Spaces of directed paths on pre-cubical sets II

For a given pre-cubical set (□\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}-set) K with two distinguished vertices 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {0}$$\end{document}, 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {1}$$\end{document}, we prove that the space P→(K)01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {P}(K)_\mathbf {0}^\mathbf {1}$$\end{document} of d-paths on the geometric realization of K with source 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {0}$$\end{document} and target 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {1}$$\end{document} is homotopy equivalent to its subspace P→t(K)01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {P}^t(K)_\mathbf {0}^\mathbf {1}$$\end{document} of tame d-paths. When K is the underlying □\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}-set of a Higher Dimensional Automaton A, tame d-paths on K represent step executions of A. Then, we define the cube chain category of K and prove that its nerve is weakly homotopy equivalent to P→(K)01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {P}(K)_\mathbf {0}^\mathbf {1}$$\end{document}.


Introduction
A directed space, or a d-space (Grandis 2003), is a topological space X with a distinguished family of paths P(X ), called d-paths, that contains all constant paths and is closed with respect to concatenations and non-decreasing reparametrizations. Directed spaces serve as models in concurrency: points of a given directed space represent possible states of a concurrent program, while d-paths represent possible partial executions. It is important to know the homotopy type of the space P(X ) y x of d-paths beginning at the point x and ending at the point y. If x and y are the initial and the final state of the program, respectively, it represents the "execution space" of the program modeled by X . Also, calculation of some invariants of d-spaces, e.g. component categories The author thanks the Hausdorff Research Institute for Mathematics for its hospitality during his visit in Bonn as a part of the programme Applied and Computational Algebraic Topology in September, 2017. The author also thanks the referees for many valuable comments and remarks. c Fig. 1 The d-path in the left-hand picture is tame; it represents an execution which can be divided into two steps: in the first, a and c are performed and, in the second, b is performed. No such division is possible for the d-paths in the right-hand picture (Raussen 2007;Ziemiański 2019b) and natural homology (Dubut et al. 2015), requires knowledge of the homotopy types of d-path spaces between two particular points.
In this paper, we consider this problem for d-spaces that are geometric realizations of pre-cubical sets, called also -sets. -sets play an important role in concurrency: Higher Dimensional Automata introduced by Pratt (1991) are -sets equipped with a labeling of edges, and then executions of a Higher Dimensional Automaton can be interpreted as d-paths on the geometric realization of the underlying -set. van Glabbeek (2006) has shown that many other models for concurrency (e.g. Petri nets) can be translated to Higher Dimensional Automata and, therefore, to -sets.
The problem of calculating the homotopy types of d-path spaces between two vertices of a -set was studied in several papers, e.g. Raussen (2010Raussen ( , 2012 and Ziemiański (2017Ziemiański ( , 2019a. All these results work only for special classes of -sets, like Euclidean complexes or proper -sets, i.e., those whose triangulations are simplicial complexes. There are some interesting examples of -sets that do not fall into any of these classes; notably, the universal labeling -sets ! introduced by Goubault (2002), see also Fahrenberg and Legay (2013). In this paper, we consider -sets in their full generality.
For an arbitrary -set K with two distinguished vertices 0, 1, we prove that the space of d-paths P(|K |) 1 0 with source 0 and target 1 is homotopy equivalent to its subspace P t (|K |) 1 0 of tame d-paths (Theorem 6.1). A d-path is tame if it can be divided into segments each of which runs from the initial to the final vertex of some cube (see Fig. 1). Then we define the cube chain category of K , denoted Ch(K ), and prove that the geometric realization of the nerve of Ch(K ) is weakly homotopy equivalent to the space of tame d-paths on K (Theorem 7.5). This provides a combinatorial model for the execution space of K (Theorem 7.6), which can be used for explicit calculations of its homotopy type. All these constructions are functorial with respect to K , regarded as an object in the category of bi-pointed -sets. These theorems generalize the results of Ziemiański (2017).
