nLab
Bousfield localization of model categories

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Locality and descent

Contents

Idea

Bousfield localization is a procedure that to a model category structure CC 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.

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

Definition

First definition

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

cof C loc=cof c cof_{C_{loc}} = cof_c

but more weak equivalences

W C locW C. 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.

Observation

It follows directly that

  • C locC_{loc} has as fibrations a subset of fibrations of CC

    fib C loc=rlp(cof C locW C loc)rlp(cof C locW C)=fib C. fib_{C_{loc}} = rlp(cof_{C_{loc}} \cap W_{C_{loc}}) \subset rlp(cof_{C_{loc}} \cap W_C) = fib_{C} \,.
  • C locC_{loc} has the same acyclic fibrations as CC

    fib C locW C loc=rlp(cof C loc)=rlp(cof C)=fib CW 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:CC locId : C \to C_{loc} preserves cofibrations and weak equivalences

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

    so that this pair of functors is a Quillen adjunction

    C locC. C_{loc} \stackrel{\leftarrow}{\to} C \,.

And a very special one: the category C C^\circ modeled by a model category CC is its full subcategory on fibrant-cofibrant objects. Under left Bousfield localization the fibrant-cofibrant objects of C locC_{loc} are a subcollection of those of CC, so that we have the full subcategory

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

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

Bousfield localization is a model category version of reflecting onto a reflective subcategory.

Indeed – at least when CC 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 lexC {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 SS of local weak equivalences in CC, and alternatively by the collection of SS-local objects in CC. Indeed, C loc {C_{loc}}^\circ is the full (,1)(\infty,1)-subcategory on the cofibrant and fibrant and SS-local objects of CC.

Localization at SS-local weak equivalences

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

Let CC be a

We want to characterize objects in CC that “see elements of SS as weak equivalences”. Notice that

Observation

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

Hom C(f,X):Hom C(B,X)Hom C(A,X) 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

(SS-local objects and SS-local weak equivalences)

Say that

  • a fibrant object XX is an SS-local object if for all s:ABs : A \hookrightarrow B in SS the morphism

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

    is a trivial Kan fibration;

  • conversely, say that a cofibration f:ABf : A \hookrightarrow B is an SS-local weak equivalence if for all SS-local fibrant objects XX the morphism C(f,X):C(B,X)C(A,X)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 SS and not that the local objects are fibrant.

Definition

(SS-local objects and SS-local weak equivalences)

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

  • an object XX is an SS-local object if for all s:ABs : A \hookrightarrow B in SS the morphism

    C(Qs,PX):C(QB,PX)C(QA,PX) 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:ABf : A \hookrightarrow B is an SS-local weak equivalence if for all SS-local objects XX the morphism C(Qf,PX):C(QB,PX)C(QA,PX)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 SW_S for the collection of SS-local weak equivalences.

Remark

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

C(f,X):C(B,X)C(A,X) 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

Observation

Every ordinary weak equivalence is also SS-local weak equivalence.

WW S. W \subset W_S \,.

Therefore, for any set SS, we can consider the left Bousfield localization at the SS-local weak equivalences W SW_S:

Definition

(left Bousfield localization)

The left Bousfield localization L SCL_S C of CC at SS is, if it exists, the new model category structure on CC with

  • cofibrations are the same as before, cof L SC=cof Ccof_{L_S C } = cof_C;

  • acyclic cofibrations are the cofibrations that are SS-local weak equivalences.

Properties

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

Fibrants in L SCL_S C are the SS-local fibrants in CC

Proposition

The fibrant objects in L SCL_S C are precisely the fibrant objects in CC that are SS-local.

Proof

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

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

  • all generating acyclic cofibrations of CC, i.e. JSJ \subset S;

  • for every original morphism f:ABf : A \to B in SS and for every nn \in \mathbb{N} also the canonical morphism

    f˜:(Q f:=AΔ n AΔ nBΔ n)BΔ n, \tilde f : (Q_f := A \cdot \Delta^n \coprod_{A \cdot \partial \Delta^n} B \cdot \partial \Delta^n) \to B \cdot \Delta^n \,,

    where AΔ nA \cdot \Delta^n etc. denotes the tensoring of CC over SSet.

Proof of the Lemma

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

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

Q f X fib CW C BΔ n Y \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

Δ n C(B,X) C(B,Y) Δ n C(A,X) C(A,Y) \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 Δ nC(B,X)\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

Δ n C(B,X) cof SSetW SSet fib SSet Δ n C(A,X). \array{ \partial \Delta^n &\to& C(B,X) \\ \downarrow^{\mathrlap{\in cof_{SSet} \cap W_{SSet}}} && \downarrow^{\in fib_{SSet}} \\ \Delta^n &\to& C(A,X) } \,.

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

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

Notice that in these arguments we made use of

Next, again using the Quillen bifunctor property of the tensoring of CC over SSet we find that with AA cofibrant in CC and Δ n\Delta^n being cofibrant in SSet it follows that AΔ nA \cdot \Delta^n is cofibrant. Similarly for the other cases. And the coproduct of two cofibrants is cofibrant because cofibrations are preserved under pushout. Therefore Q fQ_f is indeed a cofibrant domain of our cofibration.

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

So let XX be a fibrant and SS-local object of CC. Homming the defining pushout diagram for Q fQ_f into XX produces the pullback diagram

[Δ n,[A,X]] W [Δ n,[B,X]] fib [Δ n,[A,X]] W [AΔ n AΔ nBΔ n,X] W [BΔ n,X] \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][A,X][B,X] \to [A,X] is an acyclic Kan fibration by the characterization of SS-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 SS contain the generating acyclic cofibrations and the morphism called f˜\tilde f.

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

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

has a lift.

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

In one direction, if XX that has the extension property with respect to SS it has it in particular with respect to the generating acyclic cofibrations JSJ \subset S and hence is fibrant.

And it in particular has the extension property with respect to f˜:AΔ n AΔ nBΔ nBΔ n\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 fQ_f a morphism

Q fX Q_f \to X

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

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

And in terms of this a lift

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

consists of a lift

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

Since {Δ nΔ n|n}\{\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 SS with respect to all f˜\tilde f means that all C(s,X):C(B,X)C(A,X)C(s,X) : C(B,X) \to C(A,X) are acyclic Kan fibrations.

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

Now every fibrant object XX in L SWL_S W has the extension property with respect to cof CW Scof_C \cap W_S hence in particular with respect to Scof cW SS \subset cof_c \cap W_S, so is SS-local and fibrant in CC.

Conversely, if it is SS-local and fibrant in CC; then, as mentioned before, for all fcof CW Sf \in cof_C \cap W_S the map [f,X][f,X] is an acyclic Kan fibration in SSet so that in particular Hom C(f,X)Hom_C(f,X) is a surjection, which means that XX has the extension property with respect to all ff and is hence fibrant in L SCL_S C.

Fibrant replacement in L SWL_S W – localization of objects

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

Xη Xcell(S)W STXinj(S)*. 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 SS, i.e. objects whose morphism to the terminal object is in inj(S)inj(S), are fibrant as well as SS-local in CC.

Therefore TT is in particular a fibrant approximation functor in L SWL_S W and η S\eta_S is the weak equivalence

η S:X W STX \eta_S : X \stackrel{\simeq_{W_S}}{\to} T X

in L SCL_S C relating an object to its fibrant approximation.

SS-local weak equivalences between SS-local objects are weak equivalences

Proposition

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

Proof

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

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

C loc=L SC. C_{loc} = L_S C \,.
Proof

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

First, the cofibrations of C locC_{loc} and L SCL_S C coincide. Moreover, the acyclic cofibrations of L SCL_S C contain all the acyclic cofibrations of C locC_{loc} because

cof L SCW L SC=llp(rlp(J L SC))llp(rlp(S))=llp(rlp(J C loc)). 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:XYf : X \to Y in L SCL_S C is an acyclic cofibration in C locC_{loc}.

Choose a cofibrant replacement for XX and YY

X cof C Y W C W C X f 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 SCW CW_{L_S C} \supset W_C the morphism f:XYf' : X' \to Y' is still an acyclic cofibration on L SCL_S C. Again by 2-out-of-3 and since W C locW CW_{C_{loc}} \supset W_C, it is sufficient to show that ff' is an acyclic cofibration in C locC_{loc}.

To show that it is an acyclic cofibration in C locC_{loc} it suffices to show that for every fibrant object ZC locZ \in C_{loc} the morphism

C(Y,Z)C(X,Z) 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 ZZ is SS-local and fibrant in CC. The first statement is one of the direct consequences of the definition of C locC_{loc} and the second follows because S=J C locS = J_{C_{loc}}.

Functoriality of localization

Proposition

Let CC and DD be categories for which left Bousfield localization exists, and let

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

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

(LR):C L(S)RLD S. (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 SCL_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:CL SC j : C \to L_S C

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

such that the total left derived functor

Lj:HoCHoL SC L j : Ho C \to Ho L_S C

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

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

Moreover, the identity functor Id CId_C on the underlying category is a Quillen adjunction

Id C:L SCC:Id C 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 ModLoc . 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 CC is a

then the left Bousfield localization L SCL_S C does exist as a combinatorial model category.

Moreover, it satisfies the following conditions:

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 κ\kappa is regular if it is not the cardinality of a union of <κ\lt \kappa sets of size <κ\lt \kappa.

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

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

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

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

A locally small but possibly non-small category CC is an accessible category if it has a small sub-set of generating κ\kappa-compact objects such that every other object is a κ\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:CCP,Q : C \to C.

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

RHom C:=C(Q(),P()):C op×CSSet \mathbf{R}Hom_C := C(Q(-),P(-)): C^{op} \times C \to SSet

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

Local objects

An ordinary reflective subcategory C loc]TCC_{loc} \stackrel{\stackrel{T}{\leftarrow]}}{\hookrightarrow} C is specified by the preimages S=T 1(isos)S = T^{-1}(isos) of the isomorphisms under TT as the full subcategory on the SS-local objects XX: those such that Hom C(AsSB,X)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 SMor(C)S \subset Mor(C) an object XX is called an SS-local object if RHom C(AsSB,X)\mathbf{R}Hom_C(A \stackrel{s \in S}{\to} B, X) are weak equivalences.

Similarly, the collection W SW_S of morphisms f:EFf : E \to F such that for all SS-local objects XX RHom C(f,X)\mathbf{R}Hom_C(f,X) is a weak equivalence is called the collection of SS-local weak equivalences.

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

Homotopy (co)limits

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

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

Since these are the two operations under which cell(cof cW C)cell(cof_c \cap W_C) is closed, this facilitates finding this closure given that by the above the elements of cof CW Scof_C \cap W_S are characterized by their images under RHom C(,X)\mathbf{R}Hom_C(-,X) for SS-local XX.

Size issues

The following proof uses the small object argument several times. In particular, at one point it is applied relative to the collection SS of morphisms at which we localize. It is at this point that we need that assumption that SS 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 SS. These won’t in general exist if SS is not a set.

The collection of SS-local weak equivalences W SW_S, however, won’t be a small set in general even if SS is. But for Smith’s recognition theorem to apply we need to check that the full subcategory of Arr(C)Arr(C) on W SW_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 SS 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 II of cofibrations of L SCL_S C such that inj(I)W L SCinj(I) \subset W_{L_S C} and such that cof(I)W L SCcof(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:=I CI := I_C with I CI_C any set of generating cofibrations of CC, that exists by assumption on CC. Then inj(I)=inj(I C)=fib CW CW CW L SCinj(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)W L SC=cof CW L SCcof(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:

And by their definition in terms of the derived hom space functor, SS-local weak equivalences in CC are preserved under homotopy colimits:

for KLK \stackrel{}{\to}L an SS-local morphism – a morphism in W L SCW_{L_S C} – and for

K S L K L \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 ZZ – in particular for every every SS-local object ZZ – a homotopy pullback

RHom(L,Z) RHom(K,Z) RHom(L,Z) RHom(K,Z), \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 SS-locality of ZZ and (KL)(K \to L). But then also the top horizontal morphism is a weak equivalence for all SS-local ZZ and therefore KLK' \to L' is in W L SCW_{L_S C}.

Similarly for transfinite composition colimits.

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

Accessibility of the SS-local weak equivalences

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

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

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

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

By one of the above propositions, SS-local weak equivalence between SS-local objects are precisely the ordinary weak equivalences. This means that the inverse image under TT of the weak equivalences in CC are all SS-local weak equivalences

T 1(Arr W C(C)=Arr S(C). 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 SS of morphisms.

Further properties

Lemma (localization at cofibrations is sufficient)

Every combinatorial localization B=L RAB = L_{R} A of AA is already of the form L SAL_{S}A for SS a set of just cofibrations.

Proof

See HTT, prop. A.3.7.4

We demonstrate that S:=J BS := 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 CC be a left proper combinatorial model category and ZMor(C)Z \in Mor(C) a class of morphisms. Then the left Bousfield localization L ZWL_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

(all with respect to a fixed Grothendieck universe).

Then the left enriched Bousfield localization L S/VCL_{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 CC are left proper cellular/combinatorial, so that the above theorem applies and for every set SMor(C)S \subset Mor(C) the Bousfield localization L SCL_S C does exist.

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

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 (,1)(\infty,1)-categories

As described at presentable (∞,1)-category, an (∞,1)-category C\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 (,1)(\infty,1)-categories are modeled by model categories.

For notice that

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

    CS 1Funct(K,Grpd)=S 1PSh (,1)(K) \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 (,1)(K)([K,SSet] inj) 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[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] injL_S [K,SSet]_{inj} is literally the same: in both cases one passes to the full subcategory on the SS-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

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

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

So we find the diagram

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

Sh (,1)(K) lex PSh (,1)(K) abstractnonsensedef Luriestheorem ([K op,SSet] inj loc) Bousfieldloc. ([K op,SSet] inj) concreterealization. \array{ Sh_{(\infty,1)}(K) &\stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} & PSh_{(\infty,1)}(K) &&& abstract\;nonsense\;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 CC a model category whose homotopy category Ho(C)Ho(C) is a triangulated category the homotopy category of a left Bousfield localization of CC is the left Bousfield localization of Ho(C)Ho(C). See this answer on MO.

References

Bousfield localization appears as definition 3.3.1 in

  • ModLoc, 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

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

  • Expanded version: On left and right model categories and left and right 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).

Revised on June 29, 2014 05:16:29 by ilan barnea (78.50.77.95)