Trees in Partial Higher Dimensional Automata
Abstract
In this paper, we give a new definition of partial Higher Dimension Automata using lax functors. This definition is simpler and more natural from a categorical point of view, but also matches more clearly the intuition that pHDA are Higher Dimensional Automata with some missing faces. We then focus on trees. Originally, for example in transition systems, trees are defined as those systems that have a unique path property. To understand what kind of unique property is needed in pHDA, we start by looking at trees as colimits of paths. This definition tells us that trees are exactly the pHDA with the unique path property modulo a notion of homotopy, and without any shortcuts. This property allows us to prove two interesting characterisations of trees: trees are exactly those pHDA that are an unfolding of another pHDA; and trees are exactly the cofibrant objects, much as in the language of Quillen’s model structure. In particular, this last characterisation gives the premisses of a new understanding of concurrency theory using homotopy theory.
Keywords
Higher Dimensional Automata Trees Homotopy theories1 Introduction
Higher Dimensional Automata (HDA, for short), introduced by Pratt in [23], are a geometric model of true concurrency. Geometric, because they are defined very similarly to simplicial sets, and can be interpreted as glueings of geometric objects, here hypercubes of any dimension. Similarly to other models of concurrency much as event structures [21], asynchronous systems [1, 25], or transition systems with independence [22], they model true concurrency, in the sense that they distinguish interleaving behaviours from simultaneous behaviours. In [12], van Glabbeek proved that they form the most powerful models of a hierarchy of concurrent models. In [6], Fahrenberg described a notion of bisimilarity of HDA using the general framework of open maps from [17]. If this work is very natural, it is confronted with a design problem: paths (or executions) cannot be nicely encoded as HDA. Indeed, in a HDA, it is impossible to model the fact that two actions must be executed at the same time, or that two actions are executed at the same time but one must start before the other. From a geometric point of view, those impossibilities are expressed by the fact that we deal with closed cubes, that is, cubes that must contain all of their faces. Motivated by those examples, Fahrenberg, in [7], extended HDA to partial HDA, intuitively, HDA with cubes with some missing faces. If the intuition is clear, the formalisation is still complicated to achieve: the definition from [7] misses the point that faces can be not uniquely defined. This comes from the fact that Fahrenberg wanted to stick to the ‘local’ definition of precubical sets, that is, that cubes must satisfy some local conditions about faces. As we will show, those local equations are not enough in the partial case. Another missed point is the notion of morphisms of partial HDA: as defined in [7], the natural property that morphisms map executions to executions is not satisfied. In Sect. 2, we address those issues by giving a new definition of partial HDA in terms of lax functors. This definition, similar to the presheaf theoretic definition of HDA, avoid the issues discussed above by considering global inclusions, instead of local equations. This illustrates more clearly the intuition of partial HDA being HDA with missing faces: we coherently replace sets and total functions by sets and partial functions. From this similarity with the original definition of HDA, we can prove that it is possible to complete a partial HDA to turn it into a HDA, by adding the missing faces, and from this completion, it is possible to define a geometric realisation of pHDA (which was impossible with Fahrenberg’s definition).

Open maps from [17] can be understood as trivial fibrations, namely weak equivalences (here, bisimulations) that have the right lifting properties with respect to some morphisms.

Those morphisms are precisely extensions of executions, which means that they can be seen as cofibration generators (in the language of cofibrantly generated model structures [15]).

Cofibrations are then morphisms that have the left lifting property with respect to open maps. In particular, this allows us to define cofibrant objects as those objects whose unique morphisms from the initial object is a cofibration. In a way, cofibrant objects are those objects that are constructed by just using extensions of paths, and should correspond to trees.

