Magnitude cohomology

TL;DR

This paper introduces magnitude cohomology, a dual theory to magnitude homology, and demonstrates its ability to fully determine finite metric spaces, with applications to graphs and enriched categories.

Contribution

It defines magnitude cohomology as a graded ring, proves it determines finite metric spaces, and explores conditions for its commutativity in enriched categories.

Findings

Magnitude cohomology ring fully determines finite metric spaces.

The ring is generally non-commutative, but commutative in certain categories.

Complete computations of magnitude cohomology rings for various graphs.

Abstract

Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual theory, magnitude cohomology, which we equip with the structure of an associative unital graded ring. Our first main result is a 'recovery theorem' showing that the magnitude cohomology ring of a finite metric space completely determines the space itself. The magnitude cohomology ring is non-commutative in general, for example when applied to finite metric spaces, but in some settings it is commutative, for example when applied to ordinary categories. Our second main result explains this situation by proving that the magnitude cohomology ring of an enriched category is graded-commutative whenever the enriching category is cartesian. We end the paper by giving complete computations of magnitude cohomology rings for several large classes of graphs.