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Volume 2007, Article ID 87404,20pages doi:10.1155/2007/87404

Research Article

T-Homotopy and Refinement of Observation—Part II:

Adding New T-Homotopy Equivalences

Philippe Gaucher

Received 11 October 2006; Revised 24 January 2007; Accepted 31 March 2007 Recommended by Monica Clapp

This paper is the second part of a series of papers about a new notion ofT-homotopy of flows. It is proved that the old definition of T-homotopy equivalence does not al- low the identification of the directed segment with the 3-dimensional cube. This contra- dicts a paradigm of dihomotopy theory. A new definition ofT-homotopy equivalence is proposed, following the intuition of refinement of observation. And it is proved that up to weakS-homotopy, an oldT-homotopy equivalence is a newT-homotopy equiv- alence. The left properness of the weakS-homotopy model category of flows is also es- tablished in this part. The latter fact is used several times in the next papers of this se- ries.

Copyright © 2007 Philippe Gaucher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Outline of the paper

The first part [1] of this series was an expository paper about the geometric intuition underlying the notion ofT-homotopy. The purpose of this second paper is to prove that the class of oldT-homotopy equivalences introduced in [2,3] is actually not big enough.

Indeed, the only kind of oldT-homotopy equivalence consists of the deformations which locally act like inFigure 1.1. So it becomes impossible with this old definition to identify the directed segment ofFigure 1.1with the full 3-cube ofFigure 1.2by a zig-zag sequence of weakS-homotopy and ofT-homotopy equivalences preserving the initial state and the final state of the 3-cube since every point of the 3-cube is related to three distinct edges.

This contradicts the fact that concurrent execution paths cannot be distinguished by ob- servation. The end of the paper proposes a new definition ofT-homotopy equivalence following the paradigm of invariance by refinement of observation. It will be checked that the preceding drawback is then overcome.

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A 0 0

U¼ U¼¼ 1 1 U

Figure 1.1. The simplest example ofT-homotopy equivalence.

Figure 1.2. The full 3-cube.

This second part gives only a motivation for the new definition ofT-homotopy. Fur- ther developments and applications are given in [4–6]. The left properness of the model category structure of [7] is also established in this paper. The latter result is used several times in the next papers of this series (e.g., [4, Theorem 11.2], [5, Theorem 9.2]).

Section 4collects some facts about globular complexes and their relationship with the category of flows. Indeed, it is not known how to establish the limitations of the old form ofT-homotopy equivalence without using globular complexes together with a com- pactness argument.Section 5recalls the notion of oldT-homotopy equivalence of flows which is a kind of morphism between flows coming from globular complexes (the class of flows cell(Flow)).Section 6presents elementary facts about relativeI+gl-cell complexes which will be used later in the paper.Section 7proves that the model category of flows is left proper. This technical fact is used in the proof of the main theorem of the paper, and it was not established in [7].Section 8proves the first main theorem of the paper.

Theorem 1.1 (Theorem 8.5). Let n3. There does not exist any zig-zag sequence ofS- homotopy equivalences and of oldT-homotopy equivalences between the flow associated with then-cube and the flow associated with the directed segment.

FinallySection 9proposes a new definition ofT-homotopy equivalence and the sec- ond main theorem of the paper is proved.

Theorem 1.2 (Theorem 9.3). EveryT-homotopy in the old sense is the composite of anS- homotopy equivalence with aT-homotopy equivalence in the new sense. (Since aT-homotopy in the old sense is aT-homotopy in the new sense only up toS-homotopy, the terminology

“generalizedT-homotopy” used inSection 9may not be the best one. However, this termi- nology is used in the other papers of this series, so it is kept to avoid any confusion.)

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2. Prerequisites and notations

The initial object (resp., the terminal object) of a categoryᏯ, if it exists, is denoted by∅ (resp., 1).

