aa r X i v : . [ m a t h . A T ] F e b An introduction to Bousfield localization
Tyler LawsonFebruary 11, 2020
Bousfield localization encodes a wide variety of constructions in homotopy theory,analogous to localization and completion in algebra. Our goal in this chapter isto give an overview of Bousfield localization, sketch how basic results in this areaare proved, and illustrate some applications of these techniques. Near the end wewill give more details about how localizations are constructed using the small objectargument. The underlying methods apply in many contexts, and we have attemptedto provide a variety of examples that exhibit different behavior.We will begin by discussing categorical localizations. Given a collection of mapsin a category, the corresponding localization of that category is formed by makingthese maps invertible in a universal way; this technique is often applied to discardirrelevant information and focus on a particular type of phenomenon. In certaincases, localization can be carried out internally to the category itself: this happenswhen there is a sufficiently ample collection of objects that already see these mapsas isomorphisms. This leads naturally to the study of reflective localizations.Bousfield localization generalizes this by taking place in a category where thereare spaces of functions, rather than sets, with uniqueness only being true up tocontractible choice. Bousfield codified these properties, for spaces in [Bou75] and forspectra in [Bou79]. The definitions are straightforward, but proving that localizationsexist takes work, some of it of a set-theoretic nature.Our presentation is close in spirit to Bousfield’s work, but the reader should go tothe books of Farjoun [Far96] and Hirschhorn [Hir03] for more advanced informationon this material. We will focus, for the most part, on left
Bousfield localization, sincethe techniques there are easier and is where most of our applications lie. In [Bar10]right Bousfield localization is discussed at more length.
The story of localization techniques in algebraic topology probably begins with Serreclasses of abelian groups [Ser53]. After choosing a class C of abelian groups thatis closed under subobjects, quotients, and extensions, Serre showed that one couldeffectively ignore groups in C when studying the homology and homotopy of a simply-connected space X . In particular, he proved mod- C versions of the Hurewicz andWhitehead theorems, showed the equivalence between finite generation of homology1nd homotopy groups, determined the rational homotopy groups of spheres, andsignificantly reduced the technical overhead in computing the torsion in homotopygroups by allowing one to work with only one prime at a time. His techniques forcomputing rational homotopy groups only require rational homology groups; p -localhomotopy groups only require p -local homology groups; p -completed homotopygroups only require mod- p homology groups.These techniques received a significant technical upgrade in the late 1960’s andearly 1970’s, starting with the work of Quillen on rational homotopy theory [Qui69b]and work of Sullivan and Bousfield–Kan on localization and completion of spaces[Sul05, Sul74, BK72]. Rather than using Serre’s algebraic techniques to break upthe homotopy groups π ∗ X and homology groups H ∗ X into localizations and com-pletions, their insight was that space-level versions of these constructions provideda more robust theory. For example, a simply-connected space X has an associ-ated space X Q whose homotopy groups and (positive-degree) homology groups are,themselves, rational homotopy and homology groups of X ; similarly for Sullivan’s p -localization X ( p ) and p -completion X ∧ p . Without this, each topological tool re-quires a proof that it is compatible with Serre’s mod- C -theory, such as Serre’s mod- C Hurewicz and Whitehead theorems or mod- C cup products. Now these are simplyconsequences of the Hurewicz and Whitehead theorems applied to X Q , and any sub-sequent developments will automatically come along. Moreover, Sullivan pioneeredarithmetic fracture techniques that allowed X to be recovered from its rationalization X Q and its p -adic completions X ∧ p via a homotopy pullback diagram: X / / (cid:15) (cid:15) Q p X ∧ p (cid:15) (cid:15) X Q α / / ( Q p X ∧ p ) Q This allows us to reinterpret homotopy theory. We are no longer using rationalizationand completion just to understand algebraic invariants of X : instead, knowledge of X is equivalent to knowledge of its localizations, completions, and an “arithmeticattaching map” α . This entirely changed both the way theorems are proved and theway that we think about the subject. Later, work of Morava, Ravenel, and othersmade extensive use of localization techniques [Mor85, Rav84], which today gives anexplicit decomposition of the stable homotopy category into layers determined byQuillen’s relation to the structure theory of formal group laws [Qui69a].Many of the initial definitions of localization and completion were constructive.One can build X Q from X by showing that one can replace the basic cells S n in a CW-decomposition with rationalized spheres S n Q , or by showing that the Eilenberg–MacLane spaces K ( A, n ) in a Postnikov decomposition can be replaced by rationalizedversions K ( A ⊗ Q , n ) . One can instead use Bousfield and Kan’s more functorial,but also more technical, construction as the homotopy limit of a cosimplicial space.Quillen’s work gives more, in the form of a model structure whose weak equivalencesare isomorphisms on rational homology groups. In his work, the map X → X Q is afibrant replacement, and the essential uniqueness of fibrant replacements means that2 Q has a form of universality. It is this universal property that Bousfield localizationmakes into a definition. We will use S to denote an appropriately convenient category of spaces (one can usesimplicial sets, but with appropriate modifications throughout) with internal functionobjects. We similarly write S p for a category of spectra.Throughout this paper we will often be working in categories enriched in spaces:for any X and Y in C we will write Map C ( X, Y ) for the mapping space, or just Map(
X, Y ) if the ambient category is understood. Letting [ X, Y ] = π Map C ( X, Y ) ,we obtain an ordinary category called the homotopy category h C . Two objects in C are homotopy equivalent if and only if they become isomorphic in h C .For us, homotopy limits and colimits in the category of spaces are given by thedescriptions of Vogt or Bousfield–Kan [Vog73, BK72]. A homotopy limit or homotopycolimit in C is characterized by having a natural weak equivalence of spaces: Map C ( X, holim J Y j ) ≃ holim J Map C ( X, Y j )Map C (hocolim I X i , Y ) ≃ holim I Map C ( X i , Y ) In particular, since homotopy limit constructions on spaces preserve objectwise weakequivalences of diagrams, homotopy limits and colimits also preserve objectwisehomotopy equivalences in C .Some set theory is unavoidable, but we will not spend a great deal of time withit. For us, a collection or family may be a proper class, rather than a set. Categorieswill be what are sometimes called locally small categories: the collection of objectsmay be large, but there is a set of maps between any pair of objects. The author would like to thank Clark Barwick and Thomas Nikolaus for discussionsrelated to localizations of spaces.Th author was partially supported by NSF grant 1610408 and a grant from theSimons Foundation. The author would like to thank the Isaac Newton Institutefor Mathematical Sciences for support and hospitality during the programme HHHwhen work on this paper was undertaken. This work was supported by: EPSRCgrant numbers EP/K032208/1 and EP/R014604/1.
In general, we recall that for an ordinary category A and a class W of the mapscalled weak equivalences (or simply equivalences ), we can attempt to construct a cat-egorical localization A → A [ W − ] . This localization is universal among functors3 → D that send the maps in W to isomorphisms. The category A [ W − ] is uniqueup to isomorphism if it exists. Example . We will begin by remembering the case of the category S of spaces,with W the class of weak homotopy equivalences. The projection p : X × [0 , → X is always a weak equivalence with homotopy inverses i t given by i t ( x ) = ( x, t ) . In thelocalization, we find that homotopic maps are equal: for a homotopy H from f to g , we have f = Hi = Hp − = Hi = g . Therefore, localization factors through thehomotopy category h S .However, within the category of spaces we have a collection with special prop-erties: the subcategory S CW of CW-complexes. For any CW-complex K , weakequivalences X → Y induce bijections [ K, X ] → [ K, Y ] —this can be proved, for ex-ample, inductively on the cells of K —and any space X has a CW-complex K witha weak homotopy equivalence K → X . These two properties show, respectively, thatthe composite h S CW → h S → S [ W − ] is fully faithful and essentially surjective. Within the homotopy category h S we havefound a large enough library of special objects, and localization can be done byforcibly moving objects into this subcategory. Example . A similar example to the above occurs in the category K R of nonneg-atively graded cochain complexes of modules over a commutative ring R , with W the class of quasi-isomorphisms. Within K R there is a subcategory K InjR of com-plexes of injective modules. Fundamental results of homological algebra show thatfor a quasi-isomorphism A → B and a complex Q of injectives, there is a bijection [ B, Q ] → [ A, Q ] of chain homotopy classes of maps, and that any complex A has aquasi-isomorphism A → Q to a complex of injectives. This similarly shows that thecomposite functor h K InjR → h K R → K R [ W − ] is an equivalence of categories.These examples are at the foundation of Quillen’s theory of model categories,and we will return to examples like them when we discuss localization of modelcategories. In this section we will fix an ordinary category A . For the record, this category also satisfies a 2-categorical universal property: for any D , the map offunctor categories Fun( A [ W − ] , D ) → Fun( A , D ) is fully faithful, and the image consists of those functors sending W to isomorphisms. If we replace “image”with “essential image” in this description, we recover a universal property characterizing A → A [ W − ] up to equivalence of categories rather than up to isomorphism. Technically speaking, we often use a result like this to actually show that S [ W − ] exists. efinition 3.1. Let S be a class of morphisms in A . An object Y ∈ A is S -local if,for all f : A → B in S , the map Hom A ( B, Y ) f ∗ −−→ Hom A ( A, Y ) is a bijection. We write L S ( A ) for the full subcategory of S -local objects.If S = { f : A → B } consists of just one map, we simply refer to this property asbeing f -local and write L f ( A ) for the category of f -local objects. Remark . If S = { f α : A α → B α } is a set and A has coproducts indexed by S , thenby defining f = ` α f α : ` A α → ` B α we find that S -local objects are equivalent to f -local objects.A special case of localization is when our maps in S are maps to a terminalobject. Definition 3.3.
Suppose S is a class of maps { W α → ∗} , where ∗ is a terminal object.In this case, we refer to such a localization as a nullification of the objects W α . Remark . Nullification often takes place when A is pointed. If S is a set, A ispointed, and A has coproducts, then any coproduct of copies of ∗ is again ∗ and wecan again replace nullification of a set of objects with nullification of an individualobject. Definition 3.5.
A map A → B in A is an S -equivalence if, for all S -local objects Y ,the map Hom A ( B, Y ) → Hom A ( A, Y ) is a bijection.The class of S -equivalences contains S by definition. Definition 3.6.
A map X → Y is an S -localization if it is an S -equivalence and Y is S -local, and under these conditions we say that X has an S -localization . If all objectsin A have S -localizations, we say that A has S -localizations. Proposition 3.7.
Any two S -localizations f : X → Y and f : X → Y are isomorphicunder X in A .Proof. Because Y i are S -local, Hom(
B, Y i ) → Hom(
A, Y i ) is always an isomorphismfor any S -equivalence A → B . Applying this to the S -equivalences X → Y j , we getisomorphisms Hom( Y j , Y i ) → Hom(
X, Y i ) in A : any map X → Y i has a uniqueextension to a map Y j → Y i . Existence allows us to find maps Y → Y and Y → Y under X , and uniqueness allows us to conclude that these two maps are inverse toeach other in A .More concisely, Y and Y are both initial objects in the comma category of S -local objects under X in A , and this universal property forces them to be isomorphic.5s a result, it is reasonable to call such an object the S -localization of X andwrite it as L S X (or simply LX if S is understood). More generally than this, if X → LX and X ′ → LX ′ are S -localization maps, any map X → X ′ in A extendsuniquely to a commutative square. This is encoded by the following result. Proposition 3.8.
Let
Loc S ( A ) be the category of localization morphisms , whose objectsare S -localization maps X → LX in A and whose morphisms are commuting squares.Then the forgetful functor Loc S ( A ) → A , sending ( X → LX ) to X , is fully faithful. The image consists of those objects X that have S -localizations. Proposition 3.9.
The collection of S -local objects is closed under limits, and the collec-tion of S -equivalences is closed under colimits.Proof. If f : A → B is in S and { Y j } is a diagram of S -local objects, then Hom(
B, Y j ) → Hom(
A, Y j ) is a diagram of isomorphisms, and taking limits we find that we have an isomorphism Hom( B, lim J Y j ) → Hom( A, lim J Y j ) . Since A → B was an arbitrary map in S , this shows that lim J Y j is S -local.Similarly, if { A i → B i } is a diagram of S -equivalences and Y is S -local, then Hom( B i , Y ) → Hom( A i , Y ) is a diagram of isomorphisms, and taking limits we find that Hom(colim I B i , Y ) → Hom(colim I A i , Y ) is also an isomorphism. Since Y was an arbitrary local object, this shows that themap colim I A i → colim I B i is an S -equivalence. Example . Consider the map f : N → Z in the category of monoids. A monoid M is f -local if and only if any monoid homomorphism N → M automaticallyextends to a homomorphism Z → M , which is the same as asking that every ele-ment in M has an inverse. Therefore, f -local monoids are precisely groups . Thenatural transformation M → M gp , from a monoid to its group completion, is an f -localization. Example . Consider the map f : F → Z , from a free group on two generators x and y to its abelianization. A group G is f -local if and only if every homomorphism F → G , equivalent to choosing a pair of elements x and y of G , can be factoredthrough Z , which happens exactly when the commutator [ x, y ] is sent to the triv-ial element. Therefore, f -local groups are precisely abelian groups. The naturaltransformation G → G ab , from a group to its abelianization, is an f -localization.6hese two localizations are left adjoints to the inclusion of a subcategory, andthis phenomenon is completely general. Proposition 3.12.
