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arXiv · 2510.17417

Causal Coverage in Ordered Locales and Spacetimes

Chris Heunen·Nesta van der Schaaf
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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.

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Chris Heunen, Nesta van der Schaaf. 2025-10-20. Causal Coverage in Ordered Locales and Spacetimes. https://arxiv.org/abs/2510.17417

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