Notes on abelianity of categories of finitely encoded persistence modules

TL;DR

This paper investigates conditions under which categories of finitely encoded persistence modules become abelian, showing that restricting to certain topologically well-behaved modules restores abelian properties.

Contribution

It proves that categories of finitely encoded persistence modules become abelian when restricted to topologically closed and constructible modules.

Findings

Abelianity can be restored for finitely encoded persistence modules under topological restrictions.

Finitely encodable persistence modules do not form an abelian category in general.

Restricting to semi-algebraic or piecewise linear modules achieves abelianity.

Abstract

When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability, which roughly states that a persistence module can be obtained by pulling back a finite dimensional persistence module over a finite poset. From the perspective of homological algebra, finitely encodable persistence have an inconvenient property: They do not form an abelian category. Here, we prove that if one restricts to such persistence modules which can be constructed in terms of topologically closed and sufficiently constructible (piecewise linear, semi-algebraic, etc.) upsets then abelianity can be restored.