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Causal Coverage in Ordered Locales and Spacetimes

Chris Heunen, Nesta van der Schaaf

TL;DR

The paper develops a point-free causal framework for spacetimes by formalizing causal coverage on ordered locales, yielding a generalized domain of dependence and connecting this causality structure to a generalized Grothendieck topology. It introduces parallel ordered locales, monotone localic paths, and refined notions of past/future coverage, proving that these satisfy Gauss-like coverage axioms and induce monads that recover localic cones. By comparing localic coverages with curve-based notions in spacetimes, the work clarifies when localic domains of dependence dominate their curve-wise counterparts and highlights important distinctions in the presence of holes. Abstracting these ideas, the authors sketch a causal-site–like correspondence and propose deterministic sheaves as a promising direction, with holes in spacetime offering a rich area for future exploration and potential applications to relativity and concurrency theory.

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.

Causal Coverage in Ordered Locales and Spacetimes

TL;DR

The paper develops a point-free causal framework for spacetimes by formalizing causal coverage on ordered locales, yielding a generalized domain of dependence and connecting this causality structure to a generalized Grothendieck topology. It introduces parallel ordered locales, monotone localic paths, and refined notions of past/future coverage, proving that these satisfy Gauss-like coverage axioms and induce monads that recover localic cones. By comparing localic coverages with curve-based notions in spacetimes, the work clarifies when localic domains of dependence dominate their curve-wise counterparts and highlights important distinctions in the presence of holes. Abstracting these ideas, the authors sketch a causal-site–like correspondence and propose deterministic sheaves as a promising direction, with holes in spacetime offering a rich area for future exploration and potential applications to relativity and concurrency theory.

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.
Paper Structure (13 sections, 31 theorems, 59 equations, 17 figures)

This paper contains 13 sections, 31 theorems, 59 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Ordered locales
  3. Parallel ordered locales
  4. Paths in ordered locales
  5. Causal coverages in ordered locales
  6. Causal coverages as Grothendieck topologies
  7. Causal coverages in spacetimes
  8. Abstract causal coverages
  9. Domains of dependence
  10. Domains of dependence in spacetime
  11. Conclusions
  12. Deterministic sheaves
  13. Holes in spacetime

Key Result

Lemma 2.1

If $(S,\leqslant)$ is a preordered set, then for any subsets $A,B\subseteq S$: Further, for any family $(A_i)_{i\in I}$ of subsets of $S$:

Figures (17)

  • Figure 1: Transition from order on points to order on regions.
  • Figure 2: Illustration of the space in \ref{['example:LV can fail in spaces']}.
  • Figure 3: Illustration of \ref{['axiom:wedge+']} and \ref{['lemma:wedge iff strong wedge']}.
  • Figure 4: Illustration of \ref{['axiom:frobenius-']} in a Minkowski-like space.
  • Figure 5: Intuition of non-parallel vs. parallel cones.
  • ...and 12 more figures

Theorems & Definitions (103)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • ...and 93 more