many homotopy categories are homotopy categories

l

ries

05

of com-rning thewhich

ions andegoriesMay, the

iant

admitshomo-re thetegoryd weak

nces of

Topology and its Applications 153 (2006) 1084–1099

www.elsevier.com/locate/topo

Many homotopy categories are homotopy catego

Michael Cole

Department of Mathematics, Hofstra University, Hempstead, NY 11549, USA

Received 15 November 2004; received in revised form 25 February 2005; accepted 25 February 20

Abstract

We show that any category that is enriched, tensored, and cotensored over the categorypactly generated weak Hausdorff spaces, and that satisfies an additional hypothesis concebehavior of colimits of sequences of cofibrations, admits a Quillen closed model structure inthe weak equivalences are the homotopy equivalences. The fibrations are the Hurewicz fibratthe cofibrations are a subclass of the Hurewicz cofibrations. This result applies to various catof spaces, unbased or based, categories of prespectra and spectra in the sense of Lewis andcategories ofL-spectra andS-modules of Elmendorf, Kriz, Mandell and May, and the equivaranalogues of all the afore-mentioned categories. 2005 Published by Elsevier B.V.

MSC:55P42; 55U35

Keywords:Model category; Homotopy category

1. Introduction

In a classic paper [9] Arne Strøm proved that the category of topological spacesa Quillen closed model structure in which the weak equivalences are the genuinetopy equivalences, the fibrations are the Hurewicz fibrations, and the cofibrations aclosed Hurewicz cofibrations. In this paper we extend his result and prove that any cathat is enriched, tensored, and cotensored over the category of compactly generateHausdorff spaces, and that satisfies a mild hypothesis concerning colimits of seque

E-mail address:[email protected] (M. Cole).

0166-8641/$ – see front matter 2005 Published by Elsevier B.V.doi:10.1016/j.topol.2005.02.006

M. Cole / Topology and its Applications 153 (2006) 1084–1099 1085

quiv-ns. Theult ap-d,

egoriesy ofmotopy

dorffg

wing

n-s

ory

ed as a

ry is

s

cofibrations, admits a Strøm-type Quillen closed model structure in which the weak ealences are the homotopy equivalences and the fibrations are the Hurewicz fibratiocofibrations are a (generally proper) subclass of the Hurewicz cofibrations. Our resplies to categories of spaces, unbased or based, categories ofG-spaces, unbased or basecategories of prespectra and spectra in the sense of Lewis and May [5], the catof L-spectra andS-modules of Elmendorf et al. [2], and diagram categories over anthese categories. Thus all of these homotopy categories in the classical sense are hocategories in the sense of Quillen.

2. Topological categories

Let U denote the category of (unbased) compactly generated spaces (weak Hausk

spaces).U is bicomplete with limits formed using thek functor and colimits formed usinthe weak Hausdorff functor. Recall that there is an adjunction isomorphism

U (X × Y,Z) ∼= U(X,ZY

)

where× is the compactly generated product andZY is the space of mapsY → Z with thecompactly generated compact-open topology. The best reference forU is Lewis [4].

Throughout this paper we shall use the phrase “topological category” in the follosense.

Definition 2.1. A topological categoryA is a bicomplete category that is enriched, tesored, and cotensored overU . Thus the hom setsA (X,Y ) are naturally topologized aspaces inU and composition of morphisms is continuous as a map

− ◦ − :A (Y,Z) × A (X,Y ) → A (X,Z).

We have natural homeomorphisms

A (X ⊗ K,Y) ∼= A(X,YK

) ∼= U(K,A (X,Y )

)

for suitable bifunctors⊗ :A × U → A (the tensors) and(−)(−) :U op × A → A (thecotensors).

It follows formally that⊗ defines a (right) action of the symmetric monoidal categU onA . Thus we have isomorphisms

(X ⊗ K) ⊗ L ∼= X ⊗ (K × L),

X ⊗ ∗ ∼= X

that satisfy the necessary coherence conditions. Dually the cotensors, reinterpretfunctorA op × U → A op, define an action ofU onA op.