Theorem 6.1 is interesting in itself. In terms of Higher Dimensional Automata, tame d-paths represent synchronized executions: at every step, a number of processes performs one complete step while the others remain idle. As a consequence of Theorem 6.1, not only homotopy classes of synchronized executions of a given Higher Dimensional Automaton are the same as homotopy classes of all executions but also the respective execution spaces are homotopy equivalent. Thus, one may expect that all phenomena concerning executions of Higher Dimensional Automata can be observed as well after restricting to synchronized executions only. Organization of the paper and relationship with Ziemiański (2017). The paper consists of two parts. The main goal of the first part  is to prove the tamification theorem (Theorem 6.1), that of the second part (Sects. 7-10) is to prove Theorems 7.5 and 7.6. The general outline of this paper resembles that of Ziemiański (2017) but we need to use more subtle arguments here.
Fix a bi-pointed -set K with initial vertex 0 and final vertex 1. All d-paths α ∈ P(K ) 1 0 have integral L 1 -lengths (Raussen 2009), and the spaces of d-paths having length n, denoted P(K ; n) 1 0 , will be handled separately for every n ∈ Z ≥0 . In Sect. 3 we define tracks, which are sequences of cubes such that for two subsequent cubes, some upper face of the preceding cube is a lower face of the succeeding one. Tracks were investigated by Fahrenberg and Legay (2013); they called them cube paths. Then we prove that every d-path α ∈ P(K ) 1 0 is a concatenation of d-paths contained in consecutive cubes of some track. In Sect. 4, we define the set of actions that correspond to a given track C. In Sect. 5, we introduce progress functions of tracks and investigate the relationship between progress functions of a track C and d-paths contained in C. Then in Sect. 6, we construct, in a functorial way, a self-map of the space P(K ; n) 1 0 that is homotopic to the identity map and maps all natural d-paths (i.e., parametrized by length) into tame d-paths. The proof of the latter statement makes essential use of progress functions. Since the space of natural d-paths is homotopy equivalent to the space of all d-paths, this implies Theorem 6.1.
This argument is essentially different from the one used in the proof of the tamification theorem in the previous paper (Ziemiański 2017, Theorem 5.6). That one follows from a similar result for d-simplicial complexes (Ziemiański 2012), which was proved by constructing a certain self-deformation of a given d-simplicial complex. This selfdeformation was given by a direct but complicated and non-functorial formula and it is not clear how to interpret it in terms of concurrent processes. While a general outline of the argument is similar, the tamification via progress functions is more intuitive: a d-path α contained in a track C is deformed to a tame d-path by "speeding up" the executions of the actions of C.
In the second part, we introduce cube chains: sequences of cubes in K such that the final vertex of the preceding cube is the initial vertex of the succeeding one. They constitute a special case of tracks. Then, we formulate the main result of the second part of the paper stating that the nerve of the category Ch(K ) of cube chains on K is weakly homotopy equivalent to the space of natural tame d-paths on K (Theorem 7.5). A natural d-path α ∈ N (K ) 1 0 is tame if and only it is contained in some cube chain, i.e., it admits a presentation as a concatenation of d-paths contained in consecutive cubes of a given cube chain c. Such a presentation is called a natural tame presentation of α; the difficulty that arises here is due to the fact that α may have many natural tame presentations in c. This is essentially more complicated than the situation considered in Ziemiański (2017), where there is a good cover of the space of natural tame d-paths N t (K ) 1 0 indexed by the poset of cube chains on K , and the analogue of Theorem 7.5 can be proven using the Nerve Lemma.
Instead of a good cover, we need to investigate the functor G := N [0,n] ( ∨ (−) ) 1 0 : Ch(K ) → Top, where G(c) is the space of natural tame presentations in c, equipped with a map F K n : colim G → N t [0,n] (K ) 1 0 . In Sect. 8, we investigate properties of natural tame presentations. In Sects. 9 and 10, we apply the results of Sect. 8 to prove that the middle and the right-hand map in the sequence (7.5) are weak homotopy equivalences. This is easy for the left-hand map, so Theorem 7.5 follows.

