Galois Connections in Persistent Homology

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.