Elmendorf’s Theorem via Model Categories

Marc Stephan

January 29, 2010

1 Introduction

In [2], working in the category of compactly generated spaces U , Elmendorfrelates the equivariant homotopy theory of G-spaces to a homotopy theory ofdiagrams using fixed point sets. The diagrams are indexed by a topologicalcategory OG with objects the orbit spaces {G/H}H for the closed subgroupsH ⊂ G. Although, his general assumption there is that G is a compact Liegroup, a formulation of Elmendorf’s Theorem can be found on page 44 in[10] for any topological group in U . A more modern approach has been givenby Piacenza in [11] using model categories. For any topological group G inU , he equips the category UO

opG of continuous contravariant functors from

OG to U with a model category structure, where the weak equivalencesare the objectwise weak equivalences. Concerning equivariant homotopytheory, a morphism f in the category of G-spaces UG is defined to be aweak equivalence, if for all closed subgroups H ⊂ G, the map (f)H is aweak equivalence between spaces, where (−)H : UG → U is the H-fixedpoint functor. That is, for a G-space X one has

XH = {x ∈ X;hx = x for all h ∈ H}.

Piacenza’s formulation of Elmendorf’s Theorem states in particular that thehomotopy categories Ho(UO

opG ) and Ho(UG), which are obtained by formally

inverting the weak equivalences, are equivalent. For the proof, he generalizesthe cellular theory of the category of spaces U to any category UJ op

ofcontinuous contravariant diagrams from a topological category J to U .

We will formulate and prove a generalization of Elmendorf’s Theoremand reach the following two goals. First, the model category theoreticalapproach will be emphasized by equipping the category of G-spaces UG witha model category structure and by proving that there is a pair of Quillenequivalences

Θ: UOopG � UG : Φ.

Thus, by the theory of model categories, the equivalence of the homotopycategories comes for free. Secondly, we generalize this result to the case,

1

where one considers not all closed subgroups of the topological group G,but only a subset, which contains the trivial subgroup {e}.

These two results might be well known to the experts. For instance,the generalization for a family of closed subgroups, which is closed underconjugation, is stated in Remark 1.3 of [5]. Though, the author did not finda published proof. In the case, where the group G is discrete, the first resultis outlined in [3] and generalized to model categories different from U .

Many terms used up to now will be introduced in the article. A nec-essary prerequisite is familiarity with some category theory and algebraictopology, in particular with CW-complexes and homotopy groups. Someknowledge about model categories might be an advantage, but we introducethe necessary definitions and results in section 2 and show a useful argu-ment to equip a category with a model category structure in section 3. Inthe next section, we list the basics of several subcategories of the categoryof topological spaces, for instance of the category of compactly generatedspaces, and equip these subcategories with a model category structure. Insection 5, a theorem to lift a model category structure to another categorywill be proved. It will be applied to the category of G-spaces in section 6and to an example, which will be specialized to the category UO

opG in the

following section. There, we state and prove the main theorem.I would like to thank Mike Mandell, who provided me appendix A of

[8], the people involved, making my stay as an exchange student at theUniversity of Copenhagen possible, and my supervisor, Jesper Grodal, forall the helpful discussions.

2 Model Categories

Model categories as we will define them, have been introduced by Quillen in[12] under the name closed model categories. Following [1], we recall theirbasics. For stating the definition of a model category and its homotopycategory, we introduce the following terms.

Definition 2.1. A morphism f : X → X ′ in a category C is called a retractof a morphism g : Y → Y ′ of C, if there exists a commutative diagram

Xi //

f��

Yr //

g

��

X

f��

X ′i′ // Y ′

r′ // X ′

, (1)

where the compositions in the top and bottow row are the correspondingidentity morphisms.

Lemma 2.2. Let g be an isomorphism in a category C and f a retract of g.Then f is also an isomorphism.

2

Proof. By definition, there exists a commutative diagram as in (1) withri = idX and r′i′ = idX′ . By assumption, g has an inverse g−1. One checksthat rg−1i′ is the inverse of f .

Definition 2.3. Let i : A → B, p : X → Y be morphisms in a category C.We say that i has the left lifting property (LLP) with respect to p and thatp has the right lifting property (RLP) with respect to i, if there exists a liftin any commutative diagram of the form

A //

i��

X

p

��B // Y

. (2)

Definition 2.4. Let W be a class of morphisms of a category C. A categoryD together with a functor F : C → D is called the localization of C withrespect to W , if F maps morphisms of W to isomorphisms and satisfies thefollowing universal property: Given any functor G from C to a category D′,which maps morphism of W to isomorphisms, there exists a unique functorG′ : D → D′ with G′F = G.

Definition 2.5. A model category is a category C together with three classesof morphisms of C, the class of weak equivalences, of fibrations and of cofi-brations, each of which is closed under composition and contains all identitymorphisms of C, such that the five axioms MC1-MC5 below hold. A mor-phism of C is called an acyclic fibration if it is both a weak equivalence anda fibration, it is called an acyclic cofibration if it is both a weak equivalenceand a cofibration.

MC1: Every functor from a finite category to C has a limit and a colimit.MC2: If f and g are morphisms of C such that gf is defined and if two

out of the three morphisms f , g and gf are weak equivalences, then so isthe third.

MC3: If f is a retract of a morphism g of C and g is a weak equivalence,a fibration or a cofibration, then so is f .

MC4:

i) Every cofibration has the LLP with respect to all acyclic fibrations.

ii) Every fibration has the RLP with respect to all acyclic cofibrations.

MC5: Any morphism f of C can be factored as

i) f = pi, where i is a cofibration and p is an acyclic fibration, and as

ii) f = pi, where i is an acyclic cofibration and p is a fibration.

In any model category, the class of weak equivalences together with an-other class of the model category structure determines the third one. Moreprecisely, by Proposition 3.13 of [1], the following holds.

3

Proposition 2.6. Let C be a model category.

i) The cofibrations in C are the morphisms that have the LLP with respectto the acyclic fibrations.

ii) The acyclic cofibrations in C are the morphisms that have the LLP withrespect to the fibrations.

iii) The fibrations in C are the morphisms that have the RLP with respectto the acyclic cofibrations.

iv) The acyclic fibrations in C are the morphisms that have the RLP withrespect to the cofibrations.

Remark 2.7. Since any isomorphism in category is a retract of an identitymorphism, it follows that in a model category, every isomorphism is a weakequivalence, a fibration and a cofibration.

The following objects of a model category are of particular interest.

