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A general context for Goodwillie Calculus Lu´ıs Alexandre Pereira January 15, 2013 3 1 0 The objective of this paper is to provide a fairly general model category 2 context in which one can perform Goodwille Calculus. n There are two main parts to the paper, the first establishing general condi- a tions whichguaranteethe existence ofuniversaln-excisiveapproximations,and J a second providing conditions for which the layers of the Goodwillie tower can 3 be appropriately factored through the spectra categories. 1 ] T 1 Overview A The main goalof this paper is to establish a fairly generalsetup in which to do . h Goodwillie calculus, extending the treatment of Goodwillie in [7]. t a There are two main results. The first is the existence ofuniversal n-excisive m approximations, which is Theorem 4.12. The second is the classification of n- [ homogeneous functors F: C Ñ D (at least under some finitary conditions) as havingtheformF “Ω8ppF¯˝pΣ8qˆnq q,whereF¯: StabpCqˆn ÑStabpDqisa 1 hΣn symmetric multilinear functor between the stabilizations. In the current write- v 2 up, this result corresponds to piecing together Proposition 6.7, which shows 3 such functors factor through StabpDq (provided they are determined by their 8 values on a small subcategory C1 Ă C), Proposition 5.29, showing that, pro- 2 videdthetargetcategoryisstable,ahomogeneousfunctorisrecoveredfromit’s . 1 crosseffect, andTheorem7.2, whichprovesdirectly thatmultilinear symmetric 0 functors factor through stabilizations on both source and target. 3 The following is a more detailed overview of the paper. 1 : Section 2 establishes some of the general assumptions on model categories v and functors between them used throughout the paper. Particularly important i X is Proposition 2.8, stating informally that, in any cofibrant model category, r “linear/direct”colimitsoveralargeenoughordinalcommutewithfiniteholims. a Section 3 establishes the basic definitions of excisive functors (via hocartesian/hococartesiancubes), andmanyoftheirbasic properties,whicharegener- ally straightforwardgeneralizationsofthe analogouspropertiesfound in [7]. Of some technical interest seems to be Proposition 3.15, which is currently necessary for the proof of Theorem 4.12 (check Remark 4.15 for more on this). Section 4 then establishes the first main result, Theorem 4.12, for the existence of universal n-excisive approximations P F. These are constructed via n Goodwillie’sT F,exceptthoseareiteratedatransfinitenumberoftimes. Exci- n siveness of this construction then follows by Goodwillie’s usual proof combined withProposition2.8andholdsgenerallyundertheassumptionthatC,Darecofi- brantly generated model categories. The proof of universality seems currently 1 more delicate, however, relying on Proposition 3.15, which currently requires the source category C to be either a pointed simplicial model category or a left Bousfield localization of one. Section5establishesthat,whenthetargetcategoryDisstable,homogeneous functors and multilinear functors determine each other (Propositions 5.26 and 5.29),andisafairlystraightforwardgeneralizationofthetreatmentin[7]. This requires no further assumptions on the categories. Section6establishesthathomogeneousfunctorscanbedelooped. Thisdoes requireadditionalconditionsonthecategories. Namely,itisratherunclearhow to generalizeGoodwillie’s proofofthis whenthe T areto be iteratedatransfi- n nite numberoftimes,sothatone isforcedtoassumeP canbe constructedvia n countable iterations,andthis, in turn, essentiallyamounts to assuming that, in the targetcategoryD,countabledirectedhocolimscommute withfinite holims. It also establishes the infinitely iterated result, Proposition 6.7. Finally, Section 7, deals with showing Theorem 7.2. This too imposes more conditions on the modelcategories,most obviouslyrequiringthat the stabilizations (as defined in [14]) exist, andthe further technicalconditionthat in those stabilizationsonecandetectweakequivalencesbylookingatthe Ω8í functors (see [14] for a discussion of this condition). 2 Setup for homotopy calculus Thissectionlistsassumptionsthatappearrepeatedlyinthepaper,anddeduces some of their basic consequences. Pervasivethroughoutthe paper isthe assumptionthatthe model categories used are cofibrantly generated. We recall the definition (and set notation) in 2.1, along with some results about the functoriality of hocolimits and holimits in such model categories. 2.2 deals with the general assumptions made on functors between model categories. 2.3 deals with the further assumptions made when the model categoriesare also assumed simplicial. 2.4 deals with detailing the notions of spectra in a general simplicial model category that are used in Section 7. Finally, 2.5 deals with showing that our definitions are nicely compatible withreplacingoneofthe modelcategoriesinvolvedbyaQuillenequivalentone. 2.1 Cofibrant generation Definition 2.1. AmodelcategoryC issaidtobe cofibrantlygeneratedifthere exist sets of maps I and J, and regular cardinal κ such that 1. The domains of I are small relative to I with respect to κ. 2. The trivial fibrations are precisely the maps with the RLP1 with respect to I. 3. The domains of J are small relative to J with respect to κ. 1RightLiftingProperty. 2 4. The trivial fibrations are precisely the maps with the RLP with respect to J. 5. Both the domains and targets of I and J are small relative to I with respect to κ. Remark 2.2. We refer to [15], section 10, for the precise definitions of “small- ness with respect to”, but we give a sketch definition here: X is said to be κ-small with respect to I if for any λě κ regular cardinal2 and λ-sequence3 Y ÑY ÑY Ñ¨¨¨ÑY ÑY Ñ...pβ ăλq 0 1 2 β β`1 where all maps Y ÑY are relative I-cell complexes4, the natural map β β`1 colim HompX,Y qÝ„ÑHompX,colim Y q βăλ β βăλ β is an equivalence. Remark 2.3. Though the first condition in our definition is a particular case of the last, we choose to include it so as to draw attention to the fact that that conditionisnotpresentinthedefinitionofcofibrantlygeneratedmodelcategory in [15]. Our goal is to implement Goodwillie’s construction of the polynomial approximations in this general context. Cofibrant generation allows us to do so, thanks to the following results: Proposition 2.4. Suppose I a set of maps whose domains are small relative to I with respect to a regular cardinal κ. Then there exists a functorial factorization CÓ ÑCÓˆCÓ factoring any map into a relative I-cell complex followed by an I-injective. Proof. This is just Quillen’s wellknownsmallobjectargument. A reference for this is section 2.1.2 of [13], or most other basic treatments of model categories. Notethat,accordingtothe previousresult,inanygeneralfinitely generated modelcategoryC,onecanconstructfunctorialcofibrantandfibrantreplacement functors. We will denote fixed such functors by Q and R in the remainder of this paper. We will also need to make special note of particularly nice types of diagram categories that feature prominently in Goodwillie calculus (see [3] for the following definitions): 2Here,asin[15],cardinalsaretobeviewedasordinalsbyselectingtheinitialordinalwith thatcardinality. 3Werecallthat afunctor Y.:λÑC iscalledaλ-sequence ifitis“wellbehaved” onlimit ordinals,i.e. Yα“colimβăαYβ forαăγ alimitordinal. 4Themore“simpliciallyminded”readercanlikelyignorethis,asinthosecontextsobjects areoftensmallwithrespecttoallmaps. 3 Definition 2.5. A small category A is called a direct category if there is an identity reflecting functor AÑλ, where λ is an ordinal category. AsmallcategoryAiscalledaninverse categoryifAop isadirectcategory. It is well known that when A is direct a projective model structure exists on any CA, for C any model category (check [3] for the characterization of this model structure). Dually, when A is an inverse category, CA always has an injective modelstructure. We willusethe followingbasicpropertiesaboutsuch model structures: Proposition 2.6. Let A be a direct category and A1 an initial subcategory (i.e., such that if x P A1 then all maps x1 Ñ x are in A1). Note that A1 is then itself a direct category. Then if F: A Ñ C is projective cofibrant, then so is the restriction F| . Furthermore, the analogous result holds for cofibrations A1 F ֌F1. Let A,B be direct categories. Then AˆB is a direct category and, if F: Aˆ B ÑC isprojectivecofibrant,thensoaretherestrictionsFpa,‚q,Fp‚,bqforany aPA,bPB. Furthermore, the analogous result holds for cofibrations F ֌F1. Proof. The first part is straightforwardfrom the definition of the model structure,astheconditionsrequiredforF| ÑF1| tobeacofibrationareasubset A1 A1 of the conditions for F ÑF1 to be one. As for the second part, it follows from noticing that in these projective model structures cofibrations are in particular pointwise cofibrations, and by furtherviewingtheprojectivemodelstructureonCAˆB astheprojectivemodel structure in pCAqB over the projective model structure in CA. Proposition 2.7. Let A be a direct category, and C a model category with functorial factorizations as defined in Proposition 2.4. Then so does the projective model structure on CA. Moreover, such functorial factorizations can be constructed using those in C. Furthermore, if C is assumed simplicial, and with simplicial factorization functors, then CA is itself a simplicial model category and the constructed factorizations are again simplicial functors. Proof. This statement is a variation of Proposition 6.3 of [11], and the proof is essentially the same. Proposition 2.8. Let C be a finitely generated model category with respect to some regular cardinal κ. Then κ-filtered colimits (taken over the κ ordinal category) commute with κ-small limits (taken over inverse categories). Beforegiving the proof, we recallsome definitions. For more details, see [8]. Given a functor C ÝFÑD betweenmodelcategoriesCandD,wedefinethederivedfunctorsLF,RF: HopCqÑ HopDq by LF “RanγCγD˝F, RF “LanγCγD˝F where γC: C ÑHopCq, γD: D ÑHopDq are the standard maps. It is then well known that, when F is a left Quillen (resp. right Quillen) functor, LF (resp. RF) is the functor induced by F ˝Q (resp. F ˝R). 4 Thestatementof2.8isthenthatthefollowingdiagram(whereAisassumed the ordinal category κ and B assumed inverse and κ-small) commutes up to a natural isomorphism. Rlim HopCAˆBq ÐÝB // HopCAq (2.9) Llim Llim ÝÑ ÝÑ (cid:15)(cid:15) ARlim (cid:15)(cid:15) A HopCBq ÐÝB //HopCq Note,however,thatwedonotclaimthenaturalisomorphismsuppliedinthe proofcanbechosencanonically(thisshouldbeofcontrastedwiththecanonical map lim lim Ñ lim lim ), as it depends at this level on the choice of ÝÑAÐÝB ÐÝBÝÑA bifibrant replacements used. Proof of 2.8. TheproofwillbeanadaptationofResk’sargumentin[12],Lemma 1.3.2. First suppose given a diagram X such that X is injective cofibrant for a,b ‚,b all b and X is projective fibrant for all a. Then we claim than lim X is a,‚ ÝÑa a,b still projective fibrant. By the assumption that B be an inverse category, this amountstoshowingthatthemapslim X Ñlim lim X X are ÝÑa a,b ÐÝbÝÑf b1,f‰idbÝÑa a,b a,b1 fibrations. Checking this amounts to checking the RLP of these maps with respecttothegeneratingtrivialcofibrationsJ. Supposethenj Ñj1 anelement of J. A commutative square corresponds to maps j Ñ lim X and (for each ÝÑa a,b b ÝÑf b1) j1 Ñ lim X X . But compactness allows us to factor each of ÝÑa a,b a,b1 these maps j Ñ X , etc by some a which can be assumed uniform since a,b A is κ-filtered. But then a further usage of compactness (and filteredness) tells us that, possibly rechoosing a, the commutative square factors through X Ñlim X , for some a, and hence the lifting property follows. a,b ÐÝbÝÑf b1,f‰idb a,b1 We can now prove the result. First notice that, by first taking a functorial bifibrant replacement of the Aˆ B diagram, one can assume the first maps in (2.9) are taken to be straight up limits and colimits. Furthermore, by the further part of the proof, applying lim preserves projective fibrancy, so the ÝÑA second lim can also the taken straight up. ÐÝB In other words, taking X¯ an injective cofibrant replacement to lim X , a ÐÝb a,b we will be done provided the natural map limX¯ ÑlimlimX ÝÑ a ÐÝÝÑ a,b a b a isaweakequivalence. Morespecifically,weclaimitisatrivialfibration. Indeed, letting i Ñ i1 an element of the generating cofibrations I, the techniques used inthe firstpartofthe proofallowus tofactoranycommutativesquarethrough X¯ Ñlim X for some fixed a, and the result follows. a ÐÝb a,b Remark 2.10. Noticethatthepreviousresultalsofollows(withnoalterations totheproof)forotherκ-filteredshapesofthediagramcategoryAprovidedone knows the domains and codomains of I and J satisfy compactness with respect to such shapes. 5 2.2 Functors We now indicate our assumptions on functors. Throughout we will denote F: C ÑD a functor between a simplicial model categoryC and a model categoryD (both still assumed cofibrantly generated), which is typically assumed left homotopical,i.e.,suchthatF˝Qishomotopical5. Thereasonforthisslightgeneralization isthat, evenforstraightuphomotopicalfunctors,someofourconstructionsre- quire precomposition with Q anyway. We do howevernotice that, when we allowF to be merely left homotopical, our constructions will be presenting the Goodwillie tower of F ˝Q, not of F. 2.3 Simplicial categories and functors In the situation in which we are dealing with simplicial model categories C,D, wewillsometimes(butnotalways)requirethefunctorF: C ÑDbetweenthem tobeasimplicialfunctor. Whenthatisthecasewewouldalsoliketoknowthat our constructions of the Goodwillie tower still yield simplicial functors. Since cofibrantreplacementsarenecessarywhenmakingthoseconstructions,weneed the following result, which is proposition 6.3 in [11]: Proposition2.11. SupposeC acofibrantlygeneratedsimplicial modelcategory. Then there exists simplicially functorial factorizations CÓ ÑCÓˆCÓ factoring any map into a trivial cofibration followed by fibration, or as a cofibration followed by trivial fibration. In particular, C has simplicial cofibrant and fibrant replacement functors. When dealing with simplicial model categories we will hence always further assume that the chosen replacement functors Q,R are taken to be simplicial. Remark 2.12. Notethatthedefinitionof“cofibrantlygenerated”giveninthis paper is slightly more demanding than that used in [11]. 2.4 Spectra on model categories We now explain the notions of spectra that will be used in Section 7. AssumeC apointedsimplicialmodelcategory,sothatitisalsoinparticular tensored over SSet . Denote this tensoring by ^. ˚ FollowingHoveyin[14]onecanthendefineBousfield-Friedlandertypespec- tra on C as sequences pXiqiPN, plus structure maps S1^X ÑX , n n`1 where S1 is the standard simplicial circle ∆1{δ∆1. We denote the categoryofsuchspectra bySppC,S1q,orjust SppCqwhenno confusion should arise. 5Recallthatafunctoriscalledhomotopical ifitpreservesw.e.s. 6 Then when C is cofibrantly generated one has that SppCq always has a cofi- brantlygeneratedprojectivemodelstructure(i.e. amodelstructurewhereweak equivalences and fibrationsare defined levelwise onthe underlying sequences of the spectra), with generating cofibrations and generating trivial cofibrations given respectively as Iproj “ďFnI, Jproj “ďFnJ, n n whereI,J denoterespectivelythegeneratingcofibrationsandgeneratingtrivial cofibrations of C, and F is the level n suspension spectra functor. n To get the correct notion of stable w.e.’s on spectra one then needs to left Bousfield localize with respect to the maps S “tF pS1Â qÑF A qu n`1 i n i where A ranges over the (cofibrant replacements of, when necessary) domains i and codomains of the maps in