aa r X i v : . [ m a t h . C T ] F e b HOMOTOPY THEORY OF MOORE FLOWS (II)
PHILIPPE GAUCHER
Abstract.
This paper proves that the q-model structures of Moore flows and of mul-tipointed d -spaces are Quillen equivalent. The main step is the proof that the counitand unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects(all objects are q-fibrant). As an application, we provide a new proof of the fact thatthe categorization functor from multipointed d -spaces to flows has a total left derivedfunctor which induces a category equivalence between the homotopy categories. Thenew proof sheds light on the internal structure of the categorization functor which isneither a left adjoint nor a right adjoint. It is even possible to write an inverse up tohomotopy of this functor using Moore flows. Contents
1. Introduction 12. Multipointed d -spaces 43. Moore composition and Ω-final structure 94. From multipointed d -spaces to Moore flows 145. Cellular multipointed d -spaces 186. Chains of globes 267. The unit and the counit of the adjunction on q-cofibrant objects 308. From multipointed d -spaces to flows 39Appendix A. The Reedy category P u,v ( S ): reminder 44References 451. Introduction
Presentation.
This paper is the companion paper of [Gau20a]. The purpose of thesetwo papers is to prove that the q-model structure of multipointed d -spaces introduced in[Gau09] and the q-model structure of flows introduced in [Gau03] are Quillen equivalent.The only known functor which was a good candidate for a Quillen equivalence frommultipointed d -spaces to flows (Definition 8.7) has indeed a total left derived functor inthe sense of [DHSK03] which induces an equivalence of categories between the homotopycategories. It is [Gau09, Theorem 7.5] which is proved more conceptually in Theorem 8.13.However, this functor is neither a left adjoint nor a right adjoint by Theorem 8.8.Multipointed d -spaces and flows can be used to model concurrent processes. For exam-ple, the paper [Gau08] shows how to model all process algebras for any synchronizationalgebra using flows. There are many geometric models of concurrency available in the Mathematics Subject Classification.
Key words and phrases. enriched semicategory, semimonoidal structure, combinatorial model category,Quillen equivalence, locally presentable category, topologically enriched category. iterature [Gra03] [Kri09] [Gra04a] [Gra05] [Gra04b] (the list does not pretend to be ex-haustive). Most of them are used to study the fundamental category of a concurrentprocess or any derived concept. It is something which can be also carried out with theformalisms of flows and multipointed d -spaces. The fundamental category functor iseasily calculable and it interacts very well with the underlying simplicial structures.The purpose of [Gau20a] and of this paper is to prove that there is a zig-zag of Quillenequivalences between multipointed d -spaces and flows by putting in the middle the cate-gory of Moore flows . The Quillen equivalence between flows and Moore flows is provedin [Gau20a, Theorem 10.9]. The Quillen equivalence between multipointed d -spaces andMoore flows is proved in Theorem 8.1. The latter theorem is a consequence of the struc-tural properties of the adjunction between multipointed d -spaces and Moore flows whichcan be summarized as follows: Theorem. (Corollary 7.7, Corollary 7.10 and Theorem 8.1) The adjunction M G ! ⊣ M G : G Flow ⇆ G dTop between Moore flows and multipointed d -spaces is a Quillen equivalence. The counit mapand the unit map of this Quillen adjunction are isomorphisms on q-cofibrant objects (recallthat all objects are q-fibrant). The Moore flows enable us also to write explicitly an “inverse up to homotopy” ofthe categorization functor in Definition 8.12. As an application, a new proof of [Gau09,Theorem 7.5] is provided in Theorem 8.13 which is totally independent from [Gau05] and[Gau09].
Outline of the paper. • Section 2 is a reminder about multipointed d -spaces and about its q-model structure.It also contains new results about the topology of the space of execution paths whichare either implicit in our previous papers about the subject or even new. • The functor Ω which forgets the set of execution paths of a multipointed d -space istopological. Section 3 gives an explicit description of the Ω-final structure in termof Moore composition. It culminates with Theorem 3.9. The calculations are a bitlaborious but some of them are used further in the paper. • Section 4, after a reminder about Moore flows and their q-model structure, describesthe adjunctions between multipointed d -spaces and Moore flows. The right adjointfrom multipointed d -spaces to Moore flows is quite easy to build. The existence of theleft adjoint is straightforward. • Section 5 gathers some geometric properties of cellular multipointed d -spaces concern-ing their underlying topologies, the topologies of their spaces of execution paths andsome of their structural properties like Theorem 5.19 which has many consequences.It also contains Theorem 5.21 which provides a kind of normal form for the execu-tion paths of a cellular multipointed d -space obtained as a pushout along a generatingq-cofibration. • Section 6 studies chains of globes. It is an important geometric object for the proofsof this paper. It enables us to understand what happens locally in a space of executionpaths of a cellular multipointed d -space. Section 7 is the core of the paper. It proves that the unit and the counit of theadjunction is an isomorphism on q-cofibrant objects in Corollary 7.7 and in Corol-lary 7.10. The main tool of this part is Corollary 7.5 which proves that the rightadjoint constructed in Section 4 preserves pushouts of cellular multipointed d -spacesalong q-cofibrations. It relies on Theorem 7.3 whose proof performs an analysis of theexecution paths in a pushout along a q-cofibration and on Theorem 7.4 whose proofcarries out a careful analysis of the underlying topology of the spaces of execution pathsinvolved. • Section 8 is the concluding section. It establishes that the adjunction between mul-tipointed d -spaces and Moore flows yields a Quillen equivalence between the q-modelstructures. Finally, it provides, as an application, a more conceptual proof of the factthat the categorization functor cat from multipointed d -spaces to flows has a total leftderived functor which induces a category equivalence between the homotopy categoriesof the q-model structures. Prerequisites.
We refer to [AR94] for locally presentable categories, to [Ros09] for com-binatorial model categories. We refer to [Hov99] and to [Hir03] for more general modelcategories. We refer to [Kel05] and to [Bor94, Chapter 6] for enriched categories. All en-riched categories are topologically enriched categories: the word topologically is thereforeomitted . What follows is some notations and conventions. • A := B means that B is the definition of A . • ∼ = denotes an isomorphism, ≃ denotes a weak equivalence. • f ↾ A denotes the restriction of f to A . • Set is the category of sets. • T OP is the category of general topological spaces together with the continuous maps. • 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 mor-phisms. • K I is the category of functors and natural transformations from a small category I to K . • Id is the identity map. • ∅ is the initial object, 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 mapsmay be denoted by K ( X, Y ) if it is necessary to specify that we are considering theunderlying set. • The composition of two maps f : A → B and g : B → C is denoted by gf or, if it ishelpful for the reader, by g.f ; the composition of two functors is denoted in the sameway. • The notations ℓ, ℓ ′ , ℓ i , L, . . . mean a strictly positive real number unless specified some-thing else. • For a topological space X , X δ is the topological space with the same underlying setequipped with the discrete topology. • [ ℓ, ℓ ′ ] denotes a segment. A cellular object of a combinatorial model category is an object X such that the canon-ical map ∅ → X is a transfinite composition of pushouts of generating cofibrations. • A compact space is a quasicompact Hausdorff space. • A sequentially compact space is a space such that each sequence has a limit point. • A final quotient (sometimes called a proclusion in the literature) p : A → B is asurjective continuous map between topological spaces such that B is equipped with thefinal topology. • The set of rational numbers is denoted by Q , the set of real numbers by R . • The complement of A ⊂ B is denoted by A c if there is no ambiguity. • Let n >
1. Denote by D n = { b ∈ R n , | b | } the n -dimensional disk, and by S n − = { b ∈ R n , | b | = 1 } the ( n − D = { } and S − = ∅ . Acknowledgment.
I am indebted to Tyrone Cutler for drawing my attention to thepaper [CSW14]. 2.
Multipointed d -spaces Throughout the paper, we work with the category, denoted by
Top , either of ∆ -generated spaces or of ∆ -Hausdorff ∆ -generated spaces (cf. [Gau19b, Section 2 andAppendix B]). The q-model structure (we use the terminology of [MS06]) is denotedby Top q . We summarize some basic properties of Top for the convenience of the reader: • Top is locally presentable. • All objects of
Top are sequential topological spaces. • A closed subset of a ∆-generated space equipped with the relative topology is sequential. • The ∆-kelleyfication functor k ∆ : T OP →
Top does not change the underlying set. • Let A ⊂ B be a subset of a space B of Top . Then A equipped with the ∆-kelleyficationof 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 underly-ing sets. In particular, the CW-complexes, and more generally all cellular spaces areequipped with the final topology. Note that cellular spaces are even Hausdorff (andparacompact, 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
T OP ( X, Y ) of all continuousmaps from X to Y .2.1. Proposition.
A continuous bijection f : U → V of Top is a homeomorphism ifand only if a set map g : [0 , → U is continuous if and only if the composite f g iscontinuous.Proof. If f : U → V is a homeomorphism, then the condition is satisfied since g = f − f g .Conversely, suppose the condition satisfied. Then the natural set map f ∗ : Top ([0 , , U ) −→ Top ([0 , , V ) s bijective. By Yoneda, we obtain that f is a homeomorphism. (cid:3) The enriched category G is an example of reparametrization category in the sense of[Gau20a, Definition 4.3] which is different from the terminal category. It is introduced in[Gau20a, Proposition 4.9]. Another example is given in [Gau20a, Proposition 4.11].2.2. Notation.
The notation [0 , ℓ ] ∼ = + [0 , ℓ ] for two real numbers ℓ , ℓ > means anondecreasing homeomorphism from [0 , ℓ ] to [0 , ℓ ] . It takes to and ℓ to ℓ . The enriched small category G is defined as follows: • The set of objects is the open interval ]0 , ∞ [. • The space G ( ℓ , ℓ ) is the set { [0 , ℓ ] ∼ = + [0 , ℓ ] } for all ℓ , ℓ > G ( ℓ , ℓ ) ⊂ TOP ([0 , ℓ ] , [0 , ℓ ]). In other terms, a set map [0 , → G ( ℓ , ℓ ) is continuous if andonly if the composite set map [0 , → G ( ℓ , ℓ ) ⊂ TOP ([0 , ℓ ] , [0 , ℓ ]) is continuous. • For every ℓ , ℓ , ℓ >
0, the composition map G ( ℓ , ℓ ) × G ( ℓ , ℓ ) → G ( ℓ , ℓ )is induced by the composition of continuous maps. It induces a continuous map sincethe composite set map G ( ℓ , ℓ ) × G ( ℓ , ℓ ) → G ( ℓ , ℓ ) ⊂ TOP ([0 , ℓ ] , [0 , ℓ ])corresponds by the adjunction to the continuous map[0 , ℓ ] × G ( ℓ , ℓ ) × G ( ℓ , ℓ ) → [0 , ℓ ]which takes ( t, x, y ) to y ( x ( t )).2.3. Proposition.
The topology of G ( ℓ , ℓ ) is the compact-open topology and therefore itis metrizable. A sequence ( φ n ) n > of G ( ℓ , ℓ ) converges to φ ∈ G ( ℓ , ℓ ) if and only if itconverges pointwise.Proof. The compact-open topology on G ( ℓ , ℓ ) is metrizable by [Hat02, Proposition A.13].The metric is given by the distance of the uniform convergence. Consider a ball B ( φ , ǫ )for this metric. Let ψ ∈ B ( φ , ǫ ). Then for all h ∈ [0 , | (cid:0) hψ ( t ) + (1 − h ) φ ( t ) (cid:1) − φ ( t ) | = | h ( ψ ( t ) − φ ( t )) | hd ( ψ, φ ) < hǫ ǫ. Thus, the compact-open topology is locally path-connected. The compact-open topologyis therefore ∆-generated by [CSW14, Proposition 3.11]. The last assertion is a conse-quence of the second Dini theorem. (cid:3) A multipointed space is a pair ( | X | , X ) where • | X | is a topological space called the underlying space of X . • X is a subset of | X | called the set of states of X .A morphism of multipointed spaces f : X = ( | X | , X ) → Y = ( | Y | , Y ) is a commutativesquare X f / / (cid:15) (cid:15) Y (cid:15) (cid:15) | X | | f | / / | Y | . he corresponding category is denoted by MTop .2.4.
Notation.
The maps f and | f | will be often denoted by f if there is no possibleconfusion. We have the well-known proposition:2.5.
Proposition. (The Moore composition) Let U be a topological space. Let γ i : [0 , ℓ i ] → Un continuous maps with i n with n > . Suppose that γ i ( ℓ i ) = γ i +1 (0) for i < n .Then there exists a unique continuous map γ ∗ · · · ∗ γ n : [0 , X i ℓ i ] → U such that ( γ ∗ · · · ∗ γ n )( t ) = γ i (cid:0) t − X j
Let µ ℓ : [0 , ℓ ] → [0 , be the homeomorphism defined by µ ℓ ( t ) = t/ℓ . Definition.
The map γ ∗ γ is called the Moore composition of γ and γ . Thecomposite γ ∗ N γ : [0 , ( µ ) − / / [0 , γ ∗ γ / / U is called the normalized composition . The normalized composition being not associative,a notation like γ ∗ N · · · ∗ N γ n will mean, by convention, that ∗ N is applied from the leftto the right. A multipointed d -space X is a triple ( | X | , X , P G X ) where • The pair ( | X | , X ) is a multipointed space. The space | X | is called the underlying space of X and the set X the set of states of X . • 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 . – Let γ be an execution path of X . Then any composite γφ with φ ∈ G (1 ,
1) is anexecution path of X . – Let γ and γ be two composable execution paths of X ; then the normalizedcomposition γ ∗ N γ is an execution path of X .A map f : X → Y of multipointed d -spaces is a map of multipointed spaces from ( | X | , X )to ( | Y | , Y ) such that for any execution path γ of X , the map P G f : γ f.γ is an execution path of Y .2.8. Notation.
The mapping P G f will be often denoted by f if there is no ambiguity. 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 of length ℓ >
0, which is denoted by Glob G ℓ ( Z ), is themultipointed d -space defined as follows • the underlying topological space is the quotient space { , } ⊔ ( Z × [0 , ℓ ])( z,
0) = ( z ′ ,
0) = 0 , ( z,
1) = ( z ′ ,
1) = 1 • the set of states is { , }• the set of execution paths is the set of continuous maps { δ z φ | φ ∈ G (1 , ℓ ) , z ∈ Z } with δ z ( t ) = ( z, t ). It is equal to the underlying set of G (1 , × Z .In particular, Glob G ℓ ( ∅ ) is the multipointed d -space { , } = ( { , } , { , } , ∅ ). For ℓ = 1, we set Glob G ( Z ) = Glob G ( Z ) . (3) The directed segment is the multipointed d -space −→ I G = Glob G ( { } ).The category of multipointed d -spaces is denoted by G dTop . The subset of executionpaths from α to β is the set of γ ∈ P G X such that γ (0) = α and γ (1) = β ; it is denotedby P G α,β X : α is called the initial state and β the final state . It is equipped with the ∆-kelleyfication of the relative topology induced by the inclusion P G α,β X ⊂ TOP ([0 , , | X | ).In other terms, a set map U → P G α,β X is continuous if and only if the composite set map U → P G α,β X ⊂ TOP ([0 , , | X | ) is continuous. The category G dTop is locally presentableby [Gau09, Theorem 3.5].2.9. Proposition. ( [Gau20b, Proposition 6.5] ) The mapping Ω : X ( | X | , X ) inducesa functor from G dTop to MTop which is topological and fibre-small.
