Size and shape of things: magnitude, diversity, homology

Abstract

This thesis has to do with magnitude: a numerical invariant of enriched categories which, when specialized to examples such as posets, groupoids or metric spaces, turns out to encompass a variety of size-related quantities including Euler characteristic, groupoid cardinality, and geometric data such as length, surface area and volume.

The thesis comprises two parts. The first, based on a paper co-authored with Tom Leinster, introduces and investigates the maximum diversity of a compact metric space, a quantity closely related to magnitude, whose origins (in work by Leinster and Cobbold) lie in the effort to quantify the diversity of an ecological community. The second part concerns magnitude homology, introduced in 2017 by Leinster and Shulman to categorify the magnitude of enriched categories, adding depth to the picture of magnitude as a ‘size-like’ invariant. Among the important size-like features of magnitude is its behaviour under combination of objects: multiplicativity with respect to the tensor product of enriched categories; that magnitude of metric spaces satisfies an inclusion-exclusion formula; and that magnitude of graphs is invariant under certain Whitney twists. We study magnitude homology’s behaviour under such combinations, and leverage that behaviour to extend the theory to new classes of examples.

First, we adapt a strategy used in the proof of Hepworth and Willerton’s Excision Theorem for graphs to give a homological proof of magnitude’s invariance under Whitney twists. The homological approach lets us generalize that result to encompass a substantially wider class of gluings. Next, we prove a general K¨unneth Theorem for enriched categories. This allows us to iterate the theory and study the magnitude homology of categories with a second- or higher-order enrichment: for instance, metrically enriched categories or strict n-categories for n > 1, including groups with extra structure such as a conjugation-invariant norm or an ordering. Finally, we explore the question of how to extend magnitude homology from finite to compact metric spaces in a way that recovers their magnitude: a question of particular interest, since it is in this setting that magnitude’s richest geometric properties emerge. This problem is not solved in this thesis; rather, we map its contours with a view to future work.