Stable Components of Directed Spaces

In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces.


Introduction
Directed spaces, or d-spaces, Grandis [9] are topological spaces with a distinguished family of paths, called directed paths. They can be used for modeling the behavior of concurrent programs. Points of the directed space represent possible states of a concurrent program, while directed paths represent possible partial executions. This approach allows to employ topological invariants of d-spaces to examine executions of concurrent programs which they represent. All d-spaces considered in this paper are assumed to have no loops, i.e., all directed loops are constant. d-spaces with no loops are slightly more general than partially ordered spaces (po-spaces). We will write x ≤ y if there exists a d-path from x to y.
Unfortunately, most of the classical (non-directed) homotopy invariants do not have satisfactory directed counterparts. In [9], Grandis introduces the directed fundamental category π 1 (X ) [9] of a d-space X . The objects of π 1 (X ) are points of X and the morphisms from x to y are homotopy classes of d-paths from x to y. This is a directed analogue of the fundamental groupoid of a topological space. Alas, unlike for fundamental groupoids, directed Communicated by J. Rosicky. B Krzysztof Ziemiański ziemians@mimuw.edu.pl 1 Institute of Mathematics, Polish Academy of Sciences, ul.Śniadeckich 8, 00-656 Warsaw, Poland fundamental categories are not naturally equivalent to any finite, or even countable, category, except in the most trivial cases.
To overcome this problem, several authors introduced and studied component categories [4,7,12], which are quotient categories of π 1 (X ) (or some related categories as in [12]). These papers share a common idea, which will be recalled shortly below on the example of [7]. For a po-space X , consider a class E of morphisms of π 1 (X ) that are "equivalences"; namely, this class consists of Yoneda morphisms. A morphism σ ∈ C(x, y) of a category C is a Yoneda morphism if it induces bijections: (P) C(z, x) α → σ • α ∈ C(z, y), whenever C(z, x) = ∅, and (F) C(y, z) α → α • σ ∈ C(x, z), whenever C(y, z) = ∅.
Then the authors prove that there is a unique maximal system ⊆ E with good categorical properties and define the component category of X as a quotient category of the category of fractions π 1 (X )[ −1 ]. Informally, condition (P) can be stated as "σ is a past equivalence between x and y with respect to z, for every z such x that is reachable from z".
This approach works very well in many cases. The main motivation for searching for a new definition of the component system is the following example, due to Dubut [1], for which this approach fails, i.e., the component category obtained is not finite. Consider the d-space that is the geometric realization of the precubical set X where the edges marked by e are identified. The areas A and D are closed. In this case, no non-identity morphism of π 1 (X ) represented by a path in the square A is a Yoneda morphism; for any two points x y ∈ A one can find a point z ∈ D such that there are two morphisms from x to z but only one from y to z. As a consequence, no two points in A are Yoneda equivalent, and the component category of X is uncountable. In this paper we propose a new definition of the component category of a directed space with no loops. These categories, called stable component categories, are finite for all d-spaces which are geometric realizations of finite precubical sets with no loops. The main idea is to relax conditions (P) and (F) defining when points x and y are considered equivalent. Note that, in the example above, for any x y ∈ A we have a bijection π 1 (X )(y, z) π 1 (X )(x, z) if z ∈ D is "large enough", i.e., if z ≥ z(D, x, y) for some z(D, x, y) depending on D, x and y. Thus, we no longer require that x and y are future equivalent with respect to every z ≥ y but that they are equivalent with respect to "large enough" points in every component. This leads to an axiomatic definition of component systems (Definition 3.15); a decomposition of X = A i into disjoint subsets is a stable future component system if every pair (A i , A j ) is future stable, i.e., satisfies, among others, a condition similar to the property of sets A and D formulated above. In a similar way, we define past stable component systems, and total stable component systems, which are both past and future stable. In the example above, X = A∪ B∪ C∪ D is a stable total component system on X ; see Example 7.1 for details. A similar approach is presented in [8].
Stable component systems would not allow us to define invariants of d-spaces if one would not be able to single out a unique stable component system for a given d-space. It is not true in general that every d-space without loops admits a unique coarsest stable component system (Remark 4.11). Nevertheless, we are able to prove that if X admits any finite component system, then there exists the unique coarsest one (Theorem 4.10). The class of d-spaces having a finite (and, therefore, a coarsest) component system includes the geometric realizations of finite cubical complexes with no loops (Theorem 5.4). Eventually, we define the component category associated to a stable component system (Sect. 6). These results allow to define three categories for a sufficiently good d-space X : the component categories of its coarsest stable future, past and total component system.
Apart from three possible flavors (future, past and total), the stable components systems in this paper are parametrized by some class of equivalences of topological spaces. There are many possible choices for when a d-path α from x to y should be regarded a future equivalence with respect to z. One of the possible choices is to claim that α is an equivalence if is a bijection, as in the example above; another possibility is to require that the concatenation map between the spaces of directed paths with the given endpoints is a weak homotopy equivalence. These are the two most natural, and extreme, cases but other choices of the class of equivalences are possible, like all homology equivalences, all maps inducing isomorphisms on homotopy groups up to some dimension, etc. To handle all these cases simultaneously, we introduce a class F of equivalences in the category of topological spaces Top. The most important examples are the class F ∞ of all weak homotopy equivalences, as in (1.3) and the class F 0 of maps inducing a bijection between path-connected components (1.2). The component categories of stable F-component systems will be enriched in some monoidal category S, assuming that there is given a monoidal functor L : Top → S that sends maps belonging to F into isomorphisms in S. Thus, for a d-space X admitting a coarsest component systems we obtain three families of S-enriched categories, depending on the choice of F and L: the future coarsest F-component category S P + F,L (X ), the past one S P − F,L (X ) and the total one S P ± F,L (X ). Three choices of F and L deserve a special attention: • F = F 0 is the class of maps inducing an isomorphism on π 0 , and L = π 0 : Top → Set.
Then the component category S P F,L (X , A) of a component system A is Set-enriched, i.e., it is a category in the usual sense. If A is the coarsest component system, S P μ F 0 ,π 0 (X ; A), μ ∈ {+, −, ±}, can be regarded as directed analogues of π 0 (X ). • F = F ∞ is the class of weak homotopy equivalences and L : Top → hTop is the forgetful functor into the homotopy category. The component category is enriched in the homotopy category and carries information about the homotopy types of the spaces appearing as directed path spaces in X . This can be viewed as "the directed total homotopy group". • F = H R is the class of maps inducing an isomorphism on H * (−; R) for some principal ideal domain R, and L = H * is the homology functor into the category of graded R-modules with the monoidal structure given by the graded tensor product. This choice seems to be best suited for specific calculations.

