Magnitude cohomology

Richard Hepworth* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
JournalMathematische Zeitschrift
Early online date9 May 2022
DOIs
Publication statusE-pub ahead of print - 9 May 2022

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