I. Definition 2.13. Let C be a cofibrantly generated pointed simplicial model category. Thenwe define the stable model structure onSppCqas the (cofibrantly generated) left Bousfield localization with respect to S, should it exist. Remark 2.14. For a detailed treatment of left Bousfield localizations, check [15], chapters 3 and 4. Notice thatwe cruciallydo not wantto just assume C is left propercellular, asin[14],whereitis shownthatthatissufficientforthe modelstructureabove to exist. This is because left properness generally fails in the main examples we are interestedin,thoseofalgebrasoveranoperadinsymmetricspectra. Thisturns out to be ok, however, both because one can show directly in those cases that the description above produces a model category, and because even though Hovey requires left propernessin many of his statements in [14], in the ones we shall need he is only really requiring the existence of the model structure just described. We now adapt the treatment in [15] in order to show that one has a nice localization functor in SppCq. First notice that the projective model structure on SppCq is simplicial (just apply the simplicial constructions levelwise). Then set S˜“tF pS1Â qÑF A u n`1 i n i Ć the set of maps obtained by taking a cofibrant factor of those in S. And set ∆S “ J YtF pS1 Â qb∆ F A b Ą proj n`1 i mštFn`1pS1Âiqbδ∆m Ćn i δ∆ ÑF A b∆ u. m n i m Ć Proposition 2.15. Let X PSppCq. Then the localization of X can be obtained byperforming theQuillen small object argumenton themap X Ñ˚ withrespect to the set of maps ∆S. Ą 7 Proof. This is just Proposition 4.2.4 from [15]. It should be noted that though the statement asks for the category to be left proper cellular none of those properties is used, and indeed all one needs is cofibrant generation (in order to run the small object argument). We will generally denote the by Σ8: C ÕSppCq: Ω8 the standard adjoint functors. We will also require an additional niceness property about the model struc- tureonSppCq,namely,weshallrequirethatthestableequivalencesbedetectable via the Ω˜8ń functors (these are the derived functors giving the n-th space of the spectrum after being made into an omega spectrum). In [14] Hovey determines conditions under which this follows (and, impor- tantly, those proofs do not evoke left properness): Theorem 2.16 (Hovey). Suppose C is a pointed simplicial almost finitely generated model category. Suppose further that in C sequential colimits preserve finite products, and that MappS1,´q preserves sequential colimits. Then stable equivalences in SppCq are detected by the Ω˜8ń functors. Proof. This is just a particular case of Theorem 4.9 in [14]. Remark 2.17. Weshallalsoneedtomakeuseofslightlymoregeneralkindsof spectra. Namely, we shalldenote by SppC,Aqthe spectrain C constructedwith respect to the endofunctor A^‚ (in the language of Hovey). In the particular case that A“pS1q^n these can be thought of as spectra that only include the ni-th spaces. Notice that the obvious analog of 2.16 still holds. 2.5 Composing functors with Quillen equivalences Suppose we are given a Quillen equivalence F: C ÕD: G. In this section we show that, up to a zig zag of equivalences of functors, studying homotopy functors into C is the same as studying homotopy functors into D (the analogous result for functors from C,D also holds, but we won’t be using it). This is the content of the following results. Proposition 2.18. Suppose F: C ÕD: G a Quillen equivalence (between cofibrantly generated model categories). Then F ˝ QC and G ˝ RD are homotopy inverses, i.e., there are zig zags of w.e.s G˝RD˝F ˝QC „idC, F ˝QC˝G˝RD „idD 8 Proof. By reasons of symmetry we need only to construct the zig zag G˝RD˝ F ˝QC „idC. This is given by the obvious functors idC ÐQC ÑG˝F ˝QC ÑG˝RD˝F ˝QC where the natural transformation QC Ñ idC is of course given by w.e.s, while the composite natural transformation QC Ñ G˝RD ˝F ˝QC is given by w.e.s since it is adjoint to the obvious maps6 F ˝QC ÑRD˝F ˝QC. Corollary 2.19. Suppose given a Quillen equivalence F: C ÕD: G as before and pE,W q a category with w.e.s. Then, when defined, the homotopy E categories of homotopy functors HohpE,Cq and HohpE,Dq are equivalent. 