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# Non-Hausdorff manifolds over locally ordered spaces via sheaf theory

Yorgo Chamoun, Emmanuel Haucourt

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Yorgo Chamoun, Emmanuel Haucourt. Non-Hausdorff manifolds over locally ordered spaces via sheaf theory. 2025. hal-05101222v2

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Distributed under a Creative Commons Attribution 4.0 International License
NON-HAUSDORFF MANIFOLDS
OVER LOCALLY ORDERED SPACES
VIA SHEAF THEORY

Yorgo CHAMOUN and Emmanuel HAUCOURT

**Résumé.** Les *espaces localement ordonnés* peuvent être utilisés pour modéliser les programmes concurrents. En effet, l'ordre local modélise dans ce cas l'irréversibilité du temps durant l'exécution du programme. Sous certaines conditions, on peut même se ramener à des *variétés localement ordonnées*. Dans cet article, nous construisons l'ordre local *euclidien* universel au-dessus d'un espace localement ordonné quelconque. En termes catégoriels, nous montrons que la sous-catégorie des espace localement ordonnés euclidiens est *coréflexive* dans la catégorie des espace localement ordonnés. Puis nous donnons des conditions assurant le relèvement des *traces d'exécution* du programme modélisé par l'espace, le cas échéant. Notre construction d'un adjoint se base sur la correspondance bien connue entre *faisceaux* sur un espace topologique et *espaces étalés* au-dessus de cet espace. Notre résultat est une vaste généralisation d'un résultat connu pour les *réalisations* de produits de graphes. De plus, dans le cas des ordres locaux provenant de la réalisation d'un *ensemble précubique*, nous montrons que notre construction admet une description purement combinatoire. Nous mettons à profit les lemmes prouvés dans ce but pour donner un résultat d'unicité étonnant à propos des ensembles précubiques dont la réalisation localement ordonnée est isomorphe à $\mathbb{R}^n$.

**Abstract.** *Locally ordered spaces* can be used as topological models of concurrent programs: in that setting, the local order models the irre-
versibility 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 $\mathbb{R}^n$.

**Keywords.** Locally ordered spaces, manifolds, geometric realization, precubical sets, sheaves, concurrency.

**Mathematics Subject Classification (2010).** 18F20, 54B40, 68Q85.

# 1. Introduction

One can define an automaton on a set of letters $\Sigma$ as a graph morphism $f$
whose target is a $\Sigma$-bouquet, i.e. a graph with a single vertex whose set of
arrows is $\Sigma$. This definition is motivated by the idea that each arrow $a$ rep-
resents the execution of its label $f(a)$. Given that graphs are 1-dimensional
precubical sets, one defines a higher dimensional $\Sigma$-automaton (hda for
short) as a precubical morphism $f$ whose target is a higher dimensional $\Sigma$-
bouquet, i.e. a precubical set whose $n$-dimensional elements are the words of
length $n$ on $\Sigma$. In that setting, each $n$-dimensional element $a$ represents the
simultaneous execution of instructions $i_1, \dots, i_n$ with $f(a) = (i_n, \dots, i_n)$.
Higher dimensional automata thus provide a model of 'true' concurrency
[Pra91, vG91] which was proven to be very expressive [vG06]. On the
other hand, in algebraic topology, cubical sets, i.e. precubical sets with
degeneracies, provide models for combinatorial homotopy theory, which
is related to usual homotopy theory by the geometric realization functor
[BHS11, p. 371-372]. The latter preserves 'local causality' in the sense that
the geometric realization of a precubical set morphism (taken in a certain
full subcategory of precubical sets) is a locally order preserving map. This
remarkable feature motivates the use of methods from algebraic topology in
concurrency theory [FRG06].

In a simplified setting, a single program can be represented by a graph,
and parallel programs can be represented by the tensor product of the
corresponding graphs. We can then realize this tensor product as a locally
ordered space, which is just the product of the realizations of the single
graphs. In fact, tensor products of graphs can even be represented by
locally ordered non-Hausdorff manifolds, so we can take advantage of
the differential machinery to study them [Hau25]. For example, [Hau25]
defines a pseudo-metric on this manifold, which represents the execution
time of execution traces of the parallel programs. However, tensor products
of graphs are not enough to represent parallel programs, since we often need
to ‘remove’ some cubes which represent a combination of simultaneous
operations which is not allowed, for example because of conflicts on shared
memory or resources. A bestiary of examples can be found in [FGH<sup>+</sup>16].
The main purpose of this article is to show that the class of precubical sets
that can be represented by manifolds is much larger than that of tensor
products of graphs. As an introduction, we study two prototypical examples.

The locally ordered realization of the graph $G$ is the locally ordered
subspace $|G|_{lo} = \{(x, y) \in [-1, 1]^2; xy = 0\}$ of $\mathbb{R}^2$ (Fig. 1). We observe
that $|G|_{lo}$ is locally isomorphic to $\mathbb{R}$ (as a locally ordered space) at every
point except $(-1, 0)$, $(1, 0)$, $(0, -1)$, $(0, 1)$ and the origin; these points are
said to be *singular*. Boundary points of a geometric model do not play
any significant role in the semantics of a program, so they can be dropped
[Hau25, p. 10]; as a *branching*, the role of the origin is semantically crucial.

**Definition 1.1.** Two subsets $A$ and $B$ of a locally ordered space $X$ are said to be equivalent at $p$, which we denote by $A \sim_p B$ when there exists a neighborhood $V$ of $p$ such that $A \cap V = B \cap V$; the relation $\sim_p$ is an equivalence. The germ at $p$ of $A \subseteq X$, which we denote by $A_p$, is the $\sim_p$-equivalence class of $A$. For every $n \in \mathbb{N}$, the $n$-traversals at $p$ are the germs at $p$ of the subsets $A \subseteq X$ such that $p \in A$, $A$ is a subset of some ordered open set $U$ of $X$, and $A \cong \mathbb{R}^{\otimes n}$ as ordered spaces.
<figure><img src="image_3.png" /><figcaption>Figure 1: The 4 traversals at the origin in $\{(x, y) \in [-1, 1]^2; xy = 0\}$.</figcaption></figure>

The locally ordered space $|G|_{lo}$ is 1-dimensional (i.e. the greatest $n \in \mathbb{N}$ for which $|G|_{lo}$ has a subspace isomorphic to $\mathbb{R}^{\otimes n}$ is 1). A point $p$ is singular in $|G|_{lo}$ precisely when there are at least two 1-traversals at $p$; the four 1-dimensional traversals at the origin of $|G|_{lo}$ are shown on Fig. 1.

The origin of $|G|_{lo}$ has four 1-traversal which correspond to the four paths from the left and down vertices of $G$ to its up and right ones; this can be expressed in purely combinatorial terms, by means of edges entering and leaving the origin, which is the approach followed by [Hau25] to blow up the realization of such a graph. The construction leaves the edges unchanged, and puts over a vertex the number of points corresponding the the number of traversals at this vertex. See Fig. 3 of [Hau25] for an exemple similar to ours. In fact, it is proved there that in the case of graph products, the blowup of the resulting locally ordered space exists and has the expected universal property:

**Theorem 1.2** ([Hau25, §5]). Let $X$ be a product of the locally ordered realizations of $n$ graphs $G_i$, $i \in \{1, \dots, n\}$, i.e. $X = \prod_i |G_i| \cong |\otimes_i G_i|$. There is a euclidean local order $\tilde{X}$, called the blowup of $X$, and a morphism of locally ordered spaces $\beta_X : \tilde{X} \to X$, called the blowup map, satisfying the following universal property: for any euclidean local order $E$ of dimension $n$, for any local embedding $f : E \to X$, there is a unique continuous map $\tilde{f} : E \to \tilde{X}$ making the following diagram commute.

<figure><img src="image_4.png" /></figure>
Moreover, $\tilde{f}$ is a local embedding.

In particular, the blowup map itself is a local embedding. We call this
property the (unique) lifting property (one also gets a lifting property for
directed path, under a combinatorial property of the graphs). This looks like
an adjunction involving an embedding of categories, and this will be made
precise in the present paper.

Now let us see an example which is not a product of graphs. Let $I$ be the
graph $(-1) \to 0 \to 1$. A point of $I$ is either a vertex (i.e. $-1, 0$, or $1$) or an
arrow (i.e. $(-1) \to 0$ or $0 \to 1$); the dimension of a vertex (resp. an arrow)
is $0$ (resp. $1$). The points of $C$, the 3-fold tensor product of $I$, are the 3-tuples
$(x, y, z)$ of points of $I$ with $\dim_C(x, y, z) = \dim_I(x) + \dim_I(y) + \dim_I(z)$.
One of the simplest examples of precubical set $K$ that is not a tensor
product of graphs is the precubical subset $\{(x, y, z) \in C; 0 \in \{x, y, z\}\}$
of $C$. The locally ordered realization of $K$ is the locally ordered subspace
$|K|_{lo} = \{(x, y, z) \in [0, 1]^3; xyz = 0\}$ of $\mathbb{R}^3$ (Fig. 2). The locally ordered
space $|K|_{lo}$ is 2-dimensional, and it is locally homeomorphic to $\mathbb{R}^2$ at every
point $(x, y, z)$ such that only one of the coordinates is $0$ and none of them
belongs to $\{-1, 1\}$. When exactly two coordinates of $(x, y, z)$ are $0$, $|K|_{lo}$
is locally isomorphic to $\mathbb{R} \times \{(x, y) \in \mathbb{R}^2; xy = 0\}$ (Fig. 1). The only point
at which the desingularization method from [Hau25] does not work is the
origin, which has nine 2-traversals shown on Fig. 2.

<figure><img src="image_5.png" /><figcaption>Figure 2: The 9 traversals at the origin in $\{(x, y, z) \in [0, 1]^3; xyz = 0\}$.</figcaption></figure>
Our blowup construction can handle this kind of singularities, and the points of the blowup of $K$ (a short for ‘the (2-dimensional) blowup of the locally ordered space $|K|_{lo}$’) over $(x, y, z)$ are taken to be the 2-dimensional traversals of $|K|_{lo}$ at $(x, y, z)$. The number of traversals at $(x, y, z)$ is

a) $0$ if $P(x, y, z) = 0$,

b) $1$ if $P(x, y, z)Q(x, y, z) \neq 0$,

c) $4$ if $P(x, y, z) \notin \{0, 1\}$ and $Q(x, y, z) = 0$

d) $9$ if $x = y = z = 0$.

The points of the blowup being the traversals, a base of its ordered topology is given by the subsets of the form $\{A_x | x \in A\}$ for a fixed $A \cong \mathbb{R}^2$. An exhaustive description of an *atlas* of the blowup would be fastidious, nevertheless, by way of an example, a *chart* $\phi$ at the traversal $\{yz = 0 \wedge y \ge 0 \wedge z \le 0\}$ of the origin is given by $\phi(x, 0, z) = (x, z)$ and $\phi(x, y, 0) = (x, y)$. In a similar fashion we obtain charts at the other traversals of the origin; it is easy to check that the *transition maps* between such charts are identities (in particular they are smooth and order preserving).

