THE FORMAL THEORY OF MULTIMONOIDAL MONADS

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.