  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.
