Contents

model category

for ∞-groupoids

# 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 $Q \colon \mathcal{C} \to \mathcal{C}$ on a model category $\mathcal{C}$ behaves like an idempotent monad in an appropriate model category theoretic sense, then the left Bousfield localization model category structure of $\mathcal{C}$ at the $Q$-equivalences exists.

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

## Statement

###### Definition

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

1. $Q \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}$

2. $\eta \colon id_{\mathcal{C}} \longrightarrow Q$

such that

1. (homotopical functor) $Q$ preserves weak equivalences;

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

$Q(\eta_X) \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))$

and

$\eta_{Q(X)} \;\colon\; Q(X) \overset{\in W}{\longrightarrow} Q(Q(X))$

are weak equivalences;

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

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

we have that

1. $f$ is a fibration;

2. $\eta_X$, $\eta_Y$, and $Q(h)$ are weak equivalences

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

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

###### Definition

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

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

2. a $Q$-cofibation if it is a cofibration.

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

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

###### Lemma

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

###### Proof

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

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

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

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

###### Lemma

In the situation of def. , if a morphism $f \colon X \longrightarrow Y$ is a fibration, and $\eta_X, \eta_Y$ are weak equivalences, then $f$ is a $Q$-fibration.

###### Proof

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

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

there exists a lifting.

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

$\array{ Q(A) &\overset{Q(\alpha)}{\longrightarrow}& Q(X) \\ {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\mathrlap{Q(f)}} \\ Q(B) &\underset{Q(\beta)}{\longrightarrow}& Q(Y) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ Q(A) &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}& Z &\underoverset{\in Fib}{p_\alpha}{\longrightarrow}& Q(X) \\ {}^{\mathllap{Q(i)}}\downarrow && \downarrow^{\pi} && \downarrow^{\mathrlap{Q(f)}} \\ Q(B) &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}& W &\underoverset{p_\beta}{\in Fib}{\longrightarrow}& Q(Y) }$

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

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

$\array{ \alpha \colon & A &\overset{(j_\alpha \circ \eta_A, \alpha)}{\longrightarrow}& Z \underset{Q(X)}{\times} X &\overset{}{\longrightarrow}& X \\ & {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{(\pi,f)}} && \downarrow^{\mathrlap{f}}& \\ \beta \colon & B &\underset{(j_\beta\circ\eta_B,\beta)}{\longrightarrow}& W \underset{Q(Y)}{\times} Y &\underset{}{\longrightarrow}& Y } \,,$

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

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

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

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

Consider the diagram

$\array{ Q(A) &\underoverset{\in W \cap Cof}{j_\alpha}{\longrightarrow}& Z &\underoverset{\in W}{pr_1}{\longleftarrow}& Z \underset{Q(X)}{\times} X \\ {}^{\mathllap{Q(i)}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\pi}} && \downarrow^{\mathrlap{(\pi,f)}} \\ Q(B) &\underoverset{j_\beta}{\in W \cap Cof}{\longrightarrow}& Z &\underoverset{pr_2}{\in W}{\longleftarrow}& W \underset{Q(X)}{\times} X } \,.$

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 $p_\alpha$ and $p_\beta$ of the morphisms $\eta_X$ and $\eta_Y$, respectively, where the latter are weak equivalences by assumption. Moreover $Q(i)$ is a weak equivalence, since $i$ is a $Q$-weak equivalence.

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

###### Proposition

(Bousfield-Friedlander theorem)

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

###### Proof

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

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

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

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

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

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

Here $i$ is already a $Q$-acyclic $Q$-cofibration. Now apply $Q$ to obtain

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

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

$f \;\colon\; Q(X) \underoverset{\in W_Q \cap Cof_Q}{\in i}{\longrightarrow} Z \underoverset{\in Fib_Q}{p}{\longrightarrow} Q(Y) \,.$

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

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

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

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

$f \;\colon\; \overset{W_Q \cap Cof}{\longrightarrow} \overset{W \cap Fib = W_Q \cap Fib_Q}{\longrightarrow} \overset{Fib_Q}{\longrightarrow} \,.$
###### Proposition

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

1. $f$ is a fibration;

2. the $\eta$-naturality square on $f$

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

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

$X \longrightarrow p^\ast Y$

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

###### Proof

First consider the case that $f$ is a fibration and that the square is a homotopy pullback. We need to show that then $f$ is a $Q$-fibration.

Factor $Q(f)$ as

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

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

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

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

$\tilde i \;\colon\; X \underoverset{\in W \cap Cof}{j}{\longrightarrow} \hat X \underoverset{\in W \cap Fib = W_Q \cap Fib_Q}{\pi}{\longrightarrow} Y \underset{Q(Y)}{\times} Z \,.$

This yields the situation

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

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

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

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

$\array{ X &\underoverset{\in W_Q}{\eta_X}{\longrightarrow}& Q(X) &\underoverset{\in W \subset W_Q}{\eta_{Q(X)}}{\longrightarrow}& Q(Q(X)) \\ {}^{\mathllap{\tilde i}}_{\mathllap{\in W_Q}}\downarrow && {}^{\mathllap{i}}_{\mathllap{\in W\subset W_Q}}\downarrow && \downarrow^{\mathrlap{Q(i)}}_{\mathrlap{\in W}} \\ Y \underset{Q(Y)}{\times} Z &\underoverset{\in W_Q}{p^\ast \eta_Y}{\longrightarrow}& Z &\underoverset{\in W}{\eta_Z}{\longrightarrow}& Q(Z) \\ {}^{\mathllap{\tilde p}}_{\mathllap{\in Fib_Q}}\downarrow &(pb)& \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib_Q \subset Fib}} && \downarrow^{\mathrlap{Q(p)}} \\ Y &\underoverset{\eta_Y}{\in W_Q}{\longrightarrow}& Q(Y) &\underoverset{\eta_{Q(Y)}}{\in W \subset W_Q}{\longrightarrow}& Q(Q(Y)) } \,.$

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

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

where now $Q(p^\ast \eta_Y) \in W$ since $p^\ast \eta_Y \in W_Q$, 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 $\eta$-naturality square of $\tilde p$ is a homotopy pullback. We still need to show that also the $\eta$-naturality square of $f$ is a homotopy pullback.

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

$\array{ X &\underoverset{\in W_Q \cap Cof = W_Q \cap Cof_Q}{j}{\longrightarrow}& \hat X &\underoverset{\in W \cap Fib = W_Q \cap Fib_Q }{\pi}{\longrightarrow}& Y\underset{Q(Y)}{\times} Z } \,.$

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

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

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

Observe that the $\eta$-naturality square of the weak equivalence $\pi$ is a homotopy pullback, since $Q$ 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 $\eta$-naturality square of $\tilde p \circ \pi$ is a homotopy pullback, by the pasting law for homotopy pullbacks (prop.).

In conclusion, we have exhibited $f$ as a retract (in the arrow category, rmk.) of a morphism $\tilde p \circ \pi$ whose $\eta$-naturality square is a homotopy pullback. By naturality of $\eta$, this means that the whole $\eta$-naturality square of $f$ is a retract (in the category of commuting squares in $\mathcal{C}$) 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 $SeqSpectra_{stable}$ 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 $SeqSpectra_{strict}$ transferred from the model structure on sequences (in the classical model structure on simplicial sets/on topological spaces) at $Q$ 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.)

The theorem is due to

• Aldridge Bousfield, Eric Friedlander, section A.3 of Homotopy theory of $\Gamma$-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