense of Fiore, Gambino and Kock. While $q$ ones of the pseudomonoid structures of $A^t$ are lifted along the `forgetful' 1-cell $A^t \to A$, the other $p$ ones are lifted along its left adjoint. In the particular example when $\mathcal M$ is an appropriate 2-subcategory of $\mathsf{Cat}$, this yields a conceptually different proof of some recent results due to Aguiar, Haim and L\'opez Franco.