The cofibrant replacement is then given by canonically constructing a cofibrant object, which is weakly equivalent (here, bisimilar) to a given object. That should correspond to the unfolding.
The main ingredient is to understand what trees are in this context. In the case of transition systems for semantics of CCS [19], synchronisation trees are those systems with exactly one path from the initial state to any state. Those trees are then much simpler to reason on, but they are still powerful enough to capture any bisimulation type: by unfolding, it is possible to canonically construct a tree from a system. The goal of Sects. 3 and 4 will be to understand how to generalise this to pHDA. In this context, it is not clear what kind of unique path property should be considered as, in general, in truly concurrent systems, we have to deal with homotopies, namely, equivalences of paths modulo permutation of independent actions. Following [4], we will first consider trees as colimits of paths. This will guide us to determine what kind of unique path property is needed: a tree is a pHDA with exactly one class of paths modulo a notion of homotopy, from the initial state to any state, and without any shortcuts. This will be proved by defining a suitable notion of unfolding of pHDA. Finally, in Sect. 5, we prove that those trees coincide exactly with the cofibrant objects, illustrating the first steps of this new understanding of concurrency, using homotopy theory.
2 Fixing the Definition of pHDA
In this Section, we review the definitions of HDA (Sect. 2.1), the first one using face maps, and the second one using presheaves. In Sect. 2.2, we describe the definition of partial HDA from [7] and explain why it does not give us what we are expecting. We tackle those issues by introducing a new definition in Sect. 2.3, extending the presheaf theoretic definition, using lax functors instead of strict functors. Finally, in Sect. 2.4, we prove that HDA form a reflective subcategory of partial HDA, by constructing a completion of a partial HDA.
2.1 Higher Dimensional Automata
Concrete Definition of Precubical Sets. A precubical set X is a collection of sets \((X_n)_{n\in \mathbb {N}}\) together with a collection of functions \((\partial _{i,n}^\alpha : X_n \longrightarrow X_{n1})_{n>0, 1 \le i \le n, \alpha \in \{0,1\}}\) satisfying the local equations \(\partial _{i,n}^\alpha \circ \partial _{j,n+1}^\beta = \partial _{j,n}^\beta \circ \partial _{i+1,n+1}^\alpha \) for every \(\alpha , \beta \in \{0,1\}\), \(n > 0\) and \(1 \le j \le i \le n\). A morphism of precubical sets from X to Y is a collection of functions \((f_n : X_n \longrightarrow Y_n)_{n\in \mathbb {N}}\) satisfying the equations \(f_n\circ \partial _{i,n}^\alpha = \partial _{i,n}^\alpha \circ f_{n+1}\) for every \(n\in \mathbb {N}\), \(1 \le i \le n\) and \(\alpha \in \{0,1\}\). The elements of \(X_0\) are called points, \(X_1\) segments, \(X_2\) squares, \(X_n\) ncubes. In the following, we will call past (resp. future) iface maps the \(\partial _{i,n}^0\) (resp. \(\partial _{i,n}^1\)). We denote this category of precubical sets by \({\mathbf {pCub}}\).
Higher Dimensional Automata [11]. From now on, fix a set L, called the alphabet. We can form a precubical set also noted L such that \(L_n = L^n\) and the iface maps are given by \(\delta _i^\alpha (a_1\ldots a_n) = a_1\ldots a_{i1}.a_{i+1}\ldots a_n\). We can also form the following precubical set \(*\) such that \(*_0 = \{*\}\) and \(*_n = \varnothing \) for \(n > 0\). A HDA X on L is a bialgebra \(*\rightarrow X \rightarrow L\) in \(\mathbf {pCub}\). In other words, a HDA X is a precubical set, also noted X, together with a specified point, the initial state, \(i \in X_0\) and a labelling function \(\lambda : X_1 \longrightarrow L\) satisfying the equations \(\lambda \circ \partial _{i,2}^0 = \lambda \circ \partial _{i,2}^1\) for \(i \in \{1,2\}\) (see previous figure, right). A morphism of HDA from X to Y is a morphism f of precubical sets from X to Y such that \(f_0(i_X) = i_Y\) and \(\lambda _X = \lambda _Y\circ f_1.\) HDA on L and morphisms of HDA form a category that we denote by \({\mathbf {HDA}}_\mathbf{L }\). This category can also be defined as a the double slice category \(*/\mathbf {pCub}/L\). Remark that we are only concerned with labellingpreserving morphisms, not general morphisms as described in [5].
2.2 Original Definition of Partial Higher Dimensional Automata
Originally [7], partial HDA are defined similarly to the concrete definition of HDA, except that the face maps can be partial functions and the local equations hold only when both sides are well defined. There are two reasons why it fails to give the good intuition:
 first the ‘local’ equations are not enough in the partial case. Imagine that we want to model a full cube c without its lower face, that is, \(\partial _{3,3}^0\) is not defined on c, and such that \(\partial _{1,2}^1\) is undefined on \(\partial _{1,3}^1(c)\) and \(\partial _{2,3}^1(c)\), that is, we remove an edge. We cannot prove using the local equations that \(\partial _1^1\circ \partial _2^0\circ \partial _1^1(c) = \partial _1^1\circ \partial _2^0\circ \partial _2^1(c)\), that is, that the vertices of the cube are uniquely defined. Indeed, to prove this equality using the local equations, you can only permute two consecutive \(\partial \). From \(\partial _1^1\circ \partial _2^0\circ \partial _1^1(c)\), you can:and both faces are not defined. On the other hand, those two should be equal because the comaps \(d_1^1\circ d_2^0\circ d_1^1\) and \(d_2^1\circ d_2^0\circ d_1^1\) are equal in \(\Box \), and \(\partial _1^1\circ \partial _2^0\circ \partial _1^1\) and \(\partial _1^1\circ \partial _2^0\circ \partial _2^1\) are both defined on c.

either permute the first two and you obtain \(\partial _1^1\circ \partial _1^1\circ \partial _3^0(c)\),

or permute the last two and you obtain \(\partial _1^0\circ \partial _1^1\circ \partial _1^1(c)\).