A category is said to be based if the unique map∅ → ∗ from the initial object∅ tothe final object∗ is an isomorphism. It is an easy formality that if a topological categobased, then it is enriched, tensored, and cotensored over the categoryI of based compactlygenerated spaces. Specifically, ifA is a based category that is topological with tensor⊗

1086 M. Cole / Topology and its Applications 153 (2006) 1084–1099

n

ationsositiond thestan-

ornwe

st

and cotensors(−)(−) overU , then we giveA (X,Y ) the basepointX → ∗ → Y and wedefine tensors∧ :A × I → A and cotensorsF(−,−) :I op × A → A by the squares

X ⊗ ∗ X ⊗ K

∗ X ∧ K

F(K,X) ∗

XK X∗

where the left square is a pushout and the right square is a pullback.Conversely, if a category is known to be enriched, tensored, and cotensored overI with

tensors∧ :A × I → A and cotensorsF(−,−) :I op × A → A , then the specificatioX ⊗ K = X ∧ K+ andYK = F(K+, Y ) defines tensors and cotensors overU and thecategory is topological.

There is a standard body of theory concerning homotopies, cofibrations, and fibrthat applies to arbitrary topological categories, based or unbased. An excellent expof this theory in the context of based topological categories is given in May [6] anproofs given there work perfectly well in the unbased context. We briefly recall somedard definitions and facts. LetA be an arbitrary topological category.

Definition 2.2. For objectsX andY , a homotopyis a mapX ⊗ I → Y , equivalently amapX → Y I , equivalently a pathI → A (X,Y ). Two mapsf,g :X → Y arehomotopicif there is a homotopy fromf to g. Since homotopy is an equivalence relation on hsetsA (X,Y ) and is compatible with composition, we get a quotient category whichdenotehA .

Definition 2.3. A mapf :X → Y in A is acofibrationif dashed arrow completions exifor any diagram of the forms:

Xi0

f

X ⊗ I

f ⊗id

Yi0

Y ⊗ I

Z

X

f

ZI

p0

Y Z

∗i0

A (Y,Z)

f ∗

I A (X,Z)

(By adjunction, these diagrams are equivalent.) Duallyf :X → Y is afibration if dashedarrow completions exist for diagrams of the forms:

Z

XI

f I

p0X

f

Y Ip0

Y

Z

i0

X

f

Z ⊗ I Y

∗i0

A (Z,X)

f∗

I A (Z,Y )

We will call a cofibration or fibrationacyclic if it is a homotopy equivalence.


s and

pl

a

, a

topyotopy

results 2.8r more

sorsflinder

Note that the middle diagrams imply that pushouts of cofibrations are cofibrationpullbacks of fibrations are fibrations.

Definition 2.4. For a mapf :X → Y , themapping cylinderMf and the dualmapping pathobjectNf of f are defined by the squares

Xi0

f

X ⊗ I

Y Mf

Nf X

f

Y Ip0

Y

where the left square is a pushout and the right square is a pullback. We writeJ (f ) :Mf →Y ⊗ I andQ(f ) :XI → Nf for the natural maps.

Proposition 2.5. A map f :X → Y is a cofibration if and only if the natural maJ (f ) :Mf → Y ⊗ I is a coretraction. Duallyf is a fibration if and only if the naturamapQ(f ) :XI → Nf is a retraction.

Proposition 2.6. For a mapf :X → Y , let i1 :X → Mf denote the inclusion ofX at thetop of the cylinder. Letq :Mf → Y denote the evident “squash” map. Thenf = q ◦ i1 isa factorization off as a cofibrationi1 followed by a homotopy equivalenceq. Dually, letp1 :Nf → Y denote the evident projectionNf → Y I followed by evaluation at1 and letj :X → Nf denote the evident map induced byidX and the mapX → Y I that is adjoint tothe composite mapX ⊗ I → X → Y given by projection followed byf . Thenf = p1 ◦ j

is a factorization off as a homotopy equivalencej followed by a fibrationp1.

We quote three additional well-known facts we will use in the sequel.

Proposition 2.7. An acyclic cofibration is a coretraction. Dually, an acyclic fibration isretraction.

Proposition 2.8. A pushout of an acyclic cofibration is an acyclic cofibration. Duallypullback of an acyclic fibration is an acyclic fibration.

Proposition 2.9. The pushout of a homotopy equivalence by a cofibration is a homoequivalence. Dually, the pullback of a homotopy equivalence by a fibration is a homequivalence.

