Constructing monads from cubical diagrams and homotopy colimits

This paper is the first step in a general program for defining cocalculus towers of functors via sequences of compatible monads. Goodwillie's calculus of homotopy functors inspired many new functor calculi in a wide range of contexts in algebra, homotopy theory and geometric topology. Recently, the third and fourth authors have developed a general program for constructing generalized calculi from sequences of compatible comonads. In this paper, we dualize the first step of the Hess-Johnson program, focusing on monads rather than comonads. We consider categories equipped with an action of the poset category $\mathcal{P}(n)$, called $\mathcal{P}(n)$-modules. We exhibit a functor from $\mathcal{P}(n)$-modules to the category of monads. The resulting monads act on categories of functors whose codomain is equipped with a suitable notion of homotopy colimits. In the final section of the paper, we demonstrate the monads used to construct McCarthy's dual calculus as an example of a monad arising from a $\mathcal{P}(n)$-module. This confirms that our dualization of the Hess-Johnson program generalizes McCarthy's dual calculus, and serves as a proof of concept for further development of this program.