�
EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 36, Num. 2 (2021), 157 – 239

doi:10.17398/2605-5686.36.2.157

Available online November 8, 2021

Homotopy theory of Moore flows (II)

P. Gaucher

Université de Paris, CNRS, IRIF, F-75006, Paris, France

http://www.irif.fr/~gaucher

Received August 21, 2021 Presented by A.M. Cegarra
Accepted September 29, 2021

Abstract: This paper proves that the q-model structures of Moore flows and of multipointed d-spaces

are Quillen equivalent. The main step is the proof that the counit and 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 that the categorization functor from multipointed d-spaces to

flows has a total left derived functor which induces a category equivalence between the homotopy
categories. The new proof sheds light on the internal structure of the categorization functor which

is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy

of this functor using Moore flows.

Key words: enriched semicategory, semimonoidal structure, combinatorial model category, Quillen
equivalence, locally presentable category, topologically enriched category, Moore path.

MSC (2020): 18C35, 18D20, 55U35, 68Q85.

Contents

1 Introduction 158

2 Multipointed d-spaces 162

3 Moore composition and Ω-final structure 170

4 From multipointed d-spaces to Moore flows 177

5 Cellular multipointed d-spaces 184

6 Chains of globes 198

7 The unit and the counit of the adjunction
on q-cofibrant objects 207

8 From multipointed d-spaces to flows 220

A The Reedy category Pu,v(S): reminder 230

B An explicit construction of the left adjoint MG! 231

C The setting of k-spaces 235

ISSN: 0213-8743 (print), 2605-5686 (online)

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.36.2.157
http://www.irif.fr/~gaucher
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


158 p. gaucher

1. Introduction

Presentation. This paper is the companion paper of [14]. The purpose
of these two papers is to exhibit, by means of the q-model category of Moore
flows (cf. Definition 4.6), a zig-zag of Quillen equivalences between the q-
model structure of multipointed d-spaces introduced in [11] and the q-model
structure of flows introduced in [7]. The only known functor which was a
good candidate for a Quillen equivalence from multipointed d-spaces to flows
(Definition 8.7) has indeed a total left derived functor in the sense of [4]
which induces an equivalence of categories between the homotopy categories
( [11, Theorem 7.5]). 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 pro-
cesses. For example, the paper [10] shows how to model all process algebras
for any synchronization algebra using flows. There are many geometric models
of concurrency available in the literature [17–20, 28] (the list does not pretend
to be exhaustive). Most of them are used to study the fundamental category
of a concurrent process or any derived concept. It is something which can
be also carried out with the formalisms of flows and multipointed d-spaces.
The fundamental category functor is easily calculable indeed, at least for flows
since it is a left adjoint, and for cellular multipointed d-spaces by using Corol-
lary 8.12, and it interacts very well with the underlying simplicial structures.

The Quillen equivalence between flows and Moore flows is proved in [14,
Theorem 10.9]. The Quillen equivalence between multipointed d-spaces and
Moore flows is proved in Theorem 8.1. The latter theorem is a consequence
of the structural properties of the adjunction between multipointed d-spaces
and Moore flows which can be summarized as follows:

Theorem 1.1. (Theorem 7.6, Corollary 7.9 and Theorem 8.1) The ad-
junction

MG! a M
G : GFlow �GdTop

between Moore flows and multipointed d-spaces is a Quillen equivalence. The
counit map and the unit map of this Quillen adjunction are isomorphisms on
q-cofibrant objects (recall that all objects are q-fibrant).

Another standard example of the situation of Theorem 1.1 is the Quillen
equivalence between the q-model structures of ∆-generated spaces and of k-
spaces (cf. Appendix C).



homotopy theory of moore flows (II) 159

This paper is the first use in a real practical situation of the closed semi-
monoidal category of G-spaces (Definition 4.1 and Theorem 4.3). It illustrates
the interest of this structure already for calculating spaces of execution paths
of cellular multipointed d-spaces. The interest of this structure is beyond di-
rected homotopy theory, as remarked in the introduction of [14] where possible
connections with type theory are briefly discussed. The potential of this semi-
monoidal structure is visible in the proofs of Proposition 6.3, Theorem 7.2,
Theorem 7.3 and Corollary 7.4.

The Moore flows enable us to write explicitly an “inverse up to homotopy”
of the categorization functor of Definition 8.7 in Definition 8.13. Two appli-
cations of the existence of this inverse up to homotopy are given. The first
one is a new proof of [11, Theorem 7.5] provided in Theorem 8.14 which is
totally independent from [8, 11]. The second one is a concise and very natural
definition of the underlying homotopy type of a flow in Proposition 8.16.

As a curiosity, it is also proved in passing a kind of second Dini theorem
for spaces of execution paths of finite cellular multipointed d-spaces without
loops in Corollary 6.12.

This paper is written in the setting of ∆-Hausdorff or not ∆-generated
spaces. The setting of weakly Hausdorff or not k-spaces is of very little interest
for the study of multipointed d-spaces and flows not only because all concrete
examples coming from computer science are cellular objects of the q-model
structures, and also because it is not known how to left Bousfield localize be-
cause of the cofibration identifying two states. The locally presentable setting
has many other advantages like the existence of adjoints [15, Theorem 5.10],
the smallness conditions of [25, Definition 2.1.3] always satisfied and the exis-
tence of left-determined model categories in the tractable cases [22]. It might
be interesting anyway to make some comments about k-spaces to emphasize
some topological arguments of this paper. These comments are postponed to
Appendix C.

Outline of the paper.

• Section 2 is a reminder about multipointed d-spaces and about their q-
model structure. It also contains new results about the topology of the
space of execution paths. The section starts with a short reminder about
∆-generated spaces. The notion of ∆-inclusion is introduced to clarify some
topological arguments: they are for ∆-generated spaces what k-inclusions
are for k-spaces.

• The functor Ω which forgets the set of execution paths of a multipointed
d-space is topological. Section 3 gives an explicit description of the Ω-final



160 p. gaucher

structure in term of Moore composition. It culminates with Theorem 3.9.
The calculations are a bit laborious but some of them are used further in
the paper.

• Section 4, after a reminder about Moore flows and their q-model structure,
describes the adjunctions between multipointed d-spaces and Moore flows.
The right adjoint from multipointed d-spaces to Moore flows is quite easy
to define. The existence of the left adjoint is straightforward. Appendix B
provides an explicit construction of this left adjoint MG! : GFlow →GdTop.
It uses results dating back to [7] obtained for flows, i.e., small semicategories
enriched over topological spaces, and adapted to Moore flows, i.e., small
semicategories enriched over G-spaces. This explicit construction is not
necessary to establish the results of the main part of the paper. It is the
reason why it is postponed to an appendix.

• Section 5 gathers some geometric properties of cellular multipointed d-
spaces concerning their underlying topologies, the topologies of their spaces
of execution paths and some of their structural properties like Theorem 5.18
which has important consequences. The main tools are the notion of carrier
of an execution path (Definition 5.10) and the notion of achronal slice of
a globular cell (Definition 5.15) studied in Proposition 5.16 and Proposi-
tion 5.17. It also contains Theorem 5.20 which provides a kind of normal
form for the execution paths of a cellular multipointed d-space obtained as
a pushout along a generating q-cofibration.

• Section 6 studies chains of globes. It is an important geometric object for
the proofs of this paper. It enables us to understand what happens locally
in the space of execution paths of a cellular multipointed d-space. The main
theorem is Theorem 6.11 which can be viewed as a workaround of the fact
that the space G(1, 1) of nondecreasing homeomorphisms from [0, 1] to itself
equipped with the compact-open topology is not sequentially compact. As
a byproduct, it is also proved in Corollary 6.12 a second Dini theorem for
finite cellular multipointed d-spaces without loops.

• Section 7 is the core of the paper. It proves that the unit and the counit of
the adjunction are isomorphisms on q-cofibrant objects in Theorem 7.6 and
in Corollary 7.9. The main tool of this part is Corollary 7.4 which proves
that the right adjoint constructed in Section 4 preserves pushouts of cellular
multipointed d-spaces along q-cofibrations. It relies on Theorem 7.2 whose
proof performs an analysis of the execution paths in a pushout along a q-
cofibration and on Theorem 7.3 whose proof carries out a careful analysis
of the underlying topology of the spaces of execution paths involved.



homotopy theory of moore flows (II) 161

• Section 8 is the concluding section. It establishes that the adjunction be-
tween multipointed d-spaces and Moore flows yields a Quillen equivalence
between the q-model structures. It provides, as an application, a more
conceptual proof of the fact that the categorization functor cat from mul-
tipointed d-spaces to flows has a total left derived functor which induces a
category equivalence between the homotopy categories of the q-model struc-
tures of multipointed d-spaces and flows. And finally, it is shown how to
recover the underlying homotopy type of a flow in a very intuitive way.

Prerequisites. We refer to [1] for locally presentable categories, to [33]
for combinatorial model categories. We refer to [24,25] for more general model
categories. We refer to [27] and to [2, Chapter 6] for enriched categories. All
enriched categories are topologically enriched categories: the word topologi-
cally is therefore omitted. 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.
• TOP is the category of general topological spaces together with the contin-

uous maps.

• Kop denotes the opposite category of K.
• Obj(K) is the class of objects of K.
• Mor(K) is the category of morphisms of K with the commutative squares

for the morphisms.

• KI is the category of functors and natural transformations from a small
category I to K.
• ∆I(Z) is the constant diagram over the small category I with unique

value Z.

• ∅ is the initial object, 1 is the final object, IdX is the identity of X.
• K(X,Y ) is the set of maps in a set-enriched, i.e., locally small, category K.
• K(X,Y ) is the space of maps in an enriched category K. The underlying

set of maps may be denoted by K0(X,Y ) if it is necessary to specify that
we are considering the underlying set.



162 p. gaucher

• The composition of two maps f : A → B and g : B → C is denoted by gf
or, if it is helpful for the reader, by g.f; the composition of two functors is
denoted in the same way.

• The notations `,`′,`i,L, . . . mean a strictly positive real number unless
specified something else.

• [`,`′] denotes a segment. Unless specified, it is always understood
that ` < `′.

• A cellular object of a combinatorial model category is an object X such
that the canonical map ∅ → X is a transfinite composition of pushouts of
generating cofibrations.

• The notation (−)cof denotes a cofibrant replacement functor of a combina-
torial model structure; note that all model categories of this paper contain
only fibrant objects.

• A compact space is a quasicompact Hausdorff space.
• A sequentially compact space is a space such that each sequence has a limit

point.

• The set of rational numbers is denoted by Q, the set of real numbers by R.
• The complement of A ⊂ B is denoted by Ac if there is no ambiguity.
• Let n > 1. Denote by Dn = {b ∈ Rn, |b| 6 1} the n-dimensional disk, and

by Sn−1 = {b ∈ Rn, |b| = 1} the (n−1)-dimensional sphere. By convention,
let D0 = {0} and S−1 = ∅.

Acknowledgments. I am indebted to Tyrone Cutler for drawing my
attention to the paper [3]. I thank the anonymous referee for reading this very
technical paper.

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. [15, Section 2
and Appendix B]) equipped with its q-model structure (we use the terminology
of [30]). 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.



homotopy theory of moore flows (II) 163

• A closed subset of a ∆-generated space equipped with the relative topol-
ogy is not necessarily ∆-generated (e.g., the Cantor set), but it is always
sequential.

• All locally path-connected first-countable topological spaces are ∆-gener-
ated by [3, Proposition 3.11], in particular all locally path-connected metriz-
able topological spaces are ∆-generated.

• The inclusion functor from the full subcategory of ∆-generated spaces to the
category of general topological spaces together with the continuous maps
has a right adjoint called the ∆-kelleyfication functor. The latter functor
does not change the underlying set.

• Let A ⊂ B be a subset of a space B of Top. Then A equipped with the
∆-kelleyfication of the relative topology belongs to Top.

• The colimit in Top is given by the final topology in the following situations:
– A transfinite compositions of one-to-one maps.

– A pushout along a closed inclusion.

– A quotient by a closed subset or by an equivalence relation having a
closed graph.

In these cases, the underlying set of the colimit is therefore the colimit of
the underlying sets. In particular, the CW-complexes, and more generally
all cellular spaces are equipped with the final topology.

• Cellular spaces are weakly Hausdorff. It implies that the image by any
continuous application of any compact is closed and compact, i.e., closed,
quasicompact and Hausdorff. Cellular spaces are also ∆-Hausdorff and
therefore has unique sequential limits by [15, Proposition B.17].

• Top is cartesian closed. The internal hom TOP(X,Y ) is given by taking
the ∆-kelleyfication of the compact-open topology on the set TOP(X,Y ) of
all continuous maps from X to Y .

Definition 2.1. A one-to-one map of ∆-generated spaces i : A → B is a
∆-inclusion if for all ∆-generated spaces Z, the set map Z → A is continuous
if and only if the composite set map Z → A → B is continuous.

Proposition 2.2. Let i : A → B be a one-to-one continuous map. The
following assertions are equivalent:

(1) i is a ∆-inclusion;

(2) A is homeomorphic to i(A) equipped with the ∆-kelleyfication of the
relative topology;



164 p. gaucher

(3) a set map [0, 1] → A is continuous if and only if the composite set map
[0, 1] → A → B is continuous.

Proof. The proof is similar to the same statement for k-inclusions of
k-spaces.

Corollary 2.3. A continuous bijection f : U → V of Top is a homeo-
morphism if and only if it is a ∆-inclusion.

Notation 2.4. The notation [0,`1] ∼=+ [0,`2] for two real numbers `1,`2 >
0 means a nondecreasing homeomorphism from [0,`1] to [0,`2]. It takes 0 to
0 and `1 to `2.

The enriched small category G is defined as follows:

• The set of objects is the open interval ]0,∞[.
• The space G(`1,`2) is the set {[0,`1] ∼=+ [0,`2]} for all `1,`2 > 0 equipped

with the ∆-kelleyfication of the relative topology induced by the set inclu-
sion G(`1,`2) ⊂ TOP([0,`1], [0,`2]). In other terms, a set map [0, 1] →
G(`1,`2) is continuous if and only if the composite set map [0, 1] →G(`1,`2)
⊂ TOP([0,`1], [0,`2]) is continuous.
• For every `1,`2,`3 > 0, the composition map

G(`1,`2) ×G(`2,`3) −→G(`1,`3)

is induced by the composition of continuous maps. It induces a continuous
map since the composite set map

G(`1,`2) ×G(`2,`3) −→G(`1,`3) ⊂ TOP([0,`1], [0,`3])

corresponds by the adjunction to the continuous map

[0,`1] ×G(`1,`2) ×G(`2,`3) −→ [0,`3]

which takes (t,x,y) to y(x(t)).

The enriched category G is an example of a reparametrization category in
the sense of [14, Definition 4.3] which is different from the terminal category.
It is introduced in [14, Proposition 4.9]. Another example is given in [14,
Proposition 4.11].



homotopy theory of moore flows (II) 165

Proposition 2.5. The topology of G(`1,`2) is the compact-open topol-
ogy. In particular, it is metrizable. A sequence (φn)n>0 of G(`1,`2) converges
to φ ∈G(`1,`2) if and only if it converges pointwise.

It means that the topology of the pointwise convergence of G(`1,`2) is ∆-
generated. Proposition 2.5 has an interesting generalization in Corollary 6.12.

Proof. The compact-open topology on G(`1,`2) is metrizable by [21, Propo-
sition A.13]. The metric is given by the distance of the uniform convergence.
Consider a ball B(φ,�) for this metric. Let ψ ∈ B(φ,�). Then for all h ∈ [0, 1],

|
(
hψ(t) + (1 −h)φ(t)

)
−φ(t)| = |h(ψ(t) −φ(t))| < h� 6 �.

Thus, the compact-open topology is locally path-connected. The compact-
open topology is therefore equal to its ∆-kelleyfication. The last assertion is
then a consequence of the second Dini theorem.

A multipointed space is a pair (|X|,X0) where:

• |X| is a topological space called the underlying space of X;
• X0 is a subset of |X| called the set of states of X.

A morphism of multipointed spaces f : X = (|X|,X0) → Y = (|Y |,Y 0) is a
commutative square

X0
f0

//

��

Y 0

��

|X|
|f|

// |Y |.

The corresponding category is denoted by MTop.

Notation 2.6. The maps f0 and |f| will be often denoted by f if there
is no possible confusion.

We have the well-known proposition:

Proposition 2.7. (The Moore composition) Let U be a topological space.
Let

γi : [0,`i] −→ U

be n continuous maps with 1 6 i 6 n with n > 1. Suppose that γi(`i) =
γi+1(0) for 1 6 i < n. Then there exists a unique continuous map



166 p. gaucher

γ1 ∗ · · · ∗γn :
[
0,
∑

i
`i

]
−→ U

such that

(γ1 ∗ · · · ∗γn)(t) = γi
(
t−
∑

j<i
`i

)
for

∑
j<i

`i 6 t 6
∑

j6i
`i.

In particular, there is the equality (γ1 ∗γ2) ∗γ3 = γ1 ∗ (γ2 ∗γ3).

Notation 2.8. Let ` > 0. Let µ` : [0,`] → [0, 1] be the homeomorphism
defined by µ`(t) = t/`.

Definition 2.9. The map γ1 ∗ γ2 is called the Moore composition of γ1
and γ2. The composite

γ1 ∗N γ2 : [0, 1]
(µ2)

−1
−−−−−→ [0, 2]

γ1∗γ2−−−−→ U

is called the normalized composition. One has

(γ1 ∗N γ2)(t) =

{
γ1(2t) if 0 6 t 6

1
2
,

γ2(2t− 1) if 12 6 t 6 1.

The normalized composition being not associative, a notation like γ1∗N · · ·∗N
γn will mean, by convention, that ∗N is applied from the left to the right.

A multipointed d-space X is a triple (|X|,X0,PGX) where:

• The pair (|X|,X0) is a multipointed space. The space |X| is called the
underlying space of X and the set X0 the set of states of X.

• The set PGX 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) ∈ X0.
– Let γ be an execution path of X. Then any composite γφ with φ ∈
G(1, 1) is an execution path of X.

– Let γ1 and γ2 be two composable execution paths of X; then the
normalized composition γ1 ∗N γ2 is an execution path of X.

A map f : X → Y of multipointed d-spaces is a map of multipointed
spaces from (|X|,X0) to (|Y |,Y 0) such that for any execution path γ of X,



homotopy theory of moore flows (II) 167

the map
PGf : γ 7−→ f.γ

is an execution path of Y .

Notation 2.10. The mapping PGf 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 ` > 0, which is denoted by GlobG` (Z),

is the multipointed d-space defined as follows:

• the underlying topological space is the quotient space 1

{0, 1}t (Z × [0,`])
(z, 0) = (z′, 0) = 0, (z, 1) = (z′, 1) = 1

;

• the set of states is {0, 1};
• the set of execution paths is the set of continuous maps

{δzφ : φ ∈G(1,`),z ∈ Z}

with δz(t) = (z,t). It is equal to the underlying set of G(1,`) ×Z.

In particular, GlobG` (∅) is the multipointed d-space {0, 1} =
(
{0, 1},{0, 1},

∅
)
. For ` = 1, we set

GlobG(Z) = GlobG1 (Z).

(3) The directed segment is the multipointed d-space
−→
I G = GlobG({0}).

The category of multipointed d-spaces is denoted by GdTop. The subset
of execution paths from α to β is the set of γ ∈ PGX such that γ(0) = α
and γ(1) = β; it is denoted by PGα,βX: α is called the initial state and β the
final state of such a γ. An execution path having the same initial and final
state is called a loop. The set PGα,βX is equipped with the ∆-kelleyfication of
the relative topology induced by the inclusion PGα,βX ⊂ TOP([0, 1], |X|). In
other terms, a set map U → PGα,βX is continuous if and only if the composite

1It is the suspension of Z.



168 p. gaucher

set map U → PGα,βX ⊂ TOP([0, 1], |X|) is continuous. The category GdTop
is locally presentable by [11, Theorem 3.5].

Proposition 2.11. ([16, Proposition 6.5]) The mapping Ω : X 7−→(
|X|,X0

)
induces a functor from GdTop to MTop which is topological and

fibre-small.

The Ω-final structure is generated by the finite normalized composition of
execution paths. We will come back on this point in Theorem 3.9. Note that
Proposition 2.11 holds both by working with ∆-generated spaces and with
∆-Hausdorff ∆-generated spaces.

The following proposition is implicitly assumed (for ` = 1) in all the pre-
vious papers about multipointed d-spaces:

Proposition 2.12. Let Z be a topological space. Then there is the home-
omorphism

PG0,1Glob
G
` (Z)

∼= G(1,`) ×Z.