Preliminaries
In this section we recall definitions and introduce notation that is used later on. See Fajstrup et al. (2006) or Fajstrup et al. (2016) for surveys which cover most of these topics.

d-spaces
Grandis (2003) defines a d-space as a pair (X , P(X )), where X is a topological space and P(X ) is a family of paths on X (with domain [0, 1]) that contains all constant paths and which is closed with respect to concatenations and non-decreasing reparametrizations. In this paper, it is more convenient to use a slightly different, though equivalent, definition. A d-space is a topological space X equipped with a d-structure. A d-structure on X is a collection of families of paths { P [a,b] (X )} a<b∈R , then the concatenation of α and β: is a d-path, i.e., α * β ∈ P [a,c] (X ).
This definition is equivalent to the original definition by Grandis: is a d-space, then (X , P [0,1] (X )) is a d-space in Grandis' sense. If (X , P(X )) is a dspace in Grandis' sense, then by letting we obtain the d-space as defined above. We will occasionally write P(X ) for P [0,1] (X ). The sets P [a,b] (X ) are topological spaces, with the compact-open topology inherited from the space of all paths P [a,b] [a,b] (X ) and for all a < b ∈ R. The family of d-spaces with d-maps forms the category dTop, which is complete and cocomplete.
For a d-space X and x, y ∈ X , denote by P [a,b] (X ) y x ⊆ P [a,b] (X ) the subspace of d-paths α such that α(a) = x and α(b) = y.
A bi-pointed d-space is a d-space X with two distinguished points: an initial point 0 X and a final point 1 X ∈ X . A bi-pointed d-map is a d-map that preserves the initial and the final points. The category of bi-pointed d-spaces and bi-pointed maps will be denoted by dTop * * .

Directed intervals and cubes
The directed n-cube I n is the categorical product of n copies of the directed unit interval. A path on I n is a d-path if all its coordinates are d-paths in I , i.e., they are nondecreasing functions. Points of I n will be denoted by bold letters, and their coordinates are distinguished by upper indices, so that, for example x = (x 1 , x 2 , . . . , x n ) ∈ I n . A similar convention will be used for d-paths: for β ∈ P( I n ), β i ∈ P( I ) denotes the ith coordinate of β. We will write |x| for n i=1 x i . Whenever I or I n are considered as bi-pointed d-spaces, their initial and final points are 0, 1 ∈ I and 0 = (0, . . . , 0), 1 = (1, . . . , 1) ∈ I n , respectively.

Quotient d-spaces
Let X , Y be topological spaces and let p : X → Y be a quotient map. Assume that X is equipped with a d-structure { P [a,b] (X )} a<b∈R . The quotient d-structure on the space Y is defined in the following way: a path α ∈ P [a,b] (Y ) is a d-path if and only if there exist numbers a = t 0 < · · · < t n = b and d-paths The quotient d-structure is the smallest d-structure on Y such that p is a d-map. The space Y with this quotient d-structure will be called the quotient d-space of X .

-sets
A pre-cubical set, or a -set K is a sequence of disjoint sets (K [n]) n≥0 with a collection of face maps (d ε for all ε, η ∈ {0, 1} and i < j. Elements of the sets K [n] will be called n--cubes or just cubes and 0-cubes will be called vertices. The dimension of a cube c is the integer dim(c) such that c ∈ K [dim(c)]. For -sets K , L, a -map f : K → L is a sequence of maps f [n] : K [n] → L[n] that commute with the face maps. The category of -sets and -maps will be denoted by Set.
An example of a -set is the standard n-cube n , such that n [k] is the set of functions {1, . . . , n} → {0, 1, * } that take value * for exactly k arguments. The face map d ε i converts the ith occurrence of * into ε. The only element of n [n] will be denoted by u n . For any -set K , there is a 1-1 correspondence between the set of n-cubes K [n] and the set of -maps n → K : for every c ∈ K [n] there exists a unique map f c : n → K such that f c (u n ) = c.
The kth skeleton of a -set K is the sub--set K (k) ⊆ K given by The boundary of the standard n-cube is ∂ n := n (n−1) . Given a -set K , a subset A = {a 1 < · · · < a k } ⊆ {1, . . . , n} and ε ∈ {0, 1}, define the iterated face map . For a cube c ∈ K [n], d 0 (c) and d 1 (c) will be called the initial and the final vertex of c, respectively.
A bi-pointed -set is a triple (K , 0 K , 1 k ), where K is a -set and 0 K , 1 K ∈ K [0] are distinguished vertices; we will write 0 and 1 for 0 K and 1 K whenever this does not lead to confusion. The category of bi-pointed -sets and base-points-preserving -maps will be denoted by Set * * .

