T-homotopy and refinement of observation (II) : Adding new T-homotopy equivalences

This paper is the second part of a series of papers about a new notion of T-homotopy of flows. It is proved that the old definition of T-homotopy equivalence does not allow the identification of the directed segment with the 3-dimensional cube. This contradicts a paradigm of dihomotopy theory. A new definition of T-homotopy equivalence is proposed, following the intuition of refinement of observation. And it is proved that up to weak S-homotopy, a old T-homotopy equivalence is a new T-homotopy equivalence. The left-properness of the weak S-homotopy model category of flows is also established in this second part. The latter fact is used several times in the next papers of this series.

The first part [Gau05c] of this series was an expository paper about the geometric intuition underlying the notion of T-homotopy.The purpose of this second paper is to prove that the class of old T-homotopy equivalences introduced in [GG03] and in [Gau05a] is actually not big enough.Indeed, the only kind of old T-homotopy equivalence consists of the deformations which locally act like in Figure 1.So it becomes impossible with this old definition to identify the directed segment of Figure 1 with the full 3-cube of Figure 2   state and the final state of the 3-cube since every point of the 3-cube is related to three distinct edges.This contradicts the fact that concurrent execution paths cannot be distinguished by observation.The end of the paper proposes a new definition of T-homotopy equivalence following the paradigm of invariance by refinement of observation.It will be checked that the preceding drawback is then overcome.
This second part gives only a motivation for the new definition of T-homotopy.Further developments and applications are given in [Gau06b], [Gau06c] and [Gau06a].The leftproperness of the model category structure of [Gau03] is also established in this paper.The latter result is used several times in the next papers of this series (e.g., [Gau06b] Theorem 11.2, [Gau06c] Theorem 9.2).
Section 4 collects some facts about globular complexes and their relationship with the category of flows.Indeed, it is not known how to establish the limitations of the old form of T-homotopy equivalence without using globular complexes together with a compactness argument.Section 5 recalls the notion of old T-homotopy equivalence of flows which is a kind of morphism between flows coming from globular complexes (the class of flows cell(Flow)).Section 6 presents elementary facts about relative I gl + -cell complexes which will be used later in the paper.Section 7 proves that the model category of flows is left proper.This technical fact is used in the proof of the main theorem of the paper, and it was not established in [Gau03].Section 8 proves the first main theorem of the paper: Finally Section 9 proposes a new definition of T-homotopy equivalence and the second main theorem of the paper is proved: Theorem.(Theorem 9.3) Every T-homotopy in the old sense is the composite of an Shomotopy equivalence with a T-homotopy equivalence in the new sense 1 .

Prerequisites and notations
The initial object (resp.the terminal object) of a category C, if it exists, is denoted by ∅ (resp.1).
Let C be a cocomplete category.If K is a set of morphisms of C, then the class of morphisms of C that satisfy the RLP (right lifting property) with respect to any morphism of K is denoted by inj(K) and the class of morphisms of C that are transfinite compositions of pushouts of elements of K is denoted by cell(K).Denote by cof (K) the class of morphisms of C that satisfy the LLP (left lifting property) with respect to the morphisms of inj(K).It is a purely categorical fact that cell(K) ⊂ cof (K).Moreover, every morphism of cof (K) is a retract of a morphism of cell(K) as soon as the domains of K are small relative to cell(K) ([Hov99] Corollary 2.1.15).An element of cell(K) is called a relative K-cell complex.If X is an object of C, and if the canonical morphism ∅ −→ X is a relative K-cell complex, then the object X is called a K-cell complex.
Let C be a cocomplete category with a distinguished set of morphisms I. Then let cell(C, I) be the full subcategory of C consisting of the object X of C such that the canonical morphism ∅ −→ X is an object of cell(I).In other terms, cell(C, It is obviously impossible to read this paper without a strong familiarity with model categories.Possible references for model categories are [Hov99], [Hir03] and [DS95].The original reference is [Qui67] but Quillen's axiomatization is not used in this paper.The axiomatization from Hovey's book is preferred.If M is a cofibrantly generated model category with set of generating cofibrations I, let cell(M) := cell(M, I): this is the full subcategory of cell complexes of the model category M. A cofibrantly generated model structure M comes with a cofibrant replacement functor Q : M −→ cell(M).In all usual model categories which are cellular ([Hir03] Definition 12.1.1),all the cofibrations are monomorphisms.Then for every monomorphism f of such a model category M, the morphism Q(f ) is a cofibration, and even an inclusion of subcomplexes ([Hir03] Definition 10.6.7) because the cofibrant replacement functor Q is obtained by the small object argument, starting from the identity of the initial object.This is still true in the model category of flows reminded in Section 3 since the class of cofibrations which are monomorphisms is closed under pushout and transfinite composition.
A partially ordered set (P, ) (or poset) is a set equipped with a reflexive antisymmetric and transitive binary relation .A poset is locally finite if for any (x, y) ∈ P × P , the set [x, y] = {z ∈ P, x z y} is finite.A poset (P, ) is bounded if there exist 0 ∈ P and 1 ∈ P such that P = [ 0, 1] and such that 0 = 1.For a bounded poset P , let 0 = min P (the bottom element) and 1 = max P (the top element).In a poset P , the interval ]α, −] (the sub-poset of elements of P strictly bigger than α) can also be denoted by P >α .
1 Since a T-homotopy in the old sense is a T-homotopy in the new sense only up to S-homotopy, the terminology "generalized T-homotopy" used in Section 9 may not be the best one.However, this terminology is used in the other papers of this series so we keep it to avoid any confusion.
A poset P , and in particular an ordinal, can be viewed as a small category denoted in the same way: the objects are the elements of P and there exists a morphism from x to y if and only if x y.If λ is an ordinal, a λ-sequence in a cocomplete category C is a colimit-preserving functor X from λ to C. We denote by X λ the colimit lim − → X and the morphism X 0 −→ X λ is called the transfinite composition of the morphisms X µ −→ X µ+1 .
A model category is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence.The model categories Top and Flow (see below) are both left proper (cf.Theorem 7.4 for Flow).
In this paper, the notation / / means cofibration, the notation / / / / means fibration, the notation ≃ means weak equivalence, and the notation ∼ = means isomorphism.

