Six model categories for directed homotopy

We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.


Introduction
Presentation. This paper belongs to our series of papers which aims at comparing the model category Flow of flows introduced in [Gau03] (with some updated proofs in [Gau20] using Isaev's work [Isa18]) and the model category GdTop of multipointed dspaces introduced in [Gau09]. Roughly speaking, the former is a version of the latter without underlying topological space. And the latter is a variant of Grandis' notion of dspace [Gra03]. They are topological models introduced to study concurrent processes from the point of view of homotopy theory. Even if these model categories do not yet contain enough weak equivalences (their homotopical localizations with respect to the refinement of observation remain to be understood: see the digression section in [Gau20]), the model category of flows enabled us anyway to understand homological theories detecting the nondeterministic branching and merging areas of execution paths in the framework of flows [Gau06] [Gau05b]. These homology theories are interesting because they are invariant by the refinement of observation.
Using the notion of topological graph (see Definition 5.5) and the Garner Hess Kędziorek Riehl Shipley theorem [HKRS17] [GKR20] about accessible right-induced model structures, we introduce a categorical construction which takes as input an accessible model structure on the category Top of ∆-generated spaces satisfying some mild conditions (the ones of Proposition 4.3) and which gives as output an accessible model structure on multipointed d-spaces and on flows. These mild conditions are satisfied in particular by 1 the q-model structure (the Quillen model structure) of Top, the h-model structure (also called the Cole-Ström model structure) of Top and the m-model structure (which is the mixing of the two preceding model structures in the sense of [Col06, Theorem 2.1]). The latter is characterized as the unique model structure on Top such that the weak equivalences are the weak homotopy equivalences and the fibrations the h-fibrations. We obtain the following results: • a q-model structure, a h-model structure and a m-model structure on multipointed d-spaces and on flows in one step (!) • the identity functor induces a Quillen equivalence between the q-model structure and the m-model structure on multipointed d-spaces (on flows resp.) • the two q-model structures are combinatorial and left determined and they coincide with that of [Gau09] and of [Gau03] [Gau20] respectively The two h-model structures and the two m-model structures are new. They are conjecturally not combinatorial. Even if all topological spaces are h-cofibrant, it is not true that all multipointed d-spaces and all flows are h-cofibrant as well. Intuitively, the h-cofibrant objects correspond to objects without algebraic relations in their spaces of execution paths. A rigorous characterization of the h-cofibrant multipointed d-spaces and h-cofibrant flows still remains to be find out.
The main interest of this categorical construction lies in the two m-model structures. They are better behaved than the q-model structures for the following reasons. Unlike the space of execution paths functor P : Flow → Top which preserves q-cofibrancy, it is not true that the space of execution paths functor P G : GdTop → Top does as well: see Section 8. However we have the following result which can be considered as an application of the results of this paper: Theorem. (Theorem 8.6 and Theorem 8.7) The space of execution paths functors P G : GdTop → Top and P : Flow → Top preserve m-cofibrancy.
We want to end the introduction with a remark about the notion of multipointed d-space. It is easy to prove that all theorems of this paper involving multipointed dspaces, except Proposition 8.5 coming from [Gau05a] and Theorem 8.6, are still true by replacing the topological group G of nondecreasing homeomorphisms of the segment [0, 1] by the topological monoid M of nondecreasing continuous maps from the segment [0, 1] to itself preserving the extremities. However, we do not know whether Proposition 8.5 and Theorem 8.6 hold with this new definition of multipointed d-space. Indeed, the results of [Gau05a], in particular Proposition 8.5 used in the proof of Theorem 8.6, use the fact that all elements of G are invertible and we are unable to remove completely this hypothesis by now from the proofs of [Gau05a].
Outline of the paper.
• Section 2 collects some basic facts about accessible model categories. It is expounded the theorem we are going to use to right-induce accessible model structures (Theorem 2.1). • Section 3 proves two technical elementary facts about Grothendieck bifibrations that will be used in the sequel: a first one which is a toolkit to easily prove that a functor is a bifibration (Proposition 3.1), and a second one about the accessibility of two functors arising from an accessible bifibration (Proposition 3.2). • Section 4 gathers some information about ∆-generated spaces and their three standard model structures. In particular, Proposition 4.3 makes explicit and establishes that these three model structures satisfy the mild conditions which are used in our construction. • Section 5 explains how to construct an accessible model structure on V-graphs from any accessible model category V (Theorem 5.4), with an immediate application when V is the category of ∆-generated spaces (Corollary 5.6). • Section 6 applies the constructions of Section 5 to right-induce on the category of multipointed d-spaces the three model structures (Theorem 6.14). It is also proved that there exist multipointed d-spaces which are not h-cofibrant, not q-cofibrant and not m-cofibrant (Proposition 6.19). • Section 7 applies the same constructions to right-induce on the category of flows the three model structures (Theorem 7.4). It is also proved that there exist flows which are not h-cofibrant, not q-cofibrant and not m-cofibrant (Proposition 7.9). • Section 8 explains why the m-model structures are better behaved than the q-model structures (Theorem 8.6 and Theorem 8.7).

