Title: Monads on higher monoidal categories,
Abstract: We study the action of monads on categories equipped with several monoidal
structures. We identify the structure and conditions that guarantee that the
higher monoidal structure is inherited by the category of algebras over the
monad. Monoidal monads and comonoidal monads appear as the base cases in this
hierarchy. Monads acting on duoidal categories constitute the next case. We
cover the general case of $n$-monoidal categories and discuss several naturally
occurring examples in which $n\leq 3$.