Theory and Applications of Categories, Vol. 8, No. 12, 2001, pp. 324–376.

COMBINATORICS OF BRANCHINGS IN HIGHER DIMENSIONAL AUTOMATA

PHILIPPE GAUCHER

ABSTRACT. We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular ω-category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the sub-complex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is ω-categories freely generated by precubical sets. As application, we calculate the branching homology of some ωcategories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.

1. Introduction

After [22, 14], one knows that it is possible to model higher dimensional automata (HDA) using precubical sets (Definition 2.1). In such a model, a n-cube corresponds to a ntransition, that is the concurrent execution of n 1-transitions. This theoretical idea would be implemented later. Indeed a CaML program translating programs in Concurrent Pascal into a text file coding a precubical set is presented in [10]. At this step, one does not yet consider cubical sets with or without connections since the degenerate elements have no meaning at all from the point of view of computer-scientific modeling (even if in the beginning of [12], the notion of cubical sets is directly introduced by intellectual reflex).

In [14], the following fundamental observation is made : given a precubical set (K_{n})_{n⩾0} together with its two families of face maps (∂_{i}^{α}) for α ∈ {−, +}, then both chain complexes (ℤK_{∗}, ∂^{α}), where ℤX means the free abelian group generated by X and where ∂^{α} = ∑_{i}(−1)^{i+1}∂_{i}^{α}, give rise to two homology theories H_{∗}^{α} for α ∈ {−, +} whose nontrivial elements model the branching areas of execution paths for α = − and the merging areas of execution paths for α = + in strictly positive dimension. Moreover the group H_{0}^{−} (resp. H_{0}^{+}) is the free abelian group generated by the final states (resp. the initial states) of the HDA.

Consider for instance the 1-dimensional HDA of Figure 1. Then u − w gives rise to a non-trivial homology class which corresponds to the branching which is depicted.

Then the first problem is that the category of precubical sets is not appropriate to

Received by the editors 2000 January 29 and, in revised form, 2001 June 27.

Transmitted by Ronald Brown. Published on 2001 July 3.

2000 Mathematics Subject Classification: 55U.

Key words and phrases: cubical set, thin element, globular higher dimensional category, branching,

higher dimensional automata, concurrency, homology theory.

\begin{matrix}c \\ ◯ \end{matrix} Philippe Gaucher, 2001. Permission to copy for private use granted.

u w v  x

Figure 1: A 1-dimensional branching area

u w

Figure 2: A 1-dimensional branching area

identify the HDA of Figure 1 with that of Figure 2 because there is no morphism between them preserving the initial state and both final states.

No matter : it suffices indeed to work with the category of precubical sets endowed with the +_{i} cubical composition laws satisfying the axioms of Definition 2.4 and with the morphisms obviously defined. Now for any n ⩾ 1, there are n cubical composition laws +_{1}, …, +_{n} representing the concatenation of n-cubes in the n possible directions. Let X = u +_{1} v and Y := w +_{1} x. Then there is a unique morphism f in this new category of HDA from the HDA of Figure 2 to the HDA of Figure 1 such that f : u ↦ X and f : w ↦ Y. However f is not invertible in the category of precubical sets equipped with cubical composition laws because there still does not exist any morphism from the HDA of Figure 1 to the HDA of Figure 2.

To make f invertible (recall that we would like to find a category where both HDA would be isomorphic), it still remains the possibility of formally adding inverses by the process of localization of a category with respect to a collection of morphisms. However a serious problem shows up : the non-trivial cycles u − w and X − Y of Figure 1 give rise to two distinct homology classes although these two distinct homology classes correspond to the same branching area. Indeed there is no chain in dimension 2 (i.e. K_{2} = ∅), so no way to make the required identification !

This means that something must be added in dimension 2, but without creating additional homology classes. Now consider Figure 3. The element A must be understood as a thin 2-cube such that, with our convention of orientation, ∂_{1}^{−}A = u, ∂_{2}^{−}A = u, ∂_{1}^{+}A = ϵ_{1}∂_{1}^{−}v, ∂_{2}^{+}A = ∂_{2}^{−}B = ϵ_{1}∂_{1}^{−}v. And the element B must be understood as another thin 2-cube such that ∂_{2}^{−}B = ϵ_{1}∂_{1}^{−}v, ∂_{2}^{+}B = ϵ_{1}∂_{1}^{+}v and ∂_{1}^{−}B = ∂_{1}^{+}B = v. In such a situation, ∂^{−}(A +_{2} B) = u +_{1} v − u therefore u +_{1} v and u become equal in the first homology group H_{1}^{−}. By adding this kind of thin 2-cubes to the chain complex (ℤK_{∗}, ∂^{−}), one can then identify the two cycles u − w and X − Y. One sees that there are two kinds of thin cubes which are necessary to treat the branching case. The first kind is well-known in cubical set theory : this is for example B = ϵ_{1}v or ∂_{1}^{+}A = ϵ_{1}∂_{1}^{−}v. The second kind is for example A which will be denoted by Γ_{1}^{−}u and which corresponds to extra-degeneracy maps as defined in [6].

To take into account the symmetric problem of merging areas of execution paths, a third family Γ_{i}^{+} of degeneracy maps will be necessary. In this paper, we will only treat the case of branchings. The case of mergings is similar and easy to deduce from the branching case. The solution presented in this paper to overcome the above problems is then as follows :

• One considers the free globular ω-category F (K) generated by the precubical set K : it is obtained by associating to any n-cube x of K a copy of the free globular ω-category I^{n} generated by the faces of the n-cube (paragraph 3.1) ; the faces of this n-cube are denoted by (x; k_{1} … k_{n}) ; one takes the direct sum of all these cubes and one takes the quotient by the relations

(∂_{i}^{α}y; k_{1} … k_{n}) ∼ (y; k_{1} … k_{i−1}αk_{i} … k_{n})

for any y ∈ K_{n+1}, α ∈ {−, +} and 1 ⩽ i ⩽ n + 1.

• Then we take its cubical singular nerve N^{□}(F (K)) (which is equal also to the free cubical ω-category generated by K) ; the required thin elements above described (the three families ϵ_{i}, Γ_{i}^{−} and Γ_{i}^{+}) do appear in it as components of the algebraic structure of the cubical nerve (Definition 2.4 and Definition 3.3).

• The branching homology of F (K) (Definition 3.5) is the solution for both following reasons :

1. Let x and y be two n-cubes of the cubical nerve which are in the branching complex. If x +_{j} y exists for some j with 1 ⩽ j ⩽ n, then x and x +_{j} y are equal modulo elements in the chain complex generated by the thin elements (Theorem 9.2) ;

2. The chain complex generated by the thin elements is conjecturally acyclic in this situation, and so it does not create non-trivial homology classes (Conjecture 3.6).

We have explained above the situation in dimension 1. The 2-dimensional case is depicted in Figure 8. Additional explanations are available at the end of Section 9.

The branching homology (or negative corner homology) and the merging homology (or positive corner homology) were already introduced in [12]. This invariance with respect to the cubifications of the underlying HDA was already suspected for other reasons. The branching and merging homology theories are the solution to overcome the drawback of Goubault’s constructions.

There are three key concepts in this paper which are not so common in the general literature and which we would like to draw to the reader’s attention.

+_{2}^{=}  ^{=} u  v u  v v v X u  v  u u A  B  A  B

Figure 3: Identifying u +_{1} v and u

1. the extra structure of connections Γ^{±} on cubical sets, which allow extra degenerate elements in which adjacent faces coincide. This structure was first introduced in [6].

2. the notion of folding operator. This was introduced in the groupoid context in [6], to fold down a cube to an element in a crossed complex, and in the category context in [1] to fold down a cube to an element in a globular category. Properties of this folding operator are further developed in [2]. This we call the ‘usual folding operator’.

3. the notion of thin cube, namely a multiple composition of cubes of the form ϵ_{i}y or Γ^{±}z. A crucial result is that these are exactly the elements which fold down to 1 in the contained globular category.

So there are many ways of choosing a cycle in the branching complex for a given homology class, i.e. a given branching area, according to the choice of the cubification of the considered HDA. This possibility of choice reveals an intricate combinatorics. The most appropriate tool constructed in the mathematical theory of cubical sets to study this combinatorics is not relevant here. The machinery of folding operators [6, 1] does not work indeed for the study of the branching homology because the usual folding operators are not internal to the branching chain complex (see Section 6.2). The core of this paper is the proposal of a new folding operator adapted for the study of the branching complex (Section 6.5). This operator enables us to deduce several results on the reduced branching homology, the latter being obtained by taking the quotient of the former by the sub-complex generated by its thin elements. This sub-complex is conjecturally acyclic for a wide variety of ω-categories, including that freely generated by a precubical set or a globular set (Conjecture 3.6). Our main result is that the negative folding operator induces the identity map on the reduced branching complex (Corollary 8.4). Using some relations between the branching homology of some particular ω-categories and the usual simplicial homology of some associated ω-categories (Theorem 5.5), the behaviour of the composition maps (the globular and the cubical ones) modulo thin elements is completely studied (Section 9). All these results lead us to a question about the description of the reduced branching complex using globular operations by generators and relations (Proposition 9.4 and Question 9.6) and to two invariance results for the reduced branching homology (Proposition 11.1 and Theorem 11.2).

This paper is organized as follows. Section 2 recalls some important notations and conventions for the sequel. In Section 3, the branching homology and the reduced branching homology are introduced. In Section 4, the matrix notations for connections and degeneracies are described. Next in Section 5, the branching homology of some particular ω-categories (the ω-categories of length at most 1) is completely calculated in terms of the usual simplicial homology. In Section 6, the negative folding operators are introduced. In Section 7, the negative folding operators are decomposed in terms of elementary moves. In Section 8, we prove that each elementary move appearing in the decomposition of the folding operators induces the identity map on the reduced branching complex. Therefore the folding operators induce the identity map as well. In Section 9, the behaviour of the cubical and globular composition laws in the reduced branching complex is completely studied. In the following Section 10, some facts about the differential map in the reduced branching complex are exposed. In the last Section 11, some invariance results for the reduced branching homology are exposed and the reduced branching homology is calculated for some simple globular ω-categories.

2. Preliminaries : cubical set, globular and cubical category

Here is a recall of some basic definitions, in order to make precise some notations and some conventions for the sequel.

2.1. Definition. [6] [16] A cubical set consists of a family of sets (K_{n})_{n⩾0}, of a family of

face maps \xymatrix{K_{n} \ar[r]^{∂_{i}^{α}} & K_{n−1}} for α ∈ \{−, +\} and of a family of degeneracy maps \xymatrix{K_{n−1} \ar[r]^{ϵ_{i}} & K_{n}} with 1 ⩽ i ⩽ n which satisfy the following relations

1. ∂_{i}^{α}∂_{j}^{β} = ∂_{j−1}^{β}∂_{i}^{α} for all i < j ⩽ n and α, β ∈ {−, +} (called sometimes the cube axiom)

2. ϵ_{i}ϵ_{j} = ϵ_{j+1}ϵ_{i} for all i ⩽ j ⩽ n

3. ∂_{i}^{α}ϵ_{j} = ϵ_{j−1}∂_{i}^{α} for i < j ⩽ n and α ∈ {−, +}

4. ∂_{i}^{α}ϵ_{j} = ϵ_{j}∂_{i−1}^{α} for i > j ⩽ n and α ∈ {−, +}

5. ∂_{i}^{α}ϵ_{i} = Id

A family (K_{n})_{n⩾0} only equipped with a family of face maps ∂_{i}^{α} satisfying the same axiom as above is called a precubical set. An element of K_{0} will be sometimes called a state, or a 0-cube and an element of K_{n} a n-cube, or a n-dimensional cube.

2.2. Definition. Let (K_{n})_{n⩾0} and (L_{n})_{n⩾0} be two cubical (resp. precubical) sets. Then a morphism f from (K_{n})_{n⩾0} to (L_{n})_{n⩾0} is a family f = (f_{n})_{n⩾0} of set maps f_{n} : K_{n} → L_{n} such that f_{n}∂_{i}^{α} = ∂_{i}^{α}f_{n} and f_{n}ϵ_{i} = ϵ_{i}f_{n} (resp. f_{n}∂_{i}^{α} = ∂_{i}^{α}f_{n}) for any i. The corresponding category of cubical sets is isomorphic to the category of pre-sheaves Sets^{□^{op}} over a small category □. The corresponding category of precubical sets is isomorphic to the category of pre-sheaves Sets^{□^{pre}^{op}} over a small category □^{pre}.

2.3. Definition. [5] [26] [24] A (globular) ω-category is a set A endowed with two families of maps (d_{n}^{−} = s_{n})_{n⩾0} and (d_{n}^{+} = t_{n})_{n⩾0} from A to A and with a family of partially defined 2-ary operations (∗_{n})_{n⩾0} where for any n ⩾ 0, ∗_{n} is a map from {(a, b) ∈ A × A, t_{n}(a) = s_{n}(b)} to A ((a, b) being carried over a ∗_{n} b) which satisfy the following axioms for all α and β in {−, +} :

1. d_{m}^{β}d_{n}^{α}x = \begin{cases}d_{m}^{β}x if m < n \\ d_{n}^{α}x if m ⩾ n \end{cases}

2. s_{n}x ∗_{n} x = x ∗_{n} t_{n}x = x

3. if x ∗_{n} y is well-defined, then s_{n}(x ∗_{n} y) = s_{n}x, t_{n}(x ∗_{n} y) = t_{n}y and for m ≠ n, d_{m}^{α}(x ∗_{n} y) = d_{m}^{α}x ∗_{n} d_{m}^{α}y

4. as soon as the two members of the following equality exist, then (x ∗_{n} y) ∗_{n} z = x ∗_{n} (y ∗_{n} z)

5. if m ≠ n and if the two members of the equality make sense, then (x∗_{n}y)∗_{m}(z∗_{n}w) = (x ∗_{m} z) ∗_{n} (y ∗_{m} w)

6. for any x in A, there exists a natural number n such that s_{n}x = t_{n}x = x (the smallest of these numbers is called the dimension of x and is denoted by dim(x)).

A globular set is a set A endowed with two families of maps (s_{n})_{n⩾0} and (t_{n})_{n⩾0} satisfying the same axioms as above [27, 21, 3]. We call s_{n}(x) the n-source of x and t_{n}(x) the n-target of x.

Notation. The category of all ω-categories (with the obvious morphisms) is denoted by ωCat. The corresponding morphisms are called ω-functors. The set of n-dimensional morphisms of C is denoted by C_{n}. The set of morphisms of C of dimension lower or equal than n is denoted by tr_{n}C. The element of C_{0} will be sometimes called states. An initial state (resp. final state) of C is a 0-morphism α such that α = s_{0}x (resp. α = t_{0}x) implies x = α.

2.4. Definition. [6, 1] A cubical ω-category consists of a cubical set

((K_{n})_{n⩾0}, ∂_{i}^{α}, ϵ_{i})

together with two additional families of degeneracy maps called connections

Γ_{i}^{α} : \xymatrix{K_{n} \ar[r] & K_{n+1}}

with α ∈ {−, +}, n ⩾ 1 and 1 ⩽ i ⩽ n and a family of associative operations +_{j} defined on {(x, y) ∈ K_{n} × K_{n}, ∂_{j}^{+}x = ∂_{j}^{−}y} for 1 ⩽ j ⩽ n such that

1. ∂_{i}^{α}Γ_{j}^{β} = Γ_{j−1}^{β}∂_{i}^{α} for all i < j and all α, β ∈ {−, +}

Theory and Applications of Categories, Vol. 8, No. 12 330

2. ∂_{i}^{α}Γ_{j}^{β} = Γ_{j}^{β}∂_{i−1}^{α} for all i > j + 1 and all α, β ∈ {−, +}

3. ∂_{j}^{±}Γ_{j}^{±} = ∂_{j+1}^{±}Γ_{j}^{±} = Id

4. ∂_{j}^{±}Γ_{j}^{∓} = ∂_{j+1}^{±}Γ_{j}^{∓} = ϵ_{j}∂_{j}^{±}

5. Γ_{i}^{±}Γ_{j}^{±} = Γ_{j+1}^{±}Γ_{i}^{±} if i ⩽ j

6. Γ_{i}^{±}Γ_{j}^{∓} = Γ_{j+1}^{∓}Γ_{i}^{±} if i < j

7. Γ_{i}^{±}Γ_{j}^{∓} = Γ_{j}^{∓}Γ_{i−1}^{±} if i > j + 1

8. Γ_{i}^{±}ϵ_{j} = ϵ_{j+1}Γ_{i}^{±} if i < j

9. Γ_{i}^{±}ϵ_{j} = ϵ_{i}ϵ_{i} if i = j

10. Γ_{i}^{±}ϵ_{j} = ϵ_{j}Γ_{i−1}^{±} if i > j

11. (x +_{j} y) +_{j} z = x +_{j} (y +_{j} z)

12. ∂_{j}^{−}(x +_{j} y) = ∂_{j}^{−}(x)

13. ∂_{j}^{+}(x +_{j} y) = ∂_{j}^{+}(y)

14. ∂_{i}^{α}(x +_{j} y) = \begin{cases}∂_{i}^{α}(x) +_{j−1} ∂_{i}^{α}(y) if i < j \\ ∂_{i}^{α}(x) +_{j} ∂_{i}^{α}(y) if i > j \end{cases}

15. (x +_{i} y) +_{j} (z +_{i} t) = (x +_{j} z) +_{i} (y +_{j} t).

16. ϵ_{i}(x +_{j} y) = \begin{cases}ϵ_{i}(x) +_{j+1} ϵ_{i}(y) if i ⩽ j \\ ϵ_{i}(x) +_{j} ϵ_{i}(y) if i > j \end{cases}

17. Γ_{i}^{±}(x +_{j} y) = \begin{cases}Γ_{i}^{±}(x) +_{j+1} Γ_{i}^{±}(y) if i < j \\ Γ_{i}^{±}(x) +_{j} Γ_{i}^{±}(y) if i > j \end{cases}

18. If i = j, Γ_{i}^{−}(x +_{j} y) = [ \begin{matrix}ϵ_{j+1}(y) & Γ_{j}^{−}(y) \\ Γ_{j}^{−}(x) & ϵ_{j}(y) \end{matrix} ] [figure: j+1; j]

19. If i = j, Γ_{i}^{+}(x +_{j} y) = [ \begin{matrix}ϵ_{j}(x) & Γ_{j}^{+}(y) \\ Γ_{j}^{+}(x) & ϵ_{j+1}(x) \end{matrix} ] [figure: j+1; j]

20. Γ_{j}^{+}x +_{j+1} Γ_{j}^{−}x = ϵ_{j}x and Γ_{j}^{+}x +_{j} Γ_{j}^{−}x = ϵ_{j+1}x

21. ϵ_{i}∂_{i}^{−}x +_{i} x = x +_{i} ϵ_{i}∂_{i}^{+}x = x The corresponding category with the obvious morphisms is denoted by ∞Cat.

Without further precisions, the word ω-category is always supposed to be taken in the sense of globular ω-category. In [2], it is proved that the category of cubical ω-categories and the category of globular ω-categories are equivalent.

Notation. If S is a set, the free abelian group generated by S is denoted by ℤS. By definition, an element of ℤS is a formal linear combination of elements of S.