Directed spaces
For a topological space X , let P X = map([0, 1], X ) denote the space of paths on X with the compact-open topology.
is a pair (X , P X ), where X is a topological space and P X ⊆ P X is a family of paths, called directed paths or d-paths, that satisfies the following conditions: (c) Non-decreasing reparametrizations of d-paths are d-paths, i.e., if α ∈ P X and f : For short, we will write X instead of (X , P X ). The subspace P X ⊆ P X will called the d-structure of X .
For d-spaces X , Y , a continuous map f : X → Y is a d-map if f • α ∈ P Y for every d-path α ∈ P X . Equivalently, one can require that the image of P X under the map P X → P Y induced by f is contained in P Y . A d-map is a d-homeomorphism if it is a bijection and its inverse is a d-map.
Some examples of d-spaces are described below: Fix a d-space X . For subsets A, B, C ⊆ X , denote A pair of d-paths α, β ∈ P X induces the map The monoid Sur + ([0, 1]) of non-decreasing surjective self-maps of [0, 1] acts naturally on P X . The quotient space of this action is the trace space T X of X , and elements of T X will be called traces; as above, T X (A, B) = P X (A, B)/ Sur + ([0, 1]) stands for the space of traces having the required endpoints. As shown in [12], for any points x, y ∈ X the quotient map P X (x, y) → T X (x, y) is a weak homotopy equivalence. As a result, in the considerations below, path spaces can be replaced by trace spaces without any consequences. A where σ x stands for the constant path. The relation is reflexive and transitive for every d-space X . If X is has no loops, it is also antisymmetric and then (X , ≤) is a partial order. For a subset A ⊆ X and x ∈ X , we denote The d-space X with the relation ≤ is not necessarily a partially ordered space in the sense of Nachbin [11] because this partial order is not, in general, closed, regarded as a subset of X × X .

