Non-Hausdorff manifolds over locally ordered spaces via sheaf theory

TL;DR

This paper develops a sheaf-theoretic construction of universal Euclidean local orders over locally ordered spaces, generalizing previous results and providing combinatorial descriptions for realizations of precubical sets.

Contribution

It introduces a sheaf-based method to construct universal Euclidean local orders, extending the theory to locally ordered manifolds and precubical set realizations.

Findings

Constructed a universal Euclidean local order for locally ordered spaces.

Established a categorical coreflection of Euclidean local orders within locally ordered spaces.

Proved the uniqueness of certain locally ordered realizations of $\

Abstract

Locally ordered spaces can be used as topological models of concurrent programs: 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. Our construction is based on a well-known correspondance between sheaves and \'etale bundles. 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 (up to symmetry) precubical set whose locally ordered realization is isomorphic to $\mathbb{R}^n$.