Spaces with local chronological structure and the cosheaf of fundamental semicategories

In the present work, we examine a structure that generalizes the causal structure of a Lorentzian spacetime.
In contrast to similar definitions in the literature, we define the chronological relations locally, that is, on open subsets of a topological space. This has the advantage that we do not need to employ causality conditions for the whole space.
The space of timelike homotopy classes of paths in such a space $X$ forms an algebraic structure that we call the fundamental semicategory $\Pi^\mathrm{t}(X)$.

We provide a van-Kampen theorem for fundamental semicategories, show that the isomorphism class of $\Pi^\mathrm{t}(X)$ determines the topology and isomorphism class of $X$, and put a topology on the total morphism space of $\Pi^\mathrm{t}(X)$ that is locally homeomorphic to $X\times X$.