Decomposing multipersistence modules using functor calculus

Abstract

We apply poset cocalculus, a functor calculus framework for functors out of a poset, to study the problem of decomposing multipersistence modules into simpler components. We both prove new results in this topic and offer a new perspective on already established results. In particular, we show that a pointwise finite-dimensional bipersistence module is middle-exact if and only if it is isomorphic to the homology of a homotopy degree 1 functor, from which we deduce a novel, more synthetic proof of the interval decomposability of middle-exact bipersistence modules. We also give a new decomposition theorem for middle-exact multipersistence modules indexed over a finite poset, stating that such a module can always be written as a direct sum of a projective module, an injective module, and a bidegree 1 module, even in the case where it is not pointwise finite-dimensional.