Magnitude homology


Venue   Journées homotopiques (for Clemens Berger's 60th birthday), Université Côte d'Azur, Nice, 7 December 2022

Abstract Magnitude is a numerical invariant of enriched categories. It unifies topological Euler characteristic, groupoid cardinality, and invariants of geometric measure such as volume, surface area and fractal dimension. It can also be categorified: there is a graded homology theory of enriched categories called magnitude homology, whose Euler characteristic is (sometimes) magnitude. Thus, magnitude homology categorifies magnitude in the same sense that Khovanov homology categorifies the Jones polynomial. It is particularly interesting in the case of metric spaces. For example, while topological homology detects the existence of holes, magnitude homology detects how big the holes are. I will give an overview, mentioning the original magnitude homology for graphs introduced by Hepworth and Willerton, its generalization to enriched categories by Shulman and myself, and the many results in metric geometry found by a host of others.

Slides   In this PDF file.

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