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Topology and its Applications 153 (2006) 1016–1032

www.elsevier.com/locate/topo

Mixing model structures

Michael Cole

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

Received 16 November 2004; received in revised form 18 February 2005; accepted 18 February 20

Abstract

We prove that if a category has two Quillen closed model structures (W1,F1,C1) and (W2,F2,C2)that satisfy the inclusionsW1 ⊆ W2 andF1 ⊆ F2, then there exists a “mixed model structur(Wm,Fm,Cm) for whichWm = W2 andFm = F1. This shows that there is a model structuretopological spaces (and other topological categories) for whichWm is the class of weak equivalencandFm is the class of Hurewicz fibrations. The cofibrant spaces in this model structure are thethat have CW homotopy type. 2005 Published by Elsevier B.V.

MSC:55P42; 55U35

Keywords:Model category; Homotopy category

1. Introduction

In topology and algebra it often occurs that a category one wishes to study is eqwith more than one useful Quillen closed model structure. A paradigmatic examplecategory of topological spaces: There is the Strøm structure (W1,F1,C1) consisting of thehomotopy equivalences, Hurewicz fibrations, and closed Hurewicz cofibrations [10also have the Quillen model structure(W2,F2,C2) consisting of the weak equivalenceSerre fibrations, and retracts of relative cell complexes [9]. In view of the author’s rein [2], the same situation occurs for a variety of topological categories including b

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

0166-8641/$ – see front matter 2005 Published by Elsevier B.V.doi:10.1016/j.topol.2005.02.004

M. Cole / Topology and its Applications 153 (2006) 1016–1032 1017

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spaces,G-spaces, categories of spectra, etc. An analogous situation occurs for valgebraic categories of chain complexes.

In this paper we prove that whenever (W1,F1,C1) and (W2,F2,C2) are triples ofclasses of morphisms of a categoryA that form Quillen closed model structures and tsatisfy the inclusionsW1 ⊆ W2 andF1 ⊆ F2, then there is a “mixed model structur(Wm,Fm,Cm) for which Wm = W2 andFm = F1. The mixed structure for spaces hmany advantages as compared to the usual Quillen structure. The point is that, whmust invert weak equivalences rather than just homotopy equivalences in order to dous homotopy theory, Hurewicz fibrations behave much better than Serre fibrationsa variety of functors and constructions. Also, our results in Section 3 will demonstratthe cofibrations of the mixed structure have many good properties that the Quillen cotions lack. It turns out that a space is cofibrant in the mixed structure if and only if iCW homotopy type.

The paper is organized as follows: In Section 2 we prove our main result and dexamples. In Section 3 we study the cofibrations and cofibrant objects of the mixedture. In Section 4 we prove that the mixed structure inherits right and left propernessstructure 2 and we give two results that follow from left properness of structure 1.tions 5 and 6 are devoted to showing that Quillen adjunctions and monoidal structuractions behave nicely with respect to mixed model structures.

2. The mixed model structure

Recall that a Quillen closed model structure on a bicomplete categoryA is a triple(W,F ,C) of classes of morphisms ofA (the weak equivalences, fibrations, and cofibtions, respectively) that satisfy axioms that are reminiscent of properties of the homequivalences, fibrations, and cofibrations of topological spaces. The original sourcterial is in [9]. Excellent expositions of the theory of model structures are given in [We will use the definition in [4]. We abbreviate LLP and RLP for the left and right liftproperty, respectively.

We present our main result.

Theorem 2.1. If (W1,F1,C1) and (W2,F2,C2) are model structures on the same cagoryA and ifW1 ⊆ W2 andF1 ⊆ F2, then there exists a model structure(Wm,Fm,Cm)

such thatWm = W2 andFm = F1. By duality, it is also true that ifW1 ⊆ W2 andC1 ⊆ C2,then there exists a model structure(Wm,Fm,Cm) such thatWm = W2 andCm = C1.

Proof. It suffices to consider the first statement. Thus we suppose thatW1 ⊆ W2 andF1 ⊆ F2. Note thatC2 ⊆ C1 andW2 ∩ C2 ⊆ W1 ∩ C1. Now setWm = W2 andFm = F1

and letCm be the class of maps that have the LLP with respect toW2 ∩ F1. ClearlyWm,Cm, andFm contain all identity maps and are closed under retracts and compositionsWm = W2 has the 2 out of 3 property.

1018 M. Cole / Topology and its Applications 153 (2006) 1016–1032

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We consider the lifting properties. As a matter of definition, we know thatCm has theLLP with respect toWm ∩ Fm. Now suppose thatf :X → Y is in Wm ∩ Cm = W2 ∩ Cm.Factorf as

Xf

g

Y

Z

h

with g ∈ W1 ∩ C1 andh ∈ F1. Sincef ∈ W2 andg ∈ W1 ⊆ W2, it follows thath ∈ W2.Thush ∈W2 ∩F1, sof has the LLP with respect toh. From the diagram

Xg

f

Z

h

Y Y

we see thatf is a retract ofg and hence thatf ∈W1 ∩C1. We have shown thatWm ∩Cm =W2 ∩ Cm ⊆ W1 ∩ C1 and henceWm ∩ Cm has the LLP with respect toFm = F1. Nowobserve thatW1 ∩ C1 has the LLP with respect toF1 and hence, a fortiori, has the LLwith respectW2 ∩ F1. ThusW1 ∩ C1 ⊆ Cm. Also we haveW1 ∩ C1 ⊆ W1 ⊆ W2. ThusW1 ∩ C1 ⊆ W2 ∩ Cm = Wm ∩ Cm. We may conclude thatWm ∩ Cm = W1 ∩ C1.

