2024/12/31 by Simon Willerton, Willerton, Simon
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2501.00416
This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a number -- the distance -- such that the triangle inequality is satisfied; a small category is a set of objects such that between each pair of objects there is a set -- the hom-set -- such that elements of the hom-sets can be composed. The analogy between the structures that can be made in to a common generalization of the two structures, so that both are examples of enriched categories. This gives a bridge between category theory and metric space theory. I will describe this and three examples from around mathematics where this perspective has been useful or interesting. The examples are related to the tight span, the magnitude and the Legendre-Fenchel transform.