secondly, the notion of morphism is not good (or at least, ambiguous). The equations \(f_n\circ \partial _{i,n,X}^\alpha = \partial _{i,n,Y}^\alpha \circ f_{n+1}\) hold in [7] only when both face maps are defined, which authorises many morphisms. For example, consider the segment I, and the ‘split’ segment \(I'\) which is defined as I, except that no face maps are defined (geometrically, this corresponds to two points and an open segment). The identity map from I to \(I'\) is a morphism of partial precubical sets in the sense of [7], which is unexpected. A bad consequence of that is that the notion of paths in a partial HDA does not correspond to morphisms from some particular partial HDA, and paths are not preserved by morphisms, as we will see later.
2.3 Partial Higher Dimensional Automata as Lax Functors

for every object c of \(\mathcal {C}\), a set Fc,

for every morphism \(i : c \longrightarrow c'\), a partial function \(Fi : Fc \longrightarrow Fc'\)

if \(n=0\), \(\epsilon \star (j_1< \ldots< j_m ; \beta _1, \ldots , \beta _m) = (j_1< \ldots < j_m ; \beta _1, \ldots , \beta _m)\),

if \(m = 0\), \((i_1<\ldots< i_n ; \alpha _1, \ldots , \alpha _n) \star \epsilon = (i_1<\ldots < i_n ; \alpha _1, \ldots , \alpha _n)\),

if \(i_1 \le j_1\), \((i_1<\ldots< i_n ; \alpha _1, \ldots , \alpha _n) \star (j_1< \ldots< j_m ; \beta _1, \ldots , \beta _m) = (i_1;\alpha _1).((i_2<\ldots< i_n ; \alpha _2, \ldots , \alpha _n) \star (j_1+1< \ldots < j_m+1 ; \beta _1, \ldots , \beta _m))\),

if \(i_1 > j_1\), \((i_1<\ldots< i_n ; \alpha _1, \ldots , \alpha _n) \star (j_1< \ldots< j_m ; \beta _1, \ldots , \beta _m) = (j_1;\beta _1).((i_1<\ldots< i_n ; \alpha _1, \ldots , \alpha _n) \star (j_2< \ldots < j_m ; \beta _2, \ldots , \beta _m))\).
A functionvalued oplax transformation [20] from \(F : \mathcal {C} \rightharpoonup \mathbf {pSet}\) to \(G : \mathcal {C} \rightharpoonup \mathbf {pSet}\) is a collection \((f_c)_{c\in Ob(\mathcal {C})}\) of total functions such that for every \(i : c \longrightarrow c'\), \(f_{c'}\circ F(i) \subseteq G(i)\circ f_{c}\). A morphism of partial precubical sets from X to Y is then a functionvalued oplax transformation. In other words, this is a collection of total functions \((f_n : X_n \longrightarrow Y_n)_{n\in \mathbb {N}}\) satisfying the equations \(f_n\circ \partial _{i_1< \ldots< i_k}^{\alpha _1, \dots , \alpha _k} \subseteq \partial _{i_1< \ldots < i_k}^{\alpha _1, \ldots , \alpha _k}\circ f_{n+k}\). Partial precubical sets and morphisms of partial precubical sets form a category that we denote by \(\mathbf {ppCub}\). \(\mathbf {pCub}\) is a full subcategory of \(\mathbf {ppCub}\). In particular, the precubical sets \(*\) and L are partial precubical sets. A partial HDA X on L is a partial precubical set, also noted X, together with a specified point, the initial state \(i \in X_0\) and a morphism of ppCub, the labelling functions, \((\lambda _n : X_n \longrightarrow L^n)_{n\in \mathbb {N}}\). A morphism of pHDA from X to Y is a morphism f of partial precubical sets from X to Y such that \(f_0(i_X) = i_Y\) and \(\lambda _X = \lambda _Y\circ f\). Partial HDA on L and morphisms of partial HDA form a category that we note \({\mathbf {pHDA}}_\mathbf{L }\). In other words, this is the double slice category \(*/\mathbf {ppCub}/L\).
2.4 Completion of a pHDA
 if \(\partial _{i_1< \ldots < i_k}^{\alpha _1, \ldots , \alpha _k}(x)\) is defined, thenThis means that, if a face of a cube exists in X, this face is identified with both abstract faces \((\epsilon , \partial _{i_1< \ldots < i_k}^{\alpha _1, \ldots , \alpha _k}(x))\) (i.e., the cube \(\partial _{i_1< \ldots < i_k}^{\alpha _1, \ldots , \alpha _k}(x)\) itself) and \(((i_1< \ldots < i_k ; \alpha _1, \ldots , \alpha _k), x)\) (i.e., the face of x, which consists of taking the \((i_k,\alpha _k)\) face, then the \((i_{k1},\alpha _{k1})\) face, and so on).$$((i_1< \ldots< i_k ; \alpha _1, \ldots , \alpha _k), x) \sim (\epsilon , \partial _{i_1< \ldots < i_k}^{\alpha _1, \ldots , \alpha _k}(x)).$$