By using the above procedure, we generalize the blowup construction of Theorem 1.2 to *any* locally ordered space. The construction is functorial, and provides the right adjoint to the inclusion of the full subcategory of euclidean local orders into that of locally ordered spaces (§4). Before this, we recall some well-known facts about precubical sets and locally ordered spaces (§2), and generalize the gluing of topological spaces to the locally ordered setting (§3). Then we focus on locally ordered realizations of precubical sets (which form a large class of locally ordered spaces), for which blowups have a purely combinatorial description, and which can be used in practice; this provides a direct generalization of the approach in [Hau25]. By passing, we prove a bunch of theorems which are interesting on their own, about locally ordered realizations of precubical sets (§5). We conclude with a discussion about the path lifting property, which is important since directed path represent the execution traces of a program in the context of modeling concurrent systems, and we do not want to ‘forget’ these traces; so we give conditions to enforce this property (§6).
## 2. Preliminaries

### 2.1 Sheaves and presheaves

The paper will heavily rely on the machinery of sheaves (on topological spaces) and presheaves (on an arbitrary category), for example as defined in [MM12] (§II.1 and §I.1 (vii) respectively).

Recall that given a category $\mathcal{C}$, a presheaf $P$ on $\mathcal{C}$ is a functor $P : \mathcal{C}^{op} \to \text{Set}$, i.e. a collection of sets $P(c)$ for every object $c$ of $\mathcal{C}$ and maps $P(c) \to P(d)$ for every arrow $d \to c$ of $\mathcal{C}$, satisfying $P(f \circ g) = P(g) \circ P(f)$ and $P(id_c) = \text{id}_c$ for all morphisms $f, g$ and objects $c$ (where $\text{id}_c$ denotes the identity morphism at $c$).

Recall also that a sheaf on a topological space $X$ is a presheaf on the poset category $O(X)$ of open sets of $X$ satisfying an amalgamation property with respect to open covers. Namely, a presheaf $F$ on $O(X)$ is a sheaf is and only if for every open set $U$ and every open cover $(U_i)_{i \in I}$ of $U$, the following diagram is an equalizer:

$$F(U) \longrightarrow \prod_{i \in I} F(U_i) \Longrightarrow \prod_{i,j \in I} F(U_i \cap U_j)$$

where the upper morphism is given by the $\prod_{i \in I} F(U_i) \to F(U_i) \to F(U_i \cap U_j)$ and the lower morphism is given by the $\prod_{i \in I} F(U_i) \to F(U_j) \to F(U_i \cap U_j)$. The standard example is the sheaf which associates to an open set $U$ the set of continuous functions $U \to \mathbb{R}$; the above diagram being an equalizer just says that a continuous function $f$ on $U$ is equivalent to a family of continuous functions $f_i : U_i \to \mathbb{R}$ for $(U_i)_{i \in I}$ an open cover of $U$ satisfying $(f_i)_{|U_i \cap U_j} = (f_j)_{|U_i \cap U_j}$ for all $i, j \in I$. Thus, sheaves can be very useful in situations where we want to study the interplay between local and global phenomena.

### 2.2 Graphs et precubical sets

Let $\mathcal{G}$ be the category with two objects and two parallel morphisms $0 \to 1$. A graph is a presheaf on $\mathcal{G}$. Given a graph $P$, $P(0)$ is the set of vertices, $P(1)$ the set of arrows, and the functions $P(1) \to P(0)$ are the source and
the target, noted respectively s and t.

Precubical sets can be naturally seen as the generalization of graphs in higher dimension, in the following sense. Let □ be the free monoidal category on G with unit 0. In particular, the objects are the natural numbers. A precubical set can then be defined as a presheaf on □. More explicitly, a precubical set P is a collection of sets P(n) (or P_n for convenience) which are called the set of n-cubes of P (for n a natural number), and a collection of maps P(n) → P(k), k ≤ n, which are called the k-face maps. More precisely, we note δ^ϵ_{n,i} : P(n) → P(n − 1), with ϵ ∈ {0, 1}, 0 ≤ i < n the (n − 1)-face maps corresponding to every dimension, whose interpretation is the following: δ^ϵ_{n,i} associates to an n-cube its back (ϵ = −) or front (ϵ = +) face in direction i. Any k-face is obtained by successively applying maps of the form δ^ϵ_{k,i}, and these are subject to the cocubical relations

$$ \delta_{k,j}^{\epsilon} \delta_{k+1,i}^{\eta} = \delta_{k,i}^{\eta} \delta_{k+1,j+1}^{\epsilon} $$

for 0 ≤ i ≤ j < n. This completely describes the precubical set P. A face of a cube obtained by successively applying morphisms of the form δ^+_{k,i} (respectively δ^-_{k,i}) is called an upper (respectively lower) face. A vertex which is an upper (respectively lower) face is called the maximum (respectively the minimum) of the cube. A precubical set P is called n-homogeneous if the set of k-cubes of P for k ∈ N is exactly the set of k-faces of n-cubes of P. The tensor product of two precubical sets P and Q is defined by (P ⊗ Q)_n = ⋃_{i+j=n} P_i × Q_j with the expected face maps. See for example [FGH$^+$16, §3.4.1].

## 2.3 Locally ordered spaces

An ordered space is a topological space with an order relation. A morphism of ordered spaces is a continuous order-preserving map. An isomorphism of ordered spaces is called a dihomeomorphism. However, this notion is too strict in general. For example, we would like to define a directed path in X to be a morphism [0, 1] → X, where [0, 1] has the natural topology and order, but then no directed loop exists in X, which is a problem as soon as we try to define a satisfying directed version of the circle. This motivates the following definition: a locally ordered space, or local order, is a topological
space $X$ which has an open base of ordered sets. More formally, let $X$ be a topological space. An *ordered base* $\mathcal{B}$ of $X$ is a collection of (partially) ordered sets satisfying the following conditions:

* For every $B \in \mathcal{B}$, (the underlying set of) $B$ is an open subset of $X$

* $\mathcal{B}$ generates the topology of $X$

* for every $B, B' \in \mathcal{B}$ and $x \in B \cap B'$, there is $B'' \in \mathcal{B}$ such that $x \in B'' \subseteq B \cap B'$ and $B''$ is a subposet of $B$ and $B'$.

A locally ordered space is defined to be a topological space equipped with an ordered base. The elements of $\mathcal{B}$ are then called *ordered open sets*. We can now define the directed circle to be the local order whose underlying topological space is $\mathcal{S}^1$, and whose ordered open sets are the proper open sets of $\mathcal{S}^1$ with the order induced by, say, the clockwise direction. A *morphism of local orders* (respectively a *local embedding*) $f: X \to Y$ is a continuous map which is locally order preserving (respectively an embedding), i.e. for every $x \in X$ and ordered neighborhood $V$ of $f(x)$, there exists an ordered neighborhood $U$ of $x$ such that $f(U) \subseteq V$ and $f|_U$ is order preserving (respectively order preserving and admitting a continuous order preserving inverse on its image). A locally ordered space is *euclidean* (of dimension $n$) if every point has an ordered neighborhood which is dihomeomorphic to $\mathbb{R}^n$ (with the product order). Note that euclidean local orders can be though of as 'locally ordered manifolds'. A directed path is a morphism of locally ordered spaces whose domain is euclidean of dimension 1 and connected. We note $\mathcal{L}$ the category of local orders and locally ordered maps, $\mathcal{L}_e$ the category of locally ordered spaces and local embeddings, and $\mathcal{E}_n$ the category of euclidean local orders of dimension $n$ and local embeddings. Equivalent definitions of locally ordered spaces are listed in [CH24].

## 2.4 Locally ordered realization of precubical sets

In algebraic topology, we are used to having a realization functor from combinatorial to continuous models. For example, every precubical set $P$ admits a geometric realization $|P|$, which is simply given by Kan extension. An explicit description can be found in [FRG06, definition 6.5]; we mention that the underlying set of $|P|$ is $\bigcup_{n \in \mathbb{N}} P_n \times]0, 1]^n$, and that $|P|$ is sequential since it is a quotient of sequential spaces. For $c \in P_n$, $|c|$ will mean the realization of the subprecubical set of $P$ spanned by $c$, i.e. $c$ and all its faces
in all dimensions. Note that $|c|$ is closed in $|P|$. Also note that $\{c\} \times ]0, 1[^n$ is open and included in $|c|$ the interior of $|c|$, and they are in fact equal when all the faces of $c$ are distinct. For an open set $U$ of $|P|$ and a cube $c$, we say that $c$ is a cube of $U$, or $U$ intersects $c$, if $U \cap \{c\} \times ]0, 1[^{\dim(c)} \neq \emptyset$.

In the directed case, the situation is more complicated because the category of locally ordered spaces is not cocomplete [CH24]. Given a pre-cubical set $P$, we call locally ordered realization of $P$ any locally ordered space $|P|_{lo}$ (with ordered base $\mathcal{B}$) satisfying the following conditions:

(i) The underlying topological space of $|P|_{lo}$ is the geometric realization of $P$ (in particular the underlying set is the same).

(ii) For every $B \in \mathcal{B}$, the preorder given by paths which are increasing on each cube (which we call cubewise increasing) is a partial order and coincides with the order on $B$.

(iii) For every $B \in \mathcal{B}$, $B$ is order convex in the sense that for every $x, y \in B$, if $x \in \{c\} \times ]0, 1[^{\dim(c)}$ and $y \in \{c'\} \times ]0, 1[^{\dim(c')}$ with $c$ a lower face of $c'$ and such that the segment $]x, y[$ is directed in $\{c'\} \times ]0, 1[^{\dim(c')}$, then $x \le y$ in $B$.

Note that there is at most one such realization up to isomorphism, so we can soundly talk about the locally ordered realization of a precubical set. Such a realization is known to exist for the (large) class of non-self-linked precubical sets [FRG06, §6.5]; it actually exists in even bigger generality, as will be shown in a forthcoming paper of the second author. We will also suppose that the ordered base $\mathcal{B}$ satisfies the following property, which will be of technical use (see Lemma 5.3 and the discussion after it):

(iv) For every vertex $v$ of $P$, an ordered neighborhood $B$ of $v$ only intersects cubes adjacent to $v$, and for every cube $c$ with minimum (respectively maximum) $v$, for every point $y \in \{c\} \times ]0, 1[^{\dim(c)}$, every point $y \ge x$ (respectively $y \le x$) in $B$ belongs to $\{c'\} \times ]0, 1[^{\dim(c')}$ for some cube $c'$ having $c$ as lower (respectively upper) face.

### 3. Colimits of local orders

**Definition 3.1.** Let $X$ be a locally ordered space with ordered base $\mathcal{B}$, and $U \subseteq X$ an open subset of $X$. Then $U$ can naturally be seen as a locally
ordered space via the ordered base {$B \cap U | B \in \mathcal{B}$}. We say that $U$ is equipped with the induced local order.

As mentioned before, the category $\mathcal{L}$ is not cocomplete. However, some useful colimits do exist. Let $(X_i)_{i \in I}$ be a family of local orders. Suppose that there exists a family $(X_{ij}, \phi_{ij})_{i,j \in I}$ such that (the underlying set of) $X_{ij}$ is an open subset of $X_i$ and $\phi_{ij}$ is an isomorphism of locally ordered spaces $X_{ij} \cong X_{ji}$ (where the local orders are the induced ones), subject to the following conditions:

$$1. \phi_{ij}^{-1} = \phi_{ji}$$

$$2. \phi_{ij}(X_{ij} \cap X_{ik}) = X_{ji} \cap X_{jk}$$

$$3. \phi_{jk} \circ \phi_{ij} = \phi_{ik} \text{ on } X_{ij} \cap X_{ik}$$

Then we can consider the diagram of local orders consisting of the $(X_i)_i$ and the $(X_{ij})_{ij}$ together with the inclusions $\iota_{ij}: X_{ij} \hookrightarrow X_i$ and the isomorphisms $(\phi_{ij})_{ij}$. We want to show that this diagram admits a colimit.

