Metric-like spaces as enriched categories: three vignettes

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.