Left Bousfield localization without left properness
aa r X i v : . [ m a t h . A T ] J a n LEFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS
MICHAEL BATANIN AND DAVID WHITEA bstract . Given a combinatorial (semi-)model category M and a set of mor-phisms C , we establish the existence of a semi-model category L C M satisfyingthe universal property of the left Bousfield localization in the category of semi-model categories. Our main tool is a semi-model categorical version of a result ofJe ff Smith, that appears to be of independent interest. Our main result allows forthe localization of model categories that fail to be left proper. We give numerousexamples and applications, related to the Baez-Dolan stabilization hypothesis,localizations of algebras over operads, chromatic homotopy theory, parameter-ized spectra, C ∗ -algebras, enriched categories, dg-categories, functor calculus,and Voevodsky’s work on radditive functors.
1. I ntroduction
Left Bousfield localization is a fundamental tool in abstract homotopy theory. It isused for the study of homology localizations of spaces and spectra [Bou75, Bou79],the existence of stable model structures for (classical, equivariant, and motivic)spectra [Hov01], the towers used in chromatic homotopy theory [Rav84], compu-tations in equivariant homotopy theory [GW18], computations in motivic homo-topy theory [GRSO18], the study of recollement [Gil16], in homological algebra[Bau99], representation theory [Hov02], universal algebra [WY18], graph theory[Vic15], Goodwillie calculus [CW18, Per17], the homotopy theory of homotopytheories [Ber14, Rez10], and the theory of higher categories [BW20], among manyother applications.The left Bousfield localization of a model category M relative to a class of mor-phisms C is a model structure L C M on the category M , where the morphismsin C are contained in the weak equivalences in L C M , and the identity functor Id : M → L C M satisfies the universal property that, for any model category N , any left Quillen functor F : M → N , taking the morphisms in C to weakequivalences in N , factors through L C M . Normally, to prove that L C M exists onerequires C to be a set, and M to be left proper and cellular [Hir03] or left properand combinatorial [Bar10, Bek00]. In this paper, we demonstrate that, even without The first author gratefully acknowledges the financial support of the IHES in Paris where thepaper has been finished.The second author was supported by the National Science Foundation under Grant No. IIA-1414942, and by the Australian Category Seminar. left properness, L C M still exists as a semi-model category, and satisfies the univer-sal property in the category of semi-model categories. This answers a question ofBarwick [Bar10, Remark 4.13], and was also known to Cisinski (private correspon-dence). Recently, an entirely di ff erent proof of this result has been discovered bySimon Henry [Hen ∞ ]. A related approach, for cellular model categories, appearsin [GH04, HZ19].Semi-model categories were introduced in [Hov98] and [Spi01] in the context ofalgebras over operads, and are reviewed in Definition 2.1. A semi-model cate-gory satisfies axioms similar to those of a model category, but one only knows thatmorphisms with cofibrant domain admit a factorization into a trivial cofibrationfollowed by a fibration, and one only knows that trivial cofibrations with cofibrantdomain lift against fibrations. Hence, on the subcategory of cofibrant objects, asemi-model category behaves exactly like a model category, and every semi-modelcategory admits a functorial cofibrant replacement functor. Consequently, everyresult about model categories has a version for semi-model categories, usually ob-tained by cofibrantly replacing objects as needed. This includes the usual character-ization of morphisms in the homotopy category, Quillen pairs, simplicial mappingspaces, Hammock localization, path and cylinder objects, Ken Brown’s lemma, theretract argument, the cube lemma, projective / injective / Reedy semi-model struc-tures, latching and matching objects, cosimplicial and simplicial resolutions, com-putations of homotopy limits and colimits, and more. In practice, a semi-modelstructure is just as useful as a full model structure.The main source of examples of semi-model categories arises from the theory oftransferred (also known as left-induced) model structures. If T is a monad, thetransferred structure, on the category of T -algebras in a model category M , definesweak equivalences and fibrations to be created and reflected by the forgetful func-tor to M . When T arises from an operad, this transferred structure is commonlya semi-model structure [GRSO12, Fre09, WY18], but is not always a full modelstructure [BW16, Example 2.9]. Semi-model categories have been used to proveimportant results all over homotopy theory [Bar10, BD17, Bat17, BW15, BW16,EKMM97, Fre09, GH04, GRSO12, GRSO18, GW18, HZ19, Hov98, Man01, Nui19,Ost10, Spi01, Whi14b, WY18, WY15, WY19, WY16, WY17, Yau19].In recent years, the authors have seen a large number of cases where one wishesto left Bousfield localize a model structure that is not known to be left proper[Bac13, Bac14, BBPTY16a, Bea19, Ber14, BCL18, CG16, GH04, HRY17, IJ02,JJ07, Per17, RS17, Tab15, To¨e10, Vic15, Voe10]. Our main result provides a wayto do this, and even to left Bousfield localize semi-model structures. We now stateour main result. Theorem A.
Suppose that M is a combinatorial semi-model category whose gen-erating cofibrations have cofibrant domain, and C is a set of morphisms of M .Then there is a semi-model structure L C ( M ) on M , whose weak equivalences arethe C -local equivalences, whose cofibrations are the same as M , and whose fibrant EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 3 objects are the C -local objects. Furthermore, L C ( M ) satisfies the universal prop-erty that, for any any left Quillen functor of semi-model categories F : M → N taking C into the weak equivalences of N, then F is a left Quillen functor whenviewed as F : L C ( M ) → N . Note that, if M is a model category, then M is automatically a semi-model cate-gory, and so the theorem above proves that left Bousfield localization L C M exists(as a semi-model category) for non-left proper model categories M .Our main tool to prove Theorem A is a semi-model categorical version of a famoustheorem of Je ff Smith [Bar10, Bek00]. This result appears to be of independentinterest, so we state it as well. We apply this theorem by taking W to be the classof C -local equivalences. Theorem B.
Suppose M is a locally presentable category with a class W of weakequivalences and a set of map I satisfying(1) The class W is κ -accessible.(2) The class W is closed under retracts and the two out of three property.(3) Any morphism in inj ( I ) is a weak equivalence.(4) Within the class of trivial cofibrations, defined to be the intersection of cof I and W , maps with cofibrant domain are closed under pushouts toarbitrary cofibrant objects and under transfinite composition.(5) The maps of I have cofibrant domain and the initial object in M is cofi-brant.Then there is a cofibrantly generated semi-model structure on M with generatingcofibrations I, generating trivial cofibrations J, cofibrations cof I, and fibrationsdefined by the right lifting property with respect to J. Furthermore, the generatingtrivial cofibrations J have cofibrant domains.