**Proposition 3.2.** *The diagram constructed above admits a colimit in $\mathcal{L}$.*

*Proof.* First we form its colimit in the category of topological spaces. Note $X$ this colimit. Using [Duc14, (3.2.13)], we know that the structure maps $X_i \to X$ are open immersions, in the sense that they induce an isomorphism (of topological spaces) between $X_i$ and an open subset of $X$, and $X$ is the union of (the image of) all the $X_i$, $i \in I$, the intersection of $X_i$ and $X_j$ in $X$ being (the image of) $X_{ij}$ (or equivalently $X_{ji}$). Now we have to exhibit an ordered base on $X$. For every $i \in I$, there is an ordered base $(B_k^{(i)})_{k \in K(i)}$ of $X_i$. Consider the collection of (the image of) all these families in $X$. We just need to show that if $x \in B_k^{(i)} \cap B_{k'}^{(j)}$ then there is some $B_{k'}^{(l)}$ containing $x$, included in $B_k^{(i)} \cap B_{k'}^{(j)}$ and such that the corresponding orders coincide. But this is obvious using the fact that $X_{ij}$ and $X_{ji}$ are isomorphic as locally ordered spaces under $X$, and that the $(B_k^{(i)})_{k \in K(i)}$ for a fixed $i$ form an ordered neighborhood base of $X_i$. Now clearly this is the "maximal" local order structure on $X$ such that the structure maps are local order morphisms, and in fact we get that they are local order isomorphisms on their image. The universal property of the colimit follows from this and from the universal property of the colimit in the category of topological spaces, which concludes the construction. $\square$

**Definition 3.3.** *We call the above construction the gluing of the family $(X_i)$ along $(X_{ij})_{ij}$.*
## 4. Blowing up via sheaves

As suggested in the introduction, the blowup of a local order $X$ is a bundle whose fibers over $x \in X$ are the equivalence classes of ways of embedding $\mathbb{R}^n$ into $X$ by sending $0$ to $x$. In fact, there is a construction in sheaf theory which essentially does exactly this. This will lead to a very vast generalization of the construction of Theorem 1.2, to all local orders, and we will still have the same nice universal property. However, as usual for abstract vast generalization, our construction is not always well-behaved, in the sense that it sometimes 'forgets' too many points, which may lead to the loss of the directed paths lifting property. This will be dealt with in §6.

**Definition 4.1.** Let $X$ be a local order. We define the sheaf of $n$-traversals $O_X^{(n)}$ by the setting:

$$O_X^{(n)}(U) := \{A \subseteq U \mid A \cong E \text{ with } E \text{ some } n\text{-euclidean local order}\}$$

for an open set $U \subseteq X$, the restriction map $O_X^{(n)}(V) \to O_X^{(n)}(U)$ for $U \subseteq V$ being given by intersection with $U$. For $A \in O_X^{(n)}(U)$ and $x \in U$, $A_x$ will denote the germ of $A$ at $x$ (in the sheaf-theoretic sense).

**Remark 4.2.** 1. The restriction maps are well-defined because an open subset of a euclidean local order is a euclidean local order.

2. This is a priori only a presheaf (but we will prove that it is a sheaf), and the sections are not $n$-traversals (but we will prove that the germs essentially are).

3. The empty set is trivially a euclidean local order, so we always have an element $\emptyset_x \in O_{X,x}^{(n)}$. This is in fact crucial to make $O_X^{(n)}$ into a presheaf.

We can think of $O_X^{(n)}(U)$ as the set of equivalence classes of euclidean embeddings in $U$ with respect to the relation 'having the same image'. So we may write $[f : E \to U]$ to mean $\mathrm{Im}(f)$.

**Proposition 4.3.** For every local order $X$, $O_X^{(n)}$ is a sheaf.

*Proof.* We have to prove that given a family of open sets $(U_i)_{i \in I}$ and a family of subsets $(A_i)_{i \in I}$ such that $A_i \in O_X^{(n)}(U_i)$ and $A_i \cap U_j = A_j \cap U_i$ for all $i, j \in I$, there is a unique $A \in O_X^{(n)}(U)$ such that $A \cap U_i = A_i$ for all $i$. So fix such families. Uniqueness is obvious since we have local uniqueness on
an open cover. So we just need to prove existence. Clearly if $A$ exists then
$A = \bigcup_i A_i$, so set $A$ like this. Choose a representative $[f_i : E_i \to U_i] = A_i$
for all $i$. We want to use §3 to glue the $E_i$. For every $i, j$, define $E_{ij} :=$
$f_i^{-1}(A_i \cap A_j)$. Notice that $E_{ij} = f_i^{-1}(A_i \cap U_j)$, so it is an open subset of $E_i$,
because $A_i \cap U_j$ is an open subset of $A_i$. Let $\phi_{ij} := f_j^{-1} \circ f_i : E_{ij} \to E_{ji}$.
All the conditions of §3 are trivially verified, and we note $E$ the colimit
representing the gluing of the $E_i$ along the $E_{ij}$. By the universal property of
the colimit, the family $(f_i)_{i \in I}$ defines a map $f : E \to A$. $E$ is euclidean and
$f$ is an isomorphism whose inverse is given by the morphism induced by the
family $(f_i^{-1})_{i \in I}$ (note that $A$ is the gluing of the family $A_i$), because these
properties can be checked locally. This finishes the proof.
□

**Proposition 4.4.** For a local order $X$, if $A_x$ is an element of the stalk $O_{X,x}^{(n)}$ at some point $x \in X$ such that $x \in A$, then $A_x$ is a $n$-traversal. Conversely, every $n$-traversal is of this form.

*Proof*. This is a direct consequence of the two following facts: every isomorphism of ordered spaces is an isomorphism of locally ordered spaces, and every isomorphism of locally ordered spaces is locally (i.e. up to restriction) an isomorphism of ordered spaces.

□

One could argue that, in view of definition 1.1, we are only interested in the
images of order embeddings from an open subset of $\mathbb{R}^n$ to some ordered
open set of $X$, but this would not give a sheaf, only a presheaf. In fact,
taking the sheafification of this presheaf would give exactly $O_X^{(n)}$, which
justifies this definition. This is an easy consequence of the fact that they
have the same set of germs at every point $x \in X$, and essentially expresses
the fact that a euclidean local order is an amalgamation of copies of $\mathbb{R}^n$.

An étale bundle over a space $X$ is a morphism of topological spaces $f : Y \to X$ such that for every $y \in Y$, there is a neighborhood $U$ of $y$ such that $f|_U$ is an isomorphism of topological spaces. Recall that there is an equivalence of categories between étale bundles over a topological space $X$ and sheaves on $X$ [MM12, §II.6]. This has been extended in [BW05, Theorem 5.4] to the locally ordered setting, i.e. replacing 'topological space' by 'locally ordered space'. In fact, we can reformulate this result by noticing that a structure of local order can be pulled back in a unique
way along a local homeomorphism, like for smooth manifold structures. We want to use this operation in order to blow up any local order. The next proposition is just a translation of the mentioned results in our case. We fix a natural number $n$.

**Proposition 4.5.** Let $X$ be a local order. The étale bundle corresponding to $O_X^{(n)}$, noted $\beta_X^+ : B(O_X^{(n)}) \to X$, can be described as follows:

• Its underlying set map is given by the projection $\sqcup_{x \in X} O_{X,x}^{(n)} \to X$;

• An ordered base is given by the sets $U_A^+ := \{A_y \mid y \in U\}$ for $A \in O_X^{(n)}(U)$, $U$ ordered open set, $A$ order isomorphic to an open subset of $\mathbb{R}^n$; the order relation is the one inherited from $U$.

Moreover, for every $U$ open and $A \in O_X^{(n)}(U)$, the function $U \to B(O_X^{(n)})$, $x \mapsto A_x$ is an isomorphism of local orders on its image, and $\beta_X^+$ is a local dihomeomorphism.

Now we have an étale bundle over $X$. However, the blowup of $X$ cannot possibly be an étale bundle, since it is always euclidean (and not $X$). Recall that for every ordered open $U \subseteq X$, the empty set belongs to $O_X^{(n)}(U)$. This gives a ‘ghost version’ of $X$ in $B(O_X^{(n)})$, which ‘completes’ the blowup into an étale bundle. We then just need to remove it. An easy example is shown in Fig. 3.

In fact, in light of Proposition 4.4, we have to remove all the $A_x$ with $x \notin A$. These are not necessarily of the form $\emptyset_x$, for example one can consider the germ of $]0, +\infty[$ at $0$ in $\mathbb{R}$. However, this subtlety will vanish when we will consider the combinatorial case, since all the ‘combinatorial embeddings’ will be ‘maximal’ in an appropriate way. To give a first intuition, if we take again the example of $\mathbb{R}$, we see that the germ of $]0, +\infty[$ at a negative point $x$ is $\emptyset_x$, and the germ at a positive point is a traversal, so the germ at $0$ serves as a ‘transition’ between the traversals and the germs of the empty set. But when the embedding is ‘maximal’, for example the embedding of $\mathbb{R}$ in $\mathbb{R}$, no transition is needed.

**Definition 4.6.** Let $X$ be a local order. We set $\tilde{X} := \sqcup_{x \in X} \{A_x \in O_{X,x}^{(n)} \mid x \in A\} \subseteq B(O_X^{(n)})$ (with the induced topology and local order), and we call it the $n$-blowup of $X$. The restriction of $\beta_X^+$ to $\tilde{X}$ is noted $\beta_{\tilde{X}}$ and called the $n$-blowup map. The restriction of the ordered base $(U_A^+)$$_{U,A}$ is noted $(U_A)$$_{U,A}$.
<figure><img src="image_6.png" /><figcaption>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 $A_x$, $A \neq \emptyset$, $x \notin A$ for simplification. The blowup, as described in the introduction, corresponds to the complement of the ghost version.</figcaption></figure>

We will omit the dimension $n$ when it is clear from the context. Note that in particular, $\tilde{X}$ is a euclidean local order: any point belongs to some open set $V_A$ isomorphic to an open subset of $\mathbb{R}^n$, namely $A$.

**Lemma 4.7.** With the same notations, for any ordered open $U_A$, $(\beta_X)|_{U_A}$ is a dihomeomorphism on its image $A$. In particular, $\beta_X$ is a local embedding, and $\tilde{X}$ is euclidean.

*Proof*. The inverse is given by $x \mapsto A_x$, which is continuous by Proposition 4.5, since it is the restriction of a continuous map. The orders correspond by definition. $\square$

**Theorem 4.8.** Let $X$ be any local order. Then its $n$-blowup $\tilde{X}$ satisfies the following universal property: for any euclidean local order $E$ of dimension $n$, for any local embedding $f : E \to X$, there is a unique continuous map $\tilde{f} : E \to \tilde{X}$ making the following diagram commute

<figure><img src="image_7.png" /></figure>
Moreover, $\tilde{f}$ is a local embedding.

