Magnitude homology of enriched categories and metric spaces

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.

47 pages; v2: fixed some errors in the discussion of H_2; v3: reorganized to be more accessible and concrete, based on referee suggestions; v4: journal version, to appear in Algebraic & Geometric Topology

References in corpus (5)

• Magnitude meets persistence. Homology theories for filtered simplicial sets
• Persistent Magnitude
• Magnitude homology of metric spaces and order complexes
• Graph magnitude homology via algebraic Morse theory
• Magnitude homology of geodesic metric spaces with an upper curvature bound

Cited by in corpus (7)

• Magnitude meets persistence. Homology theories for filtered simplicial sets
• Persistent Magnitude
• Graph magnitude homology via algebraic Morse theory
• Magnitude homology of metric spaces and order complexes
• Quantales, persistence, and magnitude homology
• Magnitude Homology, Diagonality, Medianness, Künneth and Mayer-Vietoris
• Practical applications of metric space magnitude and weighting vectors