LetᏯbe a cocomplete category. IfKis a set of morphisms ofᏯ, then the class of mor- phisms ofᏯthat satisfy the RLP (right lifting property) with respect to any morphism of Kis denoted by inj(K) and the class of morphisms ofᏯthat are transfinite compositions of pushouts of elements ofK is denoted by cell(K). Denote by cof(K) the class of mor- phisms ofᏯthat satisfy the LLP (left lifting property) with respect to the morphisms of inj(K). It is a purely categorical fact that cell(K)cof(K). Moreover, every morphism of cof(K) is a retract of a morphism of cell(K) as soon as the domains ofKare small rela- tive to cell(K) (see [8, Corollary 2.1.15]). An element of cell(K) is called a relativeK-cell complex. IfXis an object ofᏯ, and if the canonical morphism∅Xis a relativeK-cell complex, then the objectXis called aK-cell complex.

Let Ꮿbe a cocomplete category with a distinguished set of morphismsI. Then let cell(Ꮿ,I) be the full subcategory ofᏯconsisting of the objectXofᏯsuch that the canon- ical morphism∅Xis an object of cell(I). In other terms, cell(Ꮿ,I)=(∅Ꮿ)cell(I).

It is obviously impossible to read this paper without a strong familiarity with model categories. Possible references for model categories are [8–10]. The original reference is [11] but Quillen’s axiomatization is not used in this paper. The axiomatization from Hovey’s book is preferred. Ifᏹis a cofibrantly generated model category with set of gen- erating cofibrationsI, let cell(ᏹ) :=cell(ᏹ,I): this is the full subcategory of cell com- plexes of the model categoryᏹ. A cofibrantly generated model structureᏹcomes with a cofibrant replacement functorQ:ᏹcell(ᏹ). In all usual model categories which are cellular (see [9, Definition 12.1.1]), all the cofibrations are monomorphisms. Then for every monomorphism f of such a model categoryᏹ, the morphismQ(f) is a cofibra- tion, and even is an inclusion of subcomplexes (see [9, Definition 10.6.7]) because the cofibrant replacement functorQis obtained by the small object argument, starting from the identity of the initial object. This is still true in the model category of flows remem- bered inSection 3 since the class of cofibrations which are monomorphisms is closed under pushout and transfinite composition.

A partially ordered set (P,) (or poset) is a set equipped with a reflexive antisymmetric and transitive binary relation. A poset is locally finite if for any (x,y)P×P, the set [x,y]= {zP,xzy} is finite. A poset (P,) is bounded if there exist 0P and 1Psuch thatP=[0,1] and such that0=1. For a bounded posetP, let0=minP(the bottom element) and1=maxP(the top element). In a posetP, the interval ]α,] (the subposet of elements ofPstrictly bigger thanα) can also be denoted byP.

A posetP, and in particular an ordinal, can be viewed as a small category denoted in the same way: the objects are the elements ofPand there exists a morphism fromxtoyif and only ifxy. Ifλis an ordinal, aλ-sequence in a cocomplete categoryᏯis a colimit- preserving functorXfromλtoᏯ. We denote byXλthe colimit lim−→Xand the morphism X0Xλis called the transfinite composition of the morphismsXμXμ+1.

A model category is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. The model categories Top and Flow (see below) are both left proper (cf.Theorem 7.4for Flow).

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In this paper, the notation means cofibration, the notation means fibra- tion, the notationmeans weak equivalence, and the notation=means isomorphism.

3. Reminder about the category of flows

The category Top of compactly generated topological spaces (i.e., of weak Hausdorffk- spaces) is complete, cocomplete, and cartesian closed (more details for this kind of topo- logical spaces are in [12,13], the appendix of [14] and also the preliminaries of [7]). For the sequel, any topological space will be supposed to be compactly generated. A compact space is always Hausdorff.

The category Top is equipped with the unique model structure having the weak homo- topy equivalences as weak equivalences and having the Serre fibrations (i.e., a continuous map having the RLP with respect to the inclusion Dn×0Dn×[0, 1] for anyn0 where Dnis then-dimensional disk) as fibrations.

The time flow of a higher-dimensional automaton is encoded in an object called a flow [7]. A flowXcontains a setX0 called the 0-skeleton whose elements correspond to the states (or constant execution paths) of the higher-dimensional automaton. For each pair of states (α,β)X0×X0, there is a topological spacePα,βXwhose elements correspond to the (nonconstant) execution paths of the higher-dimensional automaton beginning atα and ending atβ. ForxPα,βX, letα=s(x) andβ=t(x). For each triple (α,β,γ)X0× X0×X0, there exists a continuous map:Pα,βX×Pβ,γXPα,γXcalled the composition law which is supposed to be associative in an obvious sense. The topological spacePX= (α,β)X0×X0Pα,βXis called the path space ofX. The category of flows is denoted by Flow.

A pointαofX0such that there are no nonconstant execution paths ending atα(resp., starting fromα) is called an initial state (resp., a final state). A morphism of flows f fromX toY consists of a set map f0:X0Y0 and a continuous mapPf :PXPY preserving the structure. A flow is therefore “almost” a small category enriched in Top. A flowXis loopless if for everyαX0, the spacePα,αXis empty.

Here are four fundamental examples of flows.

(1) LetSbe a set. The flow associated withS, still denoted byS, hasSas set of states and the empty space as path space. This construction induces a functor Set Flow from the category of sets to that of flows. The flow associated with a set is loopless.

(2) Let (P,) be a poset. The flow associated with (P,), and still denoted byP, is defined as follows: the set of states ofP is the underlying set ofP; the space of morphisms from αtoβis empty ifαβand equals{(α,β)}ifα < β and the composition law is defined by (α,β)(β,γ)=(α,γ). This construction induces a functor PoSetFlow from the category of posets together with the strictly increasing maps to the category of flows. The flow associated with a poset is loopless.

(3) The flow Glob(Z) defined by

Glob(Z)0= {0,1},

PGlob(Z)=Z withs(z)=0,t(z)=1,zZ, (3.1) and a trivial composition law (cf.Figure 3.1), is called the globe ofZ.

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X

Time

Figure 3.1. Symbolic representation of Glob(Z) for some topological spaceZ.

(4) The directed segmentIis by definition Glob({0})= {0<1}.

The category Flow is equipped with the unique model structure such that [7]

(a) the weak equivalences are the weak S-homotopy equivalences, that is, the mor- phisms of flows f :XY such that f0:X0Y0 is a bijection and such that Pf :PXPY is a weak homotopy equivalence;

(b) the fibrations are the morphisms of flows f :XY such thatPf :PXPY is a Serre fibration.

This model structure is cofibrantly generated. The set of generating cofibrations is the set I+gl=Igl∪ {R:{0, 1} → {0},C:∅→ {0}}with

Igl=

GlobSn1GlobDn,n0, (3.2) where Dnis then-dimensional disk and Sn1is the (n1)-dimensional sphere. The set of generating trivial cofibrations is

Jgl=

GlobDn× {0}

GlobDn×[0, 1],n0. (3.3)

IfXis an object of cell(Flow), then a presentation of the morphismXas a transfi- nite composition of pushouts of morphisms ofI+glis called a globular decomposition ofX.

4. Globular complex

The reference is [3]. A globular complex is a topological space together with a structure describing the sequential process of attaching globular cells. A general globular com- plex may require an arbitrary long transfinite construction. We restrict our attention in this paper to globular complexes whose globular cells are morphisms of the form Globtop(Sn1)Globtop(Dn) (cf.Definition 4.2).

Definition 4.1. A multipointed topological space (X,X0) is a pair of topological spaces such thatX0is a discrete subspace ofX. A morphism of multipointed topological spaces f : (X,X0)(Y,Y0) is a continuous map f :XYsuch that f(X0)Y0. The corresponding

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category is denoted by Topm. The setX0is called the 0-skeleton of (X,X0). The spaceX is called the underlying topological space of (X,X0).

The category of multipointed spaces is cocomplete.

Definition 4.2. Let Z be a topological space. The globe of Z, which is denoted by Globtop(Z), is the multipointed space

Globtop(Z),{0,1}

, (4.1)

where the topological space|Globtop(Z)|is the quotient of{0,1} (Z×[0, 1]) by the rela- tions (z, 0)= (z, 0)= 0 and (z, 1) =(z, 1) =1 for any z,z Z. In particular, Globtop(∅) is the multipointed space ({0,1},{0,1}).

Notation 4.3. IfZis a singleton, then the globe ofZis denoted byItop.

Definition 4.4. LetIgl,top:= {Globtop(Sn1)Globtop(Dn),n0}. A relative globular pre- complex is a relativeIgl,top-cell complex in the category of multipointed topological spaces.

