Categorification of persistent homology

TL;DR

This paper reinterprets persistent homology categorically, introducing a generalized interleaving distance and extending stability results to a broader class of persistence modules.

Contribution

It develops a categorical framework for persistent homology, generalizing the interleaving distance and stability results to diagrams in arbitrary target categories.

Findings

Generalized interleaving distance extends bottleneck distance

Stability results now apply to a wider class of persistence modules

Category of interleavings is abelian when target category is abelian

Abstract

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.