Equivalences
Let Top denote the category of topological spaces and continuous maps.
The morphisms belonging to F will be called F-equivalences. Two spaces X , Y are F-equivalent if they can be connected by a zig-zag of F-equivalences. A space X is Fcontractible if X → { * } is an F-equivalence; clearly any space that is F-equivalent to a contractible space is F-contractible.

Example 2.7
For k ∈ Z ≥0 ∪ {∞}, let F k be the family of maps that induce an isomorphism on π 0 and isomophisms on all homotopy groups π n , for every n ≤ k and for every choice of basepoints. Then F k is an equivalence system. This includes the examples mentioned in the introduction: F 0 is the class of maps inducing isomorphism on π 0 , and F ∞ is the class of weak homotopy equivalences.
Notice that conditions (c) and (d) guarantee that for every equivalence system F.

Example 2.8
For an abelian group A and k ∈ Z ≥0 ∪{∞}, the family H A,k of maps that induce an isomorphism on H n (−; A) for n ≤ k is an equivalence system.
In the remaining part of the paper we will assume that F is a fixed equivalence system.

Stable components
Fix a d-space X with no loops and an equivalence system F.
i.e., every directed path having endpoints in A is contained in A. (b) future connected if, for every x, y ∈ A, there exists z ∈ A such that P A (x, z) = ∅ = P A (y, z). (c) past connected if, for every x, y ∈ A, there exists z ∈ A such that P A (z, x) = ∅ = P A (z, y).

Remark 3.2
A future connected subset may or may not have a maximal point. If such a point exists, it is unique.

Definition 3.3
We say that future (resp. past) connected subsets A, B ⊆ X are cofinal (resp. coinitial) if there exists y ∈ A ∩ B such that A ≥y = B ≥y (resp. A ≤y = B ≤y ).
From this point we will introduce only the future versions of definitions; the past counterparts are the same as the future ones applied to the opposite d-space X op .

Definition 3.4 Let
A ⊆ X be a future connected subset and let α ∈ P X be a d-path. (d) If St + F (α; A) = ∅, then we say that α future F-stabilizes in A, or that A is future F-stable with respect to α.
If A has a maximal point x ∈ A, then α future F-stabilizes in A if and only if the map The prefixes or indices F will be skipped if it is clear which class of equivalences is considered.
Next, we collect elementary properties of stabilizers, which will be used frequently later on.
Proposition 3.5 Fix α ∈ P X and a future connected subset A ⊆ X.
(a) If x ∈ St + (α; A) and y ∈ A ≥x , then y ∈ St + (α; A). In particular, if St + (α; A) is non-empty, then A and St + (α; A) are cofinal. which commutes up to homotopy, in which all solid arrows are F-equivalences. Thus, the 2-out-of-3 property of F implies that the dotted ones are also F-equivalences.
(d) and (e): Since A is future connected, these follow from (a). (f): Assume that z ∈ St + (α, β; A) and ω ∈ P A ≥z . We have Since z stabilizes both α and β, both maps in the right-hand composition are F-equivalences.
So the map which is homotopic to their composition also is. (g): Homotopic paths induce homotopic maps between the path spaces, and two homotopic maps are either both F-equivalences or both non-F-equivalences.
Notice that (g) implies that stability is not a property of a particular path from x to y but rather of its class in π 1 (X )(x, y).
which are all F-equivalences.
As a consequence, we can define the future stable d-path space from A to B as where a ∈ A and z ∈ St + F (σ a , B). By Proposition 3.9, the space S P For a totally stable pair (A, B) and a stabilizing pair (a ∈ A, b ∈ B) we have obvious F-equivalences Next, we formulate some properties of stable pairs. The first proposition is obvious: Proof If x ∈ A, y ∈ B and P(x, y) = ∅, then every path α ∈ P(x, y) extends to a path α ∈ P(x, y ) for some y ∈ St + (σ x ; B) ≥y ; the stabilizer St + (σ x ; B) ≥y is non-empty by 3.5(d). Thus, S P

Definition 3.15 A family
The stable future (resp. past) F-component system A is closed if every A i has a final (resp. initial) point.
We will occasionally skip the adjective "stable", since this is the only kind of component systems considered in this paper.