Reminder about the category of flows
The category Top of compactly generated topological spaces (i.e. of weak Hausdorff k-spaces) is complete, cocomplete and cartesian closed (more details for this kind of topological spaces in [Bro88,May99], the appendix of [Lew78] and also the preliminaries of [Gau03]).For the sequel, any topological space will be supposed to be compactly generated.A compact space is always Hausdorff.
The category Top is equipped with the unique model structure having the weak homotopy equivalences as weak equivalences and having the Serre fibrations2 as fibrations.
The time flow of a higher dimensional automaton is encoded in an object called a flow [Gau03].A flow X contains a set X 0 called the 0-skeleton whose elements correspond to the states (or constant execution paths) of the higher dimensional automaton.For each pair of states (α, β) ∈ X 0 × X 0 , there is a topological space P α,β X whose elements correspond to the (non-constant) execution paths of the higher dimensional automaton beginning at α and ending at β.For x ∈ P α,β X, let α = s(x) and β = t(x).For each triple (α, β, γ) ∈ X 0 × X 0 × X 0 , there exists a continuous map * : P α,β X × P β,γ X −→ P α,γ X called the composition law which is supposed to be associative in an obvious sense.The topological space PX = (α,β)∈X 0 ×X 0 P α,β X is called the path space of X.The category of flows is denoted by Flow.A point α of X 0 such that there are no non-constant execution paths ending at α (resp.starting from α) is called an initial state (resp.a final state).A morphism of flows f from X to Y consists of a set map f 0 : X 0 −→ Y 0 and a continuous map Pf : PX −→ PY preserving the structure.A flow is therefore "almost" a small category enriched in Top.A flow X is loopless if for every α ∈ X 0 , the space P α,α X is empty.
Here are four fundamental examples of flows: (1) Let S be a set.The flow associated with S, still denoted by S, has S as set of states and the empty space as path space.This construction induces a functor Set → Flow from the category of sets to that of flows.The flow associated with a set is loopless.(2) Let (P, ) be a poset.The flow associated with (P, ), and still denoted by P is defined as follows: the set of states of P is the underlying set of P ; the space of morphisms from α to β is empty if α β and equals to {(α, β)} if α < β • The weak equivalences are the weak S-homotopy equivalences, i.e. the morphisms of flows f : X −→ Y such that f 0 : X 0 −→ Y 0 is a bijection and such that Pf : PX −→ PY is a weak homotopy equivalence.• The fibrations are the morphisms of flows f : X −→ Y such that Pf : PX −→ PY is a Serre fibration.This model structure is cofibrantly generated.The set of generating cofibrations is the set where D n is the n-dimensional disk and S n−1 the (n − 1)-dimensional sphere.The set of generating trivial cofibrations is If X is an object of cell(Flow), then a presentation of the morphism ∅ −→ X as a transfinite composition of pushouts of morphisms of I gl + is called a globular decomposition of X.