The Ω-final structure is generated by the finite normalized composition of executionpaths. We will come back on this point in Theorem 3.9. Note that Proposition 2.9 holdsboth by working with ∆-generated spaces and with ∆-Hausdorff ∆-generated spaces.The following proposition is implicitly assumed (for ℓ = 1) in all the previous papersabout multipointed d -spaces:2.10. Proposition.
Let Z be a topological space. Then there is the homeomorphism P G , Glob G ℓ ( Z ) ∼ = G (1 , ℓ ) × Z. Proof.
By definition of Glob G ℓ ( Z ), the underlying set of P G , Glob G ℓ ( Z ) is equal to theunderlying set of G (1 , ℓ ) × Z . There is a one-to-one set map G (1 , ℓ ) × Z −→ TOP ([0 , , | Glob G ℓ ( Z ) | )which takes ( φ, z ) to t ( δ z φ )( t ) = ( z, φ ( t )). It is continuous since, by adjunction, itcorresponds to the continuous map[0 , × G (1 , ℓ ) × Z −→ Z × [0 , ℓ ] −→ | Glob G ℓ ( Z ) | which takes ( t, φ, z ) to ( z, φ ( t )). Consider a continuous map ψ : [0 , −→ TOP ([0 , , | Glob G ℓ ( Z ) | ) It is the suspension of Z . hich factors as a set map as a composite ψ : [0 , ξ −→ G (1 , ℓ ) × Z −→ TOP ([0 , , | Glob G ℓ ( Z ) | ) . Since the continuous map Z × [0 , ℓ ] → | Glob G ℓ ( Z ) | is a trivial q-fibration of spaces, themap ψ factors as a composite of continuous maps ψ : [0 , ( ψ ,ψ ) −→ TOP ([0 , , Z × [0 , ℓ ]) ∼ = TOP ([0 , , Z ) × TOP ([0 , , [0 , ℓ ]) −→ TOP ([0 , , | Glob G ℓ ( Z ) | ) . By hypothesis, ψ : [0 , → Z is a constant map and ψ ∈ G (1 , ℓ ). Since G (1 , ℓ ) isequipped with the ∆-kelleyfication of the relative topology induced by the set inclusion G (1 , ℓ ) ⊂ TOP ([0 , , [0 , ℓ ]), we deduce that ξ : [0 , → G (1 , ℓ ) × Z is continuous. Thusthe topology of G (1 , ℓ ) × Z coincides with the ∆-kelleyfication of the relative topologyinduced by the set inclusion G (1 , ℓ ) × Z ⊂ TOP ([0 , , | Glob G ℓ ( Z ) | ). (cid:3) Definition.
Let X be a multipointed d -space X . Denote again by P G X the topolog-ical space P G X = G ( α,β ) ∈ X × X P G α,β X. A straightforward consequence of the definition of the topology of P G X is:2.12. Proposition.
Let X be a multipointed d -space. Let f : [0 , → P G X be a continuousmap. Then f factors as composite of continuous maps f : [0 , → P G α,β X → P G X forsome ( α, β ) ∈ X × X .Proof. It is due to the fact that [0 ,
1] is connected. (cid:3)
Proposition.
Let X be a multipointed d -space such that X is a totally disconnectedsubset of | X | . Then the topology of P G X is the ∆ -kelleyfication of the relative topologyinduced by the inclusion P G X ⊂ TOP ([0 , , | X | ) .Proof. Call for this proof ( P G X ) + the set P G X equipped with the ∆-kelleyfication ofthe relative topology induced by the inclusion P G X λ ⊂ TOP ([0 , , | X | ). There is acontinuous bijection P G X → ( P G X ) + . Using Proposition 2.1, the proof is complete since X a totally disconnected subset of | X | and since [0 ,
1] is connected. (cid:3)
Theorem.
The functor P G : MdTop → Top is a right adjoint. In particular, it islimit preserving and accessible.Proof.
The left adjoint is constructed in [Gau09, Proposition 4.9] in the case of ∆-generated spaces. The proof still holds for ∆-Hausdorff ∆-generated spaces. It relieson the fact that
Top is cartesian closed and that every ∆-generated space is homeomor-phic to the disjoint sum of its path-connected components which are also its connectedcomponents. The construction is similar to the construction of the left adjoint of thepath presheaf functor for P -flows [Gau20a, Theorem 6.13] and to the construction of theleft adjoint of the path functor for flows [Gau19b, Theorem 5.9]. (cid:3) The q-model structure of multipointed d -spaces ( G dTop ) q is the unique combinatorialmodel structure such that { Glob G ( S n − ) ⊂ Glob G ( D n ) | n > } ∪ { C : ∅ → { } , R : { , } → { }} s the set of generating cofibrations, the maps between globes being induced by the closedinclusion S n − ⊂ D n , and such that { Glob G ( D n × { } ) ⊂ Glob G ( D n +1 ) | n > } is the set of generating trivial cofibrations, the maps between globes being induced bythe closed inclusion ( x , . . . , x n ) ( x , . . . , x n ,
0) (e.g. [Gau20b, Theorem 6.16]). Theweak equivalences are the maps of multipointed d -spaces f : X → Y inducing a bijection f : X ∼ = Y and a weak homotopy equivalence P G f : P G X → P G Y and the fibrations arethe maps of multipointed d -spaces f : X → Y inducing a q-fibration P G f : P G X → P G Y of topological spaces.3. Moore composition and Ω -final structure Notation.
Let φ i : [0 , ℓ i ] ∼ = + [0 , ℓ ′ i ] for n > and i n . Then the map φ ⊗ . . . ⊗ φ n : [0 , X i ℓ i ] ∼ = + [0 , X i ℓ ′ i ] denotes the homeomorphism defined by ( φ ⊗ . . . ⊗ φ n )( t ) = φ ( t ) if t ℓ φ ( t − ℓ ) + ℓ ′ if ℓ t ℓ + ℓ . . .φ i ( t − P j
Let φ : [0 , ℓ ] ∼ = + [0 , ℓ ′ ] . Let n > . Consider ℓ , . . . , ℓ n > with n > such that P i = ni =1 ℓ i = ℓ . Then there exists a unique decomposition of φ of the form φ = φ ⊗ . . . ⊗ φ n such that φ i : [0 , ℓ i ] ∼ = + [0 , ℓ ′ i ] for i n .Proof. By definition of φ ⊗ . . . ⊗ φ n , we necessarily have φ (cid:0) X j i ℓ j (cid:1) = φ i ( X j i ℓ j − X j φ (cid:0) X j i ℓ j (cid:1) − X j
Let φ ∈ G (1 , . Let n > . Assume that i = n X i =1 ℓ i = i = n X i =1 ℓ ′ i = 1 and that ∀ i n, φ (cid:0) X j i ℓ j (cid:1) = X j i ℓ ′ j . Then there exist (unique) φ i : [0 , ℓ i ] ∼ = + [0 , ℓ ′ i ] for i n such that φ = φ ⊗ . . . ⊗ φ n . Proposition.
Let U be a topological space. Let γ i : [0 , → U be n continuous mapswith i n and n > . Let φ i : [0 , ℓ i ] ∼ = + [0 , ℓ ′ i ] for i n . Then we have (cid:0) ( γ µ ℓ ′ ) ∗ · · · ∗ ( γ n µ ℓ ′ n ) (cid:1) ( φ ⊗ . . . ⊗ φ n ) = ( γ µ ℓ ′ φ ) ∗ · · · ∗ ( γ n µ ℓ ′ n φ n ) . Proof.
For P j
Let U be a topological space. Let γ i : [0 , → U be n continuous mapswith i n and n > . Let ℓ i > with i n nonzero real numbers with P i ℓ i = 1 . hen for all ℓ > , we have (cid:0) ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) (cid:1) µ ℓ = ( γ µ ℓ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ℓ ) . Proof.
For all 1 j n , we have by definition of the Moore composition (cid:0) ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) (cid:1) µ ℓ ( t ) = γ j (cid:18) ℓ j (cid:18) tℓ − X i Let U be a topological space. Let γ i : [0 , → U be n continuous mapswith n > and i n such that γ ∗ N · · · ∗ N γ n exists. Then there is the equality γ ∗ N · · · ∗ N γ n = (cid:0) γ µ n − (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ (cid:1) . In particular, for n = 2 , we have γ ∗ N γ = ( γ µ / ) ∗ ( γ µ / ) .Proof. The proof is by induction on n > 2. The map µ / : [0 , / ∼ = + [0 , 1] which takes t to 2 t gives rise to a homeomorphism µ / ⊗ µ / : [0 , ∼ = + [0 , 2] which is equal to µ − : [0 , ∼ = + [0 , γ ∗ N γ = ( γ ∗ γ ) µ − by definition of ∗ N = ( γ ∗ γ )( µ / ⊗ µ / ) because µ − = µ / ⊗ µ / = ( γ µ / ) ∗ ( γ µ / ) by Proposition 3.4 . The statement is therefore proved for n = 2. Assume that the statement is proved forsome n > n = 2. Then we obtain γ ∗ N · · · ∗ N γ n +1 = (cid:0)(cid:0) γ µ n − (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ (cid:1)(cid:1) ∗ N γ n +1 = (cid:0)(cid:0)(cid:0) γ µ n − (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ (cid:1)(cid:1) µ (cid:1) ∗ (cid:0) γ n +1 µ (cid:1) = (cid:0)(cid:0) γ µ n (cid:1) ∗ (cid:0) γ µ n (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ (cid:1)(cid:1) ∗ (cid:0) γ n +1 µ (cid:1) = (cid:0) γ µ n (cid:1) ∗ (cid:0) γ µ n (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ (cid:1) ∗ (cid:0) γ n +1 µ (cid:1) , the first equality by induction hypothesis, the second equality by the case n = 2, thethird equality by Proposition 3.5, and the last equality by associativity of the Moorecomposition. We have proved the statement for n + 1. (cid:3) Proposition. Let U be a topological space. Let γ i : [0 , → U be n continuous mapswith n > and i n such that γ ∗ N · · · ∗ N γ n exists. Let φ ∈ G (1 , . Then thereexist φ : [0 , ℓ ] ∼ = + [0 , n − ] , φ : [0 , ℓ ] ∼ = + [0 , n − ] , φ : [0 , ℓ ] ∼ = + [0 , n − ] , etc... until φ n : [0 , ℓ n ] ∼ = + [0 , ] such that φ = φ ⊗ . . . ⊗ φ n (which implies P i ℓ i = 1 ) and there isthe equality (cid:0) γ ∗ N · · · ∗ N γ n (cid:1) φ = (cid:0) γ µ n − φ (cid:1) ∗ (cid:0) γ µ n − φ (cid:1) ∗ (cid:0) γ µ n − φ (cid:1) ∗ · · · ∗ (cid:0) γ n µ φ n (cid:1) . roof. Let ℓ , . . . , ℓ n > P i ℓ i = 1 and such that φ ( ℓ ) = n − φ ( ℓ + ℓ ) = n − + n − φ ( ℓ + ℓ + ℓ ) = n − + n − + n − . . .φ ( ℓ + ℓ + ℓ + · · · + ℓ n ) = n − + n − + n − + · · · + = 1 . By Proposition 3.2, there exist φ : [0 , ℓ ] ∼ = + [0 , n − ], φ : [0 , ℓ ] ∼ = + [0 , n − ], φ :[0 , ℓ ] ∼ = + [0 , n − ], etc... until φ n : [0 , ℓ n ] ∼ = + [0 , ] such that φ = φ ⊗ . . . ⊗ φ n . We obtain (cid:0) γ ∗ N · · · ∗ N γ n +1 (cid:1) φ = (cid:0)(cid:0) γ µ n (cid:1) ∗ (cid:0) γ µ n (cid:1) ∗ (cid:0) γ µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ (cid:1) ∗ (cid:0) γ n +1 µ (cid:1)(cid:1) φ = (cid:0) γ µ n − φ (cid:1) ∗ (cid:0) γ µ n − φ (cid:1) ∗ (cid:0) γ µ n − φ (cid:1) ∗ · · · ∗ (cid:0) γ n µ φ n (cid:1) , the first equality by Proposition 3.6 and the second equality by Proposition 3.4. (cid:3) Proposition. Let U be a topological space. Let γ i : [0 , → U be n continuous mapswith n > and i n such that γ ∗ N · · · ∗ N γ n exists. Let ℓ , . . . , ℓ n > be nonzeroreal numbers such that P i ℓ i = 1 . Let φ : [0 , n − ] ∼ = + [0 , ℓ ] , φ : [0 , n − ] ∼ = + [0 , ℓ ] , φ : [0 , n − ] ∼ = + [0 , ℓ ] , etc... until φ n : [0 , ] ∼ = + [0 , ℓ n ] and let φ = φ ⊗ . . . ⊗ φ n . Then φ ∈ G and there is the equality (cid:0)(cid:0) γ µ ℓ (cid:1) ∗ (cid:0) γ µ ℓ (cid:1) ∗ (cid:0) γ µ ℓ (cid:1) ∗ · · · ∗ (cid:0) γ n µ ℓ n (cid:1)(cid:1) φ = (cid:0) γ µ ℓ φ µ − n − (cid:1) ∗ N (cid:0) γ µ ℓ φ µ − n − (cid:1) ∗ N (cid:0) γ µ ℓ φ µ − n − (cid:1) ∗ N · · · ∗ N (cid:0) γ n µ ℓ n φ n µ − (cid:1) . Proof. We have (cid:0)(cid:0) γ µ ℓ (cid:1) ∗ (cid:0) γ µ ℓ (cid:1) ∗ (cid:0) γ µ ℓ (cid:1) ∗ · · · ∗ (cid:0) γ n µ ℓ n (cid:1)(cid:1) φ = (cid:0) γ µ ℓ φ (cid:1) ∗ (cid:0) γ µ ℓ φ (cid:1) ∗ (cid:0) γ µ ℓ φ (cid:1) ∗ · · · ∗ (cid:0) γ n µ ℓ n φ n (cid:1) = (cid:0) γ µ ℓ φ µ − n − µ n − (cid:1) ∗ (cid:0) γ µ ℓ φ µ − n − µ n − (cid:1) ∗ (cid:0) γ µ ℓ φ µ − n − µ n − (cid:1) ∗ · · · ∗ (cid:0) γ n µ ℓ n φ n µ − µ (cid:1) = (cid:0) γ µ ℓ φ µ − n − (cid:1) ∗ N (cid:0) γ µ ℓ φ µ − n − (cid:1) ∗ N (cid:0) γ µ ℓ φ µ − n − (cid:1) ∗ N · · · ∗ N (cid:0) γ n µ ℓ n φ n µ − (cid:1) , where the first equality is due to Proposition 3.4, the second equality is due to the factthat µ − ℓ µ ℓ is the identity of [0 , ℓ ] for all nonzero real numbers ℓ > 0, and the last equalityis a consequence of Proposition 3.6. (cid:3) Theorem. Consider a cocone (Ω( X i )) • → ( | X | , X ) of MTop . Let X be the Ω -finallift. Let f i : X i → X be the canonical maps. Then the set of execution paths of X is theset of finite Moore compositions of the form ( f γ µ ℓ ) ∗ · · · ∗ ( f n γ n µ ℓ n ) such that γ i is anexecution path of X i for all i n with P i ℓ i = 1 .Proof. Let P ( X ) be the set of execution paths of X of the form ( f γ µ ℓ ) ∗ · · · ∗ ( f n γ n µ ℓ n )such that γ i is an execution path of X i for all 1 i n with P i ℓ i = 1. The finalstructure is generated by the finite normalized composition of execution paths ( f γ ) ∗ N · · · ∗ N ( f n γ n ) (with the convention that the ∗ N are calculated from the left to the right) nd all reparametrizations by φ running over G (1 , φ : [0 , ℓ ] ∼ = + [0 , n − ], φ : [0 , ℓ ] ∼ = + [0 , n − ], φ : [0 , ℓ ] ∼ = + [0 , n − ], etc... until φ n : [0 , ℓ n ] ∼ = + [0 , ] such that φ = φ ⊗ . . . ⊗ φ n and we have (cid:0) ( f γ ) ∗ N · · · ∗ N ( f n γ n ) (cid:1) φ = (cid:0) f γ µ n − φ (cid:1) ∗ (cid:0) f γ µ n − φ (cid:1) ∗ (cid:0) f γ µ n − φ (cid:1) ∗ · · · ∗ (cid:0) f n γ n µ φ n (cid:1) = (cid:0) f γ µ n − φ µ − ℓ µ ℓ (cid:1) ∗ (cid:0) f γ µ n − φ µ − ℓ µ ℓ (cid:1) ∗ (cid:0) f γ µ n − φ µ − ℓ µ ℓ (cid:1) ∗ · · · ∗ (cid:0) f n γ n µ φ n µ − ℓ n µ ℓ n (cid:1) = (cid:0) f γ ′ µ ℓ (cid:1) ∗ (cid:0) f γ ′ µ ℓ (cid:1) ∗ (cid:0) f γ ′ µ ℓ (cid:1) ∗ · · · ∗ (cid:0) f n γ ′ n µ ℓ n (cid:1) , the first equality by Proposition 3.7, the second equality because µ − ℓ µ ℓ is the identity of[0 , ℓ ] for all ℓ > γ ′ = γ (cid:0) µ n − φ µ − ℓ (cid:1) γ ′ = γ (cid:0) µ n − φ µ − ℓ (cid:1) γ ′ = γ (cid:0) µ n − φ µ − ℓ (cid:1) . . .γ ′ n = γ n (cid:0) µ φ n µ − ℓ n (cid:1) . It implies that the set P ( X ) contains the final structure. Conversely, let ( f γ µ ℓ ) ∗ · · · ∗ ( f n γ n µ ℓ n ) be an element of P ( X ). Choose φ : [0 , n − ] ∼ = + [0 , ℓ ], φ : [0 , n − ] ∼ = + [0 , ℓ ], φ : [0 , n − ] ∼ = + [0 , ℓ ], etc... until φ n : [0 , ] ∼ = + [0 , ℓ n ] and let φ = φ ⊗ . . . ⊗ φ n . UsingProposition 3.8, we obtain (cid:0)(cid:0) f γ µ ℓ (cid:1) ∗ (cid:0) f γ µ ℓ (cid:1) ∗ (cid:0) f γ µ ℓ (cid:1) ∗ · · · ∗ (cid:0) f n γ n µ ℓ n (cid:1)(cid:1) φ = (cid:0) f γ µ ℓ φ µ − n − (cid:1) ∗ N (cid:0) f γ µ ℓ φ µ − n − (cid:1) ∗ N (cid:0) f γ µ ℓ φ µ − n − (cid:1) ∗ N · · · ∗ N (cid:0) f n γ n µ ℓ n φ n µ − (cid:1) . The continuous maps µ ℓ φ µ − n − , µ ℓ φ µ − n − , µ ℓ φ µ − n − , . . . , µ ℓ n φ n µ − from [0 , 1] to itselfbelong to G (1 , γ ′ , . . . , γ ′ n defined by the equalities γ ′ = γ (cid:0) µ ℓ φ µ − n − (cid:1) γ ′ = γ (cid:0) µ ℓ φ µ − n − (cid:1) γ ′ = γ (cid:0) µ ℓ φ µ − n − (cid:1) . . .γ ′ n = γ n (cid:0) µ ℓ n φ n µ − (cid:1) are execution paths of X , . . . , X n respectively. We obtain (cid:0)(cid:0) f γ µ ℓ (cid:1) ∗ (cid:0) f γ µ ℓ (cid:1) ∗ (cid:0) f γ µ ℓ (cid:1) ∗ · · · ∗ (cid:0) f n γ n µ ℓ n (cid:1)(cid:1) φ = (cid:0) f γ ′ (cid:1) ∗ N (cid:0) f γ ′ (cid:1) ∗ N (cid:0) f γ ′ (cid:1) ∗ N · · · ∗ N (cid:0) f n γ ′ n (cid:1) . We deduce that the set of paths P ( X ) is included in the Ω-final structure. (cid:3) . From multipointed d -spaces to Moore flows Notation. The enriched category of enriched presheaves from G op to Top is denotedby [ G op , Top ] . The underlying set-enriched category is denoted by [ G op , Top ] . Proposition. [Gau19a, Proposition 5.3 and Proposition 5.5] The category [ G op , Top ] is a full reflective and coreflective subcategory of Top G op . For every enriched presheaf F : G op → Top , every ℓ > and every topological space X , we have the natural bijectionof sets [ G op , Top ] ( F G op ℓ X, F ) ∼ = Top ( X, F ( ℓ )) . Theorem. ( [Gau20a, Theorem 5.14] ) Let D and E be two enriched presheaves of [ G op , Top ] . Let D ⊗ E := Z ( ℓ ,ℓ ) G ( − , ℓ + ℓ ) × D ( ℓ ) × E ( ℓ ) . The pair ([ G op , Top ] , ⊗ ) has the structure of a closed symmetric semimonoidal category,i.e. a closed symmetric nonunital monoidal category. Notation. Let D an enriched presheaf of [ G op , Top ] . Let φ : ℓ → ℓ ′ be a map of G .Let x ∈ D ( ℓ ′ ) . We will use the notation x.φ := D ( φ )( x ) . Intuitively, x is a path of length ℓ ′ and x.φ is a path of length ℓ which is the reparametriza-tion by φ of x . A semicategory , also called nonunital category in the literature, is a category withoutidentity maps in the structure. It is enriched over a closed symmetric semimonoidalcategory ( V , ⊗ ) if it satisfied all axioms of enriched category except the one involving theidentity maps, i.e. the enriched composition is associative and not necessarily unital.By [Gau20a, Definition 6.2], a Moore flow is a small semicategory enriched over theclosed semimonoidal category ([ G op , Top ] , ⊗ ) recalled in Theorem 4.3. The category ofMoore flows, denoted by G Flow , is locally presentable by [Gau20a, Theorem 6.11]. Everyset S can be viewed as a Moore flow with an empty presheaf of execution paths denotedin the same way. Let D : G op → Top be an enriched presheaf. We denote by Glob( D )the Moore flow defined as follows:Glob( D ) = { , } P , Glob( D ) = P , Glob( D ) = P , Glob( D ) = ∆ G ∅P , Glob( D ) = D. There is no composition law. This construction yields a functorGlob : [ G op , Top ] → G Flow . There exists a unique model structure on G Flow such that { Glob( F G op ℓ S n − ) ⊂ Glob( F G op ℓ D n ) | n > , ℓ > } ∪ { C : ∅ → { } , R : { , } → { }} is the set of generating cofibrations and such that all objects are fibrant. The set ofgenerating trivial cofibrations is { Glob( F G op ℓ D n ) ⊂ Glob( F G op ℓ D n +1 ) | n > , ℓ > } here the maps D n ⊂ D n +1 are induced by the mappings ( x , . . . , x n ) ( x , . . . , x n , f : X → Y inducing a bijection X ∼ = Y and such that for all ( α, β ) ∈ X × X , the map en enriched presheaves P α,β X → P f ( α ) ,f ( β ) Y is an objectwise weak homotopy equivalence. The fibrations are themap of Moore flows f : X → Y such that for all ( α, β ) ∈ X × X , the map of enrichedpresheaves P α,β X → P f ( α ) ,f ( β ) Y is an objectwise q-fibration of spaces. It is called theq-model structure and we will use the terminology of q-cofibration and q-fibration fornaming the cofibrations and the fibrations respectively.4.5. Definition. Let X be a multipointed d -space. Let P ℓα,β X be the subspace of continuousmaps from [0 , ℓ ] to | X | defined by P ℓα,β X = { t γµ ℓ | γ ∈ P G α,β X } . Its elements are called the execution paths of length ℓ from α to β . Let P ℓ X = G ( α,β ) ∈ X × X P ℓα,β X. A map of multipointed d -spaces f : X → Y induces for each ℓ > a continuous map P ℓ f : P ℓ X → P ℓ Y by composition by f . Note that P α,β X = P G α,β X , that there is a homeomorphism P ℓα,β X ∼ = P G α,β X for all ℓ > 0, and that for any topological space Z , we have the homeomorphism P ℓ , (Glob G ( Z )) ∼ = G ( ℓ, × Z for any ℓ > ℓ > Proposition. Let X be a multipointed d -space. Let φ : [0 , ℓ ] ∼ = + [0 , ℓ ] . Let γ ∈ P ℓ X .Then γφ ∈ P ℓ X .Proof. By definition of P ℓ X , there exists γ ∈ P G X such that γ = γµ ℓ . We obtain γφ = γµ ℓ φµ − ℓ µ ℓ . Since µ ℓ φµ − ℓ ∈ G (1 , γµ ℓ φµ − ℓ ∈ P G X and that γφ ∈ P ℓ X . (cid:3) Proposition. Let X be a multipointed d -space. Let γ and γ be two execution pathsof X with γ (1) = γ (0) . Let ℓ , ℓ > . Then (cid:0) γ µ ℓ ∗ γ µ ℓ (cid:1) µ − ℓ + ℓ is an execution path of X .Proof. Let φ : [0 , / ∼ = + [0 , ℓ ] and φ : [0 , / ∼ = + [0 , ℓ ]. Then we have φ ⊗ φ : [0 , ∼ = + [0 , ℓ + ℓ ] . e obtain the sequence of equalities (cid:0) ( γ µ ℓ ) ∗ ( γ µ ℓ ) (cid:1) µ − ℓ + ℓ = (cid:0) ( γ µ ℓ ) ∗ ( γ µ ℓ ) (cid:1)(cid:0) φ ⊗ φ (cid:1)(cid:0) φ ⊗ φ (cid:1) − µ − ℓ + ℓ = (cid:0) ( γ µ ℓ φ ) ∗ ( γ µ ℓ φ ) (cid:1)(cid:0) φ ⊗ φ (cid:1) − µ − ℓ + ℓ = (cid:0) ( γ µ ℓ φ µ − / µ / ) ∗ ( γ µ ℓ φ µ − / µ / ) (cid:1)(cid:0) φ ⊗ φ (cid:1) − µ − ℓ + ℓ = (cid:0) ( γ µ ℓ φ µ − / | {z } ∈G (1 , ) ∗ N ( γ µ ℓ φ µ − / | {z } ∈G (1 , ) (cid:1) (cid:0) φ ⊗ φ (cid:1) − µ − ℓ + ℓ | {z } ∈G (1 , , the first equality because φ ⊗ φ is invertible, the second equality by Proposition 3.4, thethird equality because µ / is invertible, and finally the last equality by Proposition 3.7.The proof is complete because the set of execution paths of X is invariant by the actionof G (1 , (cid:3) Proposition. Let X be a multipointed d -space. Let ℓ , ℓ > . The Moore composi-tion of continuous maps yields a continuous maps P ℓ X × P ℓ X → P ℓ + ℓ X .Proof. It is a consequence of Definition 4.5 and Proposition 4.7 (cid:3) Theorem. Let X be a multipointed d -space. Then the following data • The set of states X of X • For all α, β ∈ X and all real numbers ℓ > , let P ℓα,β M G ( X ) := P ℓα,β X. • For all maps [0 , ℓ ] ∼ = + [0 , ℓ ′ ] , a map f : [0 , ℓ ′ ] → | X | of P ℓ ′ α,β M G ( X ) is mapped to themap [0 , ℓ ] ∼ = + [0 , ℓ ′ ] f → | X | of P ℓα,β M G ( X ) • For all α, β, γ ∈ X and all real numbers ℓ, ℓ ′ > , the composition maps ∗ : P ℓα,β M G ( X ) × P ℓ ′ β,γ M G ( X ) → P ℓ + ℓ ′ α,γ M G ( X ) of Proposition 4.8.assemble to a Moore flow M G ( X ) . This mapping induces a functor M G : G dTop −→ G Flow which is a right adjoint. Note that the left adjoint M G ! : G Flow −→ G dTop preserves the set of states as wellas the functor M G : G dTop −→ G Flow . Proof. These data give rise to an enriched presheaf P α,β M G ( X ) of [ G op , Top ] for eachpair ( α, β ) of states of X and, thanks to Proposition 4.8, to an associative compositionlaw ∗ : P ℓ α,β M G ( X ) × P ℓ β,γ M G ( X ) → P ℓ + ℓ α,γ M G ( X ) which is natural with respect to ( ℓ , ℓ ).By [Gau20a, Section 6], these data assemble to a Moore flow. Since limits and colimits ofenriched presheaves of [ G op , Top ] are calculated objectwise, the functor M G : G dTop −→G Flow is limit-preserving and accessible by Theorem 2.14. Therefore it is a right adjointby [AR94, Theorem 1.66]. (cid:3) .10. Proposition. Let X be a multipointed d -space. Let ℓ > be a real number. Let Z be a topological space. Then there is a bijection of sets G dTop (Glob G ℓ ( Z ) , X ) ∼ = G ( α,β ) ∈ X × X Top ( Z, P ℓα,β X ) which is natural with respect to Z and X .Proof. A map f of multipointed d -spaces from Glob G ℓ ( Z ) to X is determined by • The image by f of 0 and 1 which will be denoted by α and β respectively • A continuous map (still denoted by f ) from | Glob G ℓ ( Z ) | to | X | such that for all x ∈ Z and all φ : [0 , ∼ = + [0 , ℓ ], the map t f ( x, φ ( t )) from [0 , 1] to | X | belongs to P G α,β X .By definition of P ℓα,β X , for every x ∈ Z , the continuous map f ( x, − ) from [0 , ℓ ] to | X | belongs to P ℓα,β X since f ( x, − ) = f ( x, φ ( − )) .φ − for any φ : [0 , ∼ = + [0 , ℓ ]. Since f is continuous and since Top is cartesian closed, the mapping x f ( x, − ) actuallyyields a continuous map from Z to P ℓα,β X . Conversely, starting from a continuous map g : Z → P ℓα,β X , one can define a map of multipointed d -spaces from Glob G ℓ ( Z ) to X bytaking 0 and 1 to α and β respectively and by taking ( x, t ) ∈ | Glob G ℓ ( Z ) | to g ( x )( t ). (cid:3) We want to recall for the convenience of the reader:4.11. Proposition. [Gau20a, Proposition 6.10] Let D : G op → Top be an enrichedpresheaf. Let X be a Moore flow. Then there is the natural bijection G Flow (Glob( D ) , X ) ∼ = G ( α,β ) ∈ X × X [ G op , Top ] ( D, P α,β X ) . Proposition. For all topological spaces Z and all ℓ > , there are the naturalisomorphisms M G (Glob G ℓ ( Z )) ∼ = Glob( F G op ℓ ( Z )) , M G ! (Glob( F G op ℓ ( Z ))) ∼ = Glob G ℓ ( Z ) . Proof. By definition of M G and by Proposition 2.10, the only nonempty path presheaf of M G (Glob G ℓ ( Z )) is P , M G (Glob G ℓ ( Z )) = G ( − , ℓ ) × Z and we obtain the first isomorphism. There is the sequence of natural bijections, for anymultipointed d -space X , G dTop (cid:0) M G ! (Glob( F G op ℓ ( Z ))) , X (cid:1) ∼ = G Flow (cid:0) Glob( F G op ℓ ( Z )) , M G X (cid:1) ∼ = G ( α,β ) ∈ X × X [ G op , Top ] (cid:0) F G op ℓ ( Z ) , P α,β X (cid:1) ∼ = G ( α,β ) ∈ X × X Top ( Z, P ℓα,β X ) ∼ = G dTop (Glob G ℓ ( Z ) , X ) , the first bijection by adjunction, the second bijection by Proposition 4.11, the thirdbijection by Proposition 4.2 and the last bijection by Proposition 4.10. The proof of thesecond isomorphism is then complete thanks to the Yoneda lemma. (cid:3) . Cellular multipointed d -spaces Let λ be an ordinal. In this section, we work with a colimit-preserving functor X : λ −→ G dTop such that • The multipointed d -space X is a set, in other terms X = ( X , X , ∅ ) for some set X . • For all ν < λ , there is a pushout diagram of multipointed d -spacesGlob G ( S n ν − ) (cid:15) (cid:15) g ν / / X ν (cid:15) (cid:15) Glob G ( D n ν ) c g ν / / X ν +1 with n ν > X λ = lim −→ ν<λ X ν . Note that for all ν λ , there is the equality X ν = X . Denoteby c ν = | Glob G ( D n ν ) \| Glob G ( S n ν − ) | the ν -th cell of X λ . It is called a globular cell . Like in the usual setting of CW-complexes, b g ν induces a homeomorphism from c ν to b g ν ( c ν ) equipped with the relative topology whichwill be therefore denoted in the same way. It also means that b g ν ( c ν ) equipped with therelative topology is ∆-generated. The closure of c ν in | X λ | is denoted by b c ν = b g ν ( | Glob G ( D n ν ) | ) . The boundary of c ν in | X λ | is denoted by ∂c ν = b g ν ( | Glob G ( S n ν − ) | ) . The state b g ν (0) ∈ X ( b g ν (1) ∈ X resp.) is called the initial (final resp.) state of c ν . Theinteger n ν + 1 is called the dimension of the globular cell c ν . It is denoted by dim c ν . Thestates of X are also called the globular cells of dimension Definition. The cellular multipointed d -space X λ is finite if λ is a finite ordinal and X is finite. Proposition. The space | X λ | is a cellular space and contains X as a discrete sub-space. In particular, | X λ | is Hausdorff. For every ν ν λ , the continuous map | X ν | → | X ν | is a q-cofibration of spaces, and in particular a closed T -inclusion.Proof. By [Gau06, Theorem 8.2], the continuous map | Glob G ( S n ν − ) | → | Glob G ( D n ν ) | is a q-cofibration of spaces for all ν > 0. The proof is complete because the functor X 7→ | X | is colimit-preserving. (cid:3) Corollary. All spaces P G α,β X ν and P G X ν for ( α, β ) ∈ X × X and ν λ areHausdorff. roof. There is a one-to-one continuous map TOP ([0 , , | X ν | ) −→ Y t ∈ [0 , | X ν | induced by the evaluation maps. Each | X ν | is Hausdorff by Proposition 5.2. Thereforethe product in T OP of the | X ν | is Hausdorff. The set Q t ∈ [0 , | X ν | is equipped with the ∆-kelleyfication of the product topology in T OP which adds more open subsets. Therefore Q t ∈ [0 , | X ν | is Hausdorff. And therefore TOP ([0 , , | X ν | ) is Hausdorff as well. The space P G α,β X ν is by definition equipped with the ∆-kelleyfication of the relative topology inducedby the set inclusion P G α,β X ν ⊂ TOP ([0 , , | X ν | ). Thus the space P G α,β X ν is Hausdorff aswell. We deduce that P G X ν = F ( α,β ) ∈ X × X P G α,β X ν is also Hausdorff. (cid:3) Proposition. For all ν < λ , the continuous map P G X ν → P G X ν +1 is a closed T -inclusion.Proof. All spaces P G X ν are sequential, as ∆-generated spaces, and Hausdorff for all ν λ by Proposition 5.3. Therefore it suffices to prove that for every sequence ( γ n ) n > of P G X ν converging to γ ∞ in P G X ν +1 , γ ∞ belongs to P G X ν . Since P G X ν = P G X ν +1 ∩ TOP ([0 , , | X ν | ) by definition of a globe, it suffices to prove that for all t ∈ [0 , γ ∞ ( t ) ∈| X ν | . The latter point is a consequence of the fact that | X ν | ⊂ | X ν +1 | is a closed inclusionby Proposition 5.2. (cid:3) Proposition. Let K be a compact subspace of | X λ | . Then K intersects finitely many c ν .Proof. It is an adaptation of [Hat02, Proposition A.1]. Assume that there exists aninfinite set S = { m j | j > } with m j ∈ K ∩ c ν j . Then by transfinite induction on ν > S is closed in X ν for all 0 ν λ . The same argument proves that everysubset of S is closed in X λ . Thus S has the discrete topology. But it is compact, andtherefore finite. Contradiction. (cid:3) Colimits of multipointed d -spaces are calculated by taking the colimit of the underlyingspaces and of the sets of states and by taking the Ω-final structure which is generated bythe free finite compositions of execution paths. Consequently, the composite functor G dTop P G / / Top ⊂ / / Set is finitely accessible. It is unlikely that the functor P G : G dTop → Top , which is a rightadjoint by Theorem 2.14, is finitely accessible. However, we have:5.6. Theorem. The composite functor λ X −→ G dTop P G −→ Top is colimit-preserving. In particular the continuous bijection lim −→ ( P G .X ) −→ P G lim −→ X is a homeomorphism. Moreover the topology of P G lim −→ X is the final topology. ote that Theorem 5.6 holds both for ∆-generated spaces and ∆-Hausdorff ∆-generatedspaces. The proof can be adapted to work in the categories of k -spaces and weakly Haus-dorff k -spaces: it suffices to modify Proposition 2.1 by replacing the segment [0 , 1] by acompact (meaning quasi-compact Hausdorff) K . Proof. Consider the set of ordinals (cid:26) ν λ | ν limit ordinal and lim −→ ν ′ <ν ( P G X ν ′ ) −→ P G X ν not isomorphism (cid:27) Assume this set nonempty. Let ν be its smallest element. The topology of lim −→ ν ′ <ν P G X ν ′ is the final topology because the continuous maps P G X ν ′ → P G X ν ′ +1 are one-to-one. Let f : [0 , → P G X ν be a continuous map. Therefore the composite map[0 , f −→ P G X ν ⊂ TOP ([0 , , | X ν | )is continuous. It gives rise by adjunction to a continuous map [0 , × [0 , → | X ν | .Since the functor X : λ → G dTop is colimit-preserving, there is the homeomorphism | X ν | ∼ = lim −→ ν ′ <ν | X ν ′ | . Since [0 , × [0 , 1] is a final quotient of [0 , 1] by using space-fillingcurves, the latter continuous map then factors as a composite [0 , × [0 , → | X ν ′ | → | X ν | for some ordinal ν ′ < ν by Proposition 5.2. Since P G X ν ′ = P G X ν ∩ TOP ([0 , , | X ν ′ | ), f factors as a composite [0 , → P G X ν ′ → P G X ν . Using Proposition 2.1. we obtain thehomeomorphism lim −→ ν ′ <ν P G X ν ′ −→ P G X ν : contradiction. (cid:3) Corollary. For all ν ν λ , the continuous map P G X ν → P G X ν is a closed T -inclusion.Proof. Every space P G X ν is equipped with the final topology by Theorem 5.6. We provethat P G X ν → P G X ν is a closed inclusion by transfinite induction with respect to ν > ν .There is nothing to prove for ν = ν . If P G X ν → P G X ν is a closed inclusion, then thecomposite map P G X ν → P G X ν → P G X ν +1 is a closed inclusion by Proposition 5.4. Itremains the case where ν is a limit ordinal. By Theorem 5.6, there is the homeomorphism P G X ν ∼ = lim −→ ν ν<ν P G X ν and P G X ν is equipped with the final topology. Let F be a closed subset of P G X ν . Thenfor all ν ν < ν , the image F ν of F in P G X ν is a closed subset by the inductionhypothesis. Therefore the image F ν of F in P G X ν , whose inverse image in P G X ν is F ν (because all maps are one-to-one), is closed in P G X ν . Finally, we remark that all spaces P G X ν are Hausdorff by Corollary 5.3. Consequently, these closed inclusions are T . (cid:3) Theorem. The composite functor λ X −→ G dTop M G −→ G Flow is colimit-preserving. In particular the natural map lim −→ ν<λ M G ( X ν ) −→ M G X λ is an isomorphism. roof. Theorem 5.6 states that there is the homeomorphismlim −→ ν<λ P G X ν −→ P G X λ . We have, by definition of the functor M G , the equality of functors P G = P . M G . It meansthat there is the homeomorphismlim −→ ν<λ P M G ( X ν ) −→ P M G ( X λ ) . Since all maps the reparametrization category G are isomorphisms, we obtain for all ℓ > −→ ν<λ P ℓ M G ( X ν ) −→ P ℓ M G ( X λ ) . Since colimits of enriched presheaves are calculated objectwise, we obtain the isomorphismof enriched presheaves lim −→ ν<λ PM G X ν −→ PM G X λ . The proof is complete thanks to the universal property of the colimits. (cid:3) Definition. An execution path γ of a multipointed d -space X is minimal if γ (]0 , ∩ X = ∅ . For any topological space Z (q-cofibrant or not), every execution path of a multipointed d -space of the form Glob G ( Z ) is minimal. The following theorem proves that executionpaths of cellular multipointed d -spaces have a normal form.5.10. Theorem. Let γ be an execution path of X λ . Then there exist minimal executionpaths γ , . . . , γ n and ℓ , . . . , ℓ n > with P i ℓ i = 1 such that γ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) . Moreover, if there is the equality γ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) = ( γ ′ µ ℓ ′ ) ∗ · · · ∗ ( γ n ′ µ ℓ ′ n ′ ) such that all γ ′ j are also minimal and with ℓ ′ , . . . , ℓ ′ n ′ > , then n = n ′ and γ i = γ ′ i and ℓ i = ℓ ′ i for all i n .Proof. The set of execution paths of X λ is obtained as a Ω-final structure. Using Theo-rem 3.9, we obtain γ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) . for some n > ℓ + · · · + ℓ n = 1 such that for all 1 i n , there exists a globularcell c ν i such that γ i (]0 , ⊂ c ν i ,γ i (0) = c g ν i (0) ,γ i (0) = c g ν i (1) . Therefore there exists a finite set { t , . . . , t n } with t = 0 < t < · · · < t n = 1 and n > γ ([0 , ∩ X = { γ ( t i ) | i n } . We necessarily have ℓ i = t i − t i − for1 i n . Let ℓ = 0. Then we deduce that P j
With the notations above. The sequence of globular cells Carrier( γ ) = [ c ν , . . . , c ν n ] is called the carrier of γ . The integer n is called the length of the carrier. Proposition. An execution path of X λ is minimal if and only if the length of itscarrier is .Proof. It is a consequence of Theorem 5.10. (cid:3) Proposition. All execution path of X λ are locally injective.Proof. All execution paths of globes Glob G ( Z ) are one-to-one for all topological spaces Z .Therefore all minimal execution paths are locally injective (it can be a loop). The proofis complete thanks to Theorem 5.10. (cid:3) Proposition. Consider a minimal execution path γ of X λ with Carrier( γ ) = [ c ν ] .Let c ν be a globular cell of X λ with ν = ν . Then the following two assertions areequivalent:(1) γ (]0 , ∩ b c ν = ∅ (2) γ ([0 , ⊂ ∂c ν .Moreover, when the previous assertions are satisfied, there exists an execution path γ ′ from the initial state of c ν to its final state such that γ ′ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) with γ = γ i for at least one i ∈ { , . . . , n } , γ , . . . , γ n minimal and P i ℓ i = 1 .Proof. Since ∂c ν ⊂ b c ν , we deduce (2) ⇒ (1). Assume (1). Since γ (]0 , ⊂ c ν and ν = ν ,one has ν > ν . It means that there exists a point b g ν ( z, t ) of ∂c ν which belongs to c ν .It implies that z ∈ S n ν − and, since c ν ∩ X = ∅ , that t ∈ ]0 , b g ν δ z contains the globular cell c ν . We deduce that there exists φ ∈ G (1 , 1) such that b g ν δ z φ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n )with γ = γ i for at least one i ∈ { , . . . , n } , γ , . . . , γ n minimal and P i ℓ i = 1. In particular,we deduce that γ ([0 , ⊂ ∂c ν : we have proved (1) ⇒ (2). (cid:3) Proposition. An execution path γ of X λ is non-minimal if and only if there existtwo execution paths γ and γ such that γ = γ ∗ N γ . Proposition 5.15 does not hold for non q-cofibrant multipointed d -spaces. Considere.g. the multipointed d -space X obtained by starting from the directed segment −→ I G andby adding to the set of states { , } the point 1 / 2. Then all execution paths of X are on-minimal and P , / X = P / , X = ∅ . Note that the q-cofibrant replacement of X consists of the disjoint sum −→ I G ⊔ { / } . Proof. It is a consequence of Theorem 5.10 and Proposition 5.12. (cid:3) Notation. Let c ν be a globular cell of X λ . Let < h < . Let b c ν [ h ] = (cid:26) b g ν ( z, h ) | ( z, h ) ∈ | Glob G ( D n ν ) | (cid:27) Proposition. Let c ν be a globular cell of X λ . For any minimal execution path γ and any h ∈ ]0 , , the cardinal of the set (cid:26) t ∈ ]0 , | γ ( t ) ∈ b c ν [ h ] (cid:27) is at most one.Proof. If the set γ (]0 , ∩ b c ν [ h ] is nonempty, then the minimal execution path γ has atleast one point of γ (]0 , b c ν . If [ c ν ] is the carrier of γ , then γ = δ z φ with z ∈ D n ν \ S n ν − and φ ∈ G (1 , (cid:26) t ∈ ]0 , | γ ( t ) ∈ b c ν [ h ] (cid:27) = (cid:26) φ − ( h ) (cid:27) . Otherwise, by Proposition 5.14, there is the inclusion γ ([0 , ⊂ ∂c ν and there exists anexecution path b g ν δ z φ for some z ∈ S n ν − and φ ∈ G (1 , 1) from the initial state of c ν toits final state with b g ν δ z φ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n )with all γ i minimal and γ ∈ { γ , . . . , γ n } . Since γ (]0 , ∩ b c ν [ h ] is nonempty, we have h ∈ (cid:21) φ ( X j
Let Z be a topological space. Let U be an open subset of Z . Let [ a, b ] ⊂ [0 , be a segment of the real line. Denote by W ([ a, b ] , U ) the set of continuous maps f : [0 , → X such that f ([ a, b ]) ⊂ U . Theorem. Let γ ∞ be an execution path of X λ . Let ν < λ . There exists an openneighborhood Ω of γ ∞ in P G X λ such that for all execution paths γ ∈ Ω , the number ofcopies of c ν in the carrier of γ cannot exceed the length of the carrier of γ ∞ . roof. Let Carrier( γ ∞ ) = [ c ν , . . . , c ν n ]. Consider the decomposition of Theorem 5.10 γ ∞ = ( γ ∞ µ ℓ ) ∗ · · · ∗ ( γ n ∞ µ ℓ n )with P i ℓ i = 1 and all execution paths γ i ∞ minimal for i = 1 , . . . , n . For 1 i n , let ν i < λ , φ i ∈ G (1 , 1) and z i ∈ D n νi \ S n νi − such thatCarrier( γ i ∞ ) = [ c ν i ] ,γ i ∞ (]0 , ⊂ c ν i ,γ i ∞ = δ z i φ i . Since c g ν ( | Glob G ( D n ν ) | ) is a compact subspace of the Hausdorff space | X λ | , the set c g ν ( | Glob G ( D n ν ) | ) ∩ X is finite because X is discrete in | X λ | by Proposition 5.2. Wehave c c ν = { c g ν (0) , c g ν (1) } ⊔ G h ∈ ]0 , c c ν [ h ] . Therefore, for cardinality reason, there exists h ∈ ]0 , 1[ such that c c ν [ h ] ∩ X = ∅ . For all1 i n , the set (cid:26) t ∈ ]0 , | γ i ∞ ( t ) ∈ c c ν [ h ] (cid:27) contains at most one point t i by Proposition 5.17; if the set above is empty, let t i = 1 / i n , let L i and L ′ i be two real numbers such that0 < L i < t i < L ′ i < . For 1 i n , consider the covering of the segment [ P j
1] into nonoverlapping segments because we have by definition ofthe K − i , K mi , K + i for 1 i n :[0 , 1] = i = n [ i =1 (cid:20) X j
Let ( γ k ) k > be a sequence of execution paths of X λ which converges in P G X λ . Let c ν be a globular cell of X λ . Let i k be the number of times that c ν appears in Carrier( γ k ) . Then the sequence of integers ( i k ) k > is bounded.Proof. Write γ ∞ for the limit of ( γ k ) k > in P G X λ . By Theorem 5.19, there exists an openΩ containing γ ∞ such that for all γ ∈ Ω, the number of copies of c ν in the carrier of γ doesnot exceed the length of the carrier of γ ∞ . Since the sequence ( γ k ) k > converges to γ ∞ ,there exists N > k > N , γ k belongs to Ω. The proof is complete. (cid:3) Theorem. Let ν < λ . Then every execution path of X ν +1 can be written in aunique way as a finite Moore composition ( f γ µ ℓ ) ∗ · · · ∗ ( f n γ n µ ℓ n ) with n > such that(1) P i ℓ i = 1 .(2) f i = f and γ i is an execution path of X ν or f i = b g ν and γ i = δ z i φ i with z i ∈ D n ν \ S n ν − and some φ ∈ G (1 , .(3) for all i < n , either f i γ i or f i +1 γ i +1 (or both) is (are) of the form b g ν δ z φ for some z ∈ Z \ ∂Z and some φ ∈ G (1 , : intuitively, there is no possible simplification usingthe Moore composition inside X ν .Proof. We use the normal form of Theorem 5.10 and we use Proposition 4.7 to composesuccessive execution paths of X ν . (cid:3) Chains of globes Let Z , . . . , Z p be p nonempty topological spaces with p > 1. Consider the multipointed d -space X = Glob G ( Z ) ∗ · · · ∗ Glob G ( Z p ) . with p > ∗ means that the final state of a globe is identified with the initialstate of the next one by reading from the left to the right. Let { α , α , . . . , α p } be theset of states such that the canonical map Glob G ( Z i ) → X takes the initial state 0 ofGlob G ( Z i ) to α i − and the final state 1 of Glob G ( Z i ) to α i .As a consequence of the associativity of the semimonoidal structure on presheavesrecalled in Theorem 4.3 and of [Gau20a, Proposition 5.16], we have6.1. Proposition. Let U , . . . , U p be p topological spaces with p > . Let ℓ , . . . , ℓ p > .There is the natural isomorphism of enriched presheaves F G op ℓ U ⊗ . . . ⊗ F G op ℓ p U p ∼ = F G op ℓ + ··· + ℓ p ( U × . . . × U p ) . he case p = 1 of Proposition 6.2 is treated in Proposition 2.10. An additional argu-ment is required for the case p > Proposition. With the notations of this section. There is a homeomorphism P G α ,α p X ∼ = G (1 , p ) × Z × . . . × Z p . Proof. The Moore compositions of paths induced a map of enriched presheaves P , M G Glob G ( Z ) ⊗ . . . ⊗ P , M G Glob G ( Z p ) −→ P α ,α p M G ( X ) . By Proposition 4.12 (which uses Proposition 2.10), there is the isomorphism of enrichedpresheaves P , M G Glob G ( Z ) ∼ = F G op Z for all topological spaces Z . By Proposition 6.1, and since P α ,α p M G ( X ) = P G α ,α p X bydefinition of the functor M G , we obtain a continuous mapΨ : G (1 , p ) × Z × . . . × Z p = F G op p ( Z × . . . × Z p ) −→ P G α ,α p X. The latter map takes ( φ, z , . . . , z p ) to the Moore composition ( δ z φ ) ∗ · · · ∗ ( δ z p φ p )where φ i ∈ G ( ℓ i , 1) with P i ℓ i = 1 and φ = φ ⊗ . . . ⊗ φ p being the decompositionof Proposition 3.2. The map Ψ has a section which takes ( δ z φ ) ∗ · · · ∗ ( δ z p φ p ) to( φ ⊗ . . . ⊗ φ p , z , . . . , z p ). Since all executions paths of globes are one-to-one, the map Ψabove is a continuous bijection, and hence a homeomorphism. (cid:3) Until the end of this section, we work like in Section 5 with a cellular multipointed d -space X λ , with the attaching map of the globular cell c ν for ν < λ denoted by b g ν :Glob G ( D n ν ) → X λ . Each carrier c = [ c ν , . . . , c ν n ] gives rise to a map of multipointed d -spaces from a chain of globes to X λ b g c : Glob G ( D n ν ) ∗ · · · ∗ Glob G ( D n νn ) −→ X λ by “concatenating” the attaching maps of the cells c ν , . . . , c ν n . Let α i − ( α i resp.) bethe initial state (the final state resp.) of Glob G ( D n νi ) for 1 i n in Glob G ( D n ν ) ∗ · · · ∗ Glob G ( D n νn ). It induces a continuous map P G b g c : P G α ,α n (Glob G ( D n ν ) ∗ · · · ∗ Glob G ( D n νn )) −→ P G X λ . Proposition. Let γ be an execution path of X λ . Consider a nondecreasing set map φ : [0 , → [0 , preserving the extremities such that γφ = γ . Then φ is the identity of [0 , .Proof. Suppose that there exist t < t ′ such that φ ( t ) = φ ( t ′ ). Then for t ′′ ∈ [ t, t ′ ], γ ( t ′′ ) = γ ( φ ( t ′′ )) = γ ( φ ( t )) because φ ( t ) φ ( t ′′ ) φ ( t ′ ), which contradicts the fact that γ is locally injective by Proposition 5.13. Thus the set map φ is strictly increasing. LetCarrier( γ ) = [ c ν , . . . , c ν n ]. Let γ = ( γ µ ℓ ) ∗ · · · ∗ ( γ n µ ℓ n ) with ℓ + · · · + ℓ n = 1 such thatfor all 1 i n , there exist z i ∈ D n νi \ S n νi − and φ i ∈ G (1 , 1) such that for all t ∈ ]0 , γ i ( t ) = ( z i , φ i ( t )) ∈ c ν i , γ i (0) = c g ν i (0) and γ i (1) = c g ν i (1). Then { t ∈ [0 , | γ ( t ) ∈ X } = { t < t < · · · < t n = 1 } with t i = P j i ℓ j for 0 i n . We deduce that 0 = φ ( t ) < φ ( t ) < · · · < φ ( t n ) = 1because the set map φ is strictly increasing. Since γ ( φ ( t i )) = γ ( t i ) ∈ X for 0 i n , ne obtains φ ( t i ) = t i for 0 i n and φ (] t i − , t i [) ⊂ ] t i − , t i [ for all 1 i n . Then,observe that ∀ i n, ∀ t ∈ ] t i − , t i [ , ( z i , φ i ( φ ( t ))) = ( z i , φ i ( t )) . Since φ i is bijective, it means that the restriction φ ↾ ] t i − ,t i [ is the identity of ] t i − , t i [ forall 1 i n . (cid:3) Notation. Let φ be a set map from a segment [ a, b ] to a segment [ c, d ] . Let φ ( x − ) = sup { φ ( t ) | t < x } φ ( x + ) = inf { φ ( t ) | x < t } . Theorem. Let γ and γ be two execution paths of X λ such that there exist twonondecreasing set maps φ , φ : [0 , → [0 , preserving the extremities such that ∀ t ∈ [0 , , γ ( φ ( t )) = γ ( t ) ∀ t ∈ [0 , , γ ( t ) = γ ( φ ( t )) . Then φ , φ ∈ G (1 , and φ = φ − .Proof. For all t ∈ [0 , γ ( φ ( φ ( t ))) = γ ( φ ( t )) = γ ( t ). Using Proposition 6.3,we deduce that φ φ = Id [0 , . In the same way, we have φ φ = Id [0 , . This proves that φ and φ are two bijective set maps preserving the extremities which are inverse to eachother. Suppose e.g. that there exists t ∈ [0 , 1] such that φ ( t − ) < φ ( t ). Then φ cannotbe surjective: contradiction. By using similar arguments, we deduce that for all t ∈ [0 , φ ( t − ) = φ ( t ) = φ ( t + ) and φ ( t − ) = φ ( t ) = φ ( t + ). Consequently, the set maps φ and φ are continuous. (cid:3) Proposition. Every execution path of the image of P G b g c is of the form ( c g ν δ z ∗ · · · ∗ c g ν n δ z n ) φ with φ ∈ G (1 , n ) and z i ∈ D n νi for i n .Proof. The first assertion is a consequence of the definition of b g c and of Proposition 6.2. (cid:3) Notation. Using the identification provided by the homeomorphism of Proposition 6.2,we can use the notation ( P G b g c )( φ, z , . . . , z n ) = ( c g ν δ z ∗ · · · ∗ c g ν n δ z n ) φ. Theorem. Consider a sequence ( γ k ) k > of the image of P G b g c which converges to γ ∞ in P G X λ . Let γ k = ( P G b g c )( φ k , z k , . . . , z nk ) with φ k ∈ G (1 , n ) and z ik ∈ D n νi for i n and k > . Then there exist φ ∞ ∈ G (1 , n ) and z i ∞ ∈ D n νi for i n such that γ ∞ = ( P G b g c )( φ ∞ , z ∞ , . . . , z n ∞ ) and such that ( φ ∞ , z ∞ , . . . , z n ∞ ) is a limit point of the sequence (( φ k , z k , . . . , z nk )) k > . Inparticular, the image of P G b g c is closed in P G X λ .Proof. By a diagonalization argument, we can suppose that The sequence ( z ik ) k > converges to z i ∞ ∈ D n νi for each 1 i n . • The sequence ( φ k ( r )) k > converges to a real number denoted by φ ∞ ( r ) ∈ [0 , m ] for each r ∈ [0 , ∩ Q . • The sequence ( φ − k ( r )) k > converges to a real number denoted by φ − ∞ ( r ) ∈ [0 , 1] foreach r ∈ [0 , n ] ∩ Q .The sequence of execution paths ( γ k ) k > converges also pointwise to γ ∞ . We obtain γ ∞ ( r ) = ( c g ν δ z ∞ ∗ · · · ∗ c g ν n δ z n ∞ )( φ ∞ ( r ))for all r ∈ [0 , ∩ Q and γ ∞ ( φ − ∞ ( r )) = ( c g ν δ z ∞ ∗ · · · ∗ c g ν n δ z n ∞ )( r )for all r ∈ [0 , n ] ∩ Q . For r < r ∈ [0 , ∩ Q , φ k ( r ) < φ k ( r ) for all k > 0. Therefore bypassing to the limit, we obtain φ ∞ ( r ) φ ∞ ( r ). Note that φ ∞ (0) = 0 and φ ∞ (1) = n since 0 , ∈ Q . In the same way, we see that φ − ∞ : [0 , n ] ∩ Q → [0 , 1] is nondecreasingand that φ − ∞ (0) = 0 and φ − ∞ ( n ) = 1. For t ∈ ]0 , φ ∞ asfollows: φ ∞ ( t ) = sup { φ ∞ ( r ) | r ∈ ]0 , t ] ∩ Q } . The upper bound exists since { φ ∞ ( r ) | r ∈ ]0 , t ] ∩ Q } ⊂ [0 , n ]. For each t ∈ [0 , \ Q , thereexists a nondecreasing sequence ( r k ) k > of rational numbers converging to t . Thenlim k →∞ φ ∞ ( r k ) = φ ∞ ( t ) . By continuity, we deduce that γ ∞ ( t ) = ( c g ν δ z ∞ ∗ · · · ∗ c g ν n δ z n ∞ )( φ ∞ ( t ))for all t ∈ [0 , φ ∞ : [0 , → [0 , n ] is nondecreasingand that it preserves extremities. For t ∈ ]0 , φ − ∞ as well asfollows: φ − ∞ ( t ) = sup { φ − ∞ ( r ) | r ∈ ]0 , t ] ∩ Q } . The upper bound exists since { φ − ∞ ( r ) | r ∈ ]0 , t ] ∩ Q } ⊂ [0 , t ∈ [0 , n ] \ Q , thereexists a nondecreasing sequence ( r k ) k > of rational numbers converging to t . Thenlim k →∞ φ − ∞ ( r k ) = φ − ∞ ( t ) . By continuity, we deduce that γ ∞ ( φ − ∞ ( t )) = ( c g ν δ z ∞ ∗ · · · ∗ c g ν n δ z n ∞ )( t )for all t ∈ [0 , n ]. It is easy to see that the set map φ − ∞ : [0 , n ] → [0 , 1] is nondecreasingand that it preserves extremities. We obtain for all t ∈ [0 , γ ∞ ( t ) = ( c g ν δ z ∞ ∗ · · · ∗ c g ν n δ z n ∞ )( µ − n µ n φ ∞ ( t )) γ ∞ ( φ − ∞ µ − n ( t )) = ( c g ν δ z ∞ ∗ · · · ∗ c g ν n δ z n ∞ )( µ − n ( t )) . Using Theorem 6.5, we obtain that µ n φ ∞ : [0 , → [0 , 1] and φ − ∞ µ − n : [0 , → [0 , 1] arehomeomorphisms which are inverse to each other. We deduce that φ ∞ : [0 , → [0 , n ] and φ − ∞ : [0 , n ] → [0 , 1] are homeomorphisms which are inverse to each other. Let t ∈ [0 , \ Q .Since the sequence ( φ k ( t )) k > belongs to the sequential compact [0 , n ], it has at least one G (Glob G ( S n − )) = Glob( D ) (cid:15) (cid:15) M G ( g ) / / M G ( A ) M G ( f ) (cid:18) (cid:18) f (cid:15) (cid:15) M G (Glob G ( D n )) = Glob( E ) g / / M G ( b g ) / / X ψ ■■■■ $ $ ■■■■ M G ( X ) . Figure 1. Definition of X limit point ℓ . There exists a subsequence of ( φ k ( t )) k > which converges to ℓ . We obtain ∀ r ∈ [0 , t ] ∩ Q , ∀ r ′ ∈ [ t, ∩ Q , φ ∞ ( r ) ℓ φ ∞ ( r ′ ) . Since φ ∞ ∈ G (1 , n ) and by density of Q , we deduce that ℓ = φ ∞ ( t ) necessarily. Nowsuppose that the sequence ( φ k ( t )) k > does not converge to φ ∞ ( t ). Then there exists anopen neighborhood V of φ ∞ ( t ) in [0 , n ] such that for all k > φ k ( t ) / ∈ V . We deducethat the sequence ( φ k ( t )) k > of [0 , n ] has no limit point: contradiction. We have provedthat the sequence ( φ k ) k > converges pointwise to φ ∞ . Using Proposition 2.3, we deducethat ( φ k ) k > converges uniformly to φ ∞ . (cid:3) The unit and the counit of the adjunction on q-cofibrant objects Consider in this section the following situation: a pushout diagram of multipointed d -spaces Glob G ( S n − ) (cid:15) (cid:15) g / / A f (cid:15) (cid:15) Glob G ( D n ) b g / / X with n > A cellular. Note that A = X . Let D = F G op S n − and E = F G op D n .Consider the Moore flow X defined by the pushout diagram of Figure 1 where the twoequalities M G (Glob G ( S n − )) = Glob( D ) M G (Glob G ( D n )) = Glob( E )come from Proposition 4.12 and where the map ψ is induced by the universal propertyof the pushout.The enriched presheaf of execution paths of the Moore flow X can be calculated byintroducing a diagram of enriched presheaves D f over a Reedy category P g (0) ,g (1) ( A )whose definition is recalled in Appendix A. Let T be the enriched presheaf defined by the ushout diagram of [ G op , Top ] D (cid:15) (cid:15) g / / P g (0) ,g (1) M G ( A ) f (cid:15) (cid:15) E g / / T. Consider the diagram of spaces D f : P g (0) ,g (1) ( A ) → [ G op , Top ] defined as follows: D f (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n )) = Z u ,u ⊗ Z u ,u ⊗ . . . ⊗ Z u n − ,u n with Z u i − ,u i = ( P u i − ,u i A if ǫ i = 0 T if ǫ i = 1In the case ǫ i = 1, ( u i − , u i ) = ( g (0) , g (1)) by definition of P g (0) ,g (1) ( A ). The inclusionmaps I ′ i s are induced by the map f : P g (0) ,g (1) A → T . The composition maps c ′ i s areinduced by the compositions of paths of A .7.1. Theorem. [Gau20a, Theorem 9.7] We obtain a well-defined diagram of enrichedpresheaves D f : P g (0) ,g (1) ( A ) → [ G op , Top ] . There is the isomorphism of enriched presheaves lim −→ D f ∼ = P X . By the universal property of the pushout, we obtain a canonical map of enrichedpresheaves C := P ψ : lim −→ D f −→ PM G X. Proposition. Let D , . . . , D n be n enriched presheaves of [ G op , Top ] with n > .Then the mapping ( x , . . . , x n ) (Id , x , . . . , x n ) yields a surjective continuous map G ( ℓ ,...,ℓ n ) ℓ + ··· + ℓ n = L D ( ℓ ) × . . . × D n ( ℓ n ) −→ ( D ⊗ . . . ⊗ D n )( L ) . Proof. By [Gau20a, Corollary 5.13], the space ( D ⊗ . . . ⊗ D n )( L ) is the quotient of thespace G ( ℓ ,...,ℓ n ) G ( L, ℓ + · · · + ℓ n ) × D ( ℓ ) × . . . D n ( ℓ n ) . by the identifications( ψ, x φ , . . . , x n φ n ) = (( φ ⊗ . . . ⊗ φ n ) ψ, x , . . . , x n )for all ℓ , ℓ ′ , . . . , ℓ n , ℓ ′ n > 0, all ψ ∈ G ( L, ℓ + · · · + ℓ n ), all x i ∈ D i ( ℓ ′ i ) and all φ i ∈ G ( ℓ i , ℓ ′ i ).Let ℓ ′′ , . . . , ℓ ′′ n > i by the equation ∀ i n, ℓ ′′ i = ψ − (cid:18) X j i ℓ j (cid:19) − X j
Under the hypotheses and the notations of this section. The map ofenriched presheaves C : lim −→ D f −→ PM G X is an objectwive bijection. The key point in the proof of Theorem 7.3 is the existence and uniqueness proved inTheorem 5.21 which relies on the fact that the execution paths of a globe are one-to-one. Proof. The proof is divided in several parts. The final topology . If we can prove that C : lim −→ D f −→ PM G X is an objectiwe bi-jection with the colimit lim −→ D f equipped with the final topology, then the proof will becomplete even in the category of ∆-Hausdorff ∆-generated topological spaces because ofthe following facts: • Let i : A → B be a continuous one-to-one map between ∆-generated spaces such that B is also ∆-Hausdorff, then A is ∆-Hausdorff as well by [Gau19b, Proposition B.5]. • All vertices of the enriched presheaves PM G X are homeomorphic to P G X which is, bydefinition, equipped with the ∆-kelleyfication of the relative topology induced by theinclusion of set P G X ⊂ TOP ([0 , , | X | ). • If we work in the category of ∆-Hausdorff ∆-generated topological spaces, then thespace TOP ([0 , , | X | ) will be ∆-Hausdorff, and therefore lim −→ D f equipped with thefinal topology will be ∆-Hausdorff as well.Consequently, in this proof, all colimit symbols denote the colimit in the category of∆-generated topological spaces. It then suffices to prove that the continuous map C (1) : lim −→ D f (1) −→ ( PM G X )(1) = P G X. is a bijection since all objects of the reparametrization category G are isomorphic. Surjectivity of C (1). The map ψ of Figure 1 is induced by the universal property ofthe pushout. It is bijective on states. The multipointed d -space X is equipped withthe Ω-final structure because it is defined as a colimit in the category of multipointed d -spaces. By Theorem 3.9, every execution path of X is therefore a Moore compositionof the form ( f γ µ ℓ ) ∗ · · · ∗ ( f n γ n µ ℓ n )such that f i ∈ { f, b g } for all 1 i n with ( γ i ∈ P G Glob G ( D n ) if f i = b gγ i ∈ P G A if f i = f with P i ℓ i = 1. Then for all 1 i n , γ i µ ℓ i ∈ P ℓ i Glob G ( D n ) or γ i µ ℓ i ∈ P ℓ i A . It givesrise to an execution path of length 1 of the Moore flow X . It means that ψ induces a urjective continuous map from P X to P M G ( X ) = P G X . In other terms, the map C (1)is a surjection. The map b C . Consider the diagram of topological spaces E f : P g (0) ,g (1) ( A ) → Top defined as follows: E f (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n )) = G ( ℓ ,...,ℓ n ) ℓ + ··· + ℓ n =1 Z u ,u ( ℓ ) × . . . × Z u n − ,u n ( ℓ n )with Z u i − ,u i ( ℓ i ) = ( P ℓ i u i − ,u i M G A = P ℓ i u i − ,u i A if ǫ i = 0 T ( ℓ i ) if ǫ i = 1 (necessarily, ( u i − , u i ) = ( g (0) , g (1))).The composition maps c ′ i s are induced by the Moore composition of execution paths of A .The inclusion maps I ′ i s are induced by the continuous maps P ℓg (0) ,g (1) M G A → T ( ℓ ). Weobtain a well-defined diagram of topological spaces E f : P g (0) ,g (1) ( A ) → Top and, by Proposition 7.2, there is an objectwise continuous surjective map k : E f −→ D f (1) . We deduce that lim −→ k is surjective. We want to prove that the composite map b C : lim −→ E f lim −→ k / / / / (lim −→ D f )(1) C (1) / / ( PM G X )(1) = P G X is a continuous bijection. We already know that the map C (1) is surjective, and thereforethat the map b C : lim −→ E f → P G X is surjective as well.To prove that b C : lim −→ E f → P G X is one-to-one, we must first introduce the notion of simplified element. Let x be an element of some vertex of the diagram of spaces E f . Wesay that x ∈ E f ( n ) is simplified if d ( n ) = min (cid:8) d ( m ) | ∃ m ∈ Obj( P g (0) ,g (1) ( A )) and ∃ y ∈ E f ( m ) , y = x ∈ lim −→ E f (cid:9) . Let x be a simplified element belonging to some vertex E f ( n ) of the diagram E f with n = (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n )) . Case 1 . It is impossible to have ǫ i = ǫ i +1 = 0 for some 1 i < n . Indeed, otherwise x would be of the form ( . . . , γ i µ ℓ i , γ i +1 µ ℓ i +1 , . . . )where γ i and γ i +1 would be two execution paths of A . Using the equality c i (cid:0) ( . . . , γ i µ ℓ i , γ i +1 µ ℓ i +1 , . . . ) (cid:1) = ( . . . , γ i µ ℓ i ∗ γ i +1 µ ℓ i +1 , . . . ) , d is the degree function of the Reedy category. he tuple x can then be identified in the colimit with the tuple (cid:18) . . . , (cid:0) γ i µ ℓ i ∗ γ i +1 µ ℓ i +1 (cid:1) µ − ℓ i + ℓ i +1 | {z } ∈ P G A by Proposition 4.7 µ ℓ i + ℓ i +1 , . . . (cid:19) ∈ E f ( n ′ )with d ( n ′ ) = n − X i ǫ i < d ( n ) . It is a contradiction because x is simplified by hypothesis. Case 2 . Suppose that ǫ i = 1 for some 1 i n and that x is of the form( . . . , gδ z i φ i µ ℓ i , . . . ) . If z i ∈ S n − , then using the equality I i (cid:0) ( . . . , gδ z i φ i µ ℓ i , . . . ) (cid:1) = ( . . . , gδ z i φ i µ ℓ i , . . . ) , the tuple x can then be identified in the colimit with the tuple( . . . , gδ z i φ i µ ℓ i , . . . ) ∈ E f ( n ′ )with d ( n ′ ) = n + (cid:0) X i ǫ i (cid:1) − < d ( n ) . It is a contradiction because x is simplified by hypothesis. We deduce that in this case, z i ∈ D n \ S n − . Partial conclusion . Consequently, for all simplified elements x = ( x , . . . , x n ) of E f , wehave b C ( x ) = ( f x ) ∗ · · · ∗ ( f n x n )with for all 1 i n , ( f i = f and x i ∈ P ℓ i Af i = ψ and x i = gδ z i φ i µ ℓ i with z i ∈ D n \ S n − and there are no two consecutive terms of the first form (i.e. f i = f i +1 = f for some i ).It means that it is the finite Moore composition of b C ( x ) of Theorem 5.21. Injectivity of b C . Let x and y be two elements of lim −→ E f such that b C ( x ) = b C ( y ). Wecan suppose that both x and y are simplified. Let x = ( x , . . . , x m ) and y = ( y , . . . , y n ).Then ( f x ) ∗ · · · ∗ ( f m x m ) = ( g y ) ∗ · · · ∗ ( g n y n ) . Since both members of the equality are the finite Moore composition of Theorem 5.21,we deduce that m = n and that for all 1 i m , we have f i x i = g i y i . For a given i ∈ [1 , m ], there are two mutually exclusive possibilities:(1) f i = g i = f and x i and y i are two paths of length ℓ i of A . Since f is one-to-onebecause S n − is a subset of D n , we deduce that x i = y i .(2) f i = g i = ψ , x i = gδ z i φ i µ ℓ i and y i = gδ t i ψ i µ ℓ i , with z i , t i ∈ D n \ S n − and φ i , ψ i ∈G (1 , ψx i = b gδ z i φ i µ ℓ i and ψy i = b gδ t i ψ i µ ℓ i . The restriction of b g to Glob G ( D n ) \ Glob G ( S n − ) being one-to-one, we deduce that z i = t i , φ i = ψ i andtherefore once again that x i = y i . e conclude that x = y and that the map b C : lim −→ E f → P G X is one-to-one. Informal summary . The arrows of the small category P g (0) ,g (1) ( A ) and the relationssatisfied by them prove that each element of the colimit lim −→ E f has a simplified rewritingand this simplified rewriting coincides with the normal form of Theorem 5.21. The lattertheorem relies on the fact that all execution paths of globes are one-to-one, and moregenerally that all execution paths of cellular multipointed d -spaces are locally injective. Injectivity of C (1). At this point of the proof, we have a composite continuous maplim −→ E f continuous bijection b C lim −→ k / / / / lim −→ D f (1) C (1) / / / / P G X. Let a, b ∈ lim −→ D f (1) such that C (1)( a ) = C (1)( b ). Let a, b ∈ lim −→ E f such that (lim −→ k )( a ) = a and (lim −→ k )( b ) = b . Then a = b and therefore a = b . We have proved that the continuousmap C (1) : lim −→ D f (1) → P G X is one-to-one. (cid:3) Theorem. Under the hypotheses and the notations of this section. The map ofenriched presheaves C : lim −→ D f −→ PM G X is an isomorphism of presheaves.Proof. We already know by Theorem 7.3 that the map of enriched presheaves C : lim −→ D f −→ PM G X is an objectwise continuous bijection. We want to prove that it is an objectwise homeo-morphism. Since all objects of the reparametrization category G are isomorphic, it sufficesto prove that C (1) : lim −→ D f (1) −→ P G X is a homeomorphism. Consider a set map ξ : [0 , → lim −→ D f (1) such that the compositemap ξ : [0 , ξ −→ lim −→ D f (1) C (1) −→ P G X is continuous. By Proposition 2.1, it suffices to prove that the set map ξ : [0 , −→ lim −→ D f (1)is continuous as well. First reduction . The composite continuous map ξ gives rise by adjunction to a contin-uous map b ξ : [0 , × [0 , −→ | X | . Since [0 , × [0 , 1] is compact and since | X | is Hausdorff by Proposition 5.2, the subset b ξ ([0 , × [0 , | X | . By Proposition 5.5, b ξ ([0 , × [0 , d -space X . Thereforewe can suppose that the multipointed d -space X is finite by Corollary 5.7. Write { c j | j ∈ J } for its finite set of globular cells. Second reduction . It suffices to prove that there exists a finite covering { F , . . . , F n } of [0 , 1] by closed subsets such that each restriction ξ ↾ F i factors through the colimit im −→ D f (1). Let T be the set defined as follows: T = (cid:26) Carrier (cid:0) ξ ( u ) (cid:1) | u ∈ [0 , (cid:27) . Suppose that T is infinite. Since J is finite, there exist j ∈ J and a sequence ( ξ ( u n )) n > of execution paths of X such that the numbers i n of times that c j appears in the carrierof ξ ( u n ) for n > i n ) n > . Since[0 , 1] is sequentially compact, the sequence ( u n ) n > has a convergent subsequence. Bycontinuity, the sequence ( ξ ( u n )) n > has therefore a convergent subsequence in P G X . Thiscontradicts Theorem 5.20. Consequently, the set T is finite. For each carrier c ∈ T , let U c = (cid:26) u ∈ [0 , | Carrier( ξ ( u )) = c (cid:27) . Consider the closure c U c of U c in [0 , , 1] by c U c which is compact,metrizable and therefore sequential. The carrier c = [ c j , . . . , c j m ]is fixed until the very end of the proof. Third reduction . The attaching maps c g j k : Glob G ( D n jk ) −→ X for 1 k m of the cells c j , . . . , c j m yield a map of multipointed d -spaces b g c : Glob G ( D n j ) ∗ · · · ∗ Glob G ( D n jm ) −→ X. Let α i − ( α i resp.) be the initial state (the final state resp.) of Glob G ( D n ji ) for 1 i m in Glob G ( D n j ) ∗ · · · ∗ Glob G ( D n jm ). We obtain a map of enriched presheaves F G op ( D n j ) ⊗ . . . ⊗ F G op ( D n jm ) −→ D f ( m )for some m belonging to P g (0) ,g (1) ( A ) such that D f ( m ) = Z b g c ( α ) , b g c ( α ) ⊗ . . . ⊗ Z b g c ( α m − ) , b g c ( α m ) . Using Proposition 6.1, we obtain a continuous map y c : G (1 , m ) × D n j × . . . × D n jm −→ Z c ⊂ D f ( m )(1)where Z c is, by definition, the image of y c . At this point, we have obtained that thecontinuous map ξ ↾ U c : U c −→ P G X factors as a composite of maps ξ ↾ U c : U c −→ Z c ⊂ D f ( m )(1) p m −→ lim −→ D f (1) −→ P G X. Consider a sequence ( u n ) n > of U c converging to u ∞ ∈ c U c . Then for each n > ξ ( u n )belongs to the image of P G b g c which is a closed subset of the sequential space P G X byTheorem 6.8. Thus ξ ( u ∞ ) belongs to the image of P G b g c as well. We have obtained thatthe continuous map ξ ↾ c U c : c U c −→ P G X actors as a composite of maps ξ ↾ c U c : c U c −→ Z c ⊂ D f ( m )(1) p m −→ lim −→ D f (1) −→ P G X. They are all of them continuous except maybe the left-hand one from c U c to Z c . Sinceboth c U c and lim −→ D f (1) are sequential, it remains to prove that the map ξ ↾ c U c : c U c −→ Z c ⊂ D f ( m )(1) p m −→ lim −→ D f (1)is sequentially continuous to complete the proof. Sequential continuity . Consider a sequence ( u n ) n > of c U c which converges to u ∞ ∈ c U c .Write ξ ( u n ) = p m (cid:0) y c ( φ n , z n , . . . , z mn ) (cid:1) for all n > 0. We obtain ξ ( u n ) = ( P G b g c )( φ n , z n , . . . , z mn )for all n > 0. By Theorem 6.8, the sequence (( φ n , z n , . . . , z mn )) n > has a limit point( φ ∞ , z ∞ , . . . , z m ∞ ). We deduce that the sequence ( ξ ( u n )) n > has a limit point because both y c and p m are continuous. It is necessarily equal to ξ ( u ∞ ) because the map C (1) : lim −→ D f (1) → P G X is continuous bijective by Theorem 7.3 and because ξ = C (1) .ξ. The same argument shows that every subsequence of ( ξ ( u n )) n > has a limit point which isnecessarily ξ ( u ∞ ), unlike the sequence (( φ n , z n , . . . , z mn )) n > which can have several limitpoints. Suppose that the sequence ( ξ ( u n )) n > does not converge to ξ ( u ∞ ). Then thereexists an open neighborhood V of ξ ( u ∞ ) such that for all n > ξ ( u n ) / ∈ V . Since V c isclosed in lim −→ D f (1), it means that ξ ( u ∞ ) cannot be a limit point of the sequence ( ξ ( u n )) n > .Contradiction. It implies that the sequence ( ξ ( u n )) n > converges to ξ ( u ∞ ). (cid:3) Corollary. Suppose that A is a cellular multipointed d -space. Consider a pushoutdiagram of multipointed d -spaces Glob G ( S n − ) (cid:15) (cid:15) / / A (cid:15) (cid:15) Glob G ( D n ) / / X with n > . Then there is the pushout diagram of Moore flows M G (Glob G ( S n − )) = Glob( F G op S n − ) (cid:15) (cid:15) / / M G ( A ) (cid:15) (cid:15) M G (Glob G ( D n )) = Glob( F G op D n ) / / M G ( X ) . Corollary. Let X be a q-cofibrant multipointed d -space. Then M G ( X ) is a q-cofibrantMoore flow. roof. For every q-cofibrant Moore flow X , the canonical map ∅ → X is a retract ofa transfinite composition of the q-cofibrations C : ∅ → { } , R : { , } → { } and ofthe q-cofibrations Glob( F G op ℓ S n − ) ⊂ Glob( F G op ℓ D n ) for ℓ > n > 0. The cofibration R : { , } → { } is not necessary to reach all q-cofibrant objects. Therefore, this theoremis a consequence of Theorem 5.8 and of Corollary 7.5. (cid:3) Theorem. Let X be a q-cofibrant Moore flow. Then the unit of the adjunction X → M G ( M G ! ( X )) is an isomorphism.Proof. By Proposition 4.12, the theorem holds when X is a globe. It also clearly holdsfor X = { } . For every q-cofibrant Moore flow X , the canonical map ∅ → X is a retractof a transfinite composition of the q-cofibrations C : ∅ → { } , R : { , } → { } and ofthe q-cofibrations Glob( F G op ℓ S n − ) ⊂ Glob( F G op ℓ D n ) for ℓ > n > 0. The cofibration R : { , } → { } is not necessary to reach all q-cofibrant objects. Therefore, this theoremis also a consequence of Theorem 5.8 and of Corollary 7.5. (cid:3) Corollary. Let X be a q-cofibrant Moore flow. Then there is the homeomorphism P X ∼ = P G ( M G ! ( X )) . Proof. Apply the functor P ( − ) to the isomorphism X ∼ = M G ( M G ! ( X )). (cid:3) Theorem. Let λ be a limit ordinal. Let X : λ −→ G dTop be a colimit preserving functor such that • The multipointed d -space X is a set, in other terms X = ( X , X , ∅ ) . • For all ν < λ , there is a pushout diagram of multipointed d -spaces Glob G ( S n ν − ) (cid:15) (cid:15) g ν / / X ν (cid:15) (cid:15) Glob G ( D n ν ) c g ν / / X ν +1 with n ν > .Let X λ = lim −→ ν<λ X ν . For all ν λ , the counit map κ ν : M G ! ( M G ( X ν )) −→ X ν is an isomorphism.Proof. The map κ is an isomorphism because X is a set. By Theorem 5.8, and since M G ! is a left adjoint, it suffices to prove that if κ ν is an isomorphism, then κ ν +1 is anisomorphism. Assume that κ ν is an isomorphism. By Corollary 7.5, there is the pushoutdiagram of Moore flows M G (Glob G ( S n ν − )) = Glob( F G op S n ν − ) (cid:15) (cid:15) g ν / / M G ( X ν ) (cid:15) (cid:15) M G (Glob G ( D n ν )) = Glob( F G op D n ν ) c g ν / / M G ( X ν +1 ) . pply again the left adjoint M G ! to this diagram, we obtain by using the induction hy-pothesis that κ ν +1 is an isomorphism. (cid:3) Corollary. For every q-cofibrant multipointed d -space X , the counit of the adjunc-tion M G ! ( M G ( X )) → X is an isomorphism of multipointed d -spaces.Proof. It is due to the fact that every q-cofibrant multipointed d -space X is a retract of acellular multipointed d -space (note that the cofibration R : { , } → { } is not requiredto reach all cellular multipointed d -spaces) and that a retract of an isomorphism is anisomorphism. (cid:3) From multipointed d -spaces to flows The goals of this final section are to complete the proof of the Quillen equivalence be-tween multipointed d -spaces and Moore flows, which together with the results of [Gau20a]will establish that multipointed d -spaces and flows have Quillen equivalent q-model struc-tures, and to give a new and conceptual proof of [Gau09, Theorem 7.5] in Theorem 8.13.The old proof relies on the work expounded in [Gau05], whose some proofs are fixed in[Gau19b, Section 6]. All the results of this paper (and of [Gau20a]) are independant of[Gau05].8.1. Theorem. The adjunction M G ! ⊣ M G : G Flow ⇆ G dTop induces a Quillen equiv-alence between the q-model structure of Moore flows and the q-model structure of multi-pointed d -spaces.Proof. Since the q-fibrations of Moore flows are the maps of Moore flows inducing anobjectwise q-fibration on the presheaves of execution paths, the functor M G takes q-fibrations of multipointed d -spaces to q-fibrations of Moore flows. Since M G preserves theset of states and since trivial q-fibrations of Moore flows are maps inducing a bijectionon states and an an objectwise trivial q-fibration on the presheaves of execution paths,the functor M G takes trivial q-fibrations of multipointed d -spaces to trivial q-fibrationsof Moore flows. Therefore, the functor M G : G dTop → G Flow is a right Quillen adjoint.By Corollary 7.7, the map X → M G ( M G ! ( X )) is a weak equivalence of Moore flows forevery q-cofibrant Moore flow X . Let Y be a (q-fibrant) multipointed d -space. Then thecomposite map of multipointed d -spaces M G ! ( M G ( Y cof )) ∼ = −→ Y cof ∼ −→ Y where Y cof is a q-cofibrant replacement of Y , is a weak equivalence of multipointed d -spaces: 1) the left-hand map is an isomorphism by Corollary 7.10; 2) the right-hand mapis a weak equivalence by definition of a cofibrant replacement. (cid:3) Let us give now some reminders about flows and the categorization functor cat frommultipointed d -spaces to flows.8.2. Definition. [Gau03, Definition 4.11] A flow is a small semicategory enriched overthe closed monoidal category ( Top , × ) . The corresponding category is denoted by Flow . Let us expand the definition above. A flow X consists of a topological space P X ofexecution paths, a discrete space X of states, two continuous maps s and t from P X to called the source and target map respectively, and a continuous and associative map ∗ : { ( x, y ) ∈ P X × P X ; t ( x ) = s ( y ) } −→ P X such that s ( x ∗ y ) = s ( x ) and t ( x ∗ y ) = t ( y ). A morphism of flows f : X −→ Y consistsof a set map f : X −→ Y together with a continuous map P f : P X −→ P Y such that f ( s ( x )) = s ( P f ( x )) ,f ( t ( x )) = t ( P f ( x )) , P f ( x ∗ y ) = P f ( x ) ∗ P f ( y ) . Let P α,β X = { x ∈ P X | s ( x ) = α and t ( x ) = β } . Notation. The map P f : P X −→ P Y can be denoted by f : P X → P Y is there isno ambiguity. The set map f : X −→ Y can be denoted by f : X −→ Y is there isno ambiguity. The category Flow is locally presentable. Every set can be viewed as a flow withan empty path space. The obvious functor Set ⊂ Flow is limit-preserving and colimit-preserving. One another example of flow is important for the sequel:8.4. Example. For a topological space Z , let Glob( Z ) be the flow defined by Glob( Z ) = { , } , P Glob( Z ) = P , Glob( Z ) = Z,s = 0 , t = 1 . This flow has no composition law. Notation. C : ∅ → { } ,R : { , } → { } , −→ I = Glob( { } ) . The q-model structure of flows ( Flow ) q is the unique combinatorial model structuresuch that { Glob( S n − ) ⊂ Glob( D n ) | n > } ∪ { C, R } is the set of generating cofibrations and such that { Glob( D n × { } ) ⊂ Glob( D n +1 ) | n > } is the set of generating trivial cofibrations (e.g. [Gau20b, Theorem 7.6]) where the maps D n ⊂ D n +1 are induced by the mappings ( x , . . . , x n ) ( x , . . . , x n , f : X → Y inducing a bijection f : X ∼ = Y anda weak homotopy equivalence P f : P X → P Y and the fibrations are the maps of flows f : X → Y inducing a q-fibration P f : P X → P Y of topological spaces.Let X be a multipointed d -space. Consider for every ( α, β ) ∈ X × X the coequalizerof sets P α,β X = lim −→ (cid:16) P G α,β X × P G −→ I G ⇒ P G α,β X (cid:17) here the two maps are ( c, φ ) c ◦ Id −→ I G = c and ( c, φ ) c ◦ φ . Let [ − ] α,β : P G α,β X → P α,β X be the canonical set map. The set P α,β X is equipped with the final topology.8.6. Theorem. [Gau09, Theorem 7.2] Let X be a multipointed d -space. Then thereexists a flow cat ( X ) with cat ( X ) = X , P α,β cat ( X ) = P α,β X and the composition law ∗ : P α,β X × P β,γ X → P α,γ X is for every triple ( α, β, γ ) ∈ X × X × X the unique mapmaking the following diagram commutative: P G α,β X × P G β,γ X / / [ − ] α,β × [ − ] β,γ (cid:15) (cid:15) P G α,γ X [ − ] α,γ (cid:15) (cid:15) P α,β X × P β,γ X / / P α,γ X where the map P G α,β X × P G β,γ X → P G α,γ X is the continuous map defined by the concatenationof continuous paths: ( c , c ) ∈ P G α,β X × P G β,γ X is sent to c ∗ N c . The mapping X cat ( X ) induces a functor from G dTop to Flow . Definition. The functor cat : G dTop → Flow is called the categorization functor . The motivation for the constructions of this paper and of [Gau20a] comes from thefollowing theorem which is added for completeness.8.8. Theorem. The categorization functor cat : G dTop → Flow is neither a left adjointnor a right adjoint. In particular, it cannot be a left or a right Quillen equivalence.Proof. This functor is not a left adjoint by [Gau09, Proposition 7.3]. Suppose that it is aright adjoint. Let L : Flow → G dTop be the left adjoint. Pick a nonempty topologicalspace Z . The adjunction yields the bijection of sets G dTop ( L (Glob( Z )) , −→ I G ) ∼ = Flow (Glob( Z ) , −→ I ) . Since Z is nonempty, a map of flows from Glob( Z ) to −→ I is determined by the choiceof a map from Z to { } . We deduce that there is exactly one map f of multipointed d -spaces from L (Glob( Z )) to −→ I G . Suppose that L (Glob( Z )) contains at least one executionpath φ : [0 , → |L (Glob( Z )) | . Then f.