2.5. Definition. [12] Let C be an ω-category. Let C_{n} be the set of n-dimensional morphisms of C. Two n-morphisms x and y are homotopic if there exists z ∈ ℤC_{n+1} such that s_{n}z − t_{n}z = x − y. This property is denoted by x ∼ y.

We have already observed in [12] that the corner homologies do not induce functors from ωCat to the category of abelian groups. A notion of non-contracting ω-functors was required.

2.6. Definition. [12] Let f be an ω-functor from C to D. The morphism f is noncontracting if for any 1-dimensional x ∈ C, the morphism f(x) is a 1-dimensional morphism of D.

The theoretical developments of this paper and future works in progress entail the following definitions too.

2.7. Definition. Let C be an ω-category. Then C is non-contracting if and only if for any x ∈ C of strictly positive dimension, s_{1}x and t_{1}x are 1-dimensional (they could be a priori 0-dimensional as well).

A justification of this definition among a lot of them is that if C is an ω-category which is not non-contracting, then there exists a morphism u of C such that dim(u) > 1 and such that for instance s_{1}u is 0-dimensional. For example consider the two-element set {A, α} with the rules s_{1}A = t_{1}A = s_{0}A = t_{0}A = α and s_{2}A = t_{2}A = A. This defines an ω-category which is not non-contracting. Then A is 2-dimensional though s_{1}A and t_{1}A are 0-dimensional. And in this situation □_{2}^{−}(A) defined in Section 6.5 is not an element of the branching nerve, and therefore for that C, the morphism CF_{2}^{−}(C) (see Proposition 9.4) to CR_{2}^{−}(C) is not defined.

Notation. The category of non-contracting ω-categories with the non-contracting ωfunctors is denoted by ωCat_{1}.

If f is a non-contracting ω-functor from C to D, then for any morphism x ∈ C of dimension greater than 1, f(x) is of dimension greater than one as well. This is due to the equality f(s_{1}x) = s_{1}f(x).

All globular ω-categories that will appear in this work will be non-contracting.

3. Reduced branching homology

3.1. The globular ω-category I^{n}. We need first to describe precisely the ω-category associated to the n-cube. Set \underline{n} = \{1, ..., n\} and let \underline{cub}^{n} be the set of maps from \underline{n} to {−, 0, +} (or in other terms the set of words of length n in the alphabet {−, 0, +}). We say that an element x of \underline{cub}^{n} is of dimension p if x^{−1}(0) is a set of p elements. The set \underline{cub}^{n} is supposed to be graded by the dimension of its elements. The set \underline{cub}^{0} is the set of maps from the empty set to \{−, 0, +\} and therefore it is a singleton. Let y ∈ \underline{cub}^{i}. Let r_{y} be the map from (\underline{cub}^{n})_{i} to (\underline{cub}^{n})_{dim(y)} defined as follows (with x ∈ (\underline{cub}^{n})_{i}) : for k ∈ \underline{n}, x(k) ≠ 0 implies r_{y}(x)(k) = x(k) and if x(k) is the l-th zero of the sequence x(1), ..., x(n), then r_{y}(x)(k) = y(ℓ). If for any ℓ between 1 and i, y(ℓ) ≠ 0 implies y(ℓ) = (−)^{ℓ}, then we set b_{y}(x) := r_{y}(x). If for any ℓ between 1 and i, y(ℓ) ≠ 0 implies y(ℓ) = (−)^{ℓ+1}, then we set e_{y}(x) := r_{y}(x). We have

If x is an element of \underline{cub}^{n}, let us denote by R(x) the subset of \underline{cub}^{n} consisting of y ∈ \underline{cub}^{n} such that y can be obtained from x by replacing some occurrences of 0 in x by − or +. For example, −00 + +− ∈ R(−000 + −) but +000 + − ∉ R(−000 + −). If X is a subset of \underline{cub}^{n}, then let R(X) = ⋃_{x∈X} R(x). Notice that R(X ∪ Y) = R(X) ∪ R(Y).

3.2. Theorem. There is one and only one ω-category I^{n} such that

1. the underlying set of I^{n} is included in the set of subsets of \underline{cub}^{n}

2. the underlying set of I^{n} contains all subsets like R(x) where x runs over \underline{cub}^{n}

3. all elements of I^{n} are compositions of R(x) where x runs over \underline{cub}^{n}

4. for x p-dimensional with p ⩾ 1, one has

s_{p−1}(R(x)) = R ({b_{y}(x), dim(y) = p − 1})
t_{p−1}(R(x)) = R ({e_{y}(x), dim(y) = p − 1})

5. if X and Y are two elements of I^{n} such that t_{p}(X) = s_{p}(Y) for some p, then X ∪ Y ∈ I^{n} and X ∪ Y = X ∗_{p} Y.

Moreover, all elements X of I^{n} satisfy the equality X = R(X).

The elements of I^{n} correspond to the loop-free well-formed sub pasting schemes of the pasting scheme \underline{cub}^{n} [15] [9] or to the molecules of an ω-complex in the sense of [25]. The condition “X ∗_{n} Y exists if and only if X ∩ Y = t_{n}X = s_{n}Y ” of [25] is not necessary here because the situation of [25] Figure 2 cannot appear in a composable pasting scheme.

The map which sends every ω-category C to N^{□}(C)_{∗} = ωCat(I^{∗}, C) induces a functor from ωCat to the category of cubical sets. If x is an element of ωCat(I^{n}, C), ϵ_{i}(x) is the ω-functor from I^{n+1} to C defined by ϵ_{i}(x)(k_{1}...k_{n+1}) = x(k_{1}...k̂_{i}...k_{n+1}) for all i between 1 and n + 1 and ∂_{i}^{α}(x) is the ω-functor from I^{n−1} to C defined by ∂_{i}^{α}(x)(k_{1}...k_{n−1}) = x(k_{1}...k_{i−1}αk_{i}...k_{n−1}) for all i between 1 and n.

The arrow ∂_{i}^{α} for a given i such that 1 ⩽ i ⩽ n induces a natural transformation from ωCat(I^{n}, −) to ωCat(I^{n−1}, −) and therefore, by Yoneda, corresponds to an ω-functor δ_{i}^{α} from I^{n−1} to I^{n}. This functor is defined on the faces of I^{n−1} by δ_{i}^{α}(k_{1}...k_{n−1}) = R(k_{1}...[α]_{i}...k_{n−1}). The notation [...]_{i} means that the term inside the brackets is at the i-th place.

3.3. Definition. The cubical set (ωCat(I^{∗}, C), ∂_{i}^{α}, ϵ_{i}) is called the cubical singular nerve of the ω-category C.

3.4. Remark. For α ∈ {−, +}, and x ∈ ωCat(I^{n}, C), let

∂^{α}x := ∑_{i=1}^{n}(−1)^{i+1}∂_{i}^{α}x

Because of the cube axiom, one has ∂^{α} ∘ ∂^{α} = 0.

3.5. Definition. [12] Let C be a non-contracting ω-category. The set of ω-functors x ∈ ωCat(I^{n}, C) such that for any 1-morphism u with s_{0}u = −_{n+1}, x(u) is 1-dimensional (a priori x(u) could be 0-dimensional as well) is denoted by ωCat(I^{n}, C)^{−}. Then

∂^{−}(ℤωCat(I^{∗+1}, C)^{−}) ⊂ ℤωCat(I^{∗}, C)^{−}

by construction. We set

H_{∗}^{−}(C) = H_{∗}(ℤωCat(I^{∗}, C)^{−}, ∂^{−})

and we call this homology theory the branching homology of C. The cycles are called the branchings of C. The map H_{∗}^{−} induces a functor from ωCat_{1} to Ab.

The definition of ωCat(I^{n}, C)^{−} is a little bit different from that of [12]. Both definitions coincide if C is the free ω-category generated by a precubical set or a globular set. This new definition ensures that the elementary moves introduced in Section 7 are well-defined on the branching nerve. Otherwise it is easy to find counterexample, even in the case of a non-contracting ω-category.

3.6. Conjecture. (About the thin elements of the branching complex) Let C be a globular ω-category which is either the free globular ω-category generated by a precubical set or the free globular ω-category generated by a globular set. Let x_{i} be elements of ωCat(I^{n}, C)^{−} and let λ_{i} be natural numbers, where i runs over some set I. Suppose that for any i, x_{i}(0_{n}) is of dimension strictly lower than n (one calls it a thin element). Then ∑_{i} λ_{i}x_{i} is a boundary if and only if it is a cycle.

The thin elements conjecture is not true in general. Here is a counterexample. Consider an ω-category C constructed by considering I^{2} and by dividing by the relations R(−0) = R(0−) and R(−0) ∗_{0} R(0+) = R(0−) ∗_{0} R(+0). Then the ω-functor F ∈ ωCat(I^{2}, C)^{−} induced by the identity functor from I^{2} to itself is a thin cycle in the branching homology. One can verify that this cycle would be a boundary if and only if R(0+) was homotopic to R(+0) in C. This observation suggests the following questions.

3.7. Definition. Let C be an ω-category. Then the n-th composition law is said to be left regular up to homotopy if and only if for any morphisms x, y and z such that x ∗_{n} y = x ∗_{n} z, then y ∼ z.

3.8. Question. Does the thin elements conjecture hold for an ω-category C such that all composition laws ∗_{n} for any n ⩾ 0 are left regular up to homotopy ?

3.9. Question. How may we characterize the ω-categories for which the thin elements conjecture holds ?

3.10. Definition. Let M_{n}^{−}(C) ⊂ ℤωCat(I^{n}, C)^{−} be the sub-ℤ-module generated by the thin elements (M for “mince” which means “thin” in French). Set

CR_{n}^{−}(C) = ℤωCat(I^{n}, C)^{−}/(M_{n}^{−}(C) + ∂^{−}M_{n+1}^{−}(C))

where M_{n}^{−}(C) + ∂^{−}M_{n+1}^{−}(C) is the sub-ℤ-module of ℤωCat(I^{n}, C)^{−} generated by M_{n}^{−}(C) and the image of M_{n+1}^{−}(C) by ∂^{−}. The differential map ∂^{−} induces a differential map

CR_{n+1}^{−}(C) ⟶ CR_{n}^{−}(C)

This chain complex is called the reduced branching complex of C. The homology associated to this chain complex is denoted by HR_{∗}^{−}(C) and is called the reduced branching homology of C. 3.11. Proposition. Conjecture 3.6 is equivalent to the following statement : if C is the free ω-category generated by a precubical set or by a globular set, then the canonical map from the branching chain complex to the reduced branching chain complex of C is a quasi-isomorphism. Proof. By the following short exact sequence of chain complexes

\xymatrix{0 \ar[r] & M_{∗}^{−}(C) + ∂^{−}M_{∗+1}^{−}(C) \ar[r] & ℤωCat(I^{∗}, C)^{−} \ar[r] & CR_{∗}^{−}(C) \ar[r] & 0}

the assumption H_{n}^{−}(C) ≅ HR_{n}^{−}(C) for all n is equivalent to the acyclicity of the chain complex (M_{∗}^{−} + ∂^{−}M_{∗+1}^{−}, ∂^{−}) (notice that M_{0}^{−}(C) = M_{1}^{−}(C) = 0).

Now if Conjecture 3.6 holds, then take an element x ∈ M_{n}^{−}(C) + ∂^{−}M_{n+1}^{−}(C) which is a cycle. Then x = t_{1} + ∂^{−}t_{2} where t_{1} ∈ M_{n}^{−}(C) and t_{2} ∈ M_{n+1}^{−}(C). Then t_{1} is a cycle in ℤωCat(I^{n}, C)^{−} and a linear combination of thin elements. Therefore t_{1} is a cycle in ℤωCat(I^{n}, tr_{n−1}C)^{−}. By Conjecture 3.6, t_{1} = ∂^{−}t_{3} where t_{3} ∈ ℤωCat(I^{n+1}, tr_{n−1}C)^{−}. Therefore t_{1} ∈ ∂^{−}M_{n+1}^{−}(C). Conversely, suppose that the sub-complex generated by the thin elements is acyclic. Take a cycle t of ℤωCat(I^{n}, C)^{−} which is a linear combination of thin elements. Then t is a cycle of M_{n}^{−}(C) + ∂^{−}M_{n+1}^{−}(C), therefore there exists t_{1} ∈ M_{n+1}^{−}(C) and t_{2} ∈ M_{n+2}^{−}(C) such that t = ∂^{−}(t_{1} + ∂^{−}t_{2}) = ∂^{−}t_{1}. 3.12. Definition. Let x and y be two elements of ℤωCat(I^{n}, C)^{−}. Then x and y are T-equivalent (T for thin) if the corresponding elements in the reduced branching complex are equal, that means if x−y ∈ M_{n}^{−}(C)+∂^{−}M_{n+1}^{−}(C). This defines an equivalence relation on ℤωCat(I^{n}, C)^{−} indeed.

4. Matrix notation for higher dimensional composition in the cubical singular nerve

There exists on the cubical nerve ωCat(I^{∗}, C) of an ω-category C a structure of cubical ω-categories [12] by setting

Γ_{i}^{−}(x)(k_{1} … k_{n}) = x(k_{1} … max(k_{i}, k_{i+1}) … k_{n})
Γ_{i}^{+}(x)(k_{1} … k_{n}) = x(k_{1} … min(k_{i}, k_{i+1}) … k_{n})

with the order − < 0 < + and with the proposition-definition :

4.1. Proposition. [12] Let C be a globular ω-category. For any strictly positive natural number n and any j between 1 and n, there exists one and only one natural map +_{j} from the set of pairs (x, y) of N^{□}(C)_{n} × N^{□}(C)_{n} such that ∂_{j}^{+}(x) = ∂_{j}^{−}(x) to the set N^{□}(C)_{n} which satisfies the following properties :

∂_{j}^{−}(x +_{j} y) = ∂_{j}^{−}(x) ∂_{j}^{+}(x +_{j} y) = ∂_{j}^{+}(x)

∂_{i}^{α}(x +_{j} y) = \begin{cases}∂_{i}^{α}(x) +_{j−1} ∂_{i}^{α}(y) if i < j \\ ∂_{i}^{α}(x) +_{j} ∂_{i}^{α}(y) if i > j \end{cases}

Moreover, these operations induce a structure of cubical ω-category on N^{□}(C). The sum (x +_{i} y) +_{j} (z +_{i} t) = (x +_{j} z) +_{i} (y +_{j} t) if there exists will be denoted by

[ \begin{matrix}x & z \\ y & t \end{matrix} ] [figure: j; i]

and using this notation, one can write

• If i = j, Γ_{i}^{−}(x +_{j} y) = [ \begin{matrix}ϵ_{j+1}(y) & Γ_{j}^{−}(y) \\ Γ_{j}^{−}(x) & ϵ_{j}(y) \end{matrix} ] [figure: j+1; j]

• If i = j, Γ_{i}^{+}(x +_{j} y) = [ \begin{matrix}ϵ_{j}(x) & Γ_{j}^{+}(y) \\ Γ_{j}^{+}(x) & ϵ_{j+1}(x) \end{matrix} ] [figure: j+1; j]

The matrix notation can be generalized to any composition like

(a_{11} +_{i} … +_{i} a_{1n}) +_{j} … +_{j} (a_{m1} +_{i} … +_{i} a_{mn})

whenever the sources and targets of the a_{ij} match up in an obvious sense (this is not necessarily true). In that case, the above expression is equal by the interchange law to

(a_{11} +_{j} … +_{j} a_{m1}) +_{i} … +_{i} (a_{1n} +_{j} … +_{j} a_{mn})

and we can denote the common value by

[ \begin{matrix}a_{m1} & … & a_{mn} \\ ⋮ & & ⋮ \\ a_{11} & … & a_{1n} \end{matrix} ] [figure: i; j]

In such a matrix, an element like ϵ_{i}x is denoted by . An element like ϵ_{j}x is denoted by . In a situation where i = j + 1, an element like Γ_{j}^{−}(x) is denoted by and an element like Γ_{j}^{+}(x) is denoted by . An element like ϵ_{j}ϵ_{j}x = ϵ_{j+1}ϵ_{j}x is denoted by . With i = j + 1, we can verify some of the above formulae :

Γ_{j}^{−}(x +_{j} y) = [ ] = [ \begin{matrix}ϵ_{j+1}(y) & Γ_{j}^{−}(y) \\ Γ_{j}^{−}(x) & ϵ_{j}(y) \end{matrix} ]

Γ_{j}^{+}(x +_{j} y) = [ ] = [ \begin{matrix}ϵ_{j}(x) & Γ_{j}^{+}(y) \\ Γ_{j}^{+}(x) & ϵ_{j+1}(x) \end{matrix} ]

4.2. Definition. [6][1] A n-shell in the cubical singular nerve is a family of 2(n + 1) elements x_{i}^{±} of ωCat(I^{n}, C) such that ∂_{i}^{α}x_{j}^{β} = ∂_{j−1}^{β}x_{i}^{α} for 1 ⩽ i < j ⩽ n + 1 and α, β ∈ {−, +}.

4.3. Definition. The n-shell (x_{i}^{±}) is fillable if

1. the sets {x_{i}^{(−)^{i}}, 1 ⩽ i ⩽ n + 1} and {x_{i}^{(−)^{i+1}}, 1 ⩽ i ⩽ n + 1} have each one exactly one non-thin element and if the other ones are thin.

2. if x_{i_{0}}^{(−)^{i_{0}}} and x_{i_{1}}^{(−)^{i_{1}+1}} are these two non-thin elements then there exists u ∈ C such that s_{n}(u) = x_{i_{0}}^{(−)^{i_{0}}}(0_{n}) and t_{n}(u) = x_{i_{1}}^{(−)^{i_{1}+1}}(0_{n}).

The following proposition is an analogue of [1] Proposition 2.7.3.

4.4. Proposition. [12] Let (x_{i}^{±}) be a fillable n-shell with u as above. Then there exists one and only one element x of ωCat(I^{n+1}, C) such that x(0_{n+1}) = u, and for 1 ⩽ i ⩽ n+1, and α ∈ {−, +} such that ∂_{i}^{α}x = x_{i}^{α}.

Proposition 4.4 has a very important consequence concerning the use of the above notations. In dimension 2, an expression A like (for example)

[  x  y  ] [figure: 2; 1]

is necessarily equal to

[ ^{x y} ] [figure: 2; 1]

because the labels of the interior are the same (A(00) = (x+_{2}y)(00)) and because the shells of 1-faces are equal (∂_{1}^{−}A = ∂_{1}^{−}x, ∂_{1}^{+}A = ∂_{1}^{+}x +_{1} ∂_{1}^{+}y, ∂_{2}^{−}A = ∂_{2}^{−}x, ∂_{2}^{+}A = ∂_{1}^{−}y +_{1} ∂_{2}^{+}y) : the dark lines represent degenerate elements which are like mirrors reflecting rays of light. This is a fundamental phenomenon to understand some of the calculations of this work. Notice that A ≠ x +_{2} y because ∂_{1}^{−}A ≠ ∂_{1}^{−}(x +_{2} y).

All calculations involving these matrix notations are justified because the Dawson-Paré condition holds in 2-categories due to the existence of connections (see [11] and [7]). The Dawson-Paré condition stands as follows : suppose that a square α has a decomposition of one edge a as a = a_{1} +_{1} a_{2}. Then α has a compatible composition α = α_{1} +_{i} α_{2}, i.e. such that α_{j} has edge a_{j} for j = 1, 2. This condition can be understood as a coherence condition which ensures that all “compatible” tilings represent the same object.

Let us mention that these special 2-dimensional notations for connections and degeneracies first appeared in [8] and in [23].

5. Relation between the simplicial nerve and the branching nerve

5.1. Proposition. [12] Let C be an ω-category and α ∈ {−, +}. We set

N_{n}^{−}(C) = ωCat(I^{n+1}, C)^{−}

and for all n ⩾ 0 and all 0 ⩽ i ⩽ n,

\xymatrix{∂_{i} : N_{n}^{−}(C) \ar[r] & N_{n−1}^{−}(C)}

is the arrow ∂_{i+1}^{−}, and

\xymatrix{ϵ_{i} : N_{n}^{−}(C) \ar[r] & N_{n+1}^{−}(C)}

is the arrow Γ_{i+1}^{−}. We obtain in this way a simplicial set

(N_{∗}^{−}(C), ∂_{i}, ϵ_{i})

called the branching simplicial nerve of C. The non normalized complex associated to it gives exactly the branching homology of C (in degree greater than or equal to 1). The map N^{−} induces a functor from ωCat_{1} to the category Sets^{Δ^{op}} of simplicial sets.

The globular ω-category Δ^{n}. Now let us recall the construction of the ω-category called by Street the n-th oriental [26]. We use actually the construction appearing in [17]. Let O^{n} be the set of strictly increasing sequences of elements of {0, 1, …, n}. A sequence of length p + 1 will be of dimension p. If σ = {σ_{0} < … < σ_{p}} is a p-cell of O^{n}, then we set ∂_{j}σ = {σ_{0} < … < σ̂_{j} < … < σ_{k}}. If σ is an element of O^{n}, let R(σ) be the subset of O^{n} consisting of elements τ obtained from σ by removing some elements of the sequence σ and let R(Σ) = ⋃_{σ∈Σ} R(σ). Notice that R(Σ ∪ T) = R(Σ) ∪ R(T).

5.2. Theorem. There is one and only one ω-category Δ^{n} such that

1. the underlying set of Δ^{n} is included in the set of subsets of O^{n}

2. the underlying set of Δ^{n} contains all subsets like R(σ) where σ runs over O^{n}

3. all elements of Δ^{n} are compositions of R(σ) where σ runs over O^{n}

4. for σ p-dimensional with p ⩾ 1, one has

s_{p−1}(R(σ)) = R ({∂_{j}σ, j is even})
t_{p−1}(R(σ)) = R ({∂_{j}σ, j is odd})

5. if Σ and T are two elements of Δ^{n} such that t_{p}(Σ) = s_{p}(T) for some p, then Σ ∪ T ∈ Δ^{n} and Σ ∪ T = Σ ∗_{p} T.

Moreover, all elements Σ of Δ^{n} satisfy the equality Σ = R(Σ).

If C is an ω-category and if x ∈ ωCat(Δ^{n}, C), then consider the labeling of the faces of respectively Δ^{n+1} and Δ^{n−1} defined by :

• ϵ_{i}(x)(σ_{0} < … < σ_{r}) = x(σ_{0} < … < σ_{k−1} < σ_{k} − 1 < … < σ_{r} − 1) if σ_{k−1} < i and σ_{k} > i.

• x(σ_{0} < … < σ_{k−1} < i < σ_{k+1}−1 < … < σ_{r} −1) if σ_{k−1} < i, σ_{k} = i and σ_{k+1} > i+1.

• x(σ_{0} < … < σ_{k−1} < i < σ_{k+2}−1 < … < σ_{r} −1) if σ_{k−1} < i, σ_{k} = i and σ_{k+1} = i+1.

and

∂_{i}(x)(σ_{0} < … < σ_{s}) = x(σ_{0} < … < σ_{k−1} < σ_{k} + 1 < … < σ_{s} + 1)

where σ_{k}, …, σ_{s} ⩾ i and σ_{k−1} < i.

It turns out that ϵ_{i}(x) ∈ ωCat(Δ^{n+1}, C) and ∂_{i}(x) ∈ ωCat(Δ^{n−1}, C). See [19, 28] for further information about simplicial sets. One has :

5.3. Definition. [26] The simplicial set (ωCat(Δ^{n}, C), ∂_{i}, ϵ_{i}) is called the simplicial nerve N (C) of the globular ω-category C. The corresponding homology is denoted by H_{∗}(C).

5.4. Definition. Let C be a non-contracting ω-category. By definition, C is of length at most 1 if and only if for any morphisms x and y of C such that x ∗_{0} y exists, then either x or y is 0-dimensional.

5.5. Theorem. Let C be an ω-category of length at most 1. Denote by ℙC the unique ω-category such that its set of n-morphisms is exactly the set of (n+1)-morphisms of C for any n ⩾ 0 with an obvious definition of the source and target maps and of the composition laws. Then one has the isomorphisms H_{n}(ℙC) ≅ H_{n+1}^{−}(C) for n ⩾ 1.

Proof. We give only a sketch of proof. By definition, H_{n+1}^{−}(C) = H_{n}(N^{−}(C)) for n ⩾ 1. Because of the hypothesis on C, every element x of ωCat(I^{n+1}, C)^{−} is determined by the values of the x(k_{1} … k_{n+1}) where k_{1} … k_{n+1} runs over the set of words on the alphabet {0, −}. It turns out that there is a bijective correspondence between O^{n} and the word of length n + 1 on the alphabet {0, −} : if σ_{0} < … < σ_{p} is an element of O^{n}, the associated word of length n + 1 is the word m_{0} … m_{n} such that m_{σ_{i}} = 0 and if j ∉ {σ_{0}, …, σ_{p}}, then m_{j} = −. It is then straightforward to check that the simplicial structure of N^{−}(C) is exactly the same as the simplicial structure of ωCat(Δ^{∗}, ℙC) in strictly positive dimension ^{1}.

The above proof together with Proposition 5.1 gives a new proof of the fact that if x ∈ ωCat(Δ^{n}, C), the labelings ∂_{i}(x) and ϵ_{i}(x) above defined yield ω-functors from Δ^{n−1} (resp. Δ^{n+1}) to C.

Notice that the above proof also shows that H_{n}(ℙC) ≅ H_{n+1}^{+}(C) where H_{∗}^{+} is the merging homology functor ^{2} This means that for an ω-category of length at most 1, H_{n+1}^{−}(C) ≅ H_{n+1}^{+}(C) for any n ⩾ 1. In general, this isomorphism is false as shown by

1The latter point is actually detailed in [13].

2Like the branching nerve, the definition of the merging nerve needs to be slightly change, with respect to the definition given in [12]. The correct definition is : an ω-functor x from I^{n} to a non-contracting ω-category C belongs to ωCat(I^{n}, C)^{+} if and only if for any 1-morphism γ of I^{n} such that t_{0}(γ) = R(+_{n}), then x(γ) is a 1-dimensional morphism of C.

BRANCHING MERGING

Figure 4: A case where branching and merging homologies are not equal in dimension 2

Figure 4. The precubical set we are considering in this figure is the complement of the depicted obstacle. Its branching homology is ℤ ⊕ ℤ in dimension two, and its merging homology is ℤ in the same dimension.

The result H_{n}(ℙC) ≅ H_{n+1}^{−}(C) ≅ H_{n+1}^{+}(C) for C of length at most one and for n ⩾ 1 also suggests that the program of constructing the analogue in the computer-scientific framework of usual homotopy invariants is complete for this kind of ω-categories. The simplicial set N (ℙC) together with the graph obtained by considering the 1-category generated by the 1-morphisms of C up to homotopy contain indeed all the information about the topology of the underlying automaton. Intuitively the simplicial set N (ℙC) is an orthogonal section of the automaton. Theorem 5.5 suggests that non-contracting ω-categories of length at most one play a particular role in this theory. This idea will be deepened in future works.

5.6. Corollary. With the same notation, if ℙC is the free globular ω-category generated by a composable pasting scheme in the sense of [15], then H_{n+1}^{−}(C) vanishes for n ⩾ 1.

Proof. By [25] Corollary 4.17 or by [17] Theorem 2.2, the simplicial nerve of the ωcategory of any composable pasting scheme is contractible.

5.7. Corollary. Let 2_{p} be the free ω-category generated by a p-morphism. For any p ⩾ 1 and any n ⩾ 1, H_{n}^{−}(2_{p}) = 0.

Proof. It is obvious for n = 1 and for n ⩾ 2, H_{n}^{−}(2_{p}) ≅ H_{n−1}(ℙ2_{p}). But ℙ2_{p} = 2_{p−1}, therefore it suffices to notice that the (p − 1)-simplex is contractible.

5.8. Corollary. For any n ⩾ 1, let G_{n}⟨A, B⟩ be the ω-category generated by two nmorphisms A and B satisfying s_{n−1}(A) = s_{n−1}(B) and t_{n−1}(A) = t_{n−1}(B). Then

H_{p}^{−}(G_{n}⟨A, B⟩) = 0

for 0 < p < n or p > n and

H_{0}^{−}(G_{n}⟨A, B⟩) = H_{n}^{−}(G_{n}⟨A, B⟩) = ℤ.

Proof. It suffices to calculate the simplicial homology of a simplicial set homotopic to a (n − 1)-sphere.

Let S be a composable pasting scheme (see [15] for the definition and [17] for additional explanations). A reasonable conjecture is that the branching homology of the free ω-category Cat(S) generated by any composable pasting scheme S vanishes in strictly positive dimension. By Conjecture 5.10, it would suffice for a given composable pasting scheme S to calculate the branching homology of the bilocalization Cat(S)[I, F] of Cat(S) with respect to its initial state I and its final state F, that is the sub-ω-category of Cat(S) which consists of the p-morphisms x with p ⩾ 1 such that s_{0}x ∈ I and t_{0}x ∈ F and of the 0-morphism I and F. The question of the calculation of

H_{p+1}^{−}(Cat(S)[I, F]) ≅ H_{p}(ℙCat(S)[I, F])

for p ⩾ 1 seems to be related to the existence of what Kapranov and Voevodsky call the derived pasting scheme of a composable pasting scheme [17]. It is in general not true that ℙCat(S)[I, F] (denoted by ΩCat(S) in their article) is the free ω-category generated by a composable pasting scheme. But we may wonder whether there is a “free cover” of ΩCat(S) by some Cat(T) for some composable pasting scheme T. This T would be the derived pasting scheme of S.

As for the n-cube I^{n}, its derived pasting scheme is the composable pasting scheme generated by the permutohedron [20, 4, 18]. Therefore one has

5.9. Proposition. Denote by I^{n}[−_{n}, +_{n}] the bilocalization of I^{n} with respect to its initial state −_{n} and its final state +_{n}, Then for all p ⩾ 1, H_{p}^{−}(I^{n}[−_{n}, +_{n}]) = 0.

Proof. It is clear that H_{1}^{−}(I^{n}[−_{n}, +_{n}]) = 0. For p ⩾ 2, H_{p}^{−}(I^{n}[−_{n}, +_{n}]) ≅ H_{p−1}(ΩI^{n}) by Theorem 5.5. But ΩI^{n} is the free ω-category generated by the permutohedron, and with Corollary 5.6, one gets H_{p}^{−}(I^{n}[−_{n}, +_{n}]) = 0 for p ⩾ 2.

By filtrating the 1-morphisms of I^{n} by their length, it is possible to construct a spectral sequence abutting to the branching homology of I^{n}. More precisely a 1-morphism x is of length ℓ(x) if x = R(x_{1}) ∗_{0} … ∗_{0} R(x_{ℓ(x)}) where x_{1}, …, x_{ℓ(x)} ∈ (\underline{cub}^{n})_{1}. Now let F_{p}ωCat(I^{∗}, I^{n})^{−} be the subset of x ∈ ωCat(I^{∗}, I^{n})^{−} such that for any k_{1} … k_{∗} ∈ (\underline{cub}^{∗})_{1} such that + ∈ {k_{1}, …, k_{∗}}, ℓ(x(k_{1} … k_{∗})) ⩽ p. Then one gets a filtration on the branching complex of I^{n} such that

F_{−1}ℤωCat(I^{∗}, I^{n})^{−} ⊂ F_{0}ℤωCat(I^{∗}, I^{n})^{−} ⊂ … ⊂ F_{n}ℤωCat(I^{∗}, I^{n})^{−}

with

F_{−1}ℤωCat(I^{∗}, I^{n})^{−} = 0
F_{0}ℤωCat(I^{∗}, I^{n})^{−} = ℤωCat(I^{∗}, I^{n}(−_{n}, +_{n}))^{−}
F_{n}ℤωCat(I^{∗}, I^{n})^{−} = ℤωCat(I^{∗}, I^{n})^{−}.

One has E_{pq}^{1} = H_{p+q}(F_{p}ℤωCat(I^{∗}, I^{n})^{−}/F_{p−1}ℤωCat(I^{∗}, I^{n})^{−}) ⟹ H_{p+q}^{−}(I^{n}). By Proposition 5.9, E_{0q} = 0 if q ≠ 0 and E_{00} = ℤ.

The above spectral sequence probably plays a role in the following conjecture :

5.10. Conjecture. Let C be a finite ω-category (that is such that the underlying set is finite). Let I be the set of initial states of C and let F be the set of final states of C (then H_{0}^{−}(C) = H_{0}^{−}(C[I, F]) = ℤF). If for any n > 0, H_{n}^{−}(C[I, F]) = 0, then for any n > 0, H_{n}^{−}(C) = 0.

By [17], ΩΔ^{n} = I^{n−1}, therefore the vanishing of the branching homology of I^{n−1} in strictly positive dimension and Conjecture 5.10 would enable to establish that H_{p}^{−}(Δ^{n}) = 0 for p > 0 and for any n.

6. About folding operators

The aim of this section is to introduce an analogue in our framework of the usual folding operators in cubical ω-categories. First we show how to recover the usual folding operators in our context.

The notations □_{0} or □_{0}^{−} (resp. □_{1} or □_{1}^{−}) correspond to the canonical map from C_{0} to ωCat(I^{0}, C) (resp. from tr_{1}C to ωCat(I^{1}, C)). Now let us recall the construction of the operators □_{n}^{−} of [12].

6.1. Proposition. [12] Let C be an ω-category and let n ⩾ 1. There exists one and only one natural map □_{n}^{−} from tr_{n}C to ωCat(I^{n}, C) such that the following axioms hold :

1. one has ev_{0_{n}}□_{n} = Id_{tr_{n}C} where ev_{0_{n}}(x) = x(0_{n}) is the label of the interior of x.

2. if n ⩾ 3 and 1 ⩽ i ⩽ n − 2, then ∂_{i}^{±}□_{n}^{−} = Γ_{n−2}^{−}∂_{i}^{±}□_{n−1}^{−}s_{n−1}.

3. if n ⩾ 2 and n−1 ⩽ i ⩽ n, then ∂_{i}^{−}□_{n}^{−} = □_{n−1}^{−}d_{n−1}^{(−)^{i}} and ∂_{i}^{+}□_{n}^{−} = ϵ_{n−1}∂_{n−1}^{+}□_{n−1}^{−}s_{n−1}.

Moreover for 1 ⩽ i ⩽ n, we have ∂_{i}^{±}□_{n}^{−}s_{n} = ∂_{i}^{±}□_{n}^{−}t_{n} and if x is of dimension greater or equal than 1, then □_{n}^{−}(x) ∈ ωCat(I^{n}, C)^{−}.

6.2. The usual folding operators. One defines a natural map □_{n} from C_{n} to ωCat(I^{n}, C) by induction on n ⩾ 2 as follows (compare with Proposition 6.1).

6.3. Proposition. For any natural number n greater or equal than 2, there exists a unique natural map □_{n} from C_{n} to ωCat(I^{n}, C) such that

1. the equality □_{n}(x)(0_{n}) = x holds.

2. one has ∂_{1}^{α}□_{n} = □_{n−1}d_{n−1}^{(−)^{α}} for α = ±.

3. for 1 < i ⩽ n, one has ∂_{i}^{α}□_{n} = ϵ_{1}∂_{i−1}^{α}□_{n−1}s_{n−1}.

Moreover for 1 ⩽ i ⩽ n, we have ∂_{i}^{±}□_{n}s_{n}u = ∂_{i}^{±}□_{n}t_{n}u for any (n + 1)-morphism u.

Proof. The induction equations define a fillable (n − 1)-shell (see Proposition 4.4).

6.4. Proposition. For all n ⩾ 0, the evaluation map ev_{0_{n}} : x ↦ x(0_{n}) from ωCat(I^{n}, C) to C induces a bijection from γN^{□}(C)_{n} to tr_{n}C where γ is the functor defined in [1].

Proof. Obvious for n = 0 and n = 1. Recall that γ is defined by

(γG)_{n} = {x ∈ G_{n}, ∂_{j}^{α}x ∈ ϵ_{1}^{j−1}G_{n−j} for 1 ⩽ j ⩽ n, α = 0, 1}

Let us suppose that n ⩾ 2 and let us proceed by induction on n. Since ev_{0_{n}}□_{n}(u) = u by the previous proposition, then the evaluation map ev from γN^{□}(C)_{n} to tr_{n}C is surjective. Now let us prove that x ∈ γN^{□}(C)_{n} and y ∈ γN^{□}(C)_{n} and x(0_{n}) = y(0_{n}) = u imply x = y. Since x and y are in γN^{□}(C)_{n}, then one sees immediately that the four elements ∂_{1}^{±}x and ∂_{1}^{±}y are in γN^{□}(C)_{n−1}. Since all other ∂_{i}^{α}x and ∂_{i}^{α}y are thin, then ∂_{1}^{−}x(0_{n−1}) = ∂_{1}^{−}y(0_{n−1}) = s_{n−1}u and ∂_{1}^{+}x(0_{n−1}) = ∂_{1}^{+}y(0_{n−1}) = t_{n−1}u. By induction hypothesis, ∂_{1}^{−}x = ∂_{1}^{−}y = □_{n−1}(s_{n−1}u) and ∂_{1}^{+}x = ∂_{1}^{+}y = □_{n−1}(t_{n−1}u). By hypothesis, one can set ∂_{j}^{α}x = ϵ_{1}^{j−1}x_{j}^{α} and ∂_{j}^{α}y = ϵ_{1}^{j−1}y_{j}^{α} for 2 ⩽ j ⩽ n. And one gets x_{j}^{α} = (∂_{1}^{α})^{j−1}∂_{j}^{α}x = (∂_{1}^{α})^{j}x = (∂_{1}^{α})^{j}y = y_{j}^{α}. Therefore ∂_{j}^{α}x = ∂_{j}^{α}y for all α ∈ {−, +} and all j ∈ [1, …, n]. By Proposition 4.4, one gets x = y.

The above proof shows also that the map which associates to any cube x of the cubical singular nerve of C the cube □_{dim(x)}(x(0_{dim(x)})) is exactly the usual folding operator as exposed in [1].

