Homotopy theory of Moore flows (I)

Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows.


Introduction Presentation
The q-model category 1 of flows (i.e. small semicategories or small nonunital categories enriched over topological spaces) Flow was introduced in [8]. The motivation is the study of concurrent should be possible to drop somehow the enriched structure in the definition of a M-flow as given in Definition 6.1.
The notion of P-flow of Definition 6.1 for a general reparametrization category P is primarily designed to better understand the topology of the spaces of execution paths in a concurrent process. This will become more apparent in the companion paper [17] in which a lot of new results about the topology of the spaces of execution paths are expounded in the globular setting, i.e. for cellular objects of the q-model structure of multipointed d-spaces. For the precubical setting, the reader will refer to the works of Paliga, Raussen and Ziemiański [36-38, 40, 41].
The right Quillen functor M G from multipointed d-spaces to G-flows defined in the companion paper consists of forgetting the underlying topological space. From a topological space X, on can obtain a multipointed d-space X = (X, X 0 , PX) with X 0 = X and where PX is the set of all continuous maps from [0, 1] to X. The G-flow M G ( X) is then an object which contains the same information as the so-called Moore path (semi)category of the topological space X. It means that the notion of P-flow is an abstraction of the Moore path (semi)category of a topological space. And for this reason, one of the anonymous referees noticed that there could be a connection with the papers [34,35] to be investigated, the first one introducing a new family of models of type theory based on Moore paths, the second one abstracting the notion of Moore path to characterize type-theoretic weak factorization systems.
Throughout the paper, a general reparametrization category P in the sense of Definition 4.3 is used. The use of the category of reparametrization G of Proposition 4.9 will be required only in the companion paper [17].

Outline of the paper
• Section 2 is a short reminder about ∆-generated spaces and their q-model structure.
• Section 3 is a short reminder about enriched presheaves of topological spaces. Some notations are stated and some formulae are recalled and named for helping to explain the subsequent calculations.
• Section 4 introduces the notion of reparametrization category. We give two examples of such a category (Proposition 4.9 and Proposition 4.11). The notion of reparametrization category given here is very general.
• Section 5 sketches the theory of enriched presheaves over a reparametrization category P, called P-spaces. The main result of Section 5 is that it is possible to define a structure of biclosed semimonoidal category. The word semi means that the monoidal structure does not necessarily have a unit. This section contains also various calculations which will be used in the sequel.
• Section 6 introduces the notion of P-flow and proves various basic properties, like its local presentability. It is also proved that the (execution) path P-space functor from P-flows to Pspaces is a right adjoint.
• Section 7 recalls some results about Isaev's work [27] about model categories of fibrant objects. It also provides, as an easy consequence of Isaev's paper, a characterization of the class of weak equivalences as a small injectivity class with an explicit description of the generating set. The latter result is a particular case of a more general result due to Bourke.
• Section 8 expounds and describes as completely as possible the q-model structure of P-flows. It is very similar to the q-model structure of flows in many ways. In fact, we follow the method of [16] step by step constructing the q-model structure of Flow by using [27].
• Section 9 proves that the path P-space functor from P-flows to P-spaces takes (trivial resp.) q-cofibrations of P-flows between q-cofibrant P-flows to (trivial resp.) projective q-cofibrations of P-spaces. In particular, the P-space of execution paths of a q-cofibrant P-flow is a projective q-cofibrant P-space. The material expounded in [19] is used in a crucial way.

Notations, conventions and prerequisites
We refer to [1] for locally presentable categories, to [39] for combinatorial model categories. We refer to [25,26] for more general model categories. We refer to [28] and to [3,Chapter 6] for enriched categories. All enriched categories are topologically enriched categories: the word topologically is therefore omitted. What follows is some notations and conventions.
• A := B means that B is the definition of A.
• K op denotes the opposite category of K.
• Obj(K) is the class of objects of K, Mor(K) is the category of morphisms of K with the commutative squares for the morphisms.
• K I is the category of functors and natural transformations from a small category I to K.
• ∅ is the initial object, 1 is the final object, Id X is the identity of X.
• K(X, Y ) is the set of maps in a set-enriched, i.e. locally small, category K.
• K(X, Y ) is the space of maps in an enriched category K. The underlying set of maps may be denoted by K 0 (X, Y ) if it is necessary to specify that we are considering the underlying set.
• The composition of two maps f : A → B and g : B → C is denoted by gf or, if it is helpful for the reader, by g.f ; the composition of two functors is denoted in the same way.

History, informal comments and acknowledgments
The route which would lead to this work is long. After the redaction of [13], I thought that the intermediate category to put between multipointed d-spaces and flows was a Moore version of multipointed d-spaces (see e.g [14]). The latter idea seems to be a dead-end. In fall 2017, I had the idea to use a Moore version of flows instead of multipointed d-spaces. It did not come to my mind at first because the composition of paths is already strictly associative on flows. This led me to a first attempt to define Moore flows in spring 2018. A new obstacle then arised. Due to the non-contractibility of the classifying space of the group of nondecreasing homeomorphisms from [0, 1] to itself, the behavior of the homotopy colimit of spaces (see the introduction of [15]) prevents the left Quillen functor M ! from Moore flows to flows of Proposition 10.6 from being homotopically surjective. I had fruitful email discussions with Tim Porter during summer 2018, and I thank him for that, from which a better idea emerged: I had to work in an enriched setting. The preceding obstable is then overcome thanks to [15,Theorem 7.6] which is recalled in Theorem 10.8. It was then necessary to prove that a q-cofibrant Moore flow has a projective q-cofibrant P-space of execution paths. This fact is required also for the functor M ! to be homotopically surjective. The redaction of this proof (see Theorem 9.11) led me to the formulation of a Moore flow as a semicategory enriched over the biclosed semimonoidal structure of P-spaces. Some explanations about the axioms satisfied by a general reparametrization category are given after Definition 4.3. I thank the two anonymous referees for their reports and suggestions. One of the two referees suggested two terminologies P-space or P-trace for Definition 5.1. When P is the terminal category 1, the semimonoidal category ([P op , Top] 0 , ⊗) is the (semi)monoidal category (Top, ×). It is the reason why the first suggestion is chosen.