Proposition 2.7 is a standard theorem about cofiber maps. Another standardknown as the homotopy invariance of pushouts of cofibrations implies Propositionand 2.9. The reader can easily check that the usual proofs for spaces work in ougeneral setting.

Strictly speaking, all mention of fibrations is redundant in view of duality. The tenfor A are cotensors forA op and vice versa. The cofibrations ofA are the fibrations oA op and vice versa. Similarly one interchanges pushout and pullback, mapping cy


ay with

yclic)e ex-

ifm

ut

and mapping path object, etc. Thus, as is usual in such situations, we can get awproving only half of our theorems.

For the proofs of the model structure we will need a few additional facts about (accofibrations and fibrations. These probably fall under the heading of “obvious to thperts”, but the author knows of no reference for them.

Proposition 2.10. If f :X → Y is a cofibration thenJ (f ) :Mf → Y ⊗ I is an acycliccofibration. Iff is a fibration thenQ(f ) :XI → Nf is an acyclic fibration.

Proof. By duality it suffices to prove the first statement. Clearly for anyf the morphismJ (f ) is a homotopy equivalence since we have a commutative triangle

MfJ(f )

π

Y ⊗ I

π

Y

in which the vertical arrows are homotopy equivalences. Thus we need to show thatf isa cofibration thenJ (f ) is a cofibration. This is equivalent to showing that the morphis

J(J (f )

):MJ(f ) → (Y ⊗ I ) ⊗ I

is a coretraction. By mild abuse of notation we may write

Mf = Y ⊗ {0} ∪ X ⊗ I

and

MJ(f ) = [(Y ⊗ I ) ⊗ {0}] ∪ [Mf ⊗ I ]

= [(Y ⊗ I ) ⊗ {0}] ∪ [(

Y ⊗ {0}) ⊗ I] ∪ [

(X ⊗ I ) ⊗ I]

∼= [Y ⊗ (

I × {0} ∪ {0} × I)] ∪ [

X ⊗ (I × I )].

By choosing a homeomorphism of pairs of spaces(I × I, I × {0} ∪ {0} × I

) ∼= (I × I, {0} × I

)

we see thatJ (J (f )) is equivalent to the morphism[Y ⊗ ({0} × I

)] ∪ [X ⊗ (I × I )

] → Y ⊗ (I × I ),

which is equivalent to the morphism

J (f ) ⊗ id :Mf ⊗ I → (Y ⊗ I ) ⊗ I.

This last morphism is clearly a coretraction sinceJ (f ) is a coretraction. �For morphismsf :X → Y and f ′ :X → Y ′ let M(f,f ′) be denned by the pusho

square.

X X{i0,i1}

f f ′

X ⊗ I

′ M(f,f ′)
Y Y


e:

l

We call M(f,f ′) the double mapping cylinder off andf ′. Dually, if g :Z → W andg :Z′ → W are maps, the double mapping path object is given by the pullback squar

N(g,g′) Z × Z′

g×g′

WI {p0,p1} W × W

Proposition 2.11. If f :X → Y is a cofibration then the natural map

J (f,f ) :M(f,f ) → Y ⊗ I

is a cofibration which is acyclic iff is acyclic. Ifg :Z → W is a fibration then the naturamap

Q(g,g) :ZI → N(g,g)

is a fibration which is acyclic ifg is acyclic.

Corollary 2.12. If f :X → Y is an acyclic cofibration thenJ (f,f ) is a coretraction.Dually, if g is an acyclic fibration thenQ(g,g) is a retraction.

Proof of Proposition 2.11. Assume thatf is a cofibration. We show that the morphism

J(J (f,f )

):MJ(f,f ) → (Y ⊗ I ) ⊗ I

is a coretraction. We write

M(f,f ) = Y ⊗ {0,1} ∪ X ⊗ I

and

MJ(f,f ) = [(Y ⊗ I ) ⊗ {0}] ∪ [

M(f,f ) ⊗ I]

= [(Y ⊗ I ) ⊗ {0}] ∪ [(

Y ⊗ {0,1}) ⊗ I] ∪ [

(X ⊗ I ) ⊗ I]

∼= [Y ⊗ (

I × {0} ∪ {0,1} × I)] ∪ [

X ⊗ (I × I )].