Definition 4.5. A globular precomplex is aλ-sequence of multipointed topological spaces X:λTopmsuch thatXis a relative globular precomplex and such thatX0=(X0,X0) withX0 a discrete space. Thisλ-sequence is characterized by a presentation ordinalλ, and for anyβ < λ, an integernβ0 and an attaching mapφβ: Globtop(Snβ1)Xβ. The family (nββ)β<λis called the globular decomposition ofX.

LetXbe a globular precomplex. The 0-skeleton of lim−→Xis equal toX0.

Definition 4.6. A morphism of globular precomplexes f :XY is a morphism of mul- tipointed spaces still denoted by f from lim−→Xto lim−→Y.

Notation 4.7. IfXis a globular precomplex, then the underlying topological space of the multipointed space lim−→Xis denoted by|X|and the 0-skeleton of the multipointed space lim−→Xis denoted byX0.

Definition 4.8. LetX be a globular precomplex. The space|X|is called the underlying topological space ofX. The setX0is called the 0-skeleton ofX.

Definition 4.9. LetXbe a globular precomplex. A morphism of globular precomplexes γ:ItopX is a nonconstant execution path ofX if there existst0=0< t1<···< tn=1 such that

(1)γ(ti)X0for any 0in,

(2)γ(]ti,ti+1[)Globtop(Dnβi\Snβi1) for some (nβiβi) of the globular decomposi- tion ofX,

(3) for 0i < n, there existsziγDnβi\Snβi1 and a strictly increasing continuous mapψγi: [ti,ti+1][0, 1] such thatψγi(ti)=0 andψγi(ti+1)=1, and for anyt [ti,ti+1],γ(t)=(ziγγi(t)).

In particular, the restrictionγ]ti,ti+1[ofγto ]ti,ti+1[ is one-to-one. The set of nonconstant execution paths ofXis denoted byPtop(X).

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Definition 4.10. A morphism of globular precomplexes f :XYis nondecreasing if the canonical set map Top([0, 1],|X|)Top([0, 1],|Y|) induced by composition byf yields a set mapPtop(X)Ptop(Y). In other terms, one has the commutative diagram of sets:

Ptop(X)

Ptop(Y)

Top[0, 1],|X|

Top[0, 1],|Y|

(4.2)

Definition 4.11. A globular complex (resp., a relative globular complex)Xis a globular pre- complex (resp., a relative globular precomplex) such that the attaching mapsφβare non- decreasing. A morphism of globular complexes is a morphism of globular precomplexes which is nondecreasing. The category of globular complexes, together with the mor- phisms of globular complexes as defined above, is denoted by glTop. The set glTop(X,Y) of morphisms of globular complexes fromXtoYequipped with the Kelleyfication of the compact-open topology is denoted by glTOP(X,Y).

Definition 4.12. LetX be a globular complex. A pointα ofX0 such that there are no nonconstant execution paths ending toα (resp., starting fromα) is called initial state (resp., final state). More generally, a point ofX0will be sometime called a state as well.

Theorem 4.13 (see [3, Theorem III.3.1]). There exists a unique functor cat : glTopFlow such that

(1) if X=X0 is a discrete globular complex, then cat(X) is the achronal flow X0 (“achronal” meaning with an empty path space),

(2) ifZ=Sn1orZ=Dnfor some integern0, then cat(Globtop(Z))=Glob(Z), (3) for any globular complexXwith globular decomposition (nββ)β<λ, for any limit

ordinalβλ, the canonical morphism of flows lim−→α<βcatXα

−→catXβ

(4.3) is an isomorphism of flows,

(4) for any globular complexXwith globular decomposition (nββ)β<λ, for anyβ < λ, one has the pushout of flows

GlobSnβ1 cat(φβ) catXβ

GlobDnβ catXβ+1

(4.4)

The following theorem is important for the sequel.

Theorem 4.14. The functor cat induces a functor still denoted by cat from glTop to cell(Flow)Flow since its image is contained in cell(Flow). For any flowXof cell(Flow),

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there exists a globular complexY such that cat(U)=X, which is constructed by using the globular decomposition ofX.