Reminder about topological spaces
The category Top is either the category of ∆-generated spaces or the full subcategory of ∆-Hausdorff ∆-generated spaces (cf. [19, Section 2 and Appendix B]). We summarize some basic properties of Top for the convenience of the reader: • Top is locally presentable.
Accepted in Compositionality on 2022-07-11. Click on the title to verify.
• The inclusion functor from the full subcategory of ∆-generated spaces to the category of general topological spaces together with the continuous maps has a right adjoint called the ∆kelleyfication functor. The latter functor does not change the underlying set.
• Let A ⊂ B be a subset of a space B of Top. Then A equipped with the ∆-kelleyfication of the relative topology belongs to Top.
• The colimit in Top is given by the final topology in the following situations: -A transfinite compositions of one-to-one maps.
-A pushout along a closed inclusion.
-A quotient by a closed subset or by an equivalence relation having a closed graph.
In these cases, the underlying set of the colimit is therefore the colimit of the underlying sets.
In particular, the CW-complexes, and more generally all cellular spaces are equipped with the final topology. Note that cellular spaces are even Hausdorff (and paracompact, normal, etc...).
• Top is cartesian closed. The internal hom TOP(X, Y ) is given by taking the ∆-kelleyfication of the compact-open topology on the set TOP(X, Y ) of all continuous maps from X to Y . The q-model structure 2 is denoted by Top q . It is enriched over itself using the binary product. The terminology of cartesian enrichment is used sometimes. We keep denoting the set of continuous maps from X to Y by Top(X, Y ) and the space of maps from X to Y by TOP(X, Y ) like in our previous papers (and not by Top 0 (X, Y ) and Top(X, Y ) respectively).
A topological space is connected if and only if it is path-connected and every topological space is homeomorphic to the disjoint sum of its nonempty path-connected components [13,Proposition 2.8]. The space CC(Z) is the space of path-connected components of Z equipped with the final topology with respect to the canonical map Z → CC(Z), which turns out to be the discrete topology by [19,Lemma 5.8].
Note that the paper [15], which is used several times in this work, is written in the category of ∆-generated spaces; it is still valid in the category of ∆-Hausdorff ∆-generated spaces. This point is left as an exercise for the idle mathematician: see [19,Section 2 and Appendix B] for any help about the topology of ∆-generated spaces. 3 Reminder about enriched presheaves of topological spaces and consequently, together with the enriched Yoneda lemma, we obtain the homeomorphism It implies that for any enriched functor F : P × P op → Top and any topological space U , there is a homeomorphism because Top is cartesian closed. 1. The semimonoidal structure is strict, i.e. the associator is the identity.
2. All spaces of maps P( , ) for all objects and of P are contractible.
3. For all maps φ : → of P, for all 1 , 2 ∈ Obj(P) such that 1 ⊗ 2 = , there exist two maps φ 1 : 1 → 1 and φ 2 : The semimonoidal structure enables us to have a semigroup structure on objects, to formulate the third axiom and to define the enriched functors s L and s R of Proposition 5.8. The semigroup structure on objects is important for Lemma 5.10. It plays a central role for the study of the biclosed semimonoidal structure of P-spaces defined in Definition 5.7 (see Theorem 5.14). This biclosed semimonoidal structure is required to formalize P-flows as enriched semicategories in Definition 6.1. This point of view on P-flows, which is not the initial one I had chronologically, is used mainly for the proof of the key fact that a q-cofibrant P-flow has a projective q-cofibrant P-space of execution paths (Theorem 9.10 and Theorem 9.11). The latter proof relies on the calculations made in [19] in the setting of semicategories enriched over topological spaces (a.k.a. flows). The reparametrization category must be enriched to be able to take into account the contractibility of the spaces of maps. All spaces of maps must be contractible to use Theorem 10.8. Otherwise the non-contractibility of the classifying space of the group of nondecreasing homeomorphisms from [0, 1] to itself prevents the left Quillen functor M ! of Proposition 10.6 from P-flows to flows from being homotopically surjective. The third axiom is used in the proof of Proposition 5.17. It enables us in Section 10 to define the right Quillen functor M from flows to P-flows in Proposition 10.5.

Notation 4.4
To stick to the intuition, we set + := ⊗ for all , ∈ Obj(P). Indeed, morally speaking, is the length of a path.
A reparametrization category P is an enriched category with contractible spaces of morphisms such that the set Obj(P) of objects of P has a structure of a semigroup with a composition law denoted by +, such that the set map is continuous for all 1 , 1 , 2 , 2 ∈ Obj(P), and such that every map of P is of the form φ 1 ⊗ φ 2 (not necessarily in a unique way).
denotes the continuous map defined by We have the obvious proposition: Accepted in Compositionality on 2022-07-11. Click on the title to verify.

Proposition 4.9
There exists a reparametrization category, denoted by G, such that the semigroup of objects is the open interval ]0, +∞[ equipped with the addition and such that for every 1 , 2 > 0, there is the equality where the topology is the ∆-kelleyfication of the relative topology induced by the set inclusion G( 1 , 2 ) ⊂ TOP([0, 1 ], [0, 2 ]) and such that for every 1 , 2 , 3 > 0, the composition map is induced by the composition of continuous maps.
is a nondecreasing homeomorphism. The set map is continuous because the set map which is continuous. The pentagon axiom is clearly satisfied. For  ) and such that for every 1 , 2 , 3 > 0, the composition map is induced by the composition of continuous maps.