Definition 2.8. An object X of a model category C is called fibrant, ifthe unique morphism from X to the terminal object is a fibration. It iscalled cofibrant, if the unique morphism from the initial object to X is acofibration.

For instance in section 5 and 6 of [1], these objects are used to show thatthe localization of a model category with respect to the weak equivalencesexists. This enables us to make the following definition.

Definition 2.9. The homotopy category Ho(C) of a model category C is thelocalization of C with respect to the class of weak equivalences.

To compare model categories, the following notion is used.

Definition 2.10. Let C, D be model categories and F : C � D : G a pair ofadjoint functors. The pair (F,G) is called a pair of Quillen equivalences, ifit satisfies the following two conditions:

i) G preserves fibrations and acyclic fibrations,

ii) for each cofibrant object A of C and each fibrant object X of D, amorphism A → G(X) is a weak equivalence in C if and only if itsadjoint F (A)→ X is a weak equivalence in D.

Remark 9.8 of [1] states the following.

Remark 2.11. For a pair (F,G) of adjoint functors between model cate-gories, the following three conditions are equivalent.

i) G preserves fibrations and acyclic fibrations,

4

ii) F preserves cofibrations and acyclic cofibrations and

iii) F preserves cofibrations and G preserves fibrations.

The term Quillen equivalence is justified by the following theorem, whichis proved in section 9 of [1].

Theorem 2.12. Let F : C � D : G be a pair of Quillen equivalences betweentwo model categories C and D. Then the homotopy categories Ho(C) andHo(D) are equivalent.

The proof of this theorem uses Lemma 9.9 of [1], i.e. the following.

Lemma 2.13 (K. Brown). Let F : C → D be a functor between model cate-gories, that takes acyclic cofibrations between cofibrant objects to weak equiv-alences, then F preserves all weak equivalences between cofibrant objects.

3 The small object argument

Our main tool in proving that a category together with three chosen classesof morphisms is a model category, will be the small object argument.

Definition 3.1. Let C be a category and

X0 → X1 → X2 → · · · → Xn → · · ·

a diagram in C, which has a colimit X with natural map fk : Xk → X,k ≥ 0. An object A of the category C is called small relatively to thesequential colimit X if the map

colimkHomC(A,Xk)→ HomC(A,X)

induced by the maps {HomC(A, fk)}k is an isomorphism.

Roughly speaking, depending on a given set I of morphisms of the cat-egory, whose domains are small enough, the small object argument willprovide a factorization f = p∞i∞ of any morphism f of the category, suchthat p∞ and i∞ satisfy some lifting properties.

Definition 3.2. Let C be a category and let I be a set of morphisms ofC. The class of morphisms of C, which have the RLP with respect to allmorphisms in I, is called the class of I-Injectives of C and denoted by I-Inj.

Let C be a category, which has all small colimits, and let I = {fi : Ai →Bi}i be a set of morphisms in C. Let f : X → Y be a morphism in C. SetG0(I, f) := X and p0 := f . Inductively, for k ≥ 1, we construct an objectGk(I, f) and morphisms pk : Gk(I, f)→ Y , ik : Gk−1(I, f)→ Gk(I, f) in C.

5

For all i, let S(i) = Sk(i) be the set of pairs (g, h) of morphisms in C, whichmake the diagram

Aig //

fi��

Gk−1(I, f)

pk−1

��Bi

h // Y

(3)

commute. Define the object Gk(I, f) and the morphism ik via the pushoutdiagram ∐

i

∐(g,h)∈S(i)

Ai +i+(g,h)g //

∐fi

��

Gk−1(I, f)

ik

��∐i

∐(g,h)∈S(i)

Bi // Gk(I, f)

in C. One checks that ik has the LLP with respect to all morphisms in I-Inj.Recall that for each i and each pair (g, h) ∈ S(i), the square (3) commutes.So, these squares induce a morphism pk from the pushout Gk(I, f) to Y ,which satisfies pkik = pk−1.

Define the Infinite Gluing Constrution G∞(I, f) to be the sequentialcolimit G∞(I, f) := colimk≥0G

k(I, f). Denote the natural map G0(I, f)→G∞(I, f) by i∞. Let p∞ : G∞(I, f) → Y be the morphism induced by thenatural transformation {pk : Gk(I, f) → Y }k≥0. So, we have factored f asf = p∞i∞, where the morphism i∞ has the following lifting property.

Lemma 3.3. The morphism i∞ has the LLP with respect to every morphismof I-Inj.

Proof. Let a commutative square

X //

i∞��

X ′

p′

��G∞(I, f) // Y ′

with p′ in I-Inj be given. To construct a lift h : G∞(I, f)→ X ′, let h0 : G0(I, f)→X ′ be given by the morphism in the top row of the square. Inductively, fork ≥ 1, use that ik has the LLP with respect to p′ to construct a lift hk in

Gk−1(I, f)hk−1 //

ik��

X ′

p′

��Gk(I, f) // G∞(I, f) // Y ′

.

Then the morphisms {hk}k≥0 induce the desired lift h.

6

The morphism p∞ also has a lifting property, assuming that some small-ness condition for the domains of the morphisms in I holds.

Lemma 3.4. Suppose that for all i the domain Ai of fi is small relativelyto G∞(I, f), then the map p∞ : G∞(I, f)→ Y is in I-Inj.

Proof. Let a lifting problem

Aig //

fi��

G∞(I, f)

p∞

��Bi

h // Y

be given. By the smallness assumption for Ai, there exists a k ≥ 0 and a

map g′ : Ai → Gk(I, f) such that g equals the composite Aig′→ Gk(I, f) →

G∞(I, f). Hence, the diagram

Aig′ //

fi��

Gk(I, f)ik+1 //

pk

��

Gk+1(I, f) //

pk+1

��

G∞(I, f)

p∞

��Bi

h // Y // Y // Y

commutes and so, the pair (g′, h) is in Sk+1(i). It follows from the construc-tion of Gk+1(I, f) and pk+1, that the desired lift is given by the composite

Bi →∐i

∐Sk+1(i)

Bi → Gk+1(I, f)→ G∞(I, f).

Summarizing, we have proved the following.

Proposition 3.5 (Small object argument). Let C be a category, which hasall small colimits. Let I be a set of morphisms of C. Suppose f is a morphismin C such that the domains of the morphisms in I are small relatively toG∞(I, f). Then f can be factored as f = p∞i∞, where i∞ has the LLP withrespect to every map in I-Inj and p∞ is in I-Inj.

