Enriched categories, real metrics and Lorentz manifolds

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.