Enriched categories, real metrics and Lorentz manifolds

TL;DR

This paper explores the connections between enriched category theory-based generalized metrics and Lorentz manifolds in relativity, highlighting their categorical and geometric structures.

Contribution

It unifies the concepts of real-valued metrics in relativity with enriched categories, extending Lawvere's metric space framework to Lorentzian geometry.

Findings

Real-valued metrics in relativity can be modeled as enriched categories.

The antimetric satisfies a reverse triangle inequality and relates to lightcones.

Category theory provides a solid foundation for Lorentzian geometry.

Abstract

This expository article brings together two subjects: generalised metrics based on enriched categories, on the one hand, and Lorentz manifolds, on the other, at the price of dealing with details that are well known either in category theory or in relativity. The spacetime of relativity can be given a real valued metric $\rho(x, y)$, with values in the extended real line, or better (if equivalently) a real valued `antimetric' $\gamma(x, y) = - \rho(x, y)$ (satisfying a reverse triangle inequality); the latter, as a function of $y$, is positive on the timecone of $x$, annihilates on its lightcone, and is $- \infty$ on all events which cannot be influenced by $x$. All this can be given a well-established base in category theory, extending Lawvere's notion of a metric space. In fact, a space with a real valued metric can be viewed as an enriched category on the extended real line, structured as a symmetric monoidal closed category.