On the categorical and topological structure of timelike homotopy classes of paths in smooth spacetimes
Martin Günther · arXiv (Cornell University) · 2020
For a smooth spacetime $X$, based on the timelike homotopy classes of its timelike paths, we define a topology on $X$ that refines the Alexandrov topology and always coincides with the manifold topology. We show that the space of such homotopy classes forms a semicategory which encodes enough information to reconstruct the topology and conformal structure of $X$. Its set of morphisms carries a natural topology that we prove to be locally euclidean but, in general, not Hausdorff. Our results do not require any causality conditions on $X$ and do also hold under weaker regularity assumptions.