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 the 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, [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 their 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 generalize slightly partially ordered spaces (po-spaces). We will write x ≤ y if there exists a d-path from x to y.
Unfortunately, most of 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, in contrary to fundamental groupoids, directed fundamental categories are not naturally equivalent to any finite, or even countable, category, except the most trivial cases.
To overcome this problem, several authors introduced and studied component categories [4,7,11], which are quotient categories of π 1 (X) (or some related categories as in [11]). 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 Σ ⊆ E with good categorical properties and define the component category of X as the quotient category π 1 (X)[Σ −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) saying when points x and y are considered equivalent. Note that, in the example above, for any x y ∈ A we have the 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, the condition similar to the property of sets A and D formulated above. In a similar way, we define the past stable component systems, and the total stable component systems, which are both past and future, with some additional condition. In the example above, X = A∪ B∪ C∪ D is a stable total component system on X. 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 pick the unique stable component system for a given d-space. This is not true in general that every d-space without loops admits the 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, the coarsest) component system includes geometric realizations of finite cubical complexes with no loops (Theorem 5.2). Eventually, we define the component category associated to a stable component system (Section 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), 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 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 two most natural, and extreme, cases but other choices of the class of equivalences are possible, like the homology equivalences, the maps inducing isomorphisms on the 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 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 the 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., is a category in the usual sense. If A is the coarsest component system, S P µ F0,π0 (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 the most suited for specific calculations.