4 Spaces

Several subcategories of the category of topological spaces Top will be givena model category structure. Each of them will contain the following spacesand maps. The unit interval I, the n-disk Dn and its boundary Sn−1,n ≥ 0. Furthermore, the set of bottom inclusions {Dn → Dn×I}n≥0, whichwe denote by J , and the set of inclusions {Sn−1 → Dn}n≥0, which by abuseof notation is also denoted by I.

7

4.1 Topological spaces

To equip the category Top with a model category structure, we will use thefollowing terms and results.

Definition 4.1. A map f in Top is called a Serre fibration if it is in J-Inj.

Remark 4.2. Using that for any n ≥ 0, the pairs (Dn × I,Dn × {0}) and(Dn× I,Dn×{0}∪Sn−1× I) are homeomorphic, one deduces that a map fis a Serre fibration if and only if it has the RLP with respect to any inclusionX × {0} ∪A× I → X × I, where (X,A) is a CW-pair.

Definition 4.3. A map f : X → Y in Top is called a weak homotopyequivalence if either both X and Y are empty or if both are non-empty andfor every point x ∈ X and every n ≥ 0, the morphism πn(f) : πn(X,x) →πn(Y, f(x)) between homotopy groups is an isomorphism.

The following characterization holds.

Lemma 4.4. A map p : X → Y in Top is both a Serre fibration and a weakhomotopy equivalence if and only if it is in I-Inj.

Proof. For one direction, suppose that p is a Serre fibration and a weakhomotopy equivalence. Consider a lifting problem

Sn

��

f // X

p

��Dn+1

g // Y

.

If n = −1, then the disk D0 is mapped to a point y ∈ Y . Since p is a weakhomotopy equivalence, there exists a point x ∈ X such that p(x) can beconnected to y by a path H. Since p is a Serre fibration, there exists a liftH in

D0 //

��

X

p

��D0 × I H // Y

.

The restriction of H to D0 × {1} solves our lifting problem.If n ≥ 0, factor the inclusion of the sphere over the cylinder and the cone

as Sn → Sn × I → CSn ∼= Dn+1 and consider the commutative diagram

Snf //

��

X

p

��Sn × I // Dn+1

g // Y

. (4)

8

By Remark 4.2 it has a lift h since p is a Serre fibration. But since h is notnecessarily constant on the top of the cylinder, it does not directly inducea solution of our lifting problem. We will deform it slightly. The compositeph maps Sn × {1} to some point y ∈ Y . Hence the image of the restrictedmap h | : Sn × {1} → X lies in the fiber F := p−1{y}. Denote by x ∈ Fthe point to which h | maps the basepoint of the sphere. Since p is a weakhomotopy equivalence, it follows by the long exact sequence in homotopy ofa Serre fibration, that πn(F, x) is trivial. Therefore, there exists a homotopyH : Sn × {1} × I → X from h | to the constant map cx. This map H and

the composites Sn × {0} × I → Snf→ X, Sn × I × {0} → Sn × I h→ X and

Sn × I × I → Sn × I → Dn g→ Y provide a commutative square

(Sn × {1} ∪ Sn × {0})× I ∪ Sn × I × {0}

��

// X

p

��Sn × I × I // Y

.

In this diagram, there exists a lift H by Remark 4.2 applied to the CW-pair(Sn × I, Sn × {1} ∪ Sn × {0}). The restricted map H |Sn×I×{1} is anotherlift in (4), but which satisfies H |Sn×{1}×{1} = cx. It therefore induces thesolution Dn+1 → X of the original lifting problem.

For the other direction, assume that p is in I-Inj. First we show, thatp has the RLP with respect to every inclusion A → B, where (B,A) is arelative CW-complex. Recall that in this case, the topological space B is asequential colimit B = colimk≥−1B

k, where B−1 is the given space A andinductively, for k ≥ 0, the space Bk is obtained from Bk−1 by adjoiningk-cells. Given a lifting problem

Af //

��

X

p

��B // Y

,

set h−1 := f : B−1 → X and inductively, for k ≥ 0, use that p is in I-Injand the construction of Bk to find a lift hk in

Bk−1hk−1 //

��

X

p

��Bk // B // Y

.

Then the maps {hk}k induce the desired lift.Since for any n ≥ 0 the pair (Dn × I,Dn) is a relative CW-complex,

it follows that p is a Serre fibration. We show that it is also a weak ho-motopy equivalence. Let n ≥ 0, x ∈ X. For the surjectivity of πn(p), let

9

f : (Dn, Sn−1) → (Y, p(x)) be a representative of a homotopy class [f ] ∈πn(Y, p(x)). Choose a lift h in

Sn−1cx //

��

X

p

��Dn

f // Y

.

Then h is a representative of a homotopy class in πn(X,x), which πn(p) mapsto [f ]. For the injectivity, consider two homotopy classes [f ] , [g] ∈ πn(X,x)such that there exists a homotopy of pairs H : (Dn, Sn−1) × I → (Y, p(x))between pf and pg. Since (Dn × I, Sn−1 × I ∪ Dn × {0} ∪ Dn × {1}) is arelative CW-complex, there exists a lift H in the commutative square

Sn−1 × I ∪Dn × {0} ∪Dn × {1} //

��

X

p

��Dn × I H // Y

,

where Sn−1 × I → X is the constant map cx. The homotopy H shows[f ] = [g] ∈ πn(X,x).

Furthermore, we will need a smallness result.

Lemma 4.5. Let {An → Bn}n≥1 be a family of closed embeddings in Topsuch that each Bn is a T1 space. Let X0 be a topological space. Suppose thatinductively for n ≥ 1, the space Xn is given by the pushout BntϕnXn−1 forsome attaching map ϕn : An → Xn−1. Set X := colimn≥0X

n with naturalmap fn : Xn → X, n ≥ 0.

Then any compact space K is small relatively to X, that is

colimnHomTop(K,Xn)→ HomTop(K,X)

is an isomorphism.

Proof. Since An → Bn is a closed embedding, also the natural map Xn−1 →Xn is a closed embedding, n ≥ 1. Hence, each map fn is a closed embedding.

For the surjectivity, let g ∈ HomTop(K,X) be given. It’s enough to showthat there exists an n ≥ 0 such that g(K) ⊂ fn(Xn) holds. Conversely,suppose that there exists a set S = {xn}n≥0 of points in X such that xn ∈g(K) ∩X\fn(Xn). We show that S ⊂ g(K) is a closed discrete subspace ofg(K), which contradicts the compactness of g(K). Let T ⊂ S be a subset.Using that each Bn is T1, one deduces that every point x ∈ X\f0(X0) isclosed in X. For all n ≥ 0, it follows that the finite set T ∩ fn(Xn) is closedin fn(Xn). Hence, T is closed in X and therefore also in g(K).