Presentations of d-paths
Let K be a -set. A presentation of a d-path α ∈ P [a,b] We write such a presentation as (2.7) It follows immediately that every d-path in |K | admits a presentation.

Length
Let K be a -set. The L 1 -length, or just the length of a d-path α ∈ P [a,b] (K ) is defined as for some presentation (2.7) of α. The length was introduced by Raussen (2009, Section 2). This definition does not depend on the choice of a presentation, and defines, for every a < b ∈ R, a continuous function len : P(K ) [a,b] → R ≥0 (Raussen 2009, Proposition 2.7). If x, y ∈ |K | and d-paths α, α ∈ P [a,b] (K ) y x are d-homotopic (i.e., they are contained in the same path-connected component of P [a,b] where P [a,b] (K ; n) 1 0 stands for the space of d-paths having length n.

Naturalization
We say that a d- Natural d-paths were introduced and studied by Raussen (2009). He proved that for (2.10) Furthermore, for a bi-pointed -set K , the map is a homotopy inverse of the inclusion map (Raussen 2009, Propositions 2.15 and 2.16). The map nat K n is functorial with respect to K ∈ Set * * .

Tame paths
. This definition generalizes the earlier definitions for d-paths on d-simplicial complexes (Ziemiański 2012) and on proper -sets (Ziemiański 2017).
If α is tame, then one can impose an even stronger condition on its presentation. For where β i is the path obtained from β i by skipping its jth coordinate. By repeating this operation, we obtain a presentation such that β i (t i ) = (1, . . . , 1) for all i. In a similar way, we can guarantee that β i (t i−1 ) = (0, . . . , 0). A presentation (2.7) such that β i (t i−1 ) = 0 and β i (t i ) = 1 for all i will be called a tame presentation of α.
denote the space of all tame (resp. natural tame) d-paths on K from 0 to 1.

Tracks
Let K be a bi-pointed -set. Every d-path α ∈ P(K ) 1 0 is a concatenation of d-paths having the form [c; β], where c is a cube of K and β is a d-path in I dim(c) . We will show that we can impose extra conditions on the cubes and the paths which appear in such a presentation.
Remark Tracks are equivalent to cube paths introduced by Fahrenberg and Legay (2013, Section 3).

Proposition 3.2 For every track
The integer |A i | = |B i | will be called the length of the track C and denoted len(C).

Definition 3.3 Let
We say that α is contained in C if it admits a C-presentation. The space of d-paths contained in C will be denoted by P [a,b] (K , C).
Proof. Choose a C-presentation of α, as in Definition 3.3. For every i ∈ {1, . . . , l} we have Fajstrup proved (Fajstrup 2005, 2.20) that if K is a geometric -set (Fajstrup 2005, 2.8) then every d-path α ∈ P(K ) 1 0 is contained in a track called the carrier sequence of α. Below we prove an analogue of this result for arbitrary -sets.

Proposition 3.5 Every non-constant d-path
such that the integer l + l i=1 dim(c i ) is minimal among all presentations of α. Then: which contradicts the assumption that l + dim(c i ) is minimal.  (1) and (2) that, for every i, there exist: All coordinates of both x i and y i are different from both 0 and 1. Thus, are both canonical presentations of the same point and, therefore, they are equal. As a consequence, , which proves that Definition 3.1. (c) is satisfied. Thus, C is a track. Moreover, the points x i = y i fit into the Definition 3.3; hence, α is contained in C.

