Generalized persistence and graded structures

TL;DR

This paper explores the relationship between generalized persistence modules and graded modules under monoid actions, introducing new categorical frameworks and characterizations for finitely presented modules.

Contribution

It introduces the notion of an action category over a monoid graded ring and establishes categorical isomorphisms relevant to persistence theory.

Findings

Category of additive functors is isomorphic to graded modules

Characterization of finitely presented modules in poset cases

New framework connecting persistence modules and graded modules

Abstract

We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.