# The formal theory of multimonoidal monads

Research paper by **Gabriella Böhm**

Indexed on: **26 Oct '18**Published on: **26 Oct '18**Published in: **arXiv - Mathematics - Category Theory**

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#### Abstract

Certain aspects of Street's formal theory of monads in 2-categories are
extended to multimonoidal monads in strict symmetric monoidal 2-categories.
Namely, any strict symmetric monoidal 2-category $\mathcal M$ admits a strict
symmetric monoidal 2-category of pseudomonoids, monoidal 1-cells and monoidal
2-cells in $\mathcal M$. Dually, there is a strict symmetric 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 strict symmetric 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.