The formal theory of multimonoidal monads

Gabriella Bohm

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 M admits a symmetric strict monoidal 2-category of pseudomonoids, monoidal 1-cells and monoidal 2-cells in M. Dually, there is a symmetric strict monoidal 2-category of pseudomonoids, opmonoidal 1-cells and opmonoidal 2-cells in M. Extending a construction due to Aguiar and Mahajan for M = Cat, we may apply the first construction p-times and the second one q-times (in any order). It yields a 2-category M_{pq}. A 0-cell therein is an object A of M together with p+q compatible pseudomonoid structures; it is termed a (p+q)-oidal object in M. A monad in M_{pq} is called a (p,q)-oidal monad in M; it is a monad t on A in M together with p monoidal, and q opmonoidal structures in a compatible way. If 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 Sqr(M) of squares in M, from the double category of monads in Sqr(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 -> A, the other p ones are lifted along its left adjoint. In the particular example when M is an appropriate 2-subcategory of Cat, this yields a conceptually different proof of some recent results due to Aguiar, Haim and Lopez Franco.

Keywords: monoidal 2-category, monoidal double category, pseudomonoid, (op)monoidal monad, Eilenberg-Moore construction, lifting

2010 MSC: 18C15, 18D05, 18D10

Theory and Applications of Categories, Vol. 34, 2019, No. 12, pp 295-348.

Published 2019-04-12.

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