Proof. The construction ofUis made in the proof of [3, Theorem V.4.1].

5.T-homotopy equivalence

The old notion ofT-homotopy equivalence for globular complexes was given in [2]. A notion ofT-homotopy equivalence of flows was given in [3] and it was proved in the same paper that these two notions are equivalent.

We first recall the definition of the branching and merging space functors, and then the definition of aT-homotopy equivalence of flows, exactly as given in [3] (seeDefinition 5.7), and finally a characterization ofT-homotopy of flows using globular complexes (see Theorem 5.8).

Roughly speaking, the branching space of a flow is the space of germs of nonconstant execution paths beginning in the same way.

Proposition 5.1 (see [15, Proposition 3.1]). LetX be a flow. There exists a topological spacePXunique up to homeomorphism and a continuous maph:PXPXsatisfying the following universal property.

(1) For anyxandyinPXsuch thatt(x)=s(y), the equalityh(x)=h(xy) holds.

(2) Letφ:PXY be a continuous map such that for anyxand yofPX such that t(x)=s(y), the equalityφ(x)=φ(xy) holds. Then there exists a unique continu- ous mapφ:PXYsuch thatφ=φh.

Moreover, one has the homeomorphism PX=

αX0

PαX, (5.1)

wherePαX:=hβX0Pα,βX. The mappingXPXyields a functorPfrom Flow to Top.

Definition 5.2. LetX be a flow. The topological spacePXis called the branching space of the flowX. The functorPis called the branching space functor.

Proposition 5.3 (see [15, Proposition A.1]). LetX be a flow. There exists a topological spaceP+Xunique up to homeomorphism and a continuous maph+:PXP+Xsatisfying the following universal property.

(1) For anyxandyinPXsuch thatt(x)=s(y), the equalityh+(y)=h+(xy) holds.

(2) Letφ:PXY be a continuous map such that for anyxand yofPX such that t(x)=s(y), the equalityφ(y)=φ(xy) holds. Then there exists a unique continu- ous mapφ:P+XYsuch thatφ=φh+.

Moreover, one has the homeomorphism P+X=

αX0

P+αX, (5.2)

whereP+αX:=h+βX0P+α,βX. The mappingXP+Xyields a functorP+from Flow to Top.

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Roughly speaking, the merging space of a flow is the space of germs of nonconstant execution paths ending in the same way.

Definition 5.4. LetXbe a flow. The topological spaceP+Xis called the merging space of the flowX. The functorP+is called the merging space functor.

Definition 5.5 [3]. LetX be a flow. LetAandBbe two subsets ofX0. One says thatA is surrounded byB(inX) if for anyαA, eitherαBor there exist execution pathsγ1

andγ2ofPXsuch thats(γ1)B,t(γ1)=s(γ2)=αandt(γ2)B. Denote this situation byAB.

Definition 5.6 [3]. LetXbe a flow. LetAbe a subset ofX0. Then the restrictionXAof XoverAis the unique flow such that (XA)0=A, such thatPα,β(XA)=Pα,βXfor any (α,β)A×A, and such that the inclusionsAX0 andP(XA)PX induce a mor- phism of flowsXAX.

Definition 5.7 [3]. LetXandY be two objects of cell(Flow). A morphism of flows f : XY is aT-homotopy equivalence if and only if the following conditions are satisfied.

(1) The morphism of flows f :XYf(X0)is an isomorphism of flows. In particu- lar, the set map f0:X0Y0is one-to-one.

(2) ForαY0\f(X0), the topological spacesPαY andP+αY are singletons.

(3)Y0f(X0).

We recall the following important theorem for the sequel.

Theorem 5.8 (see [3, Theorem VI.3.5]). LetXandY be two objects of cell(Flow). LetU andV be two globular complexes with cat(U)=Xand cat(V)=Y(UandV always exist byTheorem 4.14). Then a morphism of flows f :XY is aT-homotopy equivalence if and only if there exists a morphism of globular complexesg:UV such that cat(g)=f and such that the continuous map|g|:|U| → |V|between the underlying topological spaces is a homeomorphism.

This characterization was actually the first definition of a T-homotopy equivalence proposed in [2] (see [2, Definition 4.10, page 66]).