Actions
Every d-path α between vertices of a -set K having length n can be interpreted as a performance of n different actions. This is an easy observation if K is a Euclidean complex in the sense of Raussen and Ziemiański (2014). In this section we will show how to interpret actions when K is an arbitrary -set.
Fix a bi-pointed -set K and a track C We will call these pairs local C-actions (or local actions if C is clear). Let ∼ be the equivalence relation on the set of local C-actions generated by Definition 4.1 A C-action (or an action if C is clear) is an equivalence class of the relation ∼. The set of all C-actions will be denoted by T (C). The C-action represented by (i, r ) will be denoted [i, r ].
The following proposition justifies the definition above.
Proof. Since x j = 1 for all j ∈ B i and y k = 0 for all k ∈ A i+1 , we have Let us collect some basic properties of C-actions: (1) For every i ∈ {1, . . . , l − 1}, the sequences of actions are equal.
(2) Every action p ∈ T (C) has at most one representative having the form (i, r ) for a fixed i. If such a representative exists, its second coordinate will be denoted by In such a case we will say that the action p is active at the ith stage.
(3) For a given action p, the set of stages at which p is active forms a (non-empty) interval, i.e., has the form These are the sets of actions that are finished, active and unstarted, respectively, at the ith stage.
(4) For every i, the following pairs of conditions are equivalent: (4.7)

Progress functions
In this section we introduce progress functions, which provide a convenient description of d-paths contained in a given track C. Fix a bi-pointed -set K , a track Let PF [a,b] (C) be the space of progress functions of the track C, with the compact-open topology.
For every progress function f ∈ PF [a,b] (C) and an action p ∈ T (C) we have f p (a) = 0 and f p (b) = 1. Thus, f p can be regarded as a d-path in P [a,b] ( I ) 1 0 , and PF [a,b] (C), as a subspace of P [a,b] ( I T (C) ) 1 0 . The support of f p , defined by is an open interval. We will denote its endpoints by a For any sequence of numbers (t i ) satisfying Definition 5.1 we have The support of a progress function f is the set In the remaining part of this section we will describe the relationship between progress functions of C and d-paths contained in C. Let f = ( f p ) p∈T (C) be a progress function of C. Our goal is to construct a d-path α f ∈ P(K ; C) 1 0 that corresponds to f. For every i ∈ {1, . . . , l} let is the maximal one that is disjoint from the supports of functions f p for actions p that are not active at the ith stage. For every i define a d-path and This implies (a). To prove (b), it is enough to check that α f Thus, f [i,r ] (t) = 1 for all r ∈ B i . As a consequence, whereb j i and q i are defined in (4.1). Finally, In a similar way we can show that ] for all j, the conclusion follows.

Definition 5.3 The d-path associated to a progress function
Proposition 5.2 guarantees that α f exists and is determined uniquely.
The construction presented above defines the map which can be shown to be continuous. We skip a proof of this fact since it is tedious and not necessary for proving the main results of this paper. Now we will construct a progress function associated to a given d-path α ∈ P [a,b] For an arbitrary action p ∈ T (C) let us define the function and let f α = ( f p α ) p∈T (C) . Proposition 5.4 implies that this definition is valid.
Proposition 5.5 f α is a progress function on C. Moreover, α f α = α.
Proof. The sequence (t i ) from the presentation (5.6) satisfies the conditions required in Definition 5.1. Notice that The function f α will be called the progress function of α ∈ P(K ; C) with the presentation (5.6). Proposition 5.5 implies that the map R C is surjective. This map is not, in general, a bijection: f α depends not only on the d-path α but also on the choice of its presentation, as shown in the example below.
Then T (C) = {[1, 1], [1, 2]} and the progress function of α is given by The following observation, which relates the progress function of a d-path with its L 1 -length, plays an important role in the succeeding section. It is an immediate consequence of the definitions.
Proposition 5.7 Fix α ∈ P [a,b] (K ; C) and its C-presentation (5.6). For every t ∈ [a, b] we have len(α| [a,t] In particular, if α is natural, then f p α is a 1-Lipschitz function for every p ∈ T (C).