Proof:
The proof is similar to the proof of Proposition 4.9.
In the cases of (G, +) and (M, +), the functors ( , ) → + and ( , ) → + coincide on objects, but not on morphisms. The terminal category is a symmetric reparametrization category. We do not know if there exist symmetric reparametrization categories not equivalent to the terminal category.
Definition and notation 5.1 An object of [P op , Top] 0 is called a P-space. Let D be a P-space. Let φ : → be a map of P. Let x ∈ D( ). We will use the notation The motivating example is if x is a path of length , then x.φ is a path of length which is the reparametrization by φ of x. Proof: Let X and Y be two topological spaces. There is a canonical set map induced by the universal property of the binary product. This set map is clearly onto. Let (x 0 , y 0 ) and (x 1 , y 1 ) be two points of X × Y such that x 0 and x 1 (y 0 and y 1 resp.) are in the same connected components. Then there exist continuous paths φ : [0, 1] → X and ψ : [0, 1] → Y such that φ(i) = x i and ψ(i) = y i for i = 0, 1. Therefore (φ, ψ) is a continuous path from (x 0 , y 0 ) to (x 1 , y 1 ).
Thus (x 0 , y 0 ) and (x 1 , y 1 ) are in the same connected components.
Notation 5.4 Let K be an enriched category. Denote by π 0 (K) the enriched category with the same objects as K and with π 0 (K)( , ) = CC(K( , )) for all , ∈ Obj(K). The composition law is defined thanks to Lemma 5.3 by the composite map: does not actually depend of φ ∈ P( , ). We obtain the commutative diagram of functors Accepted in Compositionality on 2022-07-11. Click on the title to verify.
It turns out that the category π 0 (P op ) is equivalent to the terminal category 1 with one object 1 and one map Id 1 . We obtain a commutative diagram of functors Each element of D(1) corresponds for every object of P to a path-connected component of D( ). The above commutative diagram also tells that for every map φ : → of P, the map is the identity of D (1). Thus there exists a decomposition is called the set of (nonempty) path-connected components of D. By convention, By considering the particular case where the reparametrization category P is the terminal category, we see that this generalizes the notion of path-connected component of a topological space.
Definition 5.7 Let D and E be two P-spaces. Let Proposition 5.8 (The left shift functor and the right shift functor) The following data assemble to an enriched functor s L : P → P: The following data assemble to an enriched functor s R : P → P: Proof: The maps P( , ) → P( + , + ) and P( , ) → P( + , + ) are continuous for all , ∈ Obj(P). Accepted in Compositionality on 2022-07-11. Click on the title to verify.
Proof: Pick a P-space D. Then there is the sequence of homeomorphisms the first homeomorphism since [P op , Top] is enriched, the second the fourth and the fifth homeomorphisms by (En-Yo) and the third homeomorphism by (En-Nat). By composing with the functor Top({0}, −), we obtain the natural bijection of sets The proof of the first isomorphism is complete thanks to the Yoneda lemma. The proof of the second isomorphism is similar and is left to the reader.

Proposition 5.11
Let D 1 and D 2 be two P-spaces and L ∈ Obj(P). Then the mapping (x, y) → (Id, x, y) yields a surjective continuous map Moreover, the functor ⊗ : be a representative of an element of (D 1 ⊗ D 2 )(L). Then there exist two maps ψ i : i → i for i = 1, 2 such that ψ = ψ 1 ⊗ ψ 2 . By definition of a coend (see Corollary 5.13 for a more detailed explanation), one has (ψ, be the composite of the isomorphisms (by using Lemma 5.10 twice) in ((D⊗E)⊗F )(L). The above sequence of isomorphisms takes the latter at first to the equivalence class of ((Id Id L and by Lemma 5.10 again, to the equivalence class of satisfies the pentagon axiom thanks to the first part of the proof.
Proposition 5.12 Let D 1 , . . . , D n be n P-spaces with n 1. Then there is the natural isomorphism of P-spaces Proof: Let us prove this isomorphism by induction for n 1. The formula is satisfied for n = 1 by (En-Rep). Suppose that it is proved for n 1. Then there is the sequence of natural isomorphisms of P-spaces the first and the second isomorphisms by definition of ⊗ by Fubini and by (CC), and the third isomorphism by Lemma 5.10. The induction is complete.
Corollary 5.13 Let D 1 , . . . , D n be n P-spaces with n 1. Then for all L ∈ Obj(P), the space by the equivalence relation generated by the identifications Proof: Using (Coend) and Proposition 5.12, we see that the space Accepted in Compositionality on 2022-07-11. Click on the title to verify.
defined by and Morally speaking, the x i s are execution paths. The map f 1 composes the reparametrizations φ 1 , . . . , φ n with ψ before applying them to the Moore composition x 1 * · · · * x n . The map f 2 reparametrizes each x i by φ i and then reparametrizes the Moore composition ( When P is the terminal category with one object 1 and one map Id 1 , the tensor product (D ⊗ E)(1) is the quotient of the space D(1)×E(1) by the discrete equivalence relation by Corollary 5.13. In other terms, the tensor product coincides with the binary product and it has therefore a unit in this case. When P is the reparametrization category of Proposition 4.9 or of Proposition 4.11, it is unlikely that the corresponding tensor product of P-spaces over them has a unit but we are unable to prove it.
Theorem 5.14 Let D, E and F be three P-spaces. Let These yield two P-spaces and there are the natural homeomorphisms is continuous for all , ∈ Obj(P). Using the cartesian closedness of Top, we obtain a continuous map which is natural with respect to L ∈ Obj(P), i.e. an element f , of the space of natural transformations The latter space is homeomorphic to is enriched. We obtain, by composition, a continuous map where the right-hand map is induced by the composition of natural transformations (which is continuous). It means that the mapping → [P op , Top](E, (s L ) * F ) yields a well-defined P-space.
Accepted in Compositionality on 2022-07-11. Click on the title to verify.
This proves the first part of the statement of the theorem. There is the sequence of natural homeomorphisms the first and second homeomorphisms by (En-Nat), the third homeomorphism because Top is enriched cartesian closed, the fourth homeomorphism by (En-Yo), and finally the last homeomorphism since [P op , Top] is enriched and by definition of ⊗. By composing with the functor Top({0}, −), we obtain the desired adjunction. The proof is complete thanks to Proposition 5.11.
where U is a topological space and where is an object of P.
Proof: One has Using Lemma 5.10, we obtain Using Lemma 5.10 again, we obtain Proposition 5.17 Let U and U be two topological spaces. There is the natural isomorphism of P-spaces Proof: Since Top is cartesian closed, it suffices to consider the case where U = U is a singleton. In that case, by Corollary 5.13, the topological space Let ψ ∈ P(L, + ) for some , ∈ Obj(P). By definition of a reparametrization category, write ψ = ψ 1 ⊗ ψ 2 with ψ 1 : 1 → and ψ 2 : 2 → . Then we Proposition 5.18 Let D and E be two P-spaces. Then there is a natural homeomorphism Proof: Let Z be a topological space. There is the sequence of natural homeomorphisms the first fourth and fifth homeomorphisms by (En-Adj), the second homeomorphism by Theorem 5.14, the third homeomorphism since (s L ) * ∆ P op (Z) = ∆ P op (Z) for all ∈ Obj(P) and the last homeomorphism since Top is enriched cartesian closed. By composing with the functor Top({0}, −), we obtain the natural bijection of sets The proof is complete thanks to the Yoneda lemma.

