Non-Hausdorff manifolds over locally ordered spaces via sheaf theory

TL;DR

The paper addresses constructing a universal Euclidean local order over arbitrary locally ordered spaces and ensuring that execution traces of concurrent programs persist through the construction. It develops a blowup functor via the sheaf-étale correspondence, proves that Euclidean local orders form a coreflective subcategory of locally ordered spaces, and extends the construction to precubical sets with a combinatorial description. A key contribution is a purely combinatorial blowup using a presheaf of local precubical structures, yielding a global realization and a rigidity result: if $|P| \cong \mathbb{R}^n$ then $P \cong R^{\otimes n}$. The framework connects sheaf theory, étale bundles, and directed topology to provide practical tools for modeling and analyzing execution traces in concurrency, while exposing avenues for smooth and metric enhancements. Overall, the work offers a robust, categorical approach to blowing up locally ordered realizations and extracting computable, combinatorial descriptions in the precubical setting.

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 étale 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 $\mathbb{R}^n$.

Figures (4)

• Figure 1: The $4$ traversals at the origin in $\{(x,y)\in[-1,1]^2;xy=0\}$.
• Figure 2: The $9$ traversals at the origin in $\{(x,y,z)\in[0,1]^3;xyz=0\}$.
• Figure 3: The étale bundle in the simple case of the graph $G$ of the introduction. We highlight $A$ and $G^+_{A}$ for various $A$. We omitted the $A_x$, $A\ne\varnothing$, $x\not\in A$ for simplification. The blowup, as described in the introduction, corresponds to the complement of the ghost version.
• Figure 4: The central vertical edge $c$ is a common face of the cubes $c_1$ and $c_2$; the central grey cube is a local precubical structure of $\mathbb{R}^n$ centered at $m(c)$, and the right grey cube is its image by $\psi$, a local precubical structure of $\mathbb{R}^n$ centered at $m(c_2)$.

Theorems & Definitions (56)

• Definition 1.1
• Theorem 1.2: haucourt2024non
• Definition 3.1
• Proposition 3.2
• proof
• Definition 3.3
• Definition 4.1
• Remark 4.2
• Proposition 4.3
• proof