Localization of algebras over coloured operads

LOCALIZA TION OF AL GEBRAS O VER COLOURED O PERADS CARLES CASACUBER T A, JA VIER J. GUTI ´ ERREZ, IEKE MOERDIJK, AND RA INER M . VOGT Abstract. W e give suﬃcien t conditions for homotopical localization func- tors to preserve algebras o v er coloured op erads in monoidal mo del categories. Our ap proac h e ncompasses a n umber of previous results ab out preserv at ion of structures under localizations, such as loop spaces or inﬁnite loop spaces, and prov ides new results of the same kind. F or instance, under suitable assump- tions, homotopical localizations preserve ring sp ectra (in the s tr i ct sense, not only up to homotopy ), modules ov er ring sp ectra, and al gebras o v er commuta- tiv e ri ng spectra, as w ell as ring maps, module maps, and al gebra maps. It is principally the treatmen t of mo dule spectra and their maps that led us to the use of coloured op erads (also call ed enriche d multicategories) in this cont ext. Introduction A remark able pr op erty of lo caliz a tions in homotopy theory is the fact that they preserve many kinds of algebra ic structur e s. Tha t is, if a spa ce or a sp ectr um X is equipp e d with some structure and L is a ho motopical lo calization fun ctor (such a s , for example, lo calization at a set of primes, loca lization with re s pe ct to a homology theory , or a Postnik ov section), very often LX admits the sa me structure as X , in fact in a unique way (up to homotopy) if we imp ose the condition tha t the lo calization map X − → LX b e compatible with the structur e. F o r instance, it is known that f -lo calizations in the sense o f Bousﬁeld [B o u94], [Bou96] and F arjoun [F a r96] preser ve the classes of homotop y asso ciative H -spac es, lo op spa ces, a nd inﬁnite lo op spaces, among others. Such f -lo calizatio ns also pre- serve GEMs (i.e., pro ducts of Eilenberg– Mac Lane spaces ), a s explained in [F a r 96], as well as other classes of s pa ces deﬁned by means of algebra ic theories [Bad02]. In the stable homotopy ca tegory , f -lo calizatio ns that commut e with the s us- pens ion o p e r ator prese r ve homotopy r ing sp ectra and homotopy module sp ectra [CG05]. F urther more, if a homotopy ring spectrum R is c onnective, then the class of homotopy mo dules ov er R is preserved b y all f -lo caliza tions, not necessa rily commuting with suspensio n; see [Bou99], [CG05]. As a cons e quence, the class o f stable GEMs is preser ved by all f -loc a lizations, since sta ble GEMs are precisely homotopy mo dules ov er the integral Eilenberg–Ma c La ne sp ectrum H Z . (Note that either some connectivity condition or the a ssumption that the given lo caliza- tion commutes with susp ension is necessa ry , s ince Postnik ov se c tions o f Morav a K - theory sp ectra K ( n ) need neither be homo topy r ing sp ectra nor homotop y mod- ules ov er K ( n ), as obser ved by Rudyak in [Rud98].) 2000 Mathematics Subject Classiﬁc ation. Pri mary: 55P43; Secondary: 18D50, 55 P60. Key wor ds and ph r ases. Coloured op erad; lo calization; ring s pectrum; mo dule sp ectrum. The ﬁrst and second named authors were supported by the Spanish M inistry of Education and Science under grants MTM2004-03629, MTM2007-63277, and EX2005-0521. 1 2 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT A commo n feature of these ex amples is that they can b e describ ed in terms of algebras over ope r ads or, in some cases, algebra s ov er coloured op erads. Coloured op erads ﬁrst a ppe a red in the bo ok of Boardman and V ogt [BV73] on ho motopy inv ariant algebraic structures on to po logical spac e s. They can b e viewed as mul- ticategories [Lam69] enriched ov e r a s y mmetric monoidal categor y a nd equipp ed with a symmetric g roup action. Under suitable conditions, co loured op era ds carry a mo del structure (see [BM03], [BM07]), whic h ena bles o ne to sp eak in a systematic wa y , for a co lo ured op erad P , ab out homoto py P -alg ebras as b eing P ∞ -algebra s, for a coﬁbra nt reso lution P ∞ − → P . In this article, w e study the preserv ation of classes of algebras o v er colour ed o p- erads under the e ﬀect of localiza tions in closed symmetric monoidal