Towards Point-Free Spacetimes

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.