**Proof.** First we prove uniqueness. Suppose that $\tilde{f}$ exists. Take $x \in E$. Because of the commutation of the above diagram, $\tilde{f}(x) \in O_{X,f(x)} \setminus \{[\emptyset]\} \subseteq \tilde{X}$. Note that $\tilde{f}(x)$ corresponds to the equivalence class of a subspace $A$ of an ordered open $V$ of $X$, so $V_A$ is an ordered neighborhood of $\tilde{f}(x)$. Since $\tilde{f}$ is continuous, there is an ordered neighborhood $U$ of $E$, that we can assume isomorphic to $\mathbb{R}^n$, such that $\tilde{f}(U) \subseteq V_A$. Composing with $\beta_X$, we get that $f(U) \subseteq A$. But this means that $A \sim_{f(x)} f(U)$: $V_{f(U)}$ is an open subset of $V_A$ because they are both open, and $\beta_X$ restricted to $V_A$ is a homeomorphism on its image $A$, so $f(U)$ is open in $A$, i.e. there is an open set $W$ of $X$ such that $f(U) = A \cap W$ (one can also use the theorem of invariance of domain). So $f(x)$ is necessarily the class of $f(U)$. This proves uniqueness. Now for existence, just define $\tilde{f}$ as suggested. More precisely, since $f$ is a local embedding, we can restrict to an open subset $U$ of $E$ where it is an isomorphism (of ordered spaces, but we don't really need that) on its image. Then we define $\tilde{f}$ on $U$ by $x \mapsto f(U)_{f(x)}$ (using the notations of Proposition 4.5). Note that it is different from $[\emptyset]$ because $f(x) \in f(U)$. Now this is well-defined on all $E$ because if $U$ and $V$ are two such open subsets with $x \in U \cap V$, then $f(U) \sim_{f(x)} f(V)$ if $U$ and $V$ are neighborhoods of $x$, because they are both equivalent to $f(U \cap V)$. Then Proposition 4.5 and the fact that $f$ is a local embedding automatically gives that $\tilde{f}$ is a local embedding. $\square$

In fact, we can study the functoriality of the blowup in general.

**Proposition 4.9.** Let $f : X \to Y$ be any map between local orders. The following are equivalent:

1. There is a unique continuous map $\tilde{f} : \tilde{X} \to \tilde{Y}$ making the following square commute.

<figure><img src="image_8.png" /></figure>

Moreover, $\tilde{f}$ is a local embedding.
2. There is a local embedding $\tilde{f} : \tilde{X} \to \tilde{Y}$ making the above square commute

3. For every local embedding $g : E \to X$ with $E$ euclidean, $f \circ g$ is a local embedding.

*Proof*. Clearly (i) implies (ii). Now suppose (ii), and take a local embedding $g : E \to X$. It admits a lift $\tilde{g} : E \to \tilde{X}$ which is a local embedding, so $f \circ g = \beta_Y \circ \tilde{f} \circ \tilde{g}$ is a local embedding, so (ii) implies (iii). Finally, if we suppose (iii) then $f \circ \beta_X$ is a local embedding and $\tilde{X}$ is euclidean, so there is a unique continuous lift which is moreover a local embedding, by Theorem 4.8. □

A morphism of local orders satisfying one of the above conditions is called weakly euclidean. Notice that the uniqueness part of (i) shows that $X \mapsto \tilde{X}$ is functorial from the category of local orders and weakly euclidean morphisms to the category of euclidean local orders and local embeddings. In fact, we can weaken the definition of weakly euclidean morphism if we restrict to a subcategory of locally ordered spaces to locally strongly connected spaces. This is interesting because typical examples of locally strongly connected spaces are given by locally ordered realizations of precubical sets, essentially by definition.

**Definition 4.10.** An ordered space $X$ is called strongly connected if for all $x, y \in X$, $x \le y$ if and only if there is a directed path $\gamma : \mathbb{R} \to X$ and $a, b \in \mathbb{R}$ with $a \le b$ and $\gamma(a) = x, \gamma(b) = y$. A local order $X$ is said to be locally strongly connected if $X = \cup_i U_i$ with $U_i$ ordered strongly connected spaces.

Note that in the definition of strongly connected ordered space, one direction is superfluous. Indeed, take $x \le y$ in $X$ strongly connected, $\gamma : \mathbb{R} \to X$ directed path and $a, b \in \mathbb{R}$ with $a \le b$ and $\gamma(a) = x, \gamma(b) = y$. We have $[a, b]$ compact, so we can cover it with open intervals $[a_i, b_i]_{i \in I}$, $I$ finite and $\gamma$ order preserving on each $]a_i, b_i[$. We can suppose that $I = \{0, \dots, n\}$ and $i \le j$ implies $a_i \le a_j$, and that $]a_i, b_i[ \cap [a, b] \ne \emptyset$ for all $i$. An easy induction then shows that $\gamma(a) \le \gamma(a_i)$ for all $i > 0$ (two consecutive intervals intersect because the union of all the intervals covers $[a, b]$). Then we conclude that $\gamma(a) \le \gamma(b)$ because $b \in ]a_n, b_n[$.
**Lemma 4.11.** Let $X$ and $Y$ be local orders with $X$ locally strongly connected, $f: X \to Y$ a continuous map. If $f$ satisfies one of the equivalent conditions of Proposition 4.9, then $f$ is a morphism of locally ordered spaces (so it is weakly euclidean).

*Proof.* Let $V$ an ordered open of $Y$. $U := f^{-1}(V)$ is open in $X$, and up to restriction we can suppose that $U$ is strongly connected. Now take $x, y \in X$, $x \leq y$. Take $\gamma : \mathbb{R} \to U$ a directed path and $a, b \in \text{dom}\gamma$ with $a \leq b$ and $\gamma(a) = x, \gamma(b) = y$. Note that we can suppose that $\gamma$ is a local embedding, by collapsing the segments where it is locally constant. But $f$ satisfies 3 of Proposition 4.9, so $f \circ \gamma : \mathbb{R} \to V$ is a local embedding, in particular it is a directed path, so $f(x) \leq f(y)$. $\square$

Now we finally turn to the proof of the announced adjunction.

**Lemma 4.12.** Let $f : E \to X$ be a morphism of local orders with $E$ euclidean. Then $f$ is weakly euclidean if and only if it is a local embedding.

*Proof.* A local embedding is clearly weakly euclidean. Conversely, apply the characterization (iii) of Proposition 4.9 to the identity $E \to E$. $\square$

**Corollary 4.13.** The category $\mathcal{E}_n$ of euclidean local orders and local embeddings (of dimension $n$) is a coreflective subcategory of the category of local orders and weakly euclidean morphisms. In addition, $\mathcal{E}_n$ is coreflective in $\mathbb{L}_e$.

*Proof.* For the first part, note that $\mathcal{E}_n$ is a full subcategory by Lemma 4.12. In both cases, the right adjoint is given by $X \mapsto \tilde{X}$, by Theorem 4.8 and Proposition 4.9. $\square$

**5. A combinatorial description in the case of realizations of precubical sets**

We can give a combinatorial characterization of the embeddings of $\mathbb{R}^n$ into the locally ordered realization of a precubical set. In this section, we suppose that all precubical sets have a locally ordered realization. The first step of the proof is a lemma which is interesting on its own because it generalizes the well-known theorem of invariance of domain [Hat05, Theorem
2B.3]. At the end of the section, we give a consequence of the lemmas we proved, showing that there is a unique precubical set whose locally ordered realization is isomorphic to $\mathbb{R}^n$, namely the ‘infinite (n-)grid’. However, we have to start with a technical tool, namely subdivisions of precubical sets.

Note that the category of precubical sets can be seen as a subcategory of the category of *symmetric* precubical sets, which is the category of presheaves over the free symmetric monoidal category on $\mathcal{G}$ with unit 0, and which we note $\Box_{\text{sym}}$ (see [GM03, §6] for details). More concretely, a symmetric precubical set $P$ is a precubical set for which every $P(n)$ is equipped with an action of the symmetric group $S_n$ (compatible with the face maps in an obvious way). Every precubical set can be seen as a symmetric precubical set, by freely adding the images of these actions (this is the left adjoint to the precomposition by the inclusion $\Box \to \Box_{\text{sym}}$). In what follows, by isomorphic precubical sets (noted $\cong$), we mean precubical sets which are isomorphic as symmetric precubical sets. This just means that we allow ourselves to permute the dimensions of a cube: intuitively, an $n$-cube of a symmetric precubical set $P$ is represented by the *orbit* of some element of $P(n)$. This is essential for our purpose, since we are looking for combinatorial analogs of embeddings of ordered spaces. For example, for the ordered square, the map $(x, y) \mapsto (y, x)$ is indeed an isomorphism.

**5.1 Subdivisions**

The *s-subdivision* functor (for $s \in \mathbb{N} \setminus \{0\}$) is the unique monoidal realization functor which sends the graph $(\cdot \to \cdot)$ to the graph $G_s = (\cdot \to \cdots \to \cdot)$ made of $s$ arrows. Equivalently, it is the unique realization functor which sends the $n$-fold tensor product $(\cdot \to \cdot)^{\otimes n}$ to $G_s^{\otimes n}$. This definition makes sense because the category of precubical sets is monoidal, cocomplete, and contains the category of graphs as a full subcategory. It will be convenient to represent the vertices of $G_s$ as the fractions $\frac{i}{s}$ with $i \in \{0, \dots, s\}$ and its arrows as the fractions $\frac{2i-1}{2s}$ with $i \in \{1, \dots, s\}$; indeed, the source and target maps are $t \mapsto t - \frac{1}{2s}$ and $t \mapsto t + \frac{1}{2s}$.

The elements of the subdivision $\frac{1}{s}K$ of a precubical set $K$ are the ordered pairs $(x, (t_1, \dots, t_m))$ with $t_j \in \{\frac{1}{2s}, \frac{1}{s}, \dots, \frac{s-1}{s}, \frac{2s-1}{2s}\}$ and
$\dim x = m$; the dimension of $(x, (t_1, \dots, t_m))$ is the cardinal $n$ of the set $\{j \in \{1, \dots, m\} | t_j \notin \{\frac{1}{s}, \dots, \frac{s-1}{s}\}\}$; the $(n-1)$-face operators are defined by

$$ \delta_{i,n}^{-}(x, (t_1, \dots, t_m)) = \begin{cases} (x, (t_1, \dots, t_j - \frac{1}{2s}, \dots, t_m)) & \text{if } t_j \neq \frac{1}{2s} \\ (\delta_{j,m}^{-}x, (t_1, \dots, \hat{t}_j, \dots, t_m)) & \text{if } t_j = \frac{1}{2s} \end{cases} $$

$$ \delta_{i,n}^{+}(x, (t_1, \dots, t_m)) = \begin{cases} (x, (t_1, \dots, t_j + \frac{1}{2s}, \dots, t_m)) & \text{if } t_j \neq \frac{2s-1}{2s} \\ (\delta_{j,m}^{+}x, (t_1, \dots, \hat{t}_j, \dots, t_m)) & \text{if } t_j = \frac{2s-1}{2s} \end{cases} $$

with $j$ being the index of the $i^{th}$ occurrence of an arrow of $G_s$ in the $m$-tuple $(t_1, \dots, t_m)$. For $t = (t_1, \dots, t_m)$ we define $\int t = t_1 + \dots + t_m$.

**Proposition 5.1.** If $s$ does not divide $k \in \mathbb{N} \setminus \{0\}$, then for every element $e \in \frac{1}{s}K$ of dimension $n \ge k$, the set of lower $(n-k)$-faces of $e$ is disjoint from its set of upper $(n-k)$-faces.