Notations.
• X := Y means that Y is the definition of X.
• All categories are locally small (except the category of all locally small categories).
• K always denotes a locally presentable category.
• Set is the category of sets.
• Top is the category of ∆-generated spaces.
• R is the topological space of real numbers.
• K(X, Y ) is the set of maps in a category K.
• Mor(K) is the category of morphisms of K with the commutative squares for the morphisms. • A ⊔ B is the binary coproduct, A × B is the binary product.
• lim ← − is the limit, lim − → is the colimit. • ∅ is the initial object.
• 1 is the final object.
• Id X is the identity of X.
• g.f is the composite of two maps f : A → B and g : B → C; the composite of two functors is denoted in the same way. • f g means that f satisfies the left lifting property (LLP) with respect to g, or equivalently that g satisfies the right lifting property (RLP) with respect to f .
• cell(C) is the class of transfinite compositions of pushouts of elements of C. • A cellular object X of a combinatorial model category is an object such that the canonical map ∅ → X belongs to cell(I) where I is the set of generating cofibrations. • A model structure (C, W, F ) means that the class of cofibrations is C, that the class of weak equivalences is W and that the class of fibrations is F in this order. A model category is a category equipped with a model structure.

Accessible model category
We refer to [AR94] for locally presentable categories, to [Ros09] for combinatorial model categories. We refer to [Hov99] and to [Hir03] for more general model categories.
A weak factorization system (L, R) of a locally presentable category K is accessible if there is a functorial factorization with Lf ∈ L, Rf ∈ R such that the functor E :

Suppose that there exists a functorial factorization of the diagonal of
is a weak equivalence of M and such that U(π) is a fibration of M for all objects X of N . Then there exists a unique model structure on N such that the class of fibrations is U −1 (F ) and such that the class of weak equivalences is U −1 (W). Moreover, this model structure is accessible and all its objects are fibrant.
Sketch of proof. By the dual of [HKRS17, Theorem 2.2.1] which is also stated in [Mos19, Theorem 6.2], the hypotheses of the theorem imply that the Quillen Path Object argument holds. The latter implies the acyclicity condition for right-induced model structures, and therefore the existence of the right-induced model structure (see also [GKR20]). Since a model structure is characterized by its class of weak equivalences and its class of fibrations, we deduce the uniqueness.

Accessible Grothendieck bifibration
Let p : E → B be a functor between locally small categories. The fibre of p over X, denoted by E X , consists of the subcategory of E generated by the vertical maps f , i.e. the maps f such that p(f ) = Id X . We refer to [Jac99, Chapter 1 and Chapter 9] and [Bor94b, Chapter 8] for (Grothendieck) bifibrations (also called bifibred categories) and for (Grothendieck) fibrations (also called fibred categories, the term fibration being quite confusing because it is used in a completely different sense in this paper).
The following proposition is a toolkit to minimize the work required to prove that a functor is a bifibration: 3.1. Proposition. Let p : E → B be a functor between locally small categories. Suppose that for every map u : A → B of B, there exists an adjunction u ! : E A ⊣ E B : u * such that: (1) For all objects X of E, there exists a natural map u * X → X such that every map f : X → Y of E with p(f ) = u factors uniquely as a composite Then p : E → B is a bifibration.
Proof. In the language of [Jac99], the first condition means that the map u * X → X is weakly cartesian and the second condition implies that compositions of weakly cartesian maps are weakly cartesian. By [Jac99, Exercice 1.1.6], the functor p : E → B is a fibred category. By [Jac99, Lemma 9.1.2], the existence of the adjunctions implies that the functor p : E → B is a bifibration.
Let p : E → B be a bifibration between locally small categories. Consider the commutative square of solid arrows of E Note that the diagram above is misleading: the maps g and h are not vertical. On the contrary, the two maps X → µ(f ) and have the same image p(g) by p and since they yield two factorizations of h.f = f ′ .g and since µ(f ′ ) → Y ′ is cartesian, the left-hand square is commutative as well. For dual reasons, there exists a unique map ν (g,h) : commutative. By the usual uniqueness argument, we obtain two well-defined functors µ : Mor(E) → E and ν : Mor(E) → E.
. By passing to the colimit, we obtain the factorization of lim There are the isomorphisms We obtain the factorization of lim Since the left-hand map is vertical, we obtain the equality We have proved that µ is accessible. In the same way, by passing to the colimit, there is There are the isomorphisms We obtain the factorization of lim Since the right-hand map is vertical, we obtain the equality We have proved that ν is accessible.

Delta-generated space
We refer to [AHS06, Chapter VI] or [Bor94b, Chapter 7] for the notion of topological functor. The category Top denotes the category of ∆-generated spaces, i.e. the colimits of simplices. Let ∆ n = {(t 0 , . . . , t n ) ∈ [0, 1] n | t 0 + · · · + t n = 1} be the topological n-simplex equipped with its standard topology. Then Top is the final closure of the set of topological spaces {∆ n | n 0}. For a tutorial about these topological spaces, see for example [Gau09, Section 2]. The category Top is locally presentable by [FR08, Corollary 3.7], cartesian closed and it contains all CW-complexes. The internal hom functor is denoted by TOP(−, −). We denote by ω : TOP → Set the underlying set functor where TOP is the category of general topological spaces. It is fibre-small and topological. The restriction functor ω : Top ⊂ TOP → Set is fibre-small and topological as well. The category Top is a full coreflective subcategory of the category TOP of general topological spaces. Let k : TOP → Top be the kelleyfication functor, i.e. the right adjoint. The category Top is finally closed in TOP, which means that the final topology and the ω-final structure coincides. On the contrary, the ω-initial structure in Top is obtained by taking the kelleyfication of the initial topology in TOP. If A is a subset of a space X of Top, the initial structure in Top of the inclusion A ⊂ ωX is the kelleyfication of the relative topology with respect to the inclusion.
The category Top can be equipped at least with three model structures (we use the notations of [MS06]): • The q-model structure (C q , W q , F q ) [Hov99, Section 2.4]: the cofibrations, called qcofibrations, are the retracts of the transfinite compositions of the inclusions S n−1 ⊂ D n for n 0, the weak equivalences are the weak homotopy equivalences and the fibrations, called q-fibrations are the maps satisfying the RLP with respect to the inclusions D n ⊂ D n+1 for n 0, or equivalently with respect to the inclusions D n × {0} ⊂ D n × [0, 1] for n 0; this model structure is combinatorial. A very simple way to obtain this model structure is to use [Isa18]. Its existence dates back to [Qui67].
• The h-model structure (C h , W h , F h ): the fibrations, called the h-fibrations, are the maps satisfying the RLP with respect to the inclusions X ×{0} ⊂ X ×[0, 1] for all topological spaces X, and the weak equivalences are the homotopy equivalences; we have A modern exposition is given in [BR13, Corollary 5.23] but its construction dates back to [Str72]. All topological spaces are h-cofibrant.
tions, and the weak equivalences are the weak homotopy equivalences; we have (1) They are accessible.