It is immediately clear that any morphism admits a factorization as a member ofWm ∩Cm followed by a member ofFm since this is just the factorization problem for modstructure 1. Now letf be a morphism. Factorf asf = h◦g with g ∈ C2 andh ∈W2 ∩F2.Then factorh ash = � ◦ k with k ∈W1 ∩C1 and� ∈F1. ClearlyC2 ⊆ Cm sinceW2 ∩F1 ⊆W2 ∩F2. Thusg ∈ Cm. AlsoW1 ∩ C1 = W2 ∈ Cm ⊆ Cm and hencek ∈ Cm. It follows thatk ◦ g ∈ Cm sinceCm is closed under composition. Now sinceh ∈ W2 andk ∈ W1 ⊆ W2 itfollows from the 2 out of 3 property that� ∈ W2. Thus� ∈ W2∩F1 = Wm ∩Fm. Thereforef = � ◦ (k ◦ g) provides a factorization off as a member ofCm followed by a member oWm ∩Fm. �Example 2.2. For the category of compactly generated topological spaces let modelture 1 be the Strøm structure in whichW1 is the class of homotopy equivalences,F1 isthe class of Hurewicz fibrations, andC1 is the class of Hurewicz cofibrations. Let modstructure 2 be the Quillen structure in whichW2 is the class of weak equivalences in tusual sense (maps that induce isomorphism in homotopy groups),F2 is the class of Serrfibrations, andC2 is the class of retracts of relative cell complexes. ThenW1 ⊆ W2 andF1 ⊆ F2, so we obtain a mixed model structure.

This mixed model structure has an interesting interpretation as regards the coobjects. There is a well-known formal argument which shows that if a spaceY has CWhomotopy type (equivalently cellular homotopy type) and ifX is a retract ofY in thehomotopy category, thenX also has CW homotopy type. It follows that Quillen cofibraobjects have CW homotopy type. However, the converse is not true.


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Example 2.3. Let X ⊂ R be the setX = {0} ∪ {1/n}∞n=1 and letCX be the (unreducedconeX × I/X ×{1}. ThenCX, being contractible, has CW homotopy type, butCX is notQuillen cofibrant.

Proof. It is easy to show that a cellular space must be nondegenerately based at anThe same must be true for any retract of such a space. However, the point(0,0) ∈ CX isdegenerate. �

Our mixed model structure improves the situation. It will follow from our resultsSection 3 that a space is cofibrant in the mixed structure if and only if it has CW homtype. In many ways both the cofibrations and fibrations of the mixed structure are eaunderstand than those of the Quillen structure. These remarks apply to other topocategories including based spaces,G-spaces, based or unbased, the Lewis–May cateof spectra, with or withoutG-action [7], and the categories ofL-spectra andS-modulesof [5].

Example 2.4. Let R be a ring and consider the category of chain complexes ofR-modulesthat are bounded below. LetW1 be the class of chain homotopy equivalences, letF1 bethe class of chain maps that are split epimorphisms in each degree, and letC1 be the chainmaps that in each degree are split monomorphisms.

These classes form a model structure (see, for example, [1] or [3]). For model struc(Quillen [9]) let W2 be the class of quasi-isomorphisms,F2 the class of epimorphismsand C2 the monomorphisms with degreewise projective cokernel. ThenW1 ⊆ W2 andF1 ⊆ F2, so we obtain a mixed model structure.

For this last example, the cofibrant objects of structure 2 are the projective chainplexes. Our results in Section 3 imply that in the mixed structure a cofibrant chain cois one that is chain homotopy equivalent to a projective chain complex.

3. Cofibrations and cofibrant objects in the mixed structure

Let us return to the general situation of a categoryA with two model structures sucthatW1 ⊆ W2 andF1 ⊆ F2. To ease the constant use of notation we introduce some tnology: Elements ofW1 andW2 will be called 1-equivalences and 2-equivalences, restively. Similarly we will speak of 1-fibrations, or 2-cofibrations, etc. The mixed struccofibrations will be calledm-cofibrations and, similarly, we will speak ofm-cofibrant ob-jects.

We begin our study ofm-cofibrations. First we make a definition and record a basicaboutm-cofibrations that we shall use later.

Definition 3.1. A specialm-cofibration is anm-cofibrationf that admits a factorizatiof = h ◦ g with g ∈ C2 andh ∈ W1 ∪ C1.