Remark 3.16
In most cases, the assumption that S P But not always. Let X = R × S 1 and consider the d-structure on X given by the following condition: Thus, X has no loops, the pair (X , X ) is future stable (for any F) and S P

Definition 3.17 A stable total F-component system on X is a family
is both a stable future and a stable past component system, and such that every pair Proof Choose stabilizing pairs (a i j , b i j ) for every pair (A i , A j ). Since all components are future and past connected and there is a finite number of components, there exist points a i ∈ A i and b j ∈ A j such that a i ≤ a i j and b j ≥ b i j for all i, j. The conclusion follows from Proposition 3.11(b).

Proposition 3.19
Let {A i } i∈I be a future F-component system on X . Then the relation on I Proof The reflexivity of ≤ is obvious. If i ≤ j ≤ k ∈ I , then there exists a path α ∈ P(A i , y), for some y ∈ A j and, by 3.13, a path β ∈ P(y, A k ). From the existence of α * β we conclude that i ≤ k which proves that the relation ≤ is transitive. If i = k, then also j = i, since A i = A k is d-convex, which proves the antisymmetry.

The coarsest component systems
Fix a d-space X with no loops and an equivalence system F. We say that a stable future (resp. past, total) F-component system A on X is coarser than B if for every B ∈ B there exists A ∈ A such that B ⊆ A. The relation of being coarser is a partial order on the set of all future (resp. past, total) F-component systems on X . It is not, in general, true that X admits a coarsest future (past, total) F-component system, i.e., the system that is coarser than any other system. However, if X admits any finite future (past, total) F-component system, then a coarsest system on X exists and this will be shown in this section.
Let A = {A i } i∈I and B = {B j } j∈J be finite stable future F-component systems on X . Let ∼ be the equivalence relation on I J spanned by and let K be the set of equivalence classes of the relation ∼. For k ∈ K define The family C = A ∪ B = {C k } k∈K will be called the union of the component systems A and B. It is clear that the sets C k are pairwise disjoint and cover X .
The proof of Theorem 4.1 uses several lemmas; they will be formulated for the component system A only, since similar facts hold for B. For x ∈ X , the index of the component of A (resp. B) that contains x will be denoted by i(x) (resp. j(x)), so that x ∈ A i(x) and x ∈ B j(x) . Recall that the sets I and J are partially ordered, as shown in Proposition 3.19.

Proposition 4.2 For every i ∈ I , there exists a unique element j(i) ∈ J such that
Proof Assume otherwise. Fix x 0 ∈ A i . By the assumption, there exists x 0 < x 1 ∈ A i such that j(x 0 ) < j(x 1 ). By repeating this argument, we construct an infinite sequence of points Since the sets B j r are d-convex, all indices j(x r ) must be different, which contradicts the finiteness of J . This proves the existence of j(i) and y i . If j (i) ∈ J , y i ∈ A i is another pair satisfying the condition (*), then, since A i is future connected, there exists The following conditions are equivalent: Proof Assume that A i and B j(i) are not cofinal. Thus, by Proposition 4.2 there exists a point This contradicts the maximality of i in I ∩ k and proves that (a) implies (c).
Assume that A i and B j(i) are cofinal. Thus, there exists Since A i is future connected, every path ending at some point of A i extends to a path ending at some point of U . Furthermore, by Proposition 3.13, P X (x, Similarly we obtain (C k ) ≤U = {h∈J ∩k | h≤ j(i)} B h , which implies that (C k ) ≤U = C k . As a consequence, The unique maximal element in I ∩ k (resp. J ∩ k) will be denoted i(k, +) (resp. j(k, +)). Note that j(i(k, +)) = j(k, +) and i( j(k, +)) = i(k, +).

Proposition 4.4
For every k ∈ K , C k is future connected and the sets A i(k,+) , B j(k,+) and C k are cofinal.
The cofinality of A i(k,+) , B j(k,+) and C k is an immediate consequence of Proposition 4.3.