One shows the injectivity by using the surjectivity and that the maps{fn}n are embeddings.

10

Corollary 4.6. Suppose that in the situation of Lemma 4.5 each space Bndeformation retracts onto the image of the embedding An → Bn. Thenf0 : X0 → X is a weak homotopy equivalence.

Proof. Since −×I preserves pushouts, it follows that for all n ≥ 1, the spaceXn deformation retracts onto the image of the embedding Xn−1 → Xn.Therefore, this embedding is a weak homotopy equivalence. To show thatfor k ≥ 0 and x ∈ X0, the morphism πk(f0) is an isomorphism, use thesurjectivity of the morphism in Lemma 4.5 with K = Sk for the surjectivityand with K = Sk × I for the injectivity.

Using Lemma 4.5 and its corollary, the following factorization is con-structed.

Lemma 4.7. Every map f : X → Y in Top can be factored as f = p∞i∞,where i∞ is a weak homotopy equivalence, which has the LLP with respectto all maps in J-Inj, and p∞ is in J-Inj.

Proof. Construct the factorization f = p∞i∞ using the Infinite Gluing Con-struction G∞(J, f). By Corollary 4.6, it follows that i∞ is a weak homotopyequivalence. By Lemma 4.5, the domains of the morphisms in J are smallenough to apply the small object argument, which shows the remainingclaims.

Now, we are ready to equip Top with a model category structure.

Proposition 4.8. Call a map in Top

i) a weak equivalence if it is a weak homotopy equivalence,

ii) a fibration if it is in J-Inj and

iii) a cofibration if it has the LLP with respect to every map in I-Inj.

Then with these choices Top is a model category. Furthermore, every objectin Top is fibrant.

Proof. One checks, that each of these three classes is closed unter composi-tion and contains all identity morphisms.

To prove MC1, let a functor F from a small category D to Top be given.The colimit of F is given by the quotient of the disjoint sum

∐d∈D F (d) with

respect to the equivalence relation generated by (d, x) ∼ (d′, F (f)(x)), wheref : d→ d′ is a morphism of D and x ∈ F (d). The limit of F is given by thesubspace

{(xd)d;xd′ = F (f)(xd) for every morphism f : d→ d′ in D}

of the product space∏d∈D F (d).

11

We show the interesting case of MC2. That is when f : X → Y ,g : Y → Z are maps in Top such that f and gf are weak equivalences.Let n ≥ 0, y ∈ Y . We want to show that πn(g) : πn(Y, y) → πn(Z, g(z))is an isomorphism. Since by assumption π0(f) is surjective, there exists apoint x ∈ X such that y can be connected to f(x) via a path γ. The path γinduces via concatenation an isomorphism πn(Y, f(x)) → πn(Y, y) and thepath g ◦ γ induces an isomorphism πn(Z, gf(x))→ πn(Z, g(y)), which makethe diagram

πn(Y, f(x))

∼=��

πn(X,x)πn(f)

∼=oo πn(gf)

∼=// πn(Z, gf(x))

∼=��

πn(Y, y)πn(g) // πn(Z, g(y))

commute. It follows that πn(g) is an isomorphism.To show MC3, let a commutative diagram

Xi //

f

��

X ′r //

f ′

��

X

f

��Y

i′ // Y ′r′ // Y

(5)

in Top with ri = idX and r′i′ = idY be given.First, assume that f ′ is a weak equivalence. We want to prove that f

is a weak equivalence. Let n ≥ 0, x ∈ X. Applying the functor πn to thediagram (5) shows that πn(f) : πn(X,x) → πn(Y, f(x)) is a retract of theisomorphism πn(f ′) and hence by Lemma 2.2 an isomorphism itself. Thusf is a weak equivalence.

Secondly, assume that f ′ is fibration. Given a lifting problem

Dn //

��

X

f

��Dn × I // Y

in Top, one constructs the lifting problem

Dn //

��

X // X ′

f ′

��Dn × I // Y // Y ′

,

which has a solution h, since f ′ is a fibration. The composite rh gives thedesired lift. It follows, that f is a fibration.

The third case, where f ′ is a cofibration is similar.

12

Axiom MC4i) holds, since by Lemma 4.4, a map, which is both a fi-bration and a weak equivalence, is in I-Inj. Concerning MC4ii), we showthat any acyclic cofibration f : A → B has the LLP with respect to everyfibration. Let A i∞→ A′

p∞→ B be a factorization as in Lemma 4.7. Since fand i∞ are weak equivalences, it follows that also p∞ is a weak equivalenceby MC2. Thus p∞ is in I-Inj by Lemma 4.4. By the definition of thecofibrations, there exists a lift g : B → A′ in

A

f

��

i∞ // A′

p∞

��B

= // B

.

The diagramA

= //

f

��

A= //

i∞��

A

f

��B

g // A′p∞ // B

shows that f is a retract of i∞. Using that i∞ has the LLP with respect tofibrations, one deduces that f shares the same property.

For MC5i), let a map f : X → Y in Top be given. By Lemma 4.5,the domains of the maps in I are small enough to apply the small objectargument, providing a factorization f = p∞i∞, where i∞ has the desiredlifting property to be a cofibration, and where p∞ is in I-Inj and hence byLemma 4.4 is an acyclic fibration.

Note that by Lemma 4.4 every map in I-Inj is a fibration and hence isin J-Inj. In particular, a map in Top which has the LLP with respect toevery map in J-Inj, has also the LLP with respect to the I-Injectives. So,Lemma 4.7 provides the desired factorization of a given map in Top to showMC5ii).

We have shown, that Top is a model category. Since for any n ≥ 0, thecylinder Dn × I retracts onto Dn, it follows that every topological space isfibrant.

4.2 k-spaces and compactly generated spaces

Concerning Elmendorf’s Theorem, we will not work in the category Top,but in a full subcategory, called the category of compactly generated spaces,which is better behaved. For instance, it is cartesian closed. Compactlygenerated spaces will be defined as weak Hausdorff k-spaces. They form asubcategory of another categorically well-behaved subcategory of Top, thecategory of k-spaces. But they have the advantage of fulfilling a separationaxiom, the weak Hausdorff condition, which is stronger than T1 but weaker

13

than the Hausdorff condition T2. We summarize the for the working knowl-edge necessary definitions and facts concering these two subcategories. Forthe proofs the reader is referred to Appendix A of [8].

Definition 4.9. Let X be a topological space.

a) The space X is called weak Hausdorff if for all maps g : K → X in Topwith K compact Hausdorff, the image g(K) is closed in X.

b) A subset A ⊂ X is called compactly closed if for all maps g : K → X inTop with K compact Hausdorff, the preimage g−1(A) is closed.

c) The space X is a k-space if every compactly closed subset is closed. LetK denote the full subcategory of Top consisting of the k-spaces.

d) The space X is called compactly generated if it is both weak Hausdorffand a k-space. Let U denote the full subcategory of K consisting of thecompactly generated spaces.

e) The k-space topology on X is the topology, where the closed sets areprecisely the compactly closed sets.

Let C be either the category of k-spaces K or the category of compactlygenerated spaces U .

Example 4.10. The category C contains

a) locally compact Hausdorff spaces,

b) metric spaces,

c) closed and open subsets of spaces in C.

Proposition 4.11 (Limits, colimits, quotient maps). a) The inclusion func-tor K → Top has a right adjoint and left inverse k : Top → K, whichtakes a topological space to its underlying set equipped with the k-spacetopology.

b) The category K has all small limits and colimits. Colimits are inheritedfrom Top and limits are obtained by applying k to the limit in Top.

c) If X and Y are k-spaces and Y is locally compact Hausdorff, then theproduct X × Y in Top is a k-space and hence, it is also the product inK.

d) Let p : X → Y be a quotient map in Top, where X is a k-space. Then Yis a k-space. It is weak Hausdorff and thus in U if and only if the preimage(p× p)−1(∆) of the diagonal ∆ ⊂ Y × Y is closed in the product X ×Xin K. In particular, X is in U if and only if the diagonal in the productX ×X in K is closed.

14

e) The inclusion functor U → K has a left adjoint and left inverse wH :K → U , which takes a k-space X to the quotient X/R, where R ⊂ X×Xis the smallest closed equivalence relation.

f) The category U has all small limits and colimits. Limits are inheritedfrom K and colimits are obtained by applying wH to the colimit in K.

Proposition 4.12 (Function spaces). Let X, Y be in K. For any maph : K → U from a compact Hausdorff space K to an open subset U of Y ,denote the set of maps f : X → Y with f(h(K)) ⊂ U by N(h, U). LetC(X,Y ) be the set HomK(X,Y ) with the topology generated by the subbasis{N(h, U)}. Let Y X be the k − space kC(X,Y ).

a) For all X,Y, Z in K, there is a natural isomorphism HomK(X×Y,Z)→HomK(X,ZY ), which sends a map f to f given by f(x)(y) = f(x, y). Inparticular, K is cartesian closed.

b) For all X,Y in K, evaluation Y X×X → Y is the counit of the adjunctionabove.

c) If X is in K and Y is in U , then Y X is weak Hausdorff. Hence, forall X,Y, Z in U , there is a natural isomorphism HomU (X × Y, Z) →HomU (X,ZY ), which sends a map f to f given by f(x)(y) = f(x, y). Inparticular, U is cartesian closed.

Remark 4.13 (Subspace). Let Y be a space in C and A ⊂ Y a subset ofY . We call A equipped with the topology obtained by applying k to thesubspace A ⊂ Y in Top a subspace of Y . Note, that if X is space in C andf : X → Y is a set function with f(X) ⊂ A, then f : X → Y is in C if andonly if f : X → A is in C. Most often, the subspace A ⊂ Y in Top will beclosed and hence we won’t have to apply k to get a subspace in C.

Proposition 4.14 (Miscellaneous). We call a map f in C a closed embed-ding, if it is a closed embedding in Top. We call f a quotient map, if it isa qoutient map in Top.

a) An arbitrary product of closed embeddings in C is a closed embedding.

b) An arbitrary coprodut of closed embeddings in C is a closed embedding.

c) An arbitrary coproduct taken in Top of spaces in C is also the coproductin C.

d) The pushout taken in Top of a diagram X ← A → Y in C, where onearrow is a closed embedding, is also the pushout in C

e) The sequential colimit taken in Top of a diagram

X0 → X1 → · · · → Xn → · · · ,

where each arrow is an injective map, is also the colimit in C.

15

f) Let X,X ′, Y, Y ′ be spaces in C and p : X → Y , q : X ′ → Y ′ quotientmaps. Then p× q : X ×X ′ → Y × Y ′ in C is a quotient map.

We give C a model category structure.

Proposition 4.15. Call a morphism f in C

i) a weak equivalence if f is a weak equivalence in Top,

ii) a fibration if it is a fibration in Top.

iii) a cofibration if it is a cofibration in Top.

Then C together with these choices is a model category.

Proof. Each of the three classes is closed under composition and containsall identity morphism by the same result in Top. Axiom MC1 holds byProposition 4.11. Since MC2, MC3 and MC4 hold in Top, they also holdin C.

For MC5, we have to factor a given map f in C in two ways. Thefactorization proving MC5i) can be constructed by using the Infinite GluingConstructionG∞(I, f) in Top, since by Proposition 4.14, the spaceG∞(I, f)lies in C. Similar for MC5ii) with G∞(J, f) in Top.

Proposition 4.16. In the model category C, the fibrations are precisely theJ-Injectives of C and the cofibrations are precisely the maps which have theLLP with respect to the I-Injectives of C.

Proof. The first assertion holds by definition of the fibrations in Top. Forthe second assertion, it’s enough to show that the acyclic fibrations in C arethe I-Injectives of C by Proposition 2.6. But this follows from Proposition4.4.

5 Lifting the model category structure of spaces

Let C be either the model category Top, K or U . Let D be a category, whichhas all small limits and all small colimits. Depending on a set

{Fι : C � D : Gι}ι∈F

of adjoint functors, we show how under certain assumptions, the model cate-gory structure on C can be lifted to the category D. Set FJ :=

⋃ι{Fι(j); j ∈

J} and FI :=⋃ι{Fι(i); i ∈ I}.

Theorem 5.1. Call a morphism f in D

i) a weak equivalence if Gι(f) is a weak equivalence in C for all ι,

16

ii) a fibration if it is in FJ-Inj.

iii) a cofibration if it has the LLP with respect to every map in FI-Inj.

For all ι ∈ F , n ≥ 0 and any morphism f : X → Y in D, suppose thatthe object Fι(Sn−1) is small relative to the Infinite Gluing ConstructionG∞(FI, f), that the object Fι(Dn) is small relative to G∞(FJ, f) and thatthe factor i∞ : X → G∞(FJ, f) is a weak equivalence in D.