Unfortunately, these important operators are not internal to the branching complex, due to the fact that an n-cube x of the cubical singular nerve is in the branching complex if and only for any 1-morphism γ of I^{n} starting from the initial state −_{n} of the n-cube, x(γ) is 1-dimensional (see Definition 3.5). But for example (□_{n}(x(0_{n})))(− … − 0) is 0-dimensional.

6.5. The negative folding operators. The idea of the negative folding operator Φ_{n}^{−} is to “concentrate” a n-cube x of the cubical singular nerve of an ω-category C to the faces δ_{n−1}^{−}(0_{n−1}) and δ_{n}^{−}(0_{n−1}). Hence the following definition.

6.6. Definition. Set Φ_{n}^{−}(x) = □_{n}^{−}(x(0_{n})). This operator is called the n-dimensional negative folding operator.

It is clear that x ∈ ωCat(I^{n}, C)^{−} implies Φ_{n}^{−}(x) ∈ ωCat(I^{n}, C)^{−}. Therefore Φ_{n}^{−} yields a map from ωCat(I^{n}, C)^{−} to itself.

Since ∂_{n−1}^{−}□_{n}^{−} = □_{n−1}^{−}d_{n−1}^{(−)^{n−1}} and ∂_{n}^{−}□_{n}^{−} = □_{n−1}^{−}d_{n−1}^{(−)^{n}}, the effect of □_{n}^{−}(x(0_{n})) is indeed to concentrate the faces of the n-cube x on the faces δ_{n−1}^{−}(0_{n−1}) and δ_{n}^{−}(0_{n−1}). All the (n − 1)-cubes ∂_{i}^{α}□_{n}^{−}(x) for (i, α) ∉ {(n − 1, −), (n, −)} are thin. Of course there is not only one way of concentrating the faces of x on δ_{n−1}^{−}(0_{n−1}) and δ_{n}^{−}(0_{n−1}). But in some way, they are all equivalent in the branching complex (Corollary 8.4). We could decide also to concentrate the n-cubes for n ⩾ 2 on the faces δ_{1}^{−}(0_{n−1}) and δ_{2}^{−}(0_{n−1}), or more generally to concentrate the n-cubes on the faces δ_{p(n)}^{−}(0_{n−1}) and δ_{q(n)}^{−}(0_{n−1}) where p(n) and q(n) would be integers of opposite parity for all n ⩾ 2. Let us end this section by explaining precisely the structure of all these choices.

In an ω-category, recall that d_{n}^{−} = s_{n}, d_{n}^{+} = t_{n} and by convention, let d_{ω}^{−} = d_{ω}^{+} = Id. All the usual axioms of globular ω-categories remain true with this convention and the partial order n < ω for any natural number n.

If x is an element of an ω-category C, we denote by ⟨x⟩ the ω-category generated by x. The underlying set of ⟨x⟩ is {s_{n}x, t_{n}x, n ∈ ℕ}. We denote by 2_{n} any ω-category freely generated by one n-dimensional element.

Let R(k_{1} … k_{n}) ∈ I^{n} a face. Denote by ev_{k_{1}...k_{n}} the natural transformation from ωCat(I^{n}, −) to tr_{ω} which maps f to f(R(k_{1} … k_{n})).

6.7. Definition. Let n ∈ ℕ. Recall that tr_{n} is the forgetful functor from ω-categories to sets which associates to any ω-category its set of morphisms of dimension lower or equal than n and let i_{n} be the inclusion functor from tr_{n−1} to tr_{n}. We call cubification of dimension n, or n-cubification a natural transformation □ from tr_{n} to ωCat(I^{n}, −). If moreover, ev_{0_{n}}□ = Id, we say that the cubification is thick.

We see immediately that □_{n}^{−}, □_{n} (and □_{n}^{+} of [12]) are examples of thick n-cubifica-tions. By Yoneda the set of n-cubifications is in bijection with the set of ω-functors from I^{n} to 2_{n}. So for a given n, there is a finite number of n-cubifications.

6.8. Proposition. Let f be a natural transformation from tr_{m} to tr_{n} with m, n ∈ ℕ ∪ {ω}. Then there exists p ⩽ m and α ∈ {−, +} such that f = d_{p}^{α}. And necessarily, p ⩽ Inf(m, n).

Proof. Denote by

<A> = 2_{n} ⟶^{g} <B> = 2_{m}

the ω-functor which corresponds to f by Yoneda. Then g(A) = d_{p}^{α}(B) for some p and some α. And necessarily, p ⩽ min(m, n) (where the notation min means the smallest element).

6.9. Corollary. Let □ be a n-cubification with n ⩾ 1 a natural number. Then for any i with 1 ⩽ i ⩽ n, ∂_{i}^{±}□s_{n} = ∂_{i}^{±}□t_{n}.

Proof. We have

∂_{i}^{±}□s_{n}x(l_{1} … l_{n−1}) = ev_{l_{1}...[±]_{i}...l_{n−1}}□s_{n}(x)

But ev_{l_{1}...[±]_{i}...l_{n−1}}□ is a natural transformation from tr_{n} to tr_{n−1}. By Proposition 6.8, we get

∂_{i}^{±}□s_{n}x(l_{1} … l_{n−1}) = ev_{l_{1}...[±]_{i}...l_{n−1}}□t_{n}(x) = ∂_{i}^{±}□t_{n}x(l_{1} … l_{n−1})

We arrive at a theorem which explains the structure of all cubifications :

6.10. Theorem. Let □ be a thick n-cubification and let f be an ω-functor from I^{n+1} to I^{n} such that f(R(0_{n+1})) = R(0_{n}). Denote by f^{∗} the corresponding natural transformation from ωCat(I^{n}, −) to ωCat(I^{n+1}, −). Then there exists one and only one thick (n + 1)- cubification denoted by f^{∗}.□ such that for 1 ⩽ i ⩽ n + 1,

(f^{∗}.□)i_{n+1} = f^{∗}□

where i_{n+1} is the canonical natural transformation from tr_{n} to tr_{n+1}.

Proof. One has

∂_{i}^{α}(f^{∗}.□) = ∂_{i}^{α}(f^{∗}.□)d_{n}^{(−)^{i}}
= ∂_{i}^{α}(f^{∗}.□)i_{n+1}d_{n}^{(−)^{i}}
= ∂_{i}^{α}f^{∗}□d_{n}^{(−)^{i}}

Therefore if x ∈ C_{n+1} for some ω-category C, then ∂_{i}^{α}(f^{∗}.□)x = ∂_{i}^{α}f^{∗}□d_{n}^{(−)^{i}}x for 1 ⩽ i ⩽ n + 1 and we obtain a fillable n-shell in the sense of Proposition 4.4.

6.11. Corollary. Let u be an ω-functor from I^{n} to 2_{n} which maps R(0_{n}) to the unique n-morphism of 2_{n} (we will say that u is thick because the corresponding cubification is also thick). Let f be an ω-functor from I^{n+1} to I^{n} which maps R(0_{n+1}) to R(0_{n}). Then there exists one and only one thick ω-functor v from I^{n+1} to 2_{n+1} such that the following diagram commutes :

\xymatrix{I^{n+1} \ar[r]^{v} \ar[d]_{f} & 2_{n+1} \ar[d] \\ I^{n} \ar[r]^{u} & 2_{n}}

the arrow from 2_{n+1} to 2_{n} being the unique ω-functor which sends the (n + 1)-cell of 2_{n+1} to the n-cell of 2_{n}.

If □ is a n-cubification and f_{i} thick ω-functors from I^{n+i+1} to I^{n+i} for 0 ⩽ i ⩽ p then we can denote without ambiguity by f_{p}.f_{p−1}. . . . .f_{0}.□ the (n + p)-cubification f_{p}.(f_{p−1}.(….f_{0}.□)). Let us denote by □_{0} the unique 0-cubification. We have the following formulas :

6.12. Proposition. Let x ∈ C be a p-dimensional morphism with p ⩾ 1 and let n ⩾ p. Then

□_{n}^{−}x = Γ_{n−1}^{−} … Γ_{p}^{−}□_{p}^{−}x

(by convention, the above formula is tautological for n = p)

Proof. We are going to show the formula by induction on n. The case n = p is trivial. If i ⩽ n − 1, then ∂_{i}^{±}□_{n+1}^{−}x = Γ_{n−1}^{−}∂_{i}^{±}□_{n}^{−}x = ∂_{i}^{±}Γ_{n}^{−}Γ_{n−1}^{−} … Γ_{p}^{−}□_{p}^{−}x. And if i ⩾ n, then

∂_{i}^{−}□_{n+1}^{−}x = □_{n}^{−}x = ∂_{i}^{−}Γ_{n}^{−}Γ_{n−1}^{−} … Γ_{p}^{−}□_{p}^{−}x

and

∂_{i}^{+}□_{n+1}^{−}x = ϵ_{n}∂_{i}^{+}n□_{n}^{−}x = ∂_{i}^{+}Γ_{n}^{−}Γ_{n−1}^{−} … Γ_{p}^{−}□_{p}^{−}x.

So the labelings □_{n+1}^{−}x and Γ_{n}^{−} … Γ_{p}^{−}□_{p}^{−}x of I^{n+1} are the same ones.

6.13. Proposition. For n ⩾ 1, we have □_{1}^{−} = ϵ_{1}.□_{0} and

□_{n}^{−} = Γ_{n−1}^{−}. . . . .Γ_{1}^{−}.ϵ_{1}.□_{0}

Proof. It is an immediate consequence of Proposition 6.12 and of the uniqueness of Theorem 6.10.

The converse of Theorem 6.10 is true. That is :

6.14. Proposition. Let v be a thick ω-functor from I^{n+1} to 2_{n+1}. Then there exists an ω-functor f such that for any thick ω-functor u from I^{n} to 2_{n}, the following diagram commutes :

\xymatrix{I^{n+1} \ar[r]^{v} \ar[d]_{f} & 2_{n+1} \ar[d] \\ I^{n} \ar[r]^{u} & 2_{n}}

Proof. Set v(R(k_{1} … k_{n+1})) = d_{n_{k_{1}...k_{n+1}}}^{α_{k_{1}...k_{n+1}}}(A) where ⟨A⟩ = 2_{n+1} and set ⟨B⟩ = 2_{n}. By hypothesis, the equality f(0_{n+1}) = R(0_{n}) holds and let

f(k_{1} … k_{n+1}) = d_{n_{k_{1}...k_{n+1}}}^{α_{k_{1}...k_{n+1}}}(R(0_{n}))

Take any thick ω-functor u from I^{n} to 2_{n}. Then

u ∘ f(R(k_{1} … k_{n+1})) = u(d_{n_{k_{1}...k_{n+1}}}^{α_{k_{1}...k_{n+1}}}(R(0_{n}))) = d_{n_{k_{1}...k_{n+1}}}^{α_{k_{1}...k_{n+1}}}u(R(0_{n}))
= d_{n_{k_{1}...k_{n+1}}}^{α_{k_{1}...k_{n+1}}}(B)

By Proposition 4.4, it is clear that f induces an ω-functor.

Here is an example of cubification : if the following picture depicts the 3-cube,

[figure: 00−; 0+0; −00]⟹ [figure: +00; 0−0; 00+]

we can represent a 3-cubification □ by indexing each face k_{1}k_{2}k_{3} by the corresponding value of ev_{k_{1}k_{2}k_{3}}□i_{3} which is equal to s_{d} or t_{d} for some d between 0 and 2. So let us take □ as follows :

[figure: t_{0}; t_{0}; t_{1}; s_{0}; t_{1}; t_{1}; t_{1}; t_{1}; t_{0}; s_{0}; s_{0}; s_{2}; s_{1}]⟹ [figure: t_{0}; t_{0}; t_{0}; t_{0}; t_{0}; t_{0}; t_{1}; s_{0}; t_{1}; s_{1}; t_{1}; t_{2}; t_{0}]

We see that ∂_{1}^{−}□i_{3} = Γ_{1}^{+}.□_{1} and that ∂_{3}^{+}□i_{3} = Γ_{1}^{−}.□_{1}.

Now let us come back to our choice. It is not completely arbitrary anyway because the operator □_{n}^{−} satisfies the following important property : if u is a n-morphism with n ⩾ 2, then □_{n}^{−}(u) is a simplicial homotopy within the branching nerve between □_{n−1}^{−}(s_{n−1}u) and □_{n−1}^{−}(t_{n−1}u). Moreover, the family of cubifications (□_{n}^{−})_{n⩾0} is the only family of cubifications which satisfies this property because it is equivalent to defining a n-shell for all n. However most of the results of the sequel can be probably adapted to any family of n-cubification, provided that they yield internal operations on the branching nerve (see Conjecture 7.7 and 7.8).

6.15. Characterization of the negative folding operators. Now here is a useful property of the folding operators :

6.16. Theorem. Let C be an ω-category. Let x be an element of N_{n}^{□}(C). Then the following two conditions are equivalent :

1. the equality x = Φ_{n}^{−}(x) holds

2. for 1 ⩽ i ⩽ n, one has ∂_{i}^{+}x ∈ Im(ϵ_{1}^{n−1}), and for 1 ⩽ i ⩽ n − 2, one has ∂_{i}^{−}x ∈ Im(Γ_{n−2}^{−} … Γ_{i}^{−}).

Proof. If x = Φ_{n}^{−}(x), then x = □_{n}^{−}(x(0_{n})) and by construction of □_{n}^{−},

∂_{i}^{−}□_{n}^{−}(x(0_{n})) = Γ_{n−2}^{−} … Γ_{i}^{−}□_{i}^{−}d_{i}^{(−)^{i}}x(0_{n})

for any 1 ⩽ i ⩽ n − 2 and ∂_{i}^{+}x is 0-dimensional for any 1 ⩽ i ⩽ n. For n equal to 0, 1 or 2, the converse is obvious. Suppose the converse proved for n − 1 ⩾ 2 and let us prove it by induction for n ⩾ 3. By hypothesis, as soon as + ∈ {k_{1}, …, k_{n}}, then x(k_{1} … k_{n}) is 0-dimensional. For 1 ⩽ i ⩽ n − 3, one has

∂_{i}^{−}∂_{n}^{−}x = ∂_{n−1}^{−}∂_{i}^{−}x
= ∂_{n−1}^{−}Γ_{n−2}^{−} … Γ_{i}^{−}Y_{i} for some Y_{i} ∈ N_{i}^{□}(C)
= Γ_{n−3}^{−} … Γ_{i}^{−}Y_{i}

and

∂_{i}^{−}∂_{n−1}^{−}x = ∂_{n−2}^{−}∂_{i}^{−}x
= ∂_{n−2}^{−}Γ_{n−2}^{−} … Γ_{i}^{−}Y_{i}
= Γ_{n−3}^{−} … Γ_{i}^{−}Y_{i}

therefore ∂_{n−1}^{−}x and ∂_{n}^{−}x satisfy the induction hypothesis. So ∂_{n−1}^{−}x = Φ_{n−1}^{−}(∂_{n−1}^{−}x) and ∂_{n}^{−}x = Φ_{n−1}^{−}(∂_{n}^{−}x). Since the ∂_{i}^{−}x are thin (n − 1)-cubes for all i between 1 and n − 2, then d_{n−1}^{(−)^{n−1}}(x(0_{n})) = ∂_{n−1}^{−}x(0_{n−1}) and d_{n−1}^{(−)^{n}}(x(0_{n})) = ∂_{n}^{−}x(0_{n−1}). Therefore ∂_{n−1}^{−}x = □_{n−1}^{−}(d_{n−1}^{(−)^{n−1}}(x(0_{n}))) and ∂_{n}^{−}x = □_{n−1}^{−}(d_{n−1}^{(−)^{n}}(x(0_{n}))). For 1 ⩽ i ⩽ n − 2, one has

Y_{i} = ∂_{i}^{−} … ∂_{n−2}^{−}∂_{i}^{−}x
= ∂_{i}^{−} … ∂_{n−1}^{−}x
= ∂_{i}^{−} … ∂_{n−2}^{−}□_{n−1}^{−}(d_{n−1}^{(−)^{n−1}}(x(0_{n})))
= ∂_{i}^{−} … ∂_{n−3}^{−}□_{n−2}^{−}(d_{n−2}^{(−)^{n−2}}(x(0_{n})))
= (…)
= □_{i}^{−}(d_{i}^{(−)^{i}}(x(0_{n})))

therefore an easy calculation shows that x = □_{n}^{−}(x(0_{n})).

6.17. Corollary. The folding operator Φ_{n}^{−} is idempotent.

The end of this section is devoted to the description of Φ_{2}^{−} and Φ_{3}^{−}. Since ∂_{1}^{−}□_{2}^{−} = □_{1}s_{1} and ∂_{2}^{−}□_{2}^{−} = □_{1}t_{1}, then one has for any ω-functor x from I^{2} to C

Φ_{2}^{−}(x) = _{[ x ]} [figure: 2; 1]

If x is an ω-functor from I^{3} to C, then

s_{2}(x(000)) = \left[{∂_{3}^{−}x ∂_{2}^{+}x \atop ∂_{1}^{−}x}\right] [figure: 2; 1](00)

because the 2-source of R(000) in I^{3} looks like

\left[{R(00−) R(0 + 0) \atop R(−00)}\right]

and

t_{2}(x(000)) = \left[{∂_{1}^{+}x \atop ∂_{2}^{−}x ∂_{3}^{+}x}\right] [figure: 2; 1] (00)

because the 2-target of R(000) in I^{3} looks like

\left[{R(+00) \atop R(0 − 0) R(00+)}\right]

So by convention, an element x of ωCat(I^{3}, C) will be represented as follows

x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

where A = ∂_{3}^{−}x, B = ∂_{2}^{+}x, C = ∂_{1}^{−}x, D = ∂_{1}^{+}x, E = ∂_{2}^{−}x, F = ∂_{3}^{+}x and x(000) = G^{3}. With this convention, Γ_{1}^{−}y for y ∈ ωCat(I^{2}, C) is equal to

y \underrightarrow{y(00)} y

One has ∂_{1}^{±}□_{3}^{−} = Γ_{1}^{−}∂_{1}^{±}□_{2}^{−}s_{2}, ∂_{2}^{−}□_{3}^{−} = □_{2}^{−}t_{2}, ∂_{3}^{−}□_{3}^{−} = □_{2}^{−}s_{2}, ∂_{1}^{+}□_{3}^{−} = ∂_{2}^{+}□_{3}^{−} = ∂_{3}^{+}□_{3}^{−} = □_{2}^{−}t_{0} by definition of □_{3}^{−}. Therefore

∂_{i}^{+}Φ_{3}^{−}(x) = □_{2}^{−}t_{0}(G)
∂_{2}^{−}Φ_{3}^{−}(x) = □_{2}^{−}t_{2}(G)
∂_{3}^{−}Φ_{3}^{−}(x) = □_{2}^{−}s_{2}(G)

and

∂_{1}^{±}Φ_{3}^{−}(x) = Γ_{1}^{−}∂_{1}^{±}□_{2}^{−} \left[{A B \atop C}\right] [figure: 2; 1]

= Γ_{1}^{−}∂_{1}^{±} [ \begin{matrix}A & B \\ & C \end{matrix}  ] [figure: 2; 1]

= \{ \begin{matrix}Γ_{1}^{−}(∂_{1}^{−}C +_{1} ∂_{2}^{+}C +_{1} ∂_{2}^{+}B) in the negative case \\ Γ_{1}^{−}∂_{1}^{+}∂_{1}^{+}B in the positive case \end{matrix}

So if x is the above ω-functor from I^{3} to C, then

Φ_{3}^{−}(x) = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

3Beware of the fact that A, … F are elements of the cubical singular nerve whereas G is an element of the ω-category we are considering.

