On the categorical and topological structure of timelike and causal homotopy classes of paths in smooth spacetimes

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. The space of timelike or causal homotopy classes forms a semicategory or a category, respectively. We show that either of these algebraic structures encodes enough information to reconstruct the topology and conformal structure of $X$. Furthermore, the space of timelike homotopy classes carries a natural topology that we prove to be locally euclidean but, in general, not Hausdorff. The presented results do not require any causality conditions on $X$ and do also hold under weaker regularity assumptions.