# THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY

@article{Leinster2017THEMO, title={THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY}, author={Tom Leinster and Mark W. Meckes}, journal={arXiv: Metric Geometry}, year={2017}, pages={156-193} }

Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral geometry and geometric measure theory, including volume, capacity, dimension, and intrinsic volumes. This paper gives an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities. Some new…

## 24 Citations

Magnitude homology of enriched categories and metric spaces

- MathematicsAlgebraic & Geometric Topology
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Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a…

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Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is…

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This paper restricts the sets to finite subsets of Euclidean space and investigates its individual components, and gives an explicit formula for the corrected inclusion-exclusion principle, and defines a quantity associated with each point, called the moment, which gives an intrinsic ordering to the points.

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. Magnitude is a numerical invariant of compact metric spaces, originally in-spired by category theory and now known to be related to myriad other geometric quanti-ties. Generalizing earlier results…

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Magnitude is a numerical invariant of compact metric spaces. Its theory is most mature for spaces satisfying the classical condition of being of negative type, and the magnitude of such a space lies…

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We study the geometric significance of Leinster’s notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in R2. For a large…

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The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely…

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Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in…

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Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known…

The Willmore energy and the magnitude of Euclidean domains

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. We study the geometric signiﬁcance of Leinster’s notion of magnitude for a compact metric space. For a smooth, compact domain X in an odd-dimensional Euclidean space, we show that the asymptotic…

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