Proof. The set map

Ψ : G(1,`) ×Z −→ PG0,1Glob
G
` (Z)

(φ,z) 7−→ δzφ

is continuous because the mapping (t,φ,z) 7→ (z,φ(t)) from [0, 1]×G(1,`)×Z
to |GlobG` (Z)| is continuous. It is a bijection since, by definition of Glob

G
` (Z),

the underlying set of PG0,1Glob
G
` (Z) is equal to the underlying set of G(1,`)×Z.

The composite set map

PG0,1Glob
G
` (Z) −→ (P

G
0,1Glob

G
` (Z))co −→ Z×]0, 1[

pr1−−−→ Z
γ 7−→ pr1

(
γ
(

1
2

))
where (PG0,1Glob

G
` (Z))co is the set P

G
0,1Glob

G
` (Z) equipped with the compact-

open topology is continuous. The continuous map Z →{0} induces a contin-
uous map

PG0,1Glob
G
` (Z) −→ P

G
0,1Glob

G
` ({0}) ∼= G(1,`)

γ 7−→ p.γ,

where p : |GlobG` (Z)|→ [0, 1] is the projection map. Therefore the set map

Ψ−1 : PG0,1Glob
G
` (Z) −→ G(1,`) ×Z

γ 7−→
(
p.γ, pr1

(
γ
(

1
2

)))



homotopy theory of moore flows (II) 169

is continuous and Ψ is a homeomorphism.

Definition 2.13. Let X be a multipointed d-space X. Denote again by
PGX the topological space

PGX =
⊔

(α,β)∈X0×X0
PGα,βX.

A straightforward consequence of the definition of the topology of PGX is:

Proposition 2.14. Let X be a multipointed d-space. Let f : [0, 1] →
PGX be a continuous map. Then f factors as composite of continuous maps
f : [0, 1] → PGα,βX → P

GX for some (α,β) ∈ X0 ×X0.

Proof. It is due to the fact that [0, 1] is connected.

Proposition 2.15. Let X be a multipointed d-space such that X0

is a totally disconnected subset of |X|. Then the topology of PGX is the
∆-kelleyfication of the relative topology induced by the inclusion
PGX ⊂ TOP([0, 1], |X|).

Proof. Call for this proof (PGX)+ the set PGX equipped with the
∆-kelleyfication of the relative topology induced by the inclusion PGXλ ⊂
TOP([0, 1], |X|). There is a continuous bijection PGX → (PGX)+. Using
Corollary 2.3, the proof is complete since X0 a totally disconnected subset of
|X| and since [0, 1] is connected.

Theorem 2.16. The functor PG : MdTop → Top is a right adjoint. In
particular, it is limit preserving and accessible.

Proof. The left adjoint is constructed in [11, Proposition 4.9] in the case
of ∆-generated spaces. The proof still holds for ∆-Hausdorff ∆-generated
spaces. It relies on the fact that Top is cartesian closed and that every ∆-
generated space is homeomorphic to the disjoint sum of its path-connected
components which are also its connected components. The construction is
similar to the construction of the left adjoint of the path P-space functor for
P-flows [14, Theorem 6.13] and to the construction of the left adjoint of the
path functor for flows [15, Theorem 5.9].



170 p. gaucher

The q-model structure of multipointed d-spaces is the unique combinatorial
model structure such that{

GlobG(Sn−1) ⊂ GlobG(Dn) : n > 0
}
∪
{
C : ∅ →{0},R : {0, 1}→{0}

}
is the set of generating cofibrations, the maps between globes being induced
by the closed inclusion Sn−1 ⊂ Dn, and such that{

GlobG(Dn ×{0}) ⊂ GlobG(Dn+1) : n > 0
}

is the set of generating trivial cofibrations, the maps between globes being
induced by the closed inclusion (x1, . . . ,xn) 7→ (x1, . . . ,xn, 0) (e.g., [16, The-
orem 6.16]). The weak equivalences are the maps of multipointed d-spaces
f : X → Y inducing a bijection f0 : X0 ∼= Y 0 and a weak homotopy equiv-
alence PGf : PGX → PGY and the fibrations are the maps of multipointed
d-spaces f : X → Y inducing a q-fibration PGf : PGX → PGY of topological
spaces.

3. Moore composition and Ω-final structure

Notation 3.1. Let φi : [0,`i] ∼=+ [0,`′i] for n > 1 and 1 6 i 6 n. Then
the map

φ1 ⊗···⊗φn :
[
0,
∑

i
`i

]
∼=+

[
0,
∑

i
`′i

]
denotes the homeomorphism defined by

(φ1⊗. . .⊗φn)(t) =




φ1(t) if 0 6 t 6 `1,

φ2(t− `1) + `′1 if `1 6 t 6 `1 + `2,
...

φi
(
t−
∑

j<i `j
)

+
∑

j<i `
′
j if

∑
j<i `j 6 t 6

∑
j6i `j,

...

φn
(
t−
∑

j<n `j
)

+
∑

j<n `
′
j if

∑
j<n `j 6 t 6

∑
j6n `j.

Proposition 3.2. Let φ : [0,`] ∼=+ [0,`′]. Let n > 1. Consider `1, . . . ,`n
> 0 with n > 1 such that

∑i=n
i=1 `i = `. Then there exists a unique decompo-

sition of φ of the form
φ = φ1 ⊗···⊗φn

such that φi : [0,`i] ∼=+ [0,`′i] for 1 6 i 6 n.



homotopy theory of moore flows (II) 171

Proof. By definition of φ1 ⊗···⊗φn, we necessarily have

φ
(∑

j6i
`j

)
= φi

(∑
j6i
`j −

∑
j<i
`j

)
+
∑

j<i
`′j

= φi(`i) +
∑

j<i
`′j =

∑
j6i
`′j.

The real numbers `′i are therefore defined by induction on i > 1 by the formula

φ
(∑

j6i
`j

)
−
∑

j<i
`′j = `

′
i.

In other terms, we have

φ
(∑

j6i
`j

)
=
∑
j6i

`′j for 1 6 i 6 n.

Let
φi(t) = φ

(
t +

∑
j<i

`j

)
−
∑

j<i
`′j

for all t ∈ [0,`i]. Then, if
∑

j<i `j 6 t 6
∑

j6i `j we obtain

(φ1 ⊗···⊗φn)(t) = φi
(
t−

∑
j<i

`j

)
+
∑

j<i
`′j

= φ
(
t−
∑

j<i
`j +

∑
j<i

`j

)
−
∑

j<i
`′j +

∑
j<i

`′j

= φ(t),

the first equality by definition of φ1⊗···⊗φn, the second equality by definition
of φi and the third equality by algebraic simplification. Consider a second
decomposition

φ = φ′1 ⊗···⊗φ
′
n

such that φ′i : [0,`i]
∼=+ [0,`′i] for 1 6 i 6 n. Then, for

∑
j<i `j 6 t 6

∑
j6i `j

we have

φi

(
t−
∑

j<i
`j

)
+
∑

j<i
`′j = φ(t) = φ

′
i

(
t−
∑

j<i
`j

)
+
∑

j<i
`′j

by definition of φ1 ⊗···⊗φn and of φ′1 ⊗···⊗φ
′
n. We deduce that φi = φ

′
i

for all 1 6 i 6 n.



172 p. gaucher

Corollary 3.3. Let φ ∈G(1, 1). Let n > 1. Assume that∑i=n
i=1

`i =
∑i=n

i=1
`′i = 1

and that
φ
(∑

j6i
`j

)
=
∑

j6i
`′j for 1 6 i 6 n.

Then there exist (unique) φi : [0,`i] ∼=+ [0,`′i] for 1 6 i 6 n such that
φ = φ1 ⊗···⊗φn.

Proposition 3.4. Let U be a topological space. Let γi : [0, 1] → U be
n continuous maps with 1 6 i 6 n and n > 1. Let φi : [0,`i] ∼=+ [0,`′i] for
1 6 i 6 n. Then we have(

(γ1µ`′1 ) ∗ · · · ∗ (γnµ`′n)
)
(φ1 ⊗···⊗φn) = (γ1µ`′1φ1) ∗ · · · ∗ (γnµ`′nφn).

Proof. For
∑

j<i `j 6 t 6
∑

j6i `j, we have(
(γ1µ`′1 ) ∗ · · ·∗(γnµ`′n)

)
(φ1 ⊗···⊗φn)(t)

= (γiµ`′i
)
(

(φ1 ⊗···⊗φn)(t) −
∑

j<i
`′j

)
= (γiµ`′i

)
((
φi

(
t−

∑
j<i

`j

)
+
∑

j<i
`′j

)
−
∑

j<i
`′j

)
= (γiµ`′i

)
(
φi

(
t−
∑

j<i
`j

))
=
(
(γ1µ`′1φ1) ∗ · · · ∗ (γnµ`′nφn)

)
(t),

the first and the fourth equality by definition of the Moore composition, the
second equality by definition of φ1⊗···⊗φn, and the third equality by algebraic
simplification.

Proposition 3.5. Let U be a topological space. Let γi : [0, 1] → U be n
continuous maps with 1 6 i 6 n and n > 1. Let `i > 0 with 1 6 i 6 n be
nonzero real numbers with

∑
i `i = 1. Then for all ` > 0, we have(

(γ1µ`1 ) ∗ · · · ∗ (γnµ`n)
)
µ` = (γ1µ`1`) ∗ · · · ∗ (γnµ`n`).

Proof. For all 1 6 j 6 n, we have by definition of the Moore composition

(
(γ1µ`1 ) ∗ · · · ∗ (γnµ`n)

)
µ`(t) = γj

( 1
`j

(t
`
−
∑

i<j
`i

))



homotopy theory of moore flows (II) 173

if
∑

i<j `i 6 t/` 6
∑

i6j `i, and still by definition of the Moore composition,
we have (

(γ1µ`1`) ∗ · · · ∗ (γnµ`n`)
)
(t) = γj

(
t−
∑

i<j `i`

`j`

)
if
∑

i<j `i` 6 t 6
∑

i6j `i`.

Proposition 3.6. Let U be a topological space. Let γi : [0, 1] → U be n
continuous maps with n > 2 and 1 6 i 6 n such that γ1 ∗N · · · ∗N γn exists.
Then there is the equality

γ1 ∗N · · · ∗N γn =
(
γ1µ 1

2n−1

)
∗
(
γ2µ 1

2n−1

)
∗
(
γ3µ 1

2n−2

)
∗ · · · ∗

(
γnµ1

2

)
.

In particular, for n = 2, we have γ1 ∗N γ2 =
(
γ1µ1

2

)
∗
(
γ2µ1

2

)
.

Proof. The proof is by induction on n > 2. The map µ1
2

: [0, 1
2
] ∼=+ [0, 1]

which takes t to 2t gives rise to a homeomorphism µ1
2
⊗ µ1

2
: [0, 1] ∼=+ [0, 2]

which is equal to µ−12 : [0, 1]
∼=+ [0, 2]. We then write

γ1 ∗N γ2 = (γ1 ∗γ2)µ−12 (by definition of ∗N )

= (γ1 ∗γ2)
(
µ1

2
⊗µ1

2

)
(because µ−12 = µ1

2
⊗µ1

2
)

=
(
γ1µ1

2

)
∗
(
γ2µ1

2

)
(by Proposition 3.4).

The statement is therefore proved for n = 2. Assume that the statement is
proved for some n > 2 and for n = 2. Then we obtain

γ1∗N · · · ∗N γn+1
=
((
γ1µ 1

2n−1

)
∗
(
γ2µ 1

2n−1

)
∗
(
γ3µ 1

2n−2

)
∗ · · · ∗

(
γnµ1

2

))
∗N γn+1

=
(((

γ1µ 1
2n−1

)
∗
(
γ2µ 1

2n−1

)
∗
(
γ3µ 1

2n−2

)
∗ · · · ∗

(
γnµ1

2

))
µ1

2

)
∗
(
γn+1µ1

2

)
=
((
γ1µ 1

2n

)
∗
(
γ2µ 1

2n

)
∗
(
γ3µ 1

2n−1

)
∗ · · · ∗

(
γnµ 1

22

))
∗
(
γn+1µ1

2

)
=
(
γ1µ 1

2n

)
∗
(
γ2µ 1

2n

)
∗
(
γ3µ 1

2n−1

)
∗ · · · ∗

(
γnµ 1

22

)
∗
(
γn+1µ1

2

)
,

the first equality by induction hypothesis, the second equality by the case n =
2, the third equality by Proposition 3.5, and the last equality by associativity
of the Moore composition. We have proved the statement for n + 1.



174 p. gaucher

Proposition 3.7. Let U be a topological space. Let γi : [0, 1] → U be n
continuous maps with n > 1 and 1 6 i 6 n such that γ1∗N · · ·∗N γn exists. Let
φ ∈ G(1, 1). Then there exist φ1 : [0,`1] ∼=+ [0, 12n−1 ], φ2 : [0,`2]

∼=+ [0, 12n−1 ],
φ3 : [0,`3] ∼=+ [0, 12n−2 ], . . . , φn : [0,`n]

∼=+ [0, 12 ] such that φ = φ1 ⊗···⊗φn
(which implies

∑
i `i = 1) and there is the equality(

γ1∗N · · ·∗N γn
)
φ =

(
γ1µ 1

2n−1
φ1
)
∗
(
γ2µ 1

2n−1
φ2
)
∗
(
γ3µ 1

2n−2
φ3
)
∗· · ·∗

(
γnµ1

2
φn
)
.

Proof. Let `1, . . . ,`n > 0 such that
∑

i `i = 1 and such that

φ(`1) =
1

2n−1
,

φ(`1 + `2) =
1

2n−1
+

1

2n−1
,

φ(`1 + `2 + `3) =
1

2n−1
+

1

2n−1
+

1

2n−2
,

...

φ(`1 + `2 + `3 + · · · + `n) =
1

2n−1
+

1

2n−1
+

1

2n−2
+ · · · +

1

2
= 1.

By Proposition 3.2, there exist φ1 : [0,`1] ∼=+ [0, 12n−1 ], φ2 : [0,`2]
∼=+ [0, 12n−1 ],

φ3 : [0,`3] ∼=+ [0, 12n−2 ], . . . , φn : [0,`n]
∼=+ [0, 12 ] such that φ = φ1 ⊗···⊗φn.

We obtain(
γ1∗N · · · ∗N γn+1

)
φ

=
((
γ1µ 1

2n

)
∗
(
γ2µ 1

2n

)
∗
(
γ3µ 1

2n−1

)
∗ · · · ∗

(
γnµ 1

22

)
∗
(
γn+1µ1

2

))
φ

=
(
γ1µ 1

2n−1
φ1
)
∗
(
γ2µ 1

2n−1
φ2
)
∗
(
γ3µ 1

2n−2
φ3
)
∗ · · · ∗

(
γnµ1

2
φn
)
,

the first equality by Proposition 3.6 and the second by Proposition 3.4.

Proposition 3.8. Let U be a topological space. Let γi : [0, 1] → U be
n continuous maps with n > 2 and 1 6 i 6 n such that γ1 ∗N · · · ∗N γn
exists. Let `1, . . . ,`n > 0 be nonzero real numbers such that

∑
i `i = 1. Let

φ1 : [0,
1

2n−1
] ∼=+ [0,`1], φ2 : [0, 12n−1 ]

∼=+ [0,`2], φ3 : [0, 12n−2 ]
∼=+ [0,`3], . . . ,

φn : [0,
1
2
] ∼=+ [0,`n] and let φ = φ1 ⊗···⊗φn. Then φ ∈G(1, 1), and there is

the equality((
γ1µ`1

)
∗
(
γ2µ`2

)
∗
(
γ3µ`3

)
∗ · · · ∗

(
γnµ`n

))
φ =(

γ1µ`1φ1µ
−1

1
2n−1

)
∗N
(
γ2µ`2φ2µ

−1
1

2n−1

)
∗N
(
γ3µ`3φ3µ

−1
1

2n−2

)
∗N · · · ∗N

(
γnµ`nφnµ

−1
1
2

)
.



homotopy theory of moore flows (II) 175

Proof. We have((
γ1µ`1

)
∗
(
γ2µ`2

)
∗
(
γ3µ`3

)
∗ · · · ∗

(
γnµ`n

))
φ

=
(
γ1µ`1φ1

)
∗
(
γ2µ`2φ2

)
∗
(
γ3µ`3φ3

)
∗ · · · ∗

(
γnµ`nφn

)
=
(
γ1µ`1φ1µ

−1
1

2n−1
µ 1

2n−1

)
∗
(
γ2µ`2φ2µ

−1
1

2n−1
µ 1

2n−1

)
∗
(
γ3µ`3φ3µ

−1
1

2n−2
µ 1

2n−2

)
∗ · · · ∗

(
γnµ`nφnµ

−1
1
2

µ1
2

)
=
(
γ1µ`1φ1µ

−1
1

2n−1

)
∗N
(
γ2µ`2φ2µ

−1
1

2n−1

)
∗N
(
γ3µ`3φ3µ

−1
1

2n−2

)
∗N · · · ∗N

(
γnµ`nφnµ

−1
1
2

)
,

where the first equality is due to Proposition 3.4, the second equality is due to
the fact that µ−1` µ` is the identity of [0,`] for all nonzero real numbers ` > 0,
and the last equality is a consequence of Proposition 3.6.

Theorem 3.9. Consider a cocone (Ω(Xi))
•−−→ (|X|,X0) of MTop. Let

X be the Ω-final lift. Let fi : Xi → X be the canonical maps. Then the
set of execution paths of X is the set of finite Moore compositions of the
form (f1γ1µ`1 )∗·· ·∗(fnγnµ`n) such that γi is an execution path of Xi for all
1 6 i 6 n with

∑
i `i = 1.

Proof. Let P(X) be the set of execution paths of X of the form (f1γ1µ`1 )∗
· · · ∗ (fnγnµ`n) such that γi is an execution path of Xi for all 1 6 i 6 n
with

∑
i `i = 1. The final structure is generated by the finite normalized

composition of execution paths (f1γ1) ∗N · · · ∗N (fnγn) (with the convention
that the ∗N are calculated from the left to the right) and all reparametrizations
by φ running over G(1, 1). By Proposition 3.7, there exist φ1 : [0,`1] ∼=+
[0, 1

2n−1
], φ2 : [0,`2] ∼=+ [0, 12n−1 ], φ3 : [0,`3]

∼=+ [0, 12n−2 ], . . . , φn : [0,`n]
∼=+

[0, 1
2
] such that φ = φ1 ⊗···⊗φn and we have(

(f1γ1) ∗N · · · ∗N (fnγn)
)
φ

=
(
f1γ1µ 1

2n−1
φ1
)
∗
(
f2γ2µ 1

2n−1
φ2
)
∗
(
f3γ3µ 1

2n−2
φ3
)
∗ · · · ∗

(
fnγnµ1

2
φn
)

=
(
f1γ1µ 1

2n−1
φ1µ

−1
`1
µ`1
)
∗
(
f2γ2µ 1

2n−1
φ2µ

−1
`2
µ`2
)

∗
(
f3γ3µ 1

2n−2
φ3µ

−1
`3
µ`3
)
∗ · · · ∗

(
fnγnµ1

2
φnµ

−1
`n
µ`n
)

=
(
f1γ
′
1µ`1

)
∗
(
f2γ
′
2µ`2

)
∗
(
f3γ
′
3µ`3

)
∗ · · · ∗

(
fnγ

′
nµ`n

)
,



176 p. gaucher

the first equality by Proposition 3.7, the second equality because µ−1` µ` is the
identity of [0,`] for all ` > 0 and the third equality because of the following
notations: 



γ′1 = γ1

(
µ 1

2n−1
φ1µ

−1
`1

)
,

γ′2 = γ2

(
µ 1

2n−1
φ1µ

−1
`2

)
,

γ′3 = γ3

(
µ 1

2n−2
φ3µ

−1
`3

)
,

...

γ′n = γn

(
µ1

2
φnµ

−1
`n

)
.

It implies that the set P(X) contains the final structure. Conversely, let
(f1γ1µ`1 ) ∗ ·· · ∗ (fnγnµ`n) be an element of P(X). Choose φ1 : [0,

1
2n−1

] ∼=+
[0,`1], φ2 : [0,

1
2n−1

] ∼=+ [0,`2], φ3 : [0, 12n−2 ]
∼=+ [0,`3], . . . , φn : [0, 12 ]

∼=+
[0,`n] and let φ = φ1 ⊗···⊗φn. Using Proposition 3.8, we obtain((
f1γ1µ`1

)
∗
(
f2γ2µ`2

)
∗
(
f3γ3µ`3

)
∗ · · · ∗

(
fnγnµ`n

))
φ

=
(
f1γ1µ`1φ1µ

−1
1

2n−1

)
∗N
(
f2γ2µ`2φ2µ

−1
1

2n−1

)
∗N
(
f3γ3µ`3φ3µ

−1
1

2n−2

)
∗N . . .∗N

(
fnγnµ`nφnµ

−1
1
2

)
.