Tamification theorem
In this section we will use the results obtained above to prove that the spaces of d-paths and of tame d-paths are homotopy equivalent. The main result is the following: For the vertical maps this follows from Raussen (2009). The main idea of the proof of Theorem 6.1 is to construct a functorial self-map of P [0,n] (K ; n) 1 0 that is homotopic to the identity and maps N [0,n] (K ) 1 0 into P t [0,n] (K ; n) 1 0 . Any d-map g : I → I that preserves 0 and 1 induces a self-d-map g K of |K | such that . Furthermore, a homotopy g s between g 0 = id and g 1 = g induces a homotopy between g K and the identity map on |K |. Below we introduce a "path-length-parametrized" analogue of this construction. Let R : −−→ [0, n]× I → I be an arbitrary d-map such that R(t, 0) = 0 and R(t, 1) = 1 for all t ∈ [0, n]. For every k ≥ 0, R induces the map The maps R k are compatible with the face maps, i.e., for every ε ∈ {0, 1}, k ≥ 0 and i ∈ {1, . . . , k + 1}, the diagram commutes. As a consequence, for every K ∈ Set, the maps R k induce the continuous d-map n] × |K | → |K | induced by the maps R s is a homotopy between R K and the projection on the second factor.
These maps define, for 0 ≤ a ≤ b ≤ n, the following self-maps of d-path spaces: All these maps are homotopic to the respective identities via the families of mapsR s , R k s andR K s that are defined in a similar way. Now assume that R : −−→ [0, n] × I → I satisfies the following conditions: is an interval having length less or equal to 1 4n . Such a function exists; an example is given by the formula (Figs. 2, 3) R(t, h) = min(1, max(0, (4nt + 12n 2 h − 8n 2 ))).
Immediately from the definition follows that for all i ∈ {1, . . . , l}, t ∈ [t i−1 , t i ]. Thus, be the progress functions of α and ω associated to the presentations (6.6) and (6.8), respectively.
be the decomposition into the union of connected components, ordered increasingly. Let y 0 = 0, x r +1 = n. For every s ∈ {1, . . . , r }, let A s ⊆ T (C) be the set of actions p such that supp( f p ω ) ⊆ (x s , y s ); clearly, Our goal is to show that is a tame presentation. By Proposition 6.2, for every s ∈ {1, . . . , r } the length of the interval (x s , y s ) is less than 1 4n |A s | ≤ 1 4 . Since f p α is a 1-Lipschitz function for every action p (Proposition 5.7), by Proposition 6.3 we have Therefore, u s / ∈ supp( f p α ), which implies that p / ∈ A s . As a consequence, and then b p f ω ≤ y s−1 . Therefore, for all s we have A similar argument shows that also which shows that (*) is a tame presentation of ω.
Proof of Theorem 6.1 By Proposition 6.4, the diagram (6.1) can be completed to the diagram The following is an immediate consequence of Theorem 6.1: Corollary 6.6 For every n ≥ 0 and every bi-pointed -set K ∈ Set * * , the map Tam K n is a homotopy inverse of the inclusion i K n : N t [0,n] (K ) 1 0 ⊆ P [0,n] (K ; n) 1 0 . Furthermore, the maps Tam K n are functorial with respect to K , i.e., they define the natural transformation P [0,n]

