  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.