By choosing a homeomorphism of pairs of spaces(I × I, I × {0} ∪ {0,1} × I

) ∼= (I × I, {0} × I

)

we see thatJ (J (f,f )) is equivalent to the map[Y ⊗ ({0} × I

)] ∪ [X ⊗ (I × I )

] → Y ⊗ (I × I )

which, as we observed in the previous proof, is a coretraction. HenceJ (f,f ) is a cofibra-tion.

Now suppose the cofibrationf is acyclic. In the diagram

X X{i0,i1}

f f

X ⊗ I

αf ⊗id

Y Y M(f,f )J (f,f )

Y ⊗ I


e

t

in a

7).cked

sonssource

odeland

ch

em toass ofand

α is an acyclic cofibration by Proposition 2.8 sincef f is an acyclic cofibration and thsquare is a pushout. Since alsof ⊗ id is a homotopy equivalence it follows thatJ (f,f ) isa homotopy equivalence.

The second statement is dual.�Proposition 2.13. If f :X → Y is a cofibration, thenf is acyclic if and only if it is a retracof the morphismJ (f ) :Mf → Y ⊗ I . Dually if g :Z → W is a fibration, theng is acyclicif and only if it is a retract of the morphismQ(g) :ZI → Ng.

Proof. We prove the first statement. Clearly iff is a retract ofJ (f ) thenf is acyclic.Assume then thatf is acyclic and consider the following diagram which we explainmoment:

Xi1

f

Mf

J(f )

r1X ⊗ I

πX

f

Yi1

Y ⊗ I r2M(f,f ) r3

M ′fπ ′ Y

Explanation: The mapsi1 are inclusions at the tops of the cylinders.r2 is a retractionthat is left inverse toJ (f,f ) (such a retraction exists by Corollary 2.12).M ′f denotesthe “upside down” mapping cylinder off , Y ⊗ {1} ∪ X ⊗ I . The retractionsr1 andr3 areobtained using a retractionY ⊗{0} → X⊗{0} (such a retraction exists by Proposition 2.π andπ ′ are the evident projection maps. Commutativity of the diagram is easily cheand the horizontal composites are the respective identity maps.�

3. Strong cofibrations and fibrations

Recall that a Quillen closed model structure on a categoryA consists of three classeW , C , andF of morphisms ofA (the weak equivalences, cofibrations, and fibratirespectively). These classes are required to satisfy suitable axioms. The originalmaterial is in Quillen [7]. See [1] and [3] for good expositions of the theory of mstructures. We will use the definition of [1]. We abbreviate LLP and RLP for the leftright lifting property respectively.

Theorem 3.1 (Strøm). The categoryU admits a Quillen closed model structure for whiW is the class of homotopy equivalences,F is the class of Hurewicz fibrations, andC isthe class of Hurewicz cofibrations.

In [9] Strøm proved this result for the category of all topological spaces, withC beingthe class ofclosedcofibrations. His proofs apply equally well toU and inU cofibrationsare automatically closed inclusions. However, in our extensions of Strøm’s theorother topological categories it will again be necessary in general to restrict the clcofibrations. The following definition is essentially equivalent to that of SchwänzlVogt [8].


PPg

ect to

cat-r,rong.

will

Definition 3.2. A strong cofibrationin a topological categoryA is a map that has the LLwith respect to all acyclic fibrations. Dually, astrong fibrationis a map that has the RLwith respect to all acyclic cofibrations. We shall writeCs andFs for the classes of stroncofibrations and strong fibrations.

Note that strong cofibrations are cofibrations since they have the LLP with respmorphismsZI → Z. Similarly strong fibrations are fibrations. As we noted, inU allcofibrations and fibrations are strong. This is also true for the categoryGU of unbasedG-spaces, categories of pairs of spaces orG-spaces and diagram categories over theseegories, but for generalA the inclusionsCs ⊆ C andFs ⊆ F may be proper. Howeveit turns out that certain familiar examples of cofibrations and fibrations are always st

Proposition 3.3. For anyX ∈ A , the map∅ → X (∅ is the initial object inA ) is a strongcofibration. The mapX → ∗ (∗ the final object) is a strong fibration.

This follows from Proposition 2.7 and implies that, in the model structures wediscuss shortly, all objects are cofibrant as well as fibrant.