7. Elementary moves in the cubical singular nerve

In this section, the folding operators Φ_{n}^{−} are decomposed in elementary moves. First of all, here is a definition.

7.1. Definition. The elementary moves in the n-cube are one of the following operators (with 1 ⩽ i ⩽ n − 1 and x ∈ ωCat(I^{n}, C)) :

1. ^{v}ψ_{i}^{−}x = _{[ x ]} [figure: i+1; i]

2. ^{v}ψ_{i}^{+}x = ^{[ x ]} [figure: i+1; i]

3. ^{h}ψ_{i}^{−}x = [ x ] [figure: i+1; i]

4. ^{h}ψ_{i}^{+}x = [ x ] [figure: i+1; i]

Notation. One sets θ_{i}^{−} = ^{v}ψ_{i+1}^{−} ^{v}ψ_{i}^{+}. This operator plays a central rôle in the sequel.

Proposition 7.2 expresses the elementary moves using the notation of the previous paragraph (only the operators used in the sequel are calculated).

7.2. Proposition. Let

x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

be an element of ωCat(I^{3}, C). Then one has

vψ_{1}^{+}x = A \begin{matrix}B \\ C \end{matrix} \underrightarrow{G} D ; ^{E F}

vψ_{2}^{+}x =  A B ; C \underrightarrow{G} D ; E F

vψ_{2}^{−}x = A B ; C \underrightarrow{G} D ; E F

vψ_{1}^{−}x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

hψ_{1}^{−}x = A ; \begin{matrix}B \\ C \end{matrix} \underrightarrow{G} D ; E F

hψ_{2}^{−}x = A B ; C \underrightarrow{G} D ; E F

θ_{1}^{−}x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} D ; ^{E F}

Proof. One has ^{v}ψ_{i}^{+}x = Γ_{i}^{+}∂_{i}^{−}x +_{i} x. Therefore

∂_{1}^{−} ^{v}ψ_{1}^{+}x = ϵ_{1}∂_{1}^{−}∂_{1}^{−}x ∂_{1}^{+} ^{v}ψ_{1}^{+}(x) = ∂_{1}^{+}x ∂_{2}^{−} ^{v}ψ_{1}^{+}x = ϵ_{1}∂_{1}^{−}∂_{1}^{−}x +_{1} ∂_{2}^{−}x ∂_{2}^{+} ^{v}ψ_{1}^{+}x = ∂_{1}^{−}x +_{1} ∂_{2}^{+}x ∂_{3}^{±} ^{v}ψ_{1}^{+}x = ∂_{3}^{±}Γ_{1}^{+}∂_{1}^{−}x +_{1} ∂_{3}^{±}x

So one has

vψ_{1}^{+}x = A \begin{matrix}B \\ C \end{matrix} \underrightarrow{G} D ; ^{E F}

And

∂_{1}^{−} ^{v}ψ_{2}^{+}x = Γ_{1}^{+}∂_{1}^{−}∂_{2}^{−}x +_{1} ∂_{1}^{−}x ∂_{2}^{−} ^{v}ψ_{2}^{+}x = ϵ_{2}∂_{2}^{−}∂_{2}^{−}x ∂_{3}^{−} ^{v}ψ_{2}^{+}x = ϵ_{2}∂_{2}^{−}x +_{2} ∂_{3}^{−}x ∂_{1}^{+} ^{v}ψ_{2}^{+}x = Γ_{1}^{+}∂_{1}^{+}∂_{2}^{−}x +_{1} ∂_{1}^{+}x ∂_{2}^{+} ^{v}ψ_{2}^{+}x = ∂_{2}^{+}x ∂_{3}^{+} ^{v}ψ_{2}^{+}x = ∂_{2}^{−}x +_{2} ∂_{3}^{+}x

Consequently one has

vψ_{2}^{+}x =  A B ; C \underrightarrow{G} D ; E F

One has ^{v}ψ_{i}^{−}x = x +_{i} Γ_{i}^{−}∂_{i}^{+}x. Therefore

∂_{1}^{−} ^{v}ψ_{2}^{−}x = ∂_{1}^{−}x +_{1} ∂_{1}^{−}Γ_{2}^{−}∂_{2}^{+}x ∂_{1}^{+} ^{v}ψ_{2}^{−}x = ∂_{1}^{+}x +_{1} ∂_{1}^{+}Γ_{2}^{−}∂_{2}^{+}x ∂_{2}^{−} ^{v}ψ_{2}^{−}x = ∂_{2}^{−}x ∂_{2}^{+} ^{v}ψ_{2}^{−}x = ϵ_{2}∂_{2}^{+}∂_{2}^{+}x ∂_{3}^{−} ^{v}ψ_{2}^{−}x = ∂_{3}^{−}x +_{2} ∂_{2}^{+}x ∂_{3}^{+} ^{v}ψ_{2}^{−}x = ∂_{3}^{+}x +_{2} ϵ_{2}∂_{2}^{+}∂_{2}^{+}x

Theory and Applications of Categories, Vol. 8, No. 12 351

So

vψ_{2}^{−}x = A B ; C \underrightarrow{G} D ; E F

And

∂_{1}^{−} ^{v}ψ_{1}^{−}x = ∂_{1}^{−}x
∂_{1}^{+} ^{v}ψ_{1}^{−}x = ϵ_{1}∂_{1}^{+}∂_{1}^{+}x
∂_{2}^{−} ^{v}ψ_{1}^{−}x = ∂_{2}^{−}x +_{1} ∂_{1}^{+}x
∂_{2}^{+} ^{v}ψ_{1}^{−}x = ∂_{2}^{+}x +_{1} ϵ_{1}∂_{1}^{+}∂_{1}^{+}x
∂_{3}^{−} ^{v}ψ_{1}^{−}x = ∂_{3}^{−}x +_{1} Γ_{1}^{−}∂_{2}^{−}∂_{1}^{+}x
∂_{3}^{+} ^{v}ψ_{1}^{−}x = ∂_{3}^{+}x +_{1} Γ_{1}^{−}∂_{2}^{+}∂_{1}^{+}x

therefore

vψ_{1}^{−}x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

One has ^{h}ψ_{1}^{−}x = x +_{2} Γ_{1}^{−}∂_{2}^{+}x. Then ∂_{1}^{−} ^{h}ψ_{1}^{−}x = ∂_{1}^{−}x +_{1} ∂_{2}^{+}x

∂_{1}^{+} ^{h}ψ_{1}^{−}x = ∂_{1}^{+}x
∂_{2}^{−} ^{h}ψ_{1}^{−}x = ∂_{2}^{−}x
∂_{2}^{+} ^{h}ψ_{1}^{−}x = ϵ_{1}∂_{1}^{+}∂_{2}^{+}x
∂_{3}^{−} ^{h}ψ_{1}^{−}x = ∂_{3}^{−}x +_{2} Γ_{1}^{−}∂_{2}^{−}∂_{2}^{+}x
∂_{3}^{+} ^{h}ψ_{1}^{−}x = ∂_{3}^{+}x +_{2} Γ_{1}^{−}∂_{2}^{+}∂_{2}^{+}x

So

hψ_{1}^{−}x = A ; \begin{matrix}B \\ C \end{matrix} \underrightarrow{G} D ; E F

One has ^{h}ψ_{2}^{−}x = x +_{3} Γ_{2}^{−}∂_{3}^{+}x. Therefore ∂_{1}^{−} ^{h}ψ_{2}^{−}x = ∂_{1}^{−}x +_{2} Γ_{1}^{−}∂_{1}^{−}∂_{3}^{+}x ∂_{1}^{+} ^{h}ψ_{2}^{−}x = ∂_{1}^{+}x +_{2} Γ_{1}^{−}∂_{1}^{+}∂_{3}^{+}x ∂_{2}^{−} ^{h}ψ_{2}^{−}x = ∂_{2}^{−}x +_{2} ∂_{3}^{+}x ∂_{2}^{+} ^{h}ψ_{2}^{−}x = ∂_{2}^{+}x +_{2} ϵ_{2}∂_{2}^{+}∂_{3}^{+}x ∂_{3}^{−} ^{h}ψ_{2}^{−}x = ∂_{3}^{−}x ∂_{3}^{+} ^{h}ψ_{2}^{−}x = ∂_{3}^{+}Γ_{2}^{−}∂_{3}^{+}x = ϵ_{2}∂_{2}^{+}∂_{3}^{+}x

Theory and Applications of Categories, Vol. 8, No. 12 352

so

hψ_{2}^{−} = A B ; C \underrightarrow{G} D ; E F

Now let us calculate θ_{1}^{−}x. One has

θ_{1}^{−}x = ^{v}ψ_{2}^{−} ^{v}ψ_{1}^{+}x

= ^{v}ψ_{2}^{−} A \begin{matrix}B \\ C \end{matrix} \underrightarrow{G} D ; ^{E F}

= \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} D ; ^{E F}

The following proposition describes some of the commutation relations satisfied by the previous operators, the differential maps and the connection maps. 7.3. Proposition. The following equalities hold (with α ∈ {−, +}) :

∂_{j}^{α} ^{v}ψ_{i}^{−} = \{ \begin{matrix}^{v}ψ_{i−1}^{−}∂_{j}^{α} if j < i \\ ^{v}ψ_{i}^{−}∂_{j}^{α} if j > i + 1 \end{matrix}  (1)
∂_{j}^{α} ^{h}ψ_{i}^{−} = \{ \begin{matrix}^{h}ψ_{i−1}^{−}∂_{j}^{α} if j < i \\ ^{h}ψ_{i}^{−}∂_{j}^{α} if j > i + 1 \end{matrix}  (2)
∂_{j}^{α}θ_{i}^{−} = \{ \begin{matrix}θ_{i−1}^{−}∂_{j}^{α} if j < i \\ θ_{i}^{−}∂_{j}^{α} if j > i + 2 \end{matrix}  (3)
θ_{i}^{−}Γ_{j}^{−} = \{ \begin{matrix}Γ_{j}^{−}θ_{i−1}^{−} if j < i \\ Γ_{j}^{−}θ_{i}^{−} if j > i + 2 \end{matrix}  (4)
∂_{i}^{−} ^{v}ψ_{i}^{−} = ∂_{i}^{−}  (5)
∂_{i}^{+} ^{v}ψ_{i}^{−} = ϵ_{i}∂_{i}^{+}∂_{i}^{+}  (6)
∂_{i+1}^{−} ^{v}ψ_{i}^{−} = ∂_{i+1}^{−} +_{i} ∂_{i}^{+}  (7)
∂_{i+1}^{+} ^{v}ψ_{i}^{−} = ∂_{i+1}^{+}  (8)
∂_{i}^{−} ^{h}ψ_{i}^{−} = ∂_{i}^{−} +_{i} ∂_{i+1}^{+}  (9)
∂_{i}^{+} ^{h}ψ_{i}^{−} = ∂_{i}^{+}  (10)
∂_{i+1}^{−} ^{h}ψ_{i}^{−} = ∂_{i+1}^{−}  (11)
∂_{i+1}^{+} ^{h}ψ_{i}^{−} = ϵ_{i}∂_{i}^{+}∂_{i+1}^{+}  (12)
∂_{i}^{−}θ_{i}^{−} = Γ_{i}^{−}∂_{i}^{−}∂_{i}^{−}  (13)
∂_{i}^{+}θ_{i}^{−} = ^{v}ψ_{i}^{−}∂_{i}^{+}  (14)

∂_{i+1}^{−}θ_{i}^{−} = ∂_{i+1}^{−}  (15)
∂_{i+1}^{+}θ_{i}^{−} = ϵ_{i+1}∂_{i+1}^{+}∂_{i}^{−} +_{i} ϵ_{i+1}∂_{i+1}^{+}∂_{i+1}^{+}  (16)
∂_{i+2}^{−}θ_{i}^{−} = \left[{∂_{i+2}^{−} ∂_{i+1}^{+} \atop ∂_{i}^{−}}\right] [figure: i+1; i]  (17)
∂_{i+2}^{+}θ_{i}^{−} = ^{v}ψ_{i}^{+}∂_{i+2}^{+}  (18)
θ_{i}^{−}Γ_{i}^{−} = Γ_{i+1}^{−}  (19)
θ_{i}^{−}Γ_{i+1}^{−} = Γ_{i+1}^{−}  (20)

Proof. Equalities (1), (2), (3) and (4) are obvious. Equalities from (5) to (12) are immediate consequences of the definitions. With Proposition 7.2, one sees that

∂_{1}^{−}θ_{1}^{−} = Γ_{1}^{−}∂_{1}^{−}∂_{1}^{−} ∂_{1}^{+}θ_{1}^{−} = ^{v}ψ_{1}^{−}∂_{1}^{+} ∂_{2}^{−}θ_{1}^{−} = ∂_{2}^{−} ∂_{2}^{+}θ_{1}^{−} = ϵ_{2}∂_{2}^{+}∂_{1}^{−} +_{1} ϵ_{2}∂_{2}^{+}∂_{2}^{+}

∂_{3}^{−}θ_{1}^{−} = \left[{∂_{3}^{−} ∂_{2}^{+} \atop ∂_{1}^{−}}\right] [figure: 2; 1]

∂_{3}^{+}θ_{1}^{−} = ^{v}ψ_{1}^{+}∂_{3}^{+}

For a given x, the above equalities are equalities in the free cubical ω-category generated by x. Therefore, they depend only on the relative position of the indices 1, 2 and 3 with respect to one another. Therefore, we can replace each index 1 by i, each index 2 by i + 1 and each index 3 by i + 2 to obtain the required formulae.

In the same way, it suffices to prove the last two formulae in lower dimension and for i = 1. One has

θ_{1}^{−}Γ_{1}^{−}x = θ_{1}^{−}  x \underrightarrow{x(00)} x

=  x  \underrightarrow{x(00)} x

= Γ_{2}^{−}x

and

θ_{1}^{−}Γ_{2}^{−}x = θ_{1}^{−} x  \underrightarrow{x(00)} x

= x  \underrightarrow{x(00)} x

= Γ_{2}^{−}x

7.4. Theorem. Set ^{v}Ψ_{k}^{−} = ^{v}ψ_{k}^{−} … ^{v}ψ_{1}^{−} and ^{h}Ψ_{k}^{−} = ^{h}ψ_{k}^{−} … ^{h}ψ_{1}^{−}. Then for n ⩾ 2 and 1 ⩽ i ⩽ n, one has

∂_{i}^{+}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) = ϵ_{1}^{n−1}(∂_{1}^{+})^{n}.

Proof. It is obvious for n = 2. We are going to make an induction on n. Let n ⩾ 2 and 1 ⩽ i ⩽ n. Then

∂_{i}^{+}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n}^{−} ^{h}Ψ_{n}^{−})
= ϵ_{1}^{n−1}(∂_{1}^{+})^{n}(^{v}Ψ_{n}^{−} ^{h}Ψ_{n}^{−})
= ϵ_{1}^{n−1}(∂_{1}^{+})^{n−1} ^{v}Ψ_{n−1}^{−}ϵ_{1}∂_{1}^{+}∂_{1}^{+} ^{h}Ψ_{n}^{−}
= ϵ_{1}^{n−1}(∂_{1}^{+})^{n−2} ^{v}Ψ_{n−2}^{−}(ϵ_{1}∂_{1}^{+}∂_{1}^{+})^{2} ^{h}Ψ_{n}^{−}
= (…)
= ϵ_{1}^{n−1}(ϵ_{1}∂_{1}^{+}∂_{1}^{+})^{n} ^{h}Ψ_{n}^{−}
= ϵ_{1}^{n}(∂_{1}^{+})^{n+1} ^{h}Ψ_{n}^{−}

The equality

∂_{i}^{+}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n}^{−} ^{h}Ψ_{n}^{−})x = ϵ_{1}^{n}(∂_{1}^{+})^{n+1} ^{h}Ψ_{n}^{−}x

makes sense if x is a (n + 1)-cube. And in this case, ϵ_{1}^{n}(∂_{1}^{+})^{n+1} ^{h}Ψ_{n}^{−}x is 0-dimensional and ϵ_{1}^{n}(∂_{1}^{+})^{n+1} ^{h}Ψ_{n}^{−}x = ϵ_{1}^{n}(∂_{1}^{+})^{n+1}x. This equality holds in the free cubical ω-category generated by x, and therefore

ϵ_{1}^{n}(∂_{1}^{+})^{n+1} ^{h}Ψ_{n}^{−} = ϵ_{1}^{n}(∂_{1}^{+})^{n+1}.

Now suppose that i = n + 1. Then

∂_{n+1}^{+}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n}^{−} ^{h}Ψ_{n}^{−})
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−})∂_{n+1}^{+}(^{v}Ψ_{n}^{−} ^{h}Ψ_{n}^{−})
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−})(^{v}Ψ_{n−1}^{−}∂_{n+1}^{+} ^{h}Ψ_{n}^{−})
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) ^{v}Ψ_{n−1}^{−}∂_{n+1}^{+}(^{h}ψ_{n}^{−} … ^{h}ψ_{1}^{−})
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) ^{v}Ψ_{n−1}^{−}ϵ_{n}∂_{n}^{+}∂_{n+1}^{+}(^{h}ψ_{n−1}^{−} … ^{h}ψ_{1}^{−})
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) ^{v}Ψ_{n−1}^{−}ϵ_{n}∂_{n}^{+}(^{h}ψ_{n−1}^{−} … ^{h}ψ_{1}^{−})∂_{n+1}^{+}
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) ^{v}Ψ_{n−1}^{−}ϵ_{n}ϵ_{n−1}∂_{n−1}^{+}∂_{n}^{+}(^{h}ψ_{n−2}^{−} … ^{h}ψ_{1}^{−})∂_{n+1}^{+}

= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) ^{v}Ψ_{n−1}^{−}ϵ_{n}ϵ_{n−1}∂_{n−1}^{+}(^{h}ψ_{n−2}^{−} … ^{h}ψ_{1}^{−})∂_{n}^{+}∂_{n+1}^{+}
= (…)
= (^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−}) ^{v}Ψ_{n−1}^{−}ϵ_{n} … ϵ_{1}∂_{1}^{+} … ∂_{n+1}^{+}
= (ϵ_{1})^{n}(∂_{1}^{+})^{n+1} for the same reason as above

Why does the proof of Theorem 7.4 work. The principle of the proof of Theorem 7.4 is the following observation (see in [2]) : let f_{1}, …, f_{n} be n operators such that (the product notation means the composition)

1. for any i, one has f_{i}f_{i} = f_{i} (the operators f_{i} are idempotent)

2. |i − j| ⩾ 2 implies f_{i}f_{j} = f_{j}f_{i}

3. f_{i}f_{i+1}f_{i} = f_{i+1}f_{i}f_{i+1} for any i

Then the operator F = f_{1}(f_{2}f_{1}) … (f_{n}f_{n−1} … f_{1}) satisfies f_{i}F = F for any i. This means that F enables to apply all f_{i} a maximal number of times. It turns out that the operators vψ_{i}^{±} and ^{h}ψ_{i}^{±} satisfy the above relations :

7.5. Proposition. The operators ^{v}ψ_{i}^{α} and ^{h}ψ_{j}^{β} are idempotent. Moreover for any i ⩾ 1 and any j ⩾ 1, with |i − j| ⩾ 2, the following equalities hold :

vψ_{i}^{α} ^{h}ψ_{j}^{β} = ^{h}ψ_{j}^{β} ^{v}ψ_{i}^{α} for α ∈ {−, +}  (21)
vψ_{i}^{α} ^{h}ψ_{i}^{α} = ^{h}ψ_{i}^{α} ^{v}ψ_{i}^{α} for α ∈ {−, +}  (22)
hψ_{i+1}^{α} ^{v}ψ_{i}^{α} = ^{v}ψ_{i}^{α} ^{h}ψ_{i+1}^{α} for α ∈ {−, +}  (23)
aψ_{i}^{α} ^{a}ψ_{i+1}^{α} ^{a}ψ_{i}^{α} = ^{a}ψ_{i+1}^{α} ^{a}ψ_{i}^{α} ^{a}ψ_{i+1}^{α} for a ∈ {v, h} and α ∈ {−, +}  (24)

Proof. Equalities 21 and 22 are obvious.