Globular complex
The reference is [Gau05a].A globular complex is a topological space together with a structure describing the sequential process of attaching globular cells.A general globular complex may require an arbitrary long transfinite construction.We restrict our attention in this paper to globular complexes whose globular cells are morphisms of the form Glob top (S n−1 ) −→ Glob top (D n ) (cf.Definition 4.2).Definition 4.1.A multipointed topological space (X, X 0 ) is a pair of topological spaces such that X 0 is a discrete subspace of X.A morphism of multipointed topological spaces f The space X is called the underlying topological space of (X, X 0 ).
The category of multipointed spaces is cocomplete.Definition 4.2.Let Z be a topological space.The globe of Z, which is denoted by Glob top (Z), is the multipointed space where the topological space A relative globular precomplex is a relative I gl,top -cell complex in the category of multipointed topological spaces.
Definition 4.5.A globular precomplex is a λ-sequence of multipointed topological spaces X : λ −→ Top m such that X is a relative globular precomplex and such that X 0 = (X 0 , X 0 ) with X 0 a discrete space.This λ-sequence is characterized by a presentation ordinal λ, and for any β < λ, an integer n β 0 and an attaching map Let X be a globular precomplex.The 0-skeleton of lim Notation 4.7.If X is a globular precomplex, then the underlying topological space of the multipointed space lim − → X is denoted by |X| and the 0-skeleton of the multipointed space lim − → X is denoted by X 0 .
Definition 4.8.Let X be a globular precomplex.The space |X| is called the underlying topological space of X.The set X 0 is called the 0-skeleton of X.
Definition 4.9.Let X be a globular precomplex.A morphism of globular precomplexes γ : (3) for 0 i < n, there exists z i γ ∈ D n β i \S n β i −1 and a strictly increasing continuous map In other terms, one has the commutative diagram of sets Definition 4.11.A globular complex (resp.a relative globular complex) X is a globular precomplex (resp.a relative globular precomplex) such that the attaching maps φ β are nondecreasing.A morphism of globular complexes is a morphism of globular precomplexes which is non-decreasing.The category of globular complexes together with the morphisms of globular complexes as defined above is denoted by glTop.The set glTop(X, Y ) of morphisms of globular complexes from X to Y equipped with the Kelleyfication of the compact-open topology is denoted by glTOP(X, Y ).
Definition 4.12.Let X be a globular complex.A point α of X 0 such that there are no non-constant execution paths ending to α (resp.starting from α) is called initial state (resp.final state).More generally, a point of X 0 will be sometime called a state as well.
Theorem 4.13.([Gau05a] Theorem III.3.1)There exists a unique functor cat : glTop −→ Flow such that (1) if X = X 0 is a discrete globular complex, then cat(X) is the achronal flow X 0 ("achronal" meaning with an empty path space), (2) if Z = S n−1 or Z = D n for some integer n 0, then cat(Glob top (Z)) = Glob(Z), (3) for any globular complex X with globular decomposition (n β , φ β ) β<λ , for any limit ordinal β λ, the canonical morphism of flows is an isomorphism of flows, (4) for any globular complex X with globular decomposition (n β , φ β ) β<λ , for any β < λ, one has the pushout of flows The following theorem is important for the sequel: Theorem 4.14.The functor cat induces a functor, still denoted by cat from glTop to cell(Flow) ⊂ Flow since its image is contained in cell(Flow).For any flow X of cell(Flow), there exists a globular complex Y such that cat(U ) = X, which is constructed by using the globular decomposition of X.
Proof.The construction of U is made in the proof of [Gau05a] Theorem V.4.1.

T-homotopy equivalence
The old notion of T-homotopy equivalence for globular complexes was given in [GG03].A notion of T-homotopy equivalence of flows was given in [Gau05a] and it was proved in the same paper that these two notions are equivalent.
We first recall the definition of the branching and merging space functors, and then the definition of a T-homotopy equivalence of flows, exactly as given in [Gau05a] (Definition 5.7 below), and finally a characterization of T-homotopy of flows using globular complexes (Theorem 5.8 below).
Roughly speaking, the branching space of a flow is the space of germs of non-constant execution paths beginning in the same way.
Proposition 5.1.([Gau05b] Proposition 3.1) Let X be a flow.There exists a topological space P − X unique up to homeomorphism and a continuous map h − : PX −→ P − X satisfying the following universal property: (1) For any x and y in PX such that t(x) = s(y), the equality h − (x) = h − (x * y) holds.
(2) Let φ : PX −→ Y be a continuous map such that for any x and y of PX such that t(x) = s(y), the equality φ(x) = φ(x * y) holds.Then there exists a unique continuous map φ : Moreover, one has the homeomorphism The mapping X → P − X yields a functor P − from Flow to Top.Definition 5.2.Let X be a flow.The topological space P − X is called the branching space of the flow X.The functor P − is called the branching space functor.
Proposition 5.3.([Gau05b] Proposition A.1) Let X be a flow.There exists a topological space P + X unique up to homeomorphism and a continuous map h + : PX −→ P + X satisfying the following universal property: (1) For any x and y in PX such that t(x) = s(y), the equality h + (y) = h + (x * y) holds.
(2) Let φ : PX −→ Y be a continuous map such that for any x and y of PX such that t(x) = s(y), the equality φ(y) = φ(x * y) holds.Then there exists a unique continuous map φ : Moreover, one has the homeomorphism where P + α X := h + β∈X 0 P + α,β X .The mapping X → P + X yields a functor P + from Flow to Top.
Roughly speaking, the merging space of a flow is the space of germs of non-constant execution paths ending in the same way.
Definition 5.4.Let X be a flow.The topological space P + X is called the merging space of the flow X.The functor P + is called the merging space functor.
Definition 5.5.[Gau05a] Let X be a flow.Let A and B be two subsets of X 0 .One says that A is surrounded by B (in X) if for any α ∈ A, either α ∈ B or there exists execution paths γ 1 and γ 2 of PX such that s(γ 1 ) ∈ B, t(γ 1 ) = s(γ 2 ) = α and t(γ 2 ) ∈ B. We denote this situation by A ≪ B.
Definition 5.6.[Gau05a] Let X be a flow.Let A be a subset of X 0 .Then the restriction X ↾ A of X over A is the unique flow such that (X ↾ A ) 0 = A, such that P α,β (X ↾ A ) = P α,β X for any (α, β) ∈ A × A and such that the inclusions A ⊂ X 0 and P(X ↾ A ) ⊂ PX induces a morphism of flows X ↾ A −→ X. Definition 5.7.[Gau05a] Let X and Y be two objects of cell(Flow).A morphism of flows f : X −→ Y is a T-homotopy equivalence if and only if the following conditions are satisfied: (1) The morphism of flows f : X −→ Y ↾ f (X 0 ) is an isomorphism of flows.In particular, the set map f 0 : We recall the following important theorem for the sequel: This characterization was actually the first definition of a T-homotopy equivalence proposed in [GG03] (see Definition 4.10 p66).

Some facts about relative I gl + -cell complexes
Recall that I gl Let I g = I gl ∪ {C}.Since for any n 0, the inclusion S n−1 ⊂ D n is a closed inclusion of topological spaces, so an effective monomorphism of the category Top of compactly generated topological spaces, every morphism of I g , and therefore every morphism of cell(I g ), is an effective monomorphism of flows as well (cf.also [Gau03] Theorem 10.6).
Proposition 6.1.If f : X −→ Y is a relative I gl + -cell complex and if f induces a one-to-one set map from X 0 to Y 0 , then f : X −→ Y is a relative I g -cell subcomplex.
Proof.A pushout of R appearing in the presentation of f cannot identify two elements of X 0 since, by hypothesis, f 0 : X 0 → Y 0 is one-to-one.So either such a pushout is trivial, or it identifies two elements added by a pushout of C.
, where h : Z −→ T is a morphism of cell({C}), and where g : T −→ Y is a relative I gl -cell complex.
Proof.One can use the small object argument with {R} by [Gau03] Proposition 11.8.Therefore the morphism f : X −→ Y factors as a composite g • h where h : X −→ Z is a morphism of cell({R}) and where the morphism Z −→ Y is a morphism of inj({R}).One deduces that the set map Z 0 −→ Y 0 is one-to-one.One has the pushout diagram of flows Therefore the morphism Z −→ Y is a relative I gl + -cell complex.Proposition 6.1 implies that the morphism Z −→ Y is a relative I g -cell complex.The morphism Z −→ Y factors as a composite h : Z −→ Z ⊔ (Y 0 \Z 0 ) and the inclusion g : Z ⊔ (Y 0 \Z 0 ) −→ Y .Proposition 6.3.Let X = X 0 be a set viewed as a flow (i.e. with an empty path space).Let Y be an object of cell(Flow).Then any morphism from X to Y is a cofibration.Proof.Let f : X −→ Y be a morphism of flows.Then f factors as a composite X = X 0 −→ Y 0 −→ Y .Any set map X 0 −→ Y 0 is a transfinite composition of pushouts of C and R.So any set morphism X 0 −→ Y 0 is a cofibration of flows.And for any flow Y , the canonical morphism of flows Y 0 −→ Y is a cofibration since it is a relative I g -cell complex.Hence the result.
Then the continuous map Pg : PX −→ PY is a transfinite composition of pushouts of continuous maps of the form a finite product Id × . . .× f × . . .× Id where the symbol Id denotes identity maps.
Proposition 7.2.Let f : U −→ V be a Serre cofibration.Then the pushout of a weak homotopy equivalence along a map of the form a finite product Id X 1 × . . .× f × . . .× Id Xp with p 0 is still a weak homotopy equivalence.
If the topological spaces X i for 1 i p are cofibrant, then the continuous map Id X 1 × . . .× f × . . .× . . .× Id Xp is a cofibration since the model category of compactly generated topological spaces is monoidal with the categorical product as monoidal structure.So in this case, the result follows from the left properness of this model category ([Hir03] Theorem 13.1.10).In the general case, Id X 1 × . . .× f × . . .× . . .× Id Xp is not a cofibration anymore.But any cofibration f for the Quillen model structure of Top is a cofibration for the Strøm model structure of . In the latter model structure, any space is cofibrant.Therefore the continuous map Id X 1 × . . .× f × . . .× . . .× Id Xp is a cofibration of the Strøm model structure of Top, that is a NDR pair.So the continuous map Id X 1 × . . .× f × . . .× . . .× Id Xp is a closed T 1 -inclusion anyway.This fact will be used below.
Proof.We already know that the pushout of a weak homotopy equivalence along a cofibration is a weak homotopy equivalence.The proof of this proposition is actually an adaptation of the proof of the left properness of the model category of compactly generated topological spaces.Any cofibration is a retract of a transfinite composition of pushouts of inclusions of the form S n−1 ⊂ D n for n 0. Since the category of compactly generated topological spaces is cartesian closed, the binary product preserves colimits.Thus we are reduced to considering a diagram of topological spaces like where s is a weak homotopy equivalence and we have to prove that s is a weak homotopy equivalence as well.By [Qui67] and [Hig71], it suffices to prove that s induces a bijection between the path-connected components of U and X, a bijection between the fundamental groupoids π( U ) and π( X), and that for any local coefficient system of abelian groups A of X, one has the isomorphism s * : H * ( X, A) ∼ = H * ( U , s * A).
For n = 0, one has Therefore the mapping t is the disjoint sum s ⊔ Id X 1 ×...×Xp .So it is a weak homotopy equivalence.

Let n
1.The assertion concerning the path-connected components is clear.Let Since the pair (T n , S n−1 ) is a deformation retract, the three pairs (X 1 × . . . ) and ( X, X) are deformation retracts as well.So the continuous maps U −→ U and X −→ X are both homotopy equivalences.The Seifert-Van-Kampen theorem for the fundamental groupoid (cf.[Hig71] again) then yields the diagram of groupoids Since π( s) is an isomorphism of groupoids, then so is π( s).
) is an excisive pair of U and (B n , X) is an excisive pair of X.The Mayer-Vietoris long exact sequence then yields the commutative diagram of groups Proof.The principle of the proof is standard.If the ordinal λ is not a limit ordinal, then this is a consequence of Proposition 7.2.Assume now that λ is a limit ordinal.Then λ ℵ 0 .
Let u : S n −→ lim − → N be a continuous map.Then u factors as a composite S n −→ N µ −→ lim − → N since the n-dimensional sphere S n is compact and since any compact space is ℵ 0 -small relative to closed T 1 -inclusions ([Hov99] Proposition 2.4.2).By hypothesis, there exists a continuous map S n −→ M µ such that the composite S n −→ M µ −→ N µ is homotopic to S n −→ N µ .Hence the surjectivity of the set map π n (lim − → M, * ) −→ π n (lim − → N, * ) (where π n denotes the n-th homotopy group) for n 0 and for any base point * .
Let u, v : S n −→ lim − → M be two continuous maps such that there exists an homotopy N for some µ 0 < λ.And again since the space S n is compact, the map f (resp.g) factors as a composite for n 0 and for any base point * is one-to-one.
Theorem 7.4.The model category Flow is left proper.