Topological graph
In this section, V denotes a locally presentable category. It is supposed to be equipped with an accessible model structure (C, W, F ). We recall the enriched version of the usual notion of graph and of morphism between them [Bor94a, Definition 5.1.1]. This notion appears for example in [Web13, Definition 2.1.1] and in [KL01, Section 3]. We adapt the notations to our context.

Definition. A V-graph X consists of a pair
such that X 0 is a set and such that each P α,β X is an object of V. A map of V-graphs f : X → Y consists of a set map f 0 : X 0 → Y 0 (called the underlying set map) together with a map P α,β X → P f 0 (α),f 0 (β) Y of V for all (α, β) ∈ X 0 × X 0 . The composition is defined in an obvious way. The corresponding category is denoted by Gph(V).

Notation.
We will denote P f 0 (α),f 0 (β) Y by P f (α),f (β) Y in order not to overload the notations.
5.3. Proposition. The forgetful functor X → X 0 from Gph(V) to Set is a bifibration.
Proof. Let f : X → Y be a map of V-graphs. Let Then by definition of a map of V-graphs, every map f : X → Y factors uniquely as a composite We have the natural bijections of sets the first and the fourth isomorphisms by definition of a map of V-graphs, the second isomorphism by rearranging the product and the third isomorphism by definition of the V-graph (f 0 ) ! X. The proof is complete thanks to Proposition 3.1.
For every set S, the fibre of () 0 : Gph(V) → Set over S is the functor category V S×S which is equipped for the sequel with the only model structure such that the cofibrations (the fibrations, the weak equivalences resp.) are the pointwise ones: it is both the projective and the injective model structure on a functor category over a discrete category. This model structure is obviously accessible.

Theorem.
There exists a unique model structure on Gph(V) such that • The weak equivalences are the maps of V-graphs f : X → Y such that f 0 is a bijection and such that the map X → (f 0 ) * Y is a pointwise weak equivalence of V X 0 ×X 0 , i.e. for all (α, β) ∈ X 0 × X 0 , the map P α,β X → P f (α),f (β) Y belongs to W.
• The fibrations are the maps of V-graphs f such that the map (1) if u : S → T is a weak equivalence of Set, then it is a bijection. Therefore the functor u * : V T ×T → V S×S reflects weak equivalences since it is an equivalence of categories.
(2) if u : S → T is a trivial cofibration of Set, then it is a bijection, which means that we can suppose that S = T . In that case, both u ! and u * are the identity of V S×S and the unit of the adjunction X → u * u ! X is an isomorphism, and therefore a weak equivalence of V S×S .
This proves the existence of the model structure. By [KL01, Proposition 4.4], the category Gph(V) is locally presentable 2 . Let f : X → Y be a map of V-graphs. It factors as a composite where the factorization trivial cofibration-fibration of the vertical map X → µ(f ) is carried out in V X 0 ×X 0 . Since the map Z → µ(f ) is vertical, we have Thus the composite Z → µ(f ) → Y is a fibration of Gph(V) by definition of them. We have obtained a factorization trivial cofibration-fibration in Gph(V). The functor (−) 0 : Gph(V) → Set is colimit preserving since it has a right adjoint: the functor taking a set S to the constant diagram ∆ S×S (1) over S × S. By Proposition 3.2, the endofunctor of Mor(Gph(V)) taking f : X → Y to X → µ(f ) is accessible since colimits are calculated pointwise in Mor(Gph(V)). Since the model structure of V X 0 ×X 0 is accessible, we deduce that the factorization trivial cofibration-fibration in Gph(V) is accessible. The map f : X → Y factors as well as a composite

Thus the composite X → ν(f ) → T is a cofibration of Gph(V) by definition of them.
We have obtained a factorization cofibration-trivial fibration in Gph(V). Since colimits of maps are calculated pointwise, we deduce that the endofunctor of Mor(Gph(V)) taking f : X → Y to ν(f ) → Y is accessible by Proposition 3.2. Since the model structure of V Y 0 ×Y 0 is accessible, we deduce that the factorization cofibration-trivial fibration in Gph(V) is accessible. We have proved that the model category Gph(V) is an accessible model category.