For the sequel, one can suppose α = −. In the cubical singular nerve of an ω-category, two elements A and B of the same dimension n are equal if and only if A(0_{n}) = B(0_{n}) and for 1 ⩽ k ⩽ n and α ∈ {−, +}, one has ∂_{k}^{α}A = ∂_{k}^{α}B.

Now we want to prove Equality 23. Since (^{v}ψ_{i}^{α}x)(0_{n}) = (^{h}ψ_{i}^{α}x)(0_{n}) = x(0_{n}), then hψ_{i+1}^{α} ^{v}ψ_{i}^{α}x = ^{v}ψ_{i}^{α} ^{h}ψ_{i+1}^{α}x for any x of dimension n (P_{n}) is equivalent to ∂_{k}^{β} ^{h}ψ_{i+1}^{α} ^{v}ψ_{i}^{α}x = ∂_{k}^{β} ^{v}ψ_{i}^{α} ^{h}ψ_{i+1}^{α}x for 1 ⩽ k ⩽ n and β ∈ {−, +}(E_{k,n}). Proposition 7.3 implies that P_{n−1} ⟹ E_{k,n} for k < i or k > i + 2. For k ∈ {i, i + 1, i + 2}, proving Equality E_{k,n} is equivalent to proving it for the case i = 1 and to replacing each index 1 by i, each index 2 in by i + 1 and each index 3 by i + 2. And in the case i = 1, the equality is a calculation in the free cubical ω-category generated by x. So we can suppose that x is of dimension as low as possible. In our case, this equality makes sense if x is 3-dimensional. Therefore it suffices to verify Equality 23 in dimension 3 for i = 1. And one has

hψ_{2}^{−} ^{v}ψ_{1}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

= ^{h}ψ_{2}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

= _{A B} ; C \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

= ^{v}ψ_{1}^{−} A B ; C \underrightarrow{G} D ; E F

= ^{v}ψ_{1}^{−} ^{h}ψ_{2}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

In the same way, to prove Equality 24, it suffices to prove it for i = 1 and in the 3-dimensional case. And one has

vψ_{1}^{−} ^{v}ψ_{2}^{−} ^{v}ψ_{1}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

= ^{v}ψ_{1}^{−} ^{v}ψ_{2}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

= ^{v}ψ_{1}^{−} A B ; C \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

= A B ; C \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

and

vψ_{2}^{−} ^{v}ψ_{1}^{−} ^{v}ψ_{2}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

= ^{v}ψ_{2}^{−} ^{v}ψ_{1}^{−} A B ; C \underrightarrow{G} D ; E F

= ^{v}ψ_{2}^{−} A B ; C \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

= A B ; C \underrightarrow{G} \begin{matrix}D \\ E \end{matrix} F

In the same way, one can verify that

hψ_{2}^{−} ^{h}ψ_{1}^{−} ^{h}ψ_{2}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

= ^{h}ψ_{1}^{−} ^{h}ψ_{2}^{−} ^{h}ψ_{1}^{−} \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

= A ; \begin{matrix}B \\ C \end{matrix} \underrightarrow{G} D ; E F

7.6. Theorem. For any n ⩾ 2, Φ_{n}^{−} is a composition of ^{v}ψ_{i}^{−}, ^{h}ψ_{i}^{−} and θ_{i}^{−}.

Proof. It is easy to see that Φ_{2}^{−} = ^{v}ψ_{1}^{−} ^{h}ψ_{1}^{−} = ^{h}ψ_{1}^{−} ^{v}ψ_{1}^{−}. Now we suppose that n ⩾ 3. Set Θ_{k}^{n−2} = θ_{k}^{−} … θ_{n−2}^{−}.

We are going to prove that

Φ_{n}^{−} = Θ_{n−2}^{n−2}Θ_{n−3}^{n−2} … Θ_{1}^{n−2}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−})

by verifying that the second member satisfies the characterization of Theorem 6.16. Let x ∈ ωCat(I^{n}, C)^{−}. Theorem 7.4 implies that for 1 ⩽ i ⩽ n, the dimension of

∂_{i}^{+}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−})x

is zero (or equivalently that it belongs to the image of ϵ_{1}^{n−1}). With Proposition 7.3, one gets

∂_{i}^{+}Θ_{n−2}^{n−2}Θ_{n−3}^{n−2} … Θ_{1}^{n−2}(^{v}Ψ_{1}^{−} ^{h}Ψ_{1}^{−}) … (^{v}Ψ_{n−1}^{−} ^{h}Ψ_{n−1}^{−})x ∈ Im(ϵ_{1}^{n−1})

for 1 ⩽ i ⩽ n. It remains to prove that for 1 ⩽ k ⩽ n − 2,

∂_{k}^{−}Θ_{n−2}^{n−2}Θ_{n−1}^{n−2} … Θ_{1}^{n−2}y ∈ Im(Γ_{n−2}^{−} … Γ_{k}^{−})

for any y ∈ ωCat(I^{n}, C)^{−}. One has

∂_{k}^{−}Θ_{n−2}^{n−2}Θ_{n−3}^{n−2} … Θ_{1}^{n−2}y
= (Θ_{n−3}^{n−3} … Θ_{k}^{n−3}) ∂_{k}^{−}Θ_{k}^{n−2}z with z = Θ_{k−1}^{n−2} … Θ_{1}^{n−2}y
= (Θ_{n−3}^{n−3} … Θ_{k}^{n−3}) Γ_{k}^{−}∂_{k}^{−}∂_{k}^{−}Θ_{k+1}^{n−2}z
= Γ_{n−2}^{−} (Θ_{n−4}^{n−4} … Θ_{k}^{n−4}) ∂_{k}^{−}Θ_{k}^{n−3}∂_{k}^{−}z
= (…)
= (Γ_{n−2}^{−} … Γ_{k+1}^{−}) ∂_{k}^{−}θ_{k}^{−}(∂_{k}^{−})^{n−2−k}z
= (Γ_{n−2}^{−} … Γ_{k}^{−}) (∂_{k}^{−})^{n−k}z

The operators ^{v}ψ_{i}^{±}, ^{h}ψ_{i}^{±} and θ_{i}^{−} for 1 ⩽ i ⩽ n − 1 and Φ_{n}^{−} induce natural transformations of set-valued functors from ωCat(I^{n}, −)^{−} to itself.

7.7. Conjecture. Let f be an ω-functor from I^{n} to itself such that f(0_{n}) = 0_{n} and such that the corresponding natural transformation from ωCat(I^{n}, −) to itself induces a natural transformation Φ^{−} from ωCat(I^{n}, −)^{−} to itself. Then Φ^{−} is a composition of vψ_{i}^{−}, ^{h}ψ_{i}^{−} and θ_{i}^{−} for 1 ⩽ i ⩽ n − 1.

7.8. Conjecture. Let Φ be a natural transformation from ωCat(I^{n}, −) to itself such that the corresponding functor (Φ)^{∗} from I^{n} to itself satisfies (Φ)^{∗}(0_{n}) = 0_{n}. Then Φ is a composition of ^{v}ψ_{i}^{±} and ^{h}ψ_{i}^{±} for 1 ⩽ i ⩽ n − 1.

By Yoneda, the operators ^{v}ψ_{i}^{±} and ^{h}ψ_{i}^{±} for 1 ⩽ i ⩽ n − 1 induce ω-functors from I^{n} to itself denoted by (^{v}ψ_{i}^{±})^{∗} and (^{h}ψ_{i}^{±})^{∗}. The dual conjecture is then

7.9. Conjecture. Let f be an ω-functor from I^{n} to itself such that f(0_{n}) = 0_{n}. Then f is a composition of (^{v}ψ_{i}^{±})^{∗} and (^{h}ψ_{i}^{±})^{∗}.

8. Comparison of x and Φ_{n}^{−}(x) in the reduced branching complex

This section is devoted to proving that for any x ∈ ωCat(I^{n}, C)^{−}, x and Φ_{n}^{−}(x) are Tequivalent.

8.1. Proposition. For any i ⩾ 1 and any n ⩾ 2, if x ∈ ωCat(I^{n}, C)^{−}, then ^{h}ψ_{i}^{−}(x) and x are T-equivalent.

Proof. First let us make the proof for i = 1 and n = 2. Let us consider the following ω-functor from I^{3} to C :

y_{1} = x \underrightarrow{x(00)} x

Then ∂^{−}y_{1} = ^{h}ψ_{1}^{−}(x) − x + t_{1} where t_{1} is a thin element. Therefore x and ^{h}ψ_{1}^{−}(x) are T-equivalent.

We claim that the above construction is sufficient to prove that x and ^{h}ψ_{1}^{−}(x) are Tequivalent for any x ∈ ωCat(I^{n}, C)^{−} and for any n ⩾ 2. The labeled 3-cube y_{1} is actually a certain thin 3-dimensional element of the cubical ω-category N^{□}(C) and it corresponds to the filling of a thin 2-shell. So

y_{1} = f_{1}(ϵ_{1}x, ϵ_{2}x, ϵ_{3}x, Γ_{1}^{−}x, Γ_{2}^{−}x, Γ_{1}^{+}x, Γ_{2}^{+}x)

where f_{1} is a function which only uses the operators +_{1}, +_{2}, and +_{3}. In this particular case, f_{1} could be of course calculated. But it will not be always possible in the sequel to make such a calculation : this is the reason why no explicit formula is used here. And one has ∂_{2}^{−}f_{1}(x) = x, ∂_{3}^{−}f_{1}(x) = ^{h}ψ_{1}^{−}(x) and all other 2-faces ∂_{i}^{α}f_{1}(x) are (necessarily) thin 2-faces. The equalities ∂_{2}^{−}f_{1}(x) = x and ∂_{3}^{−}f_{1}(x) = ^{h}ψ_{1}^{−}(x) do not depend on the dimension of x. Therefore for any x ∈ ωCat(I^{n}, C)^{−} and for any n ⩾ 2, one gets ∂^{−}y_{1} = ^{h}ψ_{1}^{−}(x) − x + t where t is a linear combination of thin elements.

Now we want to explain that the above construction is also sufficient to prove that x and ^{h}ψ_{i}^{−}(x) are T-equivalent for any i ⩾ 1 and any x ∈ ωCat(I^{n}, C)^{−} and for any n ⩾ 2. The equalities ∂_{2}^{−}f_{1}(x) = x and ∂_{3}^{−}f_{1}(x) = ^{h}ψ_{1}^{−}(x) do not depend on the absolute values 1, 2 3. But only on the relative values 1 = 3 − 2, 2 = 3 − 1 and 3 = 3 − 0. So let us introduce a labeled (n + 1)-cube y_{i} = f_{i}(x) by replacing in f_{1} any index 1 in by i, any index 2 by i + 1 and any index 3 by i + 2. Then one gets a thin (n + 1)-cube y_{i} = f_{i}(x) such that ∂_{i+1}^{−}f_{i}(x) = x and ∂_{i+2}^{−}f_{i}(x) = ^{h}ψ_{i}^{−}(x).

If the reader does not like this proof and prefers explicit calculations, it suffices to notice that y_{1} = ^{h}ψ_{1}^{−}Γ_{2}^{−}x by Proposition 7.2. Set y_{i} = ^{h}ψ_{i}^{−}Γ_{i+1}^{−}x. Then

∂^{−}(y_{i}) = ∑_{j<i}(−1)^{j+1} ^{h}ψ_{i−1}^{−}Γ_{i}^{−}∂_{j}^{−}x + (−1)^{i+1}(Γ_{i}^{−}∂_{i}^{−}x +_{i} ϵ_{i+1}∂_{i+1}^{+}x) +

(−1)^{i+2}(x − ^{h}ψ_{i}^{−}x) + ∑_{j>i+2}(−1)^{j+1} ^{h}ψ_{i}^{−}Γ_{i+1}^{−}∂_{j−1}^{−}x

and one completes the proof by an easy induction on the dimension of x(0_{n}).

8.2. Proposition. For any i ⩾ 1 and any n ⩾ 2, if x ∈ ωCat(I^{n}, C)^{−}, then ^{v}ψ_{i}^{−}(x) and x are T-equivalent.

Proof. It suffices to make the proof for i = 1 and n = 2. And to consider the following thin 3-cube

x \underrightarrow{x(00)} x

Notice that the above 3-cube is exactly ^{v}ψ_{2}^{−}Γ_{1}^{−}x by Proposition 7.2.

8.3. Proposition. For any i ⩾ 1 and any n ⩾ 3, if x ∈ ωCat(I^{n}, C)^{−}, then θ_{i}^{−}(x) and x are T-equivalent.

Proof. It suffices to make the proof for i = 1 and n = 3. Set

x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} \begin{matrix}D \\ E & F \end{matrix}

One has already seen that

θ_{1}^{−}x = \begin{aligned}A & B \\ & C \end{aligned} \underrightarrow{G} D ; ^{E F}

It suffices to construct a thin 4-cube y such that ∂_{3}^{−}y = x and ∂_{2}^{−}y = θ_{1}^{−}x. If the 4-cube is conventionally represented by Figure 5, the thin labeled 4-cube of Figure 6 with 00+0 ↦ (∂_{2}^{+}∂_{1}^{−}x +_{1} ∂_{2}^{+}∂_{2}^{+}x)(0) meets the requirement. The latter labeled 4-cube can be defined as the unique thin 4-cube ω(x) which fills the 3-shell defined by

∂_{1}^{−}ω(x) = Γ_{2}^{−}∂_{1}^{−}x ∂_{2}^{−}ω(x) = θ_{1}^{−}(x) ∂_{3}^{−}ω(x) = x

∂_{4}^{−}ω(x) = [ \begin{matrix}Γ_{2}^{−}∂_{3}^{−}x & ϵ_{2}∂_{2}^{+}x \\ Γ_{1}^{−}Γ_{1}^{+}∂_{1}^{−}∂_{2}^{+}x +_{2} ϵ_{1}Γ_{1}^{−}∂_{1}^{−}∂_{2}^{+}x & Γ_{1}^{−}∂_{1}^{−}x \end{matrix} ] [figure: 3; 1]

∂_{1}^{+}ω(x) = ^{v}ψ_{2}^{−}Γ_{1}^{−}∂_{1}^{+}x ∂_{2}^{+}ω(x) = Γ_{2}^{−}∂_{2}^{+}x ∂_{3}^{+}ω(x) = ϵ_{3}(Γ_{1}^{−}∂_{2}^{+}∂_{1}^{−}x +_{1} ϵ_{2}∂_{2}^{+}∂_{2}^{+}x) ∂_{4}^{+}ω(x) = ^{v}ψ_{2}^{+}Γ_{2}^{−}∂_{3}^{+}x

8.4. Corollary. For any n ⩾ 2, for any x ∈ ωCat(I^{n}, C)^{−}, x and Φ_{n}^{−}(x) are Tequivalent and Φ_{n}^{−} is the identity map on the reduced branching complex.

We have proved that for any x ∈ ωCat(I^{n}, C)^{−}, there exists t_{1} ∈ M_{n} and t_{2} ∈ M_{n+1} such that Φ_{n}^{−}(x) − x = t_{1} + ∂^{−}t_{2}. The proofs of this section use only calculations in the free cubical ω-category generated by x. This means that t_{1} and t_{2} can be formulated in terms of expressions in the same cubical ω-category. And so this means that t_{1} and t_{2} are linear combinations of expressions which use only x as variable and the operators ∂_{i}^{±}, Γ_{i}^{±}, ϵ_{i} and +_{i}. With Theorem 7.6 which allows to consider Φ_{n}^{−} like an operator defined in any cubical ω-category, one sees that Corollary 8.4 does make sense in an appropriate cubical setting. Moreover the terms t_{1} and t_{2} being elements of the free cubical ω-category generated by x, then t_{1} and t_{2} depend in a functorial way on x.

[figure: + + 00 ; +0 − 0 +00+ ; 0 − 0+ 00 + +; +0 − 0 ; 0 − −0 00 − + 0 + 0+ ; −00+; 000+; +00− +0 + 0 ; + − 00 ; 0 − −0 0 − 0+; +000; + + 00 ; 00 − − 0 + −0 0 + 0+ ; −0 − 0; 00−0; +0 + 0 ; 0 − 0− 0 − +0 00 + + ; − − 00; 0−00; 0 + 0− 0 + +0 ; − + 00 ; −0 − 0 −00+; 0+00; +00− ; 0 − 0− 00 + − 0 + +0 ; −0 + 0; 00+0; 00 − − 0 + 0− ; −00− −0 + 0 ; − − 00; −000; 000−]

Figure 5: 2 -categorical representation of the 4 -cube

[figure: D ; F; \begin{matrix}D \\ E & F \end{matrix}; F (00); D ; ^{E F}; D(00); \begin{aligned}A & B \\ & C \end{aligned}; G; \begin{aligned}A & B \\ & C \end{aligned}; G; \begin{aligned}B \\ & C \end{aligned}; B(00); \begin{aligned}A & B \\ & C \end{aligned}; 00+0; \begin{aligned}A & B \\ & C \end{aligned}; C(00); [ \begin{matrix}A & B \\ & C \end{matrix} ](00)]

Figure 6: A labeled 4-cube

9. Folding operations and composition maps

9.1. Theorem. Let x and y be two n-morphisms of C with n ⩾ 2.

1. if x∗_{n−1} y exists, then □_{n}^{−}(x∗_{n−1} y)−□_{n}^{−}(x)−□_{n}^{−}(y) is a boundary in the normalized chain complex of the branching simplicial nerve of C. Moreover, □_{n}^{−}(x ∗_{n−1} y) is Tequivalent to □_{n}^{−}(x) + □_{n}^{−}(y).

2. if 1 ⩽ p ⩽ n − 2, then □_{n}^{−}(x ∗_{p} y) is T-equivalent to □_{n}^{−}(x) + □_{n}^{−}(y).

