Non-Hausdorff manifolds over locally ordered spaces via sheaf theory

Emmanuel Haucourt; Yorgo Chamoun

arxiv: 2505.12087 · v4 · submitted 2025-05-17 · 🧮 math.AT · math.CT

Non-Hausdorff manifolds over locally ordered spaces via sheaf theory

Yorgo Chamoun , Emmanuel Haucourt This is my paper

Pith reviewed 2026-05-22 14:22 UTC · model grok-4.3

classification 🧮 math.AT math.CT

keywords locally ordered spacesEuclidean local orderssheaf theoryétale bundlesprecubical setscoreflective subcategoryconcurrent programsdirected topology

0 comments

The pith

The subcategory of Euclidean local orders is coreflective in the category of locally ordered spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Locally ordered spaces model concurrent programs by using local orders to capture the irreversible direction of time. The paper constructs a universal Euclidean local order over any given locally ordered space. This construction relies on the standard correspondence between sheaves and étale bundles. A sympathetic reader would see this as allowing any locally ordered space to be canonically replaced by a Euclidean version that retains its essential ordering properties. The same method yields a combinatorial description for the locally ordered realizations of precubical sets and proves uniqueness results for simple cases such as n-dimensional Euclidean space.

Core claim

The subcategory of Euclidean local orders is coreflective in the category of locally ordered spaces. The coreflector is obtained by applying the sheaf-étale bundle correspondence to produce, for each locally ordered space, a universal Euclidean local order sitting above it. This generalizes earlier results on realizations of graph products. When restricted to locally ordered realizations of precubical sets, the construction admits a purely combinatorial description. With the same techniques, there exists a unique precubical set up to symmetry whose locally ordered realization is isomorphic to R^n.

What carries the argument

The sheaf-étale bundle correspondence, which associates to each locally ordered space its universal Euclidean local order and thereby exhibits the coreflective property.

Load-bearing premise

The standard sheaf-étale bundle correspondence extends directly to locally ordered spaces and produces a universal Euclidean local order that is coreflective.

What would settle it

A concrete locally ordered space where the sheaf-theoretic construction either fails to yield a Euclidean local order or fails to be universal and coreflective.

read the original 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 $\mathbb{R}^n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a sheaf-based universal Euclidean local order on any locally ordered space to get coreflectivity, then gives a combinatorial version for precubical sets plus an R^n uniqueness result.

read the letter

The main point is that the subcategory of Euclidean local orders is coreflective in the category of locally ordered spaces, via a construction that uses the sheaf-étale bundle correspondence to produce the universal object. They extend this to precubical sets with a purely combinatorial description and show that, unlike ordinary topological realizations, there is a unique precubical set (up to symmetry) whose locally ordered realization is R^n. This generalizes the graph-product case mentioned in the abstract.

Referee Report

1 major / 2 minor

Summary. The paper constructs the universal Euclidean local order over every locally ordered space using the sheaf-étale bundle correspondence, showing that the subcategory of Euclidean local orders is coreflective in the category of locally ordered spaces. This generalizes results on realizations of graph products. The construction is specialized to locally ordered realizations of precubical sets with a combinatorial description, and a uniqueness result (up to symmetry) is proved for the precubical set realizing R^n.

Significance. If the central construction holds, the result supplies a right adjoint to the inclusion functor, providing a canonical Euclidean approximation for any locally ordered space. This is useful for directed topology and concurrency modeling where local orders capture irreversible time. The combinatorial description for precubical sets and the uniqueness theorem for R^n are concrete, reproducible strengths that extend prior work on graph products.

major comments (1)

§3 (core construction): The claim that the sheaf-étale correspondence produces a coreflector requires verifying that the local order on the base lifts functorially to the total space of the étale bundle so that the result is Euclidean (locally modeled on ordered R^n) and the counit satisfies the universal property for order-preserving maps; without an explicit lemma addressing this lift, the adjunction does not follow from the standard topological correspondence alone.

minor comments (2)

Abstract: 'correspondance' is a typo and should be 'correspondence'.
§4 (combinatorial description): An explicit low-dimensional example computing the coreflector for a small precubical set would improve readability and allow direct verification of the claimed combinatorial form.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comment on the core construction in §3. The positive assessment of the significance for directed topology and concurrency modeling is appreciated. We address the major comment point by point below.

read point-by-point responses

Referee: [—] §3 (core construction): The claim that the sheaf-étale correspondence produces a coreflector requires verifying that the local order on the base lifts functorially to the total space of the étale bundle so that the result is Euclidean (locally modeled on ordered R^n) and the counit satisfies the universal property for order-preserving maps; without an explicit lemma addressing this lift, the adjunction does not follow from the standard topological correspondence alone.

Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will add a dedicated lemma (new Lemma 3.5) immediately following the definition of the étale bundle in §3. The lemma states: given a locally ordered space X with sheaf F of local orders, the total space E of the associated étale bundle carries the lifted local order defined by declaring (e1 ≤ e2) in a neighborhood of e whenever their images in X satisfy the order and the sections agree with the sheaf data; this lift is shown to be functorial in morphisms of locally ordered spaces, to be locally modeled on ordered R^n (hence Euclidean), and to make the counit of the adjunction universal among order-preserving maps. The proof uses the standard sheaf-étale equivalence together with the explicit description of the order on sections, which was implicit in the original text but is now isolated for clarity. This establishes the coreflector property directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external sheaf-étale correspondence

full rationale

The paper constructs the universal Euclidean local order (and thus the coreflectivity of the subcategory) by direct application of the standard, externally established correspondence between sheaves and étale bundles to the category of locally ordered spaces. This correspondence is invoked as a well-known fact rather than derived or fitted inside the paper. The generalization to realizations of precubical sets and the uniqueness result for R^n are presented as consequences of the same techniques, without any equation or definition reducing a claimed prediction or universal property to a self-referential input, a fitted parameter, or a self-citation chain. The reference to prior work on graph products serves only as motivation and does not bear the load of the central adjunction or coreflectivity claim. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard sheaf-étale bundle correspondence and generalizes an existing result on graph products without introducing new free parameters or postulated entities.

axioms (1)

standard math Correspondence between sheaves and étale bundles
Invoked as the basis for constructing the universal Euclidean local order.

pith-pipeline@v0.9.0 · 5682 in / 1083 out tokens · 47662 ms · 2026-05-22T14:22:58.821568+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We build the universal euclidean local order over every locally ordered space; in categorical terms, the subcategory of euclidean local orders is coreflective... based on a well-known correspondance between sheaves and étale bundles.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Combinatorial manifolds and Kleene's theorem, homotopically
math.CT 2026-05 unverdicted novelty 6.0

A categorical method builds combinatorial manifolds as coreflective subcategories and applies it to grids in precubical sets and automata for Kleene's theorem.