Definition.
A topological graph is a V-graph with V = Top. The corresponding category is denoted by Gph(Top).
5.6. Corollary. Let (C, W, F ) be one of the three model structures of Top. Then there exists a unique model structure on Gph(Top) such that:

Moreover, this model structure is accessible and all objects are fibrant.
Proof. It is a consequence of Theorem 5.4 and Proposition 4.3 (1) and (2).

Multipointed d-space
6.1. Definition. A multipointed space is a pair (|X|, X 0 ) where • |X| is a topological space called the underlying space of X.
• X 0 is a subset of |X| called the set of states of X.
6.2. Definition. The map γ 1 * γ 2 is called the composition of γ 1 and γ 2 . The composite • The set P G X is a set of continous maps from [0, 1] to |X| called the execution paths, satisfying the following axioms: -For any execution path γ, one has γ(0), γ(1) ∈ X 0 .
-Let γ be an execution path of X. Then any composite γ.φ with φ ∈ G is an execution path of X.
-Let γ 1 and γ 2 be two composable execution paths of X; then the normalized composition γ 1 * N γ 2 is an execution path of X. A map f : X → Y of multipointed d-spaces is a map of multipointed spaces from (|X|, X 0 ) to (|Y |, Y 0 ) such that for any execution path γ of X, the map f.γ is an execution path of Y . The category of multipointed d-spaces is denoted by GdTop. The subset of execution paths from α to β is the set of γ ∈ P G X such that γ(0) = α and γ(1) = β; it is denoted by P G α,β X. It is equipped with the kelleyfication of the initial topology making the inclusion P G α,β X ⊂ TOP([0, 1], |X|) is continuous. 6.4. Definition. Let X be a multipointed d-space X. Let P G X be the topological space The category of multipointed d-spaces GdTop is locally presentable and the forgetful functor X → ω(|X|) is topological and fibre-small by [Gau09, Theorem 3.5].
The following examples play an important role in the sequel.
(1) Any set E will be identified with the multipointed d-space (E, E, ∅).
(2) The topological globe of Z, which is denoted by Glob G (Z), is the multipointed d-space defined as follows • the underlying topological space is the quotient space • the set of execution paths is the set of continuous maps where ℓ 1 < ℓ 2 are two real numbers has the underlying space the segment [ℓ 1 , ℓ 2 ], the set of states {ℓ 1 , ℓ 2 } and the unique space of execution paths P G Proof. The statement is very close to the statement of [Gau09, Proposition 3.6]. The proof of the latter proposition uses the final structure. We prefer to use the Ω-initial structure because it will be reused in Corollary 6.7. Let (|X|, X 0 ) be a multipointed space. Consider a cone (which can be large) (f i : (|X|, X 0 ) → Ω(X i )) i∈I . For all (α, β) ∈ X 0 ×X 0 , consider the set of paths We deduce that γ 1 * N γ 2 ∈ P α,α ′′ by definition of P α,α ′′ . We deduce that the family of (P α,β ) yields a structure of multipointed d-space on (|X|, X 0 ) and it is clearly the biggest one because all f i must be lifted to maps of multipointed d-spaces. It is therefore the Ω-initial structure. is denoted by π u : Path G (X) → X.
6.12. Proposition. Let U be a topological space. Let X be a multipointed d-space. Then we have the natural bijection Proof. A map of multipointed d-spaces from Glob G (U) to X is characterized by the choice of two states α and β of X for the image of 0 and 1 respectively and by a continuous map f from |Glob G (U)| to X such that f (u, −) ∈ P G α,β X for all u ∈ [0, 1]. In other terms, the mapping f → (u → f (u, −)) yields a natural set map Conversely, consider an element g ∈ Top(U, P G α,β X) for some (α, β) ∈ X 0 × X 0 . Then the mapping (t, u) → g(u)(t) induces a map of multipointed d-spaces from Glob G (U) to X. The proof is complete because Top is cartesian closed. 6.13. Proposition. The mapping X → Gph G (X) induces a well-defined functor from GdTop to Gph(Top). It is a right adjoint.