P-flows
A semicategory, also called nonunital category in the literature, is a category without identity maps in the structure. It is enriched over a biclosed monoidal category (V, ⊗, I) if it satisfies all axioms of enriched category except the one involving the identity maps, i.e. the enriched composition is associative and not necessarily unital. Since the existence of a unit I is not necessary anymore, it makes sense to define the notion of semicategory enriched over a biclosed semimonoidal category. This section starts on purpose with the following very concise definition which is going to be explained right after. Definition 6.1 Let P be a reparametrization category. A P-flow X is a small semicategory enriched over the biclosed semimonoidal category ([P op , Top] 0 , ⊗). The category of P-flows is denoted by PFlow.
The following definition is used only in the companion paper [17]. We give it for completeness. Definition 6.2 A Moore flow is a G-flow where G is the reparametrization category of Proposition 4.9.
We now introduce some notations and definitions to provide more details. A P-flow X consists of a set of states X 0 , for each pair (α, β) of states a P-space P α,β X of [P op , Top] 0 and for each triple (α, β, γ) of states an associative composition law A map of P-flows f from X to Y consists of a set map (often denoted by f as well if there is no possible confusion) together for each pair of states (α, β) of X with a natural transformation The topological space P α,β X( ) is denoted by P α,β X and is called the space of execution paths of length . There is the sequence of homeomorphisms the first homeomorphism by definition of ⊗ and since [P op , Top] is enriched, the second homeomorphism by (En-Yo) and the third homeomorphism by (En-Nat).
Consequently, a P-flow consists of a set of states X 0 , for each pair (α, β) of states a P-space P α,β X of [P op , Top] 0 and for each triple (α, β, γ) of states an associative composition law * : P 1 α,β X × P 2 β,γ X → P 1 + 2 α,γ X which is natural with respect to ( 1 , 2 ). In other terms, for any map φ i : i → i of P with i = 1, 2, there is the commutative diagram of topological spaces The associativity means that Proof: Let X be a P-flow. Let S be a set. Then there are the bijections of sets Set(X 0 , S) ∼ = PFlow(X, S ) and Set(S, X 0 ) ∼ = PFlow(S , X).
Definition 6.5 Let X be a P-flow. The P-space of execution paths PX of X is by definition the P-space It yields a well-defined functor P : PFlow → [P op , Top] 0 . The image of is denoted by P U . We therefore have the equality Notation 6.6 Let X be a P-flow. Denote by #X the cardinal of X, i.e. the sum of the cardinal of the set of states of X and of the cardinals of all topological spaces P α,β X for α, β running over X 0 and for running over the set of objects of P.
We want to emphasize the following elementary fact: Lemma 6.7 Let C be a complete category. Let X : I → C and Y : I → C be two small diagrams of C. Then the natural map lim Proof: The functor lim ← − : C I → C is a right adjoint, the left adjoint being the constant diagram functor. Therefore, it preserves limits, and in particular binary product.
Theorem 6.8 The category PFlow is bicomplete.
Proof: Let X : I −→ PFlow be a functor from a small category I to PFlow. Let Y be the P-flow defined as follows: • The set of states Y 0 of Y is defined as being the limit as sets lim ← − X(i) 0 (Proposition 6.4).
) be the image of α (β, γ resp.) in X(i) 0 . Then the composition map * : P α,β Y × P β,γ Y → P + α,γ Y is taken as the limit of the * i : P αi,βi X(i) × P βi,γi X(i) → P + αi,γi X(i) (we implicitly use Lemma 6.7). We obtain a well-defined P-flow Y . It is clearly the limit lim ← − X in PFlow. The constant diagram functor ∆ I : PFlow → PFlow I commutes with limits since limits in PFlow I are calculated objectwise. Every map of PFlow I f : X → ∆ I Y certainly factors as a composite There is a set of such Z up to isomorphisms. We therefore have obtained a set of solutions. By Freyd's Adjoint Functor Theorem, the constant diagram functor ∆ I from PFlow to the category PFlow I has a left adjoint. Proof: A map of P-flows from Glob(D) to X is determined by the choice of two states α and β of X and by a map from D to P α,β X because there is no composition law in Glob(D).

Theorem 6.11
The category PFlow is locally presentable.
The category of small categories enriched over a closed monoidal category V is locally presentable as soon as V is locally presentable. It is [29, Theorem 4.5] whose proof can probably be adapted to our situation: P-flows are small semicategories enriched over a biclosed semimonoidal category which is locally presentable. It is shorter to proceed as follows.  Consider a map φ : → of P. It yields the commutative diagram of spaces We deduce that (α , β ) = (α , β ).
Since the space P( , ) is contractible for any , ∈ Obj(P), it is nonempty. The mapping → (α , β ) from Obj(P) to U 0 ×U 0 is therefore a constant which only depends on f . Consequently, the map of P-spaces f : D → PU factors uniquely as a composite for some (α, β) ∈ U 0 × U 0 . This characterizes a map of P-flow f : Glob(D) → U . We easily see that the mapping f → f is bijective. is finitely accessible since colimits are generated by free finite compositions of execution paths. On the contrary, it is unlikely that the functor P : PFlow → [P op , Top] 0 is finitely accessible. However, in the case of a transfinite tower of objectwise relative-T 1 inclusions of spaces, one has: Theorem 6.14 Let λ be a limit ordinal. Let X : λ → PFlow be a colimit-preserving functor of P-flows such that for all µ < λ, the map of P-spaces PX µ → PX µ+1 is an objectwise relative-T 1 inclusion. Then the canonical map is an isomorphism of P-spaces. Moreover the topology of P lim − → X is the final topology for all > 0.
Proof: The topology of P lim − → X is the final topology for all > 0 because the relative-T 1 inclusions are one-to-one and because colimits are calculated objectwise in [P op , Top] 0 . The rest of the proof is similar to the proof of [19,Theorem 5.5]. The key point is to prove that the set map is continuous. It suffices to prove that every composite set map is continuous as soon as the left-hand map is continuous. Since [0, 1] is finite relative to relative-T 1 -inclusions by [19,Proposition 2.5], there exists an ordinal ν < λ such that the diagram is commutative. The top arrow P 1 X ν × X 0 ν P 2 X ν → P 1+ 2 X ν is continuous because it is the composition law of a P-flow. We deduce that the bottom arrow