categories, a nd under the eﬀect o f homoto pical lo ca lizations in simplicial or top olog ical monoidal mo del categor ies (see [Qui67] or [Hov99] for background about mo del categ o ries). W e pr ove the following. Let C be any set and P a c oﬁbr ant C -coloure d op- erad in the c a tegory of simplicial sets (o r c o mpactly gener ated spaces ) acting on a simplicial (or top ologica l) monoidal mode l category M . Let L b e a homotopi- cal lo calization functor on M whose clas s of equiv alences is closed under tenso r pro ducts. If X = ( X ( c )) c ∈ C is a P -algebr a with X ( c ) co ﬁbrant for a ll c , then L X = ( LX ( c )) c ∈ C admits a homotopy unique P - a lgebra structur e such that the lo calization map X − → L X is a map of P - algebra s . See Theor em 6 .1 b elow for a more general v ariant of this statement. As an ex ample of this r esult, we mention the following fac t, whic h has b een known in slightly mo re r estrictive forms for several years. Let L b e a homotopi- cal lo caliz a tion functor on the category of simplicial sets or compactly g enerated spaces. If X is a coﬁbrant A ∞ -space, then LX has a homoto py unique A ∞ -space structure such that the lo ca liz a tion map X − → LX is a map of A ∞ -spaces. The same statement is true for E ∞ -spaces. (Here and thro ughout we denote by A ∞ a coﬁbrant replacement of the a sso ciative op era d A ss, a nd b y E ∞ a coﬁbrant replace- men t of the co mmutative op erad C o m.) Since any A ∞ -space is weakly equiv alen t to a top olo gical monoid, this result implies that homotopical lo ca lizations preserve top ologica l monoids up to ho motopy . Mor eov er, since non trivial homotopical lo cal- izations induce bijections on connected co mpo nents, they also preserve lo o p s paces (that is, gro up-like A ∞ -spaces) up to homotopy . In the stable case, our result is illustrated as follows. Let L b e a homotopica l lo calization functor o n the category of symmetric sp ectra [HSS00]. Let M b e an A ∞ -mo dule ov er an A ∞ -ring R , where bo th M and R are assumed to be coﬁbra nt as sp ectr a. Firstly , if R is connective or the functor L commutes with susp ension, then LM has a homotopy unique A ∞ -mo dule s tructure o ver R such that the lo- calization ma p M − → LM is a ma p of A ∞ -mo dules. Secondly , if L commutes with suspension, then LR has a ho motopy unique A ∞ -ring structure s uch that the lo calization ma p R − → LR is a map of A ∞ -rings, and LM then admits a homo topy unique A ∞ -mo dule structure o ver LR extending the A ∞ -mo dule structure o ver R . If L do es not commute with suspens ion, then the same holds if we assume that R , LR , and at least one of M and LM are connective. The same statemen ts are true if A ∞ is replaced by E ∞ . (W e emphasize that E ∞ -algebra s are w eakly equiv alent to commutativ e monoids in the categor y of sy mmetric sp ectra, according to [GH04] or [EM06], but not in the ca tegory of simplicial sets —since inﬁnite lo op spaces need not b e GE Ms— or in o ther monoidal mo del ca tegories .) LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 3 W e subse q uent ly deduce the pr eserv ation o f s trict ring s tructures (also commu- tative) and strict mo dule structures under homotopica l lo calizatio ns in the category of symmetric spectra . In fact, w e show in Section 7 that each lo calization of a ring morphism b etw een strict ring s p ectr a is naturally weakly equiv alen t (in the cate- gory of maps b etw een sp ectra) to a ring morphism, and simila r ly for R -mo dules, under a ppropriate connectivity assumptions. F or this, w e view such morphisms as algebras over c o loured o pe r ads, a s in [MSS02 , 2.9 ] or [BM07, 1 .5.3], a nd use the corr e sp onding functorial rec tiﬁca tions from [EM06]. (Rectiﬁcation o f algebr a s has b een studied in the context of categor ies with cartesia n pro duct by Ba dzio ch [Bad02] and Berg ne r [Ber 06].) As we show in the la st section, it is also true that, for every commutative ring sp ectrum R , the class o f R -algebra s is preser ved under homotopical lo calizations commuting with susp ension. If the lo calizatio n is homolo gical, then the lo caliz e d R -alg ebra structures co incide up