The observable structure of persistence modules

TL;DR

This paper introduces the observable category of persistence modules, ensuring that the persistence diagram becomes a complete invariant and that all q-tame modules admit an interval decomposition, addressing limitations of existing categories.

Contribution

It defines a new observable category of persistence modules that restores the completeness of persistence diagrams and guarantees interval decompositions for q-tame modules.

Findings

Persistence diagrams are complete invariants in the observable category.

All q-tame modules admit an interval decomposition in this new setting.

The observable category retains key properties of q-tame modules.

Abstract

In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered finite set or the natural numbers, such diagrams do not provide a complete invariant of q-tame modules. The purpose of this paper is to show that the category of persistence modules can be adjusted to overcome this issue. We introduce the observable category of persistence modules: a localization of the usual category, in which the classical properties of q-tame modules still hold but where the persistence diagram is a complete isomorphism invariant and all q-tame modules admit an interval decomposition.