6. Some facts about relativeI+gl-cell complexes

Recall thatI+gl=Igl∪ {R:{0, 1} → {0},C:∅→ {0}}with Igl=

GlobSn1GlobDn,n0. (6.1) LetIg=Igl∪ {C}. Since for anyn0, the inclusion Sn1Dnis a closed inclusion of topological spaces, so an effective monomorphism of the category Top of compactly generated topological spaces, every morphism ofIg, and therefore every morphism of cell(Ig), is an effective monomorphism of flows as well (cf. also [7, Theorem 10.6]).

Proposition 6.1. If f :XY is a relativeI+gl-cell complex and if f induces a one-to-one set map fromX0toY0, then f :XYis a relativeIg-cell subcomplex.

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Proof. A pushout ofRappearing in the presentation of f cannot identify two elements ofX0since, by hypothesis, f0:X0Y0is one-to-one. So either such a pushout is trivial,

or it identifies two elements added by a pushout ofC.

Proposition 6.2. If f :XY is a relativeI+gl-cell complex, then f factors as a composite ghk, wherek:XZ is a morphism of cell({R}), whereh:ZT is a morphism of cell({C}), and whereg:TY is a relativeIgl-cell complex.

Proof. One can use the small object argument with{R}by [7, Proposition 11.8]. There- fore, the morphismf :XYfactors as a compositegh, whereh:XZis a morphism of cell({R}), and where the morphismZY is a morphism of inj({R}). One deduces that the set mapZ0Y0is one-to-one. One has the pushout diagram of flows

X

f

k Z

g

Y Y

(6.2)

Therefore the morphismZYis a relativeI+gl-cell complex.Proposition 6.1implies that the morphism ZY is a relative Ig-cell complex. The morphismZY factors as a compositeh:ZZ(Y0\Z0) and the inclusiong:Z(Y0\Z0)Y. Proposition 6.3. LetX=X0be a set viewed as a flow (i.e., with an empty path space). Let Y be an object of cell(Flow). Then any morphism fromXtoY is a cofibration.

Proof. Let f :XY be a morphism of flows. Then f factors as a compositeX=X0 Y0Y. Any set mapX0Y0is a transfinite composition of pushouts ofCandR. So any set morphismX0Y0is a cofibration of flows. And for any flowY, the canonical morphism of flowsY0Y is a cofibration since it is a relativeIg-cell complex. Hence we

get the result.

7. Left properness of the weakS-homotopy model structure of Flow

Proposition 7.1 (see [7, Proposition 15.1]). Letf :UVbe a continuous map. Consider the pushout diagram of flows:

Glob(U)

Glob(f)

X

g

Glob(V) Y

(7.1)

Then the continuous mapPg:PXPY is a transfinite composition of pushouts of contin- uous maps of the form a finite product Id×··· ×f× ··· ×Id, where the symbol Id denotes identity maps.

Proposition 7.2. Let f :UVbe a Serre cofibration. Then the pushout of a weak homo- topy equivalence along a map of the form a finite product IdX1×··· ×f × ··· ×IdXpwith p0 is still a weak homotopy equivalence.

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If the topological spaces Xi for 1ip are cofibrant, then the continuous map IdX1×··· ×f × ··· × ··· ×IdXp is a cofibration since the model category of compactly generated topological spaces is monoidal with the categorical product as monoidal struc- ture. So in this case, the result follows from the left properness of this model category (see [9, Theorem 13.1.10]). In the general case, IdX1×··· ×f × ··· × ··· ×IdXpis not a cofibration anymore. But any cofibration f for the Quillen model structure of Top is, an cofibration for the Strøm model structure of Top [16–19]. In the latter model structure, any space is cofibrant. Therefore the continuous map IdX1×··· ×f× ··· × ··· ×IdXpis a cofibration of the Strøm model structure of Top, that is a NDR pair. So the continuous map IdX1×··· × f × ··· × ··· ×IdXpis a closedT1-inclusion anyway. This fact will be used below.

Proof. We already know that the pushout of a weak homotopy equivalence along a cofi- bration is a weak homotopy equivalence. The proof of this proposition is actually an adaptation of the proof of the left properness of the model category of compactly gen- erated topological spaces. Any cofibration is a retract of a transfinite composition of pushouts of inclusions of the form Sn1Dnforn0. Since the category of compactly generated topological spaces is cartesian closed, the binary product preserves colimits.