to homo topy with those obtained in [E KMM97, Theorem VI I I.2.1]. In another direction, it was pro ved in [Laz01] a nd [DS06] that Postnik ov piec es of co nnective R -algebr as admit compatible R -a lgebra structures, provided that R is itself co nnective. O ur approa ch also yields this as a s p ecia l case. When w e refer to the mo del ca tegory o f symmetric spec tra, we will under- stand it in the sense of the p o sitive stable mo del s tructure, which was discussed in [MMSS01], [Sch01], or [Shi0 4 ]. Lik ewise, when we sp eak of co mpactly generated spaces we mean k -space s without any separa tion condition, as in [V og7 1], equipped with Quillen’s mo del ca tegory structure (given b y weak homoto py equiv alences, Serre ﬁbr ations, a nd the corr esp onding coﬁbra tions). This model structure has the adv antage o f b eing coﬁbrantly g e nerated, which ensures the v alidit y of certain re- sults that co uld fa il to hold otherwise, mainly the existence of an adequa te mo del category structure on the categor y of colour ed op era ds over a ﬁxed set o f colo urs. F o r certain purp oses, ho w ev er, it is more conv enien t to consider the k -space version of the Strø m mo del catego ry structure [Str72], with genuine homotopy equiv alence s, Hurewicz ﬁbrations and closed coﬁbra tions. In this case, all space s are ﬁbrant a nd coﬁbrant. Although this mo del categor y is not known to be coﬁbrantly generated, one ca n still spea k ab out o pe rads b eing coﬁbra nt, in the s e nse of having a left lifting prop erty with r esp ect to morphisms that induce trivial ﬁbrations in the underlying category . It was pr oved in [V og03] that the W -construction yields op er a ds that are coﬁbrant in this sense . A cknowle dgements . The plausibility of an interaction be tw een lo caliza tions and op erads was seen by several p eople shor tly after the developmen t of f -lo ca liz ations, at a mo ment where the technical machinery for a broad statement and pro of was not yet fully av ailable. Some o f us had discus s ions on this topic with W. Chach´ olski, G. Gra nja, B. Ric h ter, B. Shipley , and J. H. Smith. P art of this w ork was done while the second-na med author was visiting Utrech t Univ ersit y . 1. Col o ured operads and their algebras In the ﬁrst tw o sections, E will denote a co complete close d symmetric mono idal category with tensor pro duct ⊗ , unit I , and internal hom functor Hom E ( − , − ). W e denote by 0 the initial ob ject of E , that is, a colimit o f the empt y diagr am. Let Σ n denote the s ymmetric group on n e lement s (whic h is mea nt to be the trivial group if n = 0 a nd n = 1 ), and let C be a set, whos e elemen ts will be called c olours . A C - c olour e d c ol le ction K in E consists of a set of ob jects K ( c 1 , . . . , c n ; c ) 4 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT in E for every n ≥ 0 and eac h ( n + 1)-tuple of colours ( c 1 , . . . , c n ; c ), equipp ed with a right action of Σ n by mea ns of maps σ ∗ : K ( c 1 , . . . , c n ; c ) − → K ( c σ (1) , . . . , c σ ( n ) ; c ) where σ ∈ Σ n . The ob jects corr esp onding to n = 0 are deno ted by K ( ; c ). A morphism F : K − → K ′ of C -coloured collections is a family of ma ps K ( c 1 , . . . , c n ; c ) − → K ′ ( c 1 , . . . , c n ; c ) in E , ranging ov er n ≥ 0 and all ( n + 1)-tuples of colours ( c 1 , . . . , c n ; c ), compatible with the action of the s y mmetric gro ups. W e de no te by Coll C ( E ) the ca tegory o f C -co lo ured collections in E with their morphisms , fo r a ﬁxed se t of colour s C . A C -c olour e d op er ad P in E is a C - c o loured collection equipp ed with a unit map I − → P ( c ; c ) for each c in C and, for every ( n + 1 )-tuple of colours ( c 1 , . . . , c n ; c ) and n given tuples ( a 1 , 1 , . . . , a 1 ,k 1 ; c 1 ) , . . . , ( a n, 1 , . . . , a n,k n ; c n ) , a c omp osition pr o duct map P ( c 1 , . . . , c n ; c ) ⊗ P ( a 1 , 1 , . . . , a 1 ,k 1 ; c 1 ) ⊗ · · · ⊗ P ( a n, 1 , . . . , a n,k n ; c n ) − → P ( a 1 , 1 , . . . , a 1 ,k 1 , a 2 , 1 , . . . , a 2 ,k 2 , . . . , a n, 1 , . . . , a n,k n ; c ) , compatible with the action o f the symmetric groups and sub ject to asso ciativity and unitary