Cross effects for functors from posets

We establish a precise relationship between functor calculus and the projective dimension of multipersistence modules. Specifically, we develop a new notion of functor calculus for functors from posets, which detects vanishing total fibers of cubes. We give an explicit construction of the universal approximation functors of this functor calculus. We then use these approximations to prove two new theorems, providing necessary and sufficient conditions for an $n$-parameter multipersistence module to have projective dimension at most $n-1$ and at most $n-2$.