Thus, we are reduced to considering a diagram of topological spaces like X1× ··· ×Sn1× ··· ×Xp U s X

X1× ··· ×Dn× ··· ×Xp U s X

(7.2)

wheresis a weak homotopy equivalence and we have to prove thatsis a weak homotopy equivalence as well. By [11,20], it suffices to prove thatsinduces a bijection between the path-connected components ofUandX, a bijection between the fundamental groupoids π(U) and π(X), and that for any local coefficient system of Abelian groups AofX, one has the isomorphisms:H(X,A) =H(U, sA).

Forn=0, one has Sn1=and Dn= {0}. SoX1× ··· ×Sn1× ··· ×Xp=∅and X1× ··· ×Dn× ··· ×Xp=X1× ··· ×Xp. SoU =U(X1× ··· ×Xp) andX=X (X1× ··· ×Xp). Therefore, the mappingtis the disjoint sumsIdX1×···×Xp. So it is a weak homotopy equivalence.

Letn1. The assertion concerning the path-connected components is clear. Let Tn= {xRn, 0<|x|1}. Consider the diagram of topological spaces:

X1× ··· ×Sn1× ··· ×Xp U s X X1× ··· ×Tn× ··· ×Xp U s X

(7.3)

Since the pair (Tn, Sn1) is a deformation retract, the three pairs (X1× ··· ×Tn× ··· × Xp,X1× ··· ×Sn1× ··· ×Xp), (U,U), and ( X,X ) are deformation retracts as well. So

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the continuous mapsUU andXXare both homotopy equivalences. The Seifert- Van-Kampen theorem for the fundamental groupoid (cf. [20] again) then yields the dia- gram of groupoids:

π(X1× ··· ×Tn× ··· ×Xp) π(U) π(s) π(X)

π(X1× ··· ×Dn×Xp) π(U) π(s) π(X)

(7.4)

Sinceπ(s) is an isomorphism of groupoids, then so isπ(s).

Let Bn= {xRn, 0|x|<1}. Then (Bn,U) is an excisive pair of U and (Bn,X) is an excisive pair ofX. The Mayer-Vietoris long exact sequence then yields the commutative diagram of groups:

··· Hp(X, A) Hp(X,A)HpBn,A

=

HpBn\{0},A

=

···

··· Hp U,sA HpU,sAHpBn,sA HpBn\{0},sA ···

(7.5)

A five-lemma argument completes the proof.

Proposition 7.3. Letλbe an ordinal. LetM:λTop andN:λTop be twoλ-sequences of topological spaces. Lets:MNbe a morphism ofλ-sequences which is also an objectwise weak homotopy equivalence. Finally, suppose that for allμ < λ, the continuous mapsMμ Mμ+1andNμNμ+1are of the form of a finite product IdX1×··· ×f × ··· ×IdXp with p0 and with f a Serre cofibration. Then the continuous map lim−→s: lim−→Mlim−→Nis a weak homotopy equivalence.

If for allμ < λ, the continuous mapsMμMμ+1andNμNμ+1are cofibrations, then Proposition 7.3is a consequence of [9, Proposition 17.9.3] and of the fact that the model category Top is left proper. With the same additional hypotheses,Proposition 7.3is also a consequence of [21, Theorem A.7]. Indeed, the latter states that a homotopy colimit can be calculated either in the usual Quillen model structure of Top, or in the Strøm model structure of Top [18,19].

Proof. The principle of the proof is standard. If the ordinalλis not a limit ordinal, then this is a consequence ofProposition 7.2. Assume now thatλis a limit ordinal. Thenλ 0.

Letu: Snlim−→N be a continuous map. Then ufactors as a composite SnNμ lim−→N since then-dimensional sphere Snis compact and since any compact space is0- small relative to closedT1-inclusions (see [8, Proposition 2.4.2]). By hypothesis, there exists a continuous map SnMμ such that the composite SnMμNμ is homotopic to SnNμ. Hence we have the surjectivity of the set mapπn(lim−→M,)πn(lim−→N,) (whereπndenotes then-th homotopy group) forn0 and for any base point.

