Towards Point-Free Spacetimes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914964492517376 |
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| author | van der Schaaf, Nesta |
| author_facet | van der Schaaf, Nesta |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards Point-Free Spacetimes van der Schaaf, Nesta General Mathematics 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. |
| title | Towards Point-Free Spacetimes |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2406.15406 |