3 Goodwillie calculus: polynomial functors In this section we define the generalized concepts of polynomial functor and constructthe universalpolynomialapproximationsP F to a homotopyfunctor n F. All definition and results are reasonably straightforward generalizations of those of Goodwillie in [7]. 3.1 Cartesian cubes and total homotopy fibers. We first set some notation. For S a finite set, denote by PpSq its poset of subsets, and additionally define P pSq“tAPPpAq: |A|ěiu. Of particular importance will be P pSq“ i 1 PpSq´tHu.7 Notice that allof these categoriesare havefinite nerves,and hence are both direct and inverse categories (Definition 2.5). Definition 3.1. Ann-cubeinacategoryC isafunctorX : PpSqÑC,where ‚ |S|“n. X is said to be hocartesian if the canonical map hocmppXq: XH ÑholimTPP1pSqXT is a w.e.8. We call this map the homotopy comparison map, and denote it by hocmppXq, or just hocmp when this would not cause confusion. Furthermore, X is said to be strongly hocartesian if the restrictions X|PpTq are hocartesian for any subset T ĂS with |T|ě2. There are also obvious dual definitions of hococartesian and strongly hococartesian n-cubes. 6RecallthataQuillenadjunctionisaQuillenequivalenceiffamap(withX cofibrantand Y fibrant)FpXqÑY isaw.e. ifftheadjointmapX ÑRpYqis. 7ThisdoesNOTmatchGoodwillie’snotation. 8We now use the traditional notations holim and hocolim for the derived functors Rlim ÐÝ andLlimintroducedin2.1. ÝÑ 9 Remark 3.2. Note thathocartesiannessis ahomotopicalpropertyofthe cube X, as so is the map XH Ñ holimTPP1pSqXT (which we regard as a map in HopCq). Furthermore, that map can be constructed in the following way: take X¯ aninjectivefibrantreplacementofX. ThenX¯|P1pSq is itselfinjective fibrant, since P1pSqis a terminalsubposetofPpSq(see Lemma 2.6), sothatlimP1pSqX¯ is indeed a holim, and the intended map is then givenby X¯H ÑlimTPP1pSqX¯T. Remark 3.3. Notice that any strongly hocartesian cube X is determined by its restriction to P pSq. Indeed, consider X¯ its injective fibrant replacement. n´1 tThhaetnitistsrereexsttreincstiioonn(Xi¯.e|P.,n´th1peSrqigishtitKsealfninexjetcetnisvieonfi)bXr¯a1ntt,oaPnpdSiqtistriitvsieallflyinfjoelcltoiwves fibrant and the unit map X¯ ÑX¯1 is a pointwise equivalence. We now turn to the issue of defining the total fiber of a cube, and proving its iterative properties. Suppose C is now a pointed model category with zero object ˚. Definition 3.4. Consider the Quillen adjunction C ÕCÓ (where CÓ has the injective model structure) with left adjoint X ÞÑ pX Ñ ˚q and right adjoint pX ÑYqÞÑfiberpX ÑYq. We then denote by hofiber the derived functor Rfiber: HopCÓqÑHopCq. More generally, consider the Quillen adjunctions: C ÕCPpSq (where CPpSq has the injective model structure) with left adjoint X ÞÑ pH ÞÑ X;elseÞÑ˚q and right adjoint X ÞÑfiberpXH ÑlimTPP1pSqXTq. We now denote by tothofiber the derived functor HopCPpSqqÑHopCq. Remark 3.5. Though not immediately obvious, our definition of hofiber does match the obvious alternate candidate definition: holimpX Ñ Y Ð ˚q. This follows,forinstance,fromPropositionA.2.4.4in[16]. Thattheanalogousresult also holds for the higher dimensional variants can be proven by combining the following result with an induction argument. We now show that the tothofiber can be computed iteratively. Proposition 3.6. Iterative definition of the total hofiber The following diagram commutes up to a natural isomorphism: HopCPpS>S1>WqqhofiberS// HopCPpS1>Wqq (3.7) hQoQfiQbQQerQSQ>QSQ1QQQQ(( (cid:15)(cid:15) hofiberS1 HopCPpWqq Proof. Notice first that we have obvious Quillen adjunctions (for the injective model structures) CPpWq ÕCPpS>Wq 10