if \(((i_1< \ldots< i_k ; \alpha _1, \ldots , \alpha _k), x) \sim ((j_1< \ldots < j_l ; \beta _1, \ldots , \beta _l), y)\), then \(((i_1< \ldots< i_k ; \alpha _1, \ldots , \alpha _k)\star (i,\alpha ), x) \sim ((j_1< \ldots < j_l ; \beta _1, \ldots , \beta _l)\star (i,\alpha ), y)\). This means that if two abstract faces coincide, then taking both their \((i,\alpha )\) face gives two abstract faces that also coincide.
Let \(\chi (X)_n = Y_n/\sim \) and we denote by \(\ll (i_1< \ldots < i_k ; \alpha _1, \ldots , \alpha _k), x \gg \) the equivalence class of \(((i_1< \ldots < i_k ; \alpha _1, \ldots , \alpha _k), x)\) modulo \(\sim \). We define the iface map as \(\partial _i^\alpha (\ll (i_1< \ldots< i_k ; \alpha _1, \ldots , \alpha _k), x\gg ) = ~ \ll (i_1< \ldots < i_k ; \alpha _1, \ldots , \alpha _k)\star (i,\alpha ), x\gg \), the initial state as \(\ll \epsilon ,i\gg \) and the labelling function as \(\lambda (\ll (i_1< \ldots < i_k ; \alpha _1, \ldots , \alpha _k), x \gg ) = \delta _{i_1}^{\alpha _1}\circ \ldots \circ \delta _{i_k}^{\alpha _k}(\lambda (x))\).
Theorem 1
\(\chi \) is a welldefined functor and is the left adjoint of \(\tau \), the injection of \({{\varvec{HDA}}}_{{\varvec{L}}}\) into \({{\varvec{pHDA}}}_{{\varvec{L}}}\). Furthermore, \({{\varvec{HDA}}}_{{\varvec{L}}}\) is a reflective subcategory of \({{\varvec{pHDA}}}_{{\varvec{L}}}\).
Now, we can define the geometric realisation of a pHDA X as the subspace of the realisation of \(\chi (X)\) consisting of points whose carrier is of the form \(\ll \epsilon , x \gg \) for some \(x \in X\). This really corresponds to the drawings we have been using to depict pHDA until now.
3 Paths in Partial Higher Dimensional Automata
Executions of HDA are defined using the notion of paths. Those paths describe the succession of starting and finishing of actions in a HDA. For example, a HDA can start an action then start another at the same time, and finish the two actions. This sequence is then not just a sequence of 1dimensional transitions, since some actions can be made at the same time, but a sequence of hypercubes corresponding to the evolution of the state of the system. We will formalise this idea in Sect. 3.2, and we will see in particular that those paths can be encoded in the category \({\mathbf {pHDA}}_\mathbf{L }\) (while it is not possible in the category \({\mathbf {HDA}}_\mathbf{L }\)) as morphisms from particular pHDA, called path shapes. In Sect. 3.1, let us first start by recalling the general framework of [17].
3.1 Path Category, Open Maps, Coverings
In the general framework of [17], we start with a category \(\mathcal {M}\) of systems, together with a subcategory \(\mathcal {P}\) of execution shapes. For example, keep in mind the case where \(\mathcal {M}\) is the category of transition systems and \(\mathcal {P}\) is the full subcategory of finite linear systems. One interesting remark about this case is that executions of a given systems are in bijective correspondance with morphisms from a finite linear system to this given system. This means that to reason about behaviours of such systems, it is enough to reason about morphisms and execution shapes.
This idea was formalised by describing precisely which morphisms are witnesses for the existence of a bisimulation between systems. This description uses right lifting properties: we say that a morphism \(f : X \longrightarrow Y\) has the right lifting property with respect to \(g : X' \longrightarrow Y'\) if for every \(x : X' \longrightarrow X\) and \(y : Y' \longrightarrow Y\) such that \(f\circ x = y \circ g\), there exists \(\theta : Y' \longrightarrow X\) such that \(x = \theta \circ g\) and \(f\circ \theta = y\). For example, let us assume that f is a morphism of transition systems and that \(X'\) and \(Y'\) are finite linear systems. Then x (resp. y) is the same as an execution in X (resp. Y), and \(f\circ x = y \circ g\) means that the execution y is a extension of the image of the execution x by f. The right lifting property means that the longer execution y of Y can be lifted to a longer execution \(\theta \) of X, that is, \(\theta \) is an extension of x and the image of \(\theta \) by f is y. This property of lifting longer executions is precisely the property needed on a morphism to make its graph relation a bisimulation. They are also very similar to morphisms of coalgebras [16]. We call \(\mathcal {P}\)open (or simply open when \(\mathcal {P}\) is clear), a morphism that has the right lifting property with respect to every morphism in \(\mathcal {P}\). From open maps, it is possible to describe similarity and bismilarity as the existence of a span of morphisms/open maps, and many kinds of bisimilarities can be captured in this way [17]. An open map is said to be a \(\mathcal {P}\)covering (or simply covering) if furthermore the lifts in the right lifting properties are unique. Being a covering is a very strong requirement, as they correspond to partial unfolding of a system.
3.2 Encoding Paths in pHDA
In this section, we describe the classical notion of execution of HDA from [12], extended to partial HDA in [7], defined using the notion of path. We then show that those executions can be encoded as an execution shapes subcategory, as in the general framework of [17], proving in particular that paths are in bijective correspondance with a class of morphisms. A path \(\pi \) of a HDA X is a sequence \(i = x_0 \xrightarrow {j_1,\alpha _1} x_1 \xrightarrow {j_2,\alpha _2} \ldots \xrightarrow {j_n,\alpha _n} x_n\) where \(x_k \in X\), \(j_k > 0\) and \(\alpha _k \in \{0,1\}\) are such that for every k:

if \(\alpha _k =0\), then \(x_{k1} = \partial _{j_k}^{0}(x_k)\),

if \(\alpha _k =1\), then \(x_{k} = \partial _{j_k}^{1}(x_{k1})\).
This definition can easily be extended to pHDA, by requiring that the \(j_k\)face maps are defined on \(x_k\) or \(x_{k1}\). A natural property of executions and morphisms is that morphisms map executions to executions. This is the case here (while it is not for [7], e.g., the split segment):
Proposition 1
If \(f : X \longrightarrow Y\) is a map of pHDA and if \(\pi = x_0 \xrightarrow {j_1,\alpha _1} x_1 \xrightarrow {j_2,\alpha _2} \ldots \xrightarrow {j_n,\alpha _n} x_n\) is a path in X, then \(\pi ' = f(x_0) \xrightarrow {j_1,\alpha _1} f(x_1) \xrightarrow {j_2,\alpha _2} \ldots \xrightarrow {j_n,\alpha _n} f(x_n)\) is a path in Y.

if \(\alpha _k =0\), then \(d_{k1} = d_k  1\), \(\delta _{j_k}(w_k) = w_{k1}\) and \(j_k \le d_k\),

if \(\alpha _k =1\), then \(d_{k} = d_{k1}  1\), \(\delta _{j_k}(w_{k1}) = w_k\) and \(j_k \le d_{k1}\).

\(B\sigma _p = \{k \in \{0,\ldots ,n\} \mid d_k = p\}\),
 the partial face maps \(\partial _{i_1< \ldots < i_n}^{\alpha _1, \ldots , \alpha _n}\) are the smallest (as relations ordered by inclusion) partial functions such that:

if \(\alpha _k = 0\), then \(\partial _{j_k}^{0}(k) = k1\),

if \(\alpha _k = 1\), then \(\partial _{j_k}^{1}(k1) = k\),

\(\partial _{j_1< \ldots< j_m}^{\beta _1, \ldots , \beta _m}\circ \partial _{i_1< \ldots< i_n}^{\alpha _1, \ldots , \alpha _n} \subseteq \partial _{k_1< \ldots < k_{n+m}}^{\gamma _1, \ldots , \gamma _{n+m}}\), for \((k_1, \ldots , k_{n+m} ; \gamma _1, \ldots , \gamma _{n+m}) = (i_1,\ldots , i_n ; \alpha _1, \ldots , \alpha _n) \star (j_1, \ldots , j_m ; \beta _1, \ldots , \beta _m)\).


the initial state is 0,

the labelling functions \(\lambda _n\) map k to \(w_k\).
By a path shape, we mean such a pHDA \(B\sigma \). The set \({\mathbf {Spine}}_\mathbf L \) of spines can be partially ordered by prefix. B can then be extended to an embedding from \({\mathbf {Spine}}_\mathbf L \) to \({\mathbf {pHDA}}_\mathbf{L }\). We note \({\mathbf {PS}}_\mathbf{L }\) the image of this embedding, i.e., the full subcategory of path shapes.
Proposition 2
There is a bijection between paths in a pHDA X and morphisms of pHDA from a path shape to X.
Again, this is not true with the definition of morphisms from [7] (e.g., the split segment). As an example, the red path \(\pi \) above corresponds to a morphism from the path shape \(B\sigma \) to X.
4 Trees and Unfolding in pHDA
In this section, we introduce our notion of trees. Following [4], we consider trees as colimits (or glueings of paths). Section 4.1 is dedicated to proving that those colimits actually exist, by giving an explicit construction of those. From this explicit construction, we will describe the kind of unique path properties that are satisfied by those trees in Sect. 4.2. Starting by showing, that the strict unicity of path fails, we then describe a notion of homotopy, the confluent homotopy, which is weaker than the one from [12], for which every tree has the property that there is exactly one homotopy class of paths form the initial state to any state. We will also see that, because the face maps of trees are defined in a local way, they do not have any shortcuts, that is, paths that ‘skip’ dimensions, for example, going from a point to a square without going through a segment. Finally, in Sect. 4.3, we will prove that those two properties – the unicity of paths modulo confluent homotopy, and the nonexistence of shortcuts – completely characterise trees. This proof will use a suitable notion of unfolding of pHDA, showing furthermore that trees form a coreflective subcategory of pHDA.
4.1 Trees, as Colimits of Paths in pHDA

\(X_0 = \{(u,k) \mid u \in \mathcal {C}, k \le l_u \wedge d^u_k = 0\} \sqcup \{\epsilon \}\),

\(X_n = \{(u,k) \mid u \in \mathcal {C}, k \le l_u \wedge d^u_k = n\}\).

for every u, \((u,0) \sim \epsilon \),

if \(i : u \longrightarrow v \in \mathcal {C}\), and if \(k \le l_u, l_v\), then \((u,k) \sim (v,k)\).
We denote by \(Y_n\) the quotient \(X_n/\sim \), and by \([ u,k ]\) the equivalence class of (u, k) modulo \(\sim \).