After a review of the main definitions in Section 2, we prove Theorem B in Section3. We then prove Theorem A in Section 4. As the main value of our approach isthat we do not need M to be left proper in order for its localization to exist, we nowexplain the key idea that allows this assumption to be avoided. A model categoryis left proper if any pushout of a weak equivalence f : A → B along a cofibration g : A → C is a weak equivalence h : C → P . The semi-model category versionof this statement assumes that f is a weak equivalence between cofibrant objects .With this extra assumption, h is always a weak equivalence, so left properness isautomatic.The main place where left properness is used when proving the existence of leftBousfield localization, is to prove that pushouts of trivial cofibrations are againtrivial cofibrations (see Chapter 3 of [Hir03], and note that left properness is not re-quired till Proposition 3.2.10). Crucially, in a left proper model category, a pushout MICHAEL BATANIN AND DAVID WHITE square where one leg is a cofibration is a homotopy pushout square [Bar10, Propo-sition 1.19]. Thankfully, when we establish a semi-model structure L C M , we onlyneed this for a pushout square where all objects are cofibrant, and one leg is a trivialcofibration. Such squares are always homotopy pushout squares, even when L C M is only a semi-model structure.The other main place where left properness is needed in the theory of left Bousfieldlocalization is Proposition 13.2.1 in [Hir03], which states that for any cofibration g : A → B , any fibration p : X → Y , and any cofibrant replacement Qg : QA → QB (which is a cofibration) as shown below: QA (cid:127) _ (cid:15) (cid:15) / / A (cid:127) _ (cid:15) (cid:15) / / X (cid:15) (cid:15) (cid:15) (cid:15) QB / / B / / Y then p has the right lifting property with respect to g , if p has the right lifting prop-erty with respect to Qg and if M is left proper. This is used when characterizingthe fibrant objects of a left Bousfield localization, and when verifying the univer-sal property of left Bousfield localization, because of the way Hirschhorn defineshis set of generating trivial cofibrations [Hir03, Definition 4.2.2]. The semi-modelcategory version of [Hir03, Proposition 13.2.1] assumes g is already a cofibrationbetween cofibrant objects, and hence holds in any semi-model category, or in anymodel category (even one that fails to be left proper). In our case, we side-step thisresult entirely, because for us, the domains of the generating (trivial) cofibrationsin L C M are cofibrant, and the local fibrations are defined to be morphisms with theright lifting property with respect to the generating trivial cofibrations J C providedby Theorem B.After proving Theorem A in Section 4, in Section 5, we consider several applica-tions of Theorem A and propose future directions.We open Section 5 by recalling Voevodsky’s theory of radditive functors [Voe10].Voevodsky constructs an example [Voe10, Example 3.48] of a non left proper cate-gory of radditive functors and proves that it does not admit a left Bousfield localiza-tion as a model category. This example clearly shows that our theory of semi-modellocalization is a powerful new tool which allows us to overcome many technicaldi ffi culties arising from the non-existence of model theoretical localization.We then continue in Section 5 with sample applications from di ff erent areas of ho-motopy theory. We briefly describe the results from our companion paper [BW20],where we prove a strong version of the Baez-Dolan stabilization hypothesis [BD95]for Rezk’s model of weak n -categories [Rez10]. Other applications of our resultinclude the resolution model structure in chromatic homotopy theory [GH04], T Q -homology [HZ19], Ravenel’s X ( n )-spectra [Bea19], parameterized spectra after In-termont and Johnson [IJ02], C ∗ -algebras [JJ07, Ost10], chain complexes [RS17], EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 5 inverting operations in ring theory [BCL18] and operad theory [HRY17], factor-ization algebras [CG16], the theory of weakly enriched categories [Bac14], dg-categories [To¨e10], Goodwillie calculus [Per17], graph theory [Vic15] and the the-ory of homotopy colimits of diagrams of model categories [Ber14]. We anticipatemany more applications of Theorem A in the years to come.A cknowledgments
The authors would like to thank Denis-Charles Cisinski for suggesting this prob-lem, and Clark Barwick for leaving such a nice road-map to its resolution. Thesecond author is grateful to Mark Johnson for helpful conversations, and to SimonHenry for sharing an advance version of his work. We would also like to thankMacquarie University for hosting the second author on three occasions while wecarried out this research. 2. P reliminaries
In this section, we recall definitions and useful results about semi-model categories,and about left Bousfield localization. For further details on these topics, we referthe reader to [Bar10, Fre09, GH04, Hov98, Spi01, Whi14b, WY18] and to [Hir03].We assume the reader is familiar with the basics of model categories, as recountedin [Hov99]. We begin with the definition of a semi-model category [Bar10]. Recallthat, for a set of morphisms S , inj S refers to the class of morphisms having theright lifting property with respect to S . Definition 2.1. A semi-model structure on a category M consists of classes ofweak equivalences W , fibrations F , and cofibrations Q satisfying the followingaxioms:M1 The initial object is cofibrant.M2 The class W is closed under the two out of three property.M3 W , F , Q are all closed under retracts.M4 i Cofibrations have the left lifting property with respect to trivial fibra-tions.ii Trivial cofibrations whose domain is cofibrant have the left liftingproperty with respect to fibrations.M5 i Every map in M can be functorially factored into a cofibration fol-lowed by a trivial fibration.ii Every map whose domain is cofibrant can be functorially factored intoa trivial cofibration followed by a fibration.M6 Fibrations are closed under pullback. MICHAEL BATANIN AND DAVID WHITE
If, in addition, M is bicomplete, then we call M a semi-model category . M issaid to be cofibrantly generated if there are sets of morphisms I and J in M suchthat inj I is the class of trivial fibrations, inj J is the class of fibrations in M , thedomains of I are small relative to I -cell, and the domains of J are small relative tomaps in J -cell whose domain is cofibrant. We will say M is combinatorial if it iscofibrantly generated and locally presentable.Our definition of semi-model category follows Barwick [Bar10], which was in-spired by Spitzweck’s notion of a J -semi model category [Spi01], but removingthe need for this abstract structure to be transferred from some underlying modelcategory. Many of the semi-model categories M that we have in mind are in facttransferred along a right adjoint U : M → D , so that weak equivalences and fi-brations in M are maps f such that U ( f ) is a weak equivalence or fibration in D .But the definition allows for semi-model categories to exist without reference to amodel category D , and Barwick showed how to recover Spitzweck’s results in thismore general setting [Bar10]. We note that Spitzweck originally assumed in (M6)that trivial fibrations are also closed under pullback. Barwick proved that this isredundant, since trivial fibrations are characterized as maps having the right lift-ing property with respect to cofibrations [Bar10, Lemma 1.7] and hence are closedunder pullback and composition. For a cofibrantly generated semi-model cate-gory, (M6) is entirely redundant, since fibrations are characterized as inj J . If M is combinatorial, these observations show that (trivial) fibrations are closed undertransfinite composition. Throughout this paper, we work with cofibrantly generatedsemi-model categories, so we say nothing more about (M6).We note that the assumptions we require of a semi-model category are stricter thanthose required by Fresse [Fre09], who generalized Spitzweck’s notion of an ( I , J )-semi model structure, and hence all results proven by Fresse hold in our setting.We gather a few useful results, the proofs of which are useful exercises (which mayalso be found in [Bar10, Fre09, Spi01]): Lemma 2.2.
Let M be a cofibrantly generated semi-model category. Then:(1) Cofibrations are closed under pushout and transfinite composition.(2) Relative J-cell complexes with cofibrant domain are trivial cofibrations.(3) Trivial cofibrations with cofibrant domain are retracts of relative J-cellcomplexes.
3. S mith ’ s theorem for locally presentable semi - model categories In this section, we prove a version of Smith’s theorem [Bar10, Bek00], that pro-vides a set J of generating trivial cofibrations to produce a semi-model structure ona locally presentable category with a given class of weak equivalences, and a givenset of generating cofibrations, satisfying some compatibility axioms. This result isour main tool for proving Theorem A. In the following, cof S means maps with the EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 7 left lifting property with respect to inj S . Let (cof S ) c denote the class of maps incof S that have cofibrant domains. We refer the reader to [AR94] for terminologyrelated to accessibility. Theorem 3.1.
Suppose M is a locally presentable category with a class W of weakequivalences and a set of morphisms I satisfying:(1) the class W is κ -accessible,(2) the class W is closed under retracts and the two out of three property,(3) any morphism in inj ( I ) is a weak equivalence,(4) within the class of trivial cofibrations, defined to be the intersection of cof I and W , morphisms with cofibrant domain are closed under pushoutsto arbitrary cofibrant objects and under transfinite composition, and(5) the morphisms of I have cofibrant domain and the initial object in M iscofibrant.Then there is a combinatorial semi-model structure on M with generating cofibra-tions I, generating trivial cofibrations J, cofibrations cof I, and fibrations definedby the right lifting property with respect to J. Furthermore, the generating trivialcofibrations J have cofibrant domains.
We will now prove this, following [Bar10, Lemma 2.3 and 2.4] (equivalently,[Bek00, Lemma 1.8 and 1.9]), which we restate for semi-model categories below.Let ( W ∩ cof I ) c denote the subclass of W ∩ cof I consisting of maps with cofibrantdomain (hence cofibrant codomain as well). Let (cell J ) c denote the smallest classof morphisms with cofibrant domain containing J and closed under pushouts alongmorphisms to arbitrary cofibrant objects and under transfinite composition. Lemma 3.2.
