The Representation Theorem of Persistent Homology Revisited and Generalized

TL;DR

This paper revisits and refines the Representation Theorem of persistent homology, providing a clearer statement, a complete proof, and generalizing it to modules over more complex monoids, including multidimensional persistence.

Contribution

It offers a more precise formulation, a comprehensive proof, and extends the theorem to broader algebraic contexts beyond linear sequences.

Findings

Refined and clarified the original Representation Theorem.

Provided a complete, self-contained proof.

Generalized the theorem to modules over general monoids, including multidimensional persistence.

Abstract

The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of algebraic representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of $R$-modules to $R$-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimensional persistence as a special case.