\([u_0,k] \approx [v_0,k]\),

for every \(0 \le s \le l\), \(\alpha _{k+1+s}^{u_s} = \alpha _{k+1+s}^{v_s} = 1\),

for every \(0 \le s < l\), \([u_s,k+s+1] \approx [u_{s+1},k+s+1]\) and \([v_s,k+s+1] \approx [v_{s+1},k+s+1]\),

\((j_{k+1}^{u_0};1)\star \ldots \star (j_{k+l+1}^{u_l};1) = (j_{k+1}^{v_0};1)\star \ldots \star (j_{k+l+1}^{v_l};1)\),
then, \([u_l,k+l+1] \approx [v_l,k+l+1]\). \(\text {col} \, D\) is the pHDA \(Z_N\) with the face maps being the smallest relations for inclusion such that:

if \(\alpha _k^u = 0\), then \(\partial _{j_k^u}^0(\langle u,k \rangle )\) is defined and is equal to \(\langle u,k1 \rangle \),

if \(\alpha _{k+1}^u = 1\) then \(\partial _{j_k^u}^1(\langle u,k \rangle )\) is defined and is equal to \(\langle u,k+1 \rangle \),

\(\partial _{j_1< \ldots< j_m}^{\beta _1, \ldots , \beta _m}\circ \partial _{i_1< \ldots< i_n}^{\alpha _1, \ldots , \alpha _n} \subseteq \partial _{k_1< \ldots < k_{n+m}}^{\gamma _1, \ldots , \gamma _{n+m}}\), for \((k_1, \ldots , k_{n+m} ; \gamma _1, \ldots , \gamma _{n+m}) = (i_1,\ldots , i_n ; \alpha _1, \ldots , \alpha _n) \star (j_1, \ldots , j_m ; \beta _1, \ldots , \beta _m)\).
The initial state is \(\langle \epsilon \rangle \) and the labelling \(\lambda : \text {col} \, D \longrightarrow L\) maps \(\langle u,k \rangle \) to \(w_k^u\).
Proposition 3
\(\text {col} \, D\) is the colimit of D in \({{\varvec{pHDA}}}_{{\varvec{L}}}\)
By tree we mean any pHDA that is the colimit of a diagram with values in path shapes. We denote by \({\mathbf {Tr}}_\mathbf{L }\) the full subcategory of trees.
4.2 The Unique Path Properties of Trees
Failure of the Unicity of Paths. Let us consider the pushout square above again. In particular, the pHDA on the bottomright corner is a tree, by definition. However, there are two paths from \(\alpha \) to \(\beta \) (in red and blue). This actually comes from the fact that we needed to identify \(\beta _1\) and \(\beta _2\) to be able to define the face maps. This means that trees do not have the unique path property.
Confluent Homotopy. A careful reader may have observed that the only difference between the two previous paths is that some future faces are swapped. Actually, this is the only obstacle for the unicity of paths for trees: there is a unique path modulo equivalence of paths that permutes arrows of the form \(\xrightarrow {\_,1}\). That is what we call confluent homotopy. This confluent homotopy will be defined by restricting the elementary homotopies of [12] to be of only one type out of the four possible, which means our notion of homotopy makes fewer paths equivalent than the one from [12].