Cube chains
In this section we define the cube chain category Ch(K ) of a bi-pointed -set K , and formulate the main results of the paper.
Definition 7.1 Let (K , 0 K , 1 K ), (L, 0 L , 1 L ) be bi-pointed -sets. The sequential wedge, or the wedge of K and L is a bi-pointed -set (K ∨ L, 0 K ∨L , 1 K ∨L) such that • the face maps of K ∨ L are the disjoints unions of the face maps of K and L, Remark The sequential wedge operation is associative but not commutative. Similarly, one can define the "parallel" wedge of K and L by identifying 0 K with 0 L and 1 K with 1 L in K L. This will not be needed here; in this paper we use only sequential wedges.
Let Seq(n) be the set of sequences of positive integers n = (n 1 , . . . , n l ) such that n 1 + · · · + n l = n. For n ∈ Seq(n), • |n| = n is the length of n, • l(n) = l is the number of elements of n, • t n i = i j=1 n j for i ∈ {0, . . . , l(n)} is the ith vertex of n, i=0 is the set of vertices of n. Definition 7.2 Let n ∈ Seq(n). The wedge n-cube is the bi-pointed -set ∨n := n 1 ∨ n 2 ∨ · · · ∨ n l , where n i is regarded as a bi-pointed -set by taking 0 = (0, . . . , 0), 1 = (1, . . . , 1). The geometric realization of ∨n will be denoted by I ∨n . For i ∈ {0, . . . , l(n)}, is the ith vertex of ∨n .
Let Ch n be the full subcategory of Set * * with objects ∨n for n ∈ Seq(n). We will introduce a notation for some morphisms of Ch n . Let u n ∈ n [n] be the unique top dimension cube. For a partition {1, . . . , m 1 + m 2 } = A∪ B, |A| = m 2 > 0, |B| = m 1 > 0, let ϕ A,B : m 1 ∨ m 2 → m 1 +m 2 be the unique bi-pointed -map such that ϕ A,B (u m 1 ) = d 0 B (u m 1 +m 2 ) and ϕ A,B (u m 2 ) = d 1 A (u m 1 +m 2 ). For n ∈ Seq(n), i ∈ {1, . . . , l(n)} and A∪ B = {1, . . . , n i } let δ i,A,B : n 1 ∨ · · · ∨ n i−1 ∨ |A| ∨ |B| ∨ n i+1 ∨ · · · n l id n 1 ∨···∨id n i−1 ∨ϕ A,B ∨id n i+1 ∨···∨id n l Fix a bi-pointed -set K ∈ Set * * . Definition 7.3 A cube chain in K (of length n) is a bi-pointed -map c : ∨n c → K for some n c ∈ Seq(n). The multi-index n = n c will be called the type of c. For short we denote l(c) = l(n c ), t c i = t n c i . There is a 1-1 correspondence between cube chains in K and sequences of cubes such that c(u dim(c i ) ) = c i . This shows that the notion of a cube chain defined here coincides with the definition introduced in Ziemiański (2017). Below, we will identify cube chains c and the corresponding sequences (c 1 , . . . , c l ) satisfying the conditions above.
Definition 7.4 The length n cube chain category Ch(K ; n) of K is the slice category Ch n ↓ K . In other words, objects of Ch(K ; n) are cube chains of length n, and morphisms from a to b are commutative diagrams There is a forgetful functor dom : Ch(K ; n) → Set * * that assigns to every cube chain c : ∨n c → K its domain ∨n c . This is equipped with the natural transformation dom → const K , which is induced by the chains themselves. For every n ≥ 0, this transformation induces the map Consider the sequence of maps and Q K n is the natural map from the homotopy colimit to the colimit of the functor N ( ∨n (−) ) 1 0 .
Theorem 7.5 For every K ∈ Set * * , all the maps in the sequence (7.5) are weak homotopy equivalences.
Proof It follows from Ziemiański (2017, Proposition 6.2.(1)) that the space N [0,n] ( ∨n ) 1 0 is contractible for every n ∈ Seq(n). As a consequence, the map A K n is a weak homotopy equivalence by Dugger (2008, Proposition 4.7). The maps Q K n and F K n are weak homotopy equivalences by Propositions 10.4 and 9.7, respectively.
As an immediate consequence, we obtain the main result of this paper.

Natural tame presentations
In this section we study properties of presentations of natural tame paths on an arbitrary bi-pointed -set K . Fix an integer n ≥ 0, which will be the length of all d-paths considered here.
Thus, elements of c∈Ch(K ;n) N [0,n] ( I n c ) 1 0 will be also called natural tame presentations, or nt-presentations for short.