Under the hypotheses of Theorem 3.1, suppose J ⊂ ( W ∩ cof I ) c is aset of morphisms such that any commutative squareK i (cid:15) (cid:15) / / M w (cid:15) (cid:15) L / / Nin which i ∈ I and w is in W , can be factored as a commutative diagramK (cid:15) (cid:15) / / M ′ (cid:15) (cid:15) / / M (cid:15) (cid:15) L / / N ′ / / N , in which M ′ → N ′ is in J. Then (cof J ) c = ( W ∩ cof I ) c .Proof. To show (cof J ) c ⊃ ( W ∩ cof I ) c , let f ∈ ( W ∩ cof I ) c , and recall thatthis means f has cofibrant domain. We will factor f as an element i of (cell J ) c followed by an element p of inj I . Once we do this, f has the left lifting property MICHAEL BATANIN AND DAVID WHITE with respect to p and the retract argument says f is a retract of i . Lemma 2.1.10 in[Hov99] demonstrates that i is in cof J . As cof J is defined by lifting, it is closedunder retract, so this proves f is in cof J . Since f was assumed to be a map betweencofibrant objects, f is in fact in (cof J ) c .To produce the factorization for f we follow [Bar10]. Choose a regular cardinal κ such that the codomains of maps in I are κ -presentable. Consider the set ( I / f ) ofsquares K i (cid:15) (cid:15) / / X f (cid:15) (cid:15) L / / Y , where i ∈ I ; for each such square choose an element j ( i , f ) ∈ J and a factorization K i (cid:15) (cid:15) / / M ( i ) j ( i , f ) (cid:15) (cid:15) / / X f (cid:15) (cid:15) L / / N ( i ) / / Y , and let M ( I / f ) → N ( I / f ) be the coproduct ` i ∈ ( I / f ) j ( i , f ) . Define an endofunctor Q of( W / Y ) by Q f : = X a N ( I / f ) M ( I / f ) → Y for any morphism f : X → Y in W . For any regular cardinal α , set Q α : = colim β<α Q β . This provides, for any morphism f : X → Y in W , a functorialfactorization X / / Q κ f / / Y where the map X → Q κ f is in cell J and the map Q κ f → Y is in inj J .The containment (cof J ) c ⊂ ( W ∩ cof I ) c follows from Proposition 2.1.15 in [Hov99],from the small object argument above, and from hypothesis (4) of the theorem.This is because any map in (cof J ) c is a retract of a map in (cell J ) c , via the retractargument and the factorization provided above (as well as the hypothesis that J consists of cofibrations between cofibrant objects). Next, hypothesis (4) ensuresthat (cell J ) c ⊂ ( W ∩ cof I ) c , and both W and cof I are closed under retract (theformer by the argument of Proposition A.2.6.8 in [Lur09] since W is accessible;the latter because it is defined via a lifting property). (cid:3) Observe that it is not true in general for semi-model categories that trivial cofibra-tions are closed under transfinite composition and pushout. The class of maps cell J might not be contained in W ∩ cof I , even though it is always contained in cof J .However, requiring the domains of the maps in J to be cofibrant and only consider-ing pushouts via maps to cofibrant objects will result in (cell J ) c being contained in EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 9 ( W ∩ cof I ) c by Lemma 2.2 (hence (cof J ) c ⊂ ( W ∩ cof I ) c ). Similarly, observationLemma 2.2 implies that (cof J ) c ⊃ ( W ∩ cof I ) c .Next we address the existence of a set J which factorizes squares as above. Lemma 3.3.
Under the hypotheses of Theorem 3.1, there is a set J satisfying theconditions of the lemma above.Proof.
Suppose i : K → L is in I . Since W is an accessibly embedded accessiblesubcategory of the arrow category Arr ( M ), there exists a subset W ( i ) ⊂ W suchthat for any commutative square K i (cid:15) (cid:15) / / M (cid:15) (cid:15) L / / N in which M → N is in W , there exist a morphism w : P → Q in W ( i ) and acommutative diagram K (cid:15) (cid:15) / / P (cid:15) (cid:15) / / M (cid:15) (cid:15) L / / Q / / N . It thus su ffi ces to find, for every square of the type on the left, an element of W ∩ cof I factoring it.For every i and w as above, and every commutative square K i (cid:15) (cid:15) / / P (cid:15) (cid:15) L / / Q , factor the morphism L ` K P → Q through an object R as an element of cell I followed by an element of inj I . This yields a commutative diagram K (cid:15) (cid:15) / / P (cid:15) (cid:15) P (cid:15) (cid:15) L / / R / / Q factoring the original square. Furthermore, P → R is in W because P → Q is in W and R → Q is in inj I , which is assumed to be in W . Finally, P → R is in cof I because it’s the composite P → L ` K P → R where the first map is a pushout of i (hence is a cofibration) and the second is in cell I (hence is a cofibration). Thus, P → R is in W ∩ cof I .Here we are using the fact that cofibrations are closed under transfinite compositionand pushout without any hypothesis on the domains and codomains of the maps inquestion (Lemma 2.2). (cid:3) Just as in [Bar10, Corollary 2.7], we also have a corollary which replaces the set J produced above by a set of maps with cofibrant domains. Corollary 3.4.
Under the conditions of Theorem 3.1, a set J can be constructedsatisfying the hypotheses of Lemma 3.2 and consisting of maps between cofibrantobjects.Proof.
Let J be the set of maps produced by Lemma 3.3 above. Following [Bar10,Corollary 2.7], we factorize any commutative square K i (cid:15) (cid:15) / / M j (cid:15) (cid:15) L / / N with i ∈ I and j ∈ J into a commutative diagram K (cid:15) (cid:15) / / M ′ (cid:15) (cid:15) / / M (cid:15) (cid:15) L / / N ′ / / N in which M ′ is cofibrant and M ′ → N ′ is in W ∩ cof I . To do so, factor K → M as a cofibration K → M ′ followed by a trivial fibration M ′ → M and then factor L ` K M ′ → N as a cofibration L ` K M ′ → N ′ followed by a trivial fibration N ′ → N . The map M ′ → L ` K M ′ is a pushout of K → L and so is a cofibration.The map L ` K M ′ → N ′ is constructed to be a cofibration. Furthermore, because M ′ → M , N ′ → N , and M → N are weak equivalences the two out of threeproperty implies M ′ → N ′ is a weak equivalence. That M ′ and N ′ are cofibrantfollows from the fact that K and L are cofibrant, which is part of our hypotheses on I . The set of maps M ′ → N ′ is the set required. (cid:3) Using these lemmas we may prove the theorem.
Proof of Theorem 3.1.
We check the axioms in Definition 2.1 directly. First, ob-serve that M is assumed to be locally presentable so it is certainly bicomplete. M1is part of hypothesis (5). M2 is hypothesis (1) of the theorem. For M3, the closureof W under retracts follows from the accessibility of W in Arr ( M ) (see Proposi-tion A.2.6.8 of [Lur09]). Closure of fibrations under retract follows from the factthat fibrations are defined to be inj J . Closure of cofibrations under retracts followsfrom the fact that the cofibrations are defined to be cof I . This also covers M4i. ForM5i, factor a map f into an element i of cell I followed by an element p of inj I .By construction, p is a trivial fibration. Because transfinite composites of pushoutsof cofibrations are cofibrations, i is a cofibration.We turn now to the places where the definition of a semi-model category di ff ersfrom that of a model category. For M5ii, we must show that any morphism f : X → Y with a cofibrant domain admits a factorization into a trivial cofibration EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 11 (i.e. an element of W ∩ cof I ) followed by a fibration. The set J ⊂ ( W ∩ cof I ) c produced by Corollary 3.4 has the property that (cof J ) c ⊃ ( W ∩ cof I ) c . With thisset in hand we may factor f into γ ( f ) ◦ δ ( f ) where δ ( f ) is in cell J and γ ( f ) is ininj J (equivalently, γ ( f ) is a fibration).If X is cofibrant then the proof of Lemma 3.2 demonstrates that δ ( f ) is a trivialcofibration, because the factoring objects Q β are constructed via a transfinite pro-cess beginning with X and progressing via pushouts with respect to coproducts ofthe maps in J . As each map in J is a cofibration between cofibrant objects, thesecoproducts are again trivial cofibrations between cofibrant objects, and so eachpushout is again a map of this type. Thus, hypothesis (4) guarantees us that δ ( f ) isa trivial cofibration.For M4ii, let f be a trivial cofibration whose domain is cofibrant, i.e. f ∈ ( W ∩ cof I ) c . Lemma 3.2 proves ( W ∩ cof I ) c = (cof J ) c , so f has the left lifting propertywith respect to inj J (i.e. with respect to all fibrations). (cid:3)
4. L eft B ousfield localization for locally presentable semi - model categories In this section we will use Theorem 3.1 to prove existence of left Bousfield localiza-tion for semi-model categories. We first need a few facts about locally presentablesemi-model categories, following [Bar10]. The first is the semi-model categoryanalogue of [Bar10, Proposition 2.5]:
Proposition 4.1.
Suppose M is a locally presentable cofibrantly generated semi-model category, with generating cofibrations I. For any su ffi ciently large regularcardinal κ :(1) There is a κ -accessible functorial factorization of each morphism into acofibration followed by a trivial fibration.(2) There is a κ -accessible functorial factorization of each morphism with cofi-brant domain into a trivial cofibration followed by a fibration.(3) There is a κ -accessible cofibrant replacement functor.(4) There is a κ -accessible fibrant replacement functor.(5) Arbitrary κ -filtered colimits preserve weak equivalences.(6) Arbitrary κ -filtered colimits of weak equivalences are homotopy colimits.(7) The class of weak equivalences is κ -accessible.Proof. The proof proceeds just as in [Bar10, Proposition 2.5], using the small ob-ject argument to prove (1)-(4). Note that M has an accessible fibrant replacementfunctor obtained by first applying cofibrant replacement and then applying fibrantreplacement. To prove (5) and (6), we follow [Bar10, Proposition 2.5], except that in order to factor a weak equivalence into a trivial cofibration followed by a triv-ial fibration, we rely on (1) instead of (2). This produces, for an objectwise weakequivalence F → G , a factorization F → H → G where colim H → colim G isa fibration. To prove it is a trivial fibration, we apply a lifting argument againstmorphisms in I , just as in [Bar10, Proposition 2.5], relying on the κ -presentabilityof the domains and codomains of objects in I .To prove (7), consider the functor R : Arr ( M ) → Arr ( M ) which takes a morphismto the right factor in its factorization as a cofibration followed by a trivial fibra-tion. By the two out of three property, the weak equivalences are the preimage ofthe trivial fibrations under this functor. Furthermore, this functor is accessible bythe previous paragraph. Once κ is chosen large enough that the (co)domains of I are κ -presentable, the proof that the class of trivial fibrations is accessible followsprecisely as it does in [Bar10, Proposition 2.5]. (cid:3) We turn now to left Bousfield localization. We remind the reader that cofibrantlygenerated semi-model categories have simplicial mapping spaces [GH04, Section1.1], that we will denote map ( − , − ) ∈ sS et . Given a class of morphisms C ina cofibrantly generated semi-model category M , an object W is called C -local if map ( f , W ) is a weak equivalence of simplicial sets for all f ∈ C . A morphism g in M is a C -local equivalence if map ( g , W ) is a weak equivalence for all C -localobjects W . Several properties about C -local objects and equivalences, proven in[Hir03] without reference to a model structure on M , will be used below. Becausea set of morphisms C can always be replaced by a set of cofibrations betweencofibrant objects, without changing the left Bousfield localization L C M , we willalways assume C is a set of cofibrations between cofibrant objects. For the proofthat follows, we advise the reader to have copies of [Bar10, Hir03] on hand. Theorem 4.2. If M is a locally presentable, cofibrantly generated semi-model cat-egory in which the domains of the generating cofibrations are cofibrant. For anyset of morphisms C in M , there exists a cofibrantly generated semi-model struc-ture L C ( M ) on M with weak equivalences defined to be the C -local equivalences,(generating) cofibrations defined to match the (generating) cofibrations of M , andfibrations defined by the right lifting property with respect to some set J C of C -localequivalences which are also cofibrations with cofibrant domains. Furthermore, thefibrant objects of L C M are precisely C -local fibrant objects in M .Proof. The semi-model structure on L C ( M ) will be obtained via Theorem 3.1 assoon as we check conditions (1)-(5). We begin with condition (1). First, Lemma 4.5of [Bar10] states that the set of C -local objects is an accessibly embedded, accessi-ble subcategory of M . This lemma remains true for semi-model categories, sincethe proof only requires the existence of an accessible fibrant replacement functorand the fact that the subcategory of weak equivalences is accessibly embedded andaccessible. Next, this lemma implies that the class of C -local equivalences is anaccessibly embedded, accessible subcategory of Arr ( M ) (which is Lemma 4.6 in[Bar10]). This proof only requires that κ -filtered colimits are homotopy colimits for EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 13 su ffi ciently large κ , and again this holds for semi-model categories (see Proposition4.1). This completes the verification of (1).The closure of the class of C -local equivalences under the two out of three propertyis proven in a similar way to Proposition 3.2.3 in [Hir03]. Namely, given g , h , h ◦ g one applies functorial cofibrant replacement. Given a C -local object one takes thesimplicial resolution b W . Care is required here since b W is a fibrant replacement andso W should be cofibrant in order to ensure the existence of b W in the Reedy semi-model structure on M ∆ op . We will remark on this in a moment. Once b W is in hand,the diagram (from [Hir03, Proposition 3.2.3]) featuring e g ∗ : M ( e Y , b W ) → M ( e X , b W )and e h ∗ and g h ◦ g ∗ will satisfy the two out of three property in sSet and so if any twoof g , h , h ◦ g are C -local equivalences then so is the third. In order to guarantee that b W exists one should first take cofibrant replacement of W in M . This will not e ff ectthe homotopy type of b W or of M ( e Z , b W ) for any Z , by the two out of three propertyin the Reedy semi-model structure [Bar10, Theorem 3.12] and in sS et respectively.For this reason we will tacitly assume that b W exists whenever we want it to, by firstcofibrantly replacing W if necessary.The closure of the class of C -local equivalences under retract is proven analogously,following Proposition 3.2.4 in [Hir03], which again applies cofibrant replacementto the maps in question and then considers the maps induced in sS et by M ( e f , b W )for all C -local W . This completes our proof of (2).For (3), note that if f is in inj( I ) in L C ( M ) then f is in inj( I ) in M . Thus, f isa trivial fibration in M . So all we need to show is that a weak equivalence of M is a C -local equivalence. This is true by general properties of simplicial mappingspaces, even in a semi-model category [Bar10, Corollary 3.66].We must now check (4). Suppose f : A → B is an element of ( W ∩ cof I ) c , i.e., a C -local equivalence and a cofibration between cofibrant objects. Suppose g : A → X is any map such that X is cofibrant. Then the pushout A / / (cid:15) (cid:15) u B (cid:15) (cid:15) X / / P has h : X → P a cofibration and we must show it’s a C -local equivalence. We notethat this square is a homotopy pushout square, because one leg is a cofibration, andall objects are cofibrant [Spi01, page 10]. Note that this is where left propernesswould normally be required, but we don’t need it because we assume X is cofibrant.We fix a C -local object W and, following [Bar10, Theorem 4.7], we must prove thefollowing is a homotopy pullback diagram in sS et : map ( Y , W ) / / (cid:15) (cid:15) map ( X , W ) (cid:15) (cid:15) map ( B , W ) / / map ( A , W ) . As simplicial mapping spaces are fibrant simplicial sets [Bar10, Scholium 3.64], itsu ffi ces to observe that map ( B , W ) → map ( A , W ) is a weak equivalence, since f isa C -local equivalence. Next, a transfinite composition of elements of ( W ∩ cof I ) c is an element of cof I because M is a semi-model category, and is a weak equiv-alence because κ -filtered colimits are homotopy colimits for κ su ffi ciently large(Proposition 4.1).Finally, condition (5) is part of the hypotheses, since we assume M is a cofibrantlygenerated semi-model category with domains of I cofibrant. For the last sentenceof the statement of the theorem, we refer the reader to [Bar10, Corollary 3.66].One could also follow Hirschhorn’s theory of homotopy orthogonal pairs [Hir03,Propositions 17.8.5, 17.8.9], which makes use of properties of the model category sS et , and holds for semi-model categories. (cid:3) We turn now to verifying the universal property of left Bousfield localization (withrespect to left Quillen functors of semi-model categories [Bar10, Definition 1.12]).
Theorem 4.3.
