Causal Coverage in Ordered Locales and Spacetimes

TL;DR

This paper extends relativistic causality theory to ordered locales using causal coverage, linking it with sheaf theory and generalising causal structures in spacetime topology.

Contribution

It introduces causal coverage in ordered locales, generalising causal sites and connecting to sheaf theory, with new insights into domains of dependence in spacetime.

Findings

Causal coverage generalises canonical coverage in locales.

Causal coverages can be viewed as a generalized Grothendieck topology.

Localic domains of dependence are strictly larger than classical ones.

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.