*Proof.* A lower $(n-k)$-face is of the form $f^{-}(x)$ with $f^{-}$ denoting the composite of a sequence of $k$ faces of the form $\delta_{i,n'}^{-}$ with $k < n' \le n$. The same observation applies to upper $(n-k)$-faces with $f^{+}$ composite of faces of the form $\delta_{i,n'}^{+}$. Let $e = (x, t)$ be as in the statement. Put $f^{-}(x, t) = (x^{-}, t^{-})$ and $f^{+}(x, t) = (x^{+}, t^{+})$, and suppose that $(x^{-}, t^{-}) = (x^{+}, t^{+})$. In particular we have $t^{-} = t^{+}$, therefore $\int t^{-} = \int t^{+}$. We denote $\dim t - \dim t^{-}$ (which is also $\dim t - \dim t^{+}$) by $N$. We observe that applying an operator of the form $\delta_{i,n}^{-}$ reduces the sum of the coordinates by $\frac{1}{2s}$ (no matter $t_j$ is equal to $\frac{1}{2s}$ or not), while applying an operator of the form $\delta_{i,n}^{+}$ either increases the sum of the coordinates by $\frac{1}{2s}$ (if $t_j \neq \frac{2s-1}{2s}$), or reduces it by $\frac{2s-1}{2s}$ (if $t_j = \frac{2s-1}{2s}$) because the coordinate $t_j$ is 'dropped'. So $N$ is precisely the number of indices $j$ such that $t_j = \frac{2s-1}{2s}$, and $k - N$ the number of index $j$ such that $t_j \neq \frac{2s-1}{2s}$. It follows that $\int t^{-} = \int t - \frac{k}{2s}$ while $\int t^{+} = \int t - \frac{N(2s-1)}{2s} + \frac{k-N}{2s}$. The equality $\int t^{-} = \int t^{+}$ boils down to $k = sN$. $\square$

As an immediate consequence we have:

**Corollary 5.2.** If $K$ is of finite dimension $d$ and if $s > d$, then for every element $x$ of $\frac{1}{s}K$, the set of lower faces of $x$ is disjoint from its set of upper faces.
Now we can prove the following, which justifies condition (iv) of §2.4.

**Lemma 5.3.** Let $P$ be a precubical set of bounded dimension. Then for every vertex $v$ of $P$, there is an ordered neighborhood $B$ of $v$ only intersecting cubes adjacent to $v$, and for every cube $c$ with minimum (respectively maximum) $v$, for every point $y \in \{c\} \times ]0, 1^{[\dim(c)}$, every point $y \ge x$ (respectively $y \le x$) in $B$ belongs to $\{c'\} \times ]0, 1^{[\dim(c')}$ for some cube $c'$ having $c$ as lower (respectively upper) face.

*Proof*. We can suppose by subdivision (Corollary 5.2) that $P$ has the following property: every cube has distinct lower and upper faces. Now if we consider the $\epsilon$-elementary neighborhood $W_\epsilon$ of $v$ for some $\epsilon$, all the cubes of $W_\epsilon$ are adjacent to $c$, in particular if $c$ is a cube in $W_\epsilon$ whose minimum is $v$, then $W_\epsilon$ does not intersect the upper faces of $c$. Since ordered open sets generate the topology, we can find an ordered neighborhood $U \subseteq W_\epsilon$ of $v$. In particular, $U$ has all the properties we just stated for $W_\epsilon$. Now if we consider a cubewise increasing path $\gamma$ in $U$ starting at $x \in \{c\} \times ]0, 1^{[ \cap U$ with $c$ a cube having $v$ as minimum, if there is $t \in ]0, 1]$ such that $\gamma(t) \notin \{c\} \times ]0, 1^{[$, then either $\gamma$:

1. went up in dimension, and then it went in a cube which has $c$ on its lower face;

2. went down in dimension, but then it means that it reached the upper face of the cube which contains it, which is impossible by the above mentioned properties of $U$.

This means that $\gamma$ can only go up in dimension, and stays in cubes which have $v$ as minimum. Note that in particular, this is true if $\gamma$ starts at $v$. $\square$

These properties are inherited by subset, so this is enough to prove that we can suppose that all the elements of the base have this property. Note that the lemma is still true without supposing that the precubical set is bounded in dimension, by using an explicit description of the ordered base of its locally ordered realization, but we do not want to embark in this here, and we only study bounded precubical sets in this paper anyway.

We conclude with a useful technical lemma.

**Lemma 5.4.** Let $P$ be a precubical set of dimension $n$, $v$ a vertex of $P$ seen as a vertex of $\frac{1}{s}P$ for $s > 1$. Then all the $n$-cubes adjacent to $v$ are distinct,
in the sense that there is no $n$-cube $c$ with vertex $v$ on more than one of the corners of $c$.

**Proof.** The only $n$-cubes of $\frac{1}{s}P$ having $v = (v, (\))$ as vertex are of the form $(c, (t_1, \dots, t_n))$ with $t_i \in \{1/2s, (2s-1)/2s\}$ for some cube $c$ of $P$. Now for each such cube, there is at most one 1-face map which can give $(v, (\))$, because $1/2s \neq (2s-1)/2s$ since $s > 1$. $\square$

## 5.2 Combinatorial blowup

**Definition 5.5.** Let $P$ be a precubical set, $Z$ a subset of its geometric realization $Z \subseteq |P|$. The intersection of $P$ and $Z$, noted $P \cap Z$, is the precubical set spanned by all the cubes of $Z$, i.e the cubes $c$ such that $\{c\} \times [0, 1^{[\dim(c)]\cap Z} \neq \emptyset]$.

**Definition 5.6.** Let $P$ be a precubical set, $v$ a vertex of $P$. We set $N(v)$ the set of cubes of $P$ which are adjacent to $v$, and call it the combinatorial neighborhood of $v$. For a cube $c \in N(v)$, we say that $v$ is only on one of the corners of $c$ if the evaluation map from 1-face operators to vertices of $c$ does not send two different elements to $v$.

**Definition 5.7.** Let $P$ be a precubical set, $v$ a vertex. For every $c \in P_n$ having $v$ as vertex, define the position of $v$ with respect to $c$ as the tuple $p(v, c) := (p_1, \dots, p_n)$ with $p_i \in \{0, 1\}$, such that $\delta_{1,0}^{p_1} \circ \dots \circ \delta_{n,n-1}^{p_n}(c) = v$. For $\epsilon \in ]0, 1[$, define $B(v, c, \epsilon) = \{x \in ]0, 1^\epsilon | d_\infty(p(v, c), x) < \epsilon\}$, with $d_\infty$ the distance inherited from the infinite norm on $\mathbb{R}^n$. Finally, define the $\epsilon$-elementary open neighborhood of $v$ as the subspace of the realization of $|P|$ given by $W_\epsilon := \{v\} \cup \bigcup_{c \in N(v)} \{c\} \times B(v, c, \epsilon)$.

**Remark 5.8.** With the same notations, note that $W_\epsilon$ is indeed open in $|P|$. Note also that for $x \in \{c\} \times ]0, 1^{[\dim(c)}$, a similar construction gives an open neighborhood of $x$ which only intersects cubes of higher dimension adjacent to $c$.

**Definition 5.9.** For $p \in \{<, >, =\}^n$, define the $p$-cone of $\mathbb{R}^n$ as $\mathbb{R}_p^n := \{(x_1, \dots, x_n) \in \mathbb{R}^n | \forall i x_i p_i 0\}$. We will allow ourselves to represent $p$ as an element of $\{-1, 1, 0\}^n$ in the obvious way.
**Lemma 5.10 (Invariance of domain).** Let $P$ be a precubical set of maximal dimension $n$, $v$ a vertex of $P$, $U$ an open neighborhood of $0$ in $\mathbb{R}^n$, $f: U \to |P|$ an injective continuous map such that $f(0) = v$. We further suppose that for every $n$-cube $c$ adjacent to $v$, $v$ is only on one of the corners of $c$. Then there is a neighborhood $V \subseteq U$ of $0$ such that $f(V)$ is open in $|P \cap f(V)|$.

*Proof.* Consider an elementary neighborhood $W_\epsilon$ of $v$ in $|P|$. Note that the function $d_\infty(v, -)$ is well-defined and continuous on $W_\epsilon$. Since $f$ is continuous, up to restriction we can suppose that $f(U) \subseteq W_\epsilon$. We can also suppose without loss of generality that the (closed) unit ball (for the infinite norm) $B$ of $\mathbb{R}^n$ is included in $U$. Now $B$ and $S := S^{n+1} \subseteq B$ are compact, so $f(B)$ and $f(S)$ are compact (and in particular closed because $W_\epsilon$ is Hausdorff). So $d_\infty(v, -)|_{f(S)}$ is a continuous function defined on a compact which takes values in $\mathbb{R}$, so it is bounded and reaches its bounds. Note that the minimum of this function cannot be 0, because otherwise $v = f(0) \in f(S)$ so $0 \in S$ because $f$ is injective. Let $\eta > 0$ be this minimum. We note $W := W_{\eta/2}$ and $V := \mathring{B} \cap f^{-1}(W)$.

First we claim that $P \cap f(V)$ is $n$-homogeneous. So suppose that there is a $k$-cube $c$ of $P \cap f(V)$, $k < n$ such that $f(V)$ does not intersect any of the cubes of $P$ which have $c$ as $k$-face. By Remark 5.8, we can find $x \in \{c\} \times ]0, 1^k$ and an open neighborhood $T$ of $x$ such that $T$ only intersects cubes of higher dimension adjacent to $c$, i.e. such that $T \cap f(V) \subseteq \{c\} \times ]0, 1^k$. By continuity, $f^{-1}(T)$ is open in $\mathbb{R}^n$, so we get a continuous injection $f$ from an open subset $V \cap f^{-1}(T)$ of $\mathbb{R}^n$ to an open subset of $\mathbb{R}^k$, which is absurd by (classical) invariance of domain.

As a consequence of the additional hypothesis, for every $n$-cube $c$ with vertex $v$, $\{c\} \times ]0, 1^{[n \cap W}$ is connected. Now we prove that for every $n$-cube $c$ of $P \cap f(V)$, we have $\{c\} \times ]0, 1^{[n \cap W \subseteq f(B)$. Indeed, consider the set

$$Z := \{x \in \{c\} \times ]0, 1^{[n \cap W} | x \in f(B)\}$$

Notice that

$$Z = \{x \in \{c\} \times ]0, 1^{[n \cap W} | x \in f(\mathring{B})\}$$

because if $x \in W \cap f(B)$, there is $y \in B$ such that $x = f(y)$, so $f(y) \in W$, but $W \cap f(S) = \emptyset$ by definition of $W$, and so $y \in \mathring{B}$. By the choice of $c$, $Z$
is not empty. So it remains to show that it is open and closed. Notice that by injectivity of $f$,

$$Z = f(f^{-1}(\{c\} \times ]0, 1^{[n \cap W)} \cap \hat{B})$$

but $\{c\} \times ]0, 1^{[n \cap W)}$ is homeomorphic to an open subset of $\mathbb{R}^n$ and is open in $|P|$, so $Z$ is open by classical invariance of domain. $Z$ is also closed, because $B$ is compact, which concludes.