Proposition 3.2. Everym-cofibration is a retract of a special one.


e

Proof. Let f :X → Y be anm-cofibration. Factorf asf = h ◦ g with g ∈ C2 andh ∈W2∪F2. Now factorh ash = �◦k with k ∈ W1∩C1 and� ∈F1. By the 2 out of 3 property� ∈W2. Since� ∈W2 ∩F1, f has the LLP with respect to�. This shows thatf is a retractof the specialm-cofibrationk ◦ g. �

Our main technical result about maps betweenm-cofibrations is the following. Note thsimilarity between its proof and the proof of Ken Brown’s lemma.

Proposition 3.3. Consider a commutative triangle.

Aji

Xf

Y

If i andj arem-cofibrations andf is a 2-equivalence, thenf is a 1-equivalence.

Proof. Consider the pushout square

Aj

i

Y

ιY

X ιXX A Y

and notice thatιX andιY , being pushouts ofm-cofibrations, are themselvesm-cofibrations.Now consider a factorization

X A Y(f,idy)

q

Y

Z

p

with q ∈ Cm andp ∈ W2 ∩F1. Sincep ◦ (q ◦ ιY ) = idY ∈ W2 andp ∈ W2, it follows thatq ◦ ιY ∈ W2. Thus

q ◦ ιY ∈W2 ∩ Cm = W1 ∩ C1 ⊆ W1.

Since alsop ◦ (q ◦ ιY ) = idY ∈ W1, we deduce thatp ∈ W1. Now we observe thatp ◦(q ◦ ιX) = f ∈ W2 andp ∈ W1 imply thatq ◦ ιX ∈ W2. Thus

q ◦ ιX ∈W2 ∩ Cm = W1 ∩ C1 ⊆ W1.

Thereforef = p ◦ (q ◦ ιX) ∈W1. �Corollary 3.4. A 2-equivalence betweenm-cofibrant objects is a1-equivalence.

Proof. Apply our proposition to the case thatA is the initial object∅. �


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Remark 3.5. Since 2-cofibrations arem-cofibrations, Proposition 3.3 and Corollary 3specialize to statements about 2-cofibrations and 2-cofibrant objects. Those statemcourse, can be proved without making any mention of the mixed model structure.

Proposition 3.6. The following statements about a morphismf :A → X are equivalent:

(1) f is anm-cofibration.(2) f is a 1-cofibration and there exists a diagram

Aff ′

X′ξ

X

such thatf ′ is a 2-cofibration andξ is a 1-equivalence.

Proof. Suppose first thatf ∈ Cm. Then clearlyf ∈ C1. Now factorf asf = ξ ◦ f ′ withf ′ ∈ C2 andξ ∈ W2 ∩F2. Sincef ′ andf are bothm-cofibrations, by Proposition 3.3 th2-equivalenceξ must be a 1-equivalence. Thus (1) implies (2).

Conversely suppose that (2) holds. Factorf as

Af

g

X

Y

h

with g ∈ Cm andh ∈ W2 ∩F1. In the square

Ag

f ′Y

h

X′ξ

�

X

a lift � must exist sincef ′ ∈ C2 ⊆ Cm andh ∈ W2 ∩ F1. Now sinceξ ∈ W1 ⊆ W2 andh ∈ W2, we have� ∈ W2. But sincef ′ and g are m-cofibrations, Proposition 3.3 telus that the 2-equivalence� must be a 1-equivalence. Another application of the 2 ou3 property shows thath ∈ W1. Thereforeh ∈ W1 ∩ F1 and, sincef ∈ C1, f has the LLPwith respect toh. We conclude thatf is a retract ofg and hencef ∈ Cm. �Corollary 3.7. The following statements about an objectX ∈A are equivalent.

(1) X is m-cofibrant.(2) X is 1-cofibrant and has the1-homotopy type of a2-cofibrant object.

Proof. (1) ⇒ (2) follows immediately from Proposition 3.6 forA = ∅. Conversely suppose that (2) holds. LetX′ be a 2-cofibrant object such thatX′ andX are isomorphic in thehomotopy category Ho1(A) associated to model structure 1. The 2-cofibrant objectX′ is


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1-cofibrant and hence the isomorphism can be realized by a 1-equivalenceX′ → X. Nowapply Proposition 3.6 withA = ∅. �Example 3.8. Applying these results to the mixed structure for spaces, we see that ais m-cofibrant if and only if it has CW homotopy type. Combining Proposition 3.6 wwell-known facts about cofiber maps of Hurewicz cofibrations, we may say that af :A → X is anm-cofibration if and only if it is a Hurewicz cofibration that is cofibhomotopy equivalent underA to a relative CW complex.

Example 3.9. In the mixed structure for bounded below chain complexes ofR-modules,we see that a chain complex ism-cofibrant if and only if it has the chain homotopy tyof a projective chain complex. One can work out that a chain map is anm-cofibration ifand only if it is a degreewise split monomorphism whose cokernel has the chain homtype of a projective chain complex.

A general result that summarizes some of our previous work is the following:

Proposition 3.10. Consider a diagram

Aji

X

f

Y

g

Z

in which i andj are 1-cofibrations andf andg are 2-equivalences. Then if any threethe following four statements is true, the fourth is also:

(1) i is anm-cofibration.(2) j is anm-cofibration.(3) f is a 1-equivalence.(4) g is a 1-equivalence.

Proof. By symmetry,(1), (2), (3) ⇒ (4) and (1), (2), (4) ⇒ (3) are equivalent. Let uprove the latter. Thus we assume thati, j ∈ Cm, f ∈ W2, andg ∈ W1. We must show thaf ∈W1. First factorg as

Yg

h

Z

W

k

with h ∈ Cm andk ∈ W2 ∩F1. Now g ∈ W1 andk ∈ W2 impliesh ∈ W2. Thush ∈ W2 ∩Cm = W1 ∩ C1 ⊆ W1. Thereforek ∈ W1. Now i has the LLP with respect tok, so we get


-

e

ce

sfy a

a lift � in the following square.

Ah◦j

i

W

k

Xf

�

Z

Thenf ∈W2 andk ∈W1 implies� ∈ W2. But sincei andh◦ j arem-cofibrations, Proposition 3.4 assures us that� ∈W1. Thereforef ∈ W1.

By symmetry,(1), (3), (4) ⇒ (2) and (2), (3), (4) ⇒ (1) are equivalent. Let us provthe former. Thus we assume thati ∈ Cm, j ∈ C1 andf,g ∈W1. We must show thatj ∈ Cm.Factorj as

Aj

h

Y

W

k

with h ∈ Cm andk ∈ W2 ∩F1. In the diagram

A

hi

X

f

W

g◦kZ

we havei, h ∈ Cm, f ∈ W1, andg ◦ k ∈ W2. We may apply our previous result to deduthatg ◦ k ∈ W1. Sinceg ∈ W1 we conclude thatk ∈ W1. Thereforek ∈ W1 ∩F1. Sincej

has the LLP with respect tok it follows thatj is a retract ofh and hencej ∈ Cm. �Another interesting feature of the mixed structure is that the cofibrations sati

weak (2) out of (3) property.

Proposition 3.11. Given a commutative triangle,

Aji

Xf

Y

if i andj arem-cofibrations andf is a 1-cofibration, thenf is anm-cofibration.

Proof. Factorf asf = h ◦ g with g ∈ Cm andh ∈ W2 ∩F1. Then in the diagram

Ajg◦i

Z Y
h


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ce. Let

ht

n and

re

l

3 and

both g ◦ i andj arem-cofibrations. By Proposition 3.3 the 2-equivalenceh is a 1-equi-valence. Henceh ∈ W1 ∩F1. Sincef has the LLP with respect toh, it follows thatf is aretract ofg and hence thatf ∈ Cm. �Corollary 3.12. A 1-cofibration betweenm-cofibrant objects is anm-cofibration.

4. Mixing proper model structures

Recall that a model category is said to beright proper if the pullback of a weak equivalence along a fibration is always a weak equivalence. Dually, a model category isleft properif the pushout of a weak equivalence along a cofibration is always a weak equivalenus have our usual situation of a categoryA with two model structures such thatW1 ⊆ W2andF1 ⊆ F2.

Proposition 4.1. If model structure2 is right proper, then the mixed structure is rigproper.

Proof. This is immediate sinceFm = F1 ⊆ F2 andWm = W2. Thus the pullback of a2-equivalence along a 1-fibration is the pullback of a 2-equivalence along a 2-fibratiohence is a 2-equivalence.�Proposition 4.2. The mixed model structure is left proper if and only if model structu2is left proper.

Proof. Assume first that the mixed structure is left proper. SinceW2 = Wm andC2 ⊆ Cm,it is immediate that structure 2 is left proper.

Conversely, assume that structure 2 is left proper. Letf be anm-cofibration. By Propo-sition 3.2 there exists a specialm-cofibrationf ′ such thatf is a retract off ′. It thenfollows that the pushout of a mapg alongf is a retract of the pushout ofg alongf ′.SinceWm = W2 is closed under retracts, it suffices to consider the case thatf is a speciam-cofibration. Thus letf = k ◦ h with h ∈ C2 andk ∈ W1 ∩ C1. Consider the diagram

Xh

g

Y ′ k

�′

Y

�

Z m W ′n W

in which both squares are pushouts. Sinceg ∈W2 andh ∈ C2, �′ ∈W2. Also,n ∈ W1 ∩ C1sincen is the pushout ofk. Thus�◦k = n◦�′ ∈W2. Since alsok ∈W1 ⊆ W2, we concludethat� ∈W2. �

When model structure 1 is left proper, one can extend our results in Sectiondevelop additional facts aboutm-cofibrations. We give two examples of this.


ird

ce

ndn-

:

at


Aϕ

i

B

j

Xf

Y

in which i and j are 1-cofibrations,ϕ is a 1-equivalence, andf is a 2-equivalence. Ifmodel structure1 is left proper, then any two of the following statements implies the th:

(1) i is anm-cofibration.(2) j is anm-cofibration.(3) f is a 1-equivalence.

