Refined Alexandrov-topology and the semicategory of chronological path classes
In this talk, we investigate the semicategory of homotopy classes of chronological paths in a Lorentzian manifold. We show that both the topology and the conformal structure of the manifold can be reconstructed exactly from this semicategory.
Compared to the fundamental groupoid, the chronological semicategory carries much more detailed information about the underlying manifold, and is less rigid. For example, we show that all isomorphisms between such semicategories are pushforwards of homeomorphisms.
Furthermore, we show that the morphism set of the chronological semicategory has an interesting natural non-Hausdorff manifold structure.