Proposition 4.5
For every k ∈ K and every j ∈ J , the pair (B j , A i(k,+) ) is future stable. Furthermore, if l ∈ K then the spaces , the first statement follows from 3.7. The second statement follows from 3.12.
Proof of Theorem 4.1 For every k ∈ K , C k is future connected (4.4), d-convex (4.6) and every pair (C k , C l ), l ∈ K , is future F-stable (4.8). Proposition 3.12 implies that

Proposition 4.9
If A and B are stable total F-component systems on X , then C = A ∪ B is also a stable total F-component system.
Proof Theorem 4.1 implies that C is both a future and a past F-component system. For every k, l ∈ K , there are points x ∈ A i(k,−) and y ∈ A i(l,+) such that y ∈ St + F (x; A i(l,+) ) and −) ; y). The index i(k, −) denotes the unique minimal element of I ∩ k. Since A i(l,+) and C l are cofinal and A i(k,−) and C k are coinitial, there exist x ∈ (A i(k,−) ) ≤x and y ∈ (A i(l,+) ) ≥y such that (A i(k,−) ) ≤x = (C k ) ≤x and (A i(l,+) ) ≥y = (C l ) ≥y . Now x ∈ St − F (C k ; y ) and y ∈ St + F (x ; C l ); therefore (C k , C l ) is totally stable.

Theorem 4.10
If X admits a finite future (resp., past, total) F-component system, then there exists a coarsest F-component system on X . If X admits a finite future (resp. past) closed F-component system, then the coarsest future (resp. past) F-component system is closed.
Proof The existence of a finite future F-component system implies the existence of a minimal system A, i.e., such that no future F-component system is coarser than A. If A and B are minimal future F-components systems, then, by Theorem 4.1, the sum A ∪ B is a future component system that is coarser than both A and B. Thus, A ∪ B = A = B is a unique minimal system. The same argument applies for past component systems and, by Proposition 4.9, for total systems. If A is the coarsest future F-component system on X , and B is a closed one, then A = A∪B is closed.
admits total component systems but A ∪ B = {X } is neither a future nor a past component system, since (X , X ) is neither future nor past stable. In particular, X admits no coarsest component system.
The coarsest stable future (resp. past, total) F-component system on X will be denoted by

Component systems on pre-cubical sets
In this section, we prove that every finite pre-cubical set K with no loops admits a finite total (resp. past, total) F-component system and hence, by Theorem 4.10, a coarsest one. Moreover, we show that the coarsest finite future (resp. past) F-component system on |K | is closed. Since every F ∞ -component system is an F-component system for every equivalence system F, we assume F = F ∞ .
Here we recall basic definitions; for a survey on pre-cubical sets and their applications in concurrency, see eg. [5,6].
The elements of K [n] will be called n-cubes. The geometric realization |K | of a pre-cubical set K is the quotient d-space where ∼ is generated by and the face map d ε i converts the ith occurrence of * into ε. The geometric realization of n is d-homeomorphic to the directed n-cube I n .
We say that a pre-cubical set has no loops if the following conditions are satisfied, which are equivalent: • |K | has no loops, • there is no non-empty sequence of 1-cubes e 1 , . . . , e n such that d 1 (e n ) = d 0 (e 1 ) and Every point x ∈ |K | has a unique presentation x = [c; (t 1 , . . . , t n )] such that c ∈ K [n] and 0 < t i < 1 for all i. For every n-cube c ∈ K [n], define the set
The main result is the following theorem: Theorem 5.4 Let K be a pre-cubical set with no loops. For every equivalence system F, The proof of Theorem 5.4 uses the following: is a homotopy equivalence.
Proof First we will consider the case when γ i (1) = r for all i. Denote q = min 1≤i≤n γ i (0  (1), z). We will show that F and G are homotopy inverses. We have • ω and then, by the self-deformation h K s , to the identity on P(α(1), z). Next, defines the homotopy between F • G and ω → σ α(0) * ω, and the latter map is homotopic to the identity on P(α(0), z).

Proof of 5.4 (a):
It is easy to check that the sets U c are pair-wise disjoint, future and past connected, future and past F-trivial and that they cover |K |. It remains to prove that every pair (U a , U b ) is future, past and totally F-stable for all a ∈ K [n], b ∈ K [m].
(b) and (c): Again, the only non-trivial part to check is the stability of all pairs Since α lies in a single cube, i.e., α = [c; γ ] for c ∈ K [n] and γ ∈ P I n , then this follows from Proposition 5.5.
Combining Theorems 4.10 and 5.4 we obtain Corollary 5.7 Every finite pre-cubical set K with no loops admits, for every equivalence system F: The component systems given in Theorem 5.4 may or may not be the coarsest; some examples are presented in the last section.