Proof. We consider first(1), (2) ⇒ (3). Thus we assume thati, j ∈ Cm, ϕ ∈ W1, andf ∈ W2. We must prove thatf ∈ W1. Consider the diagram

Aϕ

i

B

h

jXϕ

f

P

k

Y

in which the square is a pushout. Thenh ∈ Cm sinceh is the pushout ofi. Also sincei ∈ Cm ⊆ C1 andϕ ∈ W1, left properness of structure 1 implies thatψ ∈ W1. Since alsof ∈ W2 we havek ∈ W2. Now applying Proposition 3.3 to the right triangle we deduthatk ∈W1. Thereforef = k ◦ ψ ∈W1.

We consider now(1), (3) ⇒ (2). Thus we assume thatϕ,f ∈ W1, i ∈ Cm, andj ∈ C1.We must prove thatj ∈ Cm. In our diagramψ ∈ W1 by the properness hypothesis ahencek ∈ W1. Sinceh ∈ Cm we may apply Proposition 3.10 to the right triangle to coclude thatj ∈ Cm.

Now we show that(2), (3) ⇒ (1). Assume thatϕ,f ∈W1, j ∈ Cm, andi ∈ C1. We mustprove thati ∈ Cm. We factori = n ◦ � with � ∈ Cm andn ∈ W2 ∩F1. Consider the square

Aϕ

�

B

j

Zf ◦n Y

Sinceϕ ∈ W1, �, j ∈ Cm, andf ◦ n ∈ W2, we may apply our previous result to see thf ◦ n ∈ W1. Hencen ∈ W1. Thereforei ∈ C1 has the LLP with respect ton ∈ W1 ∩ F1.Thusf is a retract of� and hencef ∈ Cm. �


3

rs

v-een

e


Aϕ

i

B

j

Xf

Y

in which i and j are m-cofibrations,ϕ is a 1-equivalence, andf is a 1-cofibration. Ifmodel structure1 is left proper thenf is anm-cofibration.

Proof. Factorf asf = h◦g with g ∈ Cm andh ∈ W2∩F1. We may apply Proposition 4.to the square

Aϕ

g◦iB

j

Xh

Y

to conclude thath ∈ W1. Thusf has the LLP with respect toh, so it follows thatf is aretract ofg and hencef ∈ Cm. �

5. Mixing Quillen adjunctions

Recall that ifA andB are model categories and ifL :A → B andR :B → A are aleft-right adjoint pair of functors, then the adjunction is called aQuillen adjunctionif thefollowing equivalent conditions are satisfied.

(1) L preserves cofibrations and acyclic cofibrations.(2) R preserves fibrations and acyclic fibrations.

The purpose of this condition is to ensure thatL andR pass to an adjoint pair of functoon the homotopy categories Ho(A) and Ho(B). Note that whenL andR form a Quillenadjunction, it follows from Ken Brown’s lemma thatL preserves arbitrary weak equialences between cofibrant objects andR preserves arbitrary weak equivalences betwfibrant objects.

Proposition 5.1. Let A andB each have two model structures such thatW1 ⊆ W2 andF1 ⊆ F2. Let L :A → B and R :B → A be a left-right adjoint pair of functors. If thadjunction is a Quillen adjunction with respect to model structures1 and2, then the ad-junction is a Quillen adjunction with respect to the mixed structures.

Proof. ClearlyR(Fm) = R(F1) ⊆ F1 = Fm. Now observe that

Wm ∩Fm = W2 ∩F1 = (W2 ∩F2) ∩F1.

Therefore

R(Wm ∩Fm) = R((W2 ∩F2) ∩F1

) ⊆ (W2 ∩F2) ∩F1 = Wm ∩Fm. �


onuced.

n

.

d

if

t

Recall that a Quillen adjunctionL :A → B and R :B → A is said to be aQuillenequivalenceif for all cofibrant X ∈A and fibrantY ∈ B, a mapf :LX → Y is a weakequivalence if and only if the adjoint mapf :X → RY is a weak equivalence. The reasfor this terminology is that when a Quillen adjunction is a Quillen equivalence, the indadjunction between the homotopy categories is an adjoint equivalence of categories

Proposition 5.2. If categoriesA andB each have two model structures such thatW1 ⊆ W2andF1 ⊆ F2 and if an adjoint pair of functorsL :A → B andR :B → A forms a Quillenequivalence with respect to model structures1 and 2, then the adjunction is a Quilleequivalence with respect to the mixed structures.

