Magnitude homology of enriched categories and metric spaces

TL;DR

This paper develops a homology theory called magnitude homology for enriched categories, including metric spaces, which categorifies the magnitude invariant and reveals geometric features like convexity.

Contribution

It introduces magnitude homology as a categorification of magnitude, extending it to enriched categories and metric spaces, and connects it to Hochschild homology.

Findings

Magnitude homology generalizes graph magnitude homology.

Magnitude homology detects geometric properties such as convexity.

Euler characteristic of magnitude homology equals the magnitude.

Abstract

Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth--Willerton magnitude homology of graphs, and detects geometric information such as convexity.