Causal Coverage in Ordered Locales and Spacetimes

Abstract

We develop relativistic causality theory in the setting of point-free topology by introducing a notion of causal coverage in ordered locales, generalising their canonical coverage relation to incorporate causal structure. This improves Christensen and Crane's construction of `causal sites'. We connect to sheaf theory by showing that causal coverages can be interpreted as a generalised Grothendieck topology, and the sheaf condition as a type of deterministic time evolution. To develop these notions, we introduce and study parallel ordered locales. Causal coverage naturally induces a notion of domain of dependence. Comparing the localic and curve-wise definitions in spacetimes, the localic domains strictly contain the classical ones.