Proof. Consider the pushout diagram of
where i is a cofibration of Flow and s a weak S-homotopy equivalence.We have to check that t is a weak S-homotopy equivalence as well.The morphism i is a retract of a I gl then t must be a retract of u.Therefore it suffices to prove that u is a weak S-homotopy equivalence.So one can suppose that one has a diagram of flows of the form where k ∈ cell(I gl + ).By Proposition 6.2, the morphism k : A −→ B factors as a composite A −→ A ′ −→ A ′′ −→ B where the morphism A −→ A ′ is an element of cell({R}), where the morphism A ′ −→ A ′′ is an element of cell({C}), and where the morphism A ′′ −→ B is a morphism of cell(I gl ).So we have to treat the cases k ∈ cell({R}), k ∈ cell({C}) and k ∈ cell(I gl ).
The case k ∈ cell(I gl ) is a consequence of Proposition 7.1, Proposition 7.2 and Proposition 7.3.The case k ∈ cell({C}) is trivial.
) is a pair of distinct elements of U 0 = X 0 identified by k.So t is a weak S-homotopy equivalence since a binary product of weak homotopy equivalences is a weak homotopy equivalence.

T-homotopy equivalence and I gl
+ -cell complex The first step to understand the reason why Definition 5.7 is badly-behaved is the following theorem which gives a description of the T-homotopy equivalences f : X −→ Y such that the 0-skeleton of Y contains exactly one more state than the 0-skeleton of X.
Theorem 8.1.Let X and Y be two objects of cell(Flow).
By Proposition 6.3, the morphism { 0, 1} = Glob(S −1 ) −→ u f (X) is a cofibration.Therefore the morphism For any ordinal µ, the morphism of flows Y µ −→ Y µ+1 induces an isomorphism between the 0-skeletons Y 0 µ and Y 0 µ+1 .If n µ 1 for some µ, then for any β, γ ∈ Y 0 , the topological space P β,γ Y µ is non empty if and only if the topological space P β,γ Y µ+1 is non empty.Consider the set of ordinals    µ < λ; It is non-empty since f is a T-homotopy equivalence.Take its smallest element µ 0 .Consider the set of ordinals    µ < λ; Take its smallest element µ 1 .Let us suppose for instance that µ 0 < µ 1 .The ordinal µ 0 cannot be a limit ordinal.Otherwise for any µ < µ 0 , the isomorphisms of flows Z µ ⊔ {α} would hold: contradiction.Therefore µ 0 = µ 2 + 1 and n µ 2 = 0.There does not exist other ordinal µ such that φ µ ( 1) = α otherwise P + α Y could not be a singleton anymore.For a slightly different reason, the ordinal µ 1 cannot be a limit ordinal either.Otherwise if µ 1 was a limit ordinal, then the isomorphism of flows Y µ 1 ∼ = lim − →µ<µ1 Y µ would hold.The path space of a colimit of flows is in general not the colimit of the path spaces.But any element of PY µ 1 is a composite γ 1 * • • • * γ p where the γ i for 1 i p belong to lim − →µ<µ1 PY µ .By hypothesis, there exists an execution path γ 1 * • • • * γ p ∈ P α,β Y µ 1 for some β ∈ X 0 .So s(γ 1 ) = α, which contradicts the definition of µ 1 .Therefore µ 1 = µ 3 + 1 and necessarily n µ 3 = 0.There does not exist any other ordinal µ such that φ µ ( 0) = α otherwise P − α Y could not be a singleton anymore.
We are now ready to give a characterization of the old T-homotopy equivalences: where for any i ∈ I, n i is an integer with n i 1 and such that r i : − → I −→ − → I * n i is the unique morphism of flows preserving the initial and final states and where the morphisms ∅ −→ u f (X) and v f (X) −→ X are relative I g -cell complexes.
The pushout above tells us that the copy of − → I corresponding to the indexing i ∈ I is divided in the concatenation of n i copies of − → I .This intuitively corresponds to a refinement of observation.
Proof.By Theorem 4.14, there exists a globular complex U (resp.V ) such that cat(U ) = X (resp.cat(V ) = Y ).If a morphism of flows f : X −→ Y is a T-homotopy equivalence, then by Theorem 5.8, there exists a morphism of globular complexes g : U −→ V such that cat(g) = f and such that the continuous map |g| : |U | −→ |V | between the underlying topological spaces is an homeomorphism.So for any pair of points (α, β) of X 0 × X 0 , and any morphism − → I −→ X appearing in the globular decomposition of X, the set of subdivision of this segment in Y is finite since Y 0 is discrete and since the segment [0, 1] is compact.The result is then established by repeatedly applying Theorem 8.1.Now suppose that a morphism of flows f : X = cat(U ) −→ Y = cat(V ) can be written as a pushout of the form of the statement of the theorem.Then start from a globular decomposition of U which is compatible with the composite Then let us divide each segment of [0, 1] corresponding to the copy of − → I indexed by i ∈ I in n i pieces.Then one obtains a globular decomposition of V and the identity of U gives rise to a morphism of globular complexes g : U −→ V which induces an homeomorphism between the underlying topological spaces and such that cat(g) = f .Hence the result.
Proof.If (α, β) and (β, α) with α = β belong to the transitive closure, then there exists a finite sequence (x 1 , . . ., x ℓ ) of elements of X 0 with x 1 = α, x ℓ = α, ℓ > 1 and with P xm,x m+1 X non-empty for each m.Consequently, the space P α,α X is non-empty because of the existence of the composition law of X: contradiction.
Theorem 8.5.Let n 3.There does not exist any zig-zag sequence where every X i is an object of cell(Flow) and where every f i is either a S-homotopy equivalence or a T-homotopy equivalence.
Proof.By an immediate induction, one sees that each flow X i is loopless, with a finite 0-skeleton.Moreover by construction, each poset (X 0 i , ) is bounded, i.e. with one bottom element 0 and one top element 1.So the zig-zag sequence above gives rise to a zig-zag sequence of posets: where { 0 < 1} n is the product { 0 < 1} × . . .× { 0 < 1} (n times) in the category of posets.Each morphism of posets is an isomorphism if the corresponding morphism of flows is a S-homotopy equivalence because a S-homotopy equivalence induces a bijection between the 0-skeletons.Otherwise one can suppose by Theorem 8.1 that the morphism of posets P 1 −→ P 2 can be described as follows: take a segment [x, y] of P 1 such that ]x, y[= ∅; add a vertex z ∈]x, y[; then let P 2 = P 1 ∪ {z} with the partial ordering x < z < y.In such a situation, min(]z, −[) exists and is equal to y, and max(]−, z[) exists and is equal to x.
So by an immediate induction, there must exist x, y, z ∈ {0 < 1} n with x < z < y and such that min(]z, −[) = y and max(]−, z[) = x.This situation is impossible in the poset { 0 < 1} n for n 3.