Then with the choices made, the category D is a model category.

Proof. By functoriality, it follows that the class of weak equivalences of D isclosed under composition and contains all identity morphisms. One checksthat these two conditions are also satisfied by the class of fibrations and theclass of cofibrations.

By assumption, D satisfies MC1. Axiom MC2 holds in C and by func-toriality also in D. The case of MC3 in D, where a morphism is a retract ofa weak equivalence, follows by functoriality and the same axiom in C. Theother two cases are shown as for Top.

For the remaining part, we show three results, where the first two arebased on the next remark. For any ι ∈ F , any morphism i : A→ B in C andmorphism p : X → Y in D, one checks that by adjointness of the functorsFι and Gι, the lifting problems

Fι(A)

Fι(i)

��

// X

p

��Fι(B) // Y

and A

i

��

// Gι(X)

Gι(p)

��B // Gι(Y )

are equivalent. It follows the first result:

a) A morphism p in D is a fibration if and only if Gι(p) is a fibration in Cfor every ι ∈ F .

Using a) and that a map in C is both a fibration and a weak equivalence ifand only if it is in J-Inj one deduces the second result:

b) A morphism p in D is both a fibration and a weak equivalence if andonly if it is in FJ-Inj.

By the assumptions, one can use the Infinite Gluing Construction dependingon the set FJ and apply the small object argument to conclude the thirdresult:

c) Any morphism f in D can be factored as f = p∞i∞, where i∞ is a weakequivalence which has the LLP with respect to all maps in FJ-Inj and,p∞ is in FJ-Inj.

17

Using the results b), c) and the Infinite Gluing Construction dependingon the set FI for MC5i), one shows the remaining axioms as for Top.

Remark 5.2. With regard to the smallness assumptions in Theorem 5.1, it’sworth to mention that for any ι ∈ F , any sequential colimit G∞ = colimkG

k

in D and any object A of C, the diagram

colimkHomD(Fι(A), Gk) //

∼=��

HomD(Fι(A), G∞)

∼=��

colimkHomC(A,Gι(Gk)) // HomC(A,Gι(G∞))

commutes, following from adjointness.

6 Lifting model category structures: Examples

From now on, concerning spaces, we will work in the category of compactlygenerated spaces U . We give two examples of lifting the model categorystructure from U to some category of what will be called continuous functors.They will provide the model category structures involved in Elemendorf’sTheorem.

Definition 6.1. A topological category (with discrete object space) is a cat-egory C, where each hom-set HomC(X,Y ) is topologized as a space C(X,Y )in U , such that composition C(X,Y )× C(Y,Z)→ C(X,Z) is continuous.

Definition 6.2. A functor F : C → D between topological categories issaid to be continuous if each function F : C(X,Y ) → D(F (X), F (Y )) iscontinuous.

Let J be a small topological category until the end of this section. Wegive examples of topological categories.

Example 6.3. a) Since the category of compactly generated spaces U iscartesian closed, it is a topological category with U(X,Y ) = Y X forspaces X,Y in U .

b) The category of continuous functors from J to U is a topological cat-egory, which we denote by UJ . The space UJ (X,Y ) is a subspace of∏j∈J U(X(j), Y (j)) in U for any diagrams X,Y in UJ .

c) Call a groupG, which is topologized as a space in U such that compositionand taking inverses is continuous, a topological group. Then a topologicalgroup G is a topological category with only one object.

18

d) Let G be a topological group and X a space in U . Call a map G×X → X,(g, x) 7→ gx, in U with g(h(x)) = (gh)x and ex = x for all g, h ∈ G,x ∈ X and the neutral element e ∈ G, an action of G on X or a G-action. Call a space in U together with a G-action a G-space and a mapin U between G-spaces, which commutes with the actions, an equivariantmap. Then the category of G-spaces together with the equivariant mapsis a topological category, which can be identified with the category UGsince U is cartesian closed.

Axiom MC1 of a model category in mind, the following result will beused in the two examples of lifting the model category structure from U .

Proposition 6.4. The category UJ has all small limits and colimits.

Proof. Let D be a small category, let S : D → UJ be a functor. The categoryU has all small limits. So as objectwise constructed in [9], the compositefunctor D S→ UJ op → Fun(J ,U) has a limit L. Using that U is cartesianclosed, one checks that L is a continuous functor and thus in UJ . Similarly,one shows that UJ has all small colimits.

6.1 Lifting along a family of contravariant Hom-functors

Since J is a small topological category, its opposite category J op shares thesame property. We will use Theorem 5.1 to equip the topological categoryof continuous contravariant functors UJ op

with a model category structure.For any object j of J op, denote by j the continuous contravariant functor

U(−, j). Let evj : UJ op → U be the functor given by evaluation in the objectj. We will show that j × (−) : U � UJ op

: evj is an adjunction, where for Yin U , we write again Y for the corresponding constant diagram in UJ op

. For

any Z in UJ op, we denote the composite J op Z→ U (−)Y→ U by ZY . The next

lemma shows in particular that (−)×Y : UJ op� U : (−)Y is an adjunction.

Lemma 6.5. For all X,Z in UJ opand all Y in U , there is a natural iso-

morphismUJ op

(X × Y, Z)ϕ→ UJ op

(X,ZY )

in U .

Proof. Using the isomorphism from Proposition 4.12, which shows that U iscartesian closed, define ϕ by ϕ(f)j := fj : X(j) → Z(j)Y in an object j ofJ op and check that it satisfies the desired properties.

Another natural isomorphism comes from the Yoneda Lemma.

Lemma 6.6. Let Z be a diagram in UJ opand let j by an object of J op.

Then there is a natural isomorphism

UJ op(j, Z)→Z(j)

19

in U .

Proof. Regarding j and Z as functors from J op to the category of sets,the Yoneda Lemma states that the morphism ψ : Nat(j, Z)→ Z(j) given byf 7→ fj(idj) is a natural bijection of sets. Using Proposition 4.12, one checksthat its inverse takes a point in Z(j) to a natural transformation, whichlies in HomUJ op (j, Z). It follows that Nat(j, Z) = HomUJ op (j, Z). UsingProposition 4.12, one deduces that ψ is the desired natural isomorphism inU .

Combining Lemma 6.5 and 6.6, one gets in particular the following.

Corollary 6.7. For any object j of J op, there is an adjunction

j × (−) : U � UJ op: evj .

Using Theorem 5.1, we define a model category structure on UJ op.

