Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem

TL;DR

This paper introduces a categorical perspective on persistent homology barcodes, leading to simplified reformulations of the stability theorem and its generalization, enhancing theoretical understanding in topological data analysis.

Contribution

It provides a novel categorical framework for barcodes, enabling simpler proofs and insights into the stability properties of persistent homology.

Findings

Categorical structure of barcodes as functors R -> Mch

Simplified reformulations of the stability theorem

Enhanced understanding of the induced matching theorem

Abstract

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R -> Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category Mch^R whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. The aim of this note is to show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem.