On the Structural Theorem of Persistent Homology

TL;DR

This paper develops a categorical framework for persistent homology, providing a nonconstructive proof of the Matrix Structural Theorem and establishing a foundation for alternative decomposition methods beyond traditional persistence vector spaces.

Contribution

It introduces a categorical approach to persistent homology, proving the Matrix Structural Theorem via the Krull-Schmidt property, and offers a new foundation for decomposition methods.

Findings

Proves the Matrix Structural Theorem using categorical methods.

Establishes the Krull-Schmidt property for filtered chain complexes.

Provides an alternative, nonconstructive proof of the theorem.

Abstract

We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. We showthat the Matrix Structural Theorem is equivalent to the Krull-Schmidt property of the category of filtered chain complexes. We separately establish the Krull-Schmidt property by abstract categorical methods, yielding a novel nonconstructive proof of the Matrix Structural Theorem. These results provide the foundation for an alternate categorical framework for decomposition in persitent homology, bypassing the usual persistence vector spaces and quiver representations.