Preliminaries
Directed spaces.
Definition 2.1. A directed space, or a d-space [9] is a pair (X, P X ), where X is a topological space and P X ⊆ P X = map([0, 1], X) is a family of paths, called directed paths or d-paths, that satisfies the following conditions: (a) Every constant path is a d-path. (b) The concatenation of two d-paths is a d-path. Namely, if α, β ∈ P X and α(1) = β(0), then α * β ∈ P X , where For short, we will write X instead of (X, P X ).
Here follow some examples of d-spaces.
Example 2.2. The directed n-cube I n = ([0, 1] n , P I n , where Example 2.4. Every subspace A ⊆ X has the inherited structure of a d-space given by P A := P X ∩P A . 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 [11], 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 d-space X has no loops if P X (x, x) = {σ x } for every x ∈ X, where σ x stands for the constant path. The relation x ≤ y ⇔ P X (x, y) = ∅ is reflexive and transitive for every d-space X t. 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 notion of a d-space without loops is very close to the notion of partially ordered space but we do not assume that the partial order is closed 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 be a zig-zag of F-equivalences, and a space X is F-contractible if X → { * } is an F-equivalence; clearly any space that is F-equivalent to a contractible space is F-contractible.
Example 2.6. 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 Example 2.7. 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 wholly lies 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.
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.
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 final 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.
Here we gather elementary properties of stabilizers, which will be frequently used later.
Proof. (a) and (b) are obvious.
(c): Fix β ∈ P A ≥x . We have the diagram which commutes up to homotopy, in which all solid arrows are F-equivalences. Thus, the 2-out-of-3 property of F implies that such are 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).
We do not assume that all d-paths in Proof. Fix y such that B ≥y = B ′ ≥y and assume that (A, B) is a stable pair. Then for every α ∈ P A there exists x ∈ St + (α; B) ≥y (by Proposition 3.5.(d)). By 3.5.
Stable d-path spaces. Let A, B ⊆ X be future connected subsets and assume that the pair (A, B) is future F-stable.
are F-equivalences, since both z and z ′ stabilize σ a in B. Proposition 3.9. Let a, a ′ ∈ A and let z, z ′ ∈ B be stabilizers of σ a and σ a ′ , respectively. Then the spaces P (a, z) and P (a ′ , z ′ ) are F-equivalent.
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 + F (A, B) is well-defined up to (and only up to) F-equivalence.
. Such a pair (a, b) will be called a stabilizing pair of the totally stable pair (A, B).
For a totally stable pair (A, B) and a stabilizing pair (a ∈ A, b ∈ B) we have obvious F-equivalences Here we formulate some properties of stable pairs. The first proposition is obvious: Proposition 3.11. Let (A, B) be a totally F-stable pair and let (a ∈ A, b ∈ B) be a stabilizing pair.
is a stabilizing pair of (A, B). Thus, Then the following conditions are equivalent: 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. Stable component systems.
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 + F (A, A) is F-contractible is obsolete. But not always. Let X = R × S 1 and consider the d-structure on X given by the following condition: the path Then X has no loops, the pair (X, X) is stable (for any F) and S P + F (X, X) is equivalent to π 1 (S 1 ) ≃ Z, which is not contractible.
Definition 3.17. A stable total F-component system on X is a family A = {A i } i∈I that is both stable future and stable past component system, and every pair ( Proof. Choose stabilizing pairs (a ij , b ij ) 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 ij and b j ≥ b ij for all i, j. The conclusion follows from Proposition 3.11.(b).
is a partial order.
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.
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 the 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 the 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 the 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.
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 x 0 < x 1 < x 2 < . . . of A i such that j(x r+1 ) = j(x r ). Since the sets B jr are d-convex, all indices j(x r ) must be different, which leads to the contradiction with the finiteness of J. This proves the existence of j(i) and y i . Proof. Assume that A i and B j(i) are not cofinal. Thus, by Proposition 4.2 there exists a point y i ∈ A i ∩ B j(i) such that (A i ) ≥yi ⊆ B j and a point z ∈ (B j(i) ) ≥yi \ A i . Obviously i < i(z) and i(z) ∈ k, since y i ∈ B j(i) ∩ A i = ∅ and z ∈ B j(i) ∩ A i(z) = ∅. 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 y i ∈ A i ∩ B j(i) such that U := (A i ) ≥yi = (B j(i,+) ) ≥yi . 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, A i ) = ∅ if and only if i(x) ≤ i. Therefore, 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, I ∩ k = {g ∈ I ∩ k | g ≤ i} and then i is the unique maximal element of I ∩ k. Thus, (c) implies (b). Implication (b) ⇒ (c) is obvious.
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 spaces Proof. Since A i(k,+) is cofinal with B j(i(k,+)) = B j(k,+) , the first statement follows from 3.7. The second statement follows from 3.12.
Proposition 4.6. For every k ∈ K, C k is d-convex.
Proposition 4.7. For k, l ∈ K and x, y ∈ C k there exists a path α ∈ P X (x, (C k ) ≥y ) that future F-stabilizes in C l .
Proof. Induction with respect to i(x) ∈ I ∩ k. If i(x) = i(k, +), then there exists y ′ ∈ (A i(k,+) ) ≥x,y , since C k is future connected and cofinal with A i(k,+) . Any d-path α ∈ P X (x, y ′ ) is contained in A i(k,+) . Thus, α future F-stabilizes in A i(l,+) and, by cofinality, in C l . Now assume that i(x) is not maximal in I ∩ k and the proposition holds for all i ∈ (I ∩ k) >i(x) . Choose y i(x) ∈ A i(x) ∩ B j(i(x)) such that (A i(x) ) ≥y i(x) ⊆ B j(i(x)) (4.2). By Proposition 4.3, A i(x) and B j(i(x)) are not cofinal; thus, there exists an element u ∈ (B j(i(x)) ) ≥y i(x) that does not belong to A i(x) . Choose v ∈ (A i(x) ) ≥y i(x) ,x ; clearly v ∈ B j(i(x)) and we can choose x ′ ∈ (B j(i(x)) ) ≥u,v . Notice that Eventually, pick β ∈ P A i(x) (x, v) and γ ∈ P B j(i(x)) (v, x ′ ). From the inductive hypothesis, there exists a path α ′ ∈ P X (x ′ , (C k ) ≥y ) which future F-stabilizes in C l . Since β future F-stabilizes in A i(l,+) , γ future F-stabilizes in B j(l,+) and all A i(l,+) , B j(l,+) and C l are cofinal, both β and γ future F-stabilize in C l . Proposition 3.5.(f) implies that β * γ * α ′ future F-stabilizes in C l . Proposition 4.8. For every k, l ∈ K, the pair (C k , C l ) is stable.
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 If A is closed, then the final point of A i(k,+) is a final in C k . Hence, C is closed.
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 future and 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 x ∈ St − F (A i(k,−) ; 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 the 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. , n + 1 2 )} n∈Z but A ∪ B = {X} is neither future not 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 M + F (X) (resp. M − F (X), M ± F (X)).

Components systems on pre-cubical sets
In this section, we prove that every finite pre-cubical set K admits a finite total (resp. past, total) F-component system and hence, by Theorem 4.10, the 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 a quotient d-space where ∼ is generated by denote the initial and the final vertex of c, respectively. 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 d 1 (e i ) = d 0 (e i+1 ) for 1 ≤ i < n.
Every point x ∈ |K| has the 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 point v = [v; ()] is a final point of V + v and an initial point of V − v . The main result is the following theorem: Theorem 5.2. Let K be a pre-cubical set with no loops. For every equivalence system F, is a past F-component system on |K|. The proof of Theorem 5.2 uses the following: Proposition 5.3. Let K be a pre-cubical set. Let γ = (γ 1 , . . . , γ n ) ∈ P I n a path such that γ i (0) > 0 for all 1 ≤ i ≤ n and, for some 0 ≤ r ≤ 1, γ i (1) ≤ r for all i.  (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(1), z).