Stable component category
In this section we construct the component category associated to a stable (future/past/total) F-component system. The objects of the component category are the components of the component system, and the morphisms carry information about the stable path space between particular components. Component categories are enriched in some category S, i.e., their "morphism sets" are objects of S. In most typical cases, S will be the homotopy category of topological spaces, the category of graded R-modules for a principal ideal domain R or the category of sets (in the last case, we obtain categories in the usual sense).
We refer the reader to the book by Kelly [10] for the basic definitions concerning enriched categories.
Let us fix an equivalence system F, a monoidal category S and a monoidal functor L : Top → S that sends F-equivalences to isomorphisms in S. Equivalently, we require that there is a factorization Fix a d-space X with no loops and a stable future F-component system A := {A i } i∈I on X . As noted before, the indexing set I is partially ordered by i ≤ j if and only if In the case when F = F ∞ is the family of weak homotopy equivalences and all components A i have the maximal elements a i ∈ A i we can define the future component category as the Top-category with I as the set of objects and T X (a i , a j ) as the morphism object from i to j. This is a full subcategory of the trace category of T (X ) defined by Raussen [12,Sect. 3]. However, we do not want to make such an assumption. In most cases, one can restrict to closed future component systems but we want to obtain a definition of a component category that works also for total component systems. Apart from the most trivial cases, d-spaces do not admit total component systems having components with final points.
Note that if {a i } i∈I is a future system of representatives of A, then S P + F (A i , A j ) and P(a i , a j ) are F-equivalent for all i, j ∈ I . Indeed, for i < j this follows immediately from the definition, for i = j both S P

Proposition 6.3 Assume that the component system A is finite. For any collection of points
The points a i can be chosen inductively. If j ∈ I is minimal, let a j ∈ (A j ) ≥b j be an arbitrary point. If j is not minimal and all points a i ∈ A i for i < j are chosen according to Definition 6.1, then let a j be an arbitrary point of the set which is non-empty by Proposition 3.5.

Remark 6.4
There exist infinite future component systems that do not admit any system of representatives. For example, consider the d-space For any choice of points a i = (a x i , a y i ) ∈ A i we have lim i→−∞ a y i = +∞ and, therefore, the space P(a i , a j ) is empty for every j ∈ Z if i < j is small enough. On the other hand, S P Definition 6.5 Let a = {a i } i∈I be a future system of representatives of A. The future component category of A with representatives a is the S-category T a = T a A,L with • I as the set of objects, • T a (i, j) = L ( P(a i , a j )) as the morphism objects, P(a i , a i )) as the identities, • For i < j < k ∈ I , the composition is given by L ( P(a i , a j )) ⊗ L ( P(a j , a k )) → L ( P(a i , a j ) × P(a j , a k )) ( P(a i , a k )), where c i jk stands for the concatenation map. The left-hand morphism is the structure morphism of the monoidal functor L. This definition make sense; the only non-trivial thing to check is that the composition is associative and this follows immediately from properties of monoidal functors and the fact that the concatenation of d-paths is associative up to homotopy.
Equivalently, we can define T a A as L * ( T (X )| a ), where T (X )| a is the full subcategory of the trace category with the objects {a i } i∈I , and L * : Top-Cat → S-Cat is the functor changing the enriching category induced by L.
Our next goal is to prove that the S-category T a does not depend on the choice of the system of representatives a of A. Proposition 6.6 Let a = {a i } i∈I and b = {b i } i∈I be collections of points in X and let {ω i ∈ P(a i , b i )} i∈I be a collection of d-paths. Assume that both maps are F-equivalences for every i, j ∈ I . Then the formula defines an equivalence of the S-categories L * ( T (X )| a ) and L * ( T (X )| b ).
Proof Clearly, for every i, j ∈ I , the morphism F i, j is an isomorphism in S and hence we can define G i, j = F −1 i, j . It remains to prove that the collections of morphisms {F i, j } and {G i, j } define functors F and G, respectively; if so, F and G are mutual inverses. This will be shown only for F; the argument for G is similar. It is clear that identities are preserved by F. To show that F preserves compositions of morphisms, we need to check that the diagram below is commutative, for all i ≤ j ≤ k ∈ I : diagram commutes since L is a monoidal functor. To prove that the right-hand part commutes, consider the diagram of spaces: in which, again, every map is given either by concatenation or by pre-or post-composition with one of the paths ω i , ω j , ω k . All maps marked are F-equivalences for the reasons mentioned before. It is elementary to check that all triangles and squares in this diagram commute up to homotopy and then, after composing with L, they commute strictly. Thus, which shows that also the right-hand part of (6.3) commutes.

