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.
|Number of pages||35|
|Journal||Algebraic and Geometric Topology|
|Publication status||Accepted/In press - 12 Nov 2020|