compatibility relations; see , e.g., [EM06, § 2 ] for a depiction o f the diagra ms inv o lved. Thus, we may view a C -colo ured op er ad P as a m u ltic ate gory enriched ov er E , where the ho m ob jects P ( c 1 , . . . , c n ; c ) hav e n inputs and one output. A morphism of C -c o loured op erads is a mor phism of the underlying C - coloured collections that is compatible with the unit maps and the comp os ition pr o duct maps. The category of C -co loured o pe rads in E will be denoted by Oper C ( E ). As shown in [BM0 7, App endix], the categ o ry of C -co lo ured collections admits a monoidal structure in which the mono ids are precisely the C -co loured op erads. W e note, for later use, that the forg etful functor Oper C ( E ) − → Coll C ( E ) r eﬂects isomorphisms, that is, if a morphism of C -co loured o p er ads induces an isomo r phism of the underlying co llections, then it is an iso morphism. If we forg e t the symmetr ic gro up actio ns in a ll the deﬁnitions g iven so far, w e obtain non-symmetric c olour e d c ol le ctions and non-symmetric c olour e d op er ads . There is a forgetful functor fr om C - coloured ope r ads to non-s y mmetric C - coloured op erads, which has a left adjoint Σ deﬁned by a copro duct (1.1) (Σ P )( c 1 , . . . , c n ; c ) = a σ ∈ Σ n P ( c σ − 1 (1) , . . . , c σ − 1 ( n ) ; c ) . If C = { c } , then a C -colour ed o p e rad P is just a n ordinar y op erad, wher e one writes P ( n ) instead of P ( c, . . . , c ; c ) with n inputs. Here we recall tha t the (non-symmetric) asso ciative op er ad A ss is deﬁned as A ss( n ) = I fo r n ≥ 0. Its symmetric version, which we keep denoting by A s s if no confusion can aris e , is therefore g iven by A ss( n ) = I [Σ n ] for n ≥ 0, where I [Σ n ] denotes a co pro duct of copies of the unit I indexed by Σ n , on which Σ n acts freely b y p e rmutations. The (symmetric) c ommu tative op er ad C o m is deﬁned a s C om( n ) = I for n ≥ 0. Algebras o v er coloured op er ads a re deﬁned as follows (b y sp ecia lizing to a single colour, one recovers the usual notion of algebr a s ov er o p erads). Let us denote by E C LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 5 the pro duct category of copies of E indexed by the set of co lours C . F o r every ob ject X = ( X ( c )) c ∈ C in E C , a C -coloured op erad E nd( X ) in E is deﬁned by End( X )( c 1 , . . . , c n ; c ) = Hom E ( X ( c 1 ) ⊗ · · · ⊗ X ( c n ) , X ( c )) , where X ( c 1 ) ⊗ · · · ⊗ X ( c n ) is meant to b e I if n = 0. The comp osition pro duct is ordinary co mpo sition and the Σ n -action is deﬁned by p ermutation o f the factors . The C -colour e d o pe r ad End( X ) is ca lle d the endomorphism c olour e d op er ad of the ob ject X o f E C . Similarly , given a morphism f : X − → Y in E C , i.e., a C -index ed family of maps ( f c : X ( c ) − → Y ( c )) c ∈ C in E , there is a C - coloured op er ad End( f ), deﬁned as the pullback of the following diagr a m of C -colour ed collections: (1.2) End( f ) / /   End( X )   End( Y ) / / Hom( X , Y ) , where the C -coloured collection Hom( X , Y ) is deﬁned as Hom( X , Y )( c 1 , . . . , c n ; c ) = Hom E ( X ( c 1 ) ⊗ · · · ⊗ X ( c n ) , Y ( c )) , and the a rrows E nd( X ) − → Hom( X , Y ) and End( Y ) − → Hom( X , Y ) are induced by f by comp osing o n each side. The C -colo ured collection E nd( f ) inher its indeed a C -co loured op er ad structur e from the C -coloured op era ds End( X ) and End( Y ), as observed in [BM03, Theorem 3.5 ]. Given a C -co loured op erad P in E , an algeb r a over P or a P -algebr a is an ob ject X = ( X ( c )) c ∈ C of E C together with a morphism P − → End( X ) of C - c o loured op era ds. Equiv alen tly , an algebra over a C -colo ured op erad P can be deﬁned as a family of ob jects X ( c ) in E , for all c ∈ C , to g ether with maps P ( c 1 , . . . , c n ; c ) ⊗ X ( c 1 ) ⊗ · · · ⊗ X ( c n ) − → X ( c ) for ev ery ( n + 1)-tuple ( c 1 , . . . , c n ; c ), compatible with the s ymmetric group a ction, asso ciativity , a nd the unit of P . R emark 1.1 . If P is a non-symmetric C -colour e d op er ad P , then P -alg ebras are deﬁned in the same wa y , by forg etting the symmetric group action on End( X ). If Σ denotes the left adjoint (1.1) o f the forgetful functor fro m symmetric to non-symmetric C -coloured op era ds, then, for a non- symmetric C -c o loured op erad P , there is a bijective c o rresp o ndence b etw een the P -algebr a s tructures and the Σ P -alg ebra structures on an ob ject X of E C . If X = ( X ( c )) c ∈ C and Y = ( Y ( c )) c ∈ C are P -alg ebras, a map of P -algebr as f : X − → Y is a family o f maps f c : X ( c ) − → Y ( c ) in E , for all c ∈ C , that ar e compatible with the P -a lgebra structures on X and Y , i.e., the following diagram commutes for all ( n + 1 )-tuples ( c 1 , . . . , c n ; c ): P ( c 1 , . . . , c n ; c ) ⊗ X ( c 1 ) ⊗ · · · ⊗ X ( c n ) / / id ⊗ f c 1 ⊗···⊗ f c n   X ( c ) f c   P ( c 1 , . . . , c n ; c ) ⊗ Y ( c 1 ) ⊗ · · · ⊗ Y ( c n ) / / Y ( c ) . 6 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT F o r a map f : X − → Y in E C , giving a morphism of C - coloured o pe r ads P − → End( f ) is eq uiv alent by (1.2) to g iving a P -algebr a structure on X and a P - algebra str ucture on Y such that f is a map of P -a lgebras . The category of P -alg ebras in E will b e denoted by Alg P ( E ). Given a C - coloured o p erad P and an o b ject X = ( X ( c )) c ∈ C in E C , we deﬁne the r estricte d endomorphi sm op er ad End P ( X ) as follows: (1.3) End P ( X )( c 1 , . . . , c n ; c ) = ( 0 if P ( c 1 , . . . , c n ; c ) = 0, End( X )( c 1 , . . . , c n ; c ) otherwis e. Thu s, there is a canonical inclusion o f C -co lo ured op era ds End P ( X ) − → End( X ) , for which the following holds: Prop ositi on 1.2. If P is a C -c olour e d op er ad in E and X = ( X ( c )) c ∈ C is an o bje ct of E C , then every morphism P − → End( X ) of C -c olour e d op er ads factors uniquely thr ough E nd P ( X ) .  Hence, a P -algebra s tructure on X is pr ecisely g iven by a morphism of C -co loured op erads P − → End P ( X ). The same holds for non-sy mmetric coloured op era ds , b y replacing endomorphism co loured op erads by their non-symmetric version. Similarly , if X a nd Y are ob jects of E C and P is a C -coloured op erad, we denote by Ho m P ( X , Y ) the C - coloured collection deﬁned as Hom P ( X , Y )( c 1 , . . . , c n ; c ) = ( 0 if P ( c 1 , . . . , c n ; c ) = 0, Hom( X , Y )( c 1 , . . . , c n ; c ) otherwise, and, for a mor phism f : X − → Y , we denote b y End P ( f ) the pullback of the restricted endomor phism oper ads of X and Y ov er Hom P ( X , Y ), as in (1.2). 2. Id eals a n d restriction o f colours The following concepts will b e useful in our discussion of lo ca lization of mo dules ov er monoids and their ma ps . If X is an algebra ov er a C -co loured op era d P , we will need to carry out certain constructions on so me co mpo nents X ( c ), but no t on others. F or this rea son, we give a name to sp ecial subsets of the set of colours (depending on P ). Examples will b e given la ter in this section. Deﬁnition 2.1. If P is a C - coloured op erad, a subset J ⊆ C is called an ide al relative to P if P ( c 1 , . . . , c n ; c ) = 0 whenev er n ≥ 1, c ∈ J , and c i 6∈ J for some i ∈ { 1 , . . . , n } . If α : C − → D is a function b etw e e n sets of c o lours, then the following pair of adjoint functor s was discussed in [BM07, § 1.6]: (2.1) α ! : Op er C ( E ) ⇄ Ope r D ( E ) : α ∗ , where the re s triction functor α ∗ is deﬁned as ( α ∗ P )( c 1 , . . . , c n ; c ) = P ( α ( c 1 ) , . . . , α ( c n ); α ( c )) . LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 7 If α is injective (which is indeed the cas e in all our applications), then the le ft adjoint α ! can b e made explicit as follows: (2.2) ( α ! Q )( d 1 , . . . , d n ; d ) =      Q ( c 1 , . . . , c n ; c ) if α ( c i ) = d i for all i and α ( c ) = d , I if n = 1, d 1 = d , and d 6∈ α ( C ), 0 otherwise. A function α : C − → D a lso deﬁnes an adjoint pair in the co rresp onding cate- gories of algebr as: (2.3) α ! : Alg α ∗ P ( E ) ⇄ Alg P ( E ) : α ∗ for every P ∈ Oper D ( E ), where α ∗ is deﬁned a s follows. If X is a P - algebra given by a structure morphism γ : P − → E nd( X ), then ( α ∗ X )( c ) = X ( α ( c )) for all c ∈ C , with a structure morphism deﬁned by means of (2.1), (2.4) α ∗ γ : α ∗ P − → α ∗ End( X ) = End( α ∗ X ) . The following examples are illustra tive. 2.1. Mo dules o ver o p erad algebras. Let P b e a (one-coloured) o p er ad in E a nd let M o d P be a colo ured op erad with two colo ur s C = { r, m } , for which the only nonzero terms are M o d P ( r , ( n ) . . ., r ; r ) = P ( n ) for n ≥ 0 and M o d P ( c 1 , . . . , c n ; m ) = P ( n ) for n ≥ 1 when e x actly one c i is m and the rest (if any) are equal to r . Then an algebra ov er M o d P is a pair ( R, M ) of ob jects of E wher e R is a P -alg ebra and M is a mo dule over R , i.e., an ob ject equipp ed w ith a family o f maps P ( n ) ⊗ R ⊗ ( k − 1) · · · ⊗ R ⊗ M ⊗ R ⊗ ( n − k ) · · · ⊗ R − → M for n ≥ 1 and 1 ≤ k ≤ n , equiv ar iant and compa tible with a sso ciativity a nd with the unit of P . If P = A ss, then an algebra over M o d P is a pair ( R, M ) where R is a monoid in E and M is an R -bimo dule, that is, an ob ject equipped with a right R -a c tion and a left R -action that comm ute with e ach other. If P = C om, then the cor r esp onding ob ject R is a commut ative monoid in E and M is a mo dule ov er it (indistinctly left or right). The ideals relative to M od P are C , { r } , and ∅ for all P . No te also tha t, if α denotes the inclusio n o f { r } into { r, m } , then α ∗ M o d P = P for each operad P , and α ∗ ( R, M ) = R on the cor r esp onding algebras. As in [BM07], we no te that there are no n-symmetric colour ed op erads yielding the notions of left mo dule and right module. F or a (non- s ymmetric, one-co lo ured) op erad P , let L Mo d P be the non-symmetric C -coloured oper ad with C = { r , m } deﬁned by L Mod P ( r , ( n ) . . ., r ; r ) = P ( n ) , L Mod P ( r , ( n ) . . ., r, m ; m ) = P ( n + 1 ) 8 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT for n ≥ 0, and zero otherwise. Similarly , consider a non-s ymmetric coloured op erad R Mo d P with tw o colours { s, m } deﬁned b y R Mo d P ( s, ( n ) . . ., s ; s ) = P ( n ) , R Mo d P ( m, s, ( n ) . . ., s ; m ) = P ( n + 1) , for n ≥ 0, a nd zero other wise. If P = A ss (as a no n- symmetric op er ad), then the algebras o v er L Mo d P are pairs ( R, M ) o f ob jects of E where R is a monoid and M suppo rts a left a ction of R , and similar ly for R Mo d P . In order to handle R - S - bimo dules, w e co ns ider a non-s ymmetric coloured op er ad B Mo d P with three colo urs { r , s, m } and such that B Mo d P ( r , ( n ) . . ., r ; r ) = P ( n ) , B Mo d P ( s, ( n ) . . ., s ; s ) = P ( n ) , B Mo d P ( r , ( n 1 ) . . . , r , m, s, ( n 2 ) . . . , s ; m ) = P ( n 1 + n 2 + 1) , if n, n 1 , n 2 ≥ 0, and zero otherwise . The ideals relative to B Mo d P are C , { r, s } , { r } , { s } , and ∅ . Those relative to L Mod P are C , { r } , and ∅ , and similarly for R Mod P . 2.2. Maps of algebras ov er col oured op erads. Let C b e any set and P a C -co lo ured op erad. Let D = { 0 , 1 } × C and deﬁne a D - coloured op era d M or P by M or P (( i 1 , c 1 ) , . . . , ( i n , c n ); ( i, c )) = ( 0 if i = 0 a nd i k = 1 for some k , P ( c 1 , . . . , c n ; c ) otherwise. If X is an algebr a o v er M or P , then both X 0 = ( X (0 , c )) c ∈ C and X 1 = ( X (1 , c )) c ∈ C acquire a P -a lgebra structure by res tr iction o f colours , since, if α i : C − → D denotes the inclusion α i ( c ) = ( i, c ) for i = 0 and i = 1, then (2.5) ( α i ) ∗ M or P = P and ( α i ) ∗ X = X i . F ur thermore, the M o r P -algebra s tructure on X gives rise to a map o f P -algebr as f : X 0 − → X 1 as follows. F or each c ∈ C , there is a map f c : X (0 , c ) − → X (1 , c ) deﬁned as the comp os ite (2.6) X (0 , c ) − → M o r P ((0 , c ); (1 , c )) ⊗ X (0 , c ) − → X (1 , c ) , where the ﬁr s t map is obtained b y tensoring the unit u c : I − → P ( c ; c ) with X (0 , c ). Conv ersely , given tw o P -alg ebras X 0 , X 1 and a map o f P -a lg ebras f : X 0 − → X 1 , there is a uniq ue M or P -algebra structur e on X = ( X 0 ( c ) , X 1 ( c )) c ∈ C extending the given P -a lgebra structures and for whic h the disting uis hed map deﬁned b y (2.6) is the given map f . F o r example, an algebra X ov er M or A ss is determined by t w o monoids X (0) and X (1) to gether with a morphism of monoids f : X (0) − → X (1). If P is any one-colour ed o p er ad, then we can write D = { 0 , 1 } , hence recov ering [BM07, 1.5.3]; cf. also [Mar0 4, § 2]. In this case, the ideals rela tive to M or P are D , { 0 } , and ∅ . If Q = M o d P , as in Subsection 2.1, wher e P is one-colour ed, then M or Q is a D - coloured ope rad with D = { 0 , 1 } × { r, m } . An alg ebra X o v er M or Q is uniquely determined by t w o P -alg e bras A = X (0 , r ) and B = X (1 , r ), an A -module M = X (0 , m ), a B -mo dule N = X (1 , m ), a map of P -algebr as A − → B , and a map of A - mo dules M − → N , where the A -mo dule s tructure on N is deﬁned by means of the map A − → B . The ideals relative to M or Q are the following subsets: D , { (0 , r ) , (0 , m ) , (1 , r ) } , { (0 , r ) , (0 , m ) } , { (0 , r ) , (1 , r ) } , { (0 , r ) } , and ∅ . LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 9 3. P r eser v a tion of algebras under localiza tions In this section, w e study the eﬀect o f localiz ations on structures deﬁned as alge- bras ov er colo ured o pe r ads in clos e d sy mmetr ic monoidal ca teg ories, and descr ibe several sp ecia l case s. First, we recall so me g eneralities ab out lo calizations. Let C b e any c a tegory . A c o augmente d functor on C is a functor L : C − → C together with a natural tra nsformation η : Id C − → L . (This is called a p ointe d endofunctor in o ther contexts.) A coa ugmented functor ( L, η ) is idemp otent if η LX = Lη X and L η X : LX − → LLX is an isomo rphism for every ob ject X in C . Idempo tent coaugmented functors ar e called lo c alizations . If ( L, η ) is a lo c a lization, then the ob jects isomor phic to LX for some X a re called L - lo c al obj e cts and the morphisms f : X − → Y such that Lf : LX − → LY is an iso mo rphism ar e called L -e quivalenc es . Lo caliza tions are characterized by each of t wo univ ersal prope rties: (i) η X : X − → LX is initial among morphisms from X to L -lo ca l ob jects; (ii) η X : X − → LX is termina l a mong L -eq uiv alences with domain X . These universal proper ties ensure that if f : X − → Y is an L -equiv alence and Y is L -lo cal, then Y ∼ = LX . In fact, the classes of L -lo cal ob jects a nd L -equiv alences determine each other by an orthogo nality relation. A mor phism f : X − → Y a nd an ob ject Z in C ar e called ortho gonal if the induced map (3.1) C ( f , Z ) : C ( Y , Z ) − → C ( X, Z ) is a bijection. Using this termino logy , a map is an L -equiv alence if and o nly if it is ortho g onal to all L - lo cal ob jects, and an ob ject is L -lo cal if and o nly if it is orthogo nal to all L -equiv a lences. Examples of lo c a lization functor s on the homo topy category of s paces o r sp ectra are lo caliza tion at primes, homo logical lo calizatio ns, and, more gener ally , f -lo caliza tions in the sense o f [F a r96]. Here it is co nv enien t to in troduce the following conven tio n ab out extending coaugmented functor s from a categ ory E to the pr o duct catego ry E C , where C is a set o f colours . In s ome of o ur results, it will b e necessary to lo ca lize a subset of components of a n algebra o ver a C -coloured op erad, but not the rest (for exa m- ple, we may wan t to lo ca lize a n R -mo dule, but not the monoid R ). Thu s, for a coaugmented functor ( L, η ) on E , we deﬁne partial extens ions ov er E C as follows: Deﬁnition 3.1. The ex t ension of ( L, η ) over E C away fr om a subset J ⊆ C is the coaugmented functor on E C —which w e keep denoting by ( L, η ) if no confusion can ar ise— given by L X = ( L c X ( c )) c ∈ C where L c is the ident it y functor if c ∈ J and L c = L if c 6∈ J . Cor resp ondingly , η X : X − → L X is deﬁned b y declar ing that ( η X ) c is the identit y map if c ∈ J and ( η X ) c = η X ( c ) if c 6∈ J . Lemma 3. 