Proposition 6.7 The category T a does not depend, up to isomorphism, on the choice of a future representative system a of A.
Proof Given two future representative systems a = {a i } and a = {a i } of A, one can find a representative system b = {b i } and paths ω i ∈ P(a i , b i ) and ω i ∈ P(a i , b i ) such that Proposition 6.6 applies both for a, b and {ω i } and for a , b and {ω i }. This can be done inductively in a similar way as in the proof of 6.3: if j ∈ I is minimal, we choose any b j ∈ St + F (a j , a j ; A j ) and any ω j ∈ P(a j , b j ), ω j ∈ P(a j , b j ). If j is not minimal and b i , ω i , ω i are chosen for all i < j according to Proposition 6.6, let b j be an element of which is non-empty by 3.5. Now, Proposition 6.6 provides natural equivalences of Scategories Propositions 6.3 and 6.7 allow to formulate the following. Given a finite total F-component system A = {A i } i∈I , one can define the future (resp. past) component category S P + F,L (X , A) (resp. S P − F,L (X , A)). A natural requirement is that these S-categories should be equivalent.
By Proposition 3.18, there exist for all i, j ∈ I . Fix two representative systems of A: the future one b = {b i } i∈I and the past one a = {a i } i∈I such that a i ≤ c i and d i ≤ b i for all i ∈ I . The existence of such systems is guaranteed by Proposition 6.3. Choose d-paths ω i ∈ P(a i , b i ).

Proposition 6.9
For all i , j ∈ I , both maps P(ω i , b j ) and P(a i , ω j ) are F-equivalences.
We have a commutative diagram induced by the paths η j and ω i . All maps marked are F-equivalences: • The left vertical map, since b j ∈ St + F (b i ; A j ). • The right vertical map, from Proposition 3.11, since (c i , d j ) is a stabilizing pair.
The dual argument shows that P(a i , ω j ) is also an F-equivalence.

Proposition 6.10
Assume that A is a finite total F-component system on X . Then the categories S P + F,L (X , A) and S P − F,L (X , A) are isomorphic. Proof This is an immediate consequence of Propositions 6.6 and 6.9.
If X admits a coarsest future (resp. past, total) stable F-component system, we define the stable future (resp. past, total) F-component category of X as S P For X 1 and X 2 these coincide with the component categories defined in [7].

The boundary of the directed cube
The boundary of the directed n-cube, ∂ I n , is the geometric realization of the pre-cubical set ∂ n , which is obtained from n by removing the sole n-cube ( * , . . . , * ) (see Example 5.2). As a consequence of Theorem 5.4, U := {U c } c∈ k ∂ n [k] is a total stable component system on X (for any class of equivalences F). Our goal is to find the coarsest stable F k -component systems for all k. To achieve this, we need to calculate the homotopy types of the d-path spaces between any two particular points of ∂ I n . is a contractible space. A similar argument works for a n = b n = 1. If (a n , b n ) = (0, * ), then we will show that the maps s, t ∈ [0, 1], defines a homotopy between id P(∂ I n ) y x and G F but we need to check that H s (α)(t) ∈ ∂ I n for all s, t, α. This is obvious for t ≤ s(1 + s) −1 . Since α(t) ∈ ∂ I n for every t, there exists c α ∈ [0, 1) such that α n (t) = 0 for t ∈ [0, c α ] and α (t) ∈ ∂ I n−1 for t ∈ [c α , 1]. In particular, y ∈ ∂ I n−1 , which implies that H s (α)(t) ∈ ∂ I n for t ∈ [(1 + s) − Thus, the homotopy H s is well-defined. As a consequence, P(∂ I n ) y x is contractible also in this case, as well as for (a n , b n ) = ( * , 1).
Finally, assume that (a i , b i ) ∈ {(0, 1), ( * , * )} for all i, and denote The last equivalence is proven in [14, 2.6.1]. Now we are ready to calculate the coarsest future F k -component systems on ∂ I n . Since ∂ I n is d-homeomorphic to its opposite d-space, similar facts hold for past component systems.

