Cross effects for functors from posets

TL;DR

This paper introduces a new functor calculus for poset-based functors, linking it to the projective dimension of multipersistence modules and providing criteria for their bounds.

Contribution

It develops a novel functor calculus framework for posets and applies it to characterize projective dimensions of multipersistence modules.

Findings

Established conditions for multipersistence modules to have projective dimension ≤ n-1

Established conditions for multipersistence modules to have projective dimension ≤ n-2

Provided explicit constructions of universal approximation functors

Abstract

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 $n-2$.