Generalized T-homotopy equivalence
As explained in the introduction, it is not satisfactory not to be able to identify − → C 3 , and more generally − → C n for n 3, with − → I .The following definitions are going to be important for the sequel of the paper, and also for the whole series.Definition 9.1.(1) The posets P 1 and P 2 are finite and bounded.
(2) The morphism of posets f : P 1 −→ P 2 is one-to-one; in particular, if x and y are two elements of P 1 with x < y, then f (x) < f (y).

Conclusion
This new definition of T-homotopy equivalence contains the old one up to S-homotopy equivalence.The drawback of the old definition presented in [Gau05a] is overcome.It is proved in [Gau06b] that this new notion of T-homotopy equivalence does preserve the branching and merging homology theories.And it is proved in [Gau06c] that the underlying homotopy type of a flow is also preserved by this new definition of T-homotopy equivalence.Finally, the paper [Gau06a] proposes an application of this new notion of dihomotopy: that is a Whitehead theorem for the full dihomotopy relation.

Figure 1 .
Figure 1.The simplest example of T-homotopy equivalence

Figure 3 .
Figure 3. Symbolic representation of Glob(Z) for some topological space Z Theorem 5.8.([Gau05a] Theorem VI.3.5)Let X and Y be two objects of cell(Flow).Let U and V be two globular complexes with cat(U ) = X and cat(V ) = Y (U and V always exist by Theorem 4.14).Then a morphism of flows f : X −→ Y is a T-homotopy equivalence if and only if there exists a morphism of globular complexes g : U −→ V such that cat(g) = f and such that the continuous map |g| : |U | −→ |V | between the underlying topological spaces is an homeomorphism.

7.
Left properness of the weak S-homotopy model structure of Flow Proposition 7.1.([Gau03] Proposition 15.1) Let f : U −→ V be a continuous map.Consider the pushout diagram of flows

A
five-lemma argument completes the proof.Proposition 7.3.Let λ be an ordinal.Let M : λ −→ Top and N : λ −→ Top be two λ-sequences of topological spaces.Let s : M −→ N be a morphism of λ-sequences which is also an objectwise weak homotopy equivalence.Finally, let us suppose that for all µ < λ, the continuous maps M µ −→ M µ+1 and N µ −→ N µ+1 are of the form a finite product Id X 1 × . . .× f × . . .× Id Xp with p 0 and with f a Serre cofibration.Then the continuous map lim − → s : lim − → M −→ lim − → N is a weak homotopy equivalence.If for all µ < λ, the continuous maps M µ −→ M µ+1 and N µ −→ N µ+1 are cofibrations, then Proposition 7.3 above is a consequence of [Hir03] Proposition 17.9.3 and of the fact that the model category Top is left proper.With the same additional hypotheses, Proposition 7.3 above is also a consequence of [DI04] Theorem A.7. Indeed, the latter states that an homotopy colimit can be calculated either in the usual Quillen model structure of Top, or in the Strøm model structure of Top [Str72] [Col06].

