Decomposing multipersistence modules using functor calculus
Bjørnar Gullikstad Hem
TL;DR
This work develops poset cocalculus as a systematic framework for decomposing multipersistence modules. By linking codegree/degree approximations to homotopical data, it shows that middle-exact pointwise finite-dimensional bipersistence modules are captured by homotopy degree-1 objects, yielding a synthetic interval-decomposition proof, and provides a finite-poset decomposition F ≅ B ⊕ K ⊕ C with B of bidegree 1, K injective, and C projective. The paper extends these ideas to higher dimensions via layers and bidegree concepts, proving interval decomposability for bidegree-1 functors and establishing structure theorems that hold beyond p.f.d. cases. Collectively, these results offer new structural insight and a unifying, calculational approach to multipersistence with potential impact on higher-dimensional TDA analysis and the theory of persistence modules.
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.