Reminder about model categories of fibrant objects
We start first by some notations: • (−) cof denotes a cofibrant replacement functor.
• f g means that f satisfies the left lifting property (LLP) with respect to g, or equivalently when 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.
All objects of all model categories of this paper are fibrant. We will be using the following characterization of a Quillen equivalence. A Quillen adjunction F G : C D is a Quillen equivalence if and only if for all objects X of D, the natural map F (G(X) cof ) → X is a weak equivalence of D (the functor is then said homotopically surjective) and if for all cofibrant objects Y of C, the unit of the adjunction Y → G(F (Y )) is a weak equivalence of C.
We summarize in the following theorem what we want to use from [27]. For short, Isaev's paper gives a systematic way to construct model categories of fibrant objects. We already used some part of the following results in [16] to simplify the construction of the q-model category of flows. Unlike in [16], we also want to add some comments about the class of weak equivalences of such model categories.

Theorem 7.1 [27, Theorem 4.3, Proposition 4.4, Proposition 4.5 and Corollary 4.6]
Let K be a locally presentable category. Let I be a set of maps of K such that the domains D of the maps of I are I-cofibrant (i.e. such that ∅ → D belongs to cof (I)). Suppose that for every map i : U → V ∈ I, there exists an object C U (V ) such that the relative codiagonal map such that the left-hand map belongs to cof (I). Let Suppose that there exists a path functor Path : K → K, i.e. an endofunctor of K equipped with two natural transformations τ : Id ⇒ Path and π : Path ⇒ Id × Id such that the composite π.τ is the diagonal. Moreover we suppose that the path functor satisfies the following hypotheses: 1. With π = (π 0 , π 1 ), π 0 : Path(X) → X and π 1 : Path(X) → X have the RLP with respect to I.
2. The map π : Path(X) → X × X has the RLP with respect to the maps of J I .
Then there exists a unique model category structure on K such that the set of generating cofibrations is I and such that the set of generating trivial cofibrations is J I . Moreover, all objects are fibrant.
Isaev's paper contains also an interesting characterization of the class of weak equivalences which is recalled now: Proof: By Theorem 7.2, a map f : X → Y of K is a weak equivalence if and only if it satisfies the RLP up to homotopy with respect to any map of I. This definition can be reworded as follows.
For any commutative diagram of solid arrows of K of the form there exist h and k making the diagram commutative. It is exactly the injectivity condition when we regard this diagram as a diagram of Mor(K).
We can now reformulate Isaev's characterization of the class of weak equivalences of K as follows: for U → V running over I. In particular, the class of weak equivalences of K is closed under small products because it is a small injectivity class of a locally presentable category.
The idea of Corollary 7.4 comes from a passing remark due to Jeff Smith and mentioned in [5]. [4,Theorem 14] contains interesting results of the same kind for more general combinatorial model categories.

Homotopy theory of P-flows
We are going to construct a model structure on PFlow, called the q-model structure, by mimicking the method used in [16] for the construction of the q-model structure of flows.
We equip the category [P op , Top] 0 with the projective q-model structure [15,Theorem 6.2]. It is denoted by [P op , Top q ] proj 0 . The (trivial resp.) projective q-fibrations are the objectwise (trivial resp.) q-fibrations of spaces. The weak equivalences are the objectwise weak homotopy equivalences. The set of generating projective q-cofibrations is the set of maps induced by the inclusions S n−1 ⊂ D n . The set of generating trivial projective q-cofibrations is the set of maps where the maps D n ⊂ D n+1 are induced by the mappings (x 1 , . . . , x n ) → (x 1 , . . . , x n , 0).

Notation 8.2 Let
for all ∈ Obj(P) if and only if for any α, β ∈ X 0 and any ∈ Obj(P), the map of topological spaces P α,β X → P f (α),f (β) Y satisfies the RLP with respect to the continuous map U → V .
Proof: A morphism of P-flows f : X → Y satisfies the RLP with respect to for all ∈ Obj(P) if and only if the set maps are onto for all ∈ Obj(P). By Proposition 6.10, it is equivalent to saying that the set maps are onto for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P). By Proposition 8.1, it is equivalent to saying that the set maps are onto for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P). It is equivalent to saying that for any α, β ∈ X 0 and any ∈ Obj(P), the map of topological spaces P α,β X → P f (α),f (β) Y satisfies the RLP with respect to the continuous map U → V . Note that the connectedness hypothesis is necessary. Indeed, D and E being two P-spaces, the P-flow Glob(D E) has two states whereas the P-flow Glob(D) Glob(E) has four states. Proof: Let D : I → [P op , Top] 0 be a functor where I is a connected small category. We obtain the sequence of natural bijections: The proof is complete using the Yoneda lemma.
There is the obvious proposition: Proposition 8.6 Let X be a P-flow. Let U be a topological space. The following data assemble into a P-flow denoted by {U, X} S : • The set of states is X 0 .
Note that we stick to the notations of [8,Notation 7.6] and [16,Notation 3.8]. Indeed, like for the case of flows, {U, X} S could be denoted by X U only if U is connected and nonempty. The correct definition of X U in the non-connected case is as follows: The mapping (U, X) → X U induces a well-defined functor from Top op × PFlow to PFlow.
We could prove that the functor X → X U has a left adjoint X → X ⊗ U and that the axioms of tensored and cotensored categories are satisfied. One of the ingredients of the proof is that every ∆-generated space is homeomorphic to the disjoint union of its nonempty connected components.

Theorem 8.8
There exists a unique model structure on PFlow such that I P + is the set of generating cofibrations and such that all objects are fibrant. The set of generating trivial cofibrations is J P . It is called the q-model structure.
Proof: We are going to check all hypotheses of Theorem 7.1. The category PFlow is locally presentable by Theorem 6.11. By Proposition 8.1, the map of P-spaces is a projective q-cofibration for all ∈ Obj(P) and all n 0. Thus all domains of all maps of I P + are cofibrant with respect to I P + . We can factor the relative codiagonal map D n S n−1 D n → D n as a composite D n S n−1 D n ⊂ D n+1 −→ D n for all n 0. We obtain for all ∈ Obj(P) a composite map of P-spaces such that the left-hand map is a projective q-cofibration by Proposition 8.1. Thus for U → V being one of the maps Glob(F P op S n−1 ) ⊂ Glob(F P op D n ) for n 0 and ∈ Obj(P), we set For all n 0, we have a pushout diagram of topological spaces which gives rise to the pushout diagram of P-spaces for all ∈ Obj(P) by Proposition 8.1. The latter diagram gives rise to the pushout diagram of P-flows Glob(F P op S n ) for all n 0 and all ∈ Obj(P) by Proposition 8.5. This implies that for U → V being one of the maps for n 0 and ∈ Obj(P), the map V U V −→ C U (V ) belongs to cell(I P + ). The map C : ∅ → {0} gives rise to the relative codiagonal map {0} {0} → {0} . Thus we set In this case, the map In this case, the map V U V −→ C U (V ) is Id {0} which belongs to cell(I P + ). The set of generating trivial cofibrations will be therefore the set of maps for n 0 and ∈ Obj(P). The composite map {0, 1} ⊂ [0, 1] → {0} yields a natural composite map of P-flows which gives rise to the composite continuous map P α,β X → TOP([0, 1], P α,β X) → P α,β X × P α,β X on the spaces of paths for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P). We have obtained a path object in the sense of Theorem 7.1. Since the maps π 0 and π 1 are bijective on states, they satisfy the RLP with respect to are trivial q-fibrations for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P). The latter fact is a consequence of the fact that the q-model structure of Top is cartesian monoidal and from the fact that the inclusions {0} ⊂ [0, 1] and {1} ⊂ [0, 1] are trivial q-cofibrations of Top. Finally, we have to check that the map π : Path(X) → X × X satisfies the RLP with respect to the maps for all n 0 and for all ∈ Obj(P). By Proposition 8.3 again, it suffices to prove that the map is a q-fibration of topological spaces for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P). Since the q-model structure of Top is cartesian monoidal, this comes from the fact that the inclusion {0, 1} ⊂ [0, 1] is a q-cofibration of Top.
where the notation P(X) means the diagram of P-spaces defined on objects by the mapping Proof: All maps of the diagram of P-flows Glob(P ) are bijective on states. Therefore, a map Glob(P ) → X is determined by the image (α, β) ∈ X 0 ∅ × X 0 ∅ of the states 0 and 1 of the P-flow Glob(P ∅ ) and, once this choice is done, by a map P → P.X.
Proof: It is a corollary of Proposition 8.11 with the 2-object small category ∅ → 1.
The following well-known proposition is an easy consequence of [31, page 219 (2)]. It is explicitly stated for helping purpose. • The functor R : D → C induces a functor still denoted by R from the functor category D I to the functor category C I which takes Y to the functor R.Y .
We obtain an adjunction L R : C I D I .
Proof: One has the natural bijections Proposition 8.14 Let n 0. Let ∈ Obj(P). A morphism of P-flows f : X → Y satisfies the RLP up to homotopy with respect to Glob(F P op S n−1 ) → Glob(F P op D n ) if and only if for all α, β ∈ X 0 , the map of topological spaces P α,β X → P f (α),f (β) Y satisfies the RLP up to homotopy with respect to the continuous map S n−1 ⊂ D n .
Proof: For U → V being one of the maps of we have (see the proof of Theorem 8.8) Therefore, by Theorem 7.3, a morphism of P-flows f : X → Y satisfies the RLP up to homotopy with respect to the map Glob(F P op S n−1 ) → Glob(F P op D n ) if and only if it injective with respect to the map of maps in the category Mor(PFlow), in other terms if and only if the set map Accepted in Compositionality on 2022-07-11. Click on the title to verify. induced by the above map of maps is onto. However, we have the sequences of bijections by Corollary 8.12 by Proposition 8.13 and by Proposition 8.13.
Therefore a morphism of P-flows f : X → Y satisfies the RLP up to homotopy with respect to the map Glob(F P op S n−1 ) → Glob(F P op D n ) if and only if the set map is onto. By Theorem 7.3, the latter condition is equivalent to saying that the map P α,β X → P f (α),f (β) Y satisfies the RLP up to homotopy with respect to the continuous map S n−1 ⊂ D n . Proof: Obvious. Theorem 8.16 A map of P-flows f : X → Y is a weak equivalence if and only if f induces a bijection between the set of states of X and Y and for all (α, β) ∈ X 0 × X 0 , the map of P-spaces P α,β X → P f (α),f (β) Y is a weak equivalence.
Proof: By Theorem 7.2, a map of P-flows f : X → Y is a weak equivalence if and only if it satisfies the RLP up to homotopy with respect to the maps of Accepted in Compositionality on 2022-07-11. Click on the title to verify.
By Proposition 8.15 and by Proposition 8.14, a map of P-flows f : X → Y is then a weak equivalence if and only if f induces a bijection between the set of states of X and Y and for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P), the map P α,β X → P f (α),f (β) Y satisfies the RLP up to homotopy with respect to the maps S n−1 ⊂ D n for all n 0. By Theorem 7.2 applied to the q-model category Top, it implies that a map of P-flows f : X → Y is then a weak equivalence if and only if f induces a bijection between the set of states of X and Y and for all (α, β) ∈ X 0 × X 0 and all ∈ Obj(P), the map P α,β X → P f (α),f (β) Y is a weak homotopy equivalence of topological spaces.
9 Q-cofibrant P-flows have a projective q-cofibrant P-space of execution paths Definition 9.1 Let f : D → E and g : F → G be two maps of P-spaces. Then the pushout product, with respect to ⊗, denoted by f g, of f and g is the map of P-spaces induced by the universal property of the pushout.
Notation 9.2 f g denotes the pushout product with respect to the binary product of two maps f and g of a bicomplete category.
Proposition 9.3 Let f : U → V and f : U → V be two maps of Top. Let , ∈ Obj(P). Then there is the isomorphism of P-spaces the first isomorphism by Proposition 5.16 and the second isomorphism by Proposition 8.1. By Proposition 5.16 again, the codomain of F P op (f ) F P op (g) is Theorem 9.4 Let f : D → E and g : F → G be two maps of P-spaces. Then • If f and g are projective q-cofibrations, then f g is a projective q-cofibration.
• If moreover f or g is a trivial projective q-cofibration, then f g is a trivial projective qcofibration.
Proof: The projective q-cofibrations of P-spaces are generated by the set of maps induced by the inclusions S n−1 ⊂ D n . The set of generating trivial projective q-cofibrations is the set of maps where the maps D n ⊂ D n+1 are induced by the mappings (x 1 , . . . , x n ) → (x 1 , . . . , x n , 0). From Proposition 9.3, Proposition 8.1 and from the fact that the q-model structure of Top is cartesian monoidal, we deduce the inclusions I I ⊂ cof (I), The proof is complete thanks to Theorem 5.14 and [26,Lemma 4.2.4].
• There are the relations (group A) c i .c j = c j−1 .c i if i < j (which means since c i and c j may correspond to several maps that if c i and c j are composable, then there exist c j−1 and c i composable satisfying the equality).
• There are the relations (group B) I i .I j = I j .I i if i = j. By definition of these maps, I i is never composable with itself.
• There are the relations (group C) By definition of these maps, c i and I i are never composable as well as c i and I i+1 .
The maps raising the degree are the inclusion maps in the above sense. The maps decreasing the degree are the composition maps in the above sense.
Notation 9.5 The latching object at n of a diagram D over P u,v (S) is denoted by L n D.
We recall the important theorem: Accepted in Compositionality on 2022-07-11. Click on the title to verify.
If S and T are two subsets of {0, . . . , p} such that S ⊂ T , let be the morphism Then: 1. the mappings S → C p (S) and i T S → C p (i T S ) give rise to a functor from the order complex of the poset {0 < · · · < p} to [P op , Top] 0 2. there exists a canonical morphism . . . , p}). and it is equal to the morphism f 0 · · · f p .