The continuous maps µ`1φ1µ
−1

1
2n−1

,µ`2φ2µ
−1

1
2n−1

,µ`3φ3µ
−1

1
2n−2

, . . . ,µ`nφnµ
−1
1
2

from

[0, 1] to itself belong to G(1, 1). Thus γ′1, . . . ,γ
′
n defined by the equalities



γ′1 = γ1

(
µ`1φ1µ

−1
1

2n−1

)
,

γ′2 = γ2

(
µ`2φ1µ

−1
1

2n−1

)
,

γ′3 = γ3

(
µ`3φ3µ

−1
1

2n−2

)
,

...

γ′n = γn

(
µ`nφnµ

−1
1
2

)
,

are execution paths of X1, . . . ,Xn respectively. We obtain((
f1γ1µ`1

)
∗
(
f2γ2µ`2

)
∗
(
f3γ3µ`3

)
∗ · · · ∗

(
fnγnµ`n

))
φ

=
(
f1γ
′
1

)
∗N
(
f2γ
′
2

)
∗N
(
f3γ
′
3

)
∗N · · · ∗N

(
fnγ

′
n

)
.

We deduce that the set of paths P(X) is included in the Ω-final structure.



homotopy theory of moore flows (II) 177

4. From multipointed d-spaces to Moore flows

Definition and notation 4.1. The enriched category of enriched pre-
sheaves from G to Top is denoted by [Gop, Top]. The underlying set-enriched
category of enriched maps of enriched presheaves is denoted by [Gop, Top]0.
The objects of [Gop, Top]0 are called the G-spaces. Let

FG
op

` U = G(−,`) ×U ∈ [G
op, Top]0

where U is a topological space and where ` > 0.

Proposition 4.2. ([12, Proposition 5.3 and Proposition 5.5]) The cat-
egory [Gop, Top]0 is a full reflective and coreflective subcategory of TopG

op
0 .

For every G-space F : Gop → Top, every ` > 0 and every topological space X,
we have the natural bijection of sets

[Gop, Top]0(FG
op

` X,F)
∼= Top(X,F(`)).

Theorem 4.3. ([14, Theorem 5.14]) Let D and E be two G-spaces. Let

D ⊗E :=
∫ (`1,`2)

G(−,`1 + `2) ×D(`1) ×E(`2).

The pair ([Gop, Top]0,⊗) has the structure of a closed symmetric semi-
monoidal category, i.e., a closed symmetric nonunital monoidal category.

Notation 4.4. Let D be a G-space. Let φ : ` → `′ be a map of G. Let
x ∈ D(`′). We will use the notation

x.φ := D(φ)(x).

Intuitively, x is a path of length `′ and x.φ is a path of length ` which is the
reparametrization by φ of x.

Proposition 4.5 sheds light on the meaning of the tensor product of G-
spaces. It is used in the proof of Theorem 7.2. It is not in [14]. The proof
is given in this section and not in Section 7 to recall [14, Corollary 5.13]
which also helps to understand the geometric contents of the tensor product
of G-spaces.



178 p. gaucher

Proposition 4.5. Let D1, . . . ,Dn be n G-spaces with n > 1. Then the
mapping

(x1, . . . ,xn) 7→ (Id,x1, . . . ,xn)

yields a surjective continuous map

ΦD1,...,Dn :
⊔

(`1,...,`n)
`1+···+`n=L

D1(`1) ×···×Dn(`n) −→ (D1 ⊗···⊗Dn)(L).

Proof. By [14, Corollary 5.13], the space (D1⊗···⊗Dn)(L) is the quotient
of the space ⊔

(`1,...,`n)

G(L,`1 + · · · + `n) ×D1(`1) ×···×Dn(`n).

by the identifications

(ψ,x1φ1, . . . ,xnφn) = ((φ1 ⊗···⊗φn)ψ,x1, . . . ,xn)

for all `1,`
′
1, . . . ,`n,`

′
n > 0, all ψ ∈G(L,`1 + · · · + `n), all xi ∈ Di(`′i) and all

φi ∈G(`i,`′i). Let `
′′
1, . . . ,`

′′
n > 0 defined by induction on i by the equation

`′′i = ψ
−1
(∑

16j6i
`j

)
−
∑

16j<i
`′′j for 1 6 i 6 n.

Note that L = `′′1 + · · · + `
′′
n. We obtain

ψ
(∑

16j6i
`′′j

)
=
∑

16j6i
`j for 1 6 i 6 n.

By Proposition 3.2, there is a (unique) decomposition ψ = ψ1 ⊗···⊗ψn with
ψi ∈G(`′′i ,`i) for 1 6 i 6 n. Then

(ψ,x1.φ1, . . . ,xn.φn) = (Id,x1.φ1.ψ1, . . . ,xn.φn.ψn)

in (D1 ⊗···⊗Dn)(L). Therefore the continuous map⊔
(`1,...,`n)

`1+···+`n=L

D1(`1) ×···×Dn(`n) −→ (D1 ⊗···⊗Dn)(L).

induced by the mapping (x1, . . . ,xn) 7→ (Id,x1, . . . ,xn) is surjective.



homotopy theory of moore flows (II) 179

A semicategory, also called nonunital category in the literature, is a cat-
egory without identity maps in the structure. It is enriched over a closed
symmetric semimonoidal category (V,⊗) if it satisfied all axioms of enriched
category except the one involving the identity maps, i.e., the enriched com-
position is associative and not necessarily unital.

Definition 4.6. ([14, Definition 6.2]) A Moore flow is a small semicate-
gory enriched over the closed semimonoidal category ([Gop, Top]0,⊗) of
Theorem 4.3.

A Moore flow X has therefore a set of objects denoted by X0, and called
states in this context, and for each (α,β) ∈ X0 × X0 a G-space Pα,βX: the
elements of

P`α,βX = Pα,βX(`)

for ` > 0 are called the execution paths of length `.
The category of Moore flows, denoted by GFlow, is locally presentable

by [14, Theorem 6.11]. A map of Moore flows f : X → Y induces a set map
f0 : X0 → Y 0 and a map of G-spaces Pα,βf : Pα,βX → Pf(α),f(β)Y for each
(α,β) ∈ X0 ×X0. Let

PX =
⊔

(α,β)∈X0×X0
Pα,βX, PY =

⊔
(α,β)∈Y 0×Y 0

Pα,βY,

Pf =
⊔

(α,β)∈X0×X0
Pα,βf.

Notation 4.7. The map Pf : PX → PY can be denoted by f : PX → PY
is there is no ambiguity. The set map f0 : X0 → Y 0 can be denoted by
f : X0 → Y 0 is there is no ambiguity.

Every set S can be viewed as a Moore flow with an empty G-space of
execution paths denoted in the same way. Let D : Gop → Top be a G-space.
We denote by Glob(D) the Moore flow defined as follows:

Glob(D)0 = {0, 1},
P0,0Glob(D) = P1,1Glob(D) = P1,0Glob(D) = ∆G0∅,
P0,1Glob(D) = D.

There is no composition law. This construction yields a functor

Glob : [Gop, Top]0 →GFlow.



180 p. gaucher

There exists a unique model structure on GFlow such that{
Glob(FG

op

` S
n−1) ⊂ Glob(FG

op

` D
n) : n > 0, ` > 0

}
∪
{
C,R

}
is the set of generating cofibrations and such that all objects are fibrant. The
set of generating trivial cofibrations is{

Glob(FG
op

` D
n) ⊂ Glob(FG

op

` D
n+1) : n > 0, ` > 0

}
where the maps Dn ⊂ Dn+1 are induced by the mappings (x1, . . . ,xn) 7→
(x1, . . . ,xn, 0). The weak equivalences are the map of Moore flows f : X →
Y inducing a bijection X0 ∼= Y 0 and such that for all (α,β) ∈ X0 × X0,
the map of G-spaces Pα,βX → Pf(α),f(β)Y is an objectwise weak homotopy
equivalence. The fibrations are the map of Moore flows f : X → Y such that
for all (α,β) ∈ X0 × X0, the map of G-spaces Pα,βX → Pf(α),f(β)Y is an
objectwise q-fibration of spaces. It is called the q-model structure and we use
the terminology of q-cofibration and q-fibration for naming the cofibrations
and the fibrations respectively.

Definition 4.8. Let X be a multipointed d-space. Let P`α,βX be the
subspace of continuous maps from [0,`] to |X| defined by

P`α,βX =
{
t 7→ γµ` : γ ∈ PGα,βX

}
.

Its elements are called the execution paths of length ` from α to β. Let

P`X =
⊔

(α,β)∈X0×X0
P`α,βX.

A map of multipointed d-spaces f : X → Y induces for each ` > 0 a continuous
map P`f : P`X → P`Y by composition by f (in fact by |f|).

Note that P1α,βX = P
G
α,βX, that there is a homeomorphism P

`
α,βX

∼= PGα,βX
for all ` > 0, and that for any topological space Z, we have the homeomor-
phism

P`0,1(Glob
G(Z)) ∼= G(`, 1) ×Z

for any ` > 0 by Proposition 2.12.
The definition above of an execution path of length ` > 0 is not restrictive.

Indeed, we have:



homotopy theory of moore flows (II) 181

Proposition 4.9. Let X be a multipointed d-space. Let φ : [0,`] ∼=+
[0,`]. Let γ ∈ P`X. Then γφ ∈ P`X.

Proof. By definition of P`X, there exists γ ∈ PGX such that γ = γµ`.
We obtain γφ = γµ`φµ

−1
` µ`. Since µ`φµ

−1
` ∈ G(1, 1), we deduce that

γµ`φµ
−1
` ∈ P

GX and that γφ ∈ P`X.

Proposition 4.10. Let X be a multipointed d-space. Let γ1 and γ2 be
two execution paths of X with γ1(1) = γ2(0). Let `1,`2 > 0. Then(

γ1µ`1 ∗γ2µ`1
)
µ−1`1+`2

is an execution path of X.

Proof. Let φ1 : [0,
1
2
] ∼=+ [0,`1] and φ2 : [0, 12 ]

∼=+ [0,`2]. Then we have

φ1 ⊗φ2 : [0, 1] ∼=+ [0,`1 + `2].

We obtain the sequence of equalities(
(γ1µ`1 )∗(γ2µ`2 )

)
µ−1`1+`2 =

(
(γ1µ`1 ) ∗ (γ2µ`2 )

)(
φ1 ⊗φ2

)(
φ1 ⊗φ2

)−1
µ−1`1+`2

=
(
(γ1µ`1φ1) ∗ (γ2µ`1φ2)

)(
φ1 ⊗φ2

)−1
µ−1`1+`2

=
(
(γ1µ`1φ1µ

−1
1
2

µ1
2
) ∗ (γ2µ`2φ2µ

−1
1
2

µ1
2
)
)(
φ1 ⊗φ2

)−1
µ−1`1+`2

=
(
(γ1 µ`1φ1µ

−1
1
2︸ ︷︷ ︸

∈G(1,1)

) ∗N (γ2 µ`2φ2µ
−1
1
2︸ ︷︷ ︸

∈G(1,1)

)
)(
φ1 ⊗φ2

)−1
µ−1`1+`2︸ ︷︷ ︸

∈G(1,1)

,

the first equality because φ1⊗φ2 is invertible, the second equality by Proposi-
tion 3.4, the third equality because µ1

2
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 action of G(1, 1).

Proposition 4.11. Let X be a multipointed d-space. Let `1,`2 > 0.
The Moore composition of continuous maps yields a continuous maps
P`1X ×P`2X → P`1+`2X.

Proof. It is a consequence of Definition 4.8 and Proposition 4.10.

Theorem 4.12. Let X be a multipointed d-space. Then the following
data



182 p. gaucher

• the set of states X0 of X;
• for all α,β ∈ X0 and all real numbers ` > 0, let

P`α,βM
G(X) := P`α,βX;

• for all maps [0,`] ∼=+ [0,`′], a map f : [0,`′] → |X| of P`
′
α,βM

G(X) is

mapped to the map [0,`] ∼=+ [0,`′]
f
→|X| of P`α,βM

G(X);

• for all α,β,γ ∈ X0 and all real numbers `,`′ > 0, the composition maps

∗ : P`α,βM
G(X) ×P`

′
β,γM

G(X) −→ P`+`
′

α,γ M
G(X)

of Proposition 4.11;

assemble to a Moore flow MG(X). This mapping induces a functor

MG : GdTop −→GFlow

which is a right adjoint.

Note that the left adjoint MG! : GFlow → GdTop preserves the set of
states as well as the functor MG : GdTop →GFlow.

Proof. These data give rise to a G-space Pα,βMG(X) for each pair (α,β) of
states of X0 and, thanks to Proposition 4.11, to an associative composition law
∗ : P`1α,βM

G(X)×P`2β,γM
G(X) → P`1+`2α,γ MG(X) which is natural with respect to

(`1,`2). By [14, Section 6], these data assemble to a Moore flow. Since limits
and colimits of G-spaces are calculated objectwise, the functor MG : GdTop →
GFlow is limit-preserving and accessible by Theorem 2.16. Therefore it is a
right adjoint by [1, Theorem 1.66].

Proposition 4.13. Let X be a multipointed d-space. Let ` > 0 be a real
number. Let Z be a topological space. Then there is a bijection of sets

GdTop(GlobG` (Z),X) ∼=
⊔

(α,β)∈X0×X0
Top(Z,P`α,βX)

which is natural with respect to Z and X.

Proof. A map f of multipointed d-spaces from GlobG` (Z) to X is deter-
mined by:



homotopy theory of moore flows (II) 183

• 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 |GlobG` (Z)| to |X| such that for

all x ∈ Z and all φ : [0, 1] ∼=+ [0,`], the map t 7→ f(x,φ(t)) from [0, 1] to
|X| belongs to PGα,βX.

By definition of P`α,βX, for every x ∈ Z, the continuous map f(x,−) from [0,`]
to |X| belongs to P`α,βX since f(x,−) = f(x,φ(−)).φ

−1 for any φ : [0, 1] ∼=+
[0,`]. Since f is continuous and since Top is cartesian closed, the mapping
x 7→ f(x,−) actually yields a continuous map from Z to P`α,βX. Conversely,
starting from a continuous map g : Z → P`α,βX, one can define a map of
multipointed d-spaces from GlobG` (Z) to X by taking 0 and 1 to α and β
respectively and by taking (x,t) ∈ |GlobG` (Z)| to g(x)(t).

We want to recall for the convenience of the reader:

Proposition 4.14. ([14, Proposition 6.10]) Let D : Gop → Top be a
G-space. Let X be a Moore flow. Then there is the natural bijection

GFlow(Glob(D),X) ∼=
⊔

(α,β)∈X0×X0
[Gop, Top]0(D,Pα,βX).

Proposition 4.15. For all topological spaces Z and all ` > 0, there are
the natural isomorphisms

MG(GlobG` (Z)) ∼= Glob(F
Gop
` (Z)) and M

G
! (Glob(F

Gop
` (Z)))

∼= GlobG` (Z).

Proof. By definition of MG and by Proposition 2.12, the only nonempty
path G-space of MG(GlobG` (Z)) is

P0,1MG(GlobG` (Z)) = G(−,`) ×Z

and we obtain the first isomorphism. There is the sequence of natural bijec-
tions, for any multipointed d-space X,

GdTop
(
MG! (Glob(F

Gop
` (Z))),X

) ∼= GFlow(Glob(FGop` (Z)),MGX)
∼=

⊔
(α,β)∈X0×X0

[Gop, Top]0
(
FG

op

` (Z),Pα,βX
)

∼=
⊔

(α,β)∈X0×X0
Top(Z,P`α,βX)

∼= GdTop(GlobG` (Z),X),



184 p. gaucher

the first bijection by adjunction, the second bijection by Proposition 4.14, the
third bijection by Proposition 4.2 and the last bijection by Proposition 4.13.
The proof of the second isomorphism is then complete thanks to the Yoneda
lemma.

5. Cellular multipointed d-spaces

Let λ be an ordinal. In this section, we work with a colimit-preserving
functor

X : λ −→GdTop

such that:

• the multipointed d-space X0 is a set, in other terms X0 = (X0,X0,∅)
for some set X0;

• for all ν < λ, there is a pushout diagram of multipointed d-spaces

GlobG(Snν−1)

��

gν
// Xν

��

GlobG(Dnν )
ĝν

// Xν+1

with nν > 0.

Let Xλ = lim−→ν<λ Xν. Note that for all ν 6 λ, there is the equality X
0
ν =

X0. Denote by
cν =

∣∣GlobG(Dnν )∣∣\∣∣GlobG(Snν−1)∣∣
the ν-th cell of Xλ. It is called a globular cell. Like in the usual setting of
CW-complexes, ĝν induces a homeomorphism from cν to ĝν(cν) equipped with
the relative topology which will be therefore denoted in the same way. It also
means that ĝν(cν) equipped with the relative topology is ∆-generated. The
closure of cν in |Xλ| is denoted by

ĉν = ĝν
(∣∣GlobG(Dnν )∣∣).

The boundary of cν in |Xλ| is denoted by

∂cν = ĝν
(∣∣GlobG(Snν−1)∣∣).

The state ĝν(0) ∈ X0 (ĝν(1) ∈ X0 resp.) is called the initial (final resp.) state
of cν. The integer nν + 1 is called the dimension of the globular cell cν. It



homotopy theory of moore flows (II) 185

is denoted by dim cν. The states of X
0 are also called the globular cells of

dimension 0.

Definition 5.1. The cellular multipointed d-space Xλ is finite if λ is a
finite ordinal and X0 is finite.

Proposition 5.2. The space |Xλ| is a cellular space. It contains X0 as
a discrete closed subspace. The space |Xλ| is weakly Hausdorff. For every
0 6 ν1 6 ν2 6 λ, the continuous map |Xν1| → |Xν2| is a q-cofibration of
spaces, and in particular a closed T1-inclusion.

Proof. By [9, Theorem 8.2], the continuous map∣∣GlobG(Snν−1)∣∣ −→ ∣∣GlobG(Dnν )∣∣
is a q-cofibration of spaces for all ν > 0 between cellular spaces. Since the
functor X 7→ |X| is colimit-preserving, the space |Xλ| is a cellular space. It is
therefore weakly Hausdorff. For every 0 6 ν1 6 ν2 6 λ, the continuous map
|Xν1| → |Xν2| is a transfinite composition of q-cofibrations, and hence a q-
cofibration. The map X0 → Xλ is a transfinite composition of q-cofibrations,
and therefore a q-cofibration, and in particular a closed T1-inclusion. Ev-
ery subset of X0 is closed since X0 is equipped with the discrete topology.
Consequently, X0 is a discrete closed subspace of |Xλ|.

Proposition 5.3. For all 0 6 ν1 6 ν2 6 λ, there is the equality

PGXν1 = P
GXν2 ∩ TOP([0, 1], |Xν1|).

Proof. It is trivial for ν1 = ν2. For ν2 = ν1 + 1, there is a pushout diagram
of multipointed d-spaces

GlobG(Snν1−1)

��

gν1 // Xν1

��

GlobG(Dnν1 )
ĝν1 // Xν2.

The equality holds because the set of execution paths of Xν2 is obtained as a
Ω-final structure. We conclude by a transfinite induction on ν2.

Proposition 5.4. For all 0 6 ν1 6 ν2 6 λ, the continuous map
PGXν1 → PGXν2 is a ∆-inclusion.



186 p. gaucher

Proof. Consider a set map [0, 1] → PGXν1 such that the composite set map

[0, 1] −→ PGXν1 −→ P
GXν2

is continuous. Then by adjunction, we obtain a continuous map

[0, 1] × [0, 1] −→|Xν2|.

By hypothesis, it factors as a composite of set maps

[0, 1] × [0, 1] −→|Xν1| −→ |Xν2|.

By Proposition 5.2, the left-hand map is continuous since [0, 1] × [0, 1] is
compact. The proof is complete by Proposition 5.3 and Proposition 2.2.

Proposition 5.5. Let K be a compact subspace of |Xλ|. Then K inter-
sects finitely many cν.

Proof. We mimick the proof of [21, Proposition A.1] for the transfinite case.
Assume that there exists an infinite set S = {mj : j > 0} with mj ∈ K ∩
cνj , where (νj)j>0 is a sequence of mutually distinct ordinals. By transfinite
induction on ν > 0, let us prove that S ∩|Xν| is a closed subset of |Xν|. The
assertion is trivial for ν = 0. There is the pushout diagram of spaces for all
ν < λ

|GlobG(Snν−1)|

��

gν
// |Xν|

��

|GlobG(Dnν )|
ĝν

// |Xν+1|.

By induction hypothesis, g−1ν (S ∩ |Xν|) is a closed subset of |Glob
G(Snν−1)|

and ĝν
−1(S ∩ |Xν+1|) is equal to g−1ν (S ∩ |Xν|) union at most one point.

Therefore, S ∩|Xν+1| is a closed subset of |Xν+1| because the latter space is
equipped with the final topology by Proposition 5.2. Suppose that we have
proved that for all ν < ν′, S∩|Xν| is a closed subset of |Xν| where ν′ is a limit
ordinal. Then, since the topology of |Xν′| is the final topology (it is a tower
of one-to-one maps), S∩|Xν′| is a closed subset of |Xν′|. Thus, by transfinite
induction on ν > 0, we prove that S is closed in |Xν| for all 0 6 ν 6 λ. The
same argument proves that every subset of S is closed in |Xλ|. Thus S has
the discrete topology. But it is compact, being a closed subset of the compact
space K, and therefore finite. Contradiction.



homotopy theory of moore flows (II) 187

Colimits of multipointed d-spaces are calculated by taking the colimit of
the underlying spaces and of the sets of states and by taking the Ω-final
structure which is generated by the free finite compositions of execution paths.
Consequently, the composite functor

GdTop P
G

−−−→ Top ⊂−−−→ Set

is finitely accessible. It is unlikely that the functor PG : GdTop → Top, which
is a right adjoint by Theorem 2.16, is finitely accessible. However, we have:

Theorem 5.6. The composite functor

λ
X−−−→GdTop P

G
−−−→ Top

is colimit-preserving. In particular the continuous bijection

lim−→(P
G.X) −→ PG lim−→X

is a homeomorphism. Moreover the topology of PG lim−→X is the final topology.

Note that Theorem 5.6 holds both for ∆-generated spaces and ∆-Hausdorff
∆-generated spaces.

Proof. Consider the set of ordinals{
ν 6 λ : ν limit ordinal and lim−→

ν′<ν

(PGXν′) −→ PGXν not isomorphism
}

Assume this set nonempty. Let ν be its smallest element. The topology of
lim−→ν′<ν P

GXν′ is the final topology because the continuous maps PGXν′ →
PGXν′+1 are one-to-one. Let f : [0, 1] → PGXν be a continuous map. There-
fore the composite map

[0, 1]
f
−−→ PGXν ⊂ TOP([0, 1], |Xν|)

is continuous. It gives rise by adjunction to a continuous map [0, 1]× [0, 1] →
|Xν|. Since the functor X : λ → GdTop is colimit-preserving, there is the
homeomorphism |Xν| ∼= lim−→ν′<ν |Xν′|. Since [0, 1] × [0, 1] is compact, the
latter continuous map then factors as a composite [0, 1] × [0, 1] → |Xν′| →
|Xν| for some ordinal ν′ < ν by Proposition 5.2. Since PGXν′ = PGXν ∩



188 p. gaucher

TOP([0, 1], |Xν′|) by Proposition 5.3, f factors as a composite [0, 1] → PGXν′
→ PGXν. Using Corollary 2.3. we obtain the homeomorphism lim−→ν′<ν P

GXν′

→ PGXν: contradiction.

Theorem 5.7. The composite functor

λ
X−−→GdTop M

G
−−−−→GFlow

is colimit-preserving. In particular the natural map

lim−→
ν<λ

MG(Xν) −→ MGXλ

is an isomorphism.

Proof. Theorem 5.6 states that there is the homeomorphism

lim−→
ν<λ

PGXν −→ PGXλ.

We have, by definition of the functor MG, the equality of functors PG = P1.MG.
It means that there is the homeomorphism

lim−→
ν<λ

P1MG(Xν) −→ P1MG(Xλ).

Since all maps the reparametrization category G are isomorphisms, we obtain
for all ` > 0 the homeomorphism

lim−→
ν<λ

P`MG(Xν) −→ P`MG(Xλ).

Since colimits of G-spaces are calculated objectwise, we obtain the isomor-
phism of G-spaces

lim−→
ν<λ

PMGXν −→ PMGXλ.

The proof is complete thanks to the universal property of the colimits.

Definition 5.8. An execution path γ of a multipointed d-space X is min-
imal if

γ(]0, 1[) ∩X0 = ∅.



homotopy theory of moore flows (II) 189

For any (q-cofibrant or not) topological space Z, every execution path of
the multipointed d-space GlobG(Z) is minimal. The following theorem proves
that execution paths of cellular multipointed d-spaces have a normal form.

Theorem 5.9. Let γ be an execution path of Xλ. Then there exist min-
imal execution paths γ1, . . . ,γn and `1, . . . ,`n > 0 with

∑
i `i = 1 such that

γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n).

Moreover, if there is the equality

γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) = (γ
′
1µ`′1 ) ∗ · · · ∗ (γn′µ`′n′)

such that all γ′j are also minimal and with `
′
1, . . . ,`

′
n′ > 0, then n = n

′ and
γi = γ

′
i and `i = `

′
i for all 1 6 i 6 n.

Proof. The set of execution paths of Xλ is obtained as a Ω-final structure.
Using Theorem 3.9, we obtain

γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n)

for some n > 1 with `1 + · · · + `n = 1 such that for all 1 6 i 6 n, there exists
a globular cell cνi such that

γi(]0, 1[) ⊂ cνi, γi(0) = ĝνi(0), γi(0) = ĝνi(1).

Therefore there exists a finite set {t0, . . . , tn} with t0 = 0 < t1 < · · · < tn = 1
and n > 1 such that γ([0, 1])∩X0 = {γ(ti) : 0 6 i 6 n}. We necessarily have
`i = ti − ti−1 for 1 6 i 6 n. Let `0 = 0. Then we deduce that

∑
j<i `j = ti−1

and
∑

j6i `j = ti. The equality γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) therefore implies
that γ(t) = (γiµ`i)(t − ti−1) for ti−1 6 t 6 ti for all 1 6 i 6 n by definition
of the Moore composition of paths. We deduce that we necessarily have the
equalities γi(t) = γ(`it + ti−1) for t ∈ [0, 1].

Let γ be an execution path of Xλ. Consider the normal form

γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n).

of Theorem 5.9. There exists a unique sequence [cν1, . . . ,cνn] of globular cells
such that for all 1 6 i 6 n, γi(]0, 1[) ⊂ cνi, γi(0) = ĝνi(0) and γi(1) = ĝνi(1).
This leads to the following notion:



190 p. gaucher

Definition 5.10. With the notations above. The sequence of globular
cells

Carrier(γ) = [cν1, . . . ,cνn]

is called the carrier of γ. The integer n is called the length of the carrier.

Proposition 5.11. An execution path of Xλ is minimal if and only if
the length of its carrier is 1.

Proof. It is a consequence of Theorem 5.9.

Proposition 5.12. An execution path γ of Xλ is non-minimal if and only
if there exist two execution paths γ1 and γ2 such that γ = γ1 ∗N γ2.

Proposition 5.12 does not hold for non q-cofibrant multipointed d-spaces.
Consider e.g., the multipointed d-space X obtained by starting from the di-

rected segment
−→
I G and by adding to the set of states {0, 1} the point 1

2
. Then

all execution paths of X are non-minimal and P0,1
2
X = P1

2
,1X = ∅. Note that

the q-cofibrant replacement of X consists of the disjoint sum
−→
I G t{1

2
}.

Proof. It is a consequence of Theorem 5.9 and Proposition 5.11.

Proposition 5.13. All execution path of Xλ are locally injective.

In [5], the terminology of regular paths is used.

Proof. All execution paths of globes GlobG(Z) are one-to-one for all topo-
logical spaces Z. Therefore all minimal execution paths are locally injective
(it can be a loop). The proof is complete thanks to Theorem 5.9.

Proposition 5.14. Consider a minimal execution path γ of Xλ with
Carrier(γ) = [cν0 ]. Let cν be a globular cell of Xλ with ν 6= ν0. Then the
following two assertions are equivalent:

(1) γ(]0, 1[) ∩ ĉν 6= ∅;

(2) γ([0, 1]) ⊂ ∂cν.

Moreover, when the previous assertions are satisfied, there exists an execution
path γ′ from the initial state of cν to its final state such that γ

′ = (γ1µ`1 ) ∗
· · · ∗ (γnµ`n) with γ = γi for at least one i ∈ {1, . . . ,n}, γ1, . . . ,γn minimal
and

∑
i `i = 1.



homotopy theory of moore flows (II) 191

Proof. Since ∂cν ⊂ ĉν, we deduce (2) ⇒ (1). Assume (1). Since γ(]0, 1[) ⊂
cν0 and ν 6= ν0, one has ν > ν0. It means that there exists a point ĝν(z,t)
of ∂cν which belongs to cν0 with z ∈ Snν−1 and, since cν0 ∩ X0 = ∅, with
t ∈]0, 1[. Therefore the carrier of the execution path ĝνδz contains the globular
cell cν0 . We deduce that there exists φ ∈G(1, 1) such that

ĝνδzφ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n)

with γ = γi for at least one i ∈{1, . . . ,n}, γ1, . . . ,γn minimal and
∑

i `i = 1.
In particular, we deduce that γ([0, 1]) ⊂ ∂cν: we have proved (1) ⇒ (2).

Definition 5.15. Let cν be a globular cell of Xλ. Let 0 < h < 1. Let

ĉν[h] =
{
ĝν(z,h) : (z,h) ∈ |GlobG(Dnν )|

}
.

It is called an achronal slice of the globular cell cν.

Proposition 5.16. For any globular cell cν of Xλ and any minimal exe-
cution path γ and any h ∈]0, 1[, the cardinal of the set{

t ∈]0, 1[ : γ(t) ∈ ĉν[h]
}

is at most one.

In other terms, a minimal execution path of Xλ intersects any achronal
slice at most one time. Remember that execution paths of Xλ are locally
injective, i.e., they do not contain zero speed points. Proposition 5.16 does
not hold in general for a non-minimal execution path because it could go back
to the initial state of the globular cell after reaching its final state, which
moreover could be equal to the initial state of the globular cell.

Proof. If the set γ(]0, 1[) ∩ ĉν[h] is nonempty, then the minimal execution
path γ has at least one point of γ(]0, 1[) belonging to ĉν. If [cν] is the carrier
of γ, then γ = δzφ with z ∈ Dnν\Snν−1 and φ ∈G(1, 1). We then have{

t ∈]0, 1[ : γ(t) ∈ ĉν[h]
}

=
{
φ−1(h)

}
.

Otherwise, by Proposition 5.14, there is the inclusion γ([0, 1]) ⊂ ∂cν and there
exists an execution path ĝνδzφ for some z ∈ Snν−1 and φ ∈ G(1, 1) from the



192 p. gaucher

initial state of cν to its final state with

ĝνδzφ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n)

with all γi minimal and γ ∈ {γ1, . . . ,γn}. Since γ(]0, 1[) ∩ ĉν[h] is nonempty,
we have

h ∈
]
φ
(∑

j<i
`j

)
,φ
(∑

j6i
`j

)[
for some i ∈ {1, . . . ,n} and γ = γi (h belongs to the interior of the interval
because γ(]0, 1[) ∩X0 = ∅). We obtain

γ(t) = ĝν

(
z,φ
(
`it +

∑
j<i

`j
))

for all t ∈ [0, 1]. We deduce the equality{
t ∈]0, 1[ : γ(t) ∈ ĉν[h]

}
=

{
φ−1(h) −

∑
j<i `j

`i

}
.

(0 , 0)

(1 , 1)(0 , 1)

(1 , 0)

Figure 1: |X| = [0, 1] × [0, 1], X0 = {0}× [0, 1] ∪{(x,x) : x ∈ [0, 1]}, PG
(0,t),(t,t)

X =

G(1, 1) for all t ∈]0, 1], PG
(0,0),(0,0)

X = {(0, 0)} and PGα,βX = ∅ otherwise, there is no
composable execution paths.

Proposition 5.17. Let cν be a globular cell of Xλ for some ν < λ. There
exists b ∈]0, 1[ such that

ĉν[h] ∩X0 = ∅ for all h ∈]0,b].



homotopy theory of moore flows (II) 193

Proposition 5.17 means that, close enough to the initial state of a globular
cell, an achronal slice does not contain any state of X0. Similarly, it is possible
to prove that close enough to the final state of a globular cell, an achronal
slice does not contain any state of X0 either. It is due to the two following
geometric facts. The first one is that close enough to the initial state of the
globular cell, there is no other states of X0 than the initial state because X0

is discrete. The second one is that a non-constant execution path cannot be
deformed in a continuous way to a point in the space of execution paths of a
cellular multipointed d-space. It is possible in more general multipointed d-
spaces as the one depicted in Figure 1. Note that the q-cofibrant replacement
of the latter is equal to the disjoint sum of the q-cofibrant replacement of the
terminal multipointed d-space and of uncountably many directed segments.

Proof. One has cν ∩ X0 = ∅. Consequently, if ĝν(z,h) ∈ X0 for some
h ∈]0, 1[, then z ∈ Snν−1. Thus, if nν = 0, then Snν−1 = ∅ and for any
h ∈]0, 1[, one has ĉν[h] ∩X0 = ∅. Assume now that nν > 1. Consider the set

J1 =
{
h ∈]0, 1[ : ĉν[h] ∩

(
X0\

{
ĝν(0)

})
6= ∅

}
.

If J1 is nonempty, then consider a sequence (h
1
n)n>0 of J1 converging to the

greatest lower bound inf J1 of J1. For all n > 0, let z
1
n ∈ Snν−1 such that

ĝν
(
z1n,h

1
n

)
∈ X0\

{
ĝν(0)

}
.

By extracting a subsequence, we can suppose that the sequence (z1n)n>0 con-
verges to z1∞ ∈ Snν−1. Since the space |Glob

G(Dnν )| is compact, the subspace
ĉν is a compact subspace of the weakly Hausdorff space |Xλ|. The set ĉν ∩X0
is therefore finite because X0 is discrete in |Xλ| by Proposition 5.2. Since(

ĉν ∩X0
)
\
{
ĝν(0)

}
⊂ ĉν ∩X0

is discrete finite as well, the sequence (ĝν(z
1
n,h

1
n))n>0, which converges to

ĝν(z
1
∞, inf J1) by continuity of ĝν, eventually becomes constant. Thus,

ĝν
(
z1∞, inf J1

)
6= ĝν(0).

It implies that
inf J1 > 0.

It means that whether J1 is empty or not, there exists a ∈]0, 1[ such that for



194 p. gaucher

all h ∈]0,a], one has ĉν[h] ∩X0 ⊂{ĝν(0)}. Consider the set

J2 =
{
h ∈]0,a] : ĉν[h] ∩X0 = {ĝν(0)}

}
.

If J2 is nonempty, then consider a sequence (h
2
n)n>0 of J2 converging to inf J2.

For all n > 0, there exists z2n ∈ Snν−1 such that

ĝν
(
z2n,h

2
n

)
= ĝν(0).

By extracting a subsequence, one can suppose that the sequence (z2n)n>0 of
Snν−1 converges to z2∞ ∈ Snν−1. Consider the sequence of globular cells
(cνn)n>0 such that for all n > 0, the globular cell cνn is the first globular cell
appearing in Carrier(ĝνδz2n), i.e.,

Carrier
(
ĝνδz2n

)
= [cνn,cn]

where cn is a sequence of globular cells which is necessarily nonempty be-
cause h2n < 1. Using Proposition 5.5, we have that the compact subspace ĉν
intersects finitely many globular cells of Xλ. Consequently, by extracting a
subsequence again, we can suppose that the sequence of globular cells (cνn)n>0
is constant and equal to the globular cell cν′ for some ν

′ < ν. Write

ĝνδz2n =
(
ĝν′δz′nφnµtn

)
∗ (γnµ1−tn)

with, for all n > 0,

0 < tn 6 h
2
n < 1, z

′
n ∈ D

nν′\Snν′−1, φn ∈G(1, 1),
ĝν′(0) = ĝν(0) (the globular cells cν and cν′ have the same initial state),

ĝν′(1) = (ĝνδz2n)(tn), γn ∈ P
G
ĝν′(1),ĝν(1)

X with Carrier(γn) = [cn].

By extracting a subsequence, one can suppose that the sequence (z′n)n>0 of
Dnν′ converges to z′∞. Since Carrier(ĝν′δz′∞) exists (it is a sequence of globular
cells intersecting ĉν′), the execution path ĝν′δz′∞ is not constant. Thus, there
exists T ∈]0, 1[ such that

ĝν′
(
z′∞,T

)
6= ĝν′(0).

By extracting again a subsequence, one can suppose that the sequence



homotopy theory of moore flows (II) 195

(tnφ
−1
n (T))n>0 of [0, 1] converges to t∞. We have

ĝνδz2n
(
tnφ
−1
n (T)

)
= ĝν′δz′n

(
φnµtn

(
tnφ
−1
n (T)

))
= ĝν′(z

′
n,T)

for all n > 0. We obtain by passing to the limit

ĝνδz2∞(t∞) = ĝν′
(
z′∞,T

)
.

We deduce that ĝνδz2∞(t∞) 6= ĝν(0) and therefore that

0 < t∞.

From the inequalities
tnφ
−1
n (T) 6 tn 6 h

2
n

for all n > 0, we obtain by passing to the limit the inequalities

0 < t∞ 6 inf J2.

It means that whether J2 is empty or not, there exists b ∈]0, 1[ such that for
all h ∈]0,b], one has ĉν[h] ∩X0 = ∅.

Theorem 5.18. Let γ∞ be an execution path of Xλ. Let ν0 < λ. There
exists an open neighborhood Ω of γ∞ in PGXλ such that for all execution
paths γ ∈ Ω, the number of copies of cν0 in the carrier of γ cannot exceed the
length of the carrier of γ∞.

Proof. Let Carrier(γ∞) = [cν1, . . . ,cνn]. Consider the decomposition of
Theorem 5.9

γ∞ = (γ
1
∞µ`1 ) ∗ · · · ∗ (γ

n
∞µ`n)

with
∑

i `i = 1 and all execution paths γ
i
∞ minimal for i = 1, . . . ,n. For

1 6 i 6 n, let νi < λ, φi ∈G(1, 1) and zi ∈ Dnνi\Snνi−1 such that

Carrier(γi∞) = [cνi], γ
i
∞(]0, 1[) ⊂ cνi, γ

i
∞ = δziφi.

Using Proposition 5.17, pick h ∈]0, 1[ such that ĉν0 [h] ∩ X0 = ∅. For all
1 6 i 6 n, the set {

t ∈]0, 1[ : γi∞(t) ∈ ĉν0 [h]
}

contains at most one point ti by Proposition 5.16; if the set above is empty,



196 p. gaucher

let ti =
1
2
. For all 1 6 i 6 n, let Li and L

′
i be two real numbers such that

0 < Li < ti < L
′
i < 1.

For 1 6 i 6 n, consider the covering of the segment [
∑

j<i `j,
∑

j6i `j] in three
nonoverlapping segments of strictly positive length:

K−i =
[∑

j<i
`j,
∑

j<i
`j + µ

−1
`i
φ−1i (Li)

]
,

Kmi =
[∑

j<i
`j + µ

−1
`i
φ−1i (Li),

∑
j<i

`j + µ
−1
`i
φ−1i (L

′
i)
]
,

K+i =
[∑

j<i
`j + µ

−1
`i
φ−1i (L

′
i),
∑

j6i
`j

]
.

The restriction γ∞� [
∑
j<i `j,

∑
j6i `j]

goes from the initial state of the globular
cell cνi to its final state. We have therefore

γ∞(K
m
i ) ⊂ cνi.

We deduce
γ∞(K

m
i ) ∩X

0 = ∅.

We have

γ∞(K
−
i ) ∩ ĉν0 [h]

=

(
γ∞

({∑
j<i

`j

})
∪γ∞

(]∑
j<i

`j,
∑

j<i
`j + µ

−1
`i
φ−1i (Li)

]))
∩ ĉν0 [h]

=

({
ĝνi(0)

}
∪γ∞

(]∑
j<i

`j,
∑

j<i
`j + µ

−1
`i
φ−1i (Li)

]))
∩ ĉν0 [h]

⊂
({

ĝνi(0)
}
∪γ∞

(]∑
j<i

`j,
∑

j<i
`j + µ

−1
`i
φ−1i (ti)

[))
∩ ĉν0 [h] = ∅,

the first equality by formal set identities, the second equality by definition of
ĝνi(0), the inclusion because Li < ti, and the last equality because ĝνi(0) ∈ X

0

and by definition of ti. In the same way, we also have

γ∞(K
+
i ) ∩ ĉν0 [h]

=

(
γ∞

([∑
j<i

`j + µ
−1
`i
φ−1i (L

′
i),
∑

j6i
`j

[)
∪γ∞

({∑
j6i

`j

}))
∩ ĉν0 [h]



homotopy theory of moore flows (II) 197

=

(
γ∞

([∑
j<i

`j + µ
−1
`i
φ−1i (L

′
i),
∑

j6i
`j

[)
∪
{
ĝνi(1)

})
∩ ĉν0 [h]

⊂
(
γ∞

(]∑
j<i

`j + µ
−1
`i
φ−1i (ti),

∑
j6i

`j

[)
∪
{
ĝνi(1)

})
∩ ĉν0 [h] = ∅,

the first equality by formal set identities, the second equality by definition of
ĝνi(1), the inclusion because ti < L

′
i, and the last equality because ĝνi(1) ∈ X

0

and by definition of hi. Since |Xλ| is weakly Hausdorff, the set ĉν0 [h] is a closed
subset of |Xλ|. Moreover, X0 is a closed subset of the space |Xλ| as well by
Proposition 5.2. Consequently, the set

Ω =
i=n⋂
i=1

(
W
(
K−i , |Xλ|\ĉν0 [h]

)
∩W

(
Kmi , |Xλ|\X

0
)
∩W

(
K+i , |Xλ|\ĉν0 [h]

))
,

where
W([a,b],U) =

{
f ∈ PGXλ : f([a,b]) ⊂ U

}
is an open neighborhood of γ∞ of PGXλ for the compact-open topology, and
therefore for its ∆-kelleyfication which adds open subsets. For all γ ∈ Ω, one
has

γ(Kmi ) ∩X
0 = ∅

and
γ(K−i ) ∩ ĉν0 [h] = γ(K

+
i ) ∩ ĉν0 [h] = ∅.

It turns out that the segments of strictly positive length K−i ,K
m
i ,K

+
i for

1 6 i 6 n are a finite partition of [0, 1] into nonoverlapping segments because
we have by definition of the K−i ,K

m
i ,K

+
i for 1 6 i 6 n:

[0, 1] =
⋃i=n

i=1

[∑
j<i

`j,
∑

j6i
`j

]
=
⋃i=n

i=1

(
K−i ∪K

m
i ∪K

+
i

)
.

Each cν0 appearing in Carrier(γ) corresponds to a minimal execution path
from ĝν0 (0) to ĝν0 (1) (note that these two states can be equal) of the de-
composition of γ obtained using Theorem 5.9. It necessarily intersects ĉν0 [h].
Thus, the number of copies of cν0 in the carrier of γ cannot exceed the number
of Kmi , i.e., the length of the carrier of γ∞.

Theorem 5.18 does not mean that the carriers of the execution paths of
Ω are of length at most the length of the carrier of γ∞. Indeed, on the



198 p. gaucher

segments K−0 ,K
+
0 ∪K

−
1 ,K

+
1 ∪K

−
2 , . . . ,K

+
n , an execution path γ of Ω can a

priori intersect X0 an arbitrarily large number of times. However, it cannot
intersect ĉν0 [h]. Therefore these segments do not add copies of cν0 in the
carrier of γ.

Theorem 5.19. Let (γk)k>0 be a sequence of execution paths of Xλ which
converges in PGXλ. Let cν0 be a globular cell of Xλ. Let ik be the number of
times that cν0 appears in Carrier(γk). Then the sequence of integers (ik)k>0
is bounded.

Proof. Write γ∞ for the limit of (γk)k>0 in PGXλ. By Theorem 5.18, there
exists an open Ω containing γ∞ such that for all γ ∈ Ω, the number of copies
of cν0 in the carrier of γ does not exceed the length of the carrier of γ∞. Since
the sequence (γk)k>0 converges to γ∞, there exists N > 0 such that for all
k > N, γk belongs to Ω. The proof is complete.

Theorem 5.20. Let 0 6 ν < λ. Then every execution path of Xν+1 can
be written in a unique way as a finite Moore composition

(f1γ1µ`1 ) ∗ · · · ∗ (fnγnµ`n)

with n > 1 such that

(1)
∑

i `i = 1;

(2) fi = f and γi is an execution path of Xν or fi = ĝν and γi = δziφi with
zi ∈ Dnν\Snν−1 and some φ ∈G(1, 1);

(3) for all 1 6 i < n, either fiγi or fi+1γi+1 (or both) is (are) of the form
ĝνδzφ for some z ∈ Dnν\Snν−1 and some φ ∈ G(1, 1): intuitively, there is
no possible simplification using the Moore composition inside Xν.

Proof. We use the normal form of Theorem 5.9 and we use Proposition 4.10
to compose successive execution paths of Xν.

6. Chains of globes

Let Z1, . . . ,Zp be p nonempty topological spaces with p > 1. Consider the
multipointed d-space

X = GlobG(Z1) ∗ · · · ∗ GlobG(Zp),



homotopy theory of moore flows (II) 199

with p > 1 where the ∗ means that the final state of a globe is identi-
fied with the initial state of the next one by reading from the left to the
right. Let {α0,α1, . . . ,αp} be the set of states such that the canonical map
GlobG(Zi) → X takes the initial state 0 of GlobG(Zi) to αi−1 and the final
state 1 of GlobG(Zi) to αi.

As a consequence of the associativity of the semimonoidal structure on
G-spaces recalled in Theorem 4.3 and of [14, Proposition 5.16], we have

Proposition 6.1. Let U1, . . . ,Up be p topological spaces with p > 1. Let
`1, . . . ,`p > 0. There is the natural isomorphism of G-spaces

FG
op

`1
U1 ⊗···⊗FG

op

`p
Up ∼= FG

op

`1+···+`p(U1 ×···×Up).

The case p = 1 of Proposition 6.3 is treated in Proposition 2.12 and already
used in Proposition 4.15. An additional argument is required for the case
p > 1. At first, we prove a lemma which is an addition to Proposition 2.5.

Lemma 6.2. The set map (−)−1 : G(1,p) → G(p, 1) which takes f :
[0, 1] ∼=+ [0,p] to its inverse f−1 : [0,p] ∼=+ [0, 1] is continuous.

Proof. Since all ∆-generated spaces are sequential, it suffices to prove that
(−)−1 : G(1,p) →G(p, 1) is sequentially continuous. Let (fn)n>0 be a sequence
of G(1,p) which converges to f ∈ G(1,p). Let t ∈ [0,p]. Then the sequence
(f−1n (t))n>0 of [0, 1] has at least one limit point denoted by L(t). By extracting
a subsequence of the sequence (fn(f

−1
n (t)))n>0, we obtain f(L(t)) = t, which

implies L(t) = f−1(t). Thus every subsequence of (f−1n (t))n>0 has a unique
limit point f−1(t). Suppose that the sequence (f−1n (t))n>0 does not converge
to f−1(t). Then there exists an open neighborhood V of f−1(t) such that
for all n > 0, f−1n (t) ∈ V c which is compact: contradiction. Therefore the
sequence (f−1n )n>0 pointwise converges to f

−1. By Proposition 2.5, we deduce
that the sequence (f−1n )n>0 converges to f

−1.

Proposition 6.3. With the notations of this section. There is a homeo-
morphism

PGα0,αpX
∼= G(1,p) ×Z1 ×···×Zp.

Proof. The Moore composition of paths induced a map of G-spaces

P0,1MGGlobG(Z1) ⊗···⊗P0,1MGGlobG(Zp) −→ Pα0,αpM
G(X).



200 p. gaucher

By Proposition 4.15, there is the isomorphism of G-spaces

P0,1MGGlobG(Z) ∼= FG
op

1 Z

for all topological spaces Z. We obtain a map of G-spaces

FG
op

1 Z1 ⊗···⊗F
Gop
1 Zp −→ Pα0,αpM

G(X).

By Proposition 6.1, and since P1α0,αpM
G(X) = PGα0,αpX by definition of the

functor MG, we obtain a continuous map

Ψ : G(1,p) ×Z1 ×···×Zp −→ PGα0,αpX
(φ,z1, . . . ,zp) 7−→ (δz1φ1) ∗ · · · ∗ (δzpφp),

where φi ∈ G(`i, 1) with
∑

i `i = 1 and φ = φ1 ⊗ ··· ⊗ φp being the de-
composition of Proposition 3.2. Since all executions paths of globes are one-
to-one, the map Ψ above is a continuous bijection. The continuous maps
Zi → {0} for 1 6 i 6 p induce by functoriality a map of multipointed d-
spaces X →

−→
I G ∗ · · · ∗

−→
I G (p times) and then a continuous map

k : PGα0,αpX −→ P
G
α0,αp

(−→
I G ∗ · · · ∗

−→
I G
)

= G(1,p)
(δz1φ1) ∗ · · · ∗ (δzpφp) 7−→ (δ0φ1) ∗ · · · ∗ (δ0φp) = φ1 ⊗···⊗φp.

Let i ∈{1, . . . ,p}. Then we have, with γ = (δz1φ1) ∗ · · · ∗ (δzpφp),

γ
(
k(γ)−1

(
i−

1

2

))
= γ

(
φ−1

(
i−

1

2

))
= δziφiφ

−1
(
i−

1

2

)
=
(
zi,

1

2

)
,

the first equality by definition of k : PGα0,αpX → G(1,p), the second equality
since i−1 < i− 1

2
< i and by definition of γ, and the last equality by definition

of the φi’s. The set map

PGα0,αpX −→ |Glob
G(Zi)|

γ 7−→ γ
(
k(γ)−1

(
i− 1

2

))
is continuous since k : PGα0,αpX → G(1,p) and (−)

−1 : G(1,p) → G(p, 1) are
both continuous (see Lemma 6.2 for the latter map). Consequently, the set
map

k : PGα0,αpX −→ Z1 ×···×Zp
(δz1φ1) ∗ · · · ∗ (δzpφp) 7−→ (z1, . . . ,zp)



homotopy theory of moore flows (II) 201

is continuous. It implies that the set map

Ψ−1 = (k,k) : (δz1φ1) ∗ · · · ∗ (δzpφp) 7−→ (φ1 ⊗···⊗φp,z1, . . . ,zp)

is continuous as well and that Ψ is a homeomorphism.

Until the end of this section, we work like in Section 5 with a cellular
multipointed d-space Xλ, with the attaching map of the globular cell cν for
ν < λ denoted by

ĝν : Glob
G(Dnν ) −→ Xλ.

Each carrier
c = [cν1, . . . ,cνn]

gives rise to a map of multipointed d-spaces from a chain of globes to Xλ

ĝc : Glob
G(Dnν1 ) ∗ · · · ∗ GlobG(Dnνn ) −→ Xλ

by “concatenating” the attaching maps of the globular cells cν1, . . . ,cνn. Let
αi−1 (αi resp.) be the initial state (the final state resp.) of Glob

G(Dnνi ) for
1 6 i 6 n in GlobG(Dnν1 ) ∗ · · · ∗ GlobG(Dnνn ). It induces a continuous map

PGĝc : Xc := PGα0,αn
(
GlobG(Dnν1 ) ∗ · · · ∗ GlobG(Dnνn )

)
−→ PGXλ.

Proposition 6.4. Let γ be an execution path of Xλ. Consider a nonde-
creasing set map φ : [0, 1] → [0, 1] preserving the extremities such that γφ = γ.
Then φ is the identity of [0, 1].

Proof. Note that it is not assumed that φ is continuous. 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) 6 φ(t′′) 6 φ(t′), which contradicts the fact that γ is
locally injective by Proposition 5.13. Thus the set map φ is strictly increasing.
Let Carrier(γ) = [cν1, . . . ,cνn]. Let γ = (γ1µ`1 )∗· · ·∗(γnµ`n) with `1+· · ·+`n =
1 such that for all 1 6 i 6 n, there exist zi ∈ Dnνi\Snνi−1 and φi ∈G(1, 1) such
that for all t ∈]0, 1[, γi(t) = (zi,φi(t)) ∈ cνi, γi(0) = ĝνi(0) and γi(1) = ĝνi(1).
Then {

t ∈ [0, 1] : γ(t) ∈ X0
}

= {0 = t0 < t1 < · · · < tn = 1}

with ti =
∑

16j6i `j for 0 6 i 6 n. We deduce that 0 = φ(t0) < φ(t1) <
· · · < φ(tn) = 1 because the set map φ is strictly increasing. Since γ(φ(ti)) =
γ(ti) ∈ X0 for 0 6 i 6 n, one obtains φ(ti) = ti for 0 6 i 6 n and φ(]ti−1, ti[) ⊂



202 p. gaucher

]ti−1, ti[ for all 1 6 i 6 n. Then, observe that

(zi,φi(φ(t))) = (zi,φi(t)), ∀1 6 i 6 n, ∀t ∈]ti−1, ti[ .

Since φi is bijective, it means that the restriction φ� ]ti−1,ti[ is the identity of
]ti−1, ti[ for all 1 6 i 6 n.

Notation 6.5. 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 6.6. Let γ1 and γ2 be two execution paths of Xλ such that
there exist two nondecreasing set maps φ1,φ2 : [0, 1] → [0, 1] preserving the
extremities such that

γ1(φ1(t)) = γ2(t) and γ1(t) = γ2(φ2(t)) for all t ∈ [0, 1].

Then φ1,φ2 ∈G(1, 1) and φ2 = φ−11 .

Proof. Note that it is not assumed that φ1 and φ2 are continuous. For all
t ∈ [0, 1], we have γ1(φ1(φ2(t))) = γ2(φ2(t)) = γ1(t). Using Proposition 6.4,
we deduce that φ1φ2 = Id[0,1]. In the same way, we have φ2φ1 = Id[0,1]. This
proves that φ1 and φ2 are two bijective set maps preserving the extremities
which are inverse to each other. Suppose e.g., that there exists t ∈ [0, 1] such
that φ1(t

−) < φ1(t). Then φ1 cannot be surjective: contradiction. By using
similar arguments, we deduce that for all t ∈ [0, 1], φ1(t−) = φ1(t) = φ1(t+)
and φ2(t

−) = φ2(t) = φ2(t
+). Consequently, the set maps φ1 and φ2 are

continuous.

Proposition 6.7. Let c be the carrier of some execution path of Xλ.
Every execution path of the image of PGĝc is of the form(

ĝν1δz1 ∗ · · · ∗ ĝνnδzn
)
φ

with φ ∈G(1,n) and zi ∈ Dnνi for 1 6 i 6 n.

Proof. The first assertion is a consequence of the definition of ĝc and of
Proposition 6.3.



homotopy theory of moore flows (II) 203

Notation 6.8. Let c be the carrier of some execution path of Xλ. Using
the identification provided by the homeomorphism of Proposition 6.3, we can
use the notation

(PGĝc)(φ,z1, . . . ,zn) = (ĝν1δz1 ∗ · · · ∗ ĝνnδzn)φ.

Before proving the main theorem of this section, we need the following
topological lemmas:

Lemma 6.9. Let U1, . . . ,Up be p first-countable ∆-Hausdorff ∆-generated
spaces with p > 1. Then the product U1 ×···× Up in the category TOP of
general topological spaces and continuous maps coincides with the product
in Top.

Proof. Consider U1 ×···×Up equipped with the product topology in the
category TOP of general topological spaces and continuous maps. This topol-
ogy is first-countable as a finite product of first-countable topologies. Each
space Ui is locally path-connected, being ∆-generated. Thus, the finite prod-
uct U1 ×···×Up equipped with the product topology in TOP is locally path-
connected. We deduce that U1 ×···×Up equipped with the product topology
in TOP is ∆-generated: the ∆-kelleyfication functor is not required. Moreover
since each Ui is ∆-Hausdorff, the product in TOP is ∆-Hausdorff as well. It
means that U1×···×Up equipped with the product topology in TOP coincides
with the product in Top.

Lemma 6.10. Let U1, . . . ,Up be p first-countable ∆-Hausdorff ∆-gener-
ated spaces with p > 1. Let (uin)n>0 be a sequence of Ui for 1 6 i 6 p
which converges to ui∞ ∈ Ui. Then the sequence ((u1n, . . . ,u

p
n))n>0 converges

to (u1∞, . . . ,u
p
∞) ∈ U1 ×···×Up for the product calculated in Top.

Note that the converse is obvious: if the sequence ((u1n, . . . ,u
p
n))n>0 con-

verges to (u1∞, . . . ,u
p
∞) ∈ U1 ×···×Up, then the sequences (uin)n>0 converge

to ui∞ ∈ Ui for all 1 6 i 6 p because of the existence of the projection maps
U1 ×···×Up → Ui for all 1 6 i 6 p. A sequence converges to some point in a
∆-generated space if and only if the corresponding application from the one-
point compactification N = N∪{∞} of the discrete space N to the ∆-generated
space is continuous. The point is that the one-point compactification of N is
not ∆-generated: its ∆-kelleyfication is a discrete space. Therefore it does
not seem possible to use the universal property of the finite product in Top
to prove Lemma 6.10.



204 p. gaucher

Proof. Each convergent sequence gives rise to a continuous map N → Ui
for 1 6 i 6 p. We obtain a continuous map N → U1 ×···×Up by using the
universal property of the finite product in TOP thanks to Lemma 6.9 and the
proof is complete.

The sequence (φk)k>1 of G(1, 1) depicted in Figure 2 has no limit point
because the only possibility is the set map which takes 0 to 0 and the other
points of [0, 1] to 1: it does not belong to G(1, 1). Thus, the topological space
G(1,n), which is homeomorphic to G(1, 1) for all n > 1, is not sequentially
compact. However, Theorem 6.11 holds anyway.

(0 , 0)

ϕk

(1 , 1)

1 −
1

k

1

k

Figure 2: A sequence (φk)k>1 of G(1, 1) without limit point

Theorem 6.11. Let c be the carrier of some execution path of Xλ.

(1) Consider a sequence (γk)k>0 of the image of PGĝc which converges point-
wise to γ∞ in PGXλ. Let

γk =
(
PGĝc

)(
φk,z

1
k, . . . ,z

n
k

)
with φk ∈G(1,n) and zik ∈ D

nνi for 1 6 i 6 n and k > 0. Then there exist
φ∞ ∈G(1,n) and zi∞ ∈ D

nνi for 1 6 i 6 n such that

γ∞ =
(
PGĝc

)(
φ∞,z

1
∞, . . . ,z

n
∞
)

and
(
φ∞,z

1
∞, . . . ,z

n
∞
)

is a limit point of the sequence
((
φk,z

1
k, . . . ,z

n
k

))
k>0

.

(2) The image of PGĝc is closed in PGXλ.



homotopy theory of moore flows (II) 205

Proof. (1) By a Cantor diagonalization argument, we can suppose that:

• the sequence (zik)k>0 converges to z
i
∞ ∈ D

nνi for each 1 6 i 6 n;

• the sequence (φk(r))k>0 converges to a real number denoted by φ∞(r) ∈
[0,m] for each r ∈ [0, 1] ∩Q;
• the sequence (φ−1k (r))k>0 converges to a real number denoted by φ

−1
∞ (r) ∈

[0, 1] for each r ∈ [0,n] ∩Q.

Since the sequence of execution paths (γk)k>0 converges pointwise to γ∞, we
obtain

γ∞(r) =
(
ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞

)
(φ∞(r)) for all r ∈ [0, 1] ∩Q,

γ∞
(
φ−1∞ (r)

)
=
(
ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞

)
(r) for all r ∈ [0,n] ∩Q.

For r1 < r2 ∈ [0, 1]∩Q, φk(r1) < φk(r2) for all k > 0. Therefore by passing to
the limit, we obtain φ∞(r1) 6 φ∞(r2). Note that φ∞(0) = 0 and φ∞(1) = n
since 0, 1 ∈ Q. In the same way, we see that φ−1∞ : [0,n] ∩ Q → [0, 1] is
nondecreasing and that φ−1∞ (0) = 0 and φ

−1
∞ (n) = 1. For t ∈]0, 1[, let us

extend the definition of φ∞ as follows:

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, 1]\Q, there exists a nondecreasing sequence (rk)k>0 of rational numbers
converging to t. Then

lim
k→∞

φ∞(rk) = φ∞(t).

By continuity, we deduce that

γ∞(t) =
(
ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞

)
(φ∞(t))

for all t ∈ [0, 1]. It is easy to see that the set map φ∞ : [0, 1] → [0,n] is
nondecreasing and that it preserves extremities. For t ∈]0, 1[, extend the
definition of φ−1∞ as well as follows:

φ−1∞ (t) = sup
{
φ−1∞ (r) : r ∈ ]0, t] ∩Q

}
.

The upper bound exists since {φ−1∞ (r) : r ∈ ]0, t] ∩ Q} ⊂ [0, 1]. For each
t ∈ [0,n]\Q, there exists a nondecreasing sequence (rk)k>0 of rational numbers
converging to t. Then

lim
k→∞

φ−1∞ (rk) = φ
−1
∞ (t).



206 p. gaucher

By continuity, we deduce that

γ∞
(
φ−1∞ (t)

)
=
(
ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞

)
(t)

for all t ∈ [0,n]. It is easy to see that the set map φ−1∞ : [0,n] → [0, 1] is
nondecreasing and that it preserves extremities. We obtain for all t ∈ [0, 1]

γ∞(t) =
(
ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞

)(
µ−1n µnφ∞(t)

)
,

γ∞
(
φ−1∞ µ

−1
n (t)

)
=
(
ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞

)(
µ−1n (t)

)
.

Using Theorem 6.6, we obtain that µnφ∞ : [0, 1] → [0, 1] and φ−1∞ µ−1n : [0, 1] →
[0, 1] are homeomorphisms which are inverse to each other. We deduce that
φ∞ : [0, 1] → [0,n] and φ−1∞ : [0,n] → [0, 1] are homeomorphisms which are
inverse to each other. Let t ∈ [0, 1]\Q. Since the sequence (φk(t))k>0 belongs
to the sequential compact [0,n], it has at least one limit point `. There exists
a subsequence of (φk(t))k>0 which converges to `. We obtain

φ∞(r) 6 ` 6 φ∞(r
′), ∀r ∈ [0, t] ∩Q, ∀r′ ∈ [t, 1] ∩Q.

Since φ∞ ∈G(1,n) and by density of Q, we deduce that ` = φ∞(t) necessarily.
Now suppose that the sequence (φk(t))k>0 does not converge to φ∞(t). Then
there exists an open neighborhood V of φ∞(t) in [0,n] such that for all k > 0,
φk(t) /∈ V . We deduce that the sequence (φk(t))k>0 of [0,n] has no limit point:
contradiction. We have proved that the sequence (φk)k>0 converges pointwise
to φ∞. Using Proposition 2.5, we deduce that (φk)k>0 converges uniformly
to φ∞. We deduce that (φ∞,z

1
∞, . . . ,z

n
∞) is a limit point of the sequence(

(φk,z
1
k, . . . ,z

n
k )
)
k>0

in G(1,n) × Dnν1 × ···× Dnνn by Proposition 2.5 and
Lemma 6.10.

(2) Let
(
PGĝc(Γn)

)
n>0

be a sequence of
(
PGĝc

)(
Xc
)

which converges in

PGXλ. The limit γ∞ ∈ PGXλ of the sequence of execution paths
(
PGĝc(Γn)

)
n>0

is also a pointwise limit. We can suppose by extracting a subsequence that
the sequence (Γn)n>0 of Xc is convergent in Xc. Thus, by continuity of PGĝc,
we obtain γ∞ =

(
PGĝc

)
(Γ∞) for some Γ∞ ∈ Xc. We deduce that PGĝc

(
Xc
)

is
sequentially closed in PGXλ. Since PGXλ is sequential, being a ∆-generated
space, the proof is complete.

Corollary 6.12. Suppose that Xλ is a finite cellular multipointed d-
space without loops. Then the topology of PGXλ is the topology of the point-
wise convergence which is therefore ∆-generated.



homotopy theory of moore flows (II) 207

We do not know whether the “without loops” hypothesis can be removed
and whether finite can be replaced by locally finite.

Proof. Let (γn)n>0 be a sequence of execution paths of Xλ which pointwise
converges to γ∞. Since Xλ is finite and without loop, the set

T =
{

Carrier(γ) : γ ∈ PGXλ
}

is finite. We obtain a finite covering by (closed) subsets

PGXλ =
⋃

c∈T

(
Pĝc
)
(Xc)

because each execution path has a carrier by Theorem 5.9. Suppose that
(γn)n>0 does not converge to γ∞ in PGXλ. Then there exists an open neigh-
borhood V of γ∞ in PGXλ such that for n > 0 and all γn /∈ V . Since T is
finite, one can suppose by extracting a subsequence that

∃c ∈T such that γn ∈
(
Pĝc
)
(Xc) for all n > 0.

By Theorem 6.11, the sequence (γn)n>0 has a limit point which belongs to the
complement of V which is closed. This limit point is necessarily the pointwise
limit γ∞. We obtain γ∞ /∈ V : contradiction.

Corollary 6.12 can be viewed as a second Dini theorem for the space of
execution paths of a finite cellular multipointed d-space without loops. Indeed,

if Xλ =
−→
I G (the directed segment), then PGXλ = G(1, 1) and we recover the

fact that the topology of G(1, 1) coincides with the pointwise convergence
topology by Proposition 2.5.

7. 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

GlobG(Sn−1)

��

g
// A

f

��

GlobG(Dn)
ĝ

// X



208 p. gaucher

with n > 0 and A cellular. Note that A0 = X0. Let D = FG
op

1 S
n−1 and

E = FG
op

1 D
n. Consider the Moore flow X defined by the pushout diagram of

Figure 3 where the two equalities

MG(GlobG(Sn−1)) = Glob(D), MG(GlobG(Dn)) = Glob(E)

come from Proposition 4.15 and where the map ψ is induced by the universal
property of the pushout.

MG(GlobG(Sn−1)) = Glob(D)

��

MG(g)
// MG(A)

MG(f)

��

f

��

MG(GlobG(Dn)) = Glob(E)
g

//

MG(ĝ) //

X

ψ
O

O
O

''O
O

O

MG(X).

Figure 3: Definition of X

The G-space of execution paths of the Moore flow X can be calculated by
introducing a diagram of G-spaces Df over a Reedy category Pg(0),g(1)(A0)
whose definition is recalled in Appendix A. Let T be the G-space defined by
the pushout diagram of [Gop, Top]0

D

��

PMG(g)
// Pg(0),g(1)MG(A)

Pf
��

E
Pg

// T.

Consider the diagram of spaces Df : Pg(0),g(1)(A0) → [Gop, Top]0 defined
as follows:

Df ((u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)) = Zu0,u1⊗Zu1,u2⊗···⊗Zun−1,un

with

Zui−1,ui =

{
Pui−1,uiM

G(A) if �i = 0,

T if �i = 1.



homotopy theory of moore flows (II) 209

In the case �i = 1, (ui−1,ui) = (g(0),g(1)) by definition of Pg(0),g(1)(A0). The
inclusion maps I′is are induced by the map Pf : Pg(0),g(1)M

G(A) → T. The
composition maps c′is are induced by the compositions of paths of the Moore
flow MG(A).

Theorem 7.1. ([14, Theorem 9.7]) We obtain a well-defined diagram of
G-spaces

Df : Pg(0),g(1)(A0) −→ [Gop, Top]0.

There is the isomorphism of G-spaces lim−→D
f ∼= PX.

By the universal property of the pushout, we obtain a canonical map of
G-spaces

Pψ : lim−→D
f −→ PMGX.

The goal of Theorem 7.2 and of Theorem 7.3 is to prove that the canonical
map of G-spaces Pψ is an isomorphism of G-spaces. The proof is twofold: at
first, it is proved that it is an objectwise continous bijection, and then that it
is an objectwise homeomorphism.

Theorem 7.2. Under the hypotheses and the notations of this section.
The map of G-spaces

Pψ : lim−→D
f −→ PMG(X)

is an objectwive bijection.

Proof. Throughout the proof, the reader must keep in mind that for any
map of multipointed d-spaces q : X1 → X2 and for any execution path

γ ∈ P`X1 = P`MG(X1)

of length ` of the multipointed d-space X1, or equivalently of the Moore flow
MG(X1), there is by Definition 4.8 and Theorem 4.12 the tautological equality(

PMG(q)
)
(γ) = |q|.γ,

the right-hand term meaning the composite of the underlying continuous map
|q| : |X1| → |X2| between the underlying spaces of X1 and X2 with the
execution path γ : [0, 1] →|X1|. It will be denoted qγ or q.γ, as it was always
done so far. In other terms, we will be using the abuse of notation

PMG(q) = q



210 p. gaucher

for any map of multipointed d-spaces q. The reader must also keep in mind
that if γ ∈ P`X1 and γ′ ∈ P`

′
X1 are two composable execution paths of X1 of

length ` and `′ respectively, then the Moore composition of execution paths
(cf. Proposition 2.7)

γ ∗γ′ ∈ P`+`
′
X1

is also by Theorem 4.12 the composition of paths in MG(X1) for tautological
reasons.

The proof of this theorem is divided in several parts.
• Objectwise calculation. It suffices to prove that the continuous map

P1ψ : lim−→D
f (1) −→ P1MG(X) = PGX

is a bijection to complete the proof since all objects of the reparametriza-
tion category G are isomorphic and since colimits of G-spaces are calculated
objectwise.
• The final topology . If we can prove that the continuous

P1ψ : lim−→D
f (1) −→ PGX

is a bijection with the colimit lim−→D
f (1) equipped with the final topology, then

the proof will be complete even in the category of ∆-Hausdorff ∆-generated
topological spaces because of the 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: let f : [0, 1] → A
be a continuous map; then f being one-to-one, f([0, 1]) = i−1((i.f)([0, 1]))
is closed.

- The space PGX is, by definition, equipped with the ∆-kelleyfication of the
relative topology induced by the inclusion of set PGX ⊂ TOP([0, 1], |X|).

- If we work in the category of ∆-Hausdorff ∆-generated topological spaces,
then the space TOP([0, 1], |X|) will be ∆-Hausdorff, hence the space PGX
will be ∆-Hausdorff, and therefore lim−→D

f equipped with the final topology
will be ∆-Hausdorff as well.

• Surjectivity of P1ψ. The map ψ of Figure 3 is induced by the universal
property of the pushout. It is bijective on states. The multipointed d-space
X is equipped with the 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 composition of the form



homotopy theory of moore flows (II) 211

(f1γ1µ`1 ) ∗ · · · ∗ (fnγnµ`n)

such that fi ∈{f, ĝ} for all 1 6 i 6 n, where{
γi ∈ PGGlobG(Dn) if fi = ĝ,
γi ∈ PGA if fi = f,

with
∑

i `i = 1. Then for all 1 6 i 6 n

γiµ`i ∈ P
`iGlobG(Dn) = P`iMG(GlobG(Dn)) or γiµ`i ∈ P

`iA = P`iMG(A).

It gives rise to the execution path

Pf1(γ1µ`1 ) ∗ · · · ∗Pfn(γnµ`n)

with {
fi = g if fi = ĝ,

fi = f if fi = f,

of length 1 of the Moore flow X. By the commutativity of Figure 3, we obtain
the equality

(f1γ1µ`1 ) ∗ · · · ∗ (fnγnµ`n) =
(
P1ψ

)(
Pf1(γ1µ`1 ) ∗ · · · ∗Pfn(γnµ`n)

)
.

It means that the map of Moore flows ψ : X → MG(X) induces a surjective
continuous map from P1X to P1MG(X) = PGX. In other terms, the map P1ψ
is a surjection.

• The map Ĉ. Consider the diagram of topological spaces

Ef : Pg(0),g(1)(A0) −→ Top

defined as follows:

Ef
(
(u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)

)
=

⊔
(`1,...,`n)

`1+···+`n=1

Zu0,u1 (`1) ×···×Zun−1,un(`n),

with



212 p. gaucher

Zui−1,ui(`i) =

{
P`iui−1,uiM

G(A) = P`iui−1,uiA if �i = 0,
T(`i) if �i = 1 [ ⇒(ui−1, ui) = (g(0), g(1)) ].

The composition maps c′is are induced by the Moore composition of execution
paths of A. The inclusion maps I′is are induced by the continuous maps
P`f : P`

g(0),g(1)
MG(A) → T(`) for ` > 0. We obtain a well-defined diagram of

topological spaces 2

Ef : Pg(0),g(1)(A0) −→ Top

and, by Proposition 4.5, there is an objectwise continuous surjective map

k : Ef −→Df (1).

We deduce that lim−→k is surjective. We want to prove that the composite map

Ĉ : lim−→E
f

lim−→k // // ( lim−→Df)(1) P1ψ // P1MG(X) = PGX
is a continuous bijection. We already know that the map P1ψ is surjective,
and therefore that the map Ĉ : lim−→E

f → PGX is surjective as well. To prove
that Ĉ : lim−→E

f → PGX 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 Ef . We say that x ∈Ef (n) is simplified if 3

d(n) = min

{
d(m) :

∃m ∈ Obj(Pg(0),g(1)(A0)) and ∃y ∈Ef (m)
such that y = x ∈ lim−→E

f

}
.

Let x be a simplified element belonging to some vertex Ef (n) of the diagram
Ef with

n =
(
(u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)

)
.

• Case 1 . It is impossible to have �i = �i+1 = 0 for some 1 6 i < n.
Indeed, otherwise x would be of the form

(. . . ,γiµ`i,γi+1µ`i+1, . . . )

2 This point is left as an exercice; Verifying the commutativity relations is easy.
3 d is the degree function of the Reedy category, see Appendix A.



homotopy theory of moore flows (II) 213

where γi and γi+1 would be two execution paths of A. Using the equality

ci
(
(. . . ,γiµ`i,γi+1µ`i+1, . . . )

)
= (. . . ,γiµ`i ∗γi+1µ`i+1, . . . ),

the tuple x can then be identified in the colimit with the tuple(
. . . ,

(
γiµ`i ∗γi+1µ`i+1

)
µ−1`i+`i+1︸ ︷︷ ︸

∈PGA by Proposition 4.10

µ`i+`i+1, . . .

)
∈Ef (n′)

with
d(n′) = n− 1 +

∑
i
�i < d(n).

It is a contradiction because x is simplified by hypothesis.
• Case 2 . Suppose that �i = 1 for some 1 6 i 6 n and that x is of the

form
(. . . ,Pg(δziφiµ`i), . . . ).

If zi ∈ Sn−1, then using the equality

Ii
(
(. . . ,gδziφiµ`i, . . . )

)
=
(
. . . ,Pg(δziφiµ`i), . . .

)
,

the tuple x can then be identified in the colimit with the tuple

(. . . ,gδziφiµ`i, . . . ) ∈E
f (n′)

with
d(n′) = n +

(∑
i
�i
)
− 1 < d(n).

It is a contradiction because x is simplified by hypothesis. We deduce that in
this case, zi ∈ Dn\Sn−1.
• Partial conclusion. Consequently, for all simplified elements x = (x1, . . . ,

xn) of Ef , we have
Ĉ(x) = (f1x1) ∗ · · · ∗ (fnxn)

with for all 1 6 i 6 n,{
fi = f and xi ∈ P`iA,
fi = Pψ and xi = Pg(δziφiµ`i) with zi ∈ D

n\Sn−1,

and there are no two consecutive terms of the first form (i.e., fi = fi+1 = f
for some i). It means that it is the finite Moore composition of Ĉ(x) of
Theorem 5.20.



214 p. gaucher

• Injectivity of Ĉ. Let x and y be two elements of lim−→E
f such that Ĉ(x) =

Ĉ(y). We can suppose that both x and y are simplified. Let x = (x1, . . . ,xm)
and y = (y1, . . . ,yn). Then

(f1x1) ∗ · · · ∗ (fmxm) = (g1y1) ∗ · · · ∗ (gnyn).

Since both members of the equality are the finite Moore composition of
Theorem 5.20, we deduce that m = n and that for all 1 6 i 6 m, we
have fixi = giyi. For a given i ∈ [1,m], there are two mutually exclusive
possibilities:

1. fi = gi = f and xi and yi are two paths of length `i of A. Since f is
one-to-one because Sn−1 is a subset of Dn, we deduce that xi = yi.

2. fi = gi = Pψ, xi = gδziφiµ`i and yi = gδtiψiµ`i, with zi, ti ∈ D
n\Sn−1 and

φi,ψi ∈ G(1, 1). We also have Pψ(xi) = ĝδziφiµ`i and Pψ(yi) = ĝδtiψiµ`i.
The restriction of ĝ to GlobG(Dn)\GlobG(Sn−1) being one-to-one, we de-
duce that zi = ti, φi = ψi and therefore once again that xi = yi.

We conclude that x = y and that the map Ĉ : lim−→E
f → PGX is one-to-one.

• Informal summary . The arrows of the small category Pg(0),g(1)(A0) and
the relations satisfied by them prove that each element of the colimit lim−→E

f has
a simplified rewriting and this simplified rewriting coincides with the normal
form of Theorem 5.20. The latter theorem relies on the fact that all execution
paths of globes are one-to-one, and more generally that all execution paths of
cellular multipointed d-spaces are locally injective.
• Injectivity of P1ψ. At this point of the proof, we have a composite con-

tinuous map

lim−→E
f

continuous bijection

Ĉ

77

lim−→k // // lim−→D
f (1)

P1ψ
// // PGX.

Let a,b ∈ lim−→D
f (1) such that P1ψ(a) = P1ψ(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 continuous map P1ψ : lim−→D
f (1) → PGX is one-to-one.

Theorem 7.3. Under the hypotheses and the notations of this section.
The map of G-spaces

Pψ : lim−→D
f −→ PMG(X)

is an isomorphism of G-spaces.



homotopy theory of moore flows (II) 215

Proof. We already know by Theorem 7.2 that the map of G-spaces

Pψ : lim−→D
f −→ PMG(X)

is an objectwise continuous bijection. We want to prove that it is an objectwise
homeomorphism. Since all objects of the reparametrization category G are
isomorphic, it suffices to prove that

P1ψ : lim−→D
f (1) −→ PGX

is a homeomorphism. Consider a set map ξ : [0, 1] → lim−→D
f (1) such that the

composite map

ξ : [0, 1]
ξ
−−→ lim−→D

f (1)
P1ψ
−−−−−→ PGX

is continuous. By Corollary 2.3, it suffices to prove that the set map

ξ : [0, 1] −→ lim−→D
f (1)

is continuous as well.

• First reduction. The composite continuous map ξ gives rise by adjunc-
tion to a continuous map

ξ̂ : [0, 1] × [0, 1] −→|X|.

Since [0, 1] × [0, 1] is compact and since |X| is weakly Hausdorff by Proposi-
tion 5.2, the subset ξ̂

(
[0, 1] × [0, 1]

)
is a compact subset of |X|. By Propo-

sition 5.5, ξ̂
(
[0, 1] × [0, 1]

)
intersects a finite number of globular cells of the

cellular multipointed d-space X. Therefore we can suppose that the multi-
pointed d-space X is finite by Proposition 5.4. Write

{cj : j ∈ J}

for its finite set of globular cells.

• Second reduction. It suffices to prove that there exists a finite covering
{F1, . . . ,Fn} of [0, 1] by closed subsets such that each restriction ξ�Fi factors
through the colimit lim−→D

f (1). Let T be the set defined as follows:

T =
{

Carrier
(
ξ(u)

)
: u ∈ [0, 1]

}
.

Suppose that T is infinite. Since J is finite, there exist j0 ∈ J and a se-
quence

(
ξ(un)

)
n>0

of execution paths of X such that the numbers in of times



216 p. gaucher

that cj0 appears in the carrier of ξ(un) for n > 0 give rise to a strictly in-
creasing sequence of integers (in)n>0. Since [0, 1] is sequentially compact, the
sequence (un)n>0 has a convergent subsequence. By continuity, the sequence(
ξ(un)

)
n>0

has therefore a convergent subsequence in PGX. This contradicts
Theorem 5.19. Consequently, the set T is finite. For each carrier c ∈T , let

Uc =
{
u ∈ [0, 1] : Carrier(ξ(u)) = c

}
.

Consider the closure Ûc of Uc in [0, 1]. We obtain the finite covering of [0, 1]
by closed subsets

[0, 1] =
⋃

c∈T
Ûc.

We replace [0, 1] by Ûc which is compact, metrizable and therefore sequential.
The carrier

c = [cj1, . . . ,cjm]

is fixed until the very end of the proof.

• Third reduction. The attaching maps

ĝjk : Glob
G(Dnjk ) −→ X

for 1 6 k 6 m of the cells cj1, . . . ,cjm yield a map of multipointed d-spaces

ĝc : Glob
G(Dnj1 ) ∗ · · · ∗ GlobG(Dnjm ) −→ X.

Let αi−1 (αi resp.) be the initial state (the final state resp.) of Glob
G(Dnji )

for 1 6 i 6 m in GlobG(Dnj1 ) ∗ · · · ∗ GlobG(Dnjm ). We obtain a map of
G-spaces

FG
op

1 (D
nj1 ) ⊗···⊗FG

op

1 (D
njm ) −→Df (m)

for some m belonging to Pg(0),g(1)(A0) such that

Df (m) = Zĝc(α0),ĝc(α1) ⊗···⊗Zĝc(αm−1),ĝc(αm).

Using Proposition 6.1, we obtain a continuous map

yc : G(1,m) × Dnj1 ×···× Dnjm −→ Zc ⊂Df (m)(1)

where Zc is, by definition, the image of yc. At this point, we have obtained
that the continuous map

ξ�Uc : Uc −→ P
GX



homotopy theory of moore flows (II) 217

factors as a composite of maps

ξ�Uc : Uc −→ Zc ⊂D
f (m)(1)

pm−−−→ lim−→D
f (1) −→ PGX.

Consider a sequence (un)n>0 of Uc converging to u∞ ∈ Ûc. Then for each
n > 0, ξ(un) belongs to the image of PGĝc which is a closed subset of the
sequential space PGX by Theorem 6.11. Thus ξ(u∞) belongs to the image of
PGĝc as well. Since

PGĝc = P1ψ.pm.yc,

we have obtained that the continuous map

ξ�
Ûc

: Ûc −→ PGX

factors as a composite of maps

ξ�
Ûc

: Ûc −→ Zc ⊂Df (m)(1)
pm−−−→ lim−→D

f (1)
P1ψ
−−−−→ PGX.

They are all of them continuous except maybe the left-hand one from Ûc to

Zc (cf. the remark after this proof). Since Ûc is sequential, it remains to prove
that the map

ξ�
Ûc

: Ûc −→ Zc ⊂Df (m)(1)
pm−−−→ lim−→D

f (1)

is sequentially continuous to complete the proof.

• Sequential continuity . Consider a sequence (un)n>0 of Ûc which con-
verges to u∞ ∈ Ûc. Write

ξ(un) = pm
(
yc(φn,z

1
n, . . . ,z

m
n )
)

for all n > 0. We obtain

ξ(un) =
(
PGĝc

)(
φn,z

1
n, . . . ,z

m
n

)
for all n > 0. By Theorem 6.11, the sequence

(
(φn,z

1
n, . . . ,z

m
n )
)
n>0

has a limit

point (φ∞,z
1
∞, . . . ,z

m
∞). We deduce that the sequence (ξ(un))n>0 has a limit

point because both yc and pm are continuous. It is necessarily equal to ξ(u∞)
because the map

P1ψ : lim−→D
f (1) −→ PGX



218 p. gaucher

is continuous bijective by Theorem 7.2 and because

ξ = P1ψ.ξ.

The same argument shows that every subsequence of
(
ξ(un)

)
n>0

has a limit

point which is necessarily ξ(u∞). Suppose that the sequence
(
ξ(un)

)
n>0

does

not converge to ξ(u∞). Then there exists an open neighborhood V of ξ(u∞)
such that for all n > 0, ξ(un) /∈ V . Since V c is closed in lim−→D

f (1), it means

that ξ(u∞) cannot be a limit point of the sequence
(
ξ(un)

)
n>0

. Contradiction.

It implies that the sequence
(
ξ(un)

)
n>0

converges to ξ(u∞).

Before expounding the consequences of Theorem 7.2 and of Theorem 7.3,
let us add an additional remark about the proof of Theorem 7.3. It can be
proved that the inverse image p−1m (γ) for each γ ∈ lim−→D

f (1) is always finite.

When the multipointed d-space X does not contain any loop, i.e., when PGα,αX
is empty for all α ∈ X0, the map pm is even one-to-one and it is then possible
to prove that the set map Ûc → Zc is always continuous. On the contrary,
when X contains loops, the set map Ûc → Zc is not necessarily continuous
mainly because an inverse image p−1m (γ) may contain several points.

Corollary 7.4. Suppose that A is a cellular multipointed d-space. Con-
sider a pushout diagram of multipointed d-spaces

GlobG(Sn−1)

��

// A

��

GlobG(Dn) // X

with n > 0. Then there is the pushout diagram of Moore flows

MG(GlobG(Sn−1)) = Glob(FG
op

1 S
n−1)

��

// MG(A)

��

MG(GlobG(Dn)) = Glob(FG
op

1 D
n) // MG(X).

Corollary 7.5. Let X be a q-cofibrant multipointed d-space. Then
MG(X) is a q-cofibrant Moore flow.

Proof. For every q-cofibrant Moore flow X, the canonical map ∅ → X
is a retract of a transfinite composition of the q-cofibrations C : ∅ → {0},



homotopy theory of moore flows (II) 219

R : {0, 1} → {0} and of the q-cofibrations Glob(FG
op

` S
n−1) ⊂ Glob(FG

op

` D
n)

for ` > 0 and n > 0. The cofibration R : {0, 1} → {0} is not necessary
to reach all q-cofibrant objects. Therefore, this theorem is a consequence of
Theorem 5.7 and of Corollary 7.4.

Theorem 7.6. Let X be a q-cofibrant Moore flow. Then the unit of the
adjunction X → MG

(
MG! (X)

)
is an isomorphism.

Proof. By Proposition 4.15, the theorem holds when X is a globe. It also
clearly holds for X = {0}. For every q-cofibrant Moore flow X, the canonical
map ∅ → X is a retract of a transfinite composition of the q-cofibrations
C : ∅ → {0}, R : {0, 1} → {0} and of the q-cofibrations Glob(FG

op

` S
n−1) ⊂

Glob(FG
op

` D
n) for ` > 0 and n > 0. The cofibration R : {0, 1} → {0} is not

necessary to reach all q-cofibrant objects. Therefore, this theorem is also a
consequence of Theorem 5.7 and of Corollary 7.4.

Corollary 7.7. Let X be a q-cofibrant Moore flow. Then there is the
homeomorphism

P1X ∼= PG(MG! (X)).

Proof. Apply the functor P1(−) to the isomorphism X ∼= MG(MG! (X)).

Theorem 7.8. Let λ be a limit ordinal. Let

X : λ −→GdTop

be a colimit preserving functor such that:

• the multipointed d-space X is a set, in other terms X = (X0,X0,∅);
• for all ν < λ, there is a pushout diagram of multipointed d-spaces

GlobG(Snν−1)

��

gν
// Xν

��

GlobG(Dnν )
ĝν

// Xν+1

with nν > 0.

Let Xλ = lim−→ν<λ Xν. For all ν 6 λ, the counit map

κν : MG! (M
G(Xν)) −→ Xν

is an isomorphism.



220 p. gaucher

Proof. The map κ0 is an isomorphism because X0 is a set. By Theorem 5.7,
and since MG! is a left adjoint, it suffices to prove that if κν is an isomorphism,
then κν+1 is an isomorphism. Assume that κν is an isomorphism. By Corol-
lary 7.4, there is the pushout diagram of Moore flows

MG
(
GlobG(Snν−1)

)
= Glob

(
FG

op

1 S
nν−1

)
��

gν
// MG(Xν)

��

MG
(
GlobG(Dnν )

)
= Glob

(
FG

op

1 D
nν
) ĝν

// MG(Xν+1).

Apply again the left adjoint MG! to this diagram, we obtain by using the
induction hypothesis that κν+1 is an isomorphism.

Corollary 7.9. For every q-cofibrant multipointed d-space X, the counit
of the adjunction MG! (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 a cellular multipointed d-space (note that the cofibration
R : {0, 1} → {0} is not required to reach all cellular multipointed d-spaces)
and that a retract of an isomorphism is an isomorphism.

8. From multipointed d-spaces to flows

The goals of this final section are to complete the proof of the Quillen
equivalence between multipointed d-spaces and Moore flows in Theorem 8.1,
which together with the results of [14] establish that multipointed d-spaces
and flows have Quillen equivalent q-model structures, and to give a new and
conceptual proof of [11, Theorem 7.5] in Theorem 8.14 independent of [8].
We also give a new presentation of the underlying homotopy type of flow in
Proposition 8.16.

Theorem 8.1. The adjunction MG! a M
G : GFlow � GdTop induces a

Quillen equivalence between the q-model structure of Moore flows and the
q-model structure of multipointed d-spaces.

Proof. Since the q-fibrations of Moore flows are the maps of Moore flows
inducing an objectwise q-fibration on the G-spaces of execution paths, the
functor MG takes q-fibrations of multipointed d-spaces to q-fibrations of Moore



homotopy theory of moore flows (II) 221

flows. Since MG preserves the set of states and since trivial q-fibrations of
Moore flows are maps inducing a bijection on states and an an objectwise
trivial q-fibration on the G-spaces of execution paths, the functor MG takes
trivial q-fibrations of multipointed d-spaces to trivial q-fibrations of Moore
flows. Therefore, the functor MG : GdTop →GFlow is a right Quillen adjoint.

By Theorem 7.6, the map X → MG(MG! (X)) is a weak equivalence of
Moore flows for every q-cofibrant Moore flow X. Let Y be a (q-fibrant)
multipointed d-space. Then the composite map of multipointed d-spaces

MG! (M
G(Y cof ))

∼=−−→ Y cof '−−→ Y

where Y cof is a q-cofibrant replacement of Y , is a weak equivalence of multi-
pointed d-spaces because: 1) the left-hand map is an isomorphism by Corol-
lary 7.9; 2) the right-hand map is a weak equivalence by definition of a cofi-
brant replacement.

Let us give now some reminders about flows and the categorization functor
cat from multipointed d-spaces to flows.

Definition 8.2. ([7, Definition 4.11]) A flow is a small semicategory en-
riched over the closed monoidal category (Top,×). The corresponding cate-
gory is denoted by Flow.

Let us expand the definition above. A flow X consists of a topological
space PX of execution paths, a discrete space X0 of states, two continuous
maps s and t from PX to X0 called the source and target map respectively,
and a continuous and associative map

∗ : {(x,y) ∈ PX ×PX : t(x) = s(y)}−→ PX

such that s(x∗y) = s(x) and t(x∗y) = t(y). A morphism of flows f : X → Y
consists of a set map f0 : X0 → Y 0 together with a continuous map Pf :
PX → PY such that

f0(s(x)) = s(Pf(x)), f0(t(x)) = t(Pf(x)),

Pf(x∗y) = Pf(x) ∗Pf(y).

Let
Pα,βX = {x ∈ PX : s(x) = α and t(x) = β}.

Notation 8.3. The map Pf : PX → PY can be denoted by f : PX → PY



222 p. gaucher

is there is no ambiguity. The set map f0 : X0 → Y 0 can be denoted by
f : X0 → Y 0 is there is no ambiguity.

The category Flow is locally presentable. Every set can be viewed as a
flow with an empty path space. The obvious functor Set ⊂ Flow is limit-
preserving and colimit-preserving. One another example of flow is important
for the sequel:

Example 8.4. For a topological space Z, let Glob(Z) be the flow defined
by

Glob(Z)0 = {0, 1}, PGlob(Z) = P0,1Glob(Z) = Z, s = 0, t = 1.

This flow has no composition law.

Notation 8.5.

C : ∅ →{0}, R : {0, 1}→{0},
−→
I = Glob({0}).

The q-model structure of flows is the unique combinatorial model structure
such that {

Glob
(
Sn−1

)
⊂ Glob

(
Dn
)

: n > 0
}
∪{C,R}

is the set of generating cofibrations and such that{
Glob

(
Dn ×{0}

)
⊂ Glob

(
Dn+1

)
: n > 0

}
is the set of generating trivial cofibrations (e.g., [16, Theorem 7.6]) where the
maps Dn ⊂ Dn+1 are induced by the mappings (x1, . . . ,xn) 7→ (x1, . . . ,xn, 0).
The weak equivalences are the maps of flows f : X → Y inducing a bijection
f0 : X0 ∼= Y 0 and a weak homotopy equivalence Pf : PX → PY and the
fibrations are the maps of flows f : X → Y inducing a q-fibration Pf : PX →
PY of topological spaces.

Let X be a multipointed d-space. Consider for every (α,β) ∈ X0×X0 the
coequalizer of spaces

Pα,βX = lim−→
(
PGα,βX ×G(1, 1) ⇒ P

G
α,βX

)
where the two maps are (c,φ) 7→ c and (c,φ) 7→ c.φ. Let [−]α,β : PGα,βX →
Pα,βX be the canonical map.



homotopy theory of moore flows (II) 223

Theorem 8.6. ([11, Theorem 7.2]) Let X be a multipointed d-space.
Then there exists a flow cat(X) with cat(X)0 = X0, Pα,βcat(X) = Pα,βX
and the composition law ∗ : Pα,βX × Pβ,γX → Pα,γX is for every triple
(α,β,γ) ∈ X0 × X0 × X0 the unique map making the following diagram
commutative:

PGα,βX ×P
G
β,γX

∗N //

[−]α,β×[−]β,γ
��

PGα,γX

[−]α,γ
��

Pα,βX ×Pβ,γX // Pα,γX

where ∗N is the normalized composition (cf. Definition 2.9). The mapping
X 7→ cat(X) induces a functor from GdTop to Flow.

Definition 8.7. The functor cat : GdTop → Flow is called the catego-
rization functor.

The motivation for the constructions of this paper and of [14] comes from
the following theorem which is added for completeness.

Theorem 8.8. The categorization functor cat : GdTop → Flow is nei-
ther a left adjoint nor a right adjoint. In particular, it cannot be a left or a
right Quillen equivalence.

Proof. This functor is not a left adjoint by [11, Proposition 7.3]. Suppose
that it is a right adjoint. Let L : Flow →GdTop be the left adjoint. Pick a
nonempty topological space Z. The adjunction yields the bijection of sets

GdTop
(
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

choice of a map from Z to {0}. 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)) con-

tains at least one execution path φ : [0, 1] →|L(Glob(Z))|. Then f.φ is an exe-
cution path of

−→
I G. Every map g ∈GdTop(

−→
I G,
−→
I G) ∼= {[0, 1] ∼=+ [0, 1]} gives

rise to and execution path g.f.φ of
−→
I G. Since g.f ∈GdTop

(
L(Glob(Z)),

−→
I G
)
,

we deduce that g.f = f. Contradiction. We deduce that the multipointed
d-space L(Glob(Z)) does not contain any execution path. Therefore this
multipointed d-space is of the form (UZ,U

0
Z,∅). We obtain the bijection

MTop
(
(UZ,U

0
Z), ([0, 1],{0, 1})

) ∼= {f}. Suppose that UZ is nonempty. Then
for all g ∈ MTop

(
([0, 1],{0, 1}), ([0, 1],{0, 1})

)
, we have g.f = f. The only



224 p. gaucher

possibilities are that f = 0 or f = 1. Since f is the unique element, we deduce
that UZ = ∅. There are also the natural bijections of sets

GdTop(L({0}),X) ∼= Flow({0},cat(X))
∼= cat(X)0 ∼= X0 ∼= GdTop({0},X).

By the Yoneda lemma, we obtain L({0}) = {0}.
To summarize, if L : Flow →GdTop is a left adjoint to the functor cat :

GdTop → Flow, then one has L({0}) = {0} and for all nonempty topological
spaces Z, there is the equalities L(Glob(Z)) = ∅. By [7, Theorem 6.1], any
flow is a colimit of globes and points. 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:

GdTop(L(Y ),X) ∼= Flow(Y,cat(X)).

Since L(Y ) is a set, we obtain the natural bijection Set(L(Y ),X0) ∼=
Flow(Y,cat(X)). We obtain the natural bijection GdTop(L(Y ),X0) ∼=
Flow(Y,cat(X)) and by adjunction the natural bijection Flow(Y,X0) ∼=
Flow(Y,cat(X)) since cat(X0) = X0. By Yoneda, we conclude that cat(X) =
X0 for all multipointed d-spaces X, which is a contradiction.

Proposition 8.9. ([14, Proposition 5.17]) Let U and U ′ be two topolog-
ical spaces. There is the natural isomorphism of G-spaces

∆GopU ⊗ ∆GopU ′ ∼= ∆Gop(U ×U ′).

Let X be a flow. The Moore flow M(X) is the enriched semicategory
defined as follows:

• the set of states is X0;
• the G-space Pα,βM(X) is the G-space ∆Gop(Pα,βX);
• the composition law is defined, using Proposition 8.9 as the composite map

∆Gop(Pα,βX)⊗∆Gop(Pβ,γX) ∼= ∆Gop(Pα,βX×Pβ,γX)
∆Gop(∗)−−−−−→ ∆Gop(Pα,γ)X.

The construction above yields a well-defined functor

M : Flow →GFlow.



homotopy theory of moore flows (II) 225

Consider a G-flow Y . For all α,β ∈ Y 0, let Yα,β = lim−→Pα,βY . Let (α,β,γ)
be a triple of states of Y . The composition law of the G-flow Y induces a
continuous map

Yα,β ×Yβ,γ ∼= lim−→(Pα,βY ⊗Pβ,γY ) −→ lim−→Pα,γY
∼= Yα,γ

which is associative. We obtain the

Proposition 8.10. ([14, Proposition 10.6 and Proposition 10.7]) For any
G-flow Y , the data

• the set of states is Y 0;
• for all α,β ∈ Y 0, let Yα,β = lim−→Pα,βY ;
• for all α,β,γ ∈ Y 0, the composition law Yα,β ×Yβ,γ → Yα,γ;

assemble to a flow denoted by M!(Y ). It yields a well-defined functor

M! : GFlow → Flow.

There is an adjunction M! a M.

Theorem 8.11. There is the isomorphism of functors

cat ∼= M!.MG.

Proof. First, let us notice that the functors cat : GdTop → Flow (Theo-
rem 8.6), MG : GdTop → GFlow (Theorem 4.12) and M! : GFlow → Flow
(Proposition 8.10) preserve the set of states by definition of these functors.
Therefore, for every multipointed d-space X, the flows cat(X) and M!.MG(X)
have the same set of states X0. Let G1 be the full subcategory of G gener-
ated by 1: the set of objects is the singleton {1} and G1(1, 1) = G(1, 1). For
(α,β) ∈ X0 ×X0, the inclusion functor ι : G1 ⊂G induces a continuous map

lim−→G1
((

Pα,βMGX
)
.ι
)
→ lim−→G Pα,βM

GX.

It turns out that there is the natural homeomorphisms

lim−→G1
((

Pα,βMGX
)
.ι
)
∼= lim−→G1 P

1
α,βM

GX ∼= lim−→G1 P
G
α,βX

∼= Pα,βcat(X),

the first one by definition of ι, the second one by definition of MG and
the last one by definition of cat. We obtain a natural map of flows



226 p. gaucher

cat(X) → (M!.MG)(X) which is bijective on states. Let ` > 0 be an object
of G. Then the comma category (`↓ι) is characterized as follows:

• the set of objects is G(`, 1) which is always nonempty for every ` > 0;

• the set of maps from an arrow u : ` → 1 to an arrow v : ` → 1 is an
element of Mor(G)(u,v).

The comma category (`↓ι) is connected since in any diagram of G of the form

[0,`]
u // [0, 1]

k

��
�
�
�

[0,`]
v // [0, 1],

there exists a map k ∈ G([0, 1], [0, 1]) making the square commute: take
k = v.u−1. By [29, Theorem 1 p. 213], we deduce that the natural map of
flows cat(X) → (M!.MG)(X) induces a homeomorphism between the spaces
of paths.

Corollary 8.12. Suppose that A is a cellular multipointed d-space.
Consider a pushout diagram of multipointed d-spaces

GlobG(Sn−1)

��

// A

��

GlobG(Dn) // X

with n > 0. Then there is the pushout diagram of flows

Glob(Sn−1)

��

// cat(A)

��

Glob(Dn) // cat(X).

Proof. It is a consequence of Corollary 7.4, Theorem 8.11 and of the fact
that M! : GFlow → Flow is a left adjoint.



homotopy theory of moore flows (II) 227

Definition 8.13. We consider the composite functors

(Lcat) : GdTop
(−)cof
−−−−−→GdTop cat−−−−→ Flow

(Lcat)−1 : Flow
M−−−→GFlow

(−)cof
−−−−−→GFlow

MG
!−−−−→GdTop

where (−)cof is a q-cofibrant replacement functor.

We can now write down the new proof of [11, Theorem 7.5].

Theorem 8.14. The categorization functor from multipointed d-spaces
to flows

cat : GdTop −→ Flow

takes q-cofibrant multipointed d-spaces to q-cofibrant flows. Its total left
derived functor in the sense of [4] induces an equivalence of categories between
the homotopy categories of the q-model structures.