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Letu,v: Snlim−→M be two continuous maps such that there exists a homotopyH: Sn×[0, 1]lim−→N between lim−→sf and lim−→sg. Since the space Sn×[0, 1] is com- pact, the homotopyH factors as a composite Sn×[0, 1]Nμ0lim−→Nfor someμ0< λ.

And again since the space Sn is compact, the map f (resp., g) factors as a compos- ite SnMμ1lim−→M (resp., SnMμ2lim−→M) withμ1< λ(resp.,μ2< λ). Thenμ4= max(μ012)< λsinceλis a limit ordinal. And the mapH: Sn×[0, 1]Nμ4is a homo- topy between f : SnMμ4andg: SnMμ4. So the set mapπn(lim−→M,)πn(lim−→N,)

forn0, and for any base pointis one-to-one.

Theorem 7.4. The model category Flow is left proper.

Proof. Consider the pushout diagram of Flow:

U s

i

X

V t Y

(7.6)

whereiis a cofibration of Flow andsa weakS-homotopy equivalence. We have to check thattis a weakS-homotopy equivalence as well. The morphismiis a retract of aI+gl-cell complex j:UW. If one considers the pushout diagram of Flow:

U s

j

X

W u Y

(7.7)

thentmust be a retract ofu. Therefore, it suffices to prove thatuis a weakS-homotopy equivalence. So one can suppose that one has a diagram of flows of the form

A φ

k

U s

i

X

B V t Y

(7.8)

wherekcell(I+gl). ByProposition 6.2, the morphismk:ABfactors as a composite AAABwhere the morphism AA is an element of cell({R}), where the morphismAA is an element of cell({C}), and where the morphismAB is a morphism of cell(Igl). So we have to treat the cases kcell({R}),kcell({C}), and kcell(Igl).

The casekcell(Igl) is a consequence of Propositions7.1,7.2, and7.3. The casek cell({C}) is trivial.

Letkcell({R}). Let (α,β)U0×U0. ThenPi(α),i(β)V (resp.,Pi(α),i(β)Y) is a coprod- uct of terms of the formPα,u0U×Pv0,u1U× ··· ×PvpU(resp.,Pα,u0X×Pv0,u1X× ··· × PvpX) such that (ui,vi) is a pair of distinct elements ofU0=X0identified byk. Sotis a

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weakS-homotopy equivalence since a binary product of weak homotopy equivalences is

a weak homotopy equivalence.

8.T-homotopy equivalence andI+gl-cell complex

The first step to understand the reason whyDefinition 5.7is badly behaved is the follow- ing theorem which gives a description of theT-homotopy equivalences f :XY such that the 0-skeleton ofY contains exactly one more state than the 0-skeleton ofX.

Theorem 8.1. LetXandY be two objects of cell(Flow). Let f :XY be aT-homotopy equivalence. Assume thatY0=X0 {α}. Then the canonical morphismXfactors as a compositeuf(X)vf(X)Xsuch that

(1) one has the diagram

{0,1} =GlobS1 uf(X)

I=GlobD0

φ

vf(X) X

II vf(X) Y

(8.1)

(2) the morphismsuf(X) andvf(X)Xare relativeIg-cell complexes.

ByProposition 6.3, the morphism{0,1} =Glob(S1)uf(X) is a cofibration. There- fore, the morphismIvf(X) is a cofibration as well. The morphismuf(X)vf(X) is a relativeIg-cell complex as well since it is a pushout of the inclusion{0,1} ⊂IIsending 0 to the initial state ofIIand1 to the final state ofII.

Proof. ByProposition 6.1, and sinceYis an object of cell(Flow), the canonical morphism of flowsY0Y is a relativeIg-cell complex. So there exist an ordinalλand aλ-sequence μYμ:λFlow (also denoted byY) such thatY=lim−→μ<λYμ and such that for any ordinalμ < λ, the morphismYμYμ+1is a pushout of the form

GlobSnμ1 φμ Yμ

GlobDnμ ψμ Yμ+1

(8.2)

of the inclusion of flows Glob(Snμ)Glob(Dnμ+1) for somenμ0.

参照

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