Galois Connections in Persistent Homology

TL;DR

This paper introduces a Galois connection framework for persistent homology, simplifying concepts and enabling new theoretical insights, including a streamlined proof of the bottleneck stability theorem.

Contribution

It develops a novel Galois connection language for persistent homology, unifying key concepts and leveraging Rota's theorem for easier proofs and relationships between persistence notions.

Findings

Simplifies and unifies core concepts like interleavings and matchings.

Provides a new proof of the bottleneck stability theorem.

Establishes relationships between different multiparameter persistence diagrams.

Abstract

We present a new language for persistent homology in terms of Galois connections. This language has two main advantages over traditional approaches. First, it simplifies and unifies central concepts such as interleavings and matchings. Second, it provides access to Rota's Galois connection theorem -- a powerful tool with many potential applications in applied topology. To illustrate this, we use Rota's Galois connection theorem to give a substantially easier proof of the bottleneck stability theorem. Finally, we use this language to establish relationships between various notions of multiparameter persistence diagrams.