Theorem 8. 2 .
Let X and Y be two objects of cell(Flow).Then a morphism of flows f : X −→ Y is a T-homotopy equivalence if and only if there exists a commutative diagram of flows of the form (with − → I * (n+1) Definition 8.3.Let n 1.The full n-cube − → C n is by definition the flow Q({ 0 < 1} n ), where Q is the cofibrant replacement functor.The flow − → C 3 is represented in Figure 2. Lemma 8.4.If a flow X is loopless, then the transitive closure of the set 3 A full directed ball is a flow − → D such that:• − → D is loopless (so by Lemma 8.4, the set − → D 0 is equipped with a partial ordering )• ( − → D 0 , ) is finite bounded • for all (α, β) ∈ − → D 0 × − → D 0 , the topological space P α,β − → D is weakly contractible if α < β, and empty otherwise by definition of .Let − → D be a full directed ball.Then by Lemma 8.4, the set − → D 0 can be viewed as a finite bounded poset.Conversely, if P is a finite bounded poset, let us consider the flow F (P ) associated with P : it is of course defined as the unique flow F (P ) such that F (P ) 0 = P and P α,β F (P ) = {u} if α < β and P α,β F (P ) = ∅ otherwise.Then F (P ) is a full directed ball and for any full directed ball − → D, the two flows − → D and F ( − → D 0 ) are weakly S-homotopy equivalent.Let − → E be another full directed ball.Let f : − → D −→ − → E be a morphism of flows preserving the initial and final states.Then f induces a morphism of posets from −→ D 0 to − → E 0 such that f (min − → D 0 ) = min − → E 0 and f (max − → D 0 ) = max − → E 0 .Hence the following definition:Definition 9.2.Let T be the class of morphisms of posets f : P 1 −→ P 2 such that:

( 3 )F
One has f (min P 1 ) = min P 2 and f (max P 1 ) = max P 2 .Then a generalized T-homotopy equivalence is a morphism of cof ({Q(F (f )), f ∈ T }) where Q is the cofibrant replacement functor of the model category Flow.It is of course possible to identity− → C n (n 1) with − → I by the following zig-zag sequence of S-homotopy and generalized T-homotopy equivalences (gn))/ / Q({ 0 < 1} n ), where g n : { 0 < 1} −→ { 0 < 1} n ∈ T .The relationship between the new definition of T-homotopy equivalence and the old definition is as follows:Theorem 9.3.Let X and Y be two objects of cell(Flow).Let f : X −→ Y be a Thomotopy equivalence.Then f can be written as a composite X −→ Z −→ Y where g : X −→ Z is a generalized T-homotopy equivalence and where h : Z −→ Y is a weak S-homotopy equivalence.Proof.By Theorem 8.2, there exists a pushout diagram of flows of the form (with − → I * (n+1) k∈K r k X k∈K − → I * n k / / Y where for any k ∈ K, n k is an integer with n k 1 and such that r k : − → I −→ − → I * n k is the unique morphism of flows preserving the initial and final states.Notice that each − → I * n k is a full directed ball.Thus one obtains the following commutative diagram: k / / Y. Now here are some justifications for this diagram.First of all, a morphism of flows f : M −→ N is a fibration of flows if and only if the continuous map Pf : PM −→ PN is a Serre fibration of topological spaces.Since any coproduct of Serre fibration is a Serre fibration, the morphism of flows i∈I Q( − → I ) −→ k∈K − → I is a trivial fibration of flows.Thus the underlying set map k∈K Q( − → I ) −→ k∈K − → I is surjective.So the commutative square k∈K Q( k ) / / Z is cocartesian and the morphism of flows X −→ Z is then a generalized T-homotopy equivalence.It is clear that the morphism k∈K Q( − → I * n k ) −→ k∈K − → I * n k is a weak Shomotopy equivalence.The latter morphism is even a fibration of flows, but that does not matter here.So the morphism Z −→ Y is the pushout of a weak S-homotopy equivalence along the cofibration k∈K Q( − → I * n k ) −→ Z. Since the model category Flow is left proper by Theorem 7.4, the proof is complete.