Suppose that there is a cofibrantly generated semi-model structureL C ( M ) on the semi-model category M as in Theorem 4.2. Then L C ( M ) satisfiesthe following universal property. Suppose F : M → N is any left Quillen functorof semi-model categories taking C into the weak equivalences of N. Then F is aleft Quillen functor when viewed as F : L C ( M ) → N . To prove this, we need a lemma, inspired by [Hir03, Proposition 8.5.3].
Lemma 4.4.
Let F : M ⇆ N : U be a pair of adjoint functors between semi-model categories M and N . Then the following are equivalent:(1) ( F , U ) is a Quillen pair.(2) F preserves cofibrations and preserves trivial cofibrations between cofi-brant objects.(3) U preserves fibrations and trivial fibrations.(4) F preserves cofibrations and U preserves fibrations.(5) F preserves trivial cofibrations whose domain is cofibrant and U preservestrivial fibrations.Proof. The equivalence of (1), (2), and (3) is part of the definition of a Quillen pairfor semi-model categories [Bar10, Definition 1.12], [Fre09, Section 12.1.8]. For(4), we use that the hypothesis on F implies U preserves trivial fibrations, sincetrivial fibrations are characterized as morphisms satisfying the right lifting prop-erty with respect to cofibrations [Bar10, Lemma 1.7.1]. For (5), we use that thehypothesis on U implies F preserves cofibrations, which are characterized as mor-phisms satisfying the left lifting property with respect to trivial fibrations [Bar10,Lemma 1.7.1]. (cid:3) EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 15
Proof of Theorem 4.3.
Let G : N → M be the right adjoint of F , and let U denote G viewed as a functor from N to L C ( M ), since as categories M and L C ( M ) areequal. We must prove U is right Quillen [Bar10, Definition 1.12]. The trivial fibra-tions of L C ( M ) are equal to the trivial fibrations of M , since both are characterizedas inj I . Thus, U preserves trivial fibrations. We will now prove F preserves trivialcofibrations whose domain is cofibrant, which is su ffi cient by Lemma 4.4.Let g be a trivial cofibration between cofibrant objects in L C M . We already knowthat Fg is a cofibration, since the cofibrations of M and L C M coincide. To provethat Fg is a weak equivalence in N , it su ffi ces to prove, for every fibrant X in N ,that map ( Fg , X ) is a weak equivalence of simplicial sets. Using [Bar10, Scholium3.64], we see that map ( Fg , X ) ≃ map ( g , U X ). It is therefore su ffi cient to prove that U X is a C -local object, i.e., that U takes fibrant objects of N to local objects of M .To prove this, let f be a morphism in C , and note that, by [Bar10, Scholium 3.64]again, map ( f , U X ) ≃ map ( F f , X ). Since C consists of cofibrations between cofi-brant objects, and F f is a weak equivalence by assumption, we see that map ( F f , X )is a weak equivalence of simplicial sets, proving that U X is C -local as required. (cid:3) Remark . While we have not needed further results from [Hir03], e ff ectivelyevery result in [Hir03] has an analogue for semi-model categories, sometimes withadditional cofibrancy hypotheses. As part of a longer proof of Theorem 4.3, weproved semi-model categorical analogues of [Hir03, Proposition 7.2.18] (where in(2), domains and codomains must be cofibrant), [Hir03, Proposition 8.5.4] (wherein (3) objects must be bifibrant instead of just fibrant), and [Hir03, Theorem 17.7.7](about Reedy cofibrant replacements and simplicial mapping spaces). A key pointwas that several other authors had already worked out that Reedy semi-model cat-egories behave precisely like Reedy model categories [Bar10, GH04, Spi01]. Wealso verified several useful facts about the semi-model category L C M , including[Hir03, Proposition 3.3.16] (characterization of local fibrations between local ob-jects, if the domain is cofibrant), [Hir03, Theorem 3.2.13] (characterization of localequivalences between bifibrant objects in L C M ), and [Hir03, Theorem 3.1.6] (sub-sumed by Theorem 4.3 above).We conclude this section with an application of Proposition 4.1 to prove a semi-model categorical version of a result of Dugger [Dug01, Corollary 1.2]. This hasthe pleasant property of demonstrating that the theory of combinatorial semi-modelcategories is homotopically the same as the theory of combinatorial model cate-gories (and, hence, of presentable ∞ -categories). This result is not required for therest of the paper, but is an important property of the theory of combinatorial semi-model categories as a whole, and answers a question Zhen Lin Low once asked thesecond author. Proposition 4.6.
Every combinatorial semi-model category M is Quillen equiv-alent, as a semi-model category, to a left proper, combinatorial model categorywhere all objects are cofibrant. Proof.
Let λ be the cardinal for which M is λ -locally presentable, and let M λ de-note a dense subcategory of M (such that every object of M is isomorphic to itscanonical colimit with respect to M λ ). Following [Dug01, Section 3], we first con-sider the case where M is a simplicial semi-model category (defined in [GH04]).We can produce a small category C and a left proper, simplicial, combinatorialmodel category U C : = sS et C op (the projective model structure) and a set of mor-phisms S such that L S U C is Quillen equivalent to M . Use of the injective modelstructure on sS et C op provides the model where all objects are cofibrant.The key point is that finding a homotopically surjective map U C → M is equiv-alent to finding a function γ : C → M such that for every fibrant X in M , X isweakly equivalent to the ‘canonical homotopy colimit’ hocolim( C × ∆ ↓ X ). Thecategory C is produced in [Dug01] as M co f λ , the subcategory of cofibrant objectsin M λ . Dugger’s proof that this C has the required property boils down to the exis-tence and homotopy invariance of cosimplicial resolutions (proven for semi-modelcategories in [GH04]) and the properties listed in Proposition 4.1; see [Dug01,Proposition 4.7]. It is important to note that none of Dugger’s proofs require themodel category axioms where model categories and semi-model categories dif-fer. Dugger’s results about U C work verbatim, whether M is a model category ofsemi-model category, e.g., Propositions 3.2 (noting that LA co f is cofibrant in thesemi-model category N ), 4.2, and 4.6 (which only makes reference to the subcate-gory of cofibrant objects in M ).For the case where M is not simplicial, we follow the proof in [Dug01, Section 6],replacing M by the category of cosimplicial objects c M with its Reedy semi-modelstructure [Bar10, GH04]. Dugger works in the subcategory C R of c M consistingof cosimplicial resolutions A ∗ where A n ∈ M co f λ for all n . Since everything in sightis cofibrant, Dugger’s arguments work verbatim when M is only a semi-modelcategory. (cid:3)
5. A pplications
In this section we provide numerous applications of Theorem A. Most of theseapplications are model categories that fail to be left proper. We begin with animportant example of Voevodsky [Voe10] which we mentioned in the Introduction.We then discuss our main application, to our companion paper [BW20]. Finally, weexplore applications to categories of algebras over operads, spectra / stabilization,(weakly) enriched categories, and Goodwillie calculus.5.1. Radditive functors.Example 5.1.
In [Voe10], Voevodsky introduced a model categorical frameworkfor the study of simplicial extensions of functors. He defined a functor F : C →
S et to be radditive if F ( ∅ ) = pt and F ( X ` Y ) (cid:27) F ( X ) × F ( Y ). He introduced acombinatorial model structure on the category of simplicial objects in radditivefunctors, but needed to assume it was left proper (e.g., in [Voe10, Theorem 3.46]), EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 17 in order to localize it. Indeed, [Voe10, Proposition 3.35] characterizes when thismodel structure is left proper, and [Voe10, Example 3.48] is a case where leftproperness fails and left Bousfield localization (as a model category) provably doesnot exist. The failure is that a certain pushout of a trivial cofibration is not a weakequivalence.However, this obstruction does not prevent the existence of a semi-model structure,and indeed, Theorem A provides a semi-model structure for Voevodsky’s example.Voevodsky writes his paper carefully, to prove results about the local homotopycategory, even when left properness fails. With Theorem A, the local homotopycategory may be studied via the local semi-model structure, providing many toolsbeyond those available to Voevodsky (e.g., fibrant replacement for the computationof homotopy limits of diagrams of simplicial radditive functors).5.2.
Inverting Unary Operations.
The classical theory of localization of cate-gories is concerned with inverting of a set of morphisms in a small category in auniversal way. The resulting localized category can often be studied through itscategory of presheaves. In the world of operads, we can also try to localize oper-ads by inverting a specified set of unary operations, and then study algebras overthese localized operads. In homotopy theory it is natural to require a form of weakinvertibility. This means that we want to study localized operads and their algebraswhere the operations from a specified set of unary morphisms act as weak equiv-alences on the level of algebras. This is the main subject of our companion paper[BW20]. We briefly describe the results from this paper here and refer the readerto [BW20] for the details.
Example 5.2.
The categories of presheaves with values in a model category M is a particularly simple case of the situation described above. Cisinski studiedlocalizations of the covariant presheaf categories [ C , M ] in [Cis06, Cis09] when M is a left proper combinatorial model category and [ C , M ] is equipped with theprojective or injective model structure. The resulting localized category [ C , M ] loc has as fibrant objects presheaves for which each morphism in C acts as a weakequivalence in M . In [BW20] Using Theorem A we extend the results of Cisinski to an arbitrary com-binatorial model category M and, moreover, we consider the semi-model Bousfieldlocalization [ C , M ] W whose fibrant objects are locally constant presheaves with re-spect to an arbitrary proper Grothendieck fundamental localizer W and an arbitrarysubset of morphisms of C . The case studied by Cisinski corresponds to the minimalfundamental localizer W = W ∞ . Example 5.3.
In [BW20], we extend the example above, and localize the categoryof algebras of a Σ -free colored operad P by lifting Cisinski’s localizations [ C , M ] W to the category of algebras of P with values in a combinatorial symmetric monoidalmodel category M . This is where we need the full power of our Theorem A becauseeven if M is left proper, the projective model structure on the category of algebrasof P is most often not a left proper category [HRY17] (for this category to be left proper we need M to be strongly h -monoidal and P be tame, which is a rareoccasion [BB17]). Example 5.4.
Our main application, contained in [BW20], proves a strong form ofthe Baez-Dolan stabilization hypothesis [BD95], using k -operads valued in weak n -categories [Rez10] to model k -tuply monoidal weak n -categories. We apply The-orem A to construct a semi-model categorical left Bousfield localization of thecategory of k -operads, with respect to the Grothendieck fundamental localizer of n -homotopy types, W n , as we now describe.Let M be a combinatorial monoidal model category with cofibrant unit. The cat-egory Op k ( M ) of k -operads in M is encoded as the category of algebras of a Σ -cofibrant colored operad whose underlying category is the opposite of the categoryof quasibijections of k -ordinals Q opk . Hence, Op k ( M ) has a semi-model structuretransferred from the projective model category structure on the category [ Q opk , M ],as we prove in [BW20]. Following Example 5.3 we now construct the localizationof the category of k -operads Op W n k ( M ) whose fibrant objects are W n -locally con-stant k -operads i.e. k -operads whose underlying presheaves on Q opk are W n -locallyconstant. This lifts [Bat10, Theorems 7.1 and 7.2] from the homotopy categorylevel to the semi-model category level.We then prove the following Stabilization Theorem for k -operads: if k ≥ n + Op W n k ( M ) and Op W n k + i ( M ) is a Quillen equivalence for any 1 ≤ i ≤ ∞ . This is our strong formof the Baez-Dolan stabilization hypothesis. The original Baez-Dolan stabilizationfor k -tuply monoidal weak n -categories follows from this Theorem if we take as M to be Rezk’s model category of Θ n -spaces [Rez10] and consider the value ofthe left derived suspension functor on a contractible k -operad (see [Bat17] for anexplanation how to model k -tuply monoidal weak n -categories as algebras of k -operads).Other approaches to the problem of weakly inverting of unary operations in operadswere recently proposed. We briefly describe them below and indicate where ourresults can be used for further improvement. Example 5.5.
Motivated by topological and conformal field theories, two recentpapers [BBPTY16a, BBPTY16b] provide an analogue for operads of the Dwyer-Kan hammock localization of categories, that allows one to weakly invert someunary operations in an operad. Algebras of such ‘localized’ operads can be inter-preted as algebras of the original operad where some set of unary operations areweak homotopy equivalences (see section 6 of [BBPTY16a]). This localizationcan be studied with Theorem A.
Example 5.6.
In [BCL18], localizations at the level of R -modules are comparedto localizations of dg-algebras, where R is a dg-ring. As observed in [BCL18,Remark 2.13], the category of dg- R -algebras is not left proper in general (it is if R is a field), and thus cofibrant replacements of dg-algebras are often required in EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 19 [BCL18]. Theorem A can be used to streamline the exposition of [BCL18], byproviding localizations even when left properness fails. Recalling that operads aremonoids with respect to the circle product [WY18], we could also use Theorem Ato extend the ideas in [BCL18] to the study of localization of operations in cate-gories of operads, algebras, and modules.5.3.
Localizing categories of algebras over colored operads.
One of the crucialideas in Example 5.4 is that a localization of a category of algebras has especiallynice properties if it coincides with an appropriate transferred semi-model structure,as explored in [BW16]. One of the main technical achievements of our paper[BW20] is the set of nontrivial combinatorial conditions on the operad P whensuch a coincidence does occur.In fact, this kind of situation, of wanting to localize a transferred (semi-)modelstructure, is ubiquitous. In the following examples, we always use I to denotethe generating cofibrations, and we recall that a model category is called tractable if it is combinatorial and has the domains of the generating (trivial) cofibrationscofibrant [Bar10]. The following lemma is often useful, and requires slightly lessthan tractability: Lemma 5.7.
Suppose M is a combinatorial model category with domains of thegenerating (trivial) cofibrations cofibrant. Suppose F : M ⇆ N : U is an adjointpair such that the monad T = U ◦ F is accessible (preserves λ -directed colimits).Suppose N admits a transferred model structure from M , i.e., a morphism in N is a weak equivalence (resp. fibration) if and only if U ( f ) is in M . Then N iscombinatorial and has domains of the generating (trivial) cofibrations cofibrant.Proof. First, N is locally presentable because T is accessible, by [AR94, 2.47,2.78]. It is standard (see, e.g., [WY18]) that the generating (trivial) cofibrations areof the form F ( I ) (resp. F ( J )) where I (resp. J ) are the generating (trivial) cofibra-tions of M . To see that the domains of maps in F ( I ) (resp. F ( J )) are cofibrant in N is now a simple lifting argument, using the adjunction. (cid:3) Finally, we recall that Je ff Smith’s category of ∆ -generated spaces is a tractablemodel category Quillen equivalent to the usual model category of spaces, as hasbeen proven in a preprint of Dan Dugger, and in published work of Philippe Gaucher.Details of this model structure, as well as how to build a tractable model for orthog-onal spectra based on ∆ -generated spaces, may be found in [Whi14b]. Example 5.8.
The resolution model structure, also known as the E -model struc-ture, is described in [GH04]. It was introduced by Dwyer, Kan, and Stover in thecontext of pointed topological spaces, and generalized by Bousfield to the settingof general left proper model categories M . It is a model structure on cosimplicialobjects c M , with more weak equivalences than the usual Reedy model structure.The weak equivalences are morphisms that induce an isomorphism on the E -termof certain spectral sequences (or on the E -term for simplicial objects). If M is tractable, then so is the resolution model structure, as shown in [GH04, 1.4.10](see also right after Theorem 1.4.6).In [GH04], Goerss and Hopkins transfer the resolution model structure on simpli-cial spectra (any choice of S -modules, orthogonal spectra, or symmetric spectra) tosimplicial T -algebras, for a well-behaved simplicial operad T . As the positive (orpositive flat) model structure is used on symmetric spectra, the model category of T -algebras is tractable by Lemma 5.7 (at least, if a combinatorial model is used forspaces as recalled above), but not left proper. For this reason, Goerss and Hopkinsdeveloped a semi-model categorical localization technique (using the language ofcellular, rather than combinatorial, model categories) to prove the existence of E ∗ -localization, for a generalized homology theory E , on simplicial T -algebras. Theexistence of E ∗ -localization as a semi-model category also follows from TheoremA, recovering Theorem 1.5.1 of [GH04].A similar example arises in the study of T Q -localization for categories of algebrasover an operad O acting in spectra. If a combinatorial model category of spectrais used, such as the positive model structure on symmetric spectra, then for any O ,the category of O -algebras admits a transferred tractable model structure, M , byLemma 5.7 and [WY18], and hence Theorem A applies. Example 5.9.
In order to left Bousfield localize with respect to the class of
T Q -homology isomorphisms (or
T Q -homology with coe ffi cients), we must first reduceto a set C of morphisms. This is done in [HZ19]. While M is almost never leftproper, the semi-model categorical localization L C M guaranteed by Theorem Amatches that of [HZ19, Theorem 5.14], providing a faster proof of the main resultof [HZ19].We conclude with one more example of localizing categories of algebras over op-erads in spectra. Example 5.10.
In [Bea19], Beardsley initiates a program of learning about thehomotopy groups of Ravenel’s X ( n )-spectra via E k -cell attachments. The first set-ting of the paper is E - X ( n )-algebras, i.e., E -algebras in the monoidal categoryof X ( n )-modules. This category admits a transferred, tractable, left proper modelstructure (by Lemma 5.7 and [BB17]). Beardsley constructs localizations of thismodel structure with respect to a prime p , and his techniques could also be used tolocalize with respect to E ∗ -equivalences for various generalized cohomology theo-ries E , such as K ( n ). Beardsley next introduces an E k -monoidal analogue of X ( n ),denoted X ( n , k ). His work attaching E k -cells most naturally takes place in E k - A -algebras (where A is one of the spectra X ( n , k )), and as categories of E k -algebrasare not known in general to be left-proper, Theorem A is required to construct theleft Bousfield localizations for this new setting, and to prove (following [BW16])that these localizations play nicely with colimits in categories of E k - A -algebras.5.4. Stabilization and Spectra.
One of the most common applications of leftBousfield localization is to build stable model categories of spectra in some base
EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 21 model category M [Hov01]. The idea here is to first build spectra S p ( M ) as se-quences of objects of M , with levelwise weak equivalences, and then localize withrespect to stable equivalences (relative to some endofunctor G on M that gener-alizes reduced suspension on pointed spaces). If M is left proper, then so is thelevelwise model structure on S p ( M ). Otherwise, it is not known how to build thestable model structure on S p ( M ). There are many places in the literature wherevarious authors wished to build a stable model structure, but could not because M was not known to be left proper. We review several such places below, and morein Examples 5.17 and 5.18. Example 5.11.
In [IJ02], Intermont and Johnson introduced model structures forthe category of ex-spaces, suitable for the study of parameterized unstable homo-topy theory. Both their coarse model structure (which is left proper [IJ02, Proposi-tion 3.1]) and their U -model structure (which is not known to be left proper [IJ02,Remark 5.6]) have several improvements over the model structure used by Mayand Sigurdsson. However, it is left as an open problem to construct a suitable ho-motopy theory for parameterized spectra based on the U -model structure. WithTheorem A, this problem can be solved. As the U -model structure is obtainedas a transfer from T op , it will be tractable if a tractable model structure (e.g. ∆ -generated spaces) for spaces is chosen, by Lemma 5.7. With that tractable modelstructure in hand, Hovey’s stabilization machinery [Hov01] may be used, resultingin a stable semi-model structure for parameterized spectra based on the U -modelstructure. Example 5.12.
In [JJ07, Theorem 9.6], Joachim and Johnson introduced a modelstructure on a particular category of C ∗ -algebras, that can be used in the study ofKasparov’s KK -theory. This model structure is obtained as a transfer from thecategory T op ∗ of pointed topological spaces (but where the left adjoint F is onlydefined on compact spaces). The generating cofibrations have the form F ( i k ) where i k : S k − + → D K + , and hence have cofibrant domains. The category T op ∗ is not lo-cally presentable, but if a combinatorial model category of spaces is used (e.g., ∆ -generated spaces), then the Joachim-Johnson model would satisfy the conditions ofTheorem A, by Lemma 5.7. A desirable localization is pointed out in the introduc-tion to [Ost10]: namely, to study the stable C ∗ -homotopy category. Theorem A pro-vides a semi-model category whose homotopy category is the stable C ∗ -homotopycategory built from the Joachim-Johnson model, analogously to [Ost10, Theo-rem 4.12]. Analogously to [Ost10, Theorem 4.70] (following [Hov01]), one canalso build a monoidal semi-model structure for symmetric spectra in the Joachim-Johnson model, semi-model structures for modules and monoids [Ost10, Theorem4.72], and other localizations such as the exact semi-model structure [Ost10, The-orem 3.32], the matrix invariant projective semi-model structure [Ost10, Theorem3.54], or the homotopy invariant model structure [Ost10, Theorem 3.65]. In addi-tion to the study of Kasparov’s KK -theory, these model structures have applicationsto the E -theory of Connes-Higson [Ost10, Remark 3.29]. Analogously to [CW18],one can also build semi-model structures for homotopy functors between the modelcategories above [Ost10, Theorem 4.84], or the stable semi-model structure on functors [Ost10, Theorem 4.94]. This is a first step towards applying Goodwilliecalculus to C ∗ -algebras.We conclude with an example about the connection between spaces and chain com-plexes. Example 5.13.
In [RS17], Richter and Shipley construct a chain of Quillen equiv-alences between commutative algebra spectra over HR (where R is a commutativering), and E ∞ -monoids in unbounded chain complexes of R -modules. Doing sorequires lifting the Dold-Kan equivalence to commutative monoids in symmetricsequences C ( sAb Σ ) ⇆ C ( ch Σ ), and then from symmetric sequences to symmetricspectra. As pointed out in [RS17, Remark 6.4], the positive model structure on C ( ch Σ ) is not left proper (but is tractable). However, there is a long history of lift-ing localizations from the level of chain complexes to the level of spectra [Bau99],and so Theorem A is an important first step to carry this program out for local-izations of commutative HR -algebra spectra lifted from C ( ch Σ ). There are manyinteresting localization of chain complexes, catalogued in [Whi14b, WY15], thatcan be carried out for C ( ch Σ ) using Theorem A.5.5. Enriched categories.
The theory of factorization algebras provides severalexamples of desirable left Bousfield localizations in non-left proper settings [CG16].We refer the reader to [CG16] for notations and definitions.
Example 5.14.
Fix a closed symmetric monoidal category V , a V -algebra C , and a V -enriched symmetric monoidal small category D . Then the category of lax (sym-metric) monoidal V -functors from D to C is a category of algebras over a colored(symmetric) operad [BM07]. If one fixes a V -enriched site S , then the categoryof symmetric lax monoidal V -enriched functors from S to C is the category ofprefactorization algebras on S [CG16]. In just the same way, this category can berealized as a category of algebras over a colored operad. Hence, it admits a trans-ferred semi-model structure, M , which is a model structure if C is su ffi ciently nice,e.g., if C satisfies ♠ from [WY18]. Tractability follows from Lemma 5.7.Costello and Gwilliam provide several applications where it would be desirableto left Bousfield localize M . First, the category of homotopy factorization al-gebras [CG16, Definition 1.4.1] should ideally be the fibrant objects of a leftBousfield localization enforcing the homotopy cosheaf property that certain mapsˇ C ( U , F ) → F ( U ) are weak equivalences, where U is an open set and U is a Weisscover of U . Second, the category of multiplicative homotopy factorization alge-bras [CG16, Definition 1.4.2], should ideally be modeled by a further left Bous-field localization, enforcing that the maps F ( U ) ⊗ F ( V ) → F ( U ` V ) are weakequivalences. Third, the theory of locally constant factorization algebras [CG16,Definition 4.0.1] should ideally be modeled by a left Bousfield localization withrespect to maps F ( D ) → F ( D ′ ) for all inclusions D → D ′ . Finally, there shouldbe a further left Bousfield localization, where the fibrant objects are ‘weakly lax’functors (i.e., satisfy codescent with respect to a Grothendieck topology on S ).Unfortunately, M is not known to be left proper, and there are also set-theoretical EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 23 concerns regarding these localizations. Theorem A allows us to circumvent the leftproperness obstacle, answering a question of Pavlov and Scholbach.We turn now to a similar example, about weakly enriched categories.
Example 5.15.
In [Bac14], Bacard introduced a model structure for the studyof “co-Segal categories,” which are weakly enriched categories over a symmet-ric monoidal model category M , satisfying a Segal-style weak equivalence ratherthan the usual composition law. In [Bac13], Bacard sought to improve the theoryof co-Segal categories to shift from non-unital weak M -categories to unital precat-egories. To do so, Bacard needed what he called an implicit Bousfield localization (page 4 of [Bac13]), because his “easy model structure” on co-Segal precategoriesis not known to be left proper (but, it is tractable). As [Bac13] was never pub-lished, it is di ffi cult to know if the implicit Bousfield localization worked. WithTheorem A, it is possible to achieve Bacard’s goal, when M is a combinatorialmodel category with domains of the generating cofibrations cofibrant.Bacard defines a co-Segal precategory with object set X as a normal lax functorof 2-categories C : ( S X ) op → M , for a particular 2-category S X . The compo-nents are denoted C AB : S X ( A , B ) op → M . Bacard is forced to discard his firstnotion of weak equivalence of co-Segal precategories, because it does not lead toa left proper model structure. This notion defines a weak equivalence to be a map( σ, f ) : C → D such that each natural transformation σ AB is a levelwise weakequivalence in M . With Theorem A, one could carry out the program of [Bac13]for either the “easy model structure” or for his notion of strict M -categories, aslong as M is a combinatorial model category with domains of the generating cofi-brations cofibrant. Lastly, throughout [Bac13], Bacard has statements that assumethe existence of various left Bousfield localizations (e.g., 3.23, 3.44, 4.6, 4.14).Theorem A can be used to verify that these localizations exist, and also to weakenthe requirement that M be left proper.We conclude with an application to the theory of dg-categories, i.e., categoriesenriched in chain complexes over a fixed commutative ring k . Example 5.16.
The category dgcat( k ) of small dg-categories, admits a tractablemodel structure [Tab15, Theorem 1.7] that is much used in the study of derivedalgebraic geometry [To¨e10]. This model category fails to be left proper in general[Tab15, Example 1.22], unless strong conditions are placed on k . Nevertheless,To¨en is able to construct localizations of its homotopy category [To¨e10, Corollary8.7] (building on earlier work of Drinfeld, Keller, and Lyubashenko). TheoremA allows us to lift To¨en’s localizations from the homotopy category level to thesemi-model category level. Furthermore, Tabuada introduces a model structureon dgcat( k ) with weak equivalences the pretriangulated equivalences [Tab15, The-orem 1.30] as a left Bousfield localization of the model structure above [Tab15,Proposition 1.33], and then introduces the Morita model structure as a left Bous-field localization of the pretriangulated model structure [Tab15, Proposition 1.39].This is done despite the fact that all of these model structures fail to be left proper. Using Theorem A, we can actually achieve the Morita model structure as a leftBousfield localization (including the universal property), and [Tab15, Theorem1.37] tells us the local semi-model structure is in fact a full model structure. Lastly,Theorem A can be used to prove the existence of various localizations of the modelstructures above desired by Tabuada, e.g. Theorems 8.5, 8.17, and 8.25, RemarkA.10, and Section 2.2.6 of [Tab15].5.6.
Functor Calculus.
Another application of Theorem A is to Goodwillie cal-culus for general model categories. The following example is motivated by Good-willie’s work studying categories of functors between categories of spaces andspectra. Our treatment follows Pereira [Per17], who seeks a version of Goodwilliecalculus for functors between categories of algebras over operads. The main ideais to recast Goodwillie’s n -excisive approximation as a left Bousfield localization,as was done previously by Biedermann, Chorny, and R ¨ondigs, but Pereira’s settingis not left proper. Example 5.17.
Let O be a simplicial operad. Let C be a pointed simplicial modelcategory such that the stable projective model structure on spectra S p (Alg O ( C ))exists. This occurs, for example, if C has domains of the generating cofibrationscofibrant. Even if C is simplicial sets or spectra, Alg O is almost never left proper, asPereira shows (one case where is is left proper is if O is the Com operad [Whi17]).Let A and B be ring spectra, D a small subcategory of A -modules, and let M (resp. M ) be the category of simplicial functors Fun ( D , Mod B ) (resp. spectral functors).Instead of taking D to be a small subcategory, one could alternatively study smallfunctors Fun s ( Mod A , Mod B ), i.e. functors that are left Kan extensions of functorsdetermined on a small subcategory, as is done in [CW18]. Using Theorem A, theprojective model structures on M i admit several left Bousfield localizations, wherethe new fibrant objects are homotopy functors, or linear functors, or n -excisivefunctors. For the latter, the localization X → L ( X ) is Goodwillie’s n -excisive ap-proximation.Remark 4.11 in [Per17] explains why having this localization on the model (orsemi-model) level would be desirable. Pereira is able to prove an equivalence ofhomotopy categories, which can be lifted to an equivalence of semi-model cate-gories using Theorem A. An application is the characterization of the Goodwillietower of the identity in Alg O as the homotopy completion tower associated to trun-cated operads O ≤ n .Other contexts where one might wish to extend the techniques of Goodwillie cal-culus (other than to Alg O as in the example above) include the setting of graphtheory or the category of small categories. We discuss these settings now. Example 5.18.
In [Vic15], Vicinsky worked out the homotopy-theoretic founda-tions required to apply Goodwillie calculus to model categories of graphs and smallcategories. Traditionally, this requires a model structure M that is pointed, leftproper, and simplicial [Vic15, Hypothesis 2.28]. Partially, this is required to build EFT BOUSFIELD LOCALIZATION WITHOUT LEFT PROPERNESS 25 a stable model structure for spectra
S p ( M ). However, the model structure used byVicinsky on pointed directed graphs (originally due to Bisson and Tsemo) is notleft proper [Vic15, Proposition 5.8]. However, Theorem A can be used to carry outthe program laid out by Vicinsky: constructing a semi-model structure for spec-tra on graphs, then proving this category is homotopically trivial. Lastly, Theo-rem A may be used to verify Vicinsky’s Conjecture 6.10. Her model structures Cat n are transferred from n -truncated model structures on simplicial sets [Vic15,Proposition 6.8], and hence are tractable by Lemma 5.7. Theorem A provides asemi-model structure on spectra in Cat n , and hence a framework to lift the Quillenequivalence sS et n ⇆ Cat n to categories of spectra, as required to prove [Vic15,Conjecture 6.10].5.7. Other possible directions.
We conclude with some suggestions about possi-ble future applications that don’t fit into the categories above.In [Ber14], Bergner defined the notion of a homotopy colimit of a diagram of modelcategories M i , as a quotient of a coproduct ` M i . Dualizing her earlier work onhomotopy limits, which she studied as a right Bousfield localization, the quotientrequired in a homotopy colimit can be studied as a left Bousfield localization. AsBergner points out, there are a number of technical di ffi culties, but Theorem A canbe used to circumvent the requirement that the coproduct (semi-)model structurebe left proper, just as Barwick’s right Bousfield localization [Bar10] was used inthe study of homotopy limits. The upside of this approach is that having a semi-model structure for the homotopy colimit gives more structure for computationsthan simply the relative category structure constructed in [Ber14].Another source of potential applications of Theorem A would be to do left Bous-field localization after a right Bousfield localization, since right Bousfield localiza-tion often destroys left properness. This would occur, for example, if one wishedto build spectra or do Goodwillie calculus for a model structure defined as a rightBousfield localization, such as model structures used for the study of slices in theslice spectral sequence, or A -cellular model structures in spaces, chain complexes,and categories [WY15].While it may seem that we have exhaustively cataloged all situations where onewishes to do left Bousfield localization without left properness, we are confidentthat there are in fact many more cases where Theorem A will be useful. We alsobelieve Theorem B will be useful in its own right.R eferences [AR94] J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories, Cambridge Uni-versity Press, London Mathematical Society Lecture Note Series (189), 1994.[Bac13] Hugo Bacard, Pursuing Lax Diagrams and Enrichment, preprint available asarXiv:1312.7833.[Bac14] Hugo Bacard, Toward weakly enriched categories: co-Segal categories, Journal of Pure andApplied Algebra 218 (2014) 1130-1170 . [BD95] John C. Baez and James Dolan. Higher-dimensional algebra and topological quantum fieldtheory. J. Math. Phys. , 36(11):6073–6105, 1995.[Bar10] Clark Barwick. On left and right model categories and left and right Bousfield localizations.
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