Lemma 3.4. If X ∈ A , the map

{i0, i1} :X X → X ⊗ I

is a strong cofibration. Dually, the map

{p0,p1} :XI → X × X

is a strong fibration.

Proof. We prove the first statement. We identifyXX with the objectX⊗{0,1} ⊂ X⊗I .Supposeg :Z → W is an acyclic fibration. We need to prove that{i0, i1} has the LLP withrespect tog. By Proposition 2.13,g is a retract of the mapQ(g). Hence it suffices to showthat{i0, i1} has the LLP with respect to the mapQ(g) :ZI → Ng. Using adjunctions, it iseasy to see that the following diagrams are equivalent:

X ⊗ {0,1}{i0,i1}

ZI

Q(g)

X ⊗ I Ng

X ⊗ (I × {0} ∪ {0,1} × I ) Z

g

X ⊗ (I × I ) W

By choosing a homeomorphism of pairs of spaces(I × I, I × {0} ∪ {0,1} × I

) ∼= (I × I, I × {0})

we see that the left arrow in the right diagram is equivalent to the map

X ⊗ (I × {0}) → X ⊗ (I × I )

which is equivalent to the map

i0 :X ⊗ I → (X ⊗ I ) ⊗ I.


r the

tion

a

a

a

polog-

iza-

trty if

ryngt we

t

By the definition of fibration, a lift must exist for the right square and hence also foleft square. �Proposition 3.5. Let f :X → Y be a morphism inA . The cofibrationi1 :X → Mf thatincludesX at the top of the mapping cylinder is a strong cofibration. Dually, the fibrap1 :Nf → Y is a strong fibration.

Proof. We prove the first statement. Letg :Z → W be an acyclic fibration and considerdiagram

Xα

i1

Z

g

Mf

�

βW

for which we must find a lift�. By Proposition 2.7,g is a retraction. Hence there existsright inverses :W → Z with g ◦ s = idW . We regardMf as the union ofX⊗[1

2,1] and the“half cylinder” M1/2f = Y ⊗ {0} ∪ X ⊗ [0, 1

2] joined together atX ⊗ {12}. First we define

�′ :M1/2f → Z to be the restriction ofs ◦ β to M1/2f . Now we use Lemma 3.4 to findlift �" for the diagram

X ⊗ {12,1}{�

′1/2,α}

Z

g

X ⊗ [12,1]

�′′

βW

Piecing together�′ and�′′ gives a lift� for our original diagram. �We now have two possible candidates for a Strøm type model structure on a to

ical categoryA . Either or both of the triples(W ,Cs ,F ) or (W ,C ,Fs) might form amodel structure onA provided that we can prove the lifting properties and the factortion axioms. We show now that the lifting properties present no difficulty. IfJ andKare classes of morphisms in a categoryA let us say that the pair(J ,K ) has the liftingproperty ifJ has the LLP with respect toK (equivalentlyK has the RLP with respecto J ). We shall further say that the pair is complete with respect to the lifting propeJ is the class of all morphisms that have the LLP with respect toK andK is the classof all morphisms that have the RLP with respect toJ . For example, in a model categothe pairs(W ∩ C ,F ) and (C ,W ∩ F ) are always complete with respect to the liftiproperty. The following facts are proved in [8], but since our setup is slightly differenprovide alternative proofs.

Proposition 3.6. In a topological categoryA the pairs(Cs ,W ∩F ), (W ∩Cs ,F ), (W ∩C ,Fs), and(C ,W ∩ Fs) are complete with respect to the lifting property.

Proof. We consider first(Cs ,W ∩ F ). As a matter of definition, we know thatCs isprecisely the class of maps that have the LLP with respect toW ∩ F . Now suppose tha


s.

g

o

am

e

for the

fibra-c. Indeed,

cate-xplicit

a mapg :Z → W has the RLP with respect toCs . Since morphisms of the formi0 :X →X ⊗ I are strong cofibrations, it follows thatg ∈ F . To show thatg is also a homotopyequivalence we consider the diagram

Z

i1

Z

g

Mg

�

q W

By Proposition 3.5 the mapi1 is a strong cofibration and hence a lift� must exist. SinceidZ andq are homotopy equivalences, it follows thati1 andg are homotopy equivalenceHenceg ∈ W ∩ F .