Definition 8.1 A natural tame presentation
The set of equivalence classes of nt-presentations of α is just (F n K ) −1 (α). The example below shows that there may exist non-equivalent nt-presentations of a given natural tame path.
Example 8.2 Let K = 3 ∪ ∂ 3 3 be the union of two standard 3-cubes, glued along their boundaries. Denote the 3-cubes of K by c and c and choose a d-path α ∈ N (∂ I 3 ) 1 0 that is not tame. But α regarded as a d-path in K is tame; it has two different tame presentations: [c; α] and [c ; α], which are not equivalent. The two presentations are not regular.
We have d i

Proposition 8.4 Every natural tame presentation is equivalent to a minimal natural tame presentation.
Proof This follows by induction from Proposition 8.3.

Comparison of the colimit and the space of natural tame d-paths
Fix a bi-pointed -set K and an integer n ≥ 0. In this Section we investigate the sequence of surjective maps c∈Ch(K ;n) The spaces appearing in this sequence are, respectively from left to right, the space of nt-presentations, the space of equivalence classes of nt-presentations and the space of natural tame d-paths. The main goal of this section is to prove that the map F n K is a homotopy equivalence. Recall that Tam K n is the tamification map defined in (6.9).
Hence, there exists r ∈ {0, . . . , m − 1} such that Vert(Tam r (α)) = Vert(Tam r +1 (α)). By Proposition 9.1, Tam r (α) is a regular tame path and hence, by Proposition 8.7, the nt-presentations Tam  there exists a function f K ,m that makes the upper triangle commutative. Since F K n is surjective, also the lower triangle commutes. However, it is not clear whether f K ,m is continuous. We will tackle with this problem in the rest of this section, starting with the following lemma. Proof We need to prove the following statement: for every compact subset K ⊆ X , an open subset U ⊆ Y and f ∈ Lip L (X , Y ) such that f (K ) ⊆ U there exist sequences x 1 , . . . , x n of points of X and U 1 , . . . , U n of open subsets of Y such that Since f (K ) is compact, the distance d between f (K ) and Y \U is strictly positive. Let {x 1 , . . . , x n } be a family of points of K such that the family of open balls having radius d/2L and centered at the x i 's cover K . Let As a consequence, g(x) ∈ U , which ends the proof.

Proof
Since it is sufficient to prove that N [0,n] ( I n ) 1 0 is compact. But N [0,n] ( I n ) 1 0 is a closed subset of Lip 1 (I , I n ), with L 1 -metric on I n , so the statement follows from Proposition 9.4. Proposition 9.6 If K is finite, then the map f K ,m (9.2) is continuous for m ≥ n. For arbitrary K , f K ,m is continuous for m ≥ n + 1.
Proof If K is finite, then so is Ch(K ; n). Thus, colim c∈Ch(K ;n) N [0,n] ( I ∨n c ) 1 0 is compact by Proposition 9.5 and then the left-hand vertical map is a quotient map. As a consequence, f m is continuous for m ≥ n.
As a consequence, for every m ≥ n+1 (9.2) is a commutative diagram of continuous maps. Since the horizontal maps are homotopic to the identities on the respective spaces, we obtain:

Comparison of the homotopy colimit with the colimit
Fix a bi-pointed -set K and an integer n ≥ 0. In this section we show that the map is a weak homotopy equivalence. We will use the following criterion due to Dugger (2008). Let C be an upwards-directed Reedy category (Dugger 2008, Definition 13.6) and let F : C → Top be a diagram. The latching object of c ∈ Ob(C) is where ∂C ↓ c is the slice category over the object c with the identity object id c removed, and dom : C ↓ c → C is the forgetful functor. The latching object is equipped with the latching map c : L c F → F(c) that is induced by the cocone {F(ϕ)} (ϕ:a→c)∈∂C↓c .
Proposition 10.1 (Dugger 2008, Proposition 14.2) Assume that the latching map c is a cofibration for every object c ∈ Ob(C). Then F is projective-cofibrant and so hocolim C F → colim C F is a weak equivalence.
The category Ch(K ; n) admits a grading deg(a) = n − l(a), which makes it an upwards-directed Reedy category: for every non-identity morphism ϕ : a → b in Ch(K ) we have l(a) > l(b).
For every c ∈ Ch(K ; n), the functor Furthermore, n is a homeomorphism onto its image.

Compliance with ethical standards
Conflict of interest The author states that there is no conflict of interest.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.