nLab Bousfield localization of model categories

Idea

Bousfield localization is a procedure that to a model category structure $C$ assigns a new one with more weak equivalences. It is a special case of a localization of model categories, corresponding to the homotopy-version of the notion of localization of categories by reflective subcategories: reflective localization. In fact, left Bousfield localization is a Quillen reflection.

The historically original example is the Bousfield localization of spectra. But the notion is much more general.

Definition

Definition

A left Bousfield localization $C_{loc}$ of a model category $C$ is another model category structure on the same underlying category with the same cofibrations,

$$cof_{C_{loc}} = cof_c$$

but more weak equivalences

$$W_{C_{loc}} \supset W_C \,.$$

While that’s a very simple definition, it turns out that something interesting happens to the fibrations when we keep the cofibrations fixed and increase the weak equivalences.

Remark

It follows directly that

• $C_{loc}$ has as fibrations a sub-class of the fibrations of $C$

  $$fib_{C_{loc}} = rlp(cof_{C_{loc}} \cap W_{C_{loc}}) \subset rlp(cof_{C_{loc}} \cap W_C) = fib_{C} \,.$$

• $C_{loc}$ has the same acyclic fibrations as $C$

  $$fib_{C_{loc}} \cap W_{C_{loc}} = rlp(cof_{C_{loc}}) = rlp(cof_C) = fib_C \cap W_C \,.$$

• on the underlying categories

  • the identity functor $Id : C \to C_{loc}$ preserves cofibrations and weak equivalences

  • the identity functor $Id : C_{loc} \to C$ preserves fibrations and acyclic fibrations

  so that this pair of functors is a Quillen adjunction

  $$C_{loc} \stackrel{\leftarrow}{\to} C \,,$$

and a very special one: With $C^\circ$ the full subcategory on fibrant-cofibrant objects, under left Bousfield localization the fibrant-cofibrant objects of $C_{loc}$ are a subcollection of those of $C$, so that we have the full subcategory

$$(C_{loc})^\circ \subset C^\circ \,.$$

Moreover, as we shall see, every object in $C$ is weakly equivalent in $C_{loc}$ to one in $C_{loc}$: it reflects into $C_{loc}$ .

Bousfield localization is a model category version of reflection onto local objects, in the sense discussed at reflective localization.

Indeed – at least when $C$ is a combinatorial simplicial model category – Bousfield localization is an example of a localization of a simplicial model category and under passage to the sub-category of fibrant-cofibrant objects this Quillen adjunction becomes the inclusion of a reflective (∞,1)-subcategory

$${C_{loc}}^\circ \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} C^\circ$$

hence of a localization of an (∞,1)-category.

Such a localization is determined by the collection $S$ of local weak equivalences in $C$, and alternatively by the collection of $S$-local objects in $C$. Indeed, ${C_{loc}}^\circ$ is the full $(\infty,1)$-subcategory on the cofibrant and fibrant and $S$-local objects of $C$.

Localization at $S$-local weak equivalences

More in detail, the weak equivalences that are added under Bousfield localization are ‘’$S$-local weak equivalences“ for some set $S \subset Mor(C)$. We will see below why this is necessarily the case if $C$ is a cofibrantly generated model category. For the moment, we take the following to be a refined definition of left Bousfield localization.

Let $C$ be a

• left proper

• cofibrantly generated

• simplicial model category.

• $S \subset cof_C \subset Mor(C)$ be a subclass of cofibrations with cofibrant domain.

We want to characterize objects in $C$ that “see elements of $S$ as weak equivalences”. Notice that

Lemma

In an ordinary category $C$, by the Yoneda lemma a morphism $f : A \to B$ is an isomorphism precisely if for all objects $X$ the morphism

$$Hom_C(f,X) : Hom_C(B,X) \to Hom_C(A,X)$$

is an isomorphism (of sets, i.e. a bijection).

So we can “test isomorphism by homming them into objects”. This phenomenon we use now the other way round, to characterize new weak equivalences:

Definition

($S$-local objects and $S$-local weak equivalences)

Say that

• a fibrant object $X$ is an $S$-local object if for all $s : A \hookrightarrow B$ in $S$ the morphism

  $$C(s,X) : C(B,X) \to C(A,X)$$

  is a trivial Kan fibration;

• conversely, say that a cofibration $f : A \hookrightarrow B$ is an $S$-local weak equivalence if for all $S$-local fibrant objects $X$ the morphism $C(f,X) : C(B,X) \to C(A,X)$ is a trivial Kan fibration.

This is a slightly simplified version of a more general definition using derived hom spaces, where we do not have to assume that the domains and codomains of elements are $S$ and not that the local objects are fibrant.

Definition

($S$-local objects and $S$-local weak equivalences)

Assume that we have fibrant and cofibrant replacement functors $P,Q : C \to C$. Then say

• an object $X$ is an $S$-local object if for all $s : A \hookrightarrow B$ in $S$ the morphism

  $$C(Q s,P X) : C(Q B,P X) \to C(Q A,P X)$$

  is a weak equivalence of simplicial sets;

• conversely, say that a cofibration $f : A \hookrightarrow B$ is an $S$-local weak equivalence if for all $S$-local objects $X$ the morphism $C(Q f,P X) : C(Q B,P X) \to C(Q A,P X)$ is a weak equivalence.

That this second condition is indeed compatible with the first one is shown here.

We write $W_S$ for the collection of $S$-local weak equivalences.

Remark

For every weak equivalence $f : A \stackrel{\simeq}{\to} B$ between cofibrant objects and every fibrant object $X$ it follows generally from the fact that $C$ is an SSet-enriched category that

$$C(f,X) : C(B,X) \to C(A,X)$$

is a weak equivalence of simplicial sets. This is described in detail at enriched homs from cofibrants to fibrants.

Here this implies in particular

Lemma

Every ordinary weak equivalence is also $S$-local weak equivalence.

$$W \subset W_S \,.$$

Therefore, for any set $S$, we can consider the left Bousfield localization at the $S$-local weak equivalences $W_S$:

Definition

(left Bousfield localization)

The left Bousfield localization $L_S C$ of $C$ at $S$ is, if it exists, the new model category structure on $C$ with

• cofibrations are the same as before, $cof_{L_S C } = cof_C$;

• acyclic cofibrations are the cofibrations that are $S$-local weak equivalences.

Properties

Assume that the left Bousfield localization $L_S C$ of a given model category at a class $S$ of cofibrations with cofibrant domain exists. Then it has the following properties.

Fibrants in $L_S C$ are the $S$-local fibrants in $C$

Proposition

The fibrant objects in $L_S C$ are precisely the fibrant objects in $C$ that are $S$-local.

Proof

To see this, we modify, if necessary, the set $S$ in a convenient way without changing the class $W_S$ of $S$-local weak equivalences that it defines.

Lemma We may add to $S$ any set of $S$-local cofibrations without changing the collection of $S$-local objects and hence without changing the collection of $S$-local weak equivalences themselves. In particular, we may add to $S$ without changing $W_S$

• all generating acyclic cofibrations of $C$, i.e. $J \subset S$;

• for every original morphism $f : A \to B$ in $S$ and for every $n \in \mathbb{N}$ also the canonical morphism

  $$\tilde f \;\colon\; \big( Q_f \coloneqq A \cdot \Delta^n \coprod_{A \cdot \partial \Delta^n} B \cdot \partial \Delta^n \big) \to B \cdot \Delta^n \,,$$

  where $A \cdot \Delta^n$ etc. denotes the tensoring of $C$ over SSet.

Proof of the Lemma

We discuss why these morphisms of the latter type are indeed $S$-local cofibrations with cofibrant domain:

to see that $\tilde f$ is indeed a cofibration notice that for every commuting diagram

$$\array{ Q_f &\to& X \\ \downarrow && \downarrow^{\in \mathrlap{fib_C \cap W_C}} \\ B \cdot \Delta^n &\to& Y }$$

we get as components of the top morphism the left square of

$$\array{ \partial \Delta^n &\to& C(B,X) &\to& C(B,Y) \\ \downarrow && \downarrow && \downarrow \\ \Delta^n &\to& C(A,X) &\to& C(A,Y) }$$

and similarly the components of the bottom morphism consitute a morphism $\Delta^n \to C(B,X)$ which by the commutativity of the original square is a lift of the outer diagram here. The top left triangle of this lift in turn gives a square

$$\array{ \partial \Delta^n &\to& C(B,X) \\ \downarrow^{\mathllap{\in cof_{SSet} \cap W_{SSet}\quad}} && \quad \downarrow^{\in fib_{SSet}} \\ \Delta^n &\to& C(A,X) } \,.$$

So this last diagram has a lift $(\Delta^n \to C(B,X))$ and this is adjunct to the lift $B \cdot \Delta^n \to X$ of the original lifting problem that we are looking for.

Therefore $\tilde f : Q_f \to B \cdot \Delta^n$ is indeed a cofibration.

Notice that in these arguments we made use of

• the powering and copowering of the simplicially enriched category $C$ over SSet

• and of the Quillen bifunctor property of the copowering which ensures that the fibrations and cofibrations are as indicated.

Next, again using the Quillen bifunctor property of the tensoring of $C$ over SSet we find that with $A$ cofibrant in $C$ and $\Delta^n$ being cofibrant in SSet it follows that $A \cdot \Delta^n$ is cofibrant; similarly for the other cases. The coproduct of two cofibrant objects is cofibrant because cofibrations are preserved under pushout. Therefore $Q_f$ is indeed a cofibrant domain of our cofibration.

With $\tilde f$ being a cofibration, we can check $S$-locality by homming into fibrant $S$-local objects and checking if that produces an acyclic Kan fibration.

So let $X$ be a fibrant and $S$-local object of $C$. Homming the defining pushout diagram for $Q_f$ into $X$ produces the pullback diagram

$$\array{ [\partial \Delta^n,[A,X]] &\stackrel{\in W}{\leftarrow}& [\partial \Delta^n, [B,X]] \\ \uparrow^{\mathrlap{\in fib}} && \uparrow \\ [\Delta^n,[A,X]] &\stackrel{\in W}{\leftarrow}& [A \cdot \Delta^n \coprod_{A \cdot \partial \Delta^n} B \cdot \partial \Delta^n, X] \\ &{}_{\in W}\nwarrow&& \nwarrow \\ &&&& [B \cdot \Delta^n, X] }$$

in SSet. Here the top and the lowest morphisms are weak equivalences by the fact that $[B,X] \to [A,X]$ is an acyclic Kan fibration by the characterization of $S$-local cofibrations and the fact that SSet is an SSet-enriched model category. Similarly for the fibration on the left, which implies by right properness of SSet that the bottom horizontal morphism is a weak equivalence, which finally implies by 2-out-of-3 that the morphism in question is a weak equivalence.

end of the proof of the lemma

This shows that we can assume that $S$ contain the generating acyclic cofibrations and the morphism called $\tilde f$.

As usual, we say that given a set of morphisms $S$ and an object $X$ that $X$ has the extension property with respect to $S$ if every diagram

$$\array{ A &\to& X \\ \downarrow^{\mathrlap{\in S}} && \downarrow \\ B &\to& {*} }$$

has a lift.

We claim now that the the objects of $C$ that have the extension property with respect to our set $S$ are precisely the fibrant and $S$-local objects. The argument proceeds along the same lines as the proof of the above lemma.

In one direction, if $X$ that has the extension property with respect to $S$ it has it in particular with respect to the generating acyclic cofibrations $J \subset S$ and hence is fibrant, and it, in particular, has the extension property with respect to $\tilde f : A \cdot \Delta^n \coprod_{A \cdot \partial \Delta^n} B \cdot \partial \Delta^n \to B \cdot \Delta^n$. Observe that by the pushout definition of $Q_f$ a morphism

$$Q_f \to X$$

consists of two component maps $(A \cdot \Delta^n \to X)$ and $(B \cdot \partial \Delta^n \to X)$ such that

$$\array{ \partial \Delta^n &\to& C(B,X) \\ \downarrow && \downarrow \\ \Delta^n &\to& C(A,X) } \,,$$

and in terms of this a lift

$$\array{ Q_f &\to& X \\ \downarrow & \nearrow_{\exists} \\ B \cdot \Delta^n }$$

consists of a lift

$$\array{ \partial \Delta^n &\to& C(B,X) \\ \downarrow &\nearrow_\exists & \downarrow \\ \Delta^n &\to& C(A,X) } \,.$$

Since $\{\partial \Delta^n \to \Delta^n | n \in \mathbb{N}\}$ are the generating acylic fibrations in the standard model structure on simplicial sets, this shows the extension property of $S$ with respect to all $\tilde f$ means that all $C(s,X) : C(B,X) \to C(A,X)$ are acyclic Kan fibrations.

Conversely, if $X$ is fibrant and $S$-local, then for all $A \to B$ in $S$ the map $[B,X] \to [A,X]$ in $SSet$ is an acyclic Kan fibration hence in particular its underlying map of sets $Hom_C(B,X) \to Hom_C(A,X)$ is a surjection, so $X$ has the extension property.

Now every fibrant object $X$ in $L_S W$ has the extension property with respect to $cof_C \cap W_S$ hence in particular with respect to $S \subset cof_c \cap W_S$, so is $S$-local and fibrant in $C$.

Conversely, if it is $S$-local and fibrant in $C$; then, as mentioned before, for all $f \in cof_C \cap W_S$ the map $[f,X]$ is an acyclic Kan fibration in SSet so that in particular $Hom_C(f,X)$ is a surjection, which means that $X$ has the extension property with respect to all $f$ and is hence fibrant in $L_S C$.

Fibrant replacement in $L_S W$ – localization of objects

If $S$ is a small set, we may apply the small object argument to $S$. If we apply it to factor all morphisms $X \to {*}$ to the terminal object we obtain a functorial factorization componentwise of the form

$$X \stackrel{\eta_X \in cell(S)\subset W_S}{\to} T X \stackrel{inj(S)}{\to} {*} \,.$$

We had remarked already in the previous argument that objects with the extension property relative to $S$, i.e. objects whose morphism to the terminal object is in $inj(S)$, are fibrant as well as $S$-local in $C$.

Therefore $T$ is in particular a fibrant approximation functor in $L_S W$ and $\eta_S$ is the weak equivalence

$$\eta_S : X \stackrel{\simeq_{W_S}}{\to} T X$$

in $L_S C$ relating an object to its fibrant approximation.

More precisely:

Proposition

Let

$$\mathcal{C}_{loc} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot} \mathcal{C}$$

be a Quillen adjunction which exhibits a left Bousfield localization of model categories, and assume that $\mathcal{C}_{loc}$ admits functorial factorization (for instance if $\mathcal{C}$ is a combinatorial model category, whence $\mathcal{C}_{loc}$ is, then via the small object argument), hence in particular a fibrant replacement natural transformation

$$(-) \longrightarrow (-)^{fib} \,.$$

Then the derived adjunction unit, i.e. the adjunction unit $\eta^{der}$ of adjoint pair of the derived functors on the homotopy category (as discussed there)

$$Ho(\mathcal{C}_{loc}) \underoverset {\underset{}{\hookrightarrow}} {\overset{}{\longleftarrow}} {\bot} Ho(\mathcal{C})$$

is isomorphic to the image of the fibrant replacement morphism in $\mathcal{C}_{loc}$:

$$\eta^{der}_X \simeq \ell(X \to X^{fib}) \,,$$

where $\ell \;\colon\; \mathcal{C}_{loc} \to Ho(\mathcal{C}_{loc})\;$ is the localization functor (as discussed at homotopy category of a model category).

Proof

First consider the general prescription (from homotopy category of a model category) of computing the right (left) derived functors by applying the Quillen functors to fibrant (cofibrant) replacements, where we apply the fibrant replacement functorially also to the non-cofibrantly replaced object:

Let

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}$$

be a general Quillen adjunction betwen model categories which admit functorial factorization.

Let $X \in \mathcal{D}$ be any object. First consider a cofibrant replacement

$$\array{ X_{cof} \\ \downarrow^{\mathrlap{\in \mathrm{W} \cap Fib}} \\ X } \,.$$

Then apply $L$ to this

$$\array{ L(X_{cof}) \\ \downarrow^{} \\ L(X) } \,.$$

Then apply fibrant replacement functorially

$$\array{ L(X_{cof}) &\overset{\in W \cap Cof}{\longrightarrow}& (L(X_{cof}))^{fib} \\ \downarrow^{} && \downarrow \\ L(X) &\underset{\in W \cap Cof}{\longrightarrow}& (L(X))^{fib} } \,.$$

Then apply $R$

$$\array{ R L (X_{cof}) &\overset{}{\longrightarrow}& R (L X_{cof})^{fib} \\ \downarrow^{\mathrlap{}} && \downarrow \\ R L X &\underset{}{\longrightarrow}& R((L X)^{fib}) }$$

Finally, precompose with the ordinary adjunction unit. The derived adjunction unit is now modeled by the top composite morphism in the following diagram:

$$\array{ X_{cof} &\overset{\eta_{X_{cof}}}{\longrightarrow}& R L (X_{cof}) &\overset{R (j \in W \cap Cof)}{\longrightarrow}& R (L X_{cof})^{fib} \\ \downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow \\ X &\underset{\eta_X}{\longrightarrow}& R L X &\underset{}{\longrightarrow}& R((L X)^{fib}) } \,.$$

Now in the special case that $(L \dashv R)$ is a left Bousfield localization of model categories, then as plain categories $\mathcal{D} = \mathcal{C}$ and as plain functors $L = id$ and $R = id$ are trivial and so in this special case the derived adjunction unit is modeled simply by the top morphism in the following diagram

$$\array{ X_{cof} &\overset{\in Cof}{\longrightarrow}& (X_{cof})^{fib} \\ {}^{\mathllap{\in W \cap Fib}}\downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} \\ X &\underset{\in W Cof}{\longrightarrow}& X^{fib} } \,.$$

But now since the vertical morphisms are weak equivalences, this means that already the fibrant replacement $X \to X^{fib}$ is isomorphic, in the homotopy category, to the derived adjunction unit, i.e. applying the localization $\ell$ to the above diagram in $\mathcal{C}$ yields the diagram

$$\array{ \mathcal{l}(X_{cof}) &\overset{\eta^{der}_X}{\longrightarrow}& \mathcal{l}((X_{cof})^{fib}) \\ {}^{\simeq}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \mathcal{l}(X) &\underset{}{\longrightarrow}& \mathcal{l}(X^{fib}) } \,.$$

$S$-local weak equivalences between $S$-local objects are weak equivalences

Proposition

The $S$-local weak equivalences between $S$-local fibrant objects are precisely the original weak equivalences between these objects.

Proof

Consider the full subcategory $Ho_S(C) \subset Ho(C)$ of the homotopy category of $C$ on the $S$-local objects. The image of an $S$-local weak equivalence $f : A \to B$ in there satisfies for every object $X$ in there that $Hom_{Ho_S(C)}(f,X)$ is an isomorphism. By the Yoneda lemma this implies that $f$ is an isomorphism in $Ho_S(C)$. Since that is a full subcategory, it follows that $f$ is also an isomorphism in $Ho(C)$. But that means precisely that it is a weak equivalence in $C$.

It follows that also the fibrations between local objects remain the same:

Proposition

Consider a left Bousfield localization with functorial factorization (e.g. of a combinatorial model category, via the small object argument).

Then if $X, Y \in Fib_{loc}$ are local objects, a morphism $p \colon X\longrightarrow Y$ between them is a fibration with respect to the local model structure precisely already if it is a fibration with respect to the original model structure.

Proof

From remark we already know that $Fib_{loc} \subset Fib$, generally. Hence we need to show that if $p \in Fib$ with $X$ and $Y$ local, then $p \in Fib_{loc}$.

So given a lifting problem of the form

$$\array{ A &\overset{f}{\longrightarrow}& X \mathrlap{\in Fib_{loc}} \\ {}^{\mathllap{\in W_{loc} \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib}} \\ B &\underset{g}{\longrightarrow}& Y \mathrlap{\in Fib_{loc}} }$$

we need to exhibit a lift. (In labeling the arrows we use throughout that $Cof_{loc} = Cof$.)

By assumption of functorial factorization we may factor this diagram as follows:

$$\array{ A &\overset{\in W_{loc} \cap Cof }{\longrightarrow}& \widehat{A} &\overset{\in Fib_{loc}}{\longrightarrow}& X \mathrlap{\in Fib_{loc}} \\ {}^{\mathllap{\in W_{loc} \cap Cof}} \downarrow && \downarrow^{\mathrlap{ }} && \downarrow^{\mathrlap{\in Fib}} \\ B &\underset{\in W_{loc} \cap Cof}{\longrightarrow}& \widehat{B} &\underset{ \in Fib_{loc} }{\longrightarrow}& Y \mathrlap{\in Fib_{loc}} }$$

It follows that $\widehat{A}, \widehat{B} \in Fib_{loc}$.

Consider next the further factorization of the middle vertical morphism as $\overset{W_{loc} \cap Cof}{\longrightarrow} \overset{\in Fib_{loc}}{\longrightarrow}$

$$\array{ A &\overset{\in W_{loc} \cap Cof }{\longrightarrow}& \widehat{A} &\overset{\in Fib_{loc}}{\longrightarrow}& X \mathrlap{\in Fib_{loc}} \\ {}^{\mathllap{id}}\downarrow && {}^{\mathllap{\in W \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib}} \\ A &\longrightarrow& \widehat{\widehat{A}} &\longrightarrow& Y \\ {}^{\mathllap{\in W_{loc} \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib_{loc}}} && \downarrow^{\mathrlap{id}} \\ B &\underset{\in W_{loc} \cap Cof}{\longrightarrow}& \widehat{B} &\underset{ \in Fib_{loc} }{\longrightarrow}& Y \mathrlap{\in Fib_{loc}} }$$

Since it follows that $\widehat A, \widehat{\widehat A} \in Fib_{loc}$ we invoke prop. to conclude that the top middle morphism is not just in $W_{loc} \cap Cof$ but indeed in $W \cap Cof$, as shown. This means that we have lifting in the top right square. Moreover, we also evidently have lifting in the bottom left square.

$$\array{ A &\overset{\in W_{loc} \cap Cof }{\longrightarrow}& \widehat{A} &\overset{\in Fib_{loc}}{\longrightarrow}& X \mathrlap{\in Fib_{loc}} \\ {}^{\mathllap{id}}\downarrow && {}^{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{\in Fib}} \\ A &\longrightarrow& \widehat{\widehat{A}} &\longrightarrow& Y \\ {}^{\mathllap{\in W_{loc} \cap Cof}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{\in Fib_{loc}}} && \downarrow^{\mathrlap{id}} \\ B &\underset{\in W_{loc} \cap Cof}{\longrightarrow}& \widehat{B} &\underset{ \in Fib_{loc} }{\longrightarrow}& Y \mathrlap{\in Fib_{loc}} }$$

Together, these lifts constitute the desired total lift.

Every Bousfield localization is of this form

We have considered two definitions of left Bousfield localization: in the first we just required that cofibrations are kept and weak equivalences are increased. In the second we more specifically took the weak equivalences to be $S$-local weak equivalences.

We now show that every localization in the first sense is indeed of the second kind if we demand that both the original and the localized category are left proper, cofibrantly generated simplicial model categories.

Proposition

In the context of left proper, cofibrantly generated simplicial model categories,

for $C_{loc}$ a left Bousfield localization of $C$ (i.e. a structure with the same cofibrations as $C$ and more weak equivalences), there is a set $S \subset Mor(C)$ such that

$$C_{loc} = L_S C \,.$$

Proof

We show that choosing $S = J_{C_{loc}}$ to be the set of generating acyclic cofibrations does the trick.

First, the cofibrations of $C_{loc}$ and $L_S C$ coincide. Moreover, the acyclic cofibrations of $L_S C$ contain all the acyclic cofibrations of $C_{loc}$ because

$$cof_{L_S C} \cap W_{L_S C} = llp(rlp(J_{L_S C})) \subset llp(rlp(S)) = llp(rlp(J_{C_{loc}})) \,.$$

It remains to show that, conversely, every acyclic cofibration $f : X \to Y$ in $L_S C$ is an acyclic cofibration in $C_{loc}$.

Choose a cofibrant replacement for $X$ and $Y$

$$\array{ X' &\stackrel{\in cof_C}{\to}& Y' \\ \downarrow^{\mathrlap{\in W_C}} && \downarrow^{\mathrlap{\in W_C}} \\ X &\stackrel{f}{\to}& Y }$$

Then by 2-out-of-3 and since $W_{L_S C} \supset W_C$ the morphism $f' : X' \to Y'$ is still an acyclic cofibration on $L_S C$. Again by 2-out-of-3 and since $W_{C_{loc}} \supset W_C$, it is sufficient to show that $f'$ is an acyclic cofibration in $C_{loc}$.

To show that it is an acyclic cofibration in $C_{loc}$ it suffices to show that for every fibrant object $Z \in C_{loc}$ the morphism

$$C(Y',Z) \to C(X',Z)$$

is a trivial fibration. Either by assumption or by the characterization of S-local cofibrations this is the case if $Z$ is $S$-local and fibrant in $C$. The first statement is one of the direct consequences of the definition of $C_{loc}$ and the second follows because $S = J_{C_{loc}}$.

Functoriality of localization

Proposition

Let $C$ and $D$ be categories for which left Bousfield localization exists, and let

$$(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$$

be a Quillen equivalence. Then for every small set $S \subset Mor(D)$ there is an induced Quillen equivalence of left Bousfield localizations

$$(L \dashv R) : C_{L(S)} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D_{S} \,.$$

This is due to Hirschhorn.

Further properties

Proposition (Bousfield localization is indeed a localization)

If the left Bousfield localization exists, i.e. $L_S C$ is indeed a model category with the above definitions of cofibrations and weak equivalences, then it is indeed a localization of a model category in that there is a left Quillen functor

$$j \colon C \to L_S C$$

(i.e. $j$ preserves cofibrations and trivial cofibrations and has a right adjoint)

such that the total left derived functor

$$L j \colon Ho C \to Ho L_S C$$

takes the images of $S \subset Mor(C)$ in $Ho(C)$ to isomorphisms

and every other left Quillen functor with this property factors by a unique left Quillen functor through $j$.

Moreover, the identity functor $Id_C$ on the underlying category is a Quillen adjunction

$$Id_C : L_S C \stackrel{\leftarrow }{\to} C : Id_C$$

(and is itself a localization functor).

Proof

The first part is theorem 3.3.19 in Hirschhorn. The second part is prop 3.3.4, which follows directly from the following proposition.

Existence of localizations for combinatorial model categories

We discuss the existence of left Bousfield localization in the context of combinatorial model categories. A similar existence result is available in the context of cellular model categories, but for the combinatorial case a somewhat better theory is available.

By the corollary to Dugger's theorem on presentations for combinatorial model categories we have that every combinatorial model category is Quillen equivalent to a left proper simplicial combinatorial model category.

Therefore there is little loss in assuming this extra structure, which the following statement of the theorem does.

Theorem

If $C$ is a

• left proper

• simplicial

• combinatorial model category

• and $S \subset Mor(C)$ is a small set of morphisms,

then the left Bousfield localization $L_S C$ does exist as a combinatorial model category.

Moreover, it satisfies the following conditions:

• the fibrant objects of $L_S C$ are precisely the $S$-local objects of $C$ that are fibrant in $C$;

• $L_S C$ is itself a left proper model category;

• $L_S C$ is itself a simplicial model category.

Proof

A proof of this making use of Jeff Smith’s recognition theorem for combinatorial model categories appears as HTT, prop. A.3.7.3 and as theorem 2.11 in Bar07 and as theorem 4.7 in Bar.

We follow Bar for the proof that the assumptions of Smith’s recognition theorem are satisfied and follow HTT, prop. A.3.7.3 for the characterization of the fibrant objects. The details are spelled out in the following subsections.

Prerequisites for the proof

The proof we give is self-contained, except that it builds on the following notions and facts.

Small objects

A cardinal number $\kappa$ is regular if it is not the cardinality of a union of $\lt \kappa$ sets of size $\lt \kappa$.

A poset $J$ is a $\kappa$-directed set if all subsets of cardinality $\lt \kappa$ have a common upper bound. A $\kappa$-directed colimit is a colimit $\lim_\to F$ over a functor $F : J \to C$.

An object $X$ in a category $C$ is a $\kappa$-compact object if $C(X,-) : C \to C$ commutes with all $\kappa$-directed colimits. For $\lambda \gt \kappa$ every $\kappa$-compact object is also $\lambda$-compact.

This means that a morphism from a $\kappa$-compact object into an object that is a $\kappa$-directed colimit over component objects always lifts to one of these component objects.

An object is a small object if it is $\kappa$-compact for some $\kappa$.

A locally small but possibly non-small category $C$ is an accessible category if it has a small sub-set of generating $\kappa$-compact objects such that every other object is a $\kappa$-directed colimit over such generators.

If such a category has all small colimits, it is called a locally presentable category.

In particular, in a locally presentable category the small object argument for factoring of morphisms applies with respect to every set of morphisms.

Combinatorial model categories

A combinatorial model category is a locally presentable category that is equipped with a cofibrantly generated model category structure. So in particular there is a set of generating (acyclic) cofibrations that map between small objects.

Smith’s recognition theorem says that a locally presentable category has a combinatorial model category structure already if it has weak equivalences and generating cofibrations satisfying a simple condition and if weak equivalences form an accessible subcategory of the arrow category. This means that only two thirds of the data for a generic combinatorial model category needs to be checked and greatly facilitates checking model category structures.

Dugger’s theorem implies that every combinatorial model category is Quillen equivalent to a left proper simplicial combinatorial model category.

So we may assume without much restriction of generality that we are dealing with the localization of a left proper combinatorial simplicial model category.

Since the small object argument applies, a combinatorial model category has fibrant- and cofibrant-replacement functors $P,Q : C \to C$ (functorial factorization).

By the axioms of an enriched model category it follows that the functor

$$\mathbf{R}Hom_C \coloneqq C(Q(-),P(-)): C^{op} \times C \to SSet$$

takes values in Kan complexes. This is called the derived hom space functor of $C$: we think of $\mathbf{R}Hom(X,Y)$ as the ∞-groupoid of maps from $X$ to $Y$, homotopies of maps, homotopies of homotopies, etc.

Local objects

An ordinary reflective subcategory $C_{loc} \stackrel{\stackrel{T}{\leftarrow]}}{\hookrightarrow} C$ is specified by the preimages $S = T^{-1}(isos)$ of the isomorphisms under $T$ as the full subcategory on the $S$-local objects $X$: those such that $Hom_C(A \stackrel{s \in S}{\to}B, X)$ are isomorphisms.

The analogous statement in the context of model categories uses the derived hom space functor instead: given a collection $S \subset Mor(C)$ an object $X$ is called an $S$-local object if $\mathbf{R}Hom_C(A \stackrel{s \in S}{\to} B, X)$ are weak equivalences.

Similarly, the collection $W_S$ of morphisms $f : E \to F$ such that for all $S$-local objects $X$ $\mathbf{R}Hom_C(f,X)$ is a weak equivalence is called the collection of $S$-local weak equivalences.

A lemma by Lurie says that for $A \stackrel{s \in S}{\hookrightarrow} B$ a cofibration and $X$ fibrant, $X$ is $S$-local precisely if $C(s,X) : C(B,X) \to C(A,X)$ is an acyclic Kan fibration. This helps identifying the $S$-local fibrant objects.

Homotopy (co)limits

In an ordinary category, a limit diagram is one such that applying $Hom_C(X,-) : C \to C$ to it produces a limit diagram in Set, for all objects $X$. Similarly a colimit diagram is one sent to Set-limits under all $Hom_C(-,X)$.

In a model category, this has an analog with respect to the derived hom space functor $\mathbf{R}Hom_C$. A homotopy limit diagram is one sent by all $\mathbf{R}Hom_C(-,X)$ to a homotopy limit (…). Similarly for homotopy colimits.

Sometimes ordinary (co)limits in a model category are already also homotopy colimits:

• an observation by Barwick shows that in a left proper simplicial model category ordinary pushouts along cofibrations are already homotopy colimits.

• an observation by Dugger shows that transfinite composition colimits of length $\kappa$ in a combinatorial model category are automatically homotopy colimits for sufficiently large $\kappa$.

Since these are the two operations under which $cell(cof_c \cap W_C)$ is closed, this facilitates finding this closure given that by the above the elements of $cof_C \cap W_S$ are characterized by their images under $\mathbf{R}Hom_C(-,X)$ for $S$-local $X$.

Size issues

The following proof uses the small object argument several times. In particular, at one point it is applied relative to the collection $S$ of morphisms at which we localize. It is at this point that we need that assumption that $S$ is indeed a (small) set, and not a proper class.

For the small object argument itself, this requirement comes from the fact that it involves colimits indexed by $S$. These won’t in general exist if $S$ is not a set.

The collection of $S$-local weak equivalences $W_S$, however, won’t be a small set in general even if $S$ is. But for Smith’s recognition theorem to apply we need to check that the full subcategory of $Arr(C)$ on $W_S$ is, while not small, accessible.

To establish this we need two properties of accessible categories: the inverse image of an accessible subcategory under a functor is accessible, and the collections of fibrations, weak equivalences and acyclic fibrations in a combinatorial model category are accessible.

The proof itself

Beginning of the proof of the existence of the left Bousfield localization of a left proper combinatorial simplicial model category at a set $S$ of morphisms.

Recognition of the combinatorial model structure

Using Smith’s recognition theorem, for establishing the combinatorial model category structure, it is sufficient to

• exhibit a set $I$ of cofibrations of $L_S C$ such that $inj(I) \subset W_{L_S C}$ and such that $cof(I) \cap W_{L_S C}$ is closed under pushout and transfinite composition.

• check that the weak equivalences form an accessibly embedded accessible subcategory.

For the first item choose $I \coloneqq I_C$ with $I_C$ any set of generating cofibrations of $C$, that exists by assumption on $C$. Then $inj(I) = inj(I_C) = fib_C \cap W_C \subset W_C \subset W_{L_S C}$.

It remains to demonstrate closure of $cof(I) \cap W_{L_S C} = cof_C \cap W_{L_S C}$ under pushout and transfinite composition.

One elegant way to see this, following Bar, is to notice that the relevant ordinary colimits all happen to be homotopy colimits:

• all pushouts along cofibrations in a left proper model category are homotopy pushouts (details are here);

• all transfinite composition colimits in a combinatorial model category are homotopy colimits (details are here)

By their definition in terms of the derived hom space functor, $S$-local weak equivalences in $C$ are preserved under homotopy colimits:

for $K \stackrel{}{\to}L$ an $S$-local morphism – a morphism in $W_{L_S C}$ – and for

$$\array{ K &\stackrel{\simeq_S}{\to}& L \\ \downarrow && \downarrow \\ K' &\to& L' }$$

a homotopy pushout diagram, we have (by the universal property of homotopy limits) for every object $Z$ – in particular for every every $S$-local object $Z$ – a homotopy pullback

$$\array{ \mathbf{R}Hom(L',Z) &\to& \mathbf{R}Hom(K',Z) \\ \downarrow && \downarrow \\ \mathbf{R}Hom(L,Z) &\stackrel{\simeq}{\to}& \mathbf{R}Hom(K,Z) } \,,$$

of $\infty$-groupoids. where the bottom morphism is a weak equivalence by assumption of $S$-locality of $Z$ and $(K \to L)$. But then also the top horizontal morphism is a weak equivalence for all $S$-local $Z$ and therefore $K' \to L'$ is in $W_{L_S C}$.

Similarly for transfinite composition colimits.

Therefore, indeed, $cof(I) \cap W_{L_S C}$ is closed under pushouts and transfinite composition.

Accessibility of the $S$-local weak equivalences

For the Smith recognition theorem to apply we still have to check that the $S$-local weak equivalences $W_S$ span an accessible full subcategory $Arr_S(C) \subset Arr(S)$ of the arrow category of $C$.

By the general properties of accessible categories for that it is sufficient to exhibit $Arr_S(C)$ as the inverse image of under functor $T : Arr_S(C) \hookrightarrow Arr(C)$ of the accessible category $Arr_W(C)$ spanned by ordinary weak equivalences in $C$.

That functor we take to be the $S$-local fibrant replacement functor from above

$$T : (X \to Y) \mapsto (T X \stackrel{T f}{\to} T Y) \,.$$

By one of the above propositions, $S$-local weak equivalence between $S$-local objects are precisely the ordinary weak equivalences. This means that the inverse image under $T$ of the weak equivalences in $C$ are all $S$-local weak equivalences

$$T^{-1}(Arr_{W_C}(C) = Arr_S(C) \,.$$

Therefore this is an accessible category.

End of the proof of the existence of the left Bousfield localization of a left proper combinatorial simplicial model category at a set $S$ of morphisms.

Further properties

Lemma (localization at cofibrations is sufficient)

Every combinatorial localization $B = L_{R} A$ of $A$ is already of the form $L_{S}A$ for $S$ a set of just cofibrations.

Proof

See HTT, prop. A.3.7.4

We demonstrate that $S \coloneqq J_B$ does the trick.

…

Using large cardinal axioms

If one assumes large cardinal axioms then the existence of Bousfield localization follows much more generally.

Theorem

Vopěnka's principle implies the statement:

Let $C$ be a left proper combinatorial model category and $Z \in Mor(C)$ a class of morphisms. Then the left Bousfield localization $L_Z W$ exists.

This is theorem 2.3 in (RosickyTholen).

Existence of localizations for tractable ennriched model categories

The above statement is generalized to the context of enriched model category theory by the following result:

Theorem (existence of enriched Bousfield localization)

Let

• $V$ be a tractable symmetric monoidal model category;

• $C$ a tractable left proper $V$-enriched model category

• $S \subset Mor(C)$ a small set

(all with respect to a fixed Grothendieck universe).

Then the left enriched Bousfield localization $L_{S/V} C$ does exist and is left proper and tractable.

This is (Barwick, theorem 4.46).

Examples and Applications

Categories to which the general existence theorem applies

The following model categories $C$ are left proper cellular/combinatorial, so that the above theorem applies and for every set $S \subset Mor(C)$ the Bousfield localization $L_S C$ does exist.

• Top with its standard model structure on topological spaces;

• SSet with its standard model structure on simplicial sets;

• the category of symmetric spectra in a pointed left-proper cellular model category

• and so on…

If $C$ is a left proper (simplicial) cellular model category, then so is

• the functor category $[D,C]$ for any (simplicial) small category $D$ (using the global model structure on functors);

• the over category $C/s$ for any object $x \in C$ .

At derived idempotent monads

The Bousfield-Friedlander theorem gives Bousfield localizations at the derived functor-version of idempotent monads.

See there for more examples of this general construction.

Localization of spectra

see Bousfield localization of spectra

Local model structure on presheaves

Left Bousfield localization produces the local model structure on homotopical presheaves. For instance the local model structure on simplicial presheaves.

Relation to other concepts

Locally presentable $(\infty,1)$-categories

As described at presentable (∞,1)-category, an (∞,1)-category $\mathbf{C}$ is presentable precisely if, as an simplicially enriched category, it arises as the full subcategory of fibrant-cofibrant objects of a combinatorial simplicial model category.

The proof of this proceeds via Bousfield localization, and effectively exhibits Bousfield localization as the procedure that models localization of an (∞,1)-category when $(\infty,1)$-categories are modeled by model categories.

For notice that

1. a presentable $(\infty,1)$-category $\mathbf{C}$ is one arising as the localization

   $$\mathbf{C} \simeq S^{-1} Funct(K,\infty Grpd) = S^{-1}PSh_{(\infty,1)}(K)$$

   of an (∞,1)-category of (∞,1)-presheaves

2. every (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(K) \simeq ([K,SSet]_{inj})^\circ$ arises as the subcategory of fibrant-cofibrant objects of the global model structure on simplicial presheaves;

3. in terms of the simplicial model category $[K,SSet]_{inj}$ the prescription for localization as an (∞,1)-category and passing to the subcategory of fibrant-cofibrant objects of the Bousfield localization $L_S [K,SSet]_{inj}$ is literally the same: in both cases one passes to the full subcategory on the $S$-local objects.

Moreover, by Dugger’s theorem on combinatorial model categories every combinatorial simplicial model category arises this way.

This is the argument of HTT, prop A.3.7.6.

This gives a good conceptual interpretation of Bousfield localization, since the localization of an (∞,1)-category is nothing but an adjunction

$$\mathbf{C} \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} \mathbf{D}$$

that exhibits $\mathbf{C}$ as a reflective (∞,1)-subcategory of $\mathbf{D}$.

So we find the diagram

Localization of $(\infty,1)$-presheaf categories

$$\array{ Sh_{(\infty,1)}(K) &\stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} & PSh_{(\infty,1)}(K) &&& abstract\;higher\;categorical\;def \\ \uparrow^{\simeq} && \uparrow^{\simeq} &&& Lurie's \;theorem \\ ([K^{op},SSet]_{inj}^{loc})^\circ &\stackrel{\stackrel{Bousfield\;loc.}{\leftarrow}} {\to}& ([K^{op},SSet]_{inj})^\circ &&& concrete\;realization } \,.$$

Here $(-)^\circ$ denotes passing to the full simplicially enriched subcategory on the fibrant-cofibrant objects, regarding that as an (∞,1)-category. (If one wants to regard that as a quasi-category, then $(-)^\circ$ also involves taking the homotopy coherent nerve of this simplicially enriched category.)

Localization of triangulated categories

There is also a notion of Bousfield localization of triangulated categories.

Under suitable conditions it should be true that for $C$ a model category whose homotopy category $Ho(C)$ is a triangulated category the homotopy category of a left Bousfield localization of $C$ is the left Bousfield localization of $Ho(C)$. See this answer on MO.

Related entries

• localization, colocalization

• right Bousfield delocalization

• localization at an object

• semi-left-exact left Bousfield localization

References

The original reference is

• A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150. doi

Bousfield localization appears as definition 3.3.1 in

• P. Hirschhorn, Model categories and their localizations, volume 99 of Mathematical Surveys and Monographs , American Mathematical Society, 2009,

for left proper cellular model categories.

In proposition A.3.7.3 of

• Jacob Lurie, Higher Topos Theory,

it is discussed in the context of combinatorial model categories and of combinatorial simplicial model categories in particular.

The relation to localization of an (infinity,1)-category is also in HTT, for the time being see the discussion at models for ∞-stack (∞,1)-toposes for more on that.

A detailed discussion of Bousfield localization in the general context of enriched model category theory is in

• Clark Barwick, On (enriched) left Bousfield localization of model categories (arXiv:0708.2067)

• Clark Barwick, On the Dreaded Right Bousfield Localization (arXiv:0708.3435)

Bousfield localization_ (pdf)

in terms of enriched tractable model categories.

The relation to Vopěnka's principle is discussed in

• Jiří Rosický, Walter Tholen, Left-determined model categories and universal homotopy theories, Transactions of the American Mathematical Society, Vol. 355, No. 9 (Sep., 2003), pp. 3611-3623 (JSTOR).