for all \(k < s\) or \(k \ge t\), \(x_k = x'_k\),

for all \(k < s\) or \(k > t\), \(j_k = j'_k\) and \(\alpha _k = \alpha '_k\),

for all \(s \le k \le t\), \(\alpha _k = \alpha '_k = 1\),

\((j_{s},\alpha _{s})\star \ldots \star (j_t,\alpha _t) = (j'_{s},\alpha '_{s})\star \ldots \star (j'_t,\alpha '_t)\).
We denote by \(\sim _{ch}\), and call confluent homotopy, the reflexive transitive closure of Open image in new window .
Lemma 1
If X is a tree, then for every element (of any dimension) x of X, there is exactly one path modulo confluent homotopy from the initial state to x.
Shortcuts. The face maps of path shapes and of the colimits we computed in Sect. 4.1 are of a very particular form: we start by defining the \(\partial _j^\alpha \) and we extend this definition to general \(\partial _{j_1< \ldots < j_n}^{\alpha _1, \ldots , \alpha _n}\). In a way, they are locally defined, and then extended to higher face maps. This means in particular that, in addition to having unique paths modulo confluent homotopy, they also do not have any ‘shortcut’. A possible shortcut can be defined as a generalisation of paths, in which we allow to make transitions that go, for example, from a point to a square or to a cube, not only to segments, a shortcut being such a possible shortcut which is not confluently homotopic to a path. Those shortcuts may occur in a pHDA, even if it has the unique path property. Concretely, by shortcut we mean the following situation: the face \(\partial _{i_1< \ldots < i_n}^{\alpha _1, \ldots , \alpha _n}(x)\) is defined, but there is no sequence \((j_1;\beta _1)\star \ldots \star (j_n;\beta _n) = (i_1< \ldots < i_n;\alpha _1, \ldots , \alpha _n)\) such that \(\partial _{j_n}^{\alpha _n}\circ \ldots \circ \partial _{j_1}^{\alpha _1}(x)\) is defined. By localdefinedness of the face maps:
Lemma 2
Trees do not have any shortcuts.
Trees. We say that a pHDA has the unique path property modulo confluent homotopy if it has no shortcut, and there is exactly one class of paths modulo confluent homotopy from the initial state to any state. Given such a pHDA X and an element x of X, by depth of x we mean the length of a path from the initial state to x in X. Since homotopic paths have the same length, this is uniquely defined. We deduce from the previous discussions that:
Proposition 4
Trees have unique path property modulo confluent homotopy.
In the following, we will prove the converse: trees, defined as colimits of path shapes are exactly those pHDA that have the unique path property modulo confluent homotopy. This will be done by proving that such a pHDA X is isomorphic to its unfolding. A question that occurs now is the following. Much as the general framework of [4], trees are colimits of paths. Everything tends to work well when those trees have a nice property, which we called accessibility, intuitively, that the colimit process do not ‘create’ paths. This property is actually deeply related to the unicity of paths. Since this unicity fails in the case of pHDA, accessibility fails too. However, an accessibility modulo confluent homotopy holds: the colimit process in pHDA do not create confluent homotopy classes of paths.
4.3 Trees Are Unfoldings
We are now constructing our unfolding U(X) of a pHDA X by giving an explicit definition, similar to [6, 11], and proving that this is a tree. We will prove that there is a covering \(\text {unf}_X : U(X) \longrightarrow X\), which in particular means that the unfolding U(X) is \({\mathbf {PS}}_\mathbf{L }\)bisimilar (in the general sense of [17]) to X, and that this covering is actually an isomorphism when X has the unique path property modulo confluent homotopy.

\(U(X)_n\) is the set of equivalence classes \([\pi ]\) of paths modulo confluent homotopy, such that \(e(\pi )\) is of dimension n,
 the face maps are the smallest relations for inclusion such that:

\(\partial _i^1(\alpha ) = [\pi \xrightarrow {i,1} \partial _i^1(e(\pi ))]\), for any \(\pi \in \alpha \) such that \(\partial _i^1(e(\pi ))\) is defined,

\(\partial _i^0(\alpha ) = [\pi _{1}]\) for any \(\pi \in \alpha \) such that \(\pi = \pi _{1} \xrightarrow {i,0} e(\pi )\),

\(\partial _{j_1< \ldots< j_m}^{\beta _1, \ldots , \beta _m}\circ \partial _{i_1< \ldots< i_n}^{\alpha _1, \ldots , \alpha _n} \subseteq \partial _{k_1< \ldots < k_{n+m}}^{\gamma _1, \ldots , \gamma _{n+m}}\), for \((k_1, \ldots , k_{n+m} ; \gamma _1, \ldots , \gamma _{n+m}) = (i_1,\ldots , i_n ; \alpha _1, \ldots , \alpha _n) \star (j_1, \ldots , j_m ; \beta _1, \ldots , \beta _m)\).