Proof. Let X be m-cofibrant and letY be 1-fibrant(= m-fibrant). Let f :LX → Y be amap. We must show thatf is a 2-equivalence if and only iff :X → RY is a 2-equivalenceChoose a 2-cofibrant approximationγ :Γ2X → X of X. SinceΓ2X andX are bothm-co-fibrant, by Corollary 3.4 the 2-equivalenceγ must be a 1-equivalence. Thusγ is a 1-equi-valence of 1-cofibrant objects and thereforeLγ :L(Γ2X) → LX is a 1-equivalence, anhence also a 2-equivalence. From the 2 out of 3 property it follows thatf :LX → Y is a2-equivalence if and only iff ◦ Lγ :L(Γ2X) → Y is a 2-equivalence. But sinceΓ2X is

2-cofibrant andY is 2-fibrant,f ◦Lγ is a 2-equivalence if and only iff ◦ Lγ :Γ2X → RY

is a 2-equivalence. From the diagram

Γ2Xf ◦Lγ

γ

RY

X

f

we see that, sinceγ is a 2-equivalence,f ◦ Lγ is a 2-equivalence if and onlyf :X → RY is a 2-equivalence. We are done.�

6. Mixing monoidal model structures

Let A,A′, andA′′ be categories and let⊗ :A × A′ → A′′ be a functor. Assume thaA′′ is cocomplete. For morphismsf :X → Y andf ′ :X′ → Y ′ of A andA′, respectively,we will let P(f,f ′) denote the object ofA′′ defined by the pushout square:

X ⊗ X′ id⊗f ′

f ⊗id

X ⊗ Y ′

Y ⊗ X′ P(f,f ′)

We writef �f ′ for the natural mapP(f,f ′) → Y ⊗ Y ′ and we callf �f ′ thepushoutproduct of f and f ′. In many cases of interest we will have functors Homr : (A′)op ×A′′ → A and Hom� : (A)op ×A′′ → A′ that satisfy adjunction isomorphisms

A′′(X ⊗ X′,X′′) ∼= A(X,Homr (X

′,X′′)) ∼= A′(X′,Hom�(X,X′′)

).


-

.

turesofAn

s

t squareuts, the

t

c-

Definition 6.1. If A,A′ andA′′ are model categories, a colimit preserving functor⊗ :A ×A′ → A′′ is aQuillen bifunctorif the following conditions are satisfied.

(1) If f andf ′ are cofibrations, thenf �f ′ is a cofibration.(2) If f andf ′ are cofibrations, one or both acyclic, then the cofibrationf �f ′ is acyclic.

For future use we record the following well-known fact.

Proposition 6.2. If A, A′ andA′′ are model categories,⊗ :A × A′ → A′′ is a Quillenbifunctor, andX ∈A is cofibrant, then the functorX ⊗ − :A →A′ preserves weak equivalences between cofibrant objects.

Proof. If f ′ :X′ → Y ′ is a map inA′, then the mapX ⊗f ′ :X ⊗X′ → X ⊗Y ′ is the sameas the pushout productiX �f ′ whereiX is the initial map∅ → X. Thus ifX is cofibrant,X⊗− preserves acyclic cofibrations. By Ken Brown’s lemma, the conclusion follows�

We will demonstrate that if all three categories are equipped with two model strucwith W1 ⊆ W2 andF1 ⊆ F2 and if ⊗ is a Quillen bifunctor with respect to both setsmodel structures, then⊗ is a Quillen bifunctor with respect to the mixed structures.important instance is the case that the three categories are the same and⊗ is a monoidalproduct. However, there are many other cases of interest.

Lemma 6.3. Let ⊗ :A × A′ → A′′ be a bifunctor and letC be any class of morphismof A′′ that is closed under pushouts and compositions. Letf and g be morphisms ofAand letf ′ and g′ be morphisms ofA′. If f �f ′, g �f ′, f �g′ and g �g′ are all in C,then also(g ◦ f )� (g′ ◦ f ′) is in C.

Proof. First we will show thatf � (g′ ◦ f ′) ∈′ C. To see this consider the diagram:

X ⊗ X′ id⊗f ′

f ⊗id

X ⊗ Y ′ id⊗g′X ⊗ Z′

Y ⊗ X′ P(f,f ′)

f �f ′

P(f,g′ ◦ f ′)ϕ

f � (g′◦f ′)

Y ⊗ Y ′ P(f,g′)f �g′ Y ⊗ Z′

Since the upper left square and the upper rectangle are pushouts, the upper righis a pushout. Now since the upper right square and the right rectangle are pusholower right square is a pushout. Thereforeϕ ∈ C sinceϕ is a pushout off �f ′ ∈ C. Sincealsof �g′ ∈ C, it follows thatf � (g′ ◦ f ′) = (f �g′) ◦ φ ∈ C. Identical logic shows thag � (g′ ◦ f ′) ∈ C and symmetric logic now shows that(g ◦ f )� (g′ ◦ f ′) ∈ C. �Proposition 6.4. Let the categoriesA, A′, andA′′ each be equipped with two model strutures such thatW1 ⊆ W2 andF1 ⊆ F2. If ⊗ :A × A′ → A′′ is a Quillen bifunctor with


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respect to both sets of model structures, then⊗ is a Quillen bifunctor with respect to thmixed structures.

Proof. Let f ∈ Cm andf ′ ∈ Cm and let one of them, sayf ′, be acyclic. Thusf ∈ Cm ⊆ C1andf ′ ∈Wm ∩ Cm = W1 ∩ C1. (Here we are abusing notation and using the same symfor classes in different categories.) By assumption about model structures 1,f �f ′ ∈W1∩C1 = Wm ∩Cm. By symmetry it is also true that iff ∈Wm ∩Cm andf ′ ∈ Cm thenf �f ′ ∈Wm ∩ Cm.

Now suppose thatf andf ′ are specialm-cofibrations. Letf = h ◦ g with g ∈ C2 andh ∈ W1 ∩ C1 and letf ′ = h′ ◦ g′ with g′ ∈ C2 andh′ ∈ W1 ∩ C1. Theng �g′ ∈ C2 ⊆ Cm

by assumption about model structures 2. Also, by our previous result we know thatg �h′,h�g′, andh�h′ are inCm. By Lemma 6.3 it follows thatf �f ′ = (h ◦ g)� (h′ ◦ g′) isin Cm.

Now let f andf ′ be arbitrarym-cofibrations. By Proposition 3.2 there exist specm-cofibrationsg andg′ such thatf is a retract ofg andf ′ is a retract ofg′. It follows thatf �f ′ is a retract ofg �g′ ∈ Cm and hence thatf �f ′ ∈ Cm. �

Recall that amonoidal producton a categoryA is a functor⊗ :A × A → A that isassociative and unital (with respect to some unit objectS ∈A) up to coherent natural isomorphism (see [8]). In many cases of interest, there is also a commutativity isomorand further coherence conditions satisfied, in which case⊗ is called asymmetric monoidaproduct.

Definition 6.5. A monoidal model categoryis a monoidal category with product (⊗) andunit S ∈ A together with a Quillen model structure onA such that the following conditionare satisfied.

(1) The product⊗ :A×A →A is a Quillen bifunctor.(2) If Γ S → S is a cofibrant approximation to the unit objectS, then for any cofibrantX,

the mapsΓ S ⊗ X → S ⊗ X ∼= X andX ⊗ Γ S → X ⊗ S ∼= X are weak equivalences

The purpose of the second condition is to ensure that the monoidal structureApasses to a well defined monoidal structure on the homotopy category Ho(A). It is wellknown and easy to prove that the unital condition (2) does not depend on the parchoice of the cofibrant approximation toS. Thus if condition (1) is satisfied, then condtion (2) is satisfied with respect to a given choice of cofibrant approximation if and ocondition (2) is satisfied with respect to any choice of cofibrant approximation.

Proposition 6.6. Let the categoryA have two model structures withW1 ⊆ W2 andF1 ⊆ F2. Let⊗ :A × A → A be a monoidal product. IfA is a monoidal model categorwith respect to both model structures, thenA is a monoidal model category with respectthe mixed model structure.

Proof. Proposition 6.4 assures us that the product⊗ is a Quillen bifunctor with respecto the mixed structure. We must check the unital condition. By repeated factorizatio


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obtain a sequence of maps

∅ i→ Γ2Sf→ ΓmS

g→ Γ1Sh→ S

in which i ∈ C2, h ◦ g ◦ f ∈ W2 ∩F2, f ∈W1 ∩ C1, h ◦ g ∈F1, g ∈ C1, andh ∈W1 ∩F1.Observe that, as indicated by the notation,Γ2S, ΓmS, andΓ1S are cofibrant approximations toS in model structures 2,m and 1, respectively. Also, letX bem-cofibrant and leγ :Γ2X → X be a 2-cofibrant approximation toX. In the following diagram we denot1m = idΓmS , 11 = idΓsS , 1S = idS , 1Γ = idΓ2X , and 1X = idX .

Γ2S ⊗ Γ2Xf ⊗1Γ

ΓmS ⊗ Γ2Xg⊗1Γ

1m⊗γ

Γ1S ⊗ Γ2Xh⊗1Γ

11⊗γ

S ⊗ Γ2X

1S⊗γ

ΓmS ⊗ Xg⊗1X

Γ1S ⊗ Xh⊗1X

S ⊗ X

Now sinceh ◦ g ◦ f :Γ2S → S is a 2-cofibrant approximation ofS and sinceΓ2X is2-cofibrant, it follows that the top row composite(h ⊗ 1Γ ) ◦ (g ⊗ 1Γ ) ◦ (f ⊗ 1Γ ) is a2-equivalence. Also, sinceΓ2X andX are 1-cofibrant, we know thath ⊗ 1Γ andh ⊗ 1X

are 1-equivalences. By the 2 out of 3 property, it follows that(g ⊗ 1Γ ) ◦ (f ⊗ 1Γ )

is a 2-equivalence. NowΓ2X is 1-cofibrant andf :Γ2S → ΓmS is a 1-equivalence o1-cofibrant objects. Hencef ⊗ 1Γ is a 1-equivalence by Proposition 6.2. We may dedthat g ⊗ 1Γ is a 2-equivalence. Now looking at the right square, we know thath ⊗ 1Γ

andh ⊗ 1X are 1-equivalences and that 1S ⊗ γ is a 2-equivalence (1S ⊗ γ is isomorphicto γ :Γ2X → X). Therefore 11 ⊗ γ is a 2-equivalence. Now sinceΓ2X andX are bothm-cofibrant, by Corollary 3.4 the 2-equivalenceγ must be a 1-equivalence. SinceΓmS is1-cofibrant andγ is a 1-equivalence of 1-cofibrant objects, we deduce that 1m ⊗ γ is a1-equivalence. Looking at the middle square, since we have thatg ⊗ 1Γ and 11 ⊗ γ are2-equivalences and 1m ⊗ γ is a 1-equivalence, it follows thatg ⊗ 1X is a 2-equivalenceSince we previously established thath⊗ 1X is a 1-equivalence, we may now conclude tthe map

