Size and shape of things: magnitude, diversity, homology
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Date
29/11/2022Item status
Restricted AccessEmbargo end date
29/11/2023Author
Roff, Emily
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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.