Volume 21, issue 5 (2021)

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Magnitude homology of enriched categories and metric spaces

Tom Leinster and Michael Shulman

Algebraic & Geometric Topology 21 (2021) 2175–2221
Abstract

Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as [0,)–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