The formal theory of multimonoidal monads

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 López Franco.

v1: 43 pages, several LaTeX figures v2: 50 pages, some more results, references and minor corrections

References in corpus (2)

• The Classification of Two-Dimensional Extended Topological Field Theories
• Polynomial and Analytic Functors and Monads, revisited