Non-Hausdorff manifolds over locally ordered spaces via sheaf theory

Abstract: Locally ordered spaces can be used as topological models of concurrent programs: in that setting, the local order models the irreversibility of time during execution. Under certain conditions, one can even work with locally ordered manifolds. In this paper, we build the universal euclidean local order over every locally ordered space; in categorical terms, the subcategory of euclidean local orders is coreflective in the category of locally ordered spaces. Then we give conditions to ensure that it preserves the execution traces of the corresponding program. Our construction is based on a well-known correspondance between sheaves on a space and \'etale bundles over this space. This is a far reaching generalization of a result about realizations of graph products. We particularize the construction to locally ordered realization of precubical sets, and show that it admits a purely combinatorial description. With the same proof techniques, we show that, unlike for the topological realization, there is a unique precubical set whose locally ordered realization is isomorphic to $\mathbb{R}^n$.