The magnitude of a metric space: from category theory to geometric measure theory

TL;DR

This paper explores the concept of magnitude as a numerical invariant of metric spaces, linking category theory with geometric measure theory to encode various geometric invariants like volume and dimension.

Contribution

It provides a comprehensive overview of magnitude, highlighting its origins in category theory and its applications in encoding geometric invariants.

Findings

Magnitude encodes volume, capacity, and dimension.

Connections between category theory and geometric measure theory.

Magnitude as a unifying invariant for metric spaces.

Abstract

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 will give an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities.