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Bousfield-Friedlander theorem

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Modalities, Closure and Reflection

Contents

Idea

In model category-theory, the Bousfield-Friedlander theorem (Bousfield-Friedlander 78, theorem A.7, Bousfield 01, theorem 9.3) states that if an endofunctor Q:𝒞𝒞Q \colon \mathcal{C} \to \mathcal{C} on a model category 𝒞\mathcal{C} behaves like an idempotent monad in an appropriate model category theoretic sense, then the left Bousfield localization model category structure of 𝒞\mathcal{C} at the QQ-equivalences exists.

The original proof assumed that 𝒞\mathcal{C} is a right-proper model category, but it turns out that this condition is not necessary (Stanculescu 08, theorem 1.1).

Statement

Definition

Let 𝒞\mathcal{C} be a proper model category. Say that a Quillen idempotent monad on 𝒞\mathcal{C} is

  1. an endofunctor

    Q:𝒞𝒞Q \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}

  2. a natural transformation

    η:id 𝒞Q\eta \colon id_{\mathcal{C}} \longrightarrow Q

such that

  1. (homotopical functor) QQ preserves weak equivalences;

  2. (idempotency) for all X𝒞X \in \mathcal{C} the morphisms

    Q(η X):Q(X)WQ(Q(X)) Q(\eta_X) \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))

    and

    η Q(X):Q(X)WQ(Q(X)) \eta_{Q(X)} \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))

    are weak equivalences;

  3. (right-properness of the localization) if in a pullback square in 𝒞\mathcal{C}

    f *W f *h X f W h Y \array{ f^\ast W &\stackrel{f^\ast h}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ W &\stackrel{h}{\longrightarrow}& Y }

    we have that

    1. ff is a fibration;

    2. η X\eta_X, η Y\eta_Y, and Q(h)Q(h) are weak equivalences

    then Q(f *h)Q(f^\ast h) is a weak equivalence.

(Here the formulation of the third item follows Bousfield 01, def. 9.2. By lemma below this condition implies that ff is a QQ-fibration, which is the condition required in Bousfield-Friedlander 78 (A.6)).

Definition

For Q:𝒞𝒞Q \colon \mathcal{C} \longrightarrow \mathcal{C} a Quillen idempotent monad according to def. , say that a morphism ff in 𝒞\mathcal{C} is

  1. a QQ-weak equivalence if Q(f)Q(f) is a weak equivalence;

  2. a QQ-cofibation if it is a cofibration.

  3. a QQ-fibration if it has the right lifting property against the morphisms that are both (QQ-)cofibrations as well as QQ-weak equivalences.

Write 𝒞 Q\mathcal{C}_Q for 𝒞\mathcal{C} equipped with these classes of morphisms.

Lemma

In the situation of def. , a morphism is an acyclic fibration in 𝒞 Q\mathcal{C}_Q precisely if it is an acyclic fibration in 𝒞\mathcal{C}.

Proof

Let ff be a fibration and a weak equivalence. Since QQ preserves weak equivalences by condition 1 in def. , ff is also a QQ-weak equivalence. Since QQ-cofibrations are cofibrations, the acyclic fibration ff has right lifting against QQ-cofibrations, hence in particular against against QQ-acyclic QQ-cofibrations, hence is a QQ-fibration.

In the other direction, let ff be a QQ-acyclic QQ-fibration. Consider its factorization into a cofibration followed by an acyclic fibration

f:CofiWFibp. f \colon \underoverset{\in Cof}{i}{\longrightarrow} \underoverset{\in W \cap Fib}{p}{\longrightarrow} \,.

Now the fact that QQ preserves weak equivalences together with two-out-of-three implies that ii is a QQ-weak equivalence, hence a QQ-acyclic QQ-cofibration. This means by assumption that ff has the right lifting property against ii. Hence the retract argument, implies that ff is a retract of the acyclic fibration pp, and so is itself an acyclic fibration.

Lemma

In the situation of def. , if a morphism f:XYf \colon X \longrightarrow Y is a fibration, and η X,η Y\eta_X, \eta_Y are weak equivalences, then ff is a QQ-fibration.

(e.g. Goerss-Jardine 96, chapter X, lemma 4.4).

Proof

We need to show that for every commuting square of the form

A α X W QCof Q i f B β Y \array{ A &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{i}}_{\mathllap{\in W_Q \cap Cof_Q}}\downarrow && \downarrow^{\mathrlap{f}} \\ B &\underset{\beta}{\longrightarrow}& Y }

there exists a lifting.

To that end, first consider a factorization of the image under QQ of this square as follows:

Q(A) Q(α) Q(X) Q(i) Q(f) Q(B) Q(β) Q(Y)Q(A) WCofj α Z Fibp α Q(X) Q(i) π Q(f) Q(B) j βWCof W p βFib Q(Y) \array{ Q(A) &\overset{Q(\alpha)}{\longrightarrow}& Q(X) \\ {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\mathrlap{Q(f)}} \\ Q(B) &\underset{Q(\beta)}{\longrightarrow}& Q(Y) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ Q(A) &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}& Z &\underoverset{\in Fib}{p_\alpha}{\longrightarrow}& Q(X) \\ {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\pi} && \downarrow^{\mathrlap{Q(f)}} \\ Q(B) &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}& W &\underoverset{p_\beta}{\in Fib}{\longrightarrow}& Q(Y) }

(This exists even without assuming functorial factorization: factor the bottom morphism, form the pullback of the resulting p βp_\beta, observe that this is still a fibration, and then factor (through j αj_\alpha) the universal morpism from the outer square into this pullback.)

Now consider the pullback of the right square above along the naturality square of η:idQ\eta \colon id \to Q, take this to be the right square in the following diagram

α: A (j αη A,α) Z×Q(X)X X i (π,f) f β: B (j βη B,β) W×Q(Y)Y Y, \array{ \alpha \colon & A &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}& Z \underset{Q(X)}{\times} X &\overset{}{\longrightarrow}& X \\ & {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{(\pi,f)}} && \downarrow^{\mathrlap{f}}& \\ \beta \colon & B &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}& W \underset{Q(Y)}{\times} Y &\underset{}{\longrightarrow}& Y } \,,

where the left square is the universal morphism into the pullback which is induced from the naturality squares of η\eta on α\alpha and β\beta.

We claim that (π,f)(\pi,f) here is a weak equivalence. This implies that we find the desired lift by factoring (π,f)(\pi,f) into an acyclic cofibration followed by an acyclic fibration and then lifting consecutively as follows

α: A (j αη A,α) Z×Q(X)X X id WCof Fib f A AAAAAAA Y Cof i WFib id β: B (j βη B,β) W×Q(Y)Y Y. \array{ \alpha \colon & A &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}& Z \underset{Q(X)}{\times} X &\overset{}{\longrightarrow}& X \\ & {}^{\mathllap{id}}\downarrow && {}^{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}}& \\ & A &\longrightarrow& &\overset{\phantom{AAAAAAA}}{\longrightarrow}& Y \\ & {}^{\mathllap{i}}_{\mathllap{\in Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{id}}& \\ \beta \colon & B &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}& W \underset{Q(Y)}{\times} Y &\longrightarrow& Y } \,.

To see that (ϕ,f)(\phi,f) indeed is a weak equivalence:

Consider the diagram

Q(A) WCofj α Z Wpr 1 Z×Q(X)X W Q(i) π (π,f) Q(B) j βWCof Z pr 2W W×Q(X)X. \array{ Q(A) &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}& Z &\underoverset{\in W}{pr_1}{\longleftarrow}& Z \underset{Q(X)}{\times} X \\ {}^{\mathllap{Q(i)}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\pi}} && \downarrow^{\mathrlap{(\pi,f)}} \\ Q(B) &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}& Z &\underoverset{pr_2}{\in W}{\longleftarrow}& W \underset{Q(X)}{\times} X } \,.

Here the projections are weak equivalences as shown, because by assumption in def. the ambient model category is right proper and these projections are the pullbacks along the fibrations p αp_\alpha and p βp_\beta of the morphisms η X\eta_X and η Y\eta_Y, respectively, where the latter are weak equivalences by assumption. Moreover Q(i)Q(i) is a weak equivalence, since ii is a QQ-weak equivalence.

Hence now it follows by two-out-of-three (def.) that π\pi and then (π,f)(\pi,f) are weak equivalences.

Proposition

(Bousfield-Friedlander theorem)

For Q:𝒞𝒞Q \colon \mathcal{C} \longrightarrow \mathcal{C} a Quillen idempotent monad according to def. , then 𝒞 Q\mathcal{C}_Q, def. is a model category.

(Bousfield-Friedlander 78, theorem 8.7, Bousfield 01, theorem 9.3, Goerss-Jardine 96, chapter X lemma 4.5, lemma 4.6)

Proof

The existence of limits and colimits is guaranteed since 𝒞\mathcal{C} is already assumed to be a model category. The two-out-of-three poperty for QQ-weak equivalences is an immediate consequence of two-out-of-three for the original weak equivalences of 𝒞\mathcal{C}.

Moreover, according to lemma the pair of classes (Cof Q,W QFib Q)(Cof_{Q}, W_Q \cap Fib_Q) equals the pair (Cof,WFib)(Cof, W \cap Fib), and this is a weak factorization system by the model structure 𝒞\mathcal{C}.

Hence it remains to show that (W QCof Q,Fib Q)(W_Q \cap Cof_Q, \; Fib_Q) is a weak factorization system. The condition Fib Q=RLP(W QCof Q)Fib_Q = RLP(W_Q \cap Cof_Q) holds by definition of Fib QFib_Q. Once we show that every morphism factors as W QCof QW_Q \cap Cof_Q followed by Fib QFib_Q, then the condition W QCof Q=LLP(Fib Q)W_Q \cap Cof_Q = LLP(Fib_Q) follows from the retract argument (and the fact that W QW_Q and Cof QCof_Q are stable under retracts, because WW and Cof=Cof QCof = Cof_Q are).

So we may conclude by showing the existence of (W QCof Q,Fib Q)(W_Q \cap Cof_Q, \; Fib_Q) factorizations:

First we consider the case of a morphism of the form f:Q(Y)Q(Y)f \colon Q(Y) \to Q(Y). This may be factored with respect to 𝒞\mathcal{C} as

f:Q(X)WCofiZFibpQ(Y). f \;\colon\; Q(X) \underoverset{\in W \cap Cof}{\in i}{\longrightarrow} Z \underoverset{\in Fib}{p}{\longrightarrow} Q(Y) \,.

Here ii is already a QQ-acyclic QQ-cofibration. Now apply QQ to obtain

f: Q(X) WCofi Z Fibp Q(Y) W η Q(X) η Z W η Q(Y) Q(Q(X)) Q(i)W Q(Z) Q(Q(Y)), \array{ f \colon & Q(X) &\underoverset{\in W \cap Cof}{i}{\longrightarrow}& Z &\underoverset{\in Fib}{p}{\longrightarrow}& Q(Y) \\ & \downarrow^{\mathrlap{\eta_{Q(X)}}}_{\mathrlap{\in W}} && \downarrow^{\mathrlap{\eta_Z}} && \downarrow^{\mathrlap{\eta_{Q(Y)}}}_{\mathrlap{\in W}} \\ & Q(Q(X)) &\underoverset{Q(i)}{\in W}{\longrightarrow}& Q(Z) &\underset{}{\longrightarrow}& Q(Q(Y)) } \,,

where η Q(X)\eta_{Q(X)} and η Q(Y)\eta_{Q(Y)} are weak equivalences by idempotency, and Q(i)Q(i) is a weak equivalence since QQ preserves weak equivalences. Hence by two-out-of-three also η Z\eta_Z is a weak equivalence. Therefore lemma gives that pp is a QQ-fibration, and hence the above factorization is already as desired

f:Q(X)W QCof QiZFib QpQ(Y). f \;\colon\; Q(X) \underoverset{\in W_Q \cap Cof_Q}{\in i}{\longrightarrow} Z \underoverset{\in Fib_Q}{p}{\longrightarrow} Q(Y) \,.

Now for gg an arbitrary morphism g:XYg \colon X \to Y, form a factorization of Q(g)Q(g) as above and then decompose the naturality square for η\eta on gg into the pullback of the resulting QQ-fibration along η Y\eta_Y:

g: X i˜ Z×Q(Y)Y p˜ Y W Q η X η (pb) W Q η Y Q(g): Q(X) iW Q Z pFib Q Q(Y). \array{ g \colon & X &\overset{\tilde i}{\longrightarrow}& Z \underset{Q(Y)}{\times} Y &\overset{\tilde p}{\longrightarrow}& Y \\ & {}^{\mathllap{\eta_X}}_{\mathllap{\in W_Q}}\downarrow && \downarrow^{\mathrlap{\eta'}}_{\mathrlap{}} &(pb)& \downarrow^{\mathrlap{\eta_Y}}_{\mathrlap{\in W_Q}} \\ Q(g) \colon & Q(X) &\underoverset{i}{\in W_Q}{\longrightarrow}& Z &\underoverset{p}{\in Fib_Q}{\longrightarrow}& Q(Y) } \,.

This exhibits η\eta' as the pullback of a QQ-weak equivalence along a QQ-fibration, and hence itself as a QQ-weak equivalence. This way, two-out-of-three implies that i˜\tilde i is a QQ-weak equivalence.

Finally, apply factorization in (Cof,WFib)(Cof,\; W\cap Fib) to i˜\tilde i to obtain the desired factorization

f:W QCofWFib=W QFib QFib Q. f \;\colon\; \overset{W_Q \cap Cof}{\longrightarrow} \overset{W \cap Fib = W_Q \cap Fib_Q}{\longrightarrow} \overset{Fib_Q}{\longrightarrow} \,.
Proposition

For Q:𝒞𝒞Q \colon \mathcal{C} \longrightarrow \mathcal{C} a Quillen idempotent monad according to def. , then a morphism f:XYf \colon X \to Y in 𝒞\mathcal{C} is a QQ-fibration (def. ) precisely if

  1. ff is a fibration;

  2. the η\eta-naturality square on ff

    X η X Q(X) f (pb) h Q(f) Y η Y Q(Y) \array{ X &\stackrel{\eta_X}{\longrightarrow}& Q(X) \\ {}^{\mathllap{f}}\downarrow &{}^{(pb)^h}& \downarrow^{\mathrlap{Q(f)}} \\ Y &\underset{\eta_Y}{\longrightarrow}& Q(Y) }

    exhibits a homotopy pullback in 𝒞\mathcal{C} (def.), in that for any factorization of Q(f)Q(f) through a weak equivalence followed by a fibration pp, then the universally induced morphism

    Xp *Y X \longrightarrow p^\ast Y

    is weak equivalence (in 𝒞\mathcal{C}).

(e.g. Goerss-Jardine 96, chapter X, theorem 4.8)

Proof

First consider the case that ff is a fibration and that the square is a homotopy pullback. We need to show that then ff is a QQ-fibration.

Factor Q(f)Q(f) as

Q(f):Q(X)WCofiZFibpQ(Y). Q(f) \;\colon\; Q(X) \underoverset{\in W \cap Cof}{i}{\longrightarrow} Z \underoverset{\in Fib}{p}{\longrightarrow} Q(Y) \,.

By the proof of prop. , the morphism pp is also a QQ-fibration. Hence by the existence of the QQ-local model structure, also due to prop. , its pullback p˜\tilde p is also a QQ-fibration

X η X Q(X) W i˜ W i Y×Q(Y)Z p *η Y Z Fib Q p˜ (pb) Fib Q p Y η Y Q(Y). \array{ X &\overset{\eta_X}{\longrightarrow}& Q(X) \\ {}^{\mathllap{\tilde i}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{i}}_{\mathrlap{\in W}} \\ Y \underset{Q(Y)}{\times} Z &\overset{p^\ast \eta_Y}{\longrightarrow}& Z \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow &(pb)& \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q}} \\ Y &\underset{\eta_Y}{\longrightarrow}& Q(Y) } \,.

Here i˜\tilde i is a weak equivalence by assumption that the diagram exhibits a homotopy pullback. Hence it factors as

i˜:XWCofjX^WFib=W QFib QπY×Q(Y)Z. \tilde i \;\colon\; X \underoverset{\in W \cap Cof}{j}{\longrightarrow} \hat X \underoverset{\in W \cap Fib = W_Q \cap Fib_Q}{\pi}{\longrightarrow} Y \underset{Q(Y)}{\times} Z \,.

This yields the situation

X = X WCof j Fib f X^ p˜πFib Q YX j X^ X f p˜π f Y = Y = Y. \array{ X &\overset{=}{\longrightarrow}& X \\ {}^{\mathllap{j}}_{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ \hat X &\underoverset{\tilde p \circ \pi}{\in Fib_Q}{\longrightarrow}& Y } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ X &\overset{j}{\longrightarrow}& \hat X &\overset{\exists}{\longrightarrow}& X \\ {}^{\mathllap{}f}\downarrow && \downarrow^{\mathrlap{\tilde p \circ \pi}} && \downarrow^{\mathrlap{f}} \\ Y &=& Y &=& Y } \,.

As in the retract argument (prop.) this diagram exhibits ff as a retract (in the arrow category, rmk.) of the QQ-fibration p˜π\tilde p \circ \pi. Hence by the existence of the QQ-model structure (prop. ) and by the closure properties for fibrations (prop.), also ff is a QQ-fibration.

Now for the converse. Assume that ff is a QQ-fibration. Since 𝒞 Q\mathcal{C}_Q is a left Bousfield localization of 𝒞\mathcal{C} (prop. ), ff is also a fibration. We need to show that the η\eta-naturality square on ff exhibits a homotopy pullback.

So factor Q(f)Q(f) as before, and consider the pasting composite of the factorization of the given square with the naturality squares of η\eta:

X W Qη X Q(X) WW Qη Q(X) Q(Q(X)) W Q i˜ WW Q i W Q(i) Y×Q(Y)Z W Qp *η Y Z Wη Z Q(Z) Fib Q p˜ (pb) Fib QFib p Q(p) Y η YW Q Q(Y) η Q(Y)WW Q Q(Q(Y)). \array{ X &\underoverset{\in W_Q}{\eta_X}{\longrightarrow}& Q(X) &\underoverset{\in W \subset W_Q}{\eta_{Q(X)}}{\longrightarrow}& Q(Q(X)) \\ {}^{\mathllap{\tilde i}}_{\mathllap{\in W_Q}}\downarrow && {}^{\mathllap{i}}_{\mathllap{\in W\subset W_Q}}\downarrow && \downarrow^{\mathrlap{Q(i)}}_{\mathrlap{\in W}} \\ Y \underset{Q(Y)}{\times} Z &\underoverset{\in W_Q}{p^\ast \eta_Y}{\longrightarrow}& Z &\underoverset{\in W}{\eta_Z}{\longrightarrow}& Q(Z) \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow &(pb)& \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q \subset Fib}} && \downarrow^{\mathrlap{Q(p)}} \\ Y &\underoverset{\eta_Y}{\in W_Q}{\longrightarrow}& Q(Y) &\underoverset{\eta_{Q(Y)}}{\in W \subset W_Q}{\longrightarrow}& Q(Q(Y)) } \,.

Here the top and bottom horizontal morphisms are weak (QQ-)equivalences by the idempotency of QQ, and Q(i)Q(i) is a weak equivalence since QQ preserves weak equivalences (first and second clause in def. ). Hence by two-out-of-three also η Z\eta_Z is a weak equivalence. From this, lemma gives that pp is a QQ-fibration. Then p *η Yp^\ast \eta_Y is a QQ-weak equivalence since it is the pullback of a QQ-weak equivalence along a fibration between objects whose η\eta is a weak equivalence, via the third clause in def. . Finally two-out-of-three implies that i˜\tilde i is a QQ-weak equivalence.

In particular, the bottom right square is a homotopy pullback (since two opposite edges are weak equivalences, by this prop.), and since the left square is a genuine pullback of a fibration, hence a homotopy pullback, the total bottom rectangle here exhibits a homotopy pullback by the pasting law for homotopy pullbacks (prop.).

Now by naturality of η\eta, that total bottom rectangle is the same as the following rectangle

Y×Q(Y)Z η (Q×Q(Y)Z) Q(Y×Q(Y)Z) WQ(p *η Y) Q(Z) Fib Q p˜ Q(p˜) Q(p) Y η Y Q(Y) Q(η Y)W Q(Q(Y)), \array{ Y \underset{Q(Y)}{\times} Z &\overset{\eta_{\left(Q \underset{Q(Y)}{\times} Z\right)}}{\longrightarrow}& Q(Y \underset{Q(Y)}{\times} Z) &\underoverset{\in W}{Q(p^\ast \eta_Y)}{\longrightarrow}& Q(Z) \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow && \downarrow^{\mathrlap{Q(\tilde p)}}_{\mathrlap{}} && \downarrow^{\mathrlap{Q(p)}} \\ Y &\underset{\eta_Y}{\longrightarrow}& Q(Y) &\underoverset{Q(\eta_Y)}{\in W}{\longrightarrow}& Q(Q(Y)) } \,,

where now Q(p *η Y)WQ(p^\ast \eta_Y) \in W since p *η YW Qp^\ast \eta_Y \in W_Q, as we had just established. This means again that the right square is a homotopy pullback (prop.), and since the total rectangle still is a homotopy pullback itself, by the previous remark, so is now also the left square, by the other direction of the pasting law for homotopy pullbacks (prop.).

So far this establishes that the η\eta-naturality square of p˜\tilde p is a homotopy pullback. We still need to show that also the η\eta-naturality square of ff is a homotopy pullback.

Factor i˜\tilde i as a cofibration followed by an acyclic fibration. Since i˜\tilde i is also a QQ-weak equivalence, by the above, two-out-of-three for QQ-fibrations gives that this factorization is of the form

X W QCof=W QCof Qj X^ WFib=W QFib Qπ Y×Q(Y)Z. \array{ X &\underoverset{\in W_Q \cap Cof = W_Q \cap Cof_Q}{j}{\longrightarrow}& \hat X &\underoverset{\in W \cap Fib = W_Q \cap Fib_Q }{\pi}{\longrightarrow}& Y\underset{Q(Y)}{\times} Z } \,.

As in the first part of the proof, but now with (WCof,Fib)(W \cap Cof, Fib) replaced by (W QCof Q,Fib Q)(W_Q \cap Cof_Q, Fib_Q) and using lifting in the QQ-model structure, this yields the situation

X = X W QCof Q j Fib Q f X^ p˜π YX j X^ X f p˜π f Y = Y = Y. \array{ X &\overset{=}{\longrightarrow}& X \\ {}^{\mathllap{j}}_{\mathllap{\in W_Q \cap Cof_Q}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib_Q}} \\ \hat X &\underoverset{\tilde p \circ \pi}{}{\longrightarrow}& Y } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\;\;\;\;\;\; \array{ X &\overset{j}{\longrightarrow}& \hat X &\overset{\exists}{\longrightarrow}& X \\ {}^{\mathllap{}f}\downarrow && \downarrow^{\mathrlap{\tilde p \circ \pi}} && \downarrow^{\mathrlap{f}} \\ Y &=& Y &=& Y } \,.

As in the retract argument (prop.) this diagram exhibits ff as a retract (in the arrow category, rmk.) of p˜π\tilde p \circ \pi.

Observe that the η\eta-naturality square of the weak equivalence π\pi is a homotopy pullback, since QQ preserves weak equivalences (first clause of def. ) and since a square with two weak equivalences on opposite sides is a homotopy pullback (prop.). It follows that also the η\eta-naturality square of p˜π\tilde p \circ \pi is a homotopy pullback, by the pasting law for homotopy pullbacks (prop.).

In conclusion, we have exhibited ff as a retract (in the arrow category, rmk.) of a morphism p˜π\tilde p \circ \pi whose η\eta-naturality square is a homotopy pullback. By naturality of η\eta, this means that the whole η\eta-naturality square of ff is a retract (in the category of commuting squares in 𝒞\mathcal{C}) of a homotopy pullback square. This means that it is itself a homotopy pullback square (prop.).

Examples

Stable model structure on sequential spectra

The Bousfield-Friedlander model structure SeqSpectra stableSeqSpectra_{stable} on sequential spectra (in any proper, pointed simplicial model category), modelling stable homotopy theory, arises via the Bousfield-Friedlander theorem from localizing the strict model structure SeqSpectra strictSeqSpectra_{strict} transferred from the model structure on sequences (in the classical model structure on simplicial sets/on topological spaces) at QQ being the spectrification endofunctor.

(For pre-spectra in the classical model structure on simplicial sets, spectrification is readily defined, more generally one needs to prooceed as in Schwede 97, section 2.1.)

References

The theorem is due to

  • Aldridge Bousfield, Eric Friedlander, section A.3 of Homotopy theory of Γ\Gamma-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

and in improved form due to

  • Aldridge Bousfield, section 9 of On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391–2426 (AMS, jstor)

The right-properness condition is shown to be unnecessary in

  • Alexandru E. Stanculescu, Note on a theorem of Bousfield and Friedlander, Topology Appl. 155 (2008), no. 13, 1434–1438 (arXiv:0806.4547)

Textbook accounts include

Last revised on May 18, 2018 at 04:15:30. See the history of this page for a list of all contributions to it.