Proposition 7.4
For n ≥ 3 and k ≥ n − 2, U = {U c } c∈ k≥0 ∂ n [k] (see (5.4)) is the coarsest total stable component system on ∂ I n . Proof By Theorem 5.4, U is a total stable component system on ∂ I n . Therefore, every element of M := M ± F k (∂ I n ) is a union of sets having the form U c . Let E (a 1 ,...,a n ) denote the component of M that contains U (a 1 ,...,a n ) . Assume that E (a 1 ,...,a n ) = E (b 1 ,...,b n ) for some a i , b i ∈ {0, 1, * }; we will prove that (a 1 , . . . , a n ) = (b 1 , . . . , b n ).
which follows from Proposition 7.2; r is the number of indices i such that (a i , b i ) = (0, 1). Notice that all compositions of morphisms are trivial, since no two morphisms objects of type L(S r −2 ) are composable. (b) If k < n − 2 and n ≥ 3, then M ± F k ,L (∂ I n ) = {∂ I n }, since the spaces P ∂ I n (x, (1, . . . , 1)) and P ∂ I n ((0, . . . , 0), x) are F k -contractible for all x ∈ ∂ I n . There exist pairs x, y ∈ ∂ I n such that P ∂ I n (x, y) is not F k -contractible but these non-trivial d-path spaces are "unstable": they trivialize when x (resp. y) is replaced by some smaller (resp. larger) point. (c) If n = 2, then ∂ I 2 is d-homeomorphic to the space X 1 from Example 7.1, and we have 1) , U ( * ,0) ∪ (1, 0) ∪ U (1, * ) } for every equivalence system F.

Directed suspension
Example 7.6 Let X be a d-space with no loops. The directed suspension of X is the d-space with the quotient d-structure, i.e., the minimal one such that the projection X × I → (X ) is a d-map. Denote the contracted points by 0 and 1 respectively. It is not very difficult to check that for x, y ∈ X , s, t ∈ (0, 1) ((x, s), (y, t)) P X (x, y) for s ≤ t ∅ for s > t, (7.7) and that P (X ) ((x, s), 1) P (X ) (0, (y, t)) { * }, (7.8) since the space of d-paths P (X ) ((x, s), 1) can be contracted into the subspace of d-paths with the constant first coordinate, which is contractible. Moreover, P (X ) (0, 1) X ; this statement is not quite obvious and will be proven below. Thus, if X is not F-contractible, regarded as a topological space: (a) M + F ( (X )) = {{0}, (X ) \ {0}} and S P + F,L ( (X )) has two objects, with L(X ) being the morphism object between them. (b) If X admits a coarsest total F-component system, then It remains to check that P (X ) (0, 1) is homotopy equivalent to X . This can be deduced from the calculations presented in [15, Sects. 8 and 9] but we will give a more elementary proof.
Proof We will show that the map S : P( (X )) 1 0 α = [α X , α I ] → α X (q(α I )) ∈ X is a homotopy inverse of J . Notice that S(α) does not depend on the choice of α X and α I . Clearly S • J is the identity on X . We will construct a homotopy between J • S and the identity on P( (X )) 1 0 . The idea is to paste to a path α at q(α I ), where q is the map from Proposition 7.7, a longer and longer "vertical" segment.

Future work
The stable component categories constitute a family of invariants of directed spaces with no loops. Still, they suffer some limitations; the author hopes that the following questions can be answered.
• Functoriality. Let X and Y be d-spaces with no loops that admit a coarsest (future/past/total) F-component system. Assume given a d-map f : X → Y . Is it possible to define an induced functor S P μ F,L ( f ) : S P μ F,L (X ) → S P μ F,L (Y ) in a functorial way? • Loops. The construction above essentially uses the fact that the d-spaces under consideration have no loops. However, in many cases, for example for pre-cubical complexes, loops can be avoided by passing to universal, or loop-length coverings. These coverings do not, in general, admit finite stable component systems but author hopes that it is possible to prove they admit the coarsest component systems, which can be used to define component categories also in this case. • Connections with [2]. The stable component categories S P μ H R ,H * (X ) seem to be related to natural homology, introduced by Dubut, Goubault and Goubault-Larrecq. It would be interesting to investigate connections between these two invariants.
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.