Proof:
The proof is similar to the proof of [11,Theorem B.3] for the case of (Top, ×). It is still valid here because the semimonoidal structure ⊗ is associative and biclosed by Theorem 5.14. The statement has to be slightly modified since ⊗ is not assumed to be symmetric. Proposition 9.9 With the notations above. Let n ∈ Obj(P g(0),g(1) (A 0 )) with n = ((u 0 , 1 , u 1 ), (u 1 , 2 , u 2 ), . . . , (u n−1 , n , u n )).

Then the continuous map
is the pushout product of the maps ∅ → P ui−1,ui A for i running over {i ∈ [1, n]| i = 0} and of the maps P g(0),g (1) A → T for i running over {i ∈ [1, n]| i = 1}. Moreover, if for all i ∈ [1, n], we have i = 0, then L n D f = ∅.
Proof: It is a consequence of Proposition 9.8. Theorem 9.10 With the notations above. Assume that the map of P-spaces ∂Z → Z is a (trivial resp.) projective q-cofibration. If PA is a projective q-cofibrant P-space, then Pf : PA → PX is a (trivial resp.) projective q-cofibration of P-spaces.
Proof: The particular case ∂Z = Z, f = Id A and A = X yields the isomorphism of P-spaces We have a map of diagrams D Id A → D f which induces for all n ∈ Obj(P g(0),g (1) 2 , u 2 ), . . . , (u n−1 , n , u n )).
There are two mutually exclusive cases: (a) All i for i = 1, . . . n are equal to zero.
Accepted in Compositionality on 2022-07-11. Click on the title to verify.
In the case (a), we have Moreover, by Proposition 9.9, we have L n D Id A = L n D f = ∅. We deduce that the map is isomorphic to the identity of D Id A (n). In the case (b), The map is by Proposition 9.9 a pushout product of several maps such that one of them is the identity map Id : P g(0),g(1) A → P g(0),g (1) A because i = 1 for some i. Therefore the map L n D Id A → D Id A (n) is an isomorphism. We deduce that the map is isomorphic to the map L n D f → D f (n). By Proposition 9.9, the map L n D f → D f (n) is a pushout product of maps of the form ∅ → P α,β A and of the form f : P g(0),g (1) A → T for all objects n ∈ Obj(P g(0),g(1) (A 0 )). We conclude that the map is for all n either an isomorphism, or a pushout product of maps of the form ∅ → P α,β A and of the form f : P g(0),g (1) A → T , the latter appearing at least once in the pushout product and being a pushout of the map of P-spaces ∂Z → Z. We are now ready to complete the proof.
Suppose now that PA is a projective q-cofibrant P-space. Therefore for all (α, β) ∈ A 0 × A 0 , the P-space P α,β A is projective q-cofibrant. We deduce that the map L n D f → D f (n) is always a projective q-cofibration of [P op , Top] 0 for all n by Theorem 9.4. We deduce that the map of diagrams D Id A → D f is a Reedy projective q-cofibration. Therefore by passing to the colimit which is a left Quillen adjoint by Theorem 9.6, we deduce that the map PA → PX is a projective qcofibration of [P op , Top] 0 . The case where ∂Z → Z is a trivial projective q-cofibration is similar.
We need for the proof of Theorem 9.11 some elementary information about the m-model structure and the h-model structure of Top to complete the transfinite induction. We invite the reader to look [19, Section 2] up. Theorem 9.11 Let X be a P-flow. If X is q-cofibrant, then the path P-space functor PX is projective q-cofibrant. In particular, for every (α, β) ∈ X 0 × X 0 , the P-space P α,β X is projective q-cofibrant if X is q-cofibrant.
Proof: By hypothesis, X is q-cofibrant. Consequently, the map of P-flows ∅ → X is a retract of a transfinite composition lim − → A λ of pushouts of maps of the form Glob(∂Z) → Glob(Z) such that ∂Z → Z is a projective q-cofibration of [P op , Top] 0 and of the maps C : ∅ → {0} and R : {0, 1} → {0} . Since the set map ∅ 0 = ∅ → X 0 is one-to-one, one can suppose that the map R : {0, 1} → {0} does not appear in the cellular decomposition. Moreover we can suppose that A 0 = X 0 and that for all λ 0, A λ → A λ+1 is a pushout of a map of the form Glob(∂Z) → Glob(Z) where the map ∂Z → Z is a projective q-cofibration of P-spaces.
Consider the set of ordinals {λ | PA λ not q-cofibrant}. Note that this set does not contain 0. If this set is nonempty, then it contains a smallest element µ 0 > 0. By Theorem 9.10, the ordinal µ 0 is a limit ordinal. By definition of the ordinal µ 0 and by Theorem 9.10 again, for all µ < ν µ 0 , the maps of P-spaces PA µ → PA ν are projective q-cofibrations of P-spaces. By [15, Proposition 7.1], we deduce that for all µ < ν µ 0 , the maps of P-spaces PA µ → PA ν are objectwise m-cofibrations of topological spaces, and therefore objectwise h-cofibrations of topological spaces. By [19,Proposition 2.6], we obtain that all µ < ν µ 0 , the maps of P-spaces PA µ → PA ν are objectwise relative-T 1 inclusions. By Theorem 6.14, we obtain that the canonical map is an objectwise homeomorphism. We deduce that PA µ0 is projective q-cofibrant: contradiction.
Accepted in Compositionality on 2022-07-11. Click on the title to verify. In other terms, a flow is a small semicategory enriched over the closed (semi)monoidal category (Top, ×). Let us expand the definition above. A flow X consists of a topological space PX of execution paths, a discrete space X 0 of states, two continuous maps s and t from PX to X 0 called the source and target map respectively, and a continuous and associative map * : {(x, y) ∈ PX × PX; t(x) = s(y)} −→ PX such that s(x * y) = s(x) and t(x * y) = t(y). A morphism of flows f : X −→ Y consists of a set map f 0 : X 0 −→ Y 0 together with a continuous map Pf : PX −→ PY such that Pf (x * y) = Pf (x) * Pf (y).