Proof. Let us denote by P (h) the following property :

“for any n ⩾ 2 and with p = n−h ⩾ 1, for any n-morphisms x and y of any ω-category C such that x∗_{p}y exists, there exists a thin n-cube A_{p}^{n}(x, y) and a thin (n+1)-cube B_{p}^{n}(x, y) which lie in the cubical singular nerve of the free globular ω-category generated by x and y, and even in its branching nerve, such that

□_{n}^{−}(x ∗_{p} y) = □_{n}^{−}(x) + □_{n}^{−}(y) + A_{p}^{n}(x, y) + ∂^{−}B_{p}^{n}(x, y)

in the normalized branching complex (i.e. the equality holds modulo degenerate elements of the branching simplicial nerve) and such that for any (n + 1)-morphisms u and v, A_{p}^{n}(s_{n}u, s_{n}v) = A_{p}^{n}(t_{n}u, t_{n}v).”

Since

∂^{−} (□_{n}^{−}(x ∗_{n−1} y) − □_{n}^{−}(x) − □_{n}^{−}(y))
= □_{n−1}^{−}(s_{n−1}x) − □_{n−1}^{−}(t_{n−1}y)
−□_{n−1}^{−}(s_{n−1}x) + □_{n−1}^{−}(t_{n−1}x) − □_{n−1}^{−}(s_{n−1}y) + □_{n−1}^{−}(t_{n−1}y)
= □_{n−1}^{−}(t_{n−1}x) − □_{n−1}^{−}(s_{n−1}y) = 0

in the normalized chain complex of the branching simplicial nerve, then □_{n}^{−}(x ∗_{n−1} y) − □_{n}^{−}(x) − □_{n}^{−}(y) is a cycle in the branching homology of the free globular ω-category D generated by two n-morphisms such that t_{n−1}x = s_{n−1}y. The ω-category D is of length at most one and non-contracting. Therefore its branching nerve coincides with the simplicial nerve of ℙD, the latter being the globular ω-category freely generated by the composable pasting scheme whose total composition is X ∗_{n−2} Y where X and Y are two (n − 1)- dimensional cells. Therefore this simplicial nerve is contractible. Consequently there exists B_{n−1}^{n}(x, y) lying in the cubical singular nerve of D (and also in its branching nerve) such that

□_{n}^{−}(x ∗_{n−1} y) − □_{n}^{−}(x) − □_{n}^{−}(y) = ∂^{−}B_{n−1}^{n}(x, y).

The (n+1)-cube B_{n−1}^{n}(x, y) is necessarily thin because there is no morphism of dimension n + 1 in D. By setting A_{n−1}^{n}(x, y) = 0, we obtain P (1). We are going to prove P (h) by induction on h. Suppose P (h) proved for h ⩾ 1. Then

∂^{−}(□_{n}^{−}(x ∗_{n−h−1} y) − □_{n}^{−}(x) − □_{n}^{−}(y) − B_{n−h−1}^{n−1}(s_{n−1}x, s_{n−1}y)
+B_{n−h−1}^{n−1}(t_{n−1}x, t_{n−1}y))

= (□_{n}^{−}(s_{n−1}x ∗_{n−h−1} s_{n−1}y) − □_{n}^{−}(s_{n−1}x) − □_{n}^{−}(s_{n−1}y) − ∂^{−}B_{n−h−1}^{n−1}(s_{n−1}x, s_{n−1}y))

− (□_{n}^{−}(t_{n−1}x ∗_{n−h−1} t_{n−1}y) − □_{n}^{−}(t_{n−1}x) − □_{n}^{−}(t_{n−1}y) − ∂^{−}B_{n−h−1}^{n−1}(t_{n−1}x, t_{n−1}y))

= A_{n−h−1}^{n−1}(s_{n−1}x, s_{n−1}y) − A_{n−h−1}^{n−1}(t_{n−1}x, t_{n−1}y) by induction hypothesis
= 0 again by induction hypothesis

Therefore we can set A_{n−h−1}^{n}(x, y) = B_{n−h−1}^{n−1}(s_{n−1}x, s_{n−1}y) − B_{n−h−1}^{n−1}(t_{n−1}x, t_{n−1}y) and we have

A_{n−h−1}^{n}(s_{n}u, s_{n}v) − A_{n−h−1}^{n}(t_{n}u, t_{n}v)
= B_{n−h−1}^{n−1}(s_{n−1}s_{n}u, s_{n−1}s_{n}v) − B_{n−h−1}^{n−1}(t_{n−1}s_{n}u, t_{n−1}s_{n}v)
−B_{n−h−1}^{n−1}(s_{n−1}t_{n}u, s_{n−1}t_{n}v) + B_{n−h−1}^{n−1}(t_{n−1}t_{n}u, t_{n−1}t_{n}v)
= 0

because of the globular equations. So we get a thin n-cube A_{n−h−1}^{n}(x, y) such that

□_{n}^{−}(x ∗_{n−h−1} y) − □_{n}^{−}(x) − □_{n}^{−}(y) − A_{n−h−1}^{n}(x, y)

is a cycle in the normalized chain complex associated to the branching simplicial nerve of C. This cycle lies in the branching nerve of the free ω-category generated by two nmorphisms x and y such that t_{n−h−1}x = s_{n−h−1}y. This ω-category is of length at most one and non-contracting. Therefore its branching nerve is isomorphic to the simplicial nerve of the globular ω-category freely generated by the composable pasting scheme whose total composition is X ∗_{n−h−2} Y where X and Y are two (n − 1)-dimensional cells. Therefore it is contractible. Therefore there exists B_{n−h−1}^{n}(x, y) such that

□_{n}^{−}(x ∗_{n−h−1} y) − □_{n}^{−}(x) − □_{n}^{−}(y) − A_{n−h−1}^{n}(x, y) = ∂^{−}B_{n−h−1}^{n}(x, y).

The cube B_{n−h−1}^{n}(x, y) is necessarily thin because there is no morphism of dimension n+1 in the cubical sub-ω-category generated by x and y. And P (h + 1) is proved.

It turns out that the (n + 1)-cube B_{n−1}^{n}(x, y) can be explicitly calculated. One can easily verify that

B_{n−1}^{n}(x, y)_{h}^{−} = Γ_{n−1}^{−}Γ_{n−2}^{−} … Γ_{h}^{−}□_{h}^{−}d_{h}^{(−)^{h}}x

for 1 ⩽ h ⩽ n − 2 (observe that in this case, d_{h}^{(−)^{h}}x = d_{h}^{(−)^{h}}y),

B_{n−1}^{n}(x, y)_{n−1}^{−} = □_{n}^{−}y
B_{n−1}^{n}(x, y)_{n}^{−} = □_{n}^{−}(x ∗_{n−1} y)
B_{n−1}^{n}(x, y)_{n+1}^{−} = □_{n}^{−}x

and for all i between 1 and n + 1,

B_{n−1}^{n}(x, y)_{i}^{+} = □_{n}^{−}t_{0}x

is a solution. It suffices to prove that (B_{n−1}^{n}(x, y)_{i}^{±})_{1⩽i⩽n+1} is a thin n-shell.

9.2. Theorem. Let C be a non-contracting ω-category. Let x and y be two elements of ωCat(I^{n}, C) such that x+_{j} y exists for some j between 1 and n and such that dim(x(0_{n})) ⩾ 1, dim(y(0_{n})) ⩾ 1 and dim((x +_{j} y)(0_{n})) ⩾ 1. Then Φ_{n}^{−}(x +_{j} y) is T-equivalent to Φ_{n}^{−}(x) or Φ_{n}^{−}(y) or to Φ_{n}^{−}(x) + Φ_{n}^{−}(y). If x is itself in the branching complex, then Φ_{n}^{−}(x +_{j} y) is T-equivalent to x.

Remark. The hypotheses about the dimension of x(0_{n}), y(0_{n}) and (x +_{j} y)(0_{n}) are only to ensure that Φ_{n}^{−}(x), Φ_{n}^{−}(y) and Φ_{n}^{−}(x +_{j} y) are in the branching nerve. The hypothesis about the dimension of (x +_{j} y)(0_{n}) is necessary because we do not assume that 1-morphisms in non-contracting ω-categories are not invertible. In dimension 1, the case x(0) ∗_{0} y(0) = (x +_{1} y)(0) ∈ C_{0} may happen.

Proof. By definition, one has Φ_{n}^{−}(x +_{j} y) = □_{n}^{−}((x +_{j} y)(0_{n})). If C was equal to the globular sub-ω-category generated by

X = \{x(k_{1} … k_{n}), k_{1} … k_{n} ∈ \underline{cub}^{n}\} ∪ \{y(k_{1} … k_{n}), k_{1} … k_{n} ∈ \underline{cub}^{n}\}

then x +_{j} y still would exist in the cubical singular nerve. Therefore, (x +_{j} y)(0_{n}) can be written as an expression using only the composition laws ∗_{n} of C and the variables of X and moreover, the variables x(0_{n}) and y(0_{n}) can appear at most once. By Theorem 9.1, □_{n}^{−}((x +_{j} y)(0_{n})) is therefore T-equivalent to □_{n}^{−}(x(0_{n})), □_{n}^{−}(y(0_{n})) or □_{n}^{−}(x(0_{n})) + □_{n}^{−}(y(0_{n})).

Now suppose that x ∈ ωCat(I^{n}, C)^{−}. Let z = Γ_{j}^{−}x +_{j} ϵ_{j+1}y ∈ ωCat(I^{n+1}, C)^{−}. Then ∂_{j}^{−}z = x, ∂_{j+1}^{−}z = x+_{j} y and ∂_{j+1}^{+}z = y. Since z is a thin element, then all other faces ∂_{k}^{±}z are thin (this can be verified directly by easy calculations). Therefore ∂^{−}z is T-equivalent to ±(x +_{j} y − x). As illustration, let us notice that for j = 1 and n = 2, z is equal to

y ; x \underrightarrow{(x +_{1} y)(00)} \begin{matrix}y \\ x \end{matrix}

9.3. Theorem. Let x and y be two morphisms of a non-contracting ω-category C such that x ∗_{0} y exists such that x and x ∗_{0} y are of dimension lower than n and of dimension strictly greater than 0. Then □_{n}^{−}(x ∗_{0} y) is T-equivalent to □_{n}^{−}(x).

Proof. We need, only for this proof, the operator □_{n}^{+} introduced in [12]. One has

∂_{1}^{+}□_{n}^{−}(x) = ϵ_{1}^{n−1}□_{0}(t_{0}x)

and

∂_{1}^{−}□_{n}^{+}(y) = ϵ_{1}^{n−1}□_{0}(s_{0}y).

Therefore □_{n}^{−}(x) +_{1} □_{n}^{+}(y) exists and is T-equivalent to □_{n}^{−}(x) by Theorem 9.2. If we work in the ω-category generated by x and y, then □_{n}^{−}(x) +_{1} □_{n}^{+}(y) is a well-defined element of the branching simplicial nerve of D. And D is the free ω-category generated by a composable pasting scheme whose total composition is x ∗_{0} y. Since union means composition in such a ω-category, then necessarily (□_{n}^{−}(x) +_{1} □_{n}^{+}(y)) (0_{n}) = x ∗_{0} y. Since Φ_{n}^{−} is the identity map on the reduced branching complex, then □_{n}^{−}(x) +_{1} □_{n}^{+}(y) is Tequivalent to □_{n}^{−}(x ∗_{0} y).

The preceding formulae suggest another way of defining the reduced branching homology.

9.4. Proposition. Set

CF_{n}^{−}(C) = ℤC_{n}/{x ∗_{0} y = x, x ∗_{1} y = x + y, …, x ∗_{n−1} y = x + y mod ℤtr_{n−1}C}.

Then s_{n−1} − t_{n−1} from CF_{n}^{−}(C) to CF_{n−1}^{−}(C) for n ⩾ 2 and s_{0} from CF_{1}^{−}(C) to CF_{0}^{−}(C) induce a differential map ∂_{f}^{−} on the ℕ-graded group CF_{∗}^{−}(C) and the chain complex one gets is called the formal branching complex. The associated homology is denoted by HF_{n}^{−}(C) and is called the formal branching homology.

Proof. Obvious.

A relation like x ∗_{0} y = x mod ℤtr_{n−1}C means that if x is for example a p-morphism for p < n and y a n-morphism such that x ∗_{0} y exists, then in CF_{n}^{−}(C), x ∗_{0} y = 0.

9.5. Proposition. Let C be a non-contracting ω-category. The linear map □_{n}^{−} from ℤC_{n} to CR_{n}^{−}(C) induces a surjective morphism of chain complexes and therefore a morphism from HF_{∗}^{−}(C) to HR_{∗}^{−}(C).

Proof. One has in the reduced branching complex □_{n}^{−}(x∗_{0}y) = □_{n}^{−}(x) and □_{n}^{−}(x∗_{p}y−x− y) = 0 therefore □_{n}^{−} induces a linear map from CF_{n}^{−}(C) to CR_{n}^{−}(C). And □_{n−1}^{−}(∂_{f}^{−}(x)) = □_{n−1}^{−}(s_{n−1} − t_{n−1})(x) = ∂^{−}□_{n}^{−}(x). Since Φ_{n}^{−} is the identity map on CR_{n}^{−}(C), then CR_{n}^{−}(C) is generated by the □_{n}^{−}(x) where x runs over C_{n}. Therefore the induced morphism of chain complexes is surjective.

9.6. Question. When is the preceding map a quasi-isomorphism ?

The meaning of the results of this section is that one homology class in branching homology corresponds really to one branching area. Here are some simple examples to understand this fact.

Figure 1 represents a 1-dimensional branching area. This branching area corresponds to one element in the reduced branching homology, that is

□_{1}(u) − □_{1}(w) = □_{1}(u ∗_{0} v) − □_{1}(w) = □_{1}(u) − □_{1}(w ∗_{0} x) = □_{1}(u ∗_{0} v) − □_{1}(w ∗_{0} x)

in homology. In fact, it even corresponds to one cycle in the reduced branching complex. The reason why it is more appropriate to work anyway with cycles modulo boundaries, and not only with cycles modulo boundaries of thin elements is illustrated in Figure 7. The two cycles □_{1}(u) − □_{1}(v) and □_{1}(u) − □_{1}(w) are in the same homology class as soon as u is homotopic to v.

These observations can be generalized in higher dimension but they are more difficult to draw. If u is a n-morphism, then, by definition of □_{n}^{−}, □_{n}^{−}(u) is an homotopy between

u v w

Figure 7: Another 1-dimensional branching area

□_{n−1}^{−}s_{n−1}u and □_{n−1}^{−}t_{n−1}u in the branching simplicial nerve. Figure 8 is an analogue of Figure 1 in dimension 2. Figure 8 represents a 2-dimensional branching area. In the branching complex, it corresponds to the cycles (A) − (F) + (I), (A, B, C, D) − (E, F, G, H) + (I, J, K, L), (A) − (F, H) + (I, K), etc. In the reduced branching complex, there are even more possible cycles which correspond to this branching area. For example (A, D)−(E, F)+(I, J, K, L), (A)−(E, F)+(I, J), etc. In the branching homology, all these cycles are equivalent and therefore there is really one homology class which corresponds to one branching area. Or in other terms, the homology class does not depend on a cubification of the HDA.

10. Folding operations and differential map

Now we explore the relations between the folding operators and the differential map of the branching complex.

10.1. Proposition. Let x be an element of ωCat(I^{n}, C)^{−}. Then

□_{n−1}^{−}(s_{n−1} − t_{n−1})(x(0_{n})) = □_{n−1}^{−}(∂^{−}x)(0_{n−1}) = ∑_{p=1}^{n}(∂_{p}^{−}x)(0_{n−1})

in CR_{n−1}^{−}(C).

Proof. Since Φ_{n}^{−} induces the identity map on CR_{n}^{−}(C), then Φ_{n−1}^{−}∂^{−} = ∂^{−}Φ_{n}^{−} = ∂^{−}. Therefore

□_{n−1}^{−}(∂^{−}x)(0_{n−1}) = Φ_{n−1}^{−}∂^{−}x = ∂^{−}Φ_{n}^{−}x = ∂^{−}□_{n}^{−}(x(0_{n})) = □_{n−1}^{−}(s_{n−1} − t_{n−1})(x(0_{n})).

10.2. Proposition. In the reduced branching homology of a given ω-category C, one has

1. if x ∈ ωCat(I^{2}, C)^{−}, then □_{1}^{−}(s_{1}x(00)) = □_{1}^{−}x(−0) and □_{1}^{−}(t_{1}x(00)) = □_{1}^{−}x(0−)

F E  G  H  K  L I  J A B  C  D

Figure 8: A 2-dimensional branching area

2. if x ∈ ωCat(I^{3}, C)^{−}, then □_{2}^{−}(s_{2}x(000)) = □_{2}^{−}x(−00) + □_{2}^{−}x(−0−)

□_{2}^{−}(t_{2}x(000)) = □_{2}^{−}x(0 − 0).

Proof. One has

□_{1}^{−}(s_{1}x(00)) = □_{1}^{−}(s_{1}(x(−0) ∗_{0} x(0+))) = □_{1}^{−}x(−0)

and

□_{1}^{−}(t_{1}x(00)) = □_{1}^{−}(s_{1}(x(0−) ∗_{0} x(+0))) = □_{1}^{−}x(0−).

Now suppose that x ∈ ωCat(I^{3}, C)^{−}. Then □_{2}^{−}(s_{2}x(000)) = □_{2}^{−} ((x(−00) ∗_{0} x(0 + +)) ∗_{1} (x(−0−) ∗_{0} x(0 + 0)) ∗_{1} (x(00−) ∗_{0} x(+ + 0)))) = □_{2}^{−}(x(−00) ∗_{0} x(0 + +)) + □_{2}^{−}(x(−0−) ∗_{0} x(0 + 0)) + □_{2}^{−}(x(00−) ∗_{0} x(+ + 0))

So □_{2}^{−}(s_{2}x(000)) = □_{2}^{−}(x(−00)) + □_{2}^{−}(x(00−)). In the same way, one has

□_{2}^{−}(t_{2}x(000)) = □_{2}^{−} ((x(− − 0) ∗_{0} x(00+)) ∗_{1} (x(0 − 0) ∗_{0} x(+0+)) ∗_{1} (x(0 − −) ∗_{0} x(+00)))) = □_{2}^{−}(x(− − 0) ∗_{0} x(00+)) + □_{2}^{−}(x(0 − 0) ∗_{0} x(+0+)) + □_{2}^{−}(x(0 − −) ∗_{0} x(+00)) = □_{2}^{−}(x(0 − 0))

The preceding propositions can be in fact generalized as follows :

10.3. Theorem. Let x be an element of ωCat(I^{n}, C)^{−} with n ⩾ 2. Then in the reduced branching complex, one has

□_{n−1}^{−}(s_{n−1}x(0_{n})) = ∑_{1⩽2i+1⩽n} □_{n−1}^{−}((∂_{2i+1}^{−}x)(0_{n−1}))

□_{n−1}^{−}(t_{n−1}x(0_{n})) = ∑_{1⩽2i⩽n} □_{n−1}^{−}((∂_{2i}^{−}x)(0_{n−1}))

Proof. For all n, we have seen that Φ_{n}^{−} induces the identity map on the reduced branching complex. Therefore for all x ∈ ωCat(I^{n}, C)^{−}, Φ_{n−1}^{−}∂^{−}x = ∂^{−}Φ_{n}^{−}x. The latter equality can be translated into

∑_{1⩽2i+1⩽n} Φ_{n−1}^{−}(∂_{2i+1}^{−}x) − ∑_{1⩽2i⩽n} Φ_{n−1}^{−}(∂_{2i}^{−}x) = □_{n−1}^{−}s_{n−1}x(0_{n}) − □_{n−1}^{−}t_{n−1}x(0_{n}).