Proof. The functor cat ∼= M!.MG takes q-cofibrant multipointed d-spaces
to q-cofibrant flows by Corollary 7.5 and because M! is a left Quillen adjoint.
The rest of the proof is divided in four parts.
• X ' Y ⇒ (Lcat)(X) ' (Lcat)(Y ). Let X ' Y be two weakly equiva-

lent multipointed d-spaces in the q-model structure. Then there is the weak
equivalence Xcof ' Y cof . Since all multipointed d-spaces are q-fibrant, the
right Quillen functor MG takes weak equivalences of multipointed d-spaces to
weak equivalences of Moore flows. Since M! is a left Quillen functor and since
MG preserves q-cofibrancy by Corollary 7.5, we deduce using Theorem 8.11
the sequence of isomorphisms and weak equivalences

(Lcat)(X) ∼= M!MG
(
Xcof

)
' M!MG

(
Y cof

) ∼= (Lcat)(Y ).
•X ' Y ⇒ (Lcat)−1(X) ' (Lcat)−1(Y ). Let X ' Y be two weakly equiv-

alent flows in the q-model structure. Since M is a right Quillen functor and
since all flows are q-fibrant, we obtain the weak equivalence of Moore flows
M(X) ' M(Y ). By definition of (Lcat)−1 and since MG! is a left Quillen
adjoint, we deduce the sequence of isomorphisms and weak equivalences

(Lcat)−1(X) ∼= MG! (M(X))
cof ' MG! (M(Y ))

cof ∼= (Lcat)−1(Y ).

The functors (Lcat) and (Lcat)−1 therefore induce functors between the ho-
motopy categories.



228 p. gaucher

• (Lcat)−1(Lcat)(X) ' X. Let X be a multipointed d-space. Then we
have the sequence of isomorphisms and of weak equivalences

(Lcat)−1(Lcat)(X) ∼= MG!
(
MM!

q-cofibrant
by Corollary 7.5

MG
(
Xcof

) )cof
' MG!

(
MG
(
Xcof

))cof
' MG! M

G(Xcof ) ∼= Xcof ' X,

the first isomorphism by definition of (Lcat) and (Lcat)−1 and by Theo-
rem 8.11, the first weak equivalence since the adjunction M! a M is a Quillen
equivalence by [14, Theorem 10.9] and since MG! is a left Quillen adjoint, the
second weak equivalence by Corollary 7.5 and again since MG! is a left Quillen
adjoint, the second isomorphism by Corollary 7.9, and the last weak equiva-
lence by definition of a q-cofibrant replacement.
• (Lcat)(Lcat)−1(Y ) ' Y . Let Y be a flow. Then we have the sequence

of isomorphisms and of weak equivalences

(Lcat)(Lcat)−1(Y ) ∼=
(
M!MG

)(
MG! (MY )

cof
)cof

'
(
M!MG

)(
MG! (MY )

cof
)
∼= M!(MY )cof ' Y,

the first isomorphism by definition of (Lcat) and (Lcat)−1 and by Theo-
rem 8.11, the first weak equivalence because MG is a right Quillen adjoint,
because MG! (MY )

cof is q-cofibrant, because MG preserves q-cofibrancy by
Corollary 7.5 and finally because M! is a left Quillen adjoint, the second
isomorphism by Theorem 7.6, and finally the last weak equivalence since the
adjunction M! a M is a Quillen equivalence by [14, Theorem 10.9]. The proof
is complete.

The underlying homotopy type of a flow is, morally speaking, the underly-
ing space of states of a flow after removing the execution paths. It is defined
only up to homotopy, not up to homeomorphism. We conclude the section
and the paper by recovering it in a very intuitive way by using (Lcat)−1.

It is proved in [9, Theorem 6.1] that for every cellular flow X, there ex-
ists a cellular multipointed d-space Xtop such that there is an isomorphism
cat(Xtop) ∼= Xcof .

Definition 8.15. ([9, Section 6]) The underlying homotopy type of a
flow X is the topological space ||X|| :=

∣∣Xtop∣∣, where |Xtop| is the underlying



homotopy theory of moore flows (II) 229

topological space of the cellular multipointed d-space Xtop. This yields a well
defined functor

||− || : Flow −→ Ho(Top)

from the category of flows to the homotopy category of the q-model structure
of Top.

Proposition 8.16. For any flow X, there is the homotopy equivalence of
topological spaces

||X|| '
∣∣(Lcat)−1(X)∣∣.

Proof. One has

cat
(
MG!
(
M(X)cof

))
=
(
M!MG

)(
MG!
(
M(X)cof

)) ∼= M!(M(X)cof),
the equality by Theorem 8.11 and the isomorphism by Theorem 7.6. Using
the Quillen equivalence M! a M of [14, Theorem 10.9], we obtain the weak
equivalences of flows

cat
(
MG!
(
M(X)cof

))
' X ' cat

(
Xtop

)
.

Thanks to Theorem 8.14, we obtain the weak equivalence of q-cofibrant multi-
pointed d-spaces MG!

(
M(X)cof

)
' Xtop. We deduce the homotopy equivalence

between the underlying q-cofibrant spaces∣∣MG! (M(X)cof)∣∣ ' ∣∣Xtop∣∣
because the underlying topological space functor |−| is a left Quillen functor
by [11, Proposition 8.1].

The composite functor

GFlow
MG

!−−−−→GdTop
|−|
−−−→ Top

is the composite of two left Quillen functors. Therefore the mapping

X 7−→ |MG! (X
cof )|

induces a functor from GFlow to Ho(Top). For all G-flows X, there is the iso-
morphism Xcof ∼= MG

(
MG!
(
Xcof

))
by Theorem 7.6. Consequently, the functor∣∣MG! ((−)cof)∣∣ can be regarded as the underlying homotopy type functor for

G-flows.



230 p. gaucher

A. The Reedy category Pu,v(S): reminder

Let S be a nonempty set. Let Pu,v(S) be the small category defined by
generators and relations as follows (see [15, Section 3]):

• u,v ∈ S (u and v may be equal).
• The objects are the tuples of the form

m =
(
(u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)

)
with n > 1, u0, . . . ,un ∈ S, �1, . . . ,�n ∈{0, 1} and

for i such that 1 6 i 6 n, �i = 1 ⇒ (ui−1,ui) = (u,v).

• There is an arrow

cn+1 :
(
m, (x, 0,y), (y, 0,z),n

)
−→

(
m, (x, 0,z),n

)
for every tuple m =

(
(u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)

)
with n > 1

and every tuple n =
(
(u′0,�

′
1,u
′
1), (u

′
1,�
′
2,u
′
2), . . . , (u

′
n′−1,�

′
n′,u

′
n′)
)

with n′ >
1. It is called a composition map.

• There is an arrow

In+1 :
(
m, (u, 0,v),n

)
−→

(
m, (u, 1,v),n

)
for every tuple m =

(
(u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)

)
with n > 1

and every tuple n =
(
(u′0,�

′
1,u
′
1), (u

′
1,�
′
2,u
′
2), . . . , (u

′
n′−1,�

′
n′,u

′
n′)
)

with n′ >
1. It is called an inclusion map.

• There are the relations (group A) ci.cj = cj−1.ci if i < j (which means since
ci and cj may correspond to several maps that if ci and cj are composable,
then there exist cj−1 and ci composable satisfying the equality).

• There are the relations (group B) Ii.Ij = Ij.Ii if i 6= j. By definition of
these maps, Ii is never composable with itself.

• There are the relations (group C)

ci.Ij =

{
Ij−1.ci if j > i + 2,

Ij.ci if j 6 i− 1.

By definition of these maps, ci and Ii are never composable as well as ci
and Ii+1.



homotopy theory of moore flows (II) 231

By [15, Proposition 3.7], there exists a structure of Reedy category on
Pu,v(S) with the N-valued degree map defined by

d((u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)) = n +
∑

i
�i.

The maps raising the degree are the inclusion maps. The maps decreasing the
degree are the composition maps.

B. An explicit construction of the left adjoint MG!

The proof of Theorem 4.12 uses a well-known characterization of right ad-
joint functors between locally presentable categories. It is possible to describe
more explicitly the functor MG! : GFlow →GdTop as follows.

Notation B.1. The composite of two natural transformations µ : F ⇒
G and ν : G ⇒ H is denoted by ν � µ to make the distinction with the
composition of maps and of functors.

The category D(GFlow) of all small diagrams of Moore flows over all
small categories is defined as follows. An object is a functor F : I → GFlow
from a small category I to GFlow. A morphism from F : I1 → GFlow to
G : I2 → GFlow is a pair (f : I1 → I2,µ : F ⇒ G.f) where f is a functor
and µ is a natural transformation. If (g,ν) is a map from G : I2 →GFlow to
H : I3 →GFlow, then the composite (g,ν).(f,µ) is defined by (g.f, (ν.f)�µ).
The identity of F : I1 → GFlow is the pair (IdI1, IdF ). If (h,ξ) : (H : I3 →
GFlow) → (I : I4 →GFlow) is another map of D(GFlow), then we have

((h,ξ).(g,ν)) .(f,µ) = (h.g,ξ.g �ν).(f,µ)
= (h.g.f,ξ.g.f �ν.f �µ)
= (h,ξ).(g.f,ν.f �µ) = (h,ξ). ((g,ν).(f,µ)) .

Thus the composition law is associative and the category D(GFlow) is well-
defined. It is well-known that the colimit of small diagrams induces a functor
lim−→ : D(GFlow) →GFlow (see e.g., [15, Proposition A.2]).

Theorem B.2. There exists a functor D : GFlow →D(GFlow) such that
the composite functor

GFlow D−−−→D(GFlow)
lim−→−−−→GFlow



232 p. gaucher

is the identity functor and such that every vertex of D(X) for any Moore flow
X is one of the following kind:

(1) the Moore flow {0},
(2) the globe Glob(D) of some G-space D,
(3) the Moore flow Glob(D)∗Glob(E) for two G-spaces D and E where the

final state of Glob(D) is identified with the initial state of Glob(E) (it
is the “composition” of the two globes, hence the notation).

Moreover, each G-space D and E used by the diagram is of the form Pα,γX
or Pα,βX ⊗Pβ,γX.

Proof. This theorem is proved in [7, Theorem 6.1] for the category of flows
which are semicategories enriched over the closed (semi)monoidal bicomplete
category (Top,×) (see Definition 8.2). The diagram is depicted in Figure 4.
We refer to the proof of [7, Theorem 6.1] for the definitions of the maps.
Since a Moore flow is a semicategory enriched over the closed semimonoidal
bicomplete category ([Gop, Top]0,⊗), the proof is complete.

Glob (Pα,γX)

Glob (Pα,βX ⊗Pβ,γX)

pα,β,γ

OO

rα,β,γ

��

Glob (Pα,βX) ∗ Glob (Pβ,γX)

{α}

i
α,β
1

��

k
α,β,γ
1

44iiiiiiiiiiiiiiiiiii

j
α,β,γ
1

33

i
α,γ
1

44

{β}

k
α,β,γ
2

OO

i
α,β
2ttiii

iiii
iiii

iiii
iiii

i

i
β,γ
1 **

UUUU
UUUU

UUUU
UUUU

UUU {γ}

k
α,β,γ
3

jjUUUUUUUUUUUUUUUUUUUU

i
β,γ
2

��

j
α,β,γ
3

kk

i
α,γ
2

jj

Glob (Pα,βX)

h
α,β,γ
1sssssssssss

99sssssssssss

Glob (Pβ,γX)

h
α,β,γ
3 KKKKKKKKKKK

eeKKKKKKKKKKK

Figure 4: The Moore flow X as a colimit of globes and points (the definition
of the maps are easily understandable, cf. the proof of [7, Theorem 6.1] for
further explanations)



homotopy theory of moore flows (II) 233

Notation B.3. Denote by B(GFlow) 4 the full subcategory of GFlow
generated by {0} and Glob(D) where D runs over the class of all G-spaces.

Theorem B.4. Let K be a bicomplete category. A functor

F : K−→GFlow

has a left adjoint
F! : GFlow −→K

if and only if there exists a functor m : B(GFlow) → K such that there are
the natural bijections

K
(
m({0}),Y

) ∼= F(Y )0,
K
(
m(D),Y

) ∼= GFlow(Glob(D),F(Y ))
for all objects Y of K and all G-spaces D.

Proof. The “only if” direction comes from the fact that there is a natural
bijection

F(X)0 ∼= GFlow({0},F(X)).

Let D and E be two G-spaces. Let m(Glob(D)∗Glob(E)) be the object of K
defined by the pushout diagram of K

m({0})
m(0 7→1)

//

m(07→0)

��

m(D)

��

m(E) // m(Glob(D) ∗ Glob(E)).

By taking the image by the functor K(−,Y ) : K→ Set, we obtain the pullback
diagram of sets

K(m(Glob(D) ∗ Glob(E)),Y ) //

��

K(m(D),Y )

��

K(m(E),Y ) // K(m({0}),Y )

4 B like “brick”: the globes and the point are the elementary bricks to build flows.



234 p. gaucher

for all objects Y of K. We therefore obtain the natural bijection of sets

K(m(Glob(D) ∗ Glob(E)),Y ) ∼= GFlow(Glob(D) ∗ Glob(E),F(Y ))

for all objects Y of K. Let

F!(X) := lim−→m(D(X)).

This defines a functor from GFlow to K. For all objects Y of K, there is the
sequence of natural bijections (note that in the calculation below, the colimits
are taken over a same small category which depends only on X)

K(F!(X),Y ) ∼= K(lim−→m(D(X)),Y )
∼= lim←−K(m(D(X)),Y )
∼= lim←−GFlow(D(X),F(Y ))
∼= GFlow(lim−→D(X),F(Y ))
∼= GFlow(X,F(Y )),

the first one by definition of F!, the second one and the fourth one by the
universal property of the (co)limit, the third one by hypothesis and by the
calculation above, and finally the last one by Theorem B.2.

After Theorem B.4, it suffices now to find a multipointed d-space denoted
by MG! ({0}) such that there is a natural bijection with respect to X

GdTop
(
MG! ({0}),X

) ∼= MG(X)0
and a multipointed d-space denoted by MG! (Glob(D)) natural with respect to
the G-space D such that there is a natural bijection with respect to D and X

GdTop
(
MG! (Glob(D)),X

) ∼= GFlow(Glob(D),MG(X)).
We have the natural bijections

GdTop
(
{0},X

) ∼= X0 ∼= MG(X)0,
and therefore

MG! ({0}) = {0}.



homotopy theory of moore flows (II) 235

We have the sequence of natural bijections

GdTop
(∫ `

GlobG` (D(`)),X

)
∼=
∫
`
GdTop

(
GlobG` (D(`)),X

)
∼=
∫
`

⊔
(α,β)∈X0×X0

Top
(
D(`),P`α,βM

G(X)
)

∼=
⊔

(α,β)∈X0×X0

∫
`
Top

(
D(`),P`α,βM

G(X)
)

∼=
⊔

(α,β)∈X0×X0
TopG

op(
D,Pα,βMG(X)

)
∼=

⊔
(α,β)∈X0×X0

[Gop, Top]0
(
D,Pα,βMG(X)

)
∼= GFlow

(
Glob(D),MG(X)

)
,

the first bijection by definition of a (co)limit, the second bijection by Propo-
sition 4.13, the third bijection because the underlying diagram of this end is
connected, the fourth bijection by [29, page 219 (2)], the fifth bijection since
[Gop, Top]0 is a full subcategory of TopG

op
, and finally the last bijection by

Proposition 4.14. We obtain

MG! (Glob(D)) =
∫ `

GlobG` (D(`)).

C. The setting of k-spaces

In this appendix, the category of k-spaces is denoted by Topk and the
category of ∆-generated spaces by Top∆. The proofs are just sketched.

We must at first prove the existence of the projective q-model structure
of [Gop, Topk]0: [12] is written in the locally presentable setting. We do not
know whether the arguments of [31] are valid here since they are written in the
category of Hausdorff k-spaces. Anyway, it is possible to give a much simpler
argument. The inclusion Top∆ ⊂ Topk has a right adjoint k∆ : Topk →
Top∆, which gives rise to a Quillen equivalence. In fact, the q-model structure
of Topk is right-induced by k∆ : Topk → Top∆ in the sense of [6, 23]. From
the functor k∆, we obtain a right adjoint U : [Gop, Topk]0 → [Gop, Top∆]0.
For an object X of [Gop, Topk]0, let Path(X) = ` 7→ TOPk([0, 1],X(`)) where



236 p. gaucher

TOPk is the internal hom of Topk. Since the composite functor k∆.TOPk
is the internal hom of Top∆, the Quillen path object argument can be used

to obtain that U : [Gop, Topk]0 → [Gop, Top∆]0 right-induces the projective
q-model structure of [Gop, Topk]0. This technique still works in the cofibrantly
generated non-combinatorial setting: it dates back to [32] (see also [24, The-

orem 11.3.2]).

The q-model category of multipointed d-spaces is constructed in [16, The-

orem 6.14] by right-inducing it and by using the Quillen path object argument

again. The q-model category of G-flows is constructed in [14, Theorem 8.8]
by mimicking the method used in [13, Theorem 3.11] which works for any

convenient category of topological spaces. Indeed, it uses Isaev’s work [26]

about model categories of fibrant objects which does not require to work in a

locally presentable setting.

Theorem 2.16 is not valid anymore. See [15, Theorem 5.10] for a treatment

of the similar problem for flows. The reason is that a k-space is not neces-

sarily homeomorphic to the disjoint sum of its path-connected components

(e.g., the Cantor space). It is used in Theorem 4.12 together with the local

presentability of the category of ∆-generated spaces to prove the existence of

the left adjoint MG! : GFlow → GdTop. In the framework of k-spaces, the
existence of MG! can be proved using Appendix B. We have therefore a Quillen
adjunction

MG! a M
G : GFlow(Topk) �GdTop(Topk)

between the q-model structures, where the notation (Topk) is to specify the

category of topological spaces which is used.

A k-space, unlike a ∆-generated space, is not necessarily sequential. It is

not clear how to adapt the proof of Theorem 6.11 by replacing sequences by

nets since there is a Cantor diagonalization argument which does not seem

to be adaptable at least with uncountable nets. It is not clear either how

to modify accordingly the last part of the proof of Theorem 7.3 about the

sequential continuity because Ûc is an arbitrary compact space now, and not

necessarily a closed subset of [0, 1] anymore. To obtain Theorem 1.1 for k-

spaces, another method must be used. For all k-spaces Z, the canonical

map k∆(Z) → Z is a weak homotopy equivalence which induces a bijection
Topk([0, 1],k∆(Z))

∼= Top∆([0, 1],Z) because [0, 1] ∈ Top∆ ⊂ Topk. From
these observations, we obtain two left Quillen equivalences GFlow(Top∆) →
GFlow(Topk) and GdTop(Top∆) → GdTop(Topk). We obtain that is



homotopy theory of moore flows (II) 237

commutative the diagram of left Quillen adjunctions

GFlow(Top∆)
⊂

//

��

GFlow(Topk)

��

GdTop(Top∆)
⊂

// GdTop(Topk)

because the G-flows {0} and Glob(D) of GFlow(Top∆) are taken to the same
multipointed d-space of GdTop(Topk) by Appendix B. Using the two-out-of-
three property, we obtain that the Quillen adjunction

MG! a M
G : GFlow(Topk) �GdTop(Topk)

is a Quillen equivalence. Let X be a q-cofibrant object of GdTop(Topk).
Then X ∈ GdTop(Top∆). By Corollary 7.9, there is the isomorphism
MG! (M

G(X)) ∼= X. Let X be a q-cofibrant object of GFlow(Topk). Then X ∈
GFlow(Top∆). By Theorem 7.6, there is the isomorphism X ∼= MG(M

G
! (X)).

At this point, it is legitimate to ask whether the main results of the com-

panion paper [14] are valid for k-spaces. The answer is that they are. The

main tool of [14] is the Quillen equivalence lim−→ : [G
op, Top∆]0 � Top∆ : ∆Gop

proved in [12, Theorem 7.6]. There is the commutative diagram of left Quillen

adjoints

[Gop, Top∆]0 //

��

Top∆

��

[Gop, Topk]0 // Topk.

All left Quillen adjoints except maybe the bottom horizontal one are left

Quillen equivalences. Therefore the bottom horizontal one is a left Quillen

equivalence as well.

As a conclusion, most of the results, but not all, of this paper and of the

companion paper [14] are still valid for k-spaces. However, there is no known

proofs avoiding to use ∆-generated spaces.

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	Introduction
	Multipointed d-spaces
	Moore composition and -final structure
	From multipointed d-spaces to Moore flows
	Cellular multipointed d-spaces
	Chains of globes
	The unit and the counit of the adjunction on q-cofibrant objects
	From multipointed d-spaces to flows
	The Reedy category Pu,v(S): reminder
	An explicit construction of the left adjoint M!G
	The setting of k-spaces