Proposition 6.8. Let F be a non-empty subset of the set of objects of J op.Then UJ op

together with the adjunctions {j × (−) : U � UJ op: evj}j∈F

satisfies the assumptions of Theorem 5.1, which therefore defines a modelcategory structure on UJ op

.

Proof. The category UJ ophas all small limits and colimits by Proposition

6.4.For the smallness assumptions, let G∞ = colimkG

k be one of the InfiniteGluing Constructions in question and corresponding to it, let the space Abe either Dn or Sn−1, n ≥ 0. By Corollary 5.2, it’s enough to show that

colimkHomC(A, evj(Gk))→ HomC(A, evj(G∞))

is an isomorphism for any j ∈ F . Recall that colimits of diagrams arecomputed objectwise. Hence, for k ≥ 1, the space (Gk)(j) is a pushout inU , which by Proposition 4.14 in fact is the pushout in Top. Furthermore,it follows that evj(G∞) = colimk(Gk(j)) in U , which in fact is also thesequential colimit in Top by Proposition 4.14. The smallness result followsnow from Lemma 4.5 since any space in U is T1.

To show that the factor i∞ in question is a weak equivalence in UJ op, i.e.

that evj(i∞) is a weak equivalence in U for any j ∈ F , one uses Corollary4.6.

6.2 Lifting to the category of G-spaces

Let G be a topological group in U . Let H ⊂ G be a closed subgroup.Deduce from Proposition 4.11d) that the quotient space G/H of cosets gHwith g ∈ G in Top is in U . We call it an orbit space. With the action

20

G×G/H → G, (g, g′H) 7→ gg′H, the orbit space G/H becomes a G-space.Thus, one gets a functor

G/H × (−) : U → UG.

Let X be a G-space. Let XH be the subset {x ∈ X;hx = x for all h ∈ H}of X. We call it the H-fixed point set. Using that the diagonal in X ×X isclosed, deduce that the fixed point set XH is closed in X and hence it is inU . One obtains a functor

(−)H : UG → U,

called fixed point functor, which by the following proposition in particularis right adjoint to G/H × (−).

Proposition 6.9. For all spaces Y in U and G-spaces Z in UG, there is anatural isomorphism

UG(G/H × Y,Z)ϕ→ U(Y,ZH)

in U .

Proof. Check that ϕ defined by ϕ(f)(y) = f(H, y) for y ∈ Y is the desirednatural isomorphism.

As a right adjoint, the functor (−)H preserves limits. It also preservessome colimits.

Lemma 6.10. The functor (−)H preserves

a) arbitrary coproducts in UG,

b) pushouts of diagrams X ← A → Y in UG, where one arrow is a closedembedding as a map in U , and

c) sequential colimits of diagrams X0 → X1 → · · · → Xn → · · · in UG,where each arrow is injective.

Proof. Recall that the colimit in UG is calculated objectwise, hence calcu-lated in U . To check the claims, use Proposition 4.14 and for b) and c), notethat (−)H preserves closed embeddings.

We are ready to lift the model category structure from U to UG.

Proposition 6.11. Let F be a non-empty subset of the closed subgroups ofG. Then UG = UG(F) together with the adjunctions {G/H × (−) : U �UG : (−)H}H∈F satisfies the assumptions of Theorem 5.1, which thereforedefines a model category structure on UG = UG(F).

21

Proof. The category UG has all small limits and colimits by Proposition 6.4.For the smallness assumptions, let G∞ = colimkG

k be one of the InfiniteGluing Constructions in question and corresponding to it, let the space Abe either Dn or Sn−1, n ≥ 0. By Corollary 5.2, it’s enough to show that

colimkHomC(A, (Gk)H)→ HomC(A, (G∞)H)

is an isomorphism for any H ∈ F . Using Lemma 6.10 and Proposition 4.14,one deduces the smallness result from Lemma 4.5 since any space in U is T1.

To show that the factor i∞ in question is a weak equivalence in UG, i.e.that (i∞)H is a weak equivalence in U for any H ∈ F , one uses Corollary4.6.

7 Elmendorf’s Theorem

We will state and prove the main theorem. Let G be a topological group inU . Let H, K be closed subgroups of G.

Definition 7.1. The subgroup H is called subconjugate to K, if there existsan element a ∈ G such that a−1Ha ⊂ K.

If a ∈ G satisfies a−1Ha ⊂ K, then G/H → G/K, gH 7→ gaK, is anequivariant map, which we denote by Ra. Conversely, given an equivariantmap f : G/H → G/K, choose a ∈ G such that f(H) = aK. Then a−1Ha iscontained in K. Thus, we have proved the next lemma.

Lemma 7.2. There exists an equivariant map G/H → G/K if and only ifH is subconjugate to K.

Furthermore, the following characterization holds.

Lemma 7.3. In U , there is an isomorphism

UG(G/H,G/K)ϕ→ (G/K)H . (6)

Proof. The isomorphism ϕ is given by ϕ(f) = f(H) for f ∈ UG(G/H,G/K)and its inverse sends aK ∈ (G/K)H to the equivariant map Ra. To checkcontinuity issues, use Proposition 4.12, Remark 4.13 and Proposition 4.14.

Let F be a set of closed subgroups of G containing the trivial subgroup{e}. Depending on F , we will show how the homotopy theory of G-spacesrelates to the homotopy theory of diagrams indexed by the following cate-gory.

Definition 7.4. The orbit category OF of G with respect to F is the fullsubcategory of UG with objects the orbit spaces {G/H}H∈F .

22

The orbit category is small and a topological category, as a subcategoryof a topological category. Setting J := OF in Proposition 6.8, the categoryUO

opF becomes a model category. It will turn out, that there is a pair of

Quillen equivalences between the model categories UOopF and UG(F).

For any G-space X, let the object X∗ of UOopF be defined by X∗(G/H) =

XH on objects and X∗(f) : XK → XH , x 7→ ax for any representative a off(H), on morphisms. Define the functor

Φ: UG → UOopF

by X 7→ X∗ on objects and f 7→ Φ(f), given by fH in an object G/H ofOopF , on morphisms.

Recall that F contains the trivial subgroup {e} by assumption. If Tis an object of UO

opF , then T (G/{e}) becomes a G-space with the action

G × T (G/{e}) → T (G/{e}) defined by (g, x) 7→ T (Rg)(x) using that {e}is subconjugate to {e}. Using this action on T evaluated in G/{e}, weconstruct a left adjoint of Φ.

