Magnitude homology of enriched categories and metric spaces

Leinster, Tom; Shulman, Michael

doi:10.2140/agt.2021.21.2175

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Magnitude homology of enriched categories and metric spaces

Tom Leinster and Michael Shulman

Algebraic & Geometric Topology 21 (2021) 2175–2221

DOI: 10.2140/agt.2021.21.2175

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.

Keywords

magnitude, magnitude homology, Euler characteristic, enriched category, metric space, categorification, Hochschild homology

Mathematical Subject Classification

Primary: 18G90

Secondary: 16E40, 51F99, 55N31

References

Publication

Received: 13 November 2017

Revised: 7 October 2020

Accepted: 12 November 2020

Published: 31 October 2021

Authors

Tom Leinster
School of Mathematics University of Edinburgh Edinburgh United Kingdom
https://www.maths.ed.ac.uk/~tl
Michael Shulman
Department of Mathematics University of San Diego San Diego, CA United States
http://home.sandiego.edu/~shulman

Volume 21, issue 5 (2021)