If the above equality was in ℤωCat(I^{n−1}, C)^{−}, the proof would be complete. Unfortunately, we are working in the reduced branching chain complex, and so there exists t_{1} ∈ M_{n−1}^{−} and t_{2} ∈ M_{n}^{−} such that, in ℤωCat(I^{n−1}, C)^{−}

∑_{1⩽2i+1⩽n} Φ_{n−1}^{−}(∂_{2i+1}^{−}x) − ∑_{1⩽2i⩽n} Φ_{n−1}^{−}(∂_{2i}^{−}x) = □_{n−1}^{−}s_{n−1}x(0_{n}) − □_{n−1}^{−}t_{n−1}x(0_{n}) + t_{1} + ∂^{−}t_{2}.

Set t_{2} = ∑_{i∈I} λ_{i}T_{i} where T_{i} are thin elements of ωCat(I^{n}, C)^{−}. Each T_{i} corresponds to a thin (n − 1)-cube in the free cubical ω-category generated by the n-cube x which will be denoted in the same way (see the last paragraph of Section 8). One can suppose that each T_{i}(0_{n}) is (n − 1)-dimensional. In the free cubical ω-category generated by x, either T_{i} is in the cubical ω-category generated by the ∂_{i}^{(−)^{i}}x for 1 ⩽ i ⩽ n (let us denote this fact by T_{i} ⩽ s_{n−1}x(0_{n})), or T_{i} is in the cubical ω-category generated by the ∂_{i}^{(−)^{i+1}}x for 1 ⩽ i ⩽ n (let us denote this fact by T_{i} ⩽ t_{n−1}x(0_{n})). Therefore one has

t_{2} = ∑_{i∈I,T_{i}⩽s_{n−1}x(0_{n})}λ_{i}T_{i} + ∑_{i∈I,T_{i}⩽t_{n−1}x(0_{n})}λ_{i}T_{i}.

and

∂^{−}t_{2} =

∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ s_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i} + ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ s_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} non-thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i}

+ ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ t_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i} + ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ t_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} non-thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i}

Because of the freeness of ℤωCat(I^{n−1}, C)^{−}, one gets

∑_{1⩽2i+1⩽n} Φ_{n−1}^{−}(∂_{2i+1}^{−}x) = □_{n−1}^{−}s_{n−1}x(0_{n}) + ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ s_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} non-thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i}

∑_{1⩽2i⩽n} Φ_{n−1}^{−}(∂_{2i}^{−}x) = □_{n−1}^{−}t_{n−1}x(0_{n}) − ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ t_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} non-thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i}

−t_{1} = ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ s_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i} + ∑ ; \begin{matrix}i ∈ I, T_{i} ⩽ t_{n−1}x(0_{n}) \\ 1 ⩽ j ⩽ n, ∂_{j}^{−}T_{i} thin \end{matrix} (−1)^{j+1}λ_{i}∂_{j}^{−}T_{i}

11. Some consequences for the reduced branching homology

The following result generalizes the invariance result of [12] for the branching homology theory.

11.1. Proposition. Let f and g be two non-contracting ω-functors from C to D satisfying the following conditions :

• for any 0-morphism x, f(x) = g(x)

• for any n-morphism x, f(x) and g(x) are two homotopic morphisms (and so of the same dimension).

Then for any n ⩾ 0, HR_{n}^{±}(f) = HR_{n}^{±}(g).

Proof. Consider the case of the reduced branching homology. Let x ∈ CR_{n}^{−}(C). If

dim(f(x(0_{n}))) = dim(g(x(0_{n})) < n,

then f(x) and g(x) are two thin elements of ωCat(I^{n}, C)^{−}. Therefore f(x) = g(x) in the reduced branching complex of D. Now suppose that

dim(f(x(0_{n}))) = dim(g(x(0_{n}))) = n.

By hypothesis, there exists z ∈ D_{n+1} such that f(x(0_{n})) − g(x(0_{n})) = (s_{n} − t_{n})(z). Therefore in the reduced branching complex, one has f(x) − g(x) = □_{n}^{−}((s_{n} − t_{n})(z)) = ∂^{−}□_{n+1}^{−}(z). So f(x) − g(x) is a boundary.

We end up this section with another invariance result for the reduced branching homology and with some results related to Question 9.6.

11.2. Theorem. Let C and D be two ω-categories. Let f and g be two non 1-contracting ω-functors from C to D which coincide for the 0-morphisms and such that for any n ⩾ 1, there exists a linear map h_{n} from CF_{n}^{−}(C) to CF_{n+1}^{−}(D) such that for any x ∈ CF_{n}^{−}(C), h_{n−1}(s_{n−1} −t_{n−1})+(s_{n} −t_{n})h_{n}(x) = f(x)−g(x). Then HR_{n}^{−}(f) = HR_{n}^{−}(g) for any n ⩾ 0.

Proof. Set h_{n}^{−}x = □_{n+1}^{−}h_{n}(x(0_{n})) for any x ∈ ωCat(I^{n}, C)^{−}. It is clear that h_{n}^{−}(M_{n}^{−}(C)) = {0} in CR_{n+1}^{−}(D). Now suppose that x = ∂^{−}y for some y ∈ M_{n+1}^{−}(C).

We already mentioned that I^{n}[−_{n}, +_{n}] is the free ω-category generated by a composable pasting scheme in the proof of Corollary 5.9. It turns out that s_{n}(R(0_{n+1})) and t_{n}(R(0_{n+1})) belong to I^{n}[−_{n}, +_{n}] and it is possible thereby to use the explicit combinatorial description of [18].

Set I = {1, 2, …, n} equipped with the total order 1 < 2 < … < n. Let C(I, k) (or C(n, k)) be the set of all subsets of I of cardinality k. Let P an arbitrary subset of C(I, k). There is a lexicographical order on C(I, k) usually defined as follows : if J = (j_{1}, …, j_{k}) with j_{1} < … < j_{k} and J′ = (j′_{1}, …, j′_{k}) with j′_{1} < … < j′_{k}, then J ⩽ J′ means that either j_{1} < j′_{1}, or j_{1} = j′_{1} and j_{2} < j′_{2}, etc. If K ∈ C(I, k + 1), a K-packet is a set like P (K) = {J, J ∈ C(I, k), J ⊂ K}. If K = (i_{1}, …, i_{k+1}) with i_{j} < i_{j+1}, then P (K) consists of the sets Kâ = K − {i_{a}} for a = 1, …, k + 1. We have lexicographically

K{ \atop k+1} < Kk̂ < … < Kˆ1.

A total order σ on C(I, k) will be denoted by σ = J_{1}J_{2} … J_{N} for N = ( \begin{matrix}n \\ k \end{matrix} ), that

is J_{i}σJ_{j} for i < j. A total order is called admissible by Manin and Schechtman if on each packet it induces either a lexicographical order or the inverse lexicographical order. The set of admissible orders of C(I, k) is denoted by A(I, k) (or A(n, k)). Two total orders σ and σ′ of A(I, k) are called elementary equivalent if they differ by an interchange of two neighbours which do not belong to a common packet. The quotient of A(I, k) by this equivalence relation is denoted by B(I, k) (or B(n, k)). Suppose that for some K ∈ C(I, k + 1), the members of the packet P (K) form a chain with respect to an admissible order σ of A(I, k), i.e. any element of C(I, k) lying between two elements of P (K) belongs to P (K). Define p_{K}(σ) the admissible order in which this chain is reversed while all the rest elements conserve their positions. Then p_{K}(σ) is still an admissible order and p_{K} passes to the quotient B(I, k). The lemma on page 300 claims that A(n, n − 1) = B(n, n − 1) = {Kn̂ … Kˆ1, Kˆ1 … Kn̂} where K = (1, …, n). And the poset B(n, n − 2) is described by the following picture :

[figure: r_{min}; p_{Kn̂}; p_{Kˆ1}; p_{K{ \atop n−1}}; p_{Kˆ2}; ⋮; p_{Kˆ2}; ⋮; p_{K{ \atop n−1}}; p_{Kˆ1}; p_{Kn̂}; r_{max}]

It turns out that in the picture B(n, n − 2), the vertices are exactly the (n − 2)- morphisms of I^{n}[−_{n}, +_{n}] and the arrows are exactly the (n − 1)-morphisms of I^{n}[−_{n}, +_{n}]. This explicit description shows therefore that s_{n}(R(0_{n+1})) is equal to a composition X_{1} ∗_{n−1} … ∗_{n−1} X_{n+1} where the only morphism of dimension n contained in X_{j} is R(δ_{j}^{(−)^{j}}(0_{n})). And the same description shows that t_{n}(R(0_{n+1})) is equal to a composition Y_{n+1} ∗_{n−1} … ∗_{n−1} Y_{1} where the only morphism of dimension n contained in Y_{j} is R(δ_{j}^{(−)^{j+1}}(0_{n})). And one has

s_{n}(y(0_{n+1})) = y(s_{n}(0_{n+1})) = y(X_{1}) ∗_{n−1} … ∗_{n−1} y(X_{n+1})
t_{n}(y(0_{n+1})) = y(t_{n}(0_{n+1})) = y(Y_{n+1}) ∗_{n−1} … ∗_{n−1} y(Y_{1})

Since y is thin, s_{n}(y(0_{n+1})) = t_{n}(y(0_{n+1})). Since h_{n} is a map from CF_{n}^{−}(C) to CF_{n+1}^{−}(D), then

∑_{p=1}^{n+1} h_{n}(y(X_{p})) = ∑_{p=1}^{n+1} h_{n}(y(Y_{p}))

in CF_{n+1}^{−}(D).

Since I^{n+1} is the free ω-category generated by the pasting scheme \underline{cub}^{n+1}, then for any p between 1 and n + 1, X_{p} is a composition of R(δ_{p}^{(−)^{p}}(0_{n})) with other R(k_{1} … k_{n+1}) of dimension strictly lower than n. Suppose that p is odd. There exists X_{p}^{(1)} and X_{p}^{(1)}′ such that X_{p} = X_{p}^{(1)} ∗_{i_{1}} X_{p}^{(1)}′ for some 0 ⩽ i_{p} ⩽ n − 2. If i_{p} > 0, then only one of the X_{p}^{(1)} or X_{p}^{(1)}′ is of dimension n therefore y(X_{p}) = y(X_{p}^{(1)}) or y(X_{p}) = y(X_{p}^{(1)}′). If i_{p} = 0, then since s_{0}X_{p} = s_{0}X_{p}^{(1)} = s_{0}R(δ_{p}^{(−)^{p}}(0_{n})) then in this case X_{p}^{(1)} is n-dimensional and X′_{p}^{(1)} is of dimension strictly lower than n. Therefore in this case h_{n}(y(X_{p})) = h_{n}(y(X_{p}^{(1)})). By repeating as many times as necessary the process, the number of cells R(k_{1} … k_{n}) included in y(X_{p}) decreases. And we obtain

h_{n}(y(X_{p})) = h_{n}(y(δ_{p}^{(−)^{p}}(0_{n}))) = h_{n}((∂_{p}^{−}y)(0_{n})).

Now suppose that p is even. Since R(−_{n}) = s_{0}(X_{p}) ≠ s_{0}R(δ_{p}^{(−)^{p}}(0_{n})), then necessarily at one step of the process, we have i_{h} = 0. Take the last h such that i_{h} = 0. Then h_{n}(y(X_{p})) = h_{n}(y(X_{p}^{(h)})) and X_{p}^{(h)} = X_{p}^{(h+1)} ∗_{0} X_{p}^{(h+1)}′. Since s_{0}X_{p}^{(h)} = s_{0}X_{p}^{(h+1)} ≠ s_{0}R(δ_{p}^{(−)^{p}}(0_{n})), then for this h, X_{p}^{(h+1)} is of dimension strictly lower than n and X_{p}^{(h+1)}′ is of dimension n. Therefore h_{n}(y(X_{p})) = 0.

In the same way, h_{n}(y(Y_{p})) = 0 if p is odd and

h_{n}(y(Y_{p})) = h_{n}(y(δ_{p}^{(−)^{p+1}}(0_{n}))) = h_{n}((∂_{p}^{−}y)(0_{n}))

if p is even. So

h_{n}^{−}(x) = □_{n+1}^{−}h_{n}(x(0_{n})) = ∑_{p=1}^{n+1}(−1)^{p+1}□_{n+1}^{−}h_{n}((∂_{p}^{−}y)(0_{n})) = 0

in CR_{n+1}^{−}(D) by Theorem 9.1. Therefore h_{n}^{−} induces a linear map from CR_{n}^{−}(C) to CR_{n+1}^{−}(D) still denoted by h_{n}^{−}. Take x ∈ ωCat(I^{n}, C)^{−}. Then in CR_{n}^{−}(D), one has

∂^{−}h_{n}^{−}(x) + h_{n−1}^{−}∂^{−}(x)
= ∂^{−}□_{n+1}^{−}h_{n}(x(0_{n})) + h_{n−1}^{−}∂^{−}Φ_{n}^{−}(x) since Φ_{n}^{−} is the identity map
= □_{n}^{−}(s_{n} − t_{n})h_{n}(x(0_{n})) + h_{n−1}^{−}∂^{−}□_{n}^{−}x(0_{n}) by definition of Φ_{n}^{−}
= □_{n}^{−}(s_{n} − t_{n})h_{n}(x(0_{n})) + h_{n−1}^{−}□_{n−1}^{−}(s_{n−1}x(0_{n}) − t_{n−1}x(0_{n}))

= □_{n}^{−}(s_{n} − t_{n})h_{n}(x(0_{n})) + □_{n}^{−}h_{n−1}(s_{n−1}x(0_{n}) − t_{n−1}x(0_{n})) by definition of h_{n}^{−}

= □_{n}^{−}(f(x)(0_{n}) − g(x)(0_{n})) by hypothesis on h_{∗}
= Φ_{n}^{−}(f(x) − g(x)) by definition of Φ_{n}^{−}
= f(x) − g(x) since Φ_{n}^{−} is the identity map

The proof of Theorem 11.2 provides another way of proving Theorem 10.3 and also establishes that Theorem 10.3 is still true for the formal branching homology.

11.3. Proposition. Let p ⩾ 1 and let 2_{p} be the ω-category generated by a p-morphism A. Then HF_{n}^{−}(2_{p}) = HR_{n}^{−}(2_{p}) = 0 for n > 0 and HF_{0}^{−}(2_{p}) = HR_{0}^{−}(2_{p}) = ℤ.

Proof. The assertions concerning the formal branching homology are obvious. Since the negative folding operator induces the identity on the reduced branching complex, then CR_{n}^{−}(2_{p}) is equal to 0 for n > p and is generated by □_{n}^{−}(s_{n}A) and □_{n}^{−}(t_{n}A) for 0 ⩽ n ⩽ p. The point is to prove that there is no relations between □_{n}^{−}(s_{n}A) and □_{n}^{−}(t_{n}A) for 1 ⩽ n < p, that is CR_{n}^{−}(2_{p}) = ℤ□_{n}^{−}(s_{n}A) ⊕ ℤ□_{n}^{−}(t_{n}A) = CF_{n}^{−}(2_{p}). Suppose that there exists a linear combination of thin n-cubes t_{1} and a linear combination of thin (n + 1)-cubes t_{2} such that for some integers λ and μ,

λ□_{n}^{−}(s_{n}A) + μ□_{n}^{−}(t_{n}A) = t_{1} + ∂^{−}t_{2}

in C_{n}^{−}(2_{p}). Then s_{n}t_{2}(0_{n+1}) = t_{n}t_{2}(0_{n+1}) and so ∂^{−}t_{2} is necessarily a linear combination of thin n-cube therefore λ = μ = 0.

Another possible proof of this proposition is to use Theorem 11.2 and to use the homotopy equivalence of [12] Proposition 8.5 between 2_{p} and 2_{1}.

11.4. Proposition. Let p ⩾ 1 and let G_{p}⟨A, B⟩ be the ω-category generated by two nonhomotopic p-morphisms A and B. Then HF_{n}^{−}(G_{p}⟨A, B⟩) = HR_{n}^{−}(G_{p}⟨A, B⟩) = 0 for 0 < n < p and HF_{0}^{−}(G_{p}⟨A, B⟩) = HR_{0}^{−}(G_{p}⟨A, B⟩) = ℤ = HF_{p}^{−}(G_{p}⟨A, B⟩) = HR_{p}^{−}(G_{p}⟨A, B⟩).

Proof. Analogous to the previous proof.

11.5. Proposition. Let n ⩾ 0. HF_{0}^{−}(I^{n}) = ℤ and for p > 0, HF_{p}^{−}(I^{n}) = 0.

Proof. We know that

CF_{p}^{−}(I^{n}) = ⨁_{R(k_{1}...k_{n}) of dimension p} ℤ□_{p}^{−}(R(k_{1} … k_{n})).

And the differential maps is also completely known. In the formal branching complex, one has

∂^{−}□_{p}^{−}(R(k_{1} … k_{n})) = ∑_{1⩽j⩽p}(−1)^{j+1}□_{p}^{−}(R(k_{1} … [−]_{n_{j}} … k_{n}))

where k_{n_{1}}, …, k_{n_{p}} are the 0’s appearing in the word k_{1} … k_{n} with n_{1} < … < n_{p}. It follows that this chain complex can be split depending on the position and the number of the +’s, and that these positions and numbers are not modified by the differential maps. If the number of the + signs is N, we are reduced to calculating the simplicial homology of the (n − N)-simplex which is known to vanish in dimension strictly greater than 0.

As for the calculation of HR_{∗}^{−}(I^{n}), the point is to prove as above for 2_{p} and G_{p}⟨A, B⟩ that there is no additional relations between the □_{p}^{−}(R(k_{1} … k_{n})) in the reduced branching complex. Unfortunately, for a thin (n + 1)-cube t_{2} of the branching nerve of I^{n}, ∂^{−}t_{2} is not necessarily a linear combination of thin n-cube. For example if a and b are two 1morphisms of I^{n} such that a∗_{0}b exists, then let t_{2} the thin 2-cube such that ∂_{1}^{−}t_{2} = □_{1}(a∗_{0} b), ∂_{2}^{+}t_{2} = □_{1}(t_{0}b), ∂_{2}^{−}t_{2} = □_{1}(a) and ∂_{1}^{+}t_{2} = □_{1}(b). Then ∂^{−}t_{2} = □_{1}(a ∗_{0} b) − □_{1}(a).

12. Acknowledgments

I would like to thank the London Mathematical Society for supporting the workshop “Geometric methods in Computer Science” held in Bangor (UK) on September 8-10th 1999 and the participants for their interest in this work.

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THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

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Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at http://www.tac.mta.ca/tac/. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: walters@fis.unico.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca