Lawvere's insight that a metric space is a category enriched over a particular monoidal category. Where an ordinary small category assigns to each ordered pair of objects a hom-set and has composition and identities, a metric space assigns to each ordered pair of points a distance in the extended non-negative reals; taking that ordered set with addition as the tensor and zero as the unit as the base monoidal category, the triangle inequality becomes the enriched composition law and zero self-distance becomes the identity. This makes metric spaces and ordinary categories two instances of one enriched-category notion, giving a bridge along which constructions such as the tight span, the magnitude of a metric space, and the Legendre-Fenchel transform can be understood category-theoretically.
An expository write-up (from a 2023 CATMI talk) introducing the category-theoretic view of metric spaces. Its organizing idea is Lawvere's observation that a metric space and a small category are ins…