We mentioned that $f(B)$ is closed. Now notice that since $|P \cap f(V)|$ is $n$-homogeneous, the closure of $(P \cap f(V))_n \times ]0, 1^{[n \cap W)}$ is equal to $|P \cap f(V)| \cap \bar{W}$, where $\bar{W}$ is the closure of $W$. So the result of the last paragraph gives $|P \cap f(V)| \cap W \subseteq f(B)$. In fact, since $W \cap f(S) = \emptyset$, this means that $|P \cap f(V)| \cap W \subseteq f(\hat{B})$. But then if $x \in |P \cap f(V)| \cap W$, there is $y \in \hat{B}$ such that $x = f(y)$, and then $f(y) \in W$ so $y \in f^{-1}(W)$, so $y \in V$ and $x \in f(V)$. Now we conclude that $f(V) = |P \cap f(V)| \cap W$, which finishes the proof. □

**Definition 5.11.** A local precubical structure of $\mathbb{R}^n$ is a pair $(P, v)$ where $P$ is a precubical set and $v$ a vertex of $P$, such that there exists an open neighborhood $W$ of $v$ in $|P|$ included in an ordered neighborhood $U$ of $v$, and such that $W$ with the induced order is dihomeomorphic to $\mathbb{R}^n$ by a dihomeomorphism sending $0$ to $v$. If $(P, v)$ is a local precubical structure of $\mathbb{R}^n$, we also say that a precubical set $P$ is locally euclidean at $v$.

We can give a purely combinatorial characterization of local precubical structures of $\mathbb{R}^n$.

**Theorem 5.12.** Let $P$ be a precubical set, $v$ a vertex. The following are equivalent:

1. $(P, v)$ is a local precubical structure of $\mathbb{R}^n$;

2. $v$ has $n$ edges in and $n$ edges out (counted 'with multiplicity', i.e. if the same edge is in and out, it counts as two edges), labeled respectively by $\{-n, -(n-1), \dots, -1\}$ and $\{1, \dots, n\}$, and such that for every $k \ge 2$, there is exactly one $k$-cube $c(p)$ adjacent to $v$ for every tuple $p$ of $k$ elements of $\{-n, -(n-1), \dots, -1\} \cup \{1, \dots, n\}$ not containing $i$ and $-i$ for some $i$, such that the $1$-faces of $c(p)$ adjacent to $v$ are given by the edges of $p$ (also up to multiplicity, i.e. we cannot have a cube
which has $v$ and the same adjacent edges at two different corners),
and such that these are no more cubes adjacent to $v$.

3. There is a precubical set $B \cong (\cdot \to \cdot \to \cdot)^{\otimes n}$ the $n$-fold tensor product (as symmetric precubical sets) and a monomorphism

$$B \to \frac{1}{3} P$$

sending the central vertex to $v$, and whose image is exactly the precubical set generated by $N(v)$, where $v$ is seen as a vertex of $\frac{1}{3}P$.

*Proof.* (3) clearly implies (1) (consider $W_e$). We start by showing that (2) implies (3). We define a monomorphism $\iota : T := (-1 \xrightarrow{a} 0 \xrightarrow{b} 1)^{\otimes n} \to \frac{1}{3}P$ as follows:

1. $\iota(0, \dots, 0) := v$;

2. $\iota(0, \dots, 0, a, 0, \dots, 0) := (-k, (2/3))$ (resp. $\iota(0, \dots, 0, b, 0, \dots, 0) := (k, (1/3))$) where $a$ (resp. $b$) is in position $k$ (recall that $-k$ (resp. $k$) is the label of an edge with target (resp. source) $v$);

3. for every tuple $p \in \{a, 0, b\}^n$, $\iota(p) = (c(p'), (t_{p_i})_{i \in \{1, \dots, n\}})$ where $p'$ is obtained from $p$ by replacing every $a$ (resp. $b$) in position $k$ by $-k$ (resp. $k$) and forgetting the 0s, and where $t_a := 2/3$, $t_b := 1/3$ and $t_0$ is omitted.

(recall that we are working up to symmetry, so we do not care about the order of the dimensions). This is enough since $T$ is $n$-homogeneous. We still need to check that this is compatible with the face maps, but this is just a formal exercise. It is also easy to check that all the vertices in the image are distinct, which proves that this is indeed a monomorphism, because if $\iota(c) = \iota(c')$ for some $k$-cubes $c, c'$, then at least $2^k + 1$ vertices are sent to $2^k$ vertices, so some vertices are identified, which contradicts what we just said. Since $(P, v)$ satisfies (2) of Theorem 5.12, then by construction all the neighboring cubes of $v$ in $\frac{1}{3}P$ are in the image of $\iota$.

We still need to prove (1) implies (2). This is probably the most technical (and least categorical) part of the paper. Fix a dihomeomorphism $f : W \to \mathbb{R}^n$. In this proof, we work up to multiplicity. First, we show that there is exactly one $n$-cube which has $v$ as minimum (resp. maximum),
which we will call the positive (respectively negative) cubes. By conditions (iii) and (iv) of §2.4 and the fact that $f$ is order preserving and order reflecting, the positive cone of $\mathbb{R}^n$ (i.e. $\mathbb{R}_+^n := \{(x_1, \dots, x_n) \in \mathbb{R}^n \mid \forall i, x_i \ge 0\}$) is exactly the image by $f$ of all the cubes (of any dimension) which have $v$ as minimum. Obviously, there is at least one such cube of dimension $n$ (by invariance of domain), so we need to prove that there is at most one. Now notice that the positive cone of $\mathbb{R}^n$ has the property that every two elements have an upper bound, which is just an upper bound pointwise. Suppose that there are more than two $n$-cubes which have $v$ as minimum. Choose two such cubes $c$ and $c'$. We take $y \in \{c\} \times ]0, 1[^n$, $y' \in \{c'\} \times ]0, 1[^n$, and we claim that they cannot have an upper bound. Indeed, since $n$ is the maximal dimension of $P$ and $U$ satisfies condition (iv) of §2.4, the elements which are bigger than $y$ (respectively $y'$) in $W$ all belong to $\{c\} \times ]0, 1[^n$ (respectively $\{c'\} \times ]0, 1[^n$), so $y$ and $y'$ cannot have an upper bound in $W$. But $f$ preserves and reflects upper bounds, so we get a contradiction. The same proof works in the negative case. We write $c_+$ and $c_-$ for the positive and negative cube respectively.

Now we prove that the only edges in $W$ are the edges of the positive and negative cubes. Indeed, suppose that there is an edge which goes out of $v$, which is not one of the edges of $c_+$. Notice that the cubes having $v$ as minimum generate a subprecubical set of $P$ which, by the same arguments as in the proof of Lemma 5.10, is $n$-homogeneous. So the existence of this additional edge implies the existence of another positive ($n$-)cube, which is a contradiction. Again the same is true in the negative case. Notice that the same argument shows that every $k$-cube $c$ whose edges in $W$ are all of positive (resp. negative), $c$ is a $k$-face of $c_+$ (respectively $c_-$).

We have proved that there is a cube $c_+$ that is exactly sent to the positive cone of $\mathbb{R}^n$. Now we claim that the edges of $c_+$ are exactly mapped to the (positive) axis of $\mathbb{R}^n$. This comes from the fact that in both cases, the corresponding points are characterized by the following property

$F(x, K)$: the set of points of $K$ that are strictly lower than $x$ is dihomeomorphic to an interval

Indeed, take a point $x$ on one of the axis of $\mathbb{R}^n$, then all its coordinates
are zero except one, say $x = (x_1, 0, \dots, 0)$, so the only points of the positive cone which are lower than $x$ are those of the form $(x'_1, 0, \dots, 0)$ with $x'_1 \le x_1$. This proves $F(x, \mathbb{R}_+^n)$. Now for $x \in \{e\} \times]0, 1[ \cap U$ for $e$ a positive edge, any element $y \in |c_+| \cap U$ which is smaller than $x$ belongs to $\{e\} \times]0, 1[ \cap U$, because otherwise a cubewise increasing path from $y$ to $x$ would necessarily go through some cube of higher dimension, which is impossible by condition (iv) of §2.4. This proves $F(x, |c_+| \cap U)$. Now for an edge $e$ of $c_+$, we need to prove that $|e| \cap W$ is connected: otherwise, there are $a, b \in |e| \cap W$ with no directed path connecting them in $|c_+| \cap W$, because the only one that does is included in $|e|$ (again by condition (iv) of §2.4), but supposing without loss of generality that $a \le b$, we get $f(a) \le f(b)$ in $\mathbb{R}^n$, so there is a directed path between them in the positive cone, which can be pulled back to a directed path in $|c_+| \cap W$, which is a contradiction. This proves $F(x, |c_+| \cap W)$. So both the points of the edges of $c_+$ and the points of the positive axis of $\mathbb{R}^n$ satisfy the property, and they are clearly the only ones that do. A dihomeomorphism respects this property (more precisely $F(x, K) \leftrightarrow F(f(x), f(K))$ if $f$ is a dihomeomorphism on $K$), so the claim is proved. Notice that by continuity, every edge of $c_+$ is mapped to exactly one axis, and by bijectivity all the axes are in the image, so all the edges of $c_+$ are distinct. As a consequence, each face is mapped to the corresponding face of $\mathbb{R}^n$: for example in dimension 3, if $c$ is a positive 2-cube of edges $e_1$ and $e_2$, the third one being $e_3$ (and supposing that $f$ sends $e_i$ to the $i$th axis of $\mathbb{R}^3$), a point $(c, (x, y))$ of the positive cone will be mapped to $(x', y', z') := f((c, (x, y)))$ in the positive cone of $\mathbb{R}^3$. Now $f((c, (x, y)))$ is the least upper bound of $(x', 0, 0)$, $(0, y', 0)$ and $(0, 0, z')$, so $(c, (x, y))$ is the least upper bound of their preimages, which are on the axes $e_i$. But then we get that these are respectively $(e_1, x)$, $(e_2, y)$ and 0, and since $f$ is injective this means that $z' = 0$, so indeed $f(x, y, 0)$ lands in the correct face. This argument works in the general case.

Now we label the edges following the axis to which they are mapped. We still need to prove the part about the $k$-cubes in definition 5.11. For better readability, we give negative coefficients to negative edges in the realization (i.e. we replace $\{c\} \times]0, 1[^k$ with $k = \dim(c)$ by $\{c\} \times]0, 1[^{k_1} \times -1, 0[^{k_2}$ (up to permutation), where $k_1$ and $k_2$ are respectively the number of positive and negative edges of the cube $c$), only in this paragraph. First we show that for
a tuple of edges containing $i$ and $-i$ for some $i$, i.e. a set of edges such that one is sent to a positive axis and another one to the corresponding negative axis, there cannot be a cube having these edges as 1-faces. It suffices to prove that there cannot be a 2-face between edges $i$ and $-i$ for some $i$. Suppose the contrary: for some label $i$, there is a 2-face $c$ between $i$ and $-i$. Consider two symmetrical points $(i, x)$ and $(-i, -x)$ on the edges $i$ and $-i$ respectively (recall that $i$ is out and $-i$ is in). The segment

$$ \gamma: t \in ]0, x[ \mapsto (c, (x-t, -t)) $$

is directed from $(i, x)$ to $(-i, -x)$, so it corresponds to a directed segment in $\mathbb{R}^n$. However, the only directed segment, up to parametrization, between two points on the same axis in $\mathbb{R}^n$ is contained in this axis, but $\gamma$ is not contained in the preimage of the axes, so we got our contradiction.