the initial state is [i],

the labelling is given by \(\lambda (\alpha ) = \lambda (e(\pi ))\) for \(\pi \in \alpha \).
Following ideas from [4] again, the unfolding can be seen as the glueing of all possible executions of a system, but with care needed to handle confluent homotopy. Concretely:
Proposition 5
The unfolding of a pHDA is a tree.
We can also define \(\text {unf}_X : U(X) \longrightarrow X\) as the function that maps \([\pi ]\) to \(e(\pi )\).
Proposition 6
\(\text {unf}_X\) is a covering, and so, U(X) is \({{{\varvec{PS}}}_{{\varvec{L}}}}\)bisimilar to X.
The Unique Path Property Characterises Trees. When X has exactly one class of paths modulo confluent homotopy from the initial state to any state, it is possible to define a function \(\eta _X : X \longrightarrow U(X)\) that maps any element x of X to the unique confluent homotopy class to x. When furthermore X does not have shortcuts, then \(\eta \) is actually a morphism of pHDA.
Proposition 7
When X has the unique path property modulo confluent homotopy, then \(\eta _X\) is the inverse of \(\text {unf}_X\). In particular, X is a tree.
Together with Proposition 4, this implies the following:
Theorem 2
Trees are exactly the pHDA that have the unique path property modulo confluent homotopy.
Another consequence is that this isomorphism \(\eta _X\) is actually natural (in the categorical sense) and is part of an adjunction, which implies that trees form a coreflective subcategory of pHDA:
Corollary 1
U extends to a functor, which is the right adjoint of the embedding \(\iota : {{{\varvec{Tr}}}_{{\varvec{L}}}} \longrightarrow {{\varvec{pHDA}}}_{{\varvec{L}}}\). Furthermore, this is a coreflection.
5 Cofibrant Objects
Cofibrant objects are another type of ‘simple objects’, coming from homotopy theory, more particularly the language of model categories from [24]. Those cofibrant objects are those whose unique morphism from the initial object is a cofibration. Intuitively (intuition which holds at least in cofibrantly generated model structures [15]), this means that cofibrant objects are those objects constructed from ‘nothing’, using only very basic constructions (generators of cofibrations). In the case of the classical model structure on topological spaces (KanQuillen), those spaces are those constructed from the empty space by adding ‘cells’, which produces what is called CWcomplexes. In this section, we want to mimic this idea with trees: trees are those pHDA constructed from an initial state by only extending paths. We also want to emphasize that much as CWcomplexes gives a kind of homotopy type of a space, trees gives a concurrency type of a pHDA, in the sense that there is a canonical way to produce an equivalent cofibrant object out of any object, which is called the cofibrant replacement in homotopy theory. In concurrency theory, this is the unfolding.
5.1 Cofibrant Objects in \({\mathbf {pHDA}}_\mathbf{L }\)
Following the language of model structures from [24], we say that a pHDA X is cofibrant if for every \({\mathbf {PS}}_\mathbf{L }\)open morphism \(f : Y \longrightarrow Z\) and every morphism \(g : X \longrightarrow Z\), there is a morphism \(h : X \longrightarrow Y\), such that \(f\circ h = g\). That is, a partial HDA X is cofibrant if and only if every \({\mathbf {PS}}_\mathbf{L }\)open morphism has the right lifting property with respect to the unique morphism from \(*\) to X.
5.2 Cofibrant Objects Are Exactly Trees
In this section, we would like to prove the following:
Theorem 3
The cofibrant objects are exactly trees.
Let us start by giving the idea of the proof of the fact that cofibrant objects are trees. By Proposition 6, \(\text {unf}_X\) is a covering, so is open. This means that for every cofibrant object X, there is a morphism \(h : X \longrightarrow U(X)\) such that \(\text {unf}_X\circ h = \text {id}_X\), that is, X is a retract of its unfolding. Since we know that the unfolding is a tree by Proposition 5, it is enough to observe the following:
Lemma 3
A retract of a tree is a tree.
Intuitively, a pHDA is the retract of a tree only when it is obtain by retracting branches. This can only produce a tree. For the converse:
Proposition 8
A tree is a cofibrant object. Furthermore, if \(f : Y \longrightarrow Z\) is a covering, then the lift \(h : X \longrightarrow Y\) is unique.
The lift h is constructed by induction as follows. We define \(X_n\) as the restriction of X to elements whose depth is smaller than n, and the face maps \(\partial _{j_1< \ldots < j_m}^{\alpha _1, \ldots , \alpha _m}(x)\) are defined if and only if \(\partial _{j_1< \ldots < j_m}^{\alpha _1, \ldots , \alpha _m}(x)\) is defined in X and belongs to \(X_n\). We then construct \(h_n : X_n \longrightarrow Y\) using the unique path property modulo confluent homotopy, in a natural way (in the categorical meaning), i.e., such that \(h_{n}\circ \kappa _n = h_{n1}\), where \(\kappa _n : X_{n1} \longrightarrow X_{n}\) is the inclusion. h is then the inductive limit of those \(h_n\). This proof can be seen as a small object argument.
5.3 The Unfolding Is Universal
As an application of the previous theorem, we would like to prove that the unfolding is universal. As in the case of covering spaces in algebraic topology, a covering corresponds to a partial unrolling of a system, in the sense that we can unroll some loops or even partially unroll a loop (imagine for example executing a few steps of a whileloop). In this sense, we can describe the fact that a covering unrolls more than another one, and that, an unfolding is a complete unrolling: since the domain is a tree, it is impossible to unroll more. Actually, much as the topological and the groupoidal cases (see [18] for example), unfoldings are the only such maximal unrollings among coverings: they are initial among coverings, that is why we call them ‘universal’. In a way, this says that our definition of unfolding is the only reasonable one. Concretely, we say that a \({\mathbf {PS}}_\mathbf{L }\)covering is universal if its domain is a tree.
Corollary 2
If \(f : Y \longrightarrow X\) is a universal covering, then for every covering \(g : Z \longrightarrow X\) there is a unique map \(h : Y \longrightarrow X\) such that \(f = g\circ h\). Furthermore, h is itself a covering. Consequently, the universal covering is unique upto isomorphism, and is given by the unfolding.
This whole story is similar to the universal covering of a topological space: just replace pHDA by spaces and trees by simplyconnected spaces [2].
6 Conclusion and Future Work
In this paper, we have given a cleaner definition of partial precubical sets and partial Higher Dimensional Automata, as they really correspond to collections of cubes with missing faces. From this categorical definition, we derived that pHDA can be completed, giving rise to a geometric realisation. We also describe the first premisses of a homotopy theory of the concurrency of pHDA where the cofibrant objects are trees, and replacement is the unfolding. As a future work, we could look at wider class of paths, typically allowing shortcuts as paths, or introducing general homotopies in the path category, which is possible because we can encode those inside the category of pHDA. Another direction would be to continue the description of this homotopy theory, to see if it corresponds to some kind of Quillen’s model structure, or at least to some weaker version (e.g., category of cofibrant objects).
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