Proof of 5.2. (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 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]. Fix a path α ∈ P Uc ; there is a presentation α = [a; γ] for γ = (γ i ) n i=1 ∈ P I n . Clearly 0 < γ i (0) ≤ γ i (1) < 1 for all i, and then r = max n i=1 γ i (1) < 1. Let y = [b; (r, . . . , r)]; for every path β ∈ P (U b ) ≥y , Proposition 5.4 implies that P |K| (α, β) is an F-equivalence. Thus, y ∈ St + F (α; U b ), which implies that the pair (U a , U b ) is future F-stable. A similar argument shows that this pair is past stable.
(b) and (c): Again, the only non-trivial part is to check is the stability of all pairs . Since w is the final point of V + w , it remains to check that P (α, w) is an F-equivalence. If α lies in a single cube, i.e., α = [c; γ] for c ∈ K[n] and γ ∈ P I n , then Proposition 5.3 implies that P (α, w) is a homotopy equivalence. In the general case, α is a finite concatenation of such paths; thus, P (α, w) is a finite composition of homotopy equivalences.
Combining Theorems 4.10 and 5.2 we obtain Corollary 5.5. Every finite pre-cubical set K with no loops admits, for every equivalence system F: (a) The coarsest total stable F-component system M ± F (K). (b) The coarsest future (resp. past) stable F-component system M + F (K) (resp. M − F (K)), which is closed.

Stable component category
In this section we construct the component category associated to a stable (future/past/total) Fcomponent 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 of 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 (6.1) 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 P X (A i , A j ) = ∅.
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 [11,Section 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. Definition 6.1. A future system of representatives of A is a collection of points {a i ∈ A i } i∈I such that a j ∈ St + F (a i , A j ) for every i < j ∈ I.
Remark 6.2. Obviously, a j ∈ St + F (a i , A j ) if i ≤ j, since P (A i , A j ) = ∅ in this case. However, we do not assume that a i ∈ St + (a i , A i ).
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 + F (A i , A i ) and P (a i , a i ) ∼ = { * } are F-contractible, and for i ≤ j both S P + F (A i , A i ) and P (a i , a i ) are empty. Proposition 6.3. Assume that the component system A is finite. For any collection of points {b i ∈ A i } i∈I , there exists a future system of representatives {a i } i∈I of A such that a i ≥ b i for all i ∈ I.
Proof. The points a i can be chosen inductively. If j ∈ I is minimal, let a j ∈ (A j ) ≥bj be an arbitrary point. If j is not minimal and all points a i ∈ A i for i < j are chosen, then let a j be an arbitrary point of a set i∈I<j St + (a i ; A j ) ≥bj , which is non-empty by Proposition 3.5.
Remark 6.4. There exist infinite future component systems that does 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 j , a i ) is empty for every i ∈ Z if j < i is small enough. On the other hand, S P + (A j , A i ) ≃ { * } is non-empty. 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, • L({ * } → {σ ai } = 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 )) where c ijk 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 points of 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 Proof. Clearly, for every i, j ∈ I, the morphism F i,j is an isomorphism in S and then we can define G i,j = F −1 i,j . It remains to prove that collections of morphisms {F i,j } and {G i,j } define the 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: All vertical morphisms are induced by the concatenation with paths ω i or are the inverses of such morphisms. They are all isomorphisms, since the products of F-equivalences are F-equivalences and L sends F-equivalences into isomorphisms. The left-hand part of this 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 the concatenation or by the pre-or post-composition with the path ω i . 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 the a, b and {ω i } and for the a ′ , b and {ω ′ i }. It 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, let b j be an element of (6.5) which is non-empty by 3.5. Now, Proposition 6.6 gives natural equivalences of S-categories Propositions 6.3 and 6.7 allow to formulate the following.
Definition 6.8. The future F-component category of a finite stable future F-component system is the S-category S P + F,L (X, A) := T a for any future system of representatives a of A. Similarly, for a past F-component system B we define its past F-component category as S P − F,L (X, B) = T b for a past system of representatives b. Given a total F-component system A = {A i } i∈I , one can define the future component category S P + F,L (X, A) and the past one S P − F,L (X, A). A natural requirement is that these S-categories should be equivalent.
By Proposition 3.18, there exist c i , d i ∈ A i , i ∈ I such that d j ∈ St + F (c i ; A j ) and c i ∈ St − F (A i ; d j ) 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 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 bottom map, since b ′ j ∈ St + F (ω i ; A j ). Thus, P (ω i , b j ) is an F-equivalence. 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 immediate consequence of Propositions 6.6 and 6.9.

Examples and final remarks
Examples. Here we provide several examples of calculations of stable component categories. In the space X 3 , the edges marked v are identified. All points in the boundaries between the areas A, B, C, D belong either to A or to D. All spaces of d-paths between particular points are homotopically discrete; therefore, the coarsest F-component system does not depend on the choice of the system of equivalences F. The coarsest systems of all these spaces are similar: • The coarsest future system is {A, B ∪ C ∪ D}, • The coarsest past system is {A ∪ B ∪ C, D}, • The coarsest total system is {A, B, C, D}.
• Functoriality. Let X and Y be d-spaces with no loops that admit the coarsest (future/past/total) F-component system. Assume that there is given a d-map f : X → Y . Is it possible to define the 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 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 the 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 HR,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.