Proof. Roughly, the left adjoint is the free multipointed d-space generated by a topological graph. The left adjoint Gph G ! : Gph(Top) → GdTop is constructed as follows. Let X = (X 0 , (X α,β )) be a topological graph. We start from the set X 0 equipped with the discrete topology. We add a topological globe Glob G (X α,β ) with 0 identified with α and 1 identified with β for each (α, β) ∈ X 0 × X 0 . We obtain a multipointed d-space Gph G ! (X). A map f of multipointed d-spaces from Gph G ! (X) to Y is equivalent to choosing a set map from Gph G ! (X) 0 = X 0 to Y 0 and for each (α, β) ∈ X 0 × X 0 a map of multipointed d-spaces from Glob G (X α,β ) to Y , which is equivalent by Proposition 6.12 to choosing a map from X α,β to P G f (α),f (β) Y . 6.14. Theorem. Let (C, W, F ) be one of the three model structures of Top. Then there exists a unique model structure on GdTop such that: • A map of multipointed d-spaces f : X → Y is a weak equivalence if and only if f 0 : X 0 → Y 0 is a bijection and for all (α, β) ∈ X 0 × X 0 , the continuous map P G α,β X → P G f (α),f (β) X belongs to W. We deduce that for all multipointed d-spaces X and all (α, β) ∈ X 0 × X 0 , the continuous map τ : P G α,β X → TOP([0, 1], P G α,β X) belongs to W and the continuous map π : TOP([0, 1], P G α,β X) → P G α,β X × P G α,β X belongs to F . By Corollary 6.7, we deduce that the factorization of the diagonal Proof. The first assertion is a consequence of [Col06, Corollary 3.7]. The second assertion is obvious. Figure 2. Symbolic representation of p : X → X source and target map respectively, and a continuous and associative map * : Proof. Roughly, the left adjoint is the free flow generated by a topological graph. The left adjoint Gph ! : Gph(Top) → Flow is constructed as follows. Let X = (X 0 , (X α,β )) be a topological graph. The set of states of Gph ! (X) is X 0 . For α, β ∈ X 0 , let The composition law is defined by concatening tuples: (x 1 , . . . , x m ) * (y 1 , . . . , y n ) = (x 1 , . . . , x m , y 1 , . . . , y n ) We obtain a flow Gph ! (X). A map f of flows from Gph ! (X) to Y is equivalent to choosing a set map from Gph ! (X) 0 = X 0 to Y 0 and for each (α, β) ∈ X 0 × X 0 a continous map from X α,β to Y f (α),f (β) . 7.4. Theorem. Let (C, W, F ) be one of the three model structures of Top. Then there exists a unique model structure on Flow such that: • A map of flows f : X → Y is a weak equivalence if and only if f 0 : X 0 → Y 0 is a bijection and for all (α, β) ∈ X 0 × X 0 , the continuous map P α,β X → P f (α),f (β) X belongs to W. • A map of multipointed d-spaces f : X → Y is a fibration if and only if for all (α, β) ∈ X 0 × X 0 , the continuous map P α,β X → P f (α),f (β) X belongs to F . Moreover, this model structure is accessible and all objects are fibrant.
Sketch of proof. The proof is similar to the proof of Theorem 6.14. Roughly speaking, it suffices to replace everywhere P G α,β X by P α,β X and to use the right adjoint Gph : Flow → Gph(Top). We also have to use the path functor Path : Flow → Flow defined on objects by Path(X) 0 := X 0 , for all (α, β) ∈ X 0 × X 0 , P α,β Path( Proof. The first assertion is a consequence of [Col06, Corollary 3.7]. The second assertion is obvious. Figure 3. Symbolic representation of q : P cof → P 8.1. Theorem. Let X be a q-cofibrant flow. Then the space of execution paths PX is q-cofibrant.
Proof. This fact, stated in various papers before this one, has a correct proof in [Gau19b].
The analogue fact for multipointed d-spaces is wrong. Indeed, the multipointed d-space Glob G (D 1 ) is q-cofibrant. Its space of paths is equal to D 1 × G which is far from being q-cofibrant in Top. However, it is a m-cofibrant space by [Col06, Corollary 3.7] because the topological group G is contractible. It turns out that this phenomenon is general. We need first to recall some results of [Gau09] and [Gau05a] to facilitate the reading of the proof for a reader who would not be familiar with our work. 8.2. Notation. Let X be a multipointed d-space. For every (α, β) ∈ X 0 ×X 0 , let P α,β X := P G α,β X/G be the quotient of the space P G α,β X by the actions of G equipped with the final structure, i.e. the final topology.
