Towards Point-Free Spacetimes

TL;DR

This thesis develops a point-free framework for spacetime causality using ordered locales, establishing dualities and new notions of causality and dependence that extend traditional Lorentzian geometry.

Contribution

It introduces ordered locales and extends Stone duality to them, enabling point-free models of spacetime causality and domain of dependence.

Findings

Established a duality between ordered topological spaces and ordered locales.

Developed point-free analogues of causality concepts like indecomposable past sets.

Proposed a new localic notion of domain of dependence.

Abstract

In this thesis we propose and study a theory of ordered locales, a type of point-free space equipped with a preorder structure on its frame of opens. It is proved that the Stone-type duality between topological spaces and locales lifts to a new adjunction between a certain category of ordered topological spaces and the newly introduced category of ordered locales. As an application, we use these techniques to develop point-free analogues of some common aspects from the causality theory of Lorentzian manifolds. In particular, we show that so-called indecomposable past sets in a spacetime can be viewed as the points of the locale of futures. This builds towards a point-free causal boundary construction. Furthermore, we introduce a notion of causal coverage that leads naturally to a generalised notion of Grothendieck topology incorporating the order structure. From this naturally emerges a localic notion of domain of dependence, which is generally distinct from the traditional notion in spacetimes.