The dual argument proves that(W ∩ C ,Fs) is complete with respect to the liftinproperty.

We consider now the pair(W ∩Cs ,F ). Letf :X → Y be inW ∩Cs and letg :Z → W

be in F . Sincef is an acyclic cofibration, it follows from Proposition 2.13 thatf is aretract of the mapJ (f ) :Mf → Y ⊗ I . Thus to show thatf has the LLP with respect tg, it suffices to prove thatJ (f ) has the LLP with respect tog. By adjunction the followingdiagrams are equivalent:

Mf

J(f )

Z

g

Y ⊗ I W

X

f

ZI

Q(g)

Y Ng

By Proposition 2.10,Q(g) is an acyclic fibration and hence the lift for the right diagr(therefore also the left) exists sincef ∈ Cs . This shows that(W ∩ Cs ,F ) has the liftingproperty.

Suppose now thatf has the LLP with respect toF . Then, a fortiori,f has the LLPwith respect toW ∩ F and hencef ∈ Cs . Now consider the following square:

Xj

f

Nf

p1

Y

�

Y

in whichj andp1 are the maps discussed in Proposition 2.6. Sincej and idY are homotopyequivalences, it follows thatf andp1 are homotopy equivalences. Hencef ∈ W ∩ Cs .

Suppose thatg has the RLP with respect toW ∩ Cs . Then since every morphism of thform i0 :X ⊗ I is in W ∩ Cs , it follows that g ∈ F . We conclude that(W ∩ Cs ,F ) iscomplete with respect to the lifting property and the dual argument proves the samepair (C ,W ∩ Fs). �Remark 3.7. Regrettably it appears to be quite difficult to characterize the strong cotions for based topological categories such as based spaces, prespectra, spectra, etthe author does not know how to prove that weak cofibrations exist for any of thesegories, though one would assume that they do. It would be interesting to have an e


ultane-xist.

s quite

m:

s

nd

topo-it the

e

example of a weak cofibration of based spaces. Presumably one would have to simously exhibit a weak acyclic fibration and construct a square for which the lift fails to eI suspect that the simplest example of a weak cofibration (assuming one exists) icomplicated and pathological.

4. The cofibration hypothesis

To obtain factorizations we introduce the following hypothesis.

Hypothesis 4.1. In a topological categoryA consider commutative diagrams of the for

X0ξ0

f0

X1

f1

ξ1 · · · ξn−1Xn

fn

ξn · · ·

Y

We have an induced map

colimfn : colimXn → Y.

We say thatA satisfies thecofibration hypothesisif for all diagrams in which the mapsξn

are strong acyclic cofibrations, the natural map

colimNfn → N(colimfn)

is an isomorphism whereNf denotes the mapping path object.

We state now our main result.

Theorem 4.2. Let A be a topological category and letW , F , andCs denote the classeof homotopy equivalences, fibrations, and strong cofibrations respectively. IfA satisfiesthe cofibration hypothesis, thenW , F , andCs are the weak equivalences, fibrations, acofibrations of a Quillen closed model structure onA .

The proof will be given in the next section. First, we wish to observe that manylogical categories of interest satisfy the cofibration hypothesis and therefore admStrøm-type model structure.

Lemma 4.3. In the categoryU suppose that for each n we are given a pullback squar:

Xn Yn

fn

Zn gnWn

Suppose that for every n we have mapsηn :Yn → Yn+1, ζn :Zn → Zn+1, andωn :Wn →Wn+1 such thatfn+1 ◦ ηn = ωn+1 ◦ fn andgn+1 ◦ ζn = ωn+1 ◦ gn. Letξn :Xn → Xn+1 bethe maps induced on the pullbacks. If the mapsηn andζn are inclusions(k-ified if necessary


,

n ind,

from

e con-ame inthe

ed

of

, map-e. Thus

otopy

the

since we are working with compactly generated spaces) and if theωn are monomorphicthen the mapsξn are inclusions and the square

colimXn colimYn

colimZn colimWn

is a pullback.

For proof see [4].

Corollary 4.4. The categoryU satisfies the cofibration hypothesis.