Let X be a multipointed d-space. Then there exists a unique flow cat(X) with cat(X) 0 = X 0 , P α,β cat(X) = P α,β X for every (α, β) ∈ X 0 × X 0 and the composition law * : P α,β X × P β,γ X → P α,γ X is for every triple (α, β, γ) ∈ X 0 × X 0 × X 0 the unique map making the following diagram commutative: The mapping X → cat(X) induces a functor from GdTop to Flow (see [Gau09, Section 7] for a complete exposition). In particular, for all topological Z, we have is a homotopy equivalence.
In fact, this proposition is a particular case of a more general theorem. In [Gau05a, Theorem IV.3.10], it is proved that Glob G (Z) can be actually replaced by any cellular object X of the q-model structure of GdTop, and Glob(Z) must then be replaced by cat(X). It is even proved in [Gau05a, Theorem IV.3.14] that this map is a h-fibration of Top. The proofs of these theorems, written down within the category of weakly Hausdorff k-spaces, are still valid in our framework since they lie on three facts: (1) All maps of G are invertible: see the introduction for a short discussion about this hypothesis.
(2) The underlying category of topological spaces must be bicomplete, cartesian closed and must contain all CW-complexes. (3) The underlying category of topological spaces must be endowed with a h-model structure which is required for the homotopical part of the proofs which uses model category techniques.
We are now able to generalize the observation above: 8.6. Theorem. Let U be a m-cofibrant multipointed d-space. Then the space of paths P G U is m-cofibrant.
Proof. By Theorem 6.17 and [Col06, Corollary 3.7], there exists a q-cofibrant multipointed d-space V and a map f : U → V which is a weak equivalence of the h-model structure of GdTop. It means that f induces a bijection from U 0 to V 0 and that for each (α, β) ∈ U 0 × U 0 , the map f : P G α,β U → P G f (α),f (β) V is a homotopy equivalence. Therefore we can suppose without loss of generality that U is q-cofibrant. Since any q-cofibrant object is a retract of a cellular one, we can suppose that U is a cellular object of the q-model structure of GdTop. From a pushout diagram of multipointed d-spaces with U 1 (and therefore U 2 ) cellular Glob G (S n−1 ) one obtains a pushout diagram of cellular flows Glob(S n−1 ) / / cat(U 1 ) This point is explained in the body of the proof of [Gau05a, Theorem IV.3.10]. It is also easily seen that the functor cat : GdTop → Flow preserves transfinite colimits of qcofibrations between cellular objects. It is even the method used in [Gau05a] to construct the mapping cat. Note that the functor cat : GdTop → Flow does not preserve colimits in general. Indeed, it does not have any right adjoint by [Gau09,Proposition 7.3] and being colimit-preserving and being a left adjoint are equivalent where the source and the target categories of a functor are locally presentable. These facts are sufficient to conclude the proof. The flow cat(U) is cellular, and therefore q-cofibrant. By Theorem 8.1, we deduce that the space Pcat(U) is q-cofibrant. By Proposition 8.5 applied with Z a singleton, the quotient map P G U → Pcat(U) is a homotopy equivalence. By [Col06, Corollary 3.7], we obtain that P G U is a m-cofibrant space and the proof is complete.
The same phenomenon holds for the category of flows: 8.7. Theorem. Let U be a m-cofibrant flow. Then the space of paths PU is m-cofibrant.

Sketch of proof.
There exists a map f : U → V which a weak equivalence of the hmodel structure of Flow towards a q-cofibrant flow V . Thus PU and PV are homotopy equivalent. By Theorem 8.1, the space PV is q-cofibrant. By [Col06, Corollary 3.7], the space PU is therefore m-cofibrant.