(h ◦ g) ⊗ idX :ΓmS ⊗ X → S ⊗ X ∼= X

is a 2-equivalence, which is what we needed to prove. Obviously an identical argestablishes that the map

idX ⊗ (h ◦ g) :X ⊗ ΓmS → X ⊗ S ∼= X

is also a 2-equivalence.�Recall that ifB is a monoidal category with product⊗ and unit objectS, then a (right)

B-structure on a categoryA is a bifunctor⊗ :A × B → A that is associative and unitaThus for objectsX ∈A andY,Y ′ ∈ B we have natural isomorphisms

(X ⊗Y) ⊗Y ′ ∼= X ⊗ (Y ⊗ Y ′),X ⊗S ∼= X

that satisfy suitable coherence conditions.


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Definition 6.7. Let B be a monoidal model category and let the bifunctor⊗ :A× B → Adefine a rightB structure on a categoryA. We say thatA is a (right)B-model category ifthe following conditions hold.

(1) ⊗ is a Quillen bifunctor.(2) If Γ S → S is a cofibrant approximation to the unit objectS ∈ B and if X ∈ A is cofi-

brant, then the mapX⊗Γ S → X⊗S is a weak equivalence.

The purpose of the second condition is to ensure that the functor⊗ passes to a Ho(B)-structure on Ho(A). If A is a category with two model structures such thatW1 ⊆ W2 andF1 ⊆ F2, we will write (A,1), (A,2), and(A,m) to denoteA with model structures 1, 2andm, respectively.

Proposition 6.8. Let categoriesA and B each have two model structures such tW1 ⊆ W2 andF1 ⊆ F2. Let ⊗ be a monoidal product forB with respect to which(B,1)

and(B,2) (and therefore, by Proposition6.6, also(B,m)) are monoidal model categorieLet⊗ :A×B → A be a bifunctor with respect to which(A,1) is a (B,1)-model categoryand (A,2) is a (B,2)-model category. Then(A,m) is a (B,m)-model category with respect to⊗.

We omit the proof since it is logically identical to the proof of Proposition 6.6. Iimportant to notice that in Propositions 6.4 and 6.8, model structures 1 and 2 might coin one or more of the relevant categories. For example, in Proposition 6.8 we can mamodel structures forB coincide. We then get the statement that if(A,1) and (A,2) areB-model categories with respect to a bifunctor⊗, then(A,m) is also aB-model structurewith respect to⊗.

We give an illustration of this. LetSSetdenote the category of simplicial sets withusual symmetric monoidal model structure (see [6] for a careful, detailed treatment)a SSet-model category is called asimplicial model category. The following is immediate.

Proposition 6.9. Let a categoryA have two model structures such thatW1 ⊆ W2 andF1 ⊆ F2. If both model structures are simplicial with respect to a bifunctor⊗ :A×SSet→A, then the mixed structure is simplicial with respect to⊗.

Example 6.10. Let U be the category of (unbased) spaces. Define the product⊗ :U ×SSet→ U by X⊗K• = X × |K•| where |K•| denotes the geometric realization of tsimplicial setK•. With respect to this bifunctor, both the Strøm and Quillen structuressimplicial. Hence the mixed structure for spaces is also simplicial.

In a similar way, atopological model categoryis usually defined to mean aU -modelstructure whereU has the Quillen structure. If a categoryA has two model structures thare topological with respect to the same bifunctorA×U → A, then the mixed structure ialso topological.


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References

[1] J.D. Christensen, M. Hovey, Quillen model structures for relative homological algebra, Math. Proc.bridge Philos. Soc. 133 (2) (2002) 261–293.

[2] M. Cole, Many homotopy categories are homotopy categories, Topology Appl. 153 (7) (2006) 1084–[3] M. Cole, The homotopy category of chain complexes is a homotopy category, Preprint.[4] W.G. Dwyer, J. Saplanski, Homotopy theories and model categories, in: I.M. James (Ed.), Handb

Algebraic Topology, Elsevier, Amsterdam, 1995.[5] A.D. Elmendorf, I. Kriz, M.A. Mandell, J.P. May, Rings, Modules, and Algebras in Stable Homotopy

ory, Math. Surveys Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, wappendix by M. Cole.

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

[7] L.G. Lewis Jr, J.P. May, M. Steinberger, Equivariant Stable Homotopy Theory, Lecture Notes in Mvol. 1213, Springer, Berlin, 1986.

[8] S. MacLane, Categories for the Working Mathematician, Springer, Berlin, 1971.[9] D.G. Quillen, Homotopical Algebra, Lecture Notes in Math., vol. 43, Springer, Berlin, 1967.

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

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