Let S be a class of morphisms in A , and suppose that A has S -localizations. Then the inclusion L S A → A is part of an adjoint pair A L ⇄ L S A . As a result, L is a reflective localization onto the subcategory L S A .Proof. In this situation, the functor
Loc S ( A ) → A is fully faithful and surjective onobjects. Therefore, it is an equivalence of categories and we can choose an inverse, functorially sending X to a pair ( X → LX ) in Loc S ( A ) . The composite functorsending X to LX is the desired left adjoint. Remark . Embedding the category A as a full subcategory of a larger categorycan change localization drastically. Consider a set S of maps in A ⊂ B . Then the S -local objects of A are simply the S -local objects of B that happen to be in A , butbecause there may be more local objects in B there may be fewer S -equivalences in B than in A . Localization in B may not preserve objects of A ; a localization map in A might not be an equivalence in B ; there might, in general, be no comparison mapbetween the two localizations.For example, consider the set S of multiplication-by- p maps Z → Z (as p rangesover primes) in the category of finitely generated abelian groups, considered as a fullsubcategory of all abelian groups. An abelian group is S -local if and only if it is arational vector space, and the only finitely generated group of this form is trivial. Amap A → B of finitely generated abelian groups is an S -equivalence in the largercategory of all abelian groups if and only if it induces an isomorphism A ⊗ Q → B ⊗ Q ,whereas it is always an equivalence within the smaller category of finitely generatedabelian groups. Within all abelian groups, S -localization is rationalization, whereaswithin finitely generated abelian groups, S -localization takes all groups to zero. We now consider the case where C is a category enriched in spaces. The previousdefinitions and results apply perfectly well to the homotopy category h C . The follow-ing illustrates that the homotopy category may be an inappropriate place to carryout such localizations. Example . Let us start with the homotopy category of spaces h S , and fix an n ≥ .Suppose that we want to invert the inclusion S n → D n +1 . We fairly readily find thatany space X has a map X → X ′ such that [ D n +1 , X ′ ] → [ S n , X ′ ] is an isomorphism: If the category A is large then we need to be a little bit more honest here, and worry about whethera fully faithful and essentially surjective map between large categories has an inverse equivalence. Thisdepends on our model for set theory: it is asking for us to make a distinguished choice of objects forour inverse functor, which may require an axiom of choice for proper classes. It is an awkward situation,because choosing these inverses isn’t categorically interesting unless we can’t do it. X ′ by attaching ( n + 1) -dimensional cells to X until the n ’th homotopygroup π n ( X ′ , x ) = 0 is trivial at any basepoint.However, this construction lacks universality . If Y is any other space whose n ’th homotopy groups are trivial, then any map X → Y can be extended to a map X ′ → Y because the attaching maps for the cells of X ′ are trivial. However, thisextension is not unique up to homotopy: any two extensions D n +1 → X ′ → Y of acell S n → X → Y glue together to an obstruction class in [ S n +1 , Y ] . As a result,if we construct two spaces X ′ and X ′′ as attempted localizations of X , we can findmaps X ′ → X ′′ and X ′′ → X ′ but cannot establish that they are mutually inverse inthe homotopy category.In short, in order for Y to have uniqueness for filling maps from n -spheres, wehave to have existence for filling maps from ( n + 1) -spheres. Thus, to make thislocalization work canonically we would need to enlarge our class S to contain S n +1 → D n +2 . The same argument then repeats, showing that a canonical local-ization for S requires that S also contain S m → D m +1 for m ≥ n .The example in the previous section leads to the following principle. In our def-initions, we must replace isomorphism on the path components of mapping spaceswith homotopy equivalence. Definition 4.2.
Let S be a class of morphisms in the category C . An object Y ∈ C is S -local if, for all f : A → B in S , the map Map C ( B, Y ) f ∗ −−→ Map C ( A, Y ) is a weak equivalence. We write L S ( C ) for the full subcategory of S -local objects.If S = { f : A → B } consists of just one map, we simply refer to this property asbeing f -local and write L f ( C ) for the category of f -local objects. Definition 4.3.
A map A → B in C is an S -equivalence if, for all S -local objects Y ,the map Map C ( B, Y ) → Map C ( A, Y ) is a weak equivalence. Definition 4.4.
A map X → Y is an S -localization if it is an S -equivalence and Y is S -local, and under these conditions we say that X has an S -localization . If all objectsin C have S -localizations, we say that C has S -localizations. By applying π to mapping spaces, we find that some of this passes to thehomotopy category. Proposition 4.5.
Let ¯ S be the image of S in the homotopy category h C . If Y is S -localin C , then its image in the homotopy category h C is ¯ S -local.Remark . An S -equivalence in C does not necessarily becomes an ¯ S -equivalencein h C because there is potentially a larger supply of ¯ S -local objects. Note that the homotopy class of the map
Map C ( B,Y ) → Map C ( A,Y ) only depends on the image of f : A → B in the homotopy category h C , and so we may simply view S as a collection of representativesfor a class of maps ¯ S in h C . roposition 4.7. Any two S -localizations f : X → Y and f : X → Y become iso-morphic under X in the homotopy category h C .Proof. This proceeds exactly as in the proof of Proposition 3.7. Applying
Map C ( − , Y i ) to the S -equivalence X → Y j , we find that the maps X → Y i extend to maps Y j → Y i which are unique up to homotopy. By first taking i , j we construct maps betweenthe Y i whose restrictions to X are homotopic to the originals, and taking i = j showsthat the double composites are homotopic under X . Remark . At this point it would be very useful to show that, if they exist, localiza-tions can be made functorial in the spirit of Proposition 3.8. There is typically noeasy way to produce a functorial localization because many choices are made up tohomotopy equivalence, and this leads to coherence issues: for example, if we have adiagram X / / (cid:15) (cid:15) X ′ (cid:15) (cid:15) LX / / LX ′ where the vertical maps are S -localization, then we can construct at best the dottedmap together with a homotopy between the two double composites. Larger diagramsdo get more extensive families of homotopies, but these take work to describe. Thisis a rectification problem and in general it is not solvable without asking for morestructure on C . The small object argument, which we will discuss in §6, can of-ten be done carefully enough to give some form of functorial construction of thelocalization. Proposition 4.9.
The following properties hold for a class S of morphisms in C .1. The collection of S -local objects is closed under equivalence in the homotopy category.2. The collection of S -equivalences is closed under equivalence in the homotopy cate-gory.3. The collection of S -local objects is closed under homotopy limits.4. The collection of S -equivalences is closed under homotopy colimits.5. The homotopy pushout of an S -equivalence is an S -equivalence.6. The S -equivalences satisfy the two-out-of-three axiom: given maps A f −→ B g −→ C , ifany two of f , g , and gf are S -equivalences then so is the third.Proof. If X → Y becomes an isomorphism in the homotopy category, then one canchoose an inverse map and homotopies between the double composites. Composingwith these makes Map C ( − , X ) → Map C ( − , Y ) a homotopy equivalence of functors on C , and so X is S -local if and only if Y is.Similarly, if two maps f : A → B and f ′ : A ′ → B ′ become isomorphic in thehomotopy category, there exist homotopy equivalences A ′ → A and B → B ′ such9hat the composite A ′ → A → B → B ′ is homotopic to f ′ , and applying Map C ( − , Y ) we obtain the desired result.If f : A → B is in S and { Y j } is a diagram of S -local objects, then Map C ( B, Y j ) → Map C ( A, Y j ) is a diagram of weak equivalences of spaces, and taking homotopy limits we findthat we have an equivalence Map C ( B, holim J Y j ) → Map C ( A, holim J Y j ) . Since A → B was an arbitrary map in S , this shows that holim J Y j is S -local.Similarly, if { A i → B i } is a diagram of S -equivalences and Y is S -local, then Map C ( B i , Y ) → Map C ( A i , Y ) is a diagram of weak equivalences of spaces, and so Map C (hocolim I B i , Y ) → Map C (hocolim I A i , Y ) is also a weak equivalence. Since Y was an arbitrary S -local object, this shows thatthe map hocolim I A i → hocolim I B i is an S -equivalence.Suppose that we have a homotopy pushout diagram A f / / (cid:15) (cid:15) B (cid:15) (cid:15) A ′ f ′ / / B ′ where f : A → B is an S -equivalence. Given any S -local object Y , we get a homotopypullback diagram Map C ( A, Y ) Map C ( B, Y ) o o Map C ( A ′ , Y ) O O Map C ( B ′ , Y ) o o O O The top arrow is an equivalence by the assumption that f is an S -equivalence, andhence the bottom arrow is an equivalence. Since Y was an arbitrary S -local object,we find that f ′ is an S -equivalence.The 2-out-of-3 property is obtained by first applying Map C ( − , Y ) to the diagram A → B → C and then using the 2-out-of-3 axiom for weak equivalences.If we expand a class S to a larger class T of equivalences, our work so far givesus an automatic relation between S -localization and T -localization. Proposition 4.10.
Suppose that S and T are classes of morphisms such that every mapin S is a T -equivalence. Then the following properties hold. . Every T -local object is also S -local.2. Every S -equivalence is also a T -equivalence.3. Suppose X → L S X is an S -localization and X → L T X is a T -localization. Thenthere exists an essentially unique factorization X → L S X → L T X , and the map L S X → L T X is a T -localization.Proof.
1. By assumption, every map f : A → B in S is a T -equivalence, and sofor any T -local object Y we get an equivalence Map C ( B, Y ) → Map C ( A, Y ) .Thus by definition Y is S -local.2. If f : A → B is an S -equivalence, and Y is any T -local object, then by theprevious point Y is also S -local, and so we get an equivalence Map C ( B, Y ) → Map C ( A, Y ) . Since Y was an arbitrary T -local object, f is therefore a T -equivalence.3. Since X → L S X is an S -equivalence, the previous point shows that it is a T -equivalence and so we have an equivalence Map C ( L S X, L T X ) → Map C ( X, L T X ) . As a result, the chosen map X → L T X has a contractible space of homotopycommuting factorizations X → L S X → L T X . As the maps X → L S X and X → L T X are both T -equivalences, the 2-out-of-3 property implies that L S X → L T X is also a T -equivalence whose target is T -local. By definition, this makes L T X into a T -localization of L S X . In this section we will observe that, if C has homotopy pushouts, we can characterizelocal objects in terms of a lifting criterion. To do so, we will need to establish a fewpreliminaries. Fix a collection S of maps in C . Proposition 5.1.
Suppose that f : A → B is an S -equivalence, and that C has homotopypushouts. Then the map hocolim( B ← A → B ) → B is an S -equivalence.Proof. The map in question is equivalent to the map of homotopy pushouts inducedby the diagram
B A f o o f / / f (cid:15) (cid:15) BB B o o / / B. However, the vertical maps are S -equivalences, and so by Proposition 4.9 the map hocolim( B ← A → B ) → B is an S -equivalence.11he lifting criterion we are about to describe rests on the following useful char-acterization of connectivity of a map. Lemma 5.2.
Suppose that f : X → Y is a map of spaces and N ≥ . Then f is N -connected if and only if the following two criteria are satisfied:1. the map π ( X ) → π ( Y ) is surjective, and2. the diagonal map X → holim( X → Y ← X ) is ( N − -connected.Proof. The map f is N -connected if and only if it is surjective on π and, for allbasepoints x ∈ X , the homotopy fiber Ff over f ( x ) is ( N − -connective.However, Ff is equivalent to the homotopy fiber of holim( X → Y ← X ) → X over x , and so this second condition is equivalent to holim( X → Y ← X ) → X being N -connected. The composite X → holim( X → Y ← X ) → X is the identity,and the map holim( X → Y ← X ) → X is N -connected if and only if the map X → holim( X → Y ← X ) is ( N − -connected. Corollary 5.3.
Suppose that C has homotopy pushouts and that we have a map f : A → B in C . Inductively define the n -fold double mapping cylinder f n as the map A n = hocolim( B ← A n − → B ) → B. Then an object Y is f -local if and only if the maps Hom h C ( B, Y ) → Hom h C ( A n , Y ) are surjective; equivalently, for any map A n → Y there is a map B → Y such that thediagram A n / / f n (cid:15) (cid:15) YB > > is homotopy commutative.Proof. We note that the definition of A n gives an identification Map C ( A n , Y ) ≃ holim h Map C ( B, Y ) → Map C ( A n − , Y ) ← Map C ( B, Y ) i . Inductive application of Lemma 5.2 shows that the map
Map C ( B, Y ) → Map C ( A , Y ) is N -connected if and only if the maps Hom h C ( B, Y ) → Hom h C ( A n , Y ) are surjective for ≤ n ≤ N . Letting N grow arbitrarily large, we find that Y is f -local if and only of the maps Hom h C ( B, Y ) → Hom h C ( A n , Y ) are surjective for all n ≥ . 12 xample . Suppose that C has homotopy pushouts and that f : W → ∗ is a mapto a homotopy terminal object of C . Then the iterated double mapping cylinders arethe maps Σ t W → ∗ , and an object of C is f -local if and only if every map Σ t W → Y factors, up to homotopy, through ∗ . Example . In the category of spaces S , the iterated double mapping cylinders f n of a cofibration f : A → B have a more familiar description as the pushout-product maps ( S n − × B ) ∪ S n − × A ( D n × A ) → D n × B → B. We now sketch how, when we have some form of colimits in our category, Bousfieldlocalizations can often be constructed using the small object argument.From the previous section we know that we can replace the mapping space cri-terion for local objects with a lifting criterion when C has homotopy colimits, asfollows. Given a map f : A → B , we construct iterated double mapping cylinders f n : A n → B , and we find that an object is Y is f -local if and only if every map g : A n → Y can be extended to a map ˜ g : B → Y up to homotopy. More generallywe can enlarge a collection of maps S to a collection T closed under double mappingcylinders, and ask whether Y satisfies an extension property with respect to T .This leads to an inductive method.1. Start with Y = Y .2. Given Y α , either Y α is local (in which case we are done) or there exists someset of maps A i → B i in T and maps g i : A i → Y α which do not extend to B i up to homotopy. Form the homotopy pushout of the diagram a i B i ← a i A i → Y α and call it Y α +1 . The map Y α → Y α +1 is an S -equivalence because it is ahomotopy pushout along an S -equivalence, and all the extension problemsthat Y α had now have solutions in Y α +1 .3. Once we have constructed Y , Y , Y , . . . , define Y ω = hocolim Y n . More gen-erally, once we have constructed Y α for all ordinals α less than some limitordinal β , we define Y β = hocolim Y α . The map Y → Y β is a homotopycolimit of S -equivalences and hence an S -equivalence.The critical thing that we need is that this procedure can be stopped at some point ,and for this we typically need to know that there will be some ordinal β which isso big that any map A i → Y β automatically factors, up to homotopy, through someobject Y α with α < β . This is a compactness property of the objects A i , and thisargument is called the small object argument . If we work on the point-set level thiscan be addressed using Smith’s theory of combinatorial model categories; if we work13n the homotopical level this can be addressed using Lurie’s theory of presentable ∞ -categories. We will discuss these approaches in §10 and §11.Another important aspect of the small object argument is that it can prove addi-tional properties about localization maps. If S is a collection of maps all satisfyingsome property P of maps in the homotopy category, and property P is preservedunder homotopy pushouts and transfinite homotopy colimits, then this process con-structs a localization Y → LY that also has property P . Since localizations areessentially unique, any localization automatically has property P as well. Remark . If our category C does not have enough colimits, the small object ar-gument may not apply. However, Bousfield localizations may still exist even if thisparticular construction cannot be applied. The classical examples of Bousfield localization are localizations of spaces. It isworthwhile first relating the localization condition to based mapping spaces.
Proposition 7.1.
Suppose f : A → B is a map of well-pointed spaces with basepoint.Then a space Y is f -local in the category of unbased spaces if and only if, for allbasepoints y ∈ Y , the restriction f ∗ : Map ∗ ( B, Y ) → Map ∗ ( A, Y ) of based mapping spaces is a weak equivalence.Proof. Evaluation at the basepoint gives a map of fibration sequences
Map ∗ ( B, Y ) (cid:15) (cid:15) / / Map(
B, Y ) (cid:15) (cid:15) / / Y Map ∗ ( A, Y ) / / Map(
A, Y ) / / Y .
The center vertical map is an isomorphism on π ∗ at any basepoint if and only if theleft-hand map is. Remark . As S -equivalences are preserved under homotopy pushouts and the 2-out-of-3 axiom, we find that any space Y local with respect to f : A → B is also localwith respect to the map B/A → ∗ from the homotopy cofiber to a point, and thusthat every path component of Y has a contractible space of based maps B/A → Y .However, we will see shortly that the converse does not hold in general. Example . Let S be the category of spaces, and take f to be the map S n → ∗ .Then a space X is f -local if and only if, for any basepoint x ∈ X , the iterated loopspace Ω n X at x is weakly contractible. Equivalently, for n ≥ the space X is f -localif and only if it is ( n − -truncated : π k ( X, x ) is trivial for all k ≥ n and all x ∈ X .A map A → B of CW-complexes, by obstruction theory, is an f -equivalenceif and only if it is ( n − -connected. Therefore, for n > a map A → B of CW-complexes is an f -localization if and only if π k ( A ) → π k ( B ) is an isomorphism for14 ≤ k < n and all basepoints, but π k B vanishes for all k ≥ n and all basepoints. This characterizes a stage P n − ( X ) in the Postnikov tower of X . Example . Let f be the inclusion S n ∨ S m → S n × S m of spaces. The Cartesianproduct is formed by attaching an ( n + m ) -cell to S n ∨ S m along an attaching mapgiven by a Whitehead product [ ι n , ι m ] ∈ π n + m − ( S n ∨ S m ) . Any map S n ∨ S m → X ,classifying a pair of elements α ∈ π n ( X ) and β ∈ π m ( X ) at some basepoint x , sendsthis attaching map to [ α, β ] . The fiber of Map( S n × S m , X ) → Map( S n ∨ S m , X ) overthe corresponding point is either empty (if [ α, β ] is nontrivial) or equivalent to theiterated loop space Ω n + m X at x (if [ α, β ] is trivial). A space X is therefore local withrespect to f if and only if, at any basepoint, the homotopy groups π k ( X ) are zerofor all k ≥ n + m and the Whitehead products π n ( X, x ) × π m ( X, x ) → π n + m − ( X, x ) vanish at any basepoint x .Consider the case n = m = 1 . For a path-connected CW-complex X with funda-mental group G , the map X → K ( G ab , is an f -localization. Example . If A is nonempty, then a space Y is local with respect to f : ∅ → A if and only if Y is weakly contractible. All maps are f -equivalences, and X → ∗ isalways an f -localization. Example . Consider a degree- p map f : S → S . A space Y is f -local if andonly if it is local for degree- p maps S n → S n , and this occurs if and only if the p ’thpower maps π n ( Y ) → π n ( Y ) are all isomorphisms.By contrast, let M ( Z /p, be the Moore space constructed as the cofiber of f , and consider the map g : M ( Z /p, → ∗ . A space Y is g -local if and only ifit satisfies the extension condition for the maps M ( Z /p, n ) → ∗ for all n ≥ , orequivalently if the mod- p homotopy sets π n ( Y ; Z /p ) vanish for all n ≥ . This isequivalent to the p ’th-power maps being isomorphisms on π n ( Y ) for all n > andinjective on π ( Y ) . Example . Let S be the set of maps { K ( Z /p, → ∗} as p ranges over theprime numbers. Then the Sullivan conjecture, as proven by Miller [Mil84], is equiva-lent to the statement that any finite CW-complex X is S -local. Since S -equivalencesare closed under products and homotopy colimits, the expression of K ( Z /p, n + 1) as the geometric realization of the bar construction { K ( Z /p, n ) q } shows inductivelythat the maps K ( Z /p, n ) → ∗ are all S -equivalences. However, if Y is any nontriv-ial 1-connected space with finitely generated homotopy groups and a finite Postnikovtower, then Y accepts a nontrivial map from some K ( Z /p, n ) and hence cannot be S -local. This argument shows that a simply-connected finite CW-complex with nonzeromod- p homology has p -torsion in infinitely many nonzero homotopy groups, which We should be careful about edge cases. When n = 0 , X is ( − -truncated if and only if it is eitherempty or weakly contractible. By convention, S − = ∅ , and X is ( − -truncated if and only if it is weaklycontractible.When n = 0 a map A → B is an f -equivalence if and only if either both A and B are empty or neitherof them is, and a map A → X is an f -localization if and only if either A is nonempty and X is weaklycontractible, or A and X are both empty. When n = − any map is an f -equivalence, and a map A → X is an f -localization if and only if X is weakly contractible. Example . Let C be the category of based spaces. A based space Y is localwith respect to the based map ∗ → S if and only if the loop space Ω Y is weaklycontractible, or equivalently if and only if the path component Y of the basepoint isweakly contractible. A model for the Bousfield localization is given by the mappingcone of the map Y → Y . Example . Fix a discrete group G , and consider the category of G -spaces: spaceswith a continuous action of a group G , with maps being continuous maps. For ex-ample, the empty space has a unique G -action, while the orbit spaces G/H havecontinuous actions under the discrete topology. Every G -space has fixed-point sub-spaces X H (cid:27) Map G ( G/H, X ) for subgroups H of G . In this context, there is anabundance of examples of localizations.A G -space Y is local with respect to ∅ → ∗ if and only if the fixed-point subspace Y G is contractible. A model for Bousfield localization is given by the mapping coneof the map Y G → Y .Fix a model for the universal contractible G -space EG . A G -space Y is localwith respect to EG → ∗ if and only if the map from the fixed point space Y G to thehomotopy fixed point space Map G ( EG, Y ) = Y hG is a weak equivalence. Since thereis a G -equivariant homotopy equivalence EG × EG → EG , a model for the Bousfieldlocalization is the space of nonequivariant maps Map(
EG, Y ) , with G acting byconjugation.A G -space Y is local with respect to ∅ → G if and only if the underlying space Y is contractible. A model for the Bousfield localization is given by the mapping coneof the map EG × Y → Y , sometimes called g EG ∧ Y + . Example . Fix a collection S of maps and a space Z , letting C be the category ofspaces over Z . We say that a map X → Y of spaces over Z is a fiberwise S -equivalence if the map of homotopy fibers over any point z ∈ Z is an S -equivalence, and refer tothe corresponding localizations as fiberwise S -localizations .A map X → Y over Z which is a weak equivalence on underlying spaces is inparticular a fiberwise S -equivalence. Applying this to the lifting characterization offibrations, we can find that for an object Y → Z of C to be fiberwise S -local the map Y → Z must be a fibration. Moreover, for fibrations Y → Z we can recharacterizebeing local. Given any map f : A → B in S and any point z ∈ Z , there is a map in C of the form f z : A → B → { z } ⊂ Z concentrated entirely over the point z ; let S Z bethe set of all such maps. A fibration Y → Z in C fiberwise S -local if and only if it is S Z -local in C .Fiberwise localizations were constructed by Farjoun in [Far96, 1.F.3]; they are alsoconstructed in [Hir03, §7] and characterized from several perspectives. Example . The category of topological monoids and continuous homomorphismshas its own homotopy theory. Consider the inclusion f : N → Z of discrete monoids.Then Map mon ( Z , M ) → Map mon ( N , M ) is isomorphic to the map M × → M from16he space of invertible elements of M to the space M . An f -local object is atopological group, and localization is a topologized version of group-completion.We note, however, that the map N → Z does not participate well with weak equivalences of topological monoids: weakly equivalent topological monoids do nothave weakly equivalent spaces of invertible elements because homomorphisms out of Z are not homotopical. We can get a version that respects weak equivalences in twoways. With model categories, we can factor the map N → Z as N ֒ → Z c ≃ −→ Z in thecategory of topological monoids, where Z c is a cofibrant topological monoid, andthere are explicit models for such. We could instead use coherent multiplications,where a map Z → M is no longer required to strictly be a homomorphism butinstead be a coherently multiplicative map.Using either correction, the space M × of strict units becomes replaced, up toequivalence, by the pullback M inv / / (cid:15) (cid:15) M (cid:15) (cid:15) π ( M ) × / / π M, the union of the components of M whose image in π ( M ) has an inverse. A localobject is then a grouplike topological monoid, and localization is homotopy-theoreticgroup completion. These play a key role the study of iterated loop spaces andalgebraic K -theory [May72, Seg74, MS76]. One of the great benefits of the stable homotopy category, and stable settings ingeneral, is that a map f : X → Y becoming an equivalence is roughly the same asthe cofiber Y /X becoming trivial.We recall the definition of stability from [Lur17, §1.1.1].
Definition 8.1.
The category C is stable if it satisfies the following properties:1. C is (homotopically) pointed : there is an object ∗ such that, for all X ∈ C , thespaces Map C ( X, ∗ ) and Map C ( ∗ , X ) are contractible.2. C has homotopy pushouts of diagrams ∗ ← X → Y and homotopy pullbacksof diagrams ∗ → Y ← X .As a special case, we have suspension and loop objects: Σ X = hocolim( ∗ ← X → ∗ ) Ω X = holim( ∗ → X ← ∗ ) As a point-set digression the reader should, as usual, be warned that the source may not have thesubspace topology. The space of invertible elements is, instead, homeomorphic to the subspace of M × M of pairs of elements ( x,y ) such that xy = yx = 1 .
17. Suppose that we have a homotopy coherent diagram X / / (cid:15) (cid:15) Y (cid:15) (cid:15) ∗ / / Z, meaning maps as given and a homotopy between the double composites. Thenthe induced map hocolim( ∗ ← X → Y ) → Z is a homotopy equivalence if and only if the map X → holim( ∗ → Z ← Y ) is a homotopy equivalence.Taking Y = ∗ , we find that a map X → Ω Z is an equivalence if and only if thehomotopical adjoint Σ X → Z is an equivalence. Example . The category of (cofibrant–fibrant) spectra is the canonical example ofa stable category.
Example . For any ring R , there is a category K R of chain complexes of R -modules. Any two complexes C and D have a Hom-complex Hom R ( C, D ) , and theDold–Kan correspondence produces a simplicial set Map K R ( C, D ) whose homotopygroups satisfy π n Map K R ( C, D ) (cid:27) H n Hom R ( C, D ) for n ≥ . This gives the category K R of complexes an enrichment in simplicial sets,and these mapping spaces make the category K R stable. Within this category thereare many stable subcategories: categories of complexes which are bounded above orbelow or both, with homology groups bounded above or below or both, which aremade up of projectives or injectives, and so on.We will write C R be the category of cofibrant objects in the projective modelstructure on R , whose homotopy category is the derived category D ( R ) . Theorem 8.4 (see [Lur17, Theorem 1.1.2.14]) . If C is stable, then the homotopy category h C has the structure of a triangulated category. In a stable category, every object Y has an equivalence Y → ΩΣ Y . However,there is a natural weak equivalence Map C ( X, Ω Z ) ≃ holim h Map C ( X, ∗ ) → Map C ( X, Z ) ← Map C ( X, ∗ ) i ≃ holim( ∗ → Map C ( X, Z ) ← ∗ ) ≃ Ω Map C ( X, Z ) , More generally, if R [ m ] is the complex equal to R in degree n and zero elsewhere, then for allcomplexes C we have [ R [ m ] ,C ] h K R (cid:27) H m ( C ) . Map C ( X, Y ) ≃ Ω n Map C ( X, Σ n Y ) can be extended to be valued in Ω -spectra. This makes it much easier to detectequivalences: we only need to check the homotopy groups of Ω t Map C ( X, Y ) at thebasepoint. Definition 8.5.
Suppose that C is stable and S is a class of maps in C . We say that S is shift-stable if the image ¯ S in h C is closed under suspension and desuspension,up to isomorphism. Proposition 8.6.
Suppose that C is stable and S is a shift-stable class of maps { f α : A α → B α } . Then an object Y in C is S -local if and only if the homotopy classes of maps [ B α /A α , X ] h C are trivial.Proof. The individual fiber sequences Ω t Map C ( B α /A α , Y ) → Ω t Map C ( B α , Y ) → Ω t Map C ( A α , Y ) , on homotopy classes classes of maps, are part of a long exact sequence · · · → [ Σ t B α /A α , Y ] h C → π t Map C ( B α , Y ) → π t Map C ( A α , Y ) → [ Σ t − B α /A α , Y ] h C → . . . from the triangulated structure. We get an isomorphism on homotopy groups if andonly if the terms [ Σ t B α /A α , Y ] h C vanish for all values of t .By contrast with the unstable case where basepoints are a continual issue, theseshift-stable localizations in a stable category are always nullifications, and they are equivalent to nullifications of the triangulated homotopy category by a class S that isclosed under shift operations. Definition 8.7.
Suppose that D is a triangulated category. A full subcategory T iscalled a thick subcategory if its objects are closed under closed under isomorphism,shifts, cofibers, and retracts. If D has coproducts, a thick subcategory T is localizing if it is also closed under coproducts. Proposition 8.8.
Suppose that D is a triangulated category and that T ⊂ D is a thicksubcategory. Then there exists a triangulated category D / T called the Verdier quotient of D by T , with a functor D → D / T . The Verdier quotient is universal among triangulatedcategories under D such that the objects of T map to trivial objects. This universal characterization allows us to strongly relate Bousfield localizationof stable categories to localization of the homotopy category.
Proposition 8.9.
Suppose that C is stable, and that S is a shift-stable collection of mapsin C .1. An object in C is S -local if and only if its image in the homotopy category h C is S -local. . A map in C is an S -equivalence if and only if its image in the homotopy categoryis an S -equivalence.3. The subcategories L S C of S -local objects and T of S -trivial objects are thick sub-categories of C .4. The subcategory T of S -trivial objects is closed under all coproducts that exist in C .If C has small coproducts then it is a localizing subcategory.5. If all objects in C have S -localizations, then the left adjoint to the inclusion hL S C → h C has a factorization h C → h C /h T → hL S C . The latter functor is an equivalence of categories.Remark . The fact that Bousfield localization of C is determined by a construc-tion purely in terms of h C is special to the stable setting. Remark . This relates Verdier quotients in a stable category to Bousfield localiza-tion, but only quotients by a localizing subcategory. For a homotopical interpretationof more general Verdier quotients, see [NS18, §I.3].
Example . Let S be the collection of multiplication-by- m maps S n → S n for n ∈ Z , m > . A spectrum Y is S -local if and only if multiplication by m is anisomorphism on the homotopy groups π ∗ Y for all positive m , or equivalently if themaps π ∗ Y → Q ⊗ π ∗ Y are isomorphisms. Such spectra are called rational .If Y is such a spectrum, we can calculate that the natural map [ X, Y ] → Y n Hom( π n X, π n Y ) is an isomorphism for any spectrum X : because π n Y is a graded vector space, Hom( − , π n Y ) is exact and so both sides are cohomology theories in X that satisfythe wedge axiom and agree on spheres. Therefore, A → B is an S -equivalence if andonly if Q ⊗ π n ( A ) → Q ⊗ π n ( B ) is an isomorphism for all n , and such maps are called rational equivalences . In this case, this is the same as the map H ∗ ( A ; Q ) → H ∗ ( B ; Q ) being an isomorphism.This analysis allows us to conclude that X → H Q ∧ X = X Q is a rationalizationfor all X . Example . In the above, we can make S smaller. If S is the set of multiplication-by- p maps S n → S n , we similarly find that S -local spectra are those whose homotopygroups are Z [1 /p ] -modules, and that equivalences are those maps which induceisomorphisms on homotopy groups after inverting p . The localization of S is thehomotopy colimit S [1 /p ] = hocolim( S p −→ S p −→ S p −→ . . . ) , which is also a Moore spectrum for Z [1 /p ] . We similarly find that X → S [1 /p ] ∧ X is an S -localization for all X .We could also let S be the set of multiplication-by- m maps for m relatively primeto p , which replaces the ring Z [1 /p ] with the local ring Z ( p ) in the above.20 xample . Fix a commutative ring R and a multiplicatively closed subset W ⊂ R ,recalling that localization with respect to W is exact. If we define S to be the set ofmaps of the form R [ n ] w −→ R [ n ] for w ∈ W , then a complex C of R -modules is S -localif and only if the multiplication-by- w maps H ∗ ( C ) → H ∗ ( C ) are isomorphisms, orequivalently if and only if H ∗ ( C ) → W − H ∗ ( C ) (cid:27) H ∗ ( W − C ) is an isomorphism. Amap A → B of complexes is an S -equivalence if and only if the map W − A → W − B is an equivalence.The natural map C → W − C (cid:27) W − R ⊗ R C is an S -localization.These examples have such nice properties that it is convenient to axiomatizethem. Definition 8.15.
A stable Bousfield localization on spectra is a smashing localization if either of the following equivalent conditions hold.1. There is a map of spectra S → L S such that, for any X , the map X → L S ∧ X is a localization.2. Local objects are closed under arbitrary homotopy colimits.The equivalence between these two characterizations is not immediately obvious.The first implies the second, because L S ∧ hocolim X i → hocolim( L S ∧ X i ) is always an equivalence and the former is always local. The converse follows be-cause the only homotopy-colimit preserving functors on spectra are all equivalentto functors of the form X A ∧ X for some A , and the resulting localization map S → A is of the desired form. Example . A spectrum Y is local for the maps S [1 /p ] ∧ S n → ∗ if and only if thehomotopy limit holim( · · · → Y p −→ Y p −→ Y ) ≃ F ( S [1 /p ] , Y ) of function spectra is weakly contractible. However, taking homotopy limits of thenatural fiber sequences . . . / / Y p / / p (cid:15) (cid:15) Y p / / p (cid:15) (cid:15) Y (cid:15) (cid:15) . . . / / Y / / (cid:15) (cid:15) Y / / (cid:15) (cid:15) Y (cid:15) (cid:15) . . . / / Y /p / / Y /p / / ∗ shows that Y is local if and only if the map Y → Y ∧ p = holim Y /p k is an equivalence.Therefore, we refer to a spectrum local for these maps as p -complete ; a Bousfield This definition extends if we have a stable category C with a symmetric monoidal structure appropri-ately compatible with the stable structure. Y will be called the p -completion ; a trivial object is called p -adicallytrivial ; an equivalence is called a p -adic equivalence . The above presents Y ∧ p as acandidate for the p -completion of Y .If we construct the fiber sequence Σ − S /p ∞ → S → S [1 /p ] , we find that we can identify Y ∧ p with the function spectrum F ( Σ − S /p ∞ , Y ) . More-over, the map Y ∧ p → ( Y ∧ p ) ∧ p is always an equivalence. Therefore, Y ∧ p is always p -complete.If multiplication-by- p is an equivalence on Z , then Z ≃ Z ∧ S [1 /p ] , and so maps Z → Y are equivalent to maps Z → F ( S [1 /p ] , Y ) . For any Y which is p -adicallycomplete, this is trivial, so such objects Z are p -adically trivial. In particular, thefiber of Y → Y ∧ p is always trivial and so Y → Y ∧ p is a p -adic equivalence. Therefore,this is a p -adic completion.If each homotopy group of Y has a bound on the order of p -power torsion, wecan further identify the homotopy groups of Y ∧ p as the ordinary p -adic completionsof the homotopy groups of Y ; if the homotopy groups of Y are finitely generated,then π ∗ ( Y ∧ p ) → π ∗ ( Y ) ⊗ Z p . Remark . Note that the previous example is not a smashing localization. Forany connective spectrum X , the map S ∧ p ∧ X → X ∧ p induces the map π ∗ ( X ) ⊗ Z p → π ∗ ( X ) ∧ p on homotopy groups; this is typically only an isomorphism if the homotopygroups π ∗ ( X ) are finitely generated. Example . For an element x in a commutative ring R , let K x be the complex · · · → → R → x − R → → . . . concentrated in degrees and − , with a map K X → R . For a sequence of elements ( x , . . . , x n ) , let K ( x ,...,x n ) = N R K x i be the stable Koszul complex . If y is in theideal generated by ( x , . . . , x n ) , then the inclusion K ( x ,...,x n ) → K ( x ,...,x n ,y ) is a quasi-isomorphism, and so up to quasi-isomorphism the Koszul complex only depends onthe ideal. Let K I be a cofibrant replacement.We say that a complex C is I -complete if and only if it is local with respect tothe shifts of the map K I → R . This is true if and only if the homology groups of C are I -complete in the derived sense. If R is Noetherian and the homology groupsof C are finitely generated, this is true if and only if the homology groups of C are I -adically complete in the ordinary sense.These frameworks for the study of localization and completion, and many gener-alizations of it, were developed by Greenlees and May [GM95]. Example . Fix a ring R , and let C be the category of unbounded complexes offinitely generated projective left R -modules that only have nonzero homology groupsin finitely many degrees. Consider the set S of maps R [ n ] → . An object C is S -local if and only if its homology groups are trivial. In general, the homotopy groups of the p -adic completion are somewhat sensitive and one needs tobe careful about derived functors of completion.
22e can inductively take mapping cones of maps R [ n ] → C to construct a lo-calization C → LC , embedding C into an unbounded complex of finitely generatedprojective modules with trivial homology groups. Therefore, localizations exist inthis category.For two such complexes C and D with trivial homology, we have Hom h C ( C, D ) (cid:27) lim n Hom R ( Z n C, Z n D ) / Hom R ( Z n C, D n +1 ) where D n +1 → Z n ( D ) is the boundary map—a surjective map from a projectivemodule.This can be interpreted in terms of the stable module category of R . Defining W n ( C ) = Z − n ( C ) , the short exact sequences → Z − n ( C ) → C − n → Z − n − ( C ) → determine isomorphisms W n ( C ) (cid:27) Ω W n +1 ( C ) in the stable module category, assem-bling the W n into an “ Ω -spectrum”. Maps C → D are then equivalent to maps of Ω -spectra in the stable module category. Definition 9.1.
Suppose E ∗ is a homology theory on spaces. Then we say that a map f : A → B of spaces is an E ∗ -equivalence if it induces an isomorphism f ∗ : E ∗ A → E ∗ B .A space is E ∗ -local if it is local with respect to the class of E ∗ -equivalences. Example . Suppose that E ∗ is integral homology H ∗ . Any Eilenberg–Mac Lanespace K ( A, n ) is H ∗ -local by the universal coefficient theorem for cohomology. More-over, any simply-connected space X is the homotopy limit of a Postnikov tower builtfrom fibration sequences P n X → P n − X → K ( π n X, n + 1) . Since local objects areclosed under homotopy limits, we find that simply-connected spaces are H ∗ -local. Remark . This example illustrates a very different approach to the constructionof localizations. Because homology isomorphisms are detected by the K ( A, n ) , thesespaces are automatically local; therefore, any object built from these using homotopylimits is automatically local. Such objects are often called nilpotent . Thus gives usa dual approach to building the Bousfield localization of X : construct a naturaldiagram of nilpotent objects that receive maps from X , and try to verify that thehomotopy limit is a localization of X . Example . Serre’s rational Hurewicz theorem implies that a map of simply-connectedspaces is an isomorphism on rational homology groups if and only if it is an isomor-phism on rational homotopy groups. A simply-connected space is local for rationalhomology if and only if it its homotopy groups are rational vector spaces. In certain cases, such as for Frobenius algebras, Ω is an autoequivalence. This definition then simplyrecovers the stable module category of R by itself. If R has finite projective dimension, Ω -spectrumobjects are necessarily trivial. This argument can be refined to show that nilpotent spaces (where π ( X ) is nilpotent, and actsnilpotently on the higher homotopy groups) are H ∗ -local. RP → ∗ is a rational homol-ogy isomorphism, and the covering map S → RP is an isomorphism on rationalhomotopy groups, but the composite S → ∗ is neither. The problem here is thefailure of a simple Postnikov tower for RP due to the action of π on the higherhomotopy groups. Example . If X is a connected space with perfect fundamental group, then Quillen’splus-construction gives a map X → X + that induces an H ∗ -isomorphism such that X + is simply-connected. This makes X + into an H ∗ -localization of X .Classically, Quillen’s plus-construction can be applied to groups with a perfectsubgroup. In order to properly identify the universal property, we need to work in arelative situation. Example . Fix a group G , and let C be the category of spaces over BG . Givenan abelian group A with G -action, there is an associated local coefficient system A on BG , and so given any object X → BG of C we can define the homology groups H ∗ ( X ; A ) . We say that a map X → Y over BG is a relative homology equivalenceif it induces isomorphisms on homology with coefficients in any A . Taking A tobe the group algebra Z [ G ] , we find that this is equivalent to the map of homotopyfibers F X → F Y being a homology isomorphism, so this is the same as a fiberwise H ∗ -equivalence . If an object Y over BG has simply-connected homotopy fiber it isautomatically local.Suppose that X is any connected space such that π ( X ) contains a perfect normalsubgroup P with quotient group G . The homomorphism π ( X ) → G lifts to a map X → BG . The plus-construction with respect to P is a fiber homology equivalence X → X + where X + → BG has simply-connected homotopy fiber, and thus is alocalization in C .Localization with respect to homology is very difficult to analyze in the casewhen a space is not simply-connected, especially if the space is not simple (eitherthe fundamental group is not nilpotent or it does not act nilpotently on the higherhomotopy groups). Many natural spaces are not local. Here are some basic tools toprove this. Lemma 9.7.
Suppose that F n is a free group on n generators and α : F n → F n is a homo-morphism, with induced map α ab : Z n → Z n . Under the identification Hom( F n , G ) (cid:27) G n for any group G , write α ∗ for the natural map of sets G n → G n .Suppose the map α ab becomes an isomorphism after tensoring with a ring R . Then, forany space X , a necessary condition for X to be H ∗ ( − ; R ) -local is that α ∗ : π ( X, x ) n → π ( X, x ) n must be a bijection at any basepoint.Proof. The map α ab , after tensoring with R , can be identified with the map H ( F n ; R ) → H ( F n ; R ) on homology induced by α . If α ab becomes an isomorphism after tensor-ing with R , then α : K ( F n , → K ( F n , is an H ∗ ( − ; R ) -equivalence.For a space X to be H ∗ ( − ; R ) -local, the induced map Map ∗ ( K ( F n , , X ) → Map ∗ ( K ( F n , , X ) ( Ω X ) n → ( Ω X ) n must be a weak equivalence. On π , this is the map α ∗ on π ( X ) n . Example . For n , , the multiplication-by- n map Z → Z is a rational isomor-phism. Therefore, for X to be rationally local, the n ’th power map π ( X ) → π ( X ) should be a bijection: every element g ∈ π ( X ) has a unique n ’th root g /n . Suchgroups are called uniquely divisible, or sometimes Q -groups. The structure of free Q -groups was studied in [Bau60]. Example . Let F be free on the generators x and y , and define α : F → F by α ( x ) = x − y − ( y x ) α ( y ) = x − y ( yx − ) − . The map α ab is the identity map. Therefore, for a space with fundamental group G to be local with respect to integral homology, any pair of elements ( z, w ) ∈ G has tobe uniquely of the form ( z, w ) = ( x − y − ( y x ) , x − y − ( yx − ) − ) for some x and y in G . Most groups do not satisfy this property.We can use this to show that any space whose fundamental group G has a surjec-tive homomorphism φ : G → A cannot be local with respect to integral homology—in particular, this applies to a free group F . Choose elements x and y in G with φ ( x ) = (123) and φ ( y ) = (12345) . Then φ ( y x ) = (14)(25) and φ ( yx − ) = (145) ,and φ ◦ α is the trivial homomorphism while φ is surjective. Several other, more easily defined, maps α can be shown to not be bijective. Forexample, the map ( x, y ) ( x [ x, y ] , y [ x, y ]) can be shown to not be a bijection, e.g.by using Fox’s free differential calculus [Fox53]. Lemma 9.10.
Let G be a group, R a ring, and β ∈ Z [ G ] an element such that thecomposite ring homomorphism Z [ G ] ǫ −→ Z → R sends β to zero.Then, for any based space X with fundamental group G , a necessary condition for X to be H ∗ ( − ; R ) -local is that π k ( X ) must be complete in the topology defined by β . Proof.
Fix the space X and basepoint and consider the space Y = X ∨ S k . The group π k ( Y ) is isomorphic to π k ( X ) ⊕ Z [ G ] , and so the element β ∈ Z [ G ] lifts to a map β : Y → Y given by the identity on X together with the map S k → Y correspondingto the element (0 , β ) ∈ π k ( X ) ⊕ Z [ G ] . The induced self-map of H ∗ ( Y ; R ) (cid:27) H ∗ ( X ; R ) ⊕ e H ∗ ( S k ; R ) is given by the identity on H ∗ ( X ; R ) together with the map ǫ ( β ) tensored with R onthe second factor. If ǫ ( β ) becomes zero after tensoring with R , then this map is zeroon the second factor. In order to use this particular technique to show that φ was not a bijection, we needed to have ahomomorphism φ whose image was a perfect group—the image of α ab is contained in the kernel of φ ab .This particular map α is complicated because it was reverse-engineered from φ . This refers to being derived complete in the sense of Example 8.18. X ′ = hocolim( Y β −→ Y β −→ · · · ) . By construction, the map H ∗ ( X ; R ) → H ∗ ( X ′ ; R ) = colim H ∗ ( Y ; R ) is an isomorphism. Therefore, X → X ′ is an H ∗ ( − ; R ) -equivalence.For X to be H ∗ ( − ; R ) -local, the induced map Map( X ′ , X ) → Map(
X, X ) must be a weak equivalence. Taking the fiber over the identity map of X , we findthat there is an induced equivalence holim( · · · β −→ Ω k X β −→ Ω k X ) ∼ −→ ∗ . Using the Milnor lim -sequence, we find that all of the homotopy groups of X mustbe derived-complete with respect to β . Remark . If R = Z , then this implies that any element s ∈ Z [ G ] with ǫ ( s ) = ± must act invertibly on the higher homotopy groups of X , and so the action mustfactor through a large localization S − Z [ G ] . Example . Consider X = S ∨ S , whose fundamental group is isomorphic to Z with generator t . The second homotopy group satisfies π ( S ∨ S ) (cid:27) Z [ t ± ] as a module over Z [ t ± ] . This is not complete with respect to the ideal generatedby β = ( t − even though ǫ ( β ) = 0 . Therefore, S ∨ S is not local with respect tointegral homology. Example . The space RP has fundamental group Z / generated by an element σ , and the second homotopy group Z satisfies σ ( y ) = − y . The element (1 − σ ) has ǫ (1 − σ ) = 0 and acts as multiplication by . Since Z is not complete in the -adictopology we find that RP is not local with respect to integral homology. Example . If R = Q , then any element S ∈ Z [ G ] with ǫ ( s ) , must act invertiblyon the higher homotopy groups of X for X to be local with respect to rationalhomology. The homotopy groups of K ( Q , ∨ ( S ) Q are Q in degree and therational group algebra Q [ Q ] in degree 3. If t is the generator of Z ⊂ Q , the element t − has ǫ (2 t −
1) = 1 and does not act invertibly on this group algebra. Therefore,this space is not local with respect to rational homology even though its homotopygroups are rational.
Remark . Bousfield localization with respect to E ∗ -equivalences leads us to someuncomfortable pressure with our previous notation. At first glance, it is not clear The homology localization of RP has, in fact, a fiber sequence ( S ) ∧ → L RP → K ( Z / , . E ∗ -homology is the same as having the same map-ping spaces into any E ∗ -local object. To prove this, one needs to prove that thereis a sufficient supply of E ∗ -local objects: for any X , we need to be able to constructan E ∗ -homology isomorphism X → L E X such that L E X is E ∗ -local. Here is howBousfield addressed this in [Bou75, Theorem 11.1]. It is essentially a cardinality ar-gument, whose general form is called the Bousfield–Smith cardinality argument in[Hir03, §2.3].Let E ∗ be a homology theory on spaces. We then have a class S of E ∗ -equivalences,which are those maps which induce equivalences on E ∗ -homology. Unfortunately, thisis a proper class of morphisms, and so we cannot immediately apply the small objectargument to construct localizations. Moreover, because we do not know anythingabout local objects we cannot assert that an S -equivalence X → Y is the same as amap inducing an isomorphism E ∗ X → E ∗ Y .Bousfield addresses this by showing the following. Suppose K → L is an inclusionof simplicial sets such that E ∗ K → E ∗ L is an isomorphism, and that we choose anysimplex σ of L . Then there exists a subcomplex L ′ ⊂ L with the following properties:1. The simplex σ is contained in L ′ .2. The map E ∗ ( K ∩ L ′ ) → E ∗ ( L ′ ) is an isomorphism on E ∗ .3. The complex L ′ has size bounded by a cardinal κ , which depends only on E .Because of the cardinality bound on L ′ , we can find a set T of E ∗ -equivalences A → B so that any such map K ∩ L ′ → L ′ must be isomorphic to one of them; an arbitrary E ∗ -equivalence K → L can then be factored as a (possibly transfinite) sequence ofpushouts along the maps in the set T followed by an equivalence. The maps in T are E ∗ -isomorphisms, and an object is S -local if and only if it is T -local. The smallobject argument then applies to T , allowing us to construct T -localizations Y → LY which are also E ∗ -isomorphisms.We will see in § 10 and § 11, in general constructions of Bousfield localization,that this verification is the key step. Definition 9.16.
For a spectrum E , a map f : X → Y is an E -homology equivalence (orsimply an E -equivalence) if the corresponding map E ∗ X → E ∗ Y is an isomorphism,and we say that Z is E -trivial if E ∗ Z = 0 . A map f is an E -equivalence if and onlyif the cofiber of f is E -trivial. This is most often employed when E is a ring spectrum. Proposition 9.17. If E has a multiplication m : E ∧ E → E with a left unit η : S → E in the homotopy category, then any spectrum Y with a unital map E ∧ Y → Y is E -local. One could, but should not, say it this way: it is not clear that an ( E ∗ -equivalence)-equivalence isautomatically an E ∗ -equivalence. Again, the definitions of this section can be applied to a stable category C with a compatible sym-metric monoidal structure. emark . Such spectra Y are sometimes called homotopy E -modules. Any spec-trum of the form E ∧ W is a homotopy E -module. Proof.
Any map f : Z → Y has the following factorization in the homotopy category: Z η ∧ −−−→ E ∧ Z ∧ f −−−→ E ∧ Y m −→ Y If Z has trivial E -homology, then E ∧ Z is trivial and so the composite Z → Y isnullhomotopic. Therefore, [ Z, Y ] = 0 for all E -trivial Z , as desired. Corollary 9.19. If E has a multiplication m : E ∧ E → E with a left unit η : S → E inthe homotopy category, then any homotopy limit of spectra that admit homotopy E -modulestructures is E -local.Example . A particular case of interest is when E = H Z . Any Eilenberg–MacLane spectrum HA is H Z -local, being of the form H Z ∧ MA for a Moore spectrumfor A .Then any connective spectrum Y is H Z -local, as follows. As H Z -local objectsform a thick subcategory, any spectrum with finitely many nonzero homotopy groupsis therefore H Z -local. If Y is connective then P n Y is H Z -local due to having a finitePostnikov tower. Therefore, Y = holim P n Y is the homotopy limit of H Z -localspectra, and is thus H Z -local.Similarly, any product of Eilenberg–Mac Lane spectra Q Σ n HA n is also H Z -local. Any rational spectrum is of this form.However, not all spectra are H Z -local. For any prime p and integer n > ,there are p -primary Morava K -theories K ( n ) such that H Z ∧ K ( n ) is trivial; theseare H Z -acyclic. The complex K -theory spectrum KU satisfies the property that H ∗ ( KU ; Z ) → H ∗ ( KU ; Q ) is an isomorphism: from this we can find that KU → KU Q is an H Z -equivalence. The target is also H Z -local because it is rational, andso KU Q is the H Z -localization of KU . Example . We can consider the case where E = H Z /p . By a similar argument,we find that any connective spectrum which is p -adically complete in the sense ofExample 8.16 is also H Z /p -complete. Again, in connective cases there is not adifference between being p -adically complete and being H Z /p -local.For nonconnective spectra, these are quite different. The Morava K -theories K ( n ) are p -adically complete but H Z /p -trivial. The periodic complex K -theoryspectrum KU has π ∗ ( KU ∧ p ) (cid:27) ( π ∗ KU ) ∧ p , but KU is also H Z /p -trivial.These localizations have the flavor of completion with respect to an ideal. Insome cases we can express them as such. Definition 9.22.
Suppose that E has a binary multiplication m with a left unit η : S → E , and let j : I → S be the fiber of η : S → E . Assemble these into theinverse system · · · → I ∧ j ∧ ∧ −−−−−→ I ∧ I j ∧ −−−→ I j −→ S The E -nilpotent completion X ∧ E is the homotopy limit holim n ( S /I ∧ n ) ∧ X, X → X ∧ E induced by the maps S → S /I ∧ n . Proposition 9.23.
The E -nilpotent completion is always E -local.If E is a finite complex, or X and I are connective and E is of finite type, then themap X → X ∧ E is an E -localization.Proof. The cofiber sequence I → S → E , after smashing with I ∧ ( n − , becomes acofiber sequence I ∧ n → I ∧ ( n − → E ∧ I ∧ ( n − , and so there are cofiber sequences S /I ∧ n ∧ X → S /I ∧ ( n − ∧ X → E ∧ I ∧ ( n − ∧ X. By induction on n we find that S /I ∧ n ∧ X is E -local, and so the homotopy limit X ∧ E is E -local.After smashing with E , the cofiber sequence E ∧ I ∧ n ∧ X → E ∧ I ∧ ( n − ∧ X → E ∧ E ∧ I ∧ ( n − ∧ X has a retraction of the second map via the (opposite) multiplication of E , and so thefirst map is nullhomotopic. Therefore, the homotopy limit holim E ∧ ( I ∧ n ∧ X ) istrivial, and from the cofiber sequences E ∧ ( I ∧ n ∧ X ) → E ∧ X → E ∧ ( S /I ∧ n ∧ X ) we find that E ∧ X → holim( E ∧ ( S /I ∧ n ∧ X ) is an equivalence.This reduces us to proving that the map E ∧ holim( S /I ∧ n ∧ X ) → holim( E ∧ S /I ∧ n ∧ X ) is an equivalence: we can move the smash product with E inside the homotopy limit.This is always true if E is finite or if E is of finite type and the homotopy limit is ofconnective objects. Remark . The spectral sequence arising from the inverse system defining X ∧ E isthe generalized Adams–Novikov spectral sequence based on E -homology . It often abuts tothe homotopy groups of the Bousfield localization with respect to E .We can generalize our construction by allowing more general towers with anilpotence property, after Bousfield in [Bou79], or by extending these methods tothe category of modules over a ring spectrum, as Baker–Lazarev did in [BL01] orCarlsson did in [Car08]. Example . For any prime p and any n > , we have the Johnson–Wilson ho-mology theories E ( n ) ∗ and the Morava K -theories K ( n ) ∗ . Associated to these wehave E ( n ) -localization functors and K ( n ) -localization functors, as well as categoriesof E ( n ) -local and K ( n ) -local spectra, which play an essential role in chomatic ho-motopy theory. Ravenel conjectured, and Devinatz–Hopkins–Smith proved, that thelocalization L E ( n ) is a smashing localization [Rav84, DHS88, Rav92]. These localiza-tions also have chromatic fractures which are built using the following result.29 roposition 9.26. Suppose that E and K are spectra such that L K L E X is always trivial.Then, for all X , there is a homotopy pullback diagram L E ∨ K X / / (cid:15) (cid:15) L E X (cid:15) (cid:15) L K X / / L E L K X. Proof.
The objects in the diagram L E X → L E L K X ← L K X are either E -local or K -local, and hence automatically E ∨ K -local; therefore, thehomotopy pullback P is E ∨ K -local. It then suffices to show that the fiber of themap X → P is E ∨ K -trivial, which is equivalent to showing that X / / (cid:15) (cid:15) L E X (cid:15) (cid:15) L K X / / L E L K X. becomes a homotopy pullback after smashing with E ∨ K . After smashing with E ,the horizontal maps become equivalences, and so the diagram is a pullback. Aftersmashing with K , the left-hand vertical map is an equivalence and the right-handvertical map is between trivial objects, so the diagram is also a pullback. Therefore,the diagram becomes a pullback after smashing with E ∨ K .
10 Model categories
The lifting characterization of local objects from §5 falls very naturally into theframework of Quillen’s model categories. The groundwork for this is in [Bou75, §10].
Definition 10.1.
Suppose that M is a category with a model structure. We say thata second model structure M ′ with the same underlying category is a left Bousfieldlocalization of M if M ′ has the same family of cofibrations but a larger family ofweak equivalences than M .As a first consequence, note that the identity functor (which is its own right andleft adjoint) preserves cofibrations and takes the weak equivalences in M to weakequivalences in M ′ . This makes it part of a Quillen adjunction M ⇄ M ′ . This has the immediate consequence that the induced adjunction on homotopy cat-egories is a reflective localization.
Proposition 10.2.
Suppose that L : M ⇄ M ′ : R is the adjunction associated to a leftBousfield localization. Then the right adjoint R identifies the homotopy category h M ′ with a full subcategory of h M . roof. It is necessary and sufficient to show that the counit ǫ : LRx → x of theadjunction on homotopy categories is always an isomorphism, for this is the same asasking that, in the factorization Hom h M ( Rx, Ry ) (cid:27) Hom h M ′ ( LRx, y ) → Hom h M ′ ( x, y ) , the second map is an isomorphism.For an object of y , the composite functor LR on homotopy categories is calculatedas follows: find a fibrant replacement y ≃ ′ −−→ y f ′ in M ′ , apply the identity functor toget to M , find a cofibrant replacement ( y f ′ ) c ≃ −→ y f ′ in M , and apply the identityfunctor to get to M ′ . The counit of the adunction is represented in the homotopycategory of M ′ by the composite ( y f ′ ) c ≃ −→ y f ′ ≃ ′ ←−− y. However, equivalences in M are automatically equivalences in M ′ , and so the counitis an isomorphism in the homotopy category of M ′ .Because fibrations and acyclic fibrations are determined by having the right lift-ing property against acyclic cofibrations and fibrations, the new model structure hasthe same acyclic fibrations but fewer fibrations. For example, a fibrant object inthe left Bousfield localization has to have a lifting property against the cofibrationswhich are weak equivalences in M ′ .The next proposition establishes the connection between left Bousfield localiza-tion and ordinary Bousfield localization when both are defined and compatible: thecase of a simplicial model category. Proposition 10.3.
Suppose that M is a simplicially enriched category with two modelstructures, making M → M ′ is a left Bousfield localization of simplicial model categories.Let S be the collection of weak equivalences between cofibrant objects in M ′ . Then, in thecategory of cofibrant-fibrant objects of M , the objects which are fibrant in M ′ are preciselythe S -local fibrant objects.Proof. Fix an object Y of M ′ . For it to be fibrant in M ′ , it must also be fibrant in M .Suppose Y is a fibrant object in M ′ . Given any acyclic cofibration A → B in M ′ , themap of simplicial sets Map M ′ ( A, Y ) → Map M ′ ( B, Y ) is an acyclic fibration by theSM7 axiom of simplicial model categories. Thus, the functor Map M ′ ( − , Y ) from M ′ to the homotopy category of spaces takes acyclic cofibrations to isomorphisms. Thus,Ken Brown’s lemma implies that it also takes weak equivalences between cofibrantobjects in M ′ to isomorphisms in the homotopy category of spaces.Suppose that we have a map f : A → B in S between cofibrant objects of M that is also a weak equivalence in M ′ . Then f is also a weak equivalence betweencofibrant objects of M ′ . The induced map Map M ( B, Y ) → Map M ( A, Y ) is a weakequivalence because the mapping spaces in M and M ′ are the same. Thus, Y is S -local. 31e would now like to establish results in the other direction. Namely, given amodel category M and a collection S of maps A i → B i in M , we would like toestablish the existence of a Bousfield localization M ′ of M . Because we want towork within the already-established homotopy theory of M , we want to use derivedmapping spaces out of A and B and replace homotopy lifting properties with strictlifting properties. We assume without loss of generality that our set S is made up ofcofibrations A i → B i between cofibrant objects. Definition 10.4.
Suppose that M is a simplicial model category, and that f : A → B is a map. Then the iterated double mapping cylinders are the maps ( B ⊗ ∂ ∆ n ) a A ⊗ ∂ ∆ n ( A ⊗ ∆ n ) → B ⊗ ∆ n . This definition is rigged so that an object Y has the right lifting property with re-spect to the iterated double mapping cylinders if and only if the map Map M ( B, Y ) → Map M ( A, Y ) is an acyclic fibration of simplicial sets. One of the equivalent formu-lations of the SM7 axioms for a simplicial model category is that double mappingcylinders are always cofibrations, as follows. Proposition 10.5.
Suppose that f : A → B is a map. If f is a cofibration, then theiterated double mapping cylinders are cofibrations. If A is also cofibrant, then the iterateddouble mapping cylinders have cofibrant source.Remark . If M does not have a simplicial model structure, we can obtain replace-ments for these objects by iteratively replacing the maps B ` A B → B with equivalentcofibrations. Definition 10.7.
Suppose that M is a simplicial model category, that S is a collectionof maps, and that T is the collection of iterated double mapping cylinders of mapsin S . We say that a map in M is an S -cofibration if it is a cofibration in M , and thatit is an S -fibration if it has the right lifting property with respect to the maps in T . Ifthese determine a new model structure M ′ , we call this the left Bousfield localizationwith respect to S .This gives us two fundamentally different approaches to the process of construct-ing a left Bousfield localization. In the first, we may try to expand our family ofweak equivalences to some new family W ; we must then prove that we can constructenough fibrations and fibrant objects to make the model structure work. In the sec-ond, we may try to start with some collection of maps S which serve as new “cells”to build acyclic cofibrations, and use them to contract our family of fibrations; wethen lose control over the weak equivalences, and typically must work to prove thatcofibrations which are weak equivalences can be built out of our new cells.The most advanced technology available for Bousfield localization is Jeff Smith’stheory of combinatorial model categories. Definition 10.8.
A model category M is cofibrantly generated if there are sets I and J of maps satisfying the following properties:32. the fibrations in M are the maps that have the right lifting property withrespect to J ;2. the acyclic fibrations in M are the maps that have the right lifting propertywith respect to I ;3. I permits the small object argument, so that from any object X we can con-struct a map X → X ′ , as a transfinite composition of pushouts along coprod-ucts of maps in I , that has the right lifting property with respect to I ;4. J also permits the small object argument.We refer to I as the set of generating cofibrations and to J as the set of generatingacyclic cofibrations respectively.The cofibrantly generated model category is also combinatorial if it is also locallypresentable, meaning there exists a regular cardinal κ and a set M of objectssatisfying the following properties:1. any small diagram in M has a colimit;2. for any object x in M , the functor Hom M ( x, − ) commutes with κ -filteredcolimits;3. every object in M is a κ -filtered colimit of objects in M . Theorem 10.9 (Dugger’s theorem [Dug01]) . Any combinatorial model category is Quillenequivalent to a left proper simplicial model category.Remark . The axioms of a cofibrantly generated model category and a locallypresentable category have nontrivial overlap. In one direction, the model categoryaxioms already ask that M has all colimits. In the other direction, being locallypresentable means that every set of maps admits the small object argument. Example . Simplicial sets are the motivating example of a combinatorial modelcategory. Fibrations and acyclic fibrations are defined as having the right lifting prop-erty with respect to the generating acyclic cofibrations Λ ni → ∆ n and the generatingcofibrations ∂ ∆ n → ∆ n . The category is also locally presentable because it is gen-erated by finite simplicial sets. Every simplicial set is the filtered colimit of its finitesubobjects; there are only countably many isomorphism classes of finite simplicialsets; for any finite simplicial set X , Hom( X, − ) commutes with filtered colimits. Theorem 10.12 (Smith’s theorem [Bek00, Bar10, Lur09]) . Suppose that M is a locallypresentable category with a family W of weak equivalences and a set I of generatingcofibrations . Call those maps which have the right lifting property with respect to I the acyclic fibrations , and those maps which have the left lifting property with respect toacyclic fibrations the cofibrations . Suppose that we have the following:1. W satisfies the 2-out-of-3 axiom;2. acyclic fibrations are in W ; . the class of cofibrations which are in W is closed under pushout and transfinitecomposition; and4. maps in W are closed under κ -filtered colimits for some regular cardinal κ , andgenerated under κ -filtered colimits by some set of maps in W .Then there exists a combinatorial model structure on M with set I of generating cofibra-tions and set W of weak equivalences. This model structure on M has cofibrant andfibrant replacement functors. Moreover, any combinatorial model structure arises in thisfashion. Corollary 10.13.
Suppose that M is a combinatorial model category with set I of gen-erating cofibrations and class W of weak equivalences. Given a functor E : M → D factoring through the homotopy category h M , define a map to be an E -equivalence if itsimage under E is an isomorphism. Then there exists a left Bousfield localization M E ,whose equivalences are the E -equivalences, if the following conditions hold:1. E -equivalence is preserved by transfinite composition along cofibrations;2. pushouts of E -acyclic cofibrations are E -equivalences; and3. there exists a set of E -acyclic cofibrations that generate all E -acyclic cofibrationsunder κ -filtered colimits.Proof. The 2-out-of-3 axiom is automatic: if two of E ( g ) , E ( f ) and E ( gf ) = E ( g ) E ( f ) are isomorphisms, then so is the third. The fact that E factors through the homotopycategory automatically implies that acyclic fibrations are taken by E to isomorphisms. Example . Let E ∗ be a homology theory on the category of simplicial sets. Theexcision and direct limit axioms for homology imply that E -equivalences are pre-served by homotopy pushouts and transfinite compositions. Therefore, the veri-fication that we have a model structure is immediately reduced to the core of theBousfield–Smith cardinality argument of Example 9.15: that there is a set of E -acycliccofibrations generating all others under filtered colimits.The great utility of combinatorial model structures is that they allow us to build new model categories: categories of diagrams and Bousfield localizations. Theorem 10.15 ([Lur09, A.2.8.2, A.3.3.2]) . Suppose that M is a combinatorial modelcategory and that I is a small category. Then there exists a projective (resp. injective )model structure on the functor category M I , where a natural transformation of diagramsis an equivalence or fibration (resp. cofibration) if and only if it is an objectwise equiva-lence or fibration (resp. cofibration).If M is a simplicial model category, then the natural simplicial enrichment on M I makes the injective and projective model structures into simplicial model categories. Theorem 10.16 ([Lur09, A.3.7.3]) . Suppose that M is a left proper combinatorial sim-plicial model category and that S is a set of cofibrations in M . Let S − M have the ame underlying category as M and the same cofibrations, but with weak equivalences the S -equivalences.Then S − M has the structure of a left proper combinatorial model category, whosefibrant objects are precisely the S -local fibrant objects of M .
11 Presentable ∞ -categories Bousfield localization for model categories has the useful property that it keeps thecategory in place and merely changes the equivalences. One cost is that makinglocalization canonical or extending monoidal structures to localized objects takeshard work. By contrast, localization for ∞ -categories has the useful property that it isgenuinely defined by a universal property , automatically making localization canonicaland making it much easier to extend a monoidal structure to local objects withoutrectifying structure. Of course, this comes at the cost of coming to grips withcoherent category theory itself.The homotopy theory of presentable ∞ -categories is equivalent, in a precisesense, to the homotopy theory of combinatorial model categories [Lur09, A.3.7.6].However, by contrast with our techniques for Bousfield localization using modelcategories and fibrant replacement functors, it allows us to rephrase some of ourlocalization techniques in a way that connects more directly with the homotopicaltechniques that we originally used in §5.In this section, we will let C be an ∞ -category in the sense of [Lur09]. It isoutside our scope to give a technically correct discussion of these. However, thestudy of ∞ -categories is equivalent to the study of categories with morphism spaces,and where possible we will attempt to make connection with classical techniques.With this in mind, if C is an enriched category we will say that a coherent diagram I → C is a coherent functor in the sense of Vogt [Vog73]. This is equivalent to eitherthe notion of a functor C [ I ] → C from a certain simplicially enriched category or tothe notion of a functor I → N C of simplicial sets to the coherent nerve in the senseof [Lur09]. As before a homotopy colimit for such a diagram is based on classicalhomotopy limits and colimits in spaces, and is characterized by having natural weakequivalences Map C (hocolim I F ( i ) , Y ) ≃ holim I Map C ( F ( i ) , Y ) . Definition 11.1 ([Lur09, 5.5.1.1]) . An ∞ -category C is presentable if there there existsa regular cardinal κ and a set C of objects satisfying the following properties:1. any small diagram in C has a homotopy colimit;2. for any object x in C , the functor Hom C ( x, − ) commutes with κ -filtered ho-motopy colimits;3. every object in C is a κ -filtered homotopy colimit of objects in C .This definition is precisely parallel to the definition of local presentability in anordinary category (see Definition 10.8). In essence, C is a large category that isformally generated under colimits by a small category.35iven such an ∞ -category C and a collection S of morphisms in C , it makessense to define the S -local objects and S -equivalences just as in §4: an object Y is S -local if and only if the mapping spaces Map C ( − , Y ) take maps in S to equivalencesof spaces. Definition 11.2 ([Lur09, 5.5.4.5]) . Suppose that C is an ∞ -category with small colim-its and that W is a collection of maps in C . We say that W is strongly saturated if itsatisfies the following conditions:1. given a homotopy pushout diagram C f / / (cid:15) (cid:15) D (cid:15) (cid:15) C ′ f ′ / / D ′ , if f is in W then so is f ′ ;2. the class W is closed under homotopy colimits;3. the class W is closed under equivalence, and its image in the homotopy cate-gory satisfies the 2-out-of-3 axiom. Proposition 11.3 ([Lur09, 5.5.4.7]) . Given a set S of morphisms in C , there is a smallestsaturated class of morphisms containing S . We denote this as ¯ S . If W = ¯ S for some set S ,then we say that W is of small generation. Example . Suppose that E : C → C ′ is a functor of ∞ -categories that preserveshomotopy colimits. Then the set W E of maps in C that map to equivalences isstrongly saturated.The presentability axioms for an ∞ -category provide a homotopical version ofwhat we needed to construct localizations by ensuring that the small object argu-ment goes through. As a result, we obtain a result on the existence of Bousfieldlocalizations for presentable ∞ -categories. Theorem 11.5 ([Lur09, 5.5.4.15]) . Let C be a presentable ∞ -category and S a set ofmorphisms in C , generating the saturated class ¯ S . Let L S C be the full subcategory of S -local objects. Then the following hold:1. for every object C ∈ C , there is a map C → C ′ in ¯ S such that C ′ is S -local;2. the ∞ -category L S C is presentable;3. the inclusion L S C → C has a (homotopical) left adjoint L ;4. the class of S -equivalences coincides with both the saturated class ¯ S and the set ofmaps taken to equivalences by L . emark . The homotopical left adjoint can be rephrased as follows. If we write
Loc S ( C ) for the category of S -localizations C → C ′ , then the forgetful functor Loc S ( C ) → C , sending ( C → C ′ ) to C , is an equivalence of categories (in fact, a trivial fibrationof quasicategories). By choosing a section, given by C ( C → LC ) , we obtain alocalization functor L .As in the case of Bousfield localization of combinatorial model categories, thisconnects the two approaches to Bousfield localization. We can start with a set S ofgenerating equivalences and construct localizations from those, so for a given class W of weak equivalences we are reduced to showing that W is generated by a set S of maps. Moreover, if the maps in S all happen to be in a particular saturated class,then so are the maps in W .
12 Multiplicative properties
Many of the categories where we carry out Bousfield localization have monoidalstructures, and under good circumstances localization is compatible with them. Inthis section we will briefly discuss the circumstances under which this is true.
In order to begin to work with these definitions, we need a monoidal or symmetricmonoidal structure on C that respects morphism spaces. Definition 12.1.
Suppose C is a category enriched in spaces. The structure of an enriched monoidal category on C consists of a functor ⊗ : C × C → C of enrichedcategories, a unit object I of C , and natural associativity and commutativity isomor-phisms that satisfy the axioms for a monoidal category.A compatible symmetric monoidal structure on C is defined similarly.Throughout this section we will fix such an enriched monoidal category C . Definition 12.2.
Suppose that S is a class of morphisms in C . We say that S -equivalences are compatible with the monoidal structure (or simply that S is compat-ible) if, for any S -equivalence f : Y → Y ′ and any object X ∈ C , the maps id X ⊗ f and f ⊗ id X are S -equivalences. Proposition 12.3.
Suppose that S is compatible with the monoidal structure. Thenlocalization respects the monoidal structure: any choices of localization give an equivalence L ( X ⊗ · · · ⊗ X n ) → L ( LX ⊗ · · · ⊗ LX n ) . Proof.
By induction, the map X ⊗ · · · ⊗ X n → LX ⊗ · · · ⊗ LX n is an S -equivalence,and therefore any S -localization of the latter is equivalent to any S -localization ofthe former. 37 orollary 12.4. The monoidal structure on the homotopy category of C induces a monoidalstructure on the homotopy catogory of the localization L S C , making any localization func-tor into a monoidal functor. If C was symmetric monoidal, then so is the localization.Remark . The inclusion L S C → C is almost never monoidal. For example, itusually does not preserve the unit.
Example . Let C be the category of spaces with cartesian product, and let E ∗ be a homology theory. Then any map X → X ′ which induces an isomorphismon E ∗ -homology also induces isomorphisms E ∗ ( X × Y ) → E ∗ ( X ′ × Y ) for any CW-complex Y : one can prove this inductively on the cells of Y . Therefore, E -homologyequivalences are compatible with the Cartesian product monoidal structure.Similarly, E -homology equivalences are compatible with the smash product onbased spaces (using that based spaces are built from S ) or the smash product onspectra (using that all spectra are built from spheres S n ). Example . Let C be the category of spectra, and f be the map S n → ∗ . Then f -equivalences are maps inducing isomorphisms in degree strictly less than n . Thisis not compatible with the smash product on spectra: for example, smashing with Σ − S does not preserve f -equivalences. If one restricts to the subcategory of connec-tive spectra, however, one finds that f -equivalences are compatible with the smashproduct. Example . Consider the map f : S n → ∗ of spaces, so that S -equivalences aremaps inducing an isomorphism on all homotopy groups in degrees less than n . Thismap is compatible with several symmetric monoidal structures, such as:1. spaces with Cartesian product;2. spaces with disjoint union;3. based spaces with wedge product; and4. based spaces with smash product.Despite the usefulness of these results, the existence of a (symmetric) monoidallocalization functor on the homotopy category does not, by itself, allow us to extendvery structured multiplication from an object X to its localization LX . To counterthis we typically require the theory of operads. Definition 12.9.
Suppose that C is (symmetric) monoidal, and that X is an objectof C . The endomorphism operad End C ( X ) is the (symmetric) sequence of spaces Map C ( X ⊗ · · · ⊗ X, X ) , with (symmetric) operad structure given by composition.Given a map f : X → Y , the endomorphism operad End C ( f ) is the (symmetric)sequence which in degree n is the pullback diagram End C ( f ) n / / (cid:15) (cid:15) Map C ( X ⊗ · · · ⊗ X, X ) (cid:15) (cid:15) Map C ( Y ⊗ · · · ⊗ Y , Y ) / / Map C ( X ⊗ · · · ⊗ X, Y ) . End C ( f ) n is the space of strictly commutative diagrams X ⊗ n / / f ⊗ n (cid:15) (cid:15) X f (cid:15) (cid:15) Y ⊗ n / / Y , and as such the operad structure is given by composition.The operad
End C ( f ) has forgetful maps to End C ( X ) and End C ( Y ) . Proposition 12.10.
Suppose that the (symmetric) monoidal structure on C is compatiblewith S and that f : X → LX is an S -localization. If the maps Map C ( LX ⊗ n , LX ) → Map C ( X ⊗ n , LX ) are fibrations for all n ≥ , then in the diagram of operads End C ( X ) ← End C ( f ) → End C ( LX ) , the left-hand arrow is an equivalence on the level of underlying spaces.Proof. This is merely the observation that
End C ( f ) → End C ( X ) is, level by level, ahomotopy pullback of the equivalences Map C ( LX ⊗ n , LX ) → Map C ( X ⊗ n , LX ) .This condition then allows us to lift structured multiplication. Corollary 12.11.
Suppose that a (symmetric) operad O acts on X via a map C →
End C ( X ) . Then there exists a weak equivalence O ′ → O of operads and an actionof O ′ on LX such that f is a map of O ′ -algebras.Proof. We define O ′ to be the fiber product of the diagram O →
End C ( X ) ← End C ( f ) . The map O ′ → O is an equivalence by the fibration condition, and themap O ′ → End C ( f ) of operads precisely states that f is a map of O ′ -algebras. This means that A ∞ and E ∞ multiplications on X extend automatically to A ∞ and E ∞ multiplications on LX . However, this is the best we can do in general: liftingmore refined multiplicative structures requires stronger assumptions.In cases where the category C has more structure, it is typically easier to verifythat S is compatible with the monoidal structure. Proposition 12.12.
Suppose that the monoidal structure on C has internal functionobjects F L ( X, Y ) and F R ( X, Y ) that are adjoint to the monoidal structure: there areisomorphisms Map C ( X, F L ( Y , Z )) (cid:27) Map( X ⊗ Y , Z ) (cid:27) Map C ( Y , F R ( X, Z )) that are natural in X , Y , and Z . Then S is compatible with the monoidal structure on C if and only if, for any f : A → B in S and any object X ∈ C , the maps id X ⊗ f and f ⊗ id X are S -equivalences. If O happens to be a cofibrant (symmetric) operad O in Berger–Moerdijk’s model structure [BM13] wecan do better. Any map O →
End C ( X ) lifts, up to homotopy, to a map O →
End C ( f ) → End C ( LX ) . roof. Suppose that for any f : A → B in S and any object X ∈ C , the maps id X ⊗ f are S -equivalences. Using the unit isomorphisms, we find that if Z is S -local themaps in the diagram Map C ( X ⊗ B, Z ) / / (cid:15) (cid:15) Map C ( X ⊗ A, Z ) (cid:15) (cid:15) Map C ( B, F R ( X, Z )) / / Map C ( A, F R ( X, Z )) are equivalences. Therefore, F R ( X, Z ) is S -local, and so for any S -equivalence f : Y → Y ′ the maps in the diagram Map C ( X ⊗ Y ′ , Z ) / / (cid:15) (cid:15) Map C ( X ⊗ Y , Z ) (cid:15) (cid:15) Map C ( Y ′ , F R ( X, Z )) / / Map C ( Y , F R ( X, Z )) are all equivalences. Similar considerations apply to F L . The necessary conditions for compatibility between model structures and monoidalstructures were determined by Schwede–Shipley [SS00] and Hovey [Hov99, §4.2], inthe symmetric and nonsymmetric cases respectively. This structure allows us, after[SS00], to construct model structures on categories of algebras and modules in M ′ such that the localization functor M → M ′ preserves this structure. Definition 12.13. A (symmetric) monoidal model category M is a model category witha (symmetric) monoidal closed structure satisfying the following axioms.1. (Pushout-product) Given cofibrations i : A → A and j : B → B ′ in M , theinduced pushout-product map i ⊠ j : ( A ⊗ B ′ ) a A ⊗ B ( A ′ ⊗ B ) → A ′ ⊗ B ′ is a cofibration, which is acyclic if either i or j is.2. (Unit) Let Q I → I be a cofibrant replacement of the unit. Then the naturalmaps Q I ⊗ X → X ← X ⊗ Q I are isomorphisms for all cofibrant X . Proposition 12.14.
Suppose that M is a monoidal model category. Then, for cofibrantobjects X , the functors X ⊗ ( − ) and ( − ) ⊗ X preserve cofibrations, acyclic cofibrations, andweak equivalences between cofibrant objects. Analogously to the previous section, this means that the symmetric monoidal structure must have leftand right function objects which are adjoints in each variable. roof. Since ⊗ has adjoints, it preserves colimits in each variable. In particular, anyobject tensored with an initial object of M is an initial object of M . Applying thepushout-product axiom to the map ∅ → X in either variable, we find that the twofunctors in question preserve cofibrations and acyclic cofibrations. By Ken Brown’slemma, they also automatically take weak equivalences between cofibrant objects toweak equivalences.This connects with our work in the the previous section, which only asked thatthe tensor product preserved equivalences in each variable. The pushout-product ax-iom for monoidal model categories looks stronger, in principle, but Proposition 12.14has a partial converse. Proposition 12.15.
Suppose that j : B → B ′ is a map such that ( − ) ⊗ B preserves acycliccofibrations and that ( − ) ⊗ B ′ preserves weak equivalences between cofibrant objects. If i is an acyclic cofibration with cofibrant source, then the pushout-product map i ⊠ j is anequivalence.Proof. Without loss of generality, let i : A → A ′ be an acyclic cofibration and j : B → B ′ a cofibration, with all four objects cofibrant. Then the pushout-product i ⊠ j ispart of the following diagram: A ′ ⊗ B (cid:15) (cid:15) $ $ ❏❏❏❏❏❏❏❏❏ A ⊗ B ∼ : : ✉✉✉✉✉✉✉✉✉ $ $ ■■■■■■■■■ P i ⊠ j / / A ′ ⊗ B ′ A ⊗ B ′∼ O O ∼ : : ttttttttt The upper-left and lower-right maps are equivalences because they are obtainedby tensoring an acyclic cofibration with the cofibrant objects B and B ′ . The map A ⊗ B ′ → P is the pushout of an acyclic cofibration, and so it is an acyclic cofibration.Therefore, by the 2-out-of-3 axiom the map i ⊠ j is an equivalence.The adunction isomorphism Hom M ( X ⊗ Y , Z ) (cid:27) Hom M ( X, F R ( Y , Z )) , and sim-ilarly for the left, allows us to rephrase the pushout-product axiom in multiple ways. Proposition 12.16 ([Hov99, 4.2.2]) . The following are equivalent for a model category M with a closed monoidal structure.1. The model category M satisfies the pushout-product axiom.2. For a cofibration i : A → B and a fibration p : X → Y in M , the induced map F R ( B, X ) → F R ( B, Y ) × F R ( A,Y ) F R ( A, X ) is a fibration, which is acyclic if either i or p are. . For a cofibration i : A → B and a fibration p : X → Y in M , the induced map F L ( B, X ) → F L ( B, Y ) × F L ( A,Y ) F L ( A, X ) is a fibration, which is acyclic if either i or p are. Corollary 12.17 ([Hov99, 4.2.5]) . Suppose that M is a cofibrantly generated model cate-gory with a closed monoidal structure, a set I of generating cofibrations and J of generat-ing acyclic cofibrations. Then the pushout-product axiom for M holds if and only if thepushout-product takes I × I to cofibrations in M and takes both I × J and J × I to acycliccofibrations. Because left Bousfield localization doesn’t change the cofibrations in a modelstructure, one is reduced to a few key verifications.
Proposition 12.18.
Suppose that M is a (symmetric) monoidal closed model categorywith left Bousfield localization M ′ . Then M ′ is compatibly a (symmetric) monoidalmodel category if and only if, for cofibrations i and j such that one is acyclic, the pushout-product map i ⊠ j is acyclic.If M ′ is cofibrantly generated, then it suffices to check that the pushout-product ofa generating acyclic cofibration with a generating cofibration, in either order, is a weakequivalence.Remark . If the generating cofibrations and generating acyclic cofibrations of M ′ have cofibrant source, then by Proposition 12.15 we only need to show thattensoring with the sources or target of any map in I or J takes generating cofibrationsin M ′ to weak equivalences. Remark . Bousfield localization of stable model categories has been more exten-sively studied by Barnes and Roitzheim [BR14, BR15]. To have homotopical controlover commutative algebra objects in a symmetric monoidal model category, one needsto obtain control over the extended power constructions; see [Whi]. ∞ -categories We will begin by giving a brief background on monoidal structures on ∞ -categorieswhich is light on technical details.Recall that a multicategory O is equivalent to the following data:1. a collection of objects of O ;2. for any object Y and indexed set of objects { X s } s ∈ S of O , a space Map O ( { X s } s ∈ S ; Y ) of multimaps; and3. for a surjection p : S → T of finite sets, natural composition maps Map O ( { Y t } t ∈ T ; Z ) × Y t ∈ T Map O ( { X s } s ∈ p − ( t ) ; Y t ) → Map O ( { X s } s ∈ S ; Z ) that are compatible with composing surjections S → T → U .42 emark . As a special case, for σ a permutation of S there is an isomorphism Map O ( { X s } s ∈ S ; Y ) → Map O ( { X σ ( s ) } s ∈ S ; Y ) , and the composition operations are ap-propriately equivariant with respect to these isomorphisms.For such a multicategory, we could give a prototype definition of an O -monoidal ∞ -category C as an enriched functor from O to ∞ -categories. This data specifies,for each object X of O , a category C X . For each object Y and indexed set { X s } s ∈ S ofobjects, there is a specified continuous map from Map O ( { X s } s ∈ S ; Y ) to the space offunctors Q s ∈ S C X s → C Y . Moreover, these maps must be compatible with composi-tion on both sides.The definition of an ∞ -operad O and an O -monoidal ∞ -category C is slightlydifferent from this [Lur17, §2.1]. Roughly, it is an unstraightened definition where thespaces of multimaps in O and the product functors on C are only specified up toa contractible space of choices; the technical details are related in spirit to Segal’swork [Seg74]. Even though the functors induced from O are specified only up tocontractible indeterminacy, it still makes sense to ask about compatibility of themonoidal structure with localization.The following result very general result encodes the situations under which ho-motopical localization is compatible with monoidal structures. Theorem 12.22 ([Lur17, 2.2.1.9]) . Let O ⊗ be an ∞ -operad and let C be an O -monoidal ∞ -category. Suppose that for all objects X of O we have a localization functor L X : C X →C X , and that for any map α : { X s } s ∈ S → Y in O ⊗ the induced functor Q s ∈ S C X s → C Y preserves L -equivalences in each variable. Then there exists an O -monoidal structure onthe category L C of local objects making the localization L : C → L C into an O -monoidalfunctor. Corollary 12.23.
Suppose that C is a (symmetric) monoidal ∞ -category and that L is alocalization functor on C such that L ( X ⊗ Y ) → L ( LX ⊗ LY ) is always an equivalence.Then the subcategory L C of local objects has the structure of a (symmetric) monoidal ∞ -category and any localization functor L has the structure of a (symmetric) monoidalfunctor.Example . In the category of spaces, we can use the mapping space adjunctionsand find that for any S -local object Z , we have Map( X × Y , Z ) ≃ Map( X, Map(
Y , Z )) ≃ Map( X, Map(
LY , Z ) ≃ Map( X × LY , Z ) and similarly on the other side, showing that LX × LY is a localization of X × Y . Thisgives the cartesian product on spaces the special property that it is compatible with all localization functors. Example . Fix an E n -operad O and an O -algebra B in spaces representing an n -fold loop space. Consider the category C of functors B → S , viewed as localsystems of spaces over B . Then the category C has a Day convolution , developed byGlasman [Gla16] in the E ∞ -case and by Lurie [Lur17, §2.2.6] in general, making C O -monoidal category. The category C is equivalent (via unstraightening) tothe category of spaces over B . In these terms the O -monoidal structure is given bymaps O ( n ) → Map( B n , B ) → Fun(( S /B ) n , S /B ) that respect composition. Here f ∈ O ( n ) first goes to f : B n → B , then to the functorsending { X i → B } to the map Q X i → B n f −→ B . An O -algebra in C is equivalent toan E n -space X with a map X → B of E n -spaces.Suppose L is a Bousfield localization on spaces, and consider the associatedpointwise localization on the functor category C (which corresponds to the fiberwiselocalization on spaces over B ). All operations in O are, up to homotopy, compositesof the binary multiplication operation, and so it suffices to show that this preserveslocalization. However, if the maps X i → B have homotopy fibers F i , then the ho-motopy fiber of the map X × X → B × B → B is, up to equivalence, the geometricrealization of the bar construction B ( F , Ω B, F ) . Since any localization preserves homotopy colimits and products of spaces, this barconstruction preserves it also. Therefore, fiberwise localization is an E n -monoidalfunctor on the category of spaces over B . References [Bar10] Clark Barwick,
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