φ is an execution path of −→ I G . Every map g ∈ G dTop ( −→ I G , −→ I G ) ∼ = { [0 , ∼ = + [0 , } gives rise to and execution path g.f.φ of −→ I G .since g.f ∈ G dTop ( L (Glob( Z )) , −→ I G ), we deduce that g.f = f . Contradiction. Wededuce that the multipointed d -space L (Glob( Z )) does not contain any execution path.Therefore this multipointed d -space is of the form ( U Z , U Z , ∅ ). We obtain the bijection MTop (( U Z , U Z ) , ([0 , , { , } )) ∼ = { f } . Suppose that U Z is nonempty. Then for all g ∈ MTop (([0 , , { , } ) , ([0 , , { , } )), we have g.f = f . The only possibilities are that f = 0 or f = 1. Since f is the unique element, we deduce that U Z = ∅ . There are alsothe natural bijections of sets G dTop ( L ( { } ) , X ) ∼ = Flow ( { } , cat ( X )) ∼ = cat ( X ) ∼ = X ∼ = G dTop ( { } , X ) . By the Yoneda lemma, we obtain L ( { } ) = { } .To summarize, if L : Flow → G dTop is a left adjoint to the functor cat : G dTop → Flow , then one has L ( { } ) = { } and for all nonempty topological spaces Z , there is theequalities L (Glob( Z )) = ∅ . By [Gau03, Theorem 6.1], any flow is a colimit of globes and oints. Since L is colimit-preserving, we deduce that for all flows Y , the multipointed d -space L ( Y ) is a set. We go back to the natural bijection given by the adjunction: G dTop ( L ( Y ) , X ) ∼ = Flow ( Y, cat ( X )) . Since L ( Y ) is a set, we obtain the natural bijection Set ( L ( Y ) , X ) ∼ = Flow ( Y, cat ( X )).We obtain the natural bijection G dTop ( L ( Y ) , X ) ∼ = Flow ( Y, cat ( X )) and by adjunctionthe natural bijection Flow ( Y, X ) ∼ = Flow ( Y, cat ( X )) since cat ( X ) = X . By Yoneda,we conclude that cat ( X ) = X for all multipointed d -spaces X , which is a contradiction. (cid:3) Proposition. [Gau20a, Proposition 5.17] Let U and U ′ be two topological spaces.There is the natural isomorphism of enriched presheaves ∆ G op U ⊗ ∆ G op U ′ ∼ = ∆ G op ( U × U ′ ) . Let X be a flow. The Moore flow M ( X ) is the enriched semicategory defined as follows: • The set of states is X . • The enriched presheaf P α,β M ( X ) is the enriched presheaf ∆ G op ( P α,β X ). • The composition law is defined, using Proposition 8.9 as the composite map∆ G op ( P α,β X ) ⊗ ∆ G op ( P β,γ X ) ∼ = ∆ G op ( P α,β X × P β,γ X ) ∆ G op ( ∗ ) / / ∆ G op ( P α,γ ) X. The construction above yields a well-defined functor M : Flow → P Flow . Consider a P -flow Y . For all α, β ∈ Y , let Y α,β = lim −→ P α,β Y . Let ( α, β, γ ) be a triple ofstates of Y . The composition law of the P -flow Y induces a continuous map Y α,β × Y β,γ ∼ = lim −→ ( P α,β Y ⊗ P β,γ Y ) −→ lim −→ P α,γ Y ∼ = Y α,γ which is associative. We obtain the8.10. Proposition. [Gau20a, Proposition 10.6 and Proposition 10.7] For any P -flow Y ,the data • The set of states is Y • For all α, β ∈ Y , let Y α,β = lim −→ P α,β Y • For all α, β, γ ∈ Y , the composition law Y α,β × Y β,γ → Y α,γ assemble to a flow denoted by M ! ( Y ) . It yields a well-defined functor M ! : P Flow → Flow . There is an adjunction M ! ⊣ M . Theorem. There is the isomorphism of functors cat ∼ = M ! . M G . Proof. First let us notice that the functors cat : G dTop → Flow (Theorem 8.6), M G : G dTop → P Flow (Theorem 4.9) and M ! : P Flow → Flow (Proposition 8.10) preservethe set of states by definition of these functors. Therefore, for every multipointed d -space X , the flows cat ( X ) and M ! . M G ( X ) have the same set of states X . Let G bethe full subcategory of G generated by 1: the set of objects is the singleton { } and (1 , 1) = G (1 , α, β ) ∈ X × X , the inclusion functor ι : G ⊂ G induces acontinuous map lim −→ G (cid:18) (cid:0) P α,β M G X (cid:1) .ι (cid:19) → lim −→ G P α,β M G X. It turns out that there is the natural homeomorphismslim −→ G (cid:18) (cid:0) P α,β M G X (cid:1) .ι (cid:19) ∼ = lim −→ G P α,β M G X ∼ = lim −→ G P G α,β X ∼ = P α,β cat ( X ) , the first one by definition of ι , the second one by definition of M G and the last one bydefinition of cat . We obtain a natural map of flows cat ( X ) → ( M ! . M G )( X ) which isbijective on states. Let ℓ > G . Then the comma category ( ℓ ↓ ι ) ischaracterized as follows: • The set of objects is G ( ℓ, 1) which is always nonempty for every ℓ > • The set of maps from an arrow u : ℓ → v : ℓ → G )( u, v ).The comma category ( ℓ ↓ ι ) is connected since in any diagram of G of the form[0 , ℓ ] u / / [0 , k (cid:15) (cid:15) ✤✤✤✤ [0 , ℓ ] v / / [0 , , there exists a map k ∈ G ([0 , , [0 , k = v.u − .By [ML98, Theorem 1 p. 213], we deduce that the natural map of flows cat ( X ) → ( M ! . M G )( X ) induces a homeomorphism between the spaces of paths. (cid:3) Definition. We consider the composite functors ( L cat ) : G dTop ( − ) cof / / G dTop cat / / Flow ( L cat ) − : Flow M / / G Flow ( − ) cof / / G Flow M G ! / / G dTop where ( − ) cof is a q-cofibrant replacement functor. We can now write down the new proof of [Gau09, Theorem 7.5].8.13. Theorem. The categorization functor from multipointed d -spaces to flows cat : G dTop −→ Flow takes q-cofibrant multipointed d -spaces to q-cofibrant flows. Its total left derived functorin the sense of [DHKS04] induces an equivalence of categories between the homotopycategories of the q-model structures.Proof. Let us recall that all objects are q-fibrant. By Corollary 7.6, every q-cofibrant mul-tipointed d -space X is taken to a q-cofibrant Moore flow M G ( X ). Thus, by Theorem 8.11,the categorization functor from multipointed d -spaces to flows preserves q-cofibrancy. Wededuce that ( L cat ) preserves q-cofibrancy as well. Any flow, q-cofibrant or not, are takenby ( L cat ) − to a q-cofibrant multipointed d -space since M G ! is a left Quillen adjoint. et X be a multipointed d -space. Then we have the sequence of isomorphisms and ofweak equivalences ( L cat ) − ( L cat )( X ) ∼ = M G ! (cid:18) MM ! q-cofibrantby Corollary 7.6 M G ( X cof ) (cid:19) cof ≃ M G ! (cid:18) M G ( X cof ) (cid:19) cof ≃ M G ! M G ( X cof ) ∼ = X cof ≃ X, the first isomorphism by definition of ( L cat ) and ( L cat ) − and by Theorem 8.11, thefirst weak equivalence since the adjunction M ! ⊢ M is a Quillen equivalence by [Gau20a,Theorem 10.9] and since M G ! is a left Quillen adjoint, the second weak equivalence byCorollary 7.6 and again since M G ! is a left Quillen adjoint, the second isomorphism byCorollary 7.10, and the last weak equivalence by definition of a q-cofibrant replacement.Let Y be a flow. Then we have the sequence of isomorphisms and of weak equivalences( L cat )( L cat ) − ( Y ) ∼ = (cid:0) M ! M G (cid:1)(cid:18) M G ! ( M Y ) cof (cid:19) cof ≃ (cid:0) M ! M G (cid:1)(cid:18) M G ! ( M Y ) cof (cid:19) ∼ = M ! ( M Y ) cof ≃ Y, the first isomorphism by definition of ( L cat ) and ( L cat ) − and by Theorem 8.11, the firstweak equivalence because M G is a right Quillen adjoint, because M G ! ( M Y ) cof is q-cofibrant,because M G preserves q-cofibrancy by Corollary 7.6 and finally because M ! is a left Quillenadjoint, the second isomorphism by Corollary 7.7, and finally the last weak equivalencesince the adjunction M ! ⊢ M is a Quillen equivalence by [Gau20a, Theorem 10.9]. Theproof is complete. (cid:3) Appendix A. The Reedy category P u,v ( S ) : reminder Let S be a nonempty set. Let P u,v ( S ) be the small category defined by generators andrelations as follows (see [Gau19b, Section 3]): • u, v ∈ S ( u and v may be equal). • The objects are the tuples of the form m = (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n ))with n > u , . . . , u n ∈ S and ∀ i such that 1 i n, ǫ i = 1 ⇒ ( u i − , u i ) = ( u, v ) . • There is an arrow c n +1 : ( m, ( x, , y ) , ( y, , z ) , n ) → ( m, ( x, , z ) , n ) or every tuple m = (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n )) with n > n = (( u ′ , ǫ ′ , u ′ ) , ( u ′ , ǫ ′ , u ′ ) , . . . , ( u ′ n ′ − , ǫ ′ n ′ , u ′ n ′ )) with n ′ > 0. It is called a compo-sition map . • There is an arrow I n +1 : ( m, ( u, , v ) , n ) → ( m, ( u, , v ) , n )for every tuple m = (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n )) with n > n = (( u ′ , ǫ ′ , u ′ ) , ( u ′ , ǫ ′ , u ′ ) , . . . , ( u ′ n ′ − , ǫ ′ n ′ , u ′ n ′ )) with n ′ > 0. It is called an inclusion map . • There are the relations (group A) c i .c j = c j − .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 − 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) c i .I j = ( I j − .c i if j > i + 2 I j .c i if j i − . By definition of these maps, c i and I i are never composable as well as c i and I i +1 .By [Gau19b, Proposition 3.7], there exists a structure of Reedy category on P u,v ( S )with the N -valued degree map defined by d (( u , ǫ , u ) , ( u , ǫ , u ) , . . . , ( u n − , ǫ n , u n )) = n + X i ǫ i . The maps raising the degree are the inclusion maps in the above sense. References [AR94] J. Ad´amek and J. Rosick´y. Locally presentable and accessible categories. Cambridge Univer-sity Press, Cambridge, 1994. 3, 16[Bor94] F. Borceux. Handbook of categorical algebra. 2. Cambridge University Press, Cambridge,1994. Categories and structures. 3[CSW14] D. Christensen, G. Sinnamon, and E. Wu. The D Gau19a] P. Gaucher. Enriched diagrams of topological spaces over locally contractible enriched cate-gories. New York J. Math., 25:1485–1510 (electronic), 2019. 14[Gau19b] P. Gaucher. Left properness of flows. arXiv:1907.01454, 2019. 4, 8, 32, 39, 44, 45[Gau20a] P. Gaucher. Homotopy theory of Moore flows (I). arXiv, 2020. 1, 2, 5, 8, 14, 16, 17, 26, 31,39, 41, 42, 44[Gau20b] P. Gaucher. Six model categories for directed homotopy, 2020. To appear in Categories andGeneral Algebraic Structures with Applications. 7, 9, 40[Gra03] M. Grandis. Directed homotopy theory. I. Cah. Topol. G´eom. Diff´er. Cat´eg., 44(4):281–316,2003. 2[Gra04a] M. Grandis. Inequilogical spaces, directed homology and noncommutative geometry.Homology Homotopy Appl., 6(1):413–437, 2004. 2[Gra04b] M. Grandis. Normed combinatorial homology and noncommutative tori. Theory Appl. Categ.,13:No. 7, 114–128, 2004. 2[Gra05] M. Grandis. Directed combinatorial homology and noncommutative tori (the breaking ofsymmetries in algebraic topology). Math. Proc. Cambridge Philos. Soc., 138(2):233–262, 2005.2[Hat02] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. 5, 19[Hir03] P. S. Hirschhorn. Model categories and their localizations, volume 99 of Mathematical Surveysand Monographs. American Mathematical Society, Providence, RI, 2003. 3[Hov99] M. Hovey. Model categories. American Mathematical Society, Providence, RI, 1999. 3[Kel05] G. M. Kelly. Basic concepts of enriched category theory. Repr. Theory Appl. Categ.,(10):vi+137 pp. (electronic), 2005. Reprint of the 1982 original [Cambridge Univ. Press, Cam-bridge; MR0651714]. 3[Kri09] S. Krishnan. A convenient category of locally preordered spaces. Appl. Categ. Structures,17(5):445–466, 2009. 2[ML98] S. Mac Lane. Categories for the working mathematician. Springer-Verlag, New York, secondedition, 1998. 43[MS06] J. P. May and J. Sigurdsson. Parametrized homotopy theory, volume 132 of MathematicalSurveys and Monographs. American Mathematical Society, Providence, RI, 2006. 4[Ros09] J. Rosick´y. On combinatorial model categories. Appl. Categ. Structures, 17(3):303–316, 2009.3 Universit´e de Paris, CNRS, IRIF, F-75006, Paris, France URL :