Saecular persistence

TL;DR

This paper introduces saecular decomposition, a natural categorical method to decompose persistence modules into simple parts, enabling generalized persistence diagrams and applications in topology and data analysis.

Contribution

It generalizes existing factorizations of 1-parameter persistence modules using categorical tools, allowing for broader applications including homotopy persistence diagrams.

Findings

Saecular decomposition exists under generic conditions.

Enables persistence diagrams in homotopy.

Applications include inverse problems and spectral sequences.

Abstract

A persistence module is a functor $f: \mathbf{I} \to \mathsf{E}$, where $\mathbf{I}$ is the poset category of a totally ordered set. This work introduces saecular decomposition: a categorically natural method to decompose $f$ into simple parts, called interval modules. Saecular decomposition exists under generic conditions, e.g., when $\mathbf{I}$ is well ordered and $\mathsf{E}$ is a category of modules or groups. This represents a substantial generalization of existing factorizations of 1-parameter persistence modules, leading to, among other things, persistence diagrams not only in homology, but in homotopy. Applications of saecular decomposition include inverse and extension problems involving filtered topological spaces, the 1-parameter generalized persistence diagram, and the Leray-Serre spectral sequence. Several examples -- including cycle representatives for generalized barcodes -- hold special significance for scientific applications. The key tools in this approach are modular and distributive order lattices, combined with Puppe exact categories.