Higher interpolation and extension for persistence modules

TL;DR

This paper introduces a category-theoretic coherence criterion enabling the extension and comparison of persistence modules across different metric spaces, enhancing topological data analysis tools.

Contribution

It presents a novel coherence criterion based on Kan extensions for extending non-expansive maps into the space of persistence modules.

Findings

Provides an isometric embedding of metric spaces into persistence modules.

Enables comparison of Vietoris-Rips and Čech complexes within persistence modules.

Facilitates higher interpolation and extension in topological data analysis.

Abstract

The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. Our main construction gives an isometric embedding of a metric space into the metric space of persistence modules with values in the spacetime of this metric space. As a consequence of such "higher interpolation", it becomes possible to compare Vietoris-Rips and \v{C}ech complexes built within the space of persistence modules.