Let
P α,β X = {x ∈ PX | s(x) = α and t(x) = β}. The category Flow is locally presentable. Every set can be viewed as a flow with an empty path space. The obvious functor Set ⊂ Flow is limit-preserving and colimit-preserving. The following example of flow is important for the sequel: The q-model structure of flows (Flow) q is the unique combinatorial model structure such that I gl ∪ {C, R} is the set of generating q-cofibrations and such that J gl is the set of generating trivial q-cofibrations (e.g. [18,Theorem 7.6] or [16,Theorem 3.11]). The weak equivalences are the maps of flows f : X → Y inducing a bijection f 0 : X 0 ∼ = Y 0 and a weak homotopy equivalence Pf : PX → PY . The q-fibrations are the maps of flows f : X → Y inducing a q-fibration Pf : PX → PY of topological spaces.
Let X be a flow. The P-flow M(X) is the enriched semicategory defined as follows: • The set of states is X 0 .
Accepted in Compositionality on 2022-07-11. Click on the title to verify.
We obtain the Proposition 10.5 The construction above yields a well-defined functor M : Flow → PFlow.
Consider a P-flow Y . For all α, β ∈ Y 0 , let Y α,β = lim − → P α,β Y . Let (α, β, γ) be a triple of states of Y . By Proposition 5.18, the composition law of the P-flow Y induces a continuous map which is associative. We obtain the for all P-flows Y and all flows X. Since the functors M ! : PFlow → Flow and M : Flow → PFlow preserve the set of states, the only problem is to verify that everything is well-behaved with the path spaces, and more specifically with the composition laws. Let f : M ! (Y ) → X be a map of flows. It induces for each pair of states (α, β) of Y 0 a map of topological spaces P α,β M ! (Y ) → P f (α),f (β) X such that for all triples (α, β, γ) of Y 0 , the following diagram is commutative (the horizontal maps being the composition laws): By adjunction, this means that the map of P-spaces is compatible with the composition laws, and then that we have a well-defined map of P-flows from Y to M(X). Let f : Y → M(X) be a map of P-flows. It induces for each pair of states (α, β) of Y 0 a map of P-spaces P α,β Y → ∆ P op (P f (α),f (β) X). We obtain by adjunction a map of topological spaces P α,β M ! (Y ) → P f (α),f (β) X and by naturality of the adjunction the commutative diagram (the horizontal maps being the composition laws): P α,β M ! (Y ) × P β,γ M ! (Y ) * / / P α,γ M ! (Y ) P f (α),f (β) X × P f (β),f (γ) X * / / P f (α),f (γ) X.
Before ending the paper, we need to recall the We can now conclude the first part of the proof of the existence of a zig-zag of Quillen equivalences between the q-model structure of multipointed d-spaces and the q-model structure of flows: By adjunction, we obtain the map of P-spaces P α,β M(X) cof −→ ∆ P op (P α,β X) = P α,β (M(X)).
The latter is a weak equivalence of the projective q-model structure of [P op , Top] 0 since the map M(X) cof → M(X) is a weak equivalence of the q-model structure of PFlow. By Theorem 9.11, the P-space P α,β M(X) cof is projective q-cofibrant. It means that P α,β M(X) cof is a cofibrant replacement of ∆ P op (P α,β X) for the projective q-model structure of [P op , Top] 0 . By Theorem 10.8, we deduce that the continuous map lim − → P α,β M(X) cof −→ P α,β X.
is a weak homotopy equivalence. In other terms, the left Quillen adjoint M ! : PFlow → Flow is homotopically surjective. Let Y be a q-cofibrant P-flow. The map of P-flows is bijective on states since the functors M and M ! preserves the set of states. For every pair (α, β) ∈ X 0 × X 0 , the map Y → M(M ! Y ) induces the map of P-spaces The P-space P α,β Y is projective q-cofibrant by Theorem 9.11. The above map is therefore an objectwise weak homotopy equivalence by Theorem 10.8. In other terms, the left Quillen adjoint M ! : PFlow → Flow is a left Quillen equivalence.