Proof. All cofibrations inU are inclusions. Thus if we have a diagram of the type giveour statement of Hypothesis 4.1, in which theξn are strong acyclic cofibrations (or, indeeany inclusions), we may apply the lemma to deduce that the square

colimNfn colimXn

fn

Y Ip0

Y

is a pullback. Thus colimNfn∼= N(colimfn). �

Using Lemma 4.3 it is easy to show that many topological categories constructedU satisfy the cofibration hypothesis.

Example 4.5. Let T denote the category of based compactly generated spaces. Thstructions of pullbacks, colimits of sequences, and mapping path objects are the sU andT and thereforeT must also satisfy the cofibration hypothesis. It follows thatbased homotopy category is a homotopy category.

Example 4.6. For a topological groupG let GU andGT denote the unbased and bascategories ofG-spaces. The cofibration hypothesis is satisfied and hence theG-homotopycategories are homotopy categories.

Example 4.7. Let PU , I U , andS U denote respectively the Lewis–May categoriesprespectra, inclusion prespectra, and spectra indexed on a universeU [5]. Gaunce Lewis[4] proved that in all these categories, cofibrations are closed inclusions. Pullbacksping path objects, and colimits of sequences of inclusions are constructed spacewisthe cofibration hypothesis is satisfied and the various homotopy categories are homcategories in the sense of Quillen.

Example 4.8. Let LS andM s denote respectively the categories ofL-spectra andS-modules of Elmendorf et al. [2]. It is not difficult to check that for these categoriescofibration hypothesis is satisfied.


is,ructed

ctively.

er

pleteod is

a of asmallon ofation.

s

Example 4.9. Let A be a topological category and letΛ be a small category. LetAΛ

be the diagram category of functorsΛ → A . If A satisfies the cofibration hypothesthenAΛ does also since pullbacks, mapping path objects, and colimits are constobjectwise.

5. The proof of the model structure

We begin the proof of Theorem 4.2. We fix a topological categoryA and we letW , F ,andCs denote the homotopy equivalences, fibrations, and strong cofibrations respeThe following is obvious.

Proposition 5.1. The classesW , F , andCs contain all identity maps and are closed undretracts and compositions. AlsoW satisfies the two out of three property.

The lifting properties have already been proved in Proposition 3.6. Hence to comthe proof of Theorem 4.2 it remains only to prove the factorization axioms. Our methreminiscent of Quillen’s small object argument. Essentially, we replace Quillen’s ideset of “generating cofibrations” that satisfy a smallness condition with the idea of a “generating functor”. In a forthcoming paper the author will present this generalizatiQuillen’s argument and study other examples. First we need the following observRecall that for any morphismf :X → Y we have a pullback square:

Nfα(f )

β(f )

X

f

Y Ip0

Y

Observation 5.2. A morphismf :X → Y is a fibration if and only if a diagonal lift existfor the diagram

Nfα(f )

i0

X

f

Nf ⊗ Iβ̂(f )

Y

whereβ̂(f ) is the morphism adjoint toβ(f ).

Proposition 5.3. If the categoryA satisfies our cofibration Hypothesis4.1, then any mor-phismf :X → Y admits a factorization

Xg

Zh

Y

with g ∈ W ∩ Cs andh ∈ F .


.

Proof. DenoteZ0 = X. Suppose inductively that we have a commutative diagram

X = Z0γ0

f =h0

Z1

h1

γ1 · · · γn−1Zn

hn

Y

in which theγk are inW ∩ Cs . We defineZn+1, γn, andhn+1 by the diagram

Nhnα(hn)

i0

Zn

γn

hnNhn ⊗ Iδ(hn)

hn+1

β̂(hn)

Zn+1

Y

where the square is a pushout andhn+1 is the unique map toY that completes the diagramLet Z = colimZn and leth :Z → Y be the map on the colimit induced by{hn}. For eachn we have a pullback square

Nhnα(hn)

β(hn)

Zn

hn

Y Ip0

Y

Applying Hypothesis 4.1 we see that we have a pullback square

colimNhn{α(hn)}

{β(hn)}colimZn

h

Y Ip0

Y

ThereforeNh ∼= colimNhn. Since the functor−⊗I preserves colimits we also haveNh⊗I ∼= colim(Nhn ⊗ I ). Therefore by Observation 5.2 we can prove thath is a fibration byconstructing a liftλ for the diagram

colimNhn{α(hn)}

i0

colimZn

{hn}

colim(Nhn ⊗ I )

λ

{β̂(hn)} Y

One easily sees that such aλ is induced on the colimit by the compositions

Nhn ⊗ Iδ(hn)

Zn+1 → colimZn.