We still need to show that given a tuple of $k$ labels not containing $i$ and $-i$ for some $i$, there is exactly one $k$-cube having these as 1-faces. Take $(i_1, \dots, i_k)$ tuple of labels, $c$ a $k$-cubes having exactly these as 1-faces. Consider a point $(c, (t_1, \dots, t_k))$. Suppose for simplicity that $i_1, \dots, i_p$ are positive edges and $i_{p+1}, \dots, i_k$ are negative edges. As remarked earlier, there is only one cube $c_1$ (respectively $c_2$) having $i_1, \dots, i_p$ (respectively $i_{p+1}, \dots, i_k$) as edges. Let $x_+$ and $x_-$ be the 'projections' of $x$ on these cubes, i.e. $x_+ = (c_1, (t_1, \dots, t_p))$ and $x_- = (c_2, (t_{p+1}, \dots, t_k))$. Since $U$ is order convex, we have $x_- < x < x_+$. Now $x_-$ and $x_+$ are sent respectively to a point $(0, \dots, 0, y_{p+1}, \dots, y_n)$, $y_{p+1}, \dots, y_n < 0$ and $(y_1, \dots, y_p, 0, \dots, 0)$, $y_1, \dots, y_p > 0$ of $\mathbb{R}^k$, seen as a subset of $\mathbb{R}^n$. Since $f$ is order preserving,

$$ f(x) \in \{(z_1, \dots, z_k) \in \mathbb{R}^k \mid z_1, \dots, z_p > 0, z_{p+1}, \dots, z_k < 0\} $$

(note that it does belong to $\mathbb{R}^k$ seen as a subset of $\mathbb{R}^n$, because for any other coordinate $j$, $0 = f(x_-)_j \le (f(x))_j \le f(x_+)_j = 0$, so $(f(x))_j = 0$ as well). This shows that all the cubes of $P$, of different dimensions and for different choices of signs of the coordinates, are exactly sent to the corresponding cone (i.e. 'with the same signs') in $\mathbb{R}^n$. So for every such choice, there is at least one cube filling the edges, else $f$ would not be surjective. Moreover, suppose that there exists $(i_1, \dots, i_k)$ tuple of labels, $c_1$ and $c_2$ distinct $k$-cubes
having exactly these as 1-faces. Take $x_1 \in \{c_1\} \times ]0, 1[^k$, $x_2 \in \{c_2\} \times ]0, 1[^k$. $f(x_1)$ and $f(x_2)$ have an upper bound $z$ in the interior of the corresponding cone of $\mathbb{R}^n$, with two directed paths $\gamma_1: f(x_1) \to z$ and $\gamma_2: f(x_2) \to z$ entirely contained in the interior of this cone, which is the preimage of $C \times ]0, 1[^k$ with $C$ the set of cubes having $(i_1, \dots, i_k)$ as edges. But $z = f(y)$ for some $y$, and $y$ necessarily belongs to $\{c_3\} \times ]0, 1[^k$ for some $c_3$ having the same edges, but there is no directed path joining for example $c_1$ and $c_3$ and staying in $C \times ]0, 1[^k$, contradiction.
□

**Remark 5.13.** 1. The last proof heavily relies on the order structure, which is reasonable since the result is absolutely not true in the non-directed case.

2. In the previous theorem, if $P$ is already a precubical subset of $\frac{1}{3}Q$ for some precubical set $Q$, there is no need to take the 3-subdivision in (3), just because the proof (2) implies (3) (and (3) implies (1)) still works.

3. Let $R$ be the countable directed linear graph without endpoints seen as a precubical set, i.e. $R_0 = \mathbb{Z}$, $R_1 = \mathbb{Z}$, $s = \text{id}$, $t = (x \mapsto x+1)$. Then for any vertex $v$ of $R^{\otimes n}$, $(R^{\otimes n}, v)$ is a local precubical structure of $\mathbb{R}^n$. In fact, this is the motivating example.

4. Let $(P, v)$ be a local precubical structure of $\mathbb{R}^n$. Then for every $\epsilon \in ]0, 1/2]$, we can define a map $\phi: W_\epsilon \to \mathbb{R}^n$ by sending $v$ to 0, the edges to the axis and the cubes accordingly to the cones. If all the adjacent cubes of $v$ are distinct, this is true for any $\epsilon \in ]0, 1]$. We say that $W_\epsilon$ is canonically isomorphic to $\mathbb{R}^n$ if the preorder on it given by cubewise increasing paths is a partial order, and $\phi$ is a dihomeomorphism. In particular, for every $\epsilon \in ]0, 1/2]$, $W_\epsilon$ is canonically isomorphic to $\mathbb{R}^n$. If all the adjacent cubes of $v$ are distinct, then this is true for any $\epsilon \in ]0, 1]$.

Now we can encapsulate the blowup of the locally ordered realization of a precubical set $P$ in a purely combinatorial presheaf. This is partly inspired by cellular sheaves [Cur14, Ghr14]. For every $c \in P_k$, we write $m(c)$ for the vertex of $\frac{1}{2}P$ which is the 'center' of $c$ (with the notations of appendix 5.1, $m(c) = (c, (1/2, \dots, 1/2))$). This allows to 'represent' each cube of $P$ by a vertex.

**Lemma 5.14.** Let $P$ be a precubical set of dimension $n$, $c$ a $k$-cube, $0 \le i < k$, $c' := \delta_{k,i}^{\epsilon}(c)$. Given $(Q, m(c'))$ a local precubical structure
ture of $\mathbb{R}^n$ in $\frac{1}{2}P$, seen as a monomorphism $(\cdot \rightarrow \cdot \rightarrow \cdot)^{\otimes n} \rightarrow \frac{1}{6}P$, let $S_i$ be the set of cubes of $Q$ whose underlying cubes in $P$ are adjacent to $c$. If $S_i$ contains the edge $e$ 'going inside $c$ in direction $i$' (namely $e = (c, (1/2, \dots, 1/2, z, 1/2, \dots, 1/2))$ where $z$ is in position $i$ and $z = 1/12$ (resp. $11/12$) if $\epsilon = -$ (resp. $+$)), then the precubical set $\psi_{P,c,i,\epsilon}(Q, m(c'))$ obtained by 'shifting $S_i$ to the center' (namely the precubical set generated by $\{(c, (t_1, \dots, t_{i-1}, t_i - \epsilon \cdot 1/6, t_{i+1} \dots, t_n)), (c, (t_1, \dots, t_{i-1}, t_i - \epsilon \cdot 1/2, t_{i+1} \dots, t_n)) | (c, (t_1, \dots, t_n)) \text{ is an } n\text{-cube of } S_i\}$) is a local precubical structure of $\mathbb{R}^n$. Else $S_i$ is empty.

*Proof.* $S_i$ contains $e$ if and only if the realization of the cubes of $Q$ adjacent to $m(c')$ intersects $\{c\} \times ]0, 1^{\dim(c')})$. In this case, for any point $x \in \{c\} \times ]0, 1^{\dim(c')}$, there exists an embedding of $\mathbb{R}^n$ centered at $x$. This is in particular true for $m(c)$. Then by Lemma 5.10 (and Lemma 5.4) and Theorem 5.12, there is a precubical subset $Q'$ such that $(Q', m(c))$ is a local precubical structure of $\mathbb{R}^n$. But the $n$-cubes of $Q'$ necessarily have the same underlying $n$-cubes in $P$ as $S_i$, and $\psi_{P,c,i,\epsilon}(Q, m(c'))$ is the only possible such (up to equivalence). □

This is illustrated in Fig. 4. Note that we could also give a proof using explicit calculations on the subdivisions.

<figure><img src="image_9.png" /><figcaption>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 $m(c_2)$.</figcaption></figure>

**Definition 5.15.** Let $P$ be a precubical set. We define a presheaf $\text{Comb}_P$ on the opposite of the category of elements [BW90, §12.2] of $P$ in the following
way: for every $c \in P_k$, $\text{Comb}_P(k, c)$ is given by the set of elements $(Q, m(c))$ where $Q$ is a subobject of $\frac{1}{6}P$ represented by a monomorphism (of symmetric precubical sets) $(\cdot \to \cdot \to \cdot)^{\otimes n} \to \frac{1}{6}P$ centered at $m(c)$, and an additional element $\perp$. The morphism $\phi_{P,c,i,\epsilon}: \text{Comb}_P(k-1, c') \to \text{Comb}_P(k, c), c' := \delta_{k,i}^{\epsilon}(c)$ is defined as follows: given $x \in \text{Comb}_P(k-1, c')$, if $\psi_{P,c,i,\epsilon}(x)$ is defined, set $\phi_{P,c,i,\epsilon}(x) := (\psi_{P,c,i,\epsilon}(x), m(c))$. Else set $\phi_{P,c,i,\epsilon}(x) := \perp$.

This is well-defined: one just needs to check that the restriction maps respect the cocubical relations, which is formal.

**Definition 5.16.** Let $P$ be a precubical set of dimension $n$. Let $O_{\min}(|P|)$ be the full subcategory of $O(|P|)$ given by open sets $U$ that have a cube which is a face of all the other cubes of $U$ (these open sets generate the topology by Remark 5.8). For $U \in O_{\min}(|P|)$, define $\min(U)$ to be this 'minimal' cube. Now we define a presheaf $C_P$ on $O_{\min}(|P|)$ by $C_P(U) := \text{Comb}_P(\dim(\min(U)), \min(U))$. For $U \subseteq V$, if $U$ intersects $\min(V)$ then the restriction map is given by identity, else it is given by the $\phi_{P,c,i}$. We can extend this presheaf to $O(|P|)$ by right Kan extension, and the resulting presheaf is also called $C_P$.

**Theorem 5.17.** Let $P$ be a precubical set of dimension $n$, $B(C_P)$ the étale bundle associated to $C_P$. Then $B(C_P) \setminus \{\perp_x | x \in |P|\}$ is the $n$-blowup of $|P|$.

*Proof.* Define a morphism $r : C_P \to O_X^{(n)}$ in the following way: for $U \in O_{\min}(|P|)$, send a local precubical structure of $\mathbb{R}^n (Q, v)$ to the 'maximal' subset of $U$ corresponding to it, that is the subset consisting of the realization of the cubes of $Q$. One readily checks that $r$ is a natural transformation, i.e. it is compatible to restriction. Since the open sets of $O_{\min}(|P|)$ form a base of $|P|$, this is enough to define $r$. Looking at the map induced by $r$ on the stalks, we see that $r$ induces a morphism $B(r) : B(C_P) \to B(O_{|P|}^{(n)})$ over $|P|$ which restricts to a morphism $B(C_P) \setminus \{\perp_x | x \in |P|\} \to |\tilde{P}|$. By Lemma 5.10 (and Lemma 5.4) and Theorem 5.12 (and Remark 5.13 (2)), this map is bijective. But this is the restriction of an étale bundle (because $g \circ f$ and $g$ étale implies $f$ étale), so it is in fact an isomorphism of locally ordered spaces. $\square$

The proof is essentially sheaf-theoretic, which is quite satisfying. The ‘non-categorical’ content of the proof is clearly identified and entirely contained
argument, all the neighboring vertices of $v$ (graph-theoretically) are distinct. In fact, this defines a monomorphism (of symmetric precubical sets) $\iota : (-1 \to 0 \to 1)^{\otimes n} \to P$ centered at $v$, such that the image of a tuple whose only nonzero term is in position $i$ is the (graph-theoretic) neighbor of $v$ which is sent to the $i$th axis of $\mathbb{R}^n$. Note that all the cubes that are adjacent to $v$ are in the image of $\iota$, because $(P, v)$ is a local precubical structure of $\mathbb{R}^n$. Note also that $\text{Im}(f \circ |\iota|)$ is a geometric cube in $\mathbb{R}^n$ centered at 0.