Lemma 7.5. Define the functor Θ: UOopF → UG by evaluation in G/{e}.

Then Θ is left inverse and left adjoint to Φ.

Proof. We show that ΘΦ = idUG . For any G-space X, one has ΘΦ(X) =Θ(X∗) = X∗(G/{e}) = X{e} and checks that the G-action agrees with theone on X. If f is a map in UG, then ΘΦ(f) = f{e}. Thus, Θ is left inverseto Φ.

For the adjunction, let T be a diagram in UOopF and X a G-space.

Note that for Re : G/{e} → G/H in OF , the image of T (Re) : T (G/H) →T (G/{e}) lies in theH-fixed point set of Θ(T ) = T (G/{e}), thus in Φ(Θ(T )).Let the morphism

ηT : T → Φ(Θ(T ))

of UOopF be given by T (Re) in the object G/H of Oop

F . One checks that ηTis a natural transformation by using the commutativity of the diagram

G/{e} Re //

Ra��

G/H

Ra��

G/{e} Re // G/K

for a ∈ G, H,K ∈ F with a−1Ha ⊂ K. If f : T → S is a morphism in UOopF ,

then in particular the diagram

T (G/H)T (Re)//

fG/H��

T (G/{e})H

fHG/{e}

��S(G/H)

S(Re)// S(G/{e})H

23

commutes for any H ∈ F . It follows that η = {ηT }T : id→ ΦΘ is a naturaltransformation. It will turn out to be the unit of the adjunction. Note thatΘ(ηT ) = idΘ(T ) and ηΦ(X) = idΦ(X). Letting ε : ΘΦ → id be the identity,it follows that η and ε satisfy the necessary equations to determine theadjunction and hence are the unit and counit, respectively.

Remark 7.6. For any diagram T in UOopF and G-space X, the natural

isomorphismHomUG(ΘT,X)→ Hom

UOopF

(T,Φ(X))

of the adjunction is given by f 7→ Φ(f)ηT with inverse g 7→ Θ(g) by thedescription of the unit and counit above.

As defined, the counit ε of the adjunction (Θ,Φ) is an isomorphism inevery object of Oop

F . The unit is an isomorphism in some objects of OopF .

Recall that for K ∈ F , we denote the contravariant functor OF (−, G/K) inUO

opF by G/K.

Lemma 7.7. Let Y be a space in U and K ∈ F . Then

ηG/K×Y : G/K × Y → ΦΘ(G/K × Y )

is an isomorphism in UOopF .

Proof. In an object G/H of OopF , the right-hand side is evaluated

(ΦΘ(G/K × Y ))(G/H) = (G/K(G/{e})× Y )H = (G/K(G/{e}))H × Y

and ηG/K×Y is given by

(G/K × idY )(Re) = G/K(Re)× idY .

Hence, it’s enough to show that ηG/K : G/K → ΦΘ(G/K), is an isomor-

phism in UOopF . In the object G/H of Oop

F , this morphism sends Ra ∈G/K(G/H) to RaRe ∈ (G/K(G/{e}))H for a ∈ G with a−1Ha ⊂ K. Oneconcludes that it is an isomorphism by noting that

G/HRa // G/K

G/{e}

Re

OORa

::vvvvvvvvv

commutes and that Ra ∈ G/K(G/{e}) is an H-fixed point if and only ifa−1Ha ⊂ K.

We are ready for the main theorem of this paper.

24

Theorem 7.8 (Elmendorf). The adjunction

Θ: UOopF � UG(F) : Φ

is a pair of Quillen equivalences. In particular, the homotopy categoriesHo(UO

opF ) and Ho(UG(F)) are equivalent.

Proof. By the result a) in the proof of Theorem 5.1, it follows that a mor-phism f in UG is an (acyclic) fibration if and only if f is an (acyclic) fi-bration for all H ∈ F and that Φ(f) is an (acyclic) fibration if and onlyif evG/H(Φ(f)) = fH is an (acyclic) fibration. Thus, Φ preserves (acyclic)fibrations.

Next, given a cofibrant object T of UOopF and a fibrant object X of UG,

we have to show that a map f ′ : T → Φ(X) in UOopF is a weak equivalence

if and only if its adjoint Θ(f ′) : Θ(T )→ X is a weak equivalence in UG. Infact, since in Top every object is fibrant, the category UG shares the sameproperty. So, X can be any G-space. Set

FI :=⋃H∈F{G/H × Sn−1 → G/H ×Dn}n≥0.

Factor the unique map i : ∅ → T from the initial object to T in UOopF as a

cofibration i∞ folllowed by an acyclic fibration p∞ using the Infinite GluingConstruction G∞(FI, i). Note that by Remark 2.11, the functor Θ pre-serves in particular acyclic cofibrations between cofibrant objects. Hence byLemma 2.13, the weak equivalence p∞ is taken to a weak equivalence by Θ.By MC2, the map f ′ is a weak equivalence if and only if f ′p∞ is a weakequivalence and Θ(f ′) is a weak equivalence if and only if Θ(f ′)Θ(p∞) is aweak equivalence. Thus, it’s enough to show that a map f : G∞ → Φ(X) inUO

opF is a weak equivalence if and only if its adjoint Θ(f) : Θ(G∞) → X is

a weak equivalence in UG, where we abbreviated G∞ := G∞(FI, i). Simi-larly, we write Gk := Gk(FI, i), k ≥ 0. Recall that by definition, f is a weakequivalence if fG/H is a weak equivalence in U for all H ∈ F and that Θ(f)is a weak equivalence if (Θ(f))H = (fG/{e})H is a weak equivalence in U forall H ∈ F . Fix H ∈ F . Since f is a natural transformation, the diagram

G∞(G/H)fG/H //

G∞(Re)

��

XH

X∗(Re)

��G∞(G/{e})

fG/{e} // X{e}

commutes. It follows that fG/H = (fG/{e})H(ηG∞)G/H . Thus, by MC2 itsuffices to show that

(ηG∞)G/H : G∞(G/H)→ (G∞(G/{e}))H

25

is a weak equivalence in U or more generally, that it is an isomorphism.Using that G0 = ∅, one deduces that (ηG0)G/H is an isomorphism in U .Recall that as a left adjoint, the functor Θ preserves colimits. Inductively,for k ≥ 1, by using Proposition 6.10 and applying Lemma 7.7, it followsthat (ηGk)G/H is an isomorphism in U . Finally, by Proposition 6.10c), wededuce that (ηG∞)G/H indeed is an isomorphism.

The second assertion holds by Theorem 2.12.

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