Let g :X → Z be the map

X = Z0 → colimZn = Z.


that inppliedgical

fcks of

wing

.

Theng is in W ∩ Cs andf = h ◦ g is our desired factorization off . �Proposition 5.4. If the categoryA satisfies Hypothesis4.1 then any morphismf :X → Y

admits a factorization

Xk

W�

Y

with k ∈ Cs and� ∈ W ∩ F .

Proof. We have a factorization

Xi1 Mf π

Y

with i1 a cofibration andπ a homotopy equivalence. By Proposition 3.5,i1 is in Cs . Letπ = h ◦ g be a factorization ofπ with g ∈ W ∩ Cs andh ∈ F . Sinceπ andg are inW ,alsoh ∈ W . Then� = h andk = g ◦ i1 is our desired factorization.�

This completes the proof of Theorem 4.2. Since an easy point set argument showsU all cofibrations are strong, we have an alternative proof of Strøm’s theorem (as atoU ). With a little extra argument our methods work also for the category of all topolospaces.

Recall that a model structure is said to beproper if it is both left proper (pushouts oweak equivalences by cofibrations are weak equivalences) and right proper (pullbaweak equivalences by fibrations are weak equivalences).

Proposition 5.5. The Strøm type model structure on a topological categoryA is proper.

Proof. This follows easily from Proposition 2.9.�Finally, we observe that Strøm type model structures are “topological” in the follo

sense.

Proposition 5.6. In a topological categoryA if f :X → Y is a strong cofibration andg :Z → W is a fibration, then the map

A�(f, g) :A (Y,Z) → A (X,Z) ×A (X,W) A (Y,W)

is a fibration inU , which is acyclic if at least one off andg is acyclic.

Proof. Let f ∈ Cs andg ∈ F . Let us writeP for the codomain ofA�(f, g). Using ad-junctions, one can work out that the following diagrams are equivalent for any spaceK .

K

i0

A (Y,Z)

A�(f,g)

K × I P

X

f

(ZK)I

Q(gK)

Y N(gK)

SincegK is a fibration, it follows from Proposition 2.10 thatQ(gK) is an acyclic fibrationHence the lift exists in the right diagram and therefore also for the left.


s

f this

ook of

topy

Provi-

1978.Math.,

sity of

(2002)

Now consider the following diagram in which the square is a pullback.

A

g∗

f ∗

PA�(f,g)

A (X,Z)

g∗

A (Y,W)f ∗ A (X,W)

Applying right properness of the Strøm structure onU and the 2 out of 3 property, it ieasily seen thatA�(f, g) is acyclic if one, or both, off andg is acyclic. �

Acknowledgement

I would like to thank Peter May and Johann Sigurdsson who read early drafts opaper and suggested improvements of exposition.

References

[1] W.G. Dwyer, J. Saplanski, Homotopy theories and model categories, in: I.M. James (Ed.), HandbAlgebraic Topology, Elsevier, Amsterdam, 1995.

[2] A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, M. Cole, Rings, Modules, and Algebras in Stable HomoTheory, Math. Surveys Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997.

[3] M. Hovey, Model Categories, Math. Surveys Monographs, vol. 63, American Mathematical Society,dence, RI, 1999.

[4] L.G. Lewis Jr, The stable category and generalized Thom spectra, PhD thesis, University of Chicago,[5] L.G. Lewis Jr, J.P. May, M. Steinberger, Equivariant Satable Homotopy Theory, Lecture Notes in

vol. 113, Springer, Berlin, 1986.[6] J.P. May, The Homotopical Foundations of Algebraic Topology, Mimeographed Lecture Notes, Univer

Chicago.[7] D.G. Quillen, Homotopical Algebra, Lecture Notes in Math., vol. 43, Springer, Berlin, 1967.[8] R. Schwanzl, R.M. Vogt, Strong cofibrations and fibrations in enriched categories, Arch. Math. 79

449–462.[9] A. Strøm, The homotopy category is a homotopy category, Arch. Math. 23 (1972) 435–441.

many homotopy categories are homotopy categories

Documents