<figure><img src="image_10.png" /></figure>

Now we want to define $\iota_p : (-1 \to 0 \to 1)^{\otimes n} \hookrightarrow P$ for every $p \in \mathbb{Z}^n$ which cover $P$. We already defined $\iota_{0...0} := \iota$. We consider every neighbor (in the graph-theoretic sense) of $v$ and do the same as in the last paragraph to get $\iota_p$ for $p$ a neighbor of $(0, \dots, 0)$ in $\mathbb{Z}^n$. We do this inductively to get $\iota_{0...0k0...0}$ where $k$ is in position $i$, for every $1 \le i \le n$ and $k \in \mathbb{Z}$. Simultaneously, we define sequences $(x_{i,j})_{j \in \mathbb{Z}}$ for $1 \le i \le n$, such that $x_{i,j}$ belongs to the $i$th axis of $\mathbb{R}^n$, and $\text{Im}(f \circ |\iota|)$ is centered at $x_{i,j}$. In the right part of the picture, we represented the axis of $\mathbb{R}^2$; the points on the axis are the $x_{i,j}$, and the dotted squares are the images of the $\iota$s. The important point is that these form the image of a precubical subset of $P$. Now for every $p \in \mathbb{Z}^n$, we build $\iota_p : (-1 \to 0 \to 1)^{\otimes n} \hookrightarrow P$ centered at $([x_{1,p_1}], \dots, [x_{n,p_n}])$, where $[x_{i,j}]$ is the value of the only nonzero coordinate of $x_{i,j}$, and whose other vertices (in $\mathbb{R}^n$) are the expected ones with respect to the $(p_i)_i$. We do this by induction, using the construction of the last paragraph and the fact that the image we get in $\mathbb{R}^n$ each time is a geometric cube, i.e. if we know 3/4 of the cube we know all the cube. More precisely, we use the lexicographic order on each cone of $\mathbb{Z}^n$ (discarding the sign). For example, see the grey cube on the picture, which is the first inductive step in the left upper cone. Now notice that, by construction, if one vertex of $P$ belongs to the image of one of the $\iota_p$, all of its adjacent cubes also are, so in fact this is the case of all the cubes of $P$, because $P$ is connected (since its realization is isomorphic to $\mathbb{R}^n$). Putting
everything together, we get an epimorphism

$$ \bigsqcup_{p \in \mathbb{Z}^n} \iota_p : \bigsqcup_{p \in \mathbb{Z}^n} (-1 \to 0 \to 1)^{\otimes n} \to P $$

which exhibits $P$ as a quotient of $\bigsqcup_{p \in \mathbb{Z}^n} (-1 \to 0 \to 1)^{\otimes n}$ by an equivalence relation $S$ (since we are in a topos), whose realization exhibits $\mathbb{R}^n$ as a quotient of $\bigsqcup_{p \in \mathbb{Z}^n} |-1 \to 0 \to 1|^{\otimes n}$. This implies that

$$ P \cong \bigsqcup_{p \in \mathbb{Z}^n} (-1 \to 0 \to 1)^{\otimes n} / S \cong R^{\otimes n} $$

□

## 6. The path lifting property

Now we have to mention that the $n$-blowup may forget a lot of points. For example, if $X$ is a euclidean local order of dimension $n$, then the $(n+1)$-blowup of $X$ is the empty space. We may consider changing the universal property we want to achieve, for example lifting a larger class of morphisms, and adapt the construction to see what this gives. In this section, we give a condition to ensure that the blowup that we defined admits the directed paths lifting property, which is a very important property since directed paths represent execution traces of programs, and we do not want to forget any of these if we want to use our blowup in the context of concurrency theory. In the following, a directed path is supposed injective. This is not very restrictive knowing [FR07, Theorem 3.6]

Notice that for every $n \in \mathbb{N} \setminus \{0\}$, $z \in O_{\mathbb{R}^{n,0}}^{(1)} \setminus \{A_0 | 0 \notin A\}$, for every local order $X$ and $x \in X$, there is a map

$$ z^* : O_{X,x}^{(n)} \setminus \{A_x | x \notin A\} \to O_{X,x}^{(1)} \setminus \{A_x | x \notin A\} $$

defined in the following way: an element of $O_{X,x}^{(n)} \setminus \{A_x | x \notin A\}$ is an equivalence class of order embeddings from an open subset of $\mathbb{R}^n$ to an ordered open of $X$. Notice that we can always take a representative $h: U \to V$ of the class such that $U$ contains $0$ and that $h(0) = x$. Let $h$ such a representative of an element of $O_{X,x}^{(n)} \setminus \{A_x | x \notin A\}$, and $\gamma$ such a representative of $z$. Then $z^*([h]) := [h \circ \gamma]$. This is clearly well-defined.
**Proposition 6.1.** Let $X$ be a local order. The following are equivalent :

1. The family $\{z^* | z \in O_{\mathbb{R}^{n,0}}^{(1)} \setminus \{A_0 | 0 \notin A\}\}$ is jointly surjective for all $x \in X$.

2. For every directed path $\gamma$ on $X$, every $x \in \text{dom}(\gamma)$, there is an open neighborhood $U$ of $x$ such that $\gamma|_U$ admits a lift to $\tilde{X}$ the $n$-blowup of $X$.

*Proof.* Suppose (i) is true, and take $\gamma$ and $x$ as in (ii). By (i), there is $[A] \in O_{X,x}^{(n)}$ and $z \in O_{\mathbb{R}^{n,0}}^{(1)}$ such that $z^*([A]) = (\text{Im } \gamma)_{\gamma(x)}$. Now this means that there is an open set $V \subseteq X$ containing $\gamma(x)$ such that $A \cap V = \text{Im } \gamma \cap V$. So $x \mapsto A_{\gamma(x)}$, which a morphism of local orders because it is the composite of two such, is a lift of $\gamma$ on $\gamma^{-1}(V) \neq \emptyset$, which is what we wanted.

Conversely, let $[a] \in O_{X,x}^{(1)} \setminus \{A_x | x \notin A\}$. Take $\gamma$ a representative of $a$, which we can take to be a directed path $\mathbb{R} \to X$ with $\gamma(0) = x$. By (ii) we can lift it to $\tilde{X}$ (up to restriction), say by $\tilde{\gamma}$, then we can take an open neighborhood $A$ of $\tilde{\gamma}(0)$. Then $U := \tilde{\gamma}^{-1}(A)$ is an open subset of $\text{dom}(\tilde{\gamma})$, so also of $\text{dom}(\gamma)$. $\tilde{X}$ is locally euclidean, so we can suppose that $A$ is isomorphic to an open subset of $\mathbb{R}^n$. Now (up to translation) composing this isomorphism with $\tilde{\gamma}$ gives an element $z$ of $O_{\mathbb{R}^{n,0}}^{(1)} \setminus \{A_0 | 0 \notin A\}$, and we clearly have $z^*(A_x) = [a]$, which concludes. $\square$

If one of the previous conditions is satisfied, we say that $X$ satisfies the local directed paths lifting property. This implies the existence of a global lift for every directed path if, for example, the set of singularities of $X$ is discrete. Note that this is not true in general. For example, one can consider the subspace $\{(x, y, 0) \in \mathbb{R}^3 | x > -1\} \cup \{(x, 0, z) \in \mathbb{R}^3 | x < 1\}$ of $\mathbb{R}^3$, which satisfies the local directed paths lifting property, but such that the directed path $x \in \mathbb{R} \mapsto (x, 0, 0)$ cannot be lifted to a (continuous) path on the corresponding blowup (notice that we can get a continuous lift for a smaller topology on $\tilde{X}$, namely the smallest topology for which the blowup map is continuous). However, we can give a global version. For every $n \in \mathbb{N} \setminus \{0\}$, for every local order $X$, for every $[f : E \to A] \in O_X^{(n)}(X)$ and $[\gamma : E' \to E] \in O_E^{(1)}(E)$, we have an element $[f \circ \gamma]$ of $O_X^{(1)}(X)$. This defines a map $l_n : \bigsqcup_{[E \to A] \in O_X^{(n)}(X)} O_E^{(1)}(E) \to O_X^{(1)}(X)$. A similar proof to the previous one then gives:
**Proposition 6.2.** For any local order X, every directed path on X can be
lifted to a directed path on $\tilde{X}$ the n-blowup of X if and only if the function
$l_n$ defined above is surjective.

This is clearly always true for $n = 1$, which explains why we always get the
directed paths lifting property for graphs. We can give combinatorial ver-
sions of these characterizations is the case of realizations of precubical sets.
In fact, it would be interesting to look at lifting properties *up to (directed) homotopy*, i.e. given a directed path, is there a dihomotopic path [Gra09]
which can be lifted to the blowup. This would be particularly interesting in
the case of *non-selfintersecting* precubical sets, since a directed path on their
realization is always dihomotopic to path on their 1-skeleton [Faj05], which
should give a simple criterion.

7. Conclusion

Note that the use of sheaf theory could have been avoided in this paper, but
we chose to present our results as we did for several reasons. Firstly, sheaf
theory significantly simplifies the proofs: notice that we got Proposition
4.5 for free; in fact, one can look at the amount of work needed in [Hau25]
to prove Theorem 1.2, and compare it to our proof of the more general
Theorem 4.8 (however, the proof of Theorem 1.2 in [Hau25] also provides
an explicit description of the blowup, which also needed a lot of work in
§5). Secondly, our work paves the way to generalizations of this procedure
outside of directed algebraic topology. In fact, the first author is working
on a generalization of the blowup in an abstract setting, which can be
particularized to various contexts, including differential geometry and
sequential spaces, and with possible applications to logic [dAM23, TV08].
One can also wonder if the sheaf $O_X^{(1)}$ for some local order X does not
represent some kind of ‘ordered tangent bundle’ of X, since its stalk at
$x \in X$ precisely represents the equivalence classes of directed path crossing
$x$. Finally, the idea of ‘completing’ the blowup into an étale bundle could be
interesting on its own, as it replaces a map with a much better behaved one.

More concretely, we wonder if the results of [Hau25, §6] can be ex-
tended to our setting; in other words, if there is a pseudometric on the
blowup of the locally ordered realization of a higher dimensional automaton
A which reflects the execution time of the traces on A. In [Hau25], this
pseudometric comes from a *smooth* manifold structure on the blowup. In
this paper, we have only defined a locally ordered manifold structure on the
blowup, but as suggested in the introduction, it should not be very hard to
get a smooth atlas in the case of locally ordered realizations of precubical
sets (we even conjecture the existence of an atlas whose related transition
maps are identities). However, this is a bit far from the core subject of the
present paper, and will be developed in a future work.

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Yorgo CHAMOUN
École Polytechnique
Laboratoire d'informatique (LIX)
Bâtiment Alan Turing
1 rue Honoré d'Estienne d'Orves
CS35003 91120 Palaiseau Cedex, France
yorgo.chamoun@polytechnique.edu

Emmanuel HAUCOURT
École Polytechnique
Laboratoire d'informatique (LIX)
Bâtiment Alan Turing
1 rue Honoré d'Estienne d'Orves
CS35003 91120 Palaiseau Cedex, France
emmanuel.haucourt@polytechnique.edu