2. L et E b e a close d symmetric monoidal c ate go ry, P a C -c olour e d op er ad in E , and X a P -algebr a. L et ( L, η ) b e any extension over E C of a c o augmente d functor on E . Supp ose t hat the morphism of C -c ol our e d c ol le ctions End P ( L X ) − → Hom P ( X , L X ) induc e d by η X is an isomorphism. Then L X has a unique P -algebr a structu r e such that η X is a map of P -algebr as. Pr o of. The ass umption made implies that the pullback morphism End P ( η X ) − → End P ( X ) 10 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT is an isomo rphism of C -coloured colle ctions, and therefo re it is an isomor phism of C -co lo ured op erads. Comp osing the inv erse of this isomorphis m with the morphism γ : P − → E nd P ( X ) that endows X with its P -algebr a structure yields a morphism P − → End P ( L X ) as depicted in the dia g ram End P ( η X ) / /   End P ( L X )   P γ / / End P ( X ) I I / / Hom P ( X , L X ) . In this w a y , L X acquires a P -algebr a str ucture. The fact that this P -alge bra structure morphism factor s throug h E nd P ( η X ) implies precis ely that η X is a map of P -algebr as. F urthermore, the P -alg e bra structure o n L X is unique with this prop erty , by the universal proper t y of the pullback.  Our main so urce of applicatio ns o f this r e sult corresp o nds to the situation where ( L, η ) is of a specia l kind. W e will ask it to satisfy an orthogona lity condition that is stronger than (3.1), but nonetheless ho lds in our e x amples in this section. Deﬁnition 3.3. W e say that a localiza tio n ( L , η ) on a closed symmetric monoidal category E is close d if, fo r every L -equiv a lence f : X − → Y and every L -lo cal ob ject Z , the map Hom E ( f , Z ) : Ho m E ( Y , Z ) − → Hom E ( X, Z ) is an isomor phis m in E . F o r such a functor L , if f 1 : X 1 − → Y 1 and f 2 : X 2 − → Y 2 are L - equiv ale nces, then the tensor pro duct f 1 ⊗ f 2 is a gain an L -equiv alence, since, by the hom-tensor adjunction, (3.2) Hom E ( Y 1 ⊗ Y 2 , Z ) ∼ = Hom E ( Y 1 , Hom E ( Y 2 , Z )) ∼ = Hom E ( Y 1 , Hom E ( X 2 , Z )) ∼ = Hom E ( Y 1 ⊗ X 2 , Z ) for every L -lo cal ob ject Z , and similarly in order to r e place Y 1 by X 1 . In the r e s t of this sectio n, we will only cons ider clos e d lo calizatio ns. The following theorem states that the assumptions o f Lemma 3.2 hold for these functors. Theorem 3.4. L et E b e a close d symmetric monoidal c ate gory, P a C -c olour e d op er ad in E , and ( L, η ) the extension over E C of a close d lo c alization away fr om an ide al J ⊆ C r elative to P . If X is a P -algebr a, then L X has a unique P -algebr a structur e such that η X is a map of P -algebr as. Pr o of. By Lemma 3.2, it is enough to show tha t the morphism of C -coloured col- lections End P ( L X ) − → Hom P ( X , L X ) induced by η X is a n iso morphism. Since J is an ideal, we need only consider the v alues of these co llections on tuples ( c 1 , . . . , c n ; c ) for which c ∈ J and c i ∈ J for all i , or c 6∈ J . In the ﬁrst case , the map Hom E ( L c 1 X ( c 1 ) ⊗ · · · ⊗ L c n X ( c n ) , L c X ( c ))   Hom E ( X ( c 1 ) ⊗ · · · ⊗ X ( c n ) , L c X ( c )) LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 11 is trivially an iso mo rphism b ecause L c and all the L c i are ident it y functors, accord- ing to Deﬁnition 3.1. If c 6∈ J , then L c = L , and the isomorphis m follo ws from the fact that the tenso r pro duct o f L -equiv alences is an L -equiv alence.  Observe that Lemma 3 .2 and Theo rem 3.4 remain true if the coloured op era d P is non-symmetric. At a ny moment, if necessary , we may replace P by its symmetric version Σ P , since b oth y ie ld the same class of algebras (see Remar k 1.1). Corollary 3 .5. L et ( L, η ) b e a close d lo c alization on a close d symmetric monoidal c ate gory E . (i) If R is a monoid in E , then LR has a unique m onoid stru ctur e su ch that η R : R − → L R is a morphism of monoids. If R is c ommutative, then LR is also c ommutative. (ii) If f : R 1 − → R 2 is a morphism of m onoids in E , then Lf : LR 1 − → LR 2 is also a morphism of m onoids. (iii) If R is a monoi d in E and M is a left R - mo dule, then LM has a u nique left R -mo dule structur e s u ch that η M : M − → LM is a morphism of R - mo dules. Mor e over, L M also has a unique left LR -mo dule structur e extending the R -mo dule st ructur e. The same statements ar e t r u e for right R - m o dules. (iv) If R and S ar e monoids in E and M is an R - S -bimo dule, then LM has a unique R - S - bimo dule struct ur e such that η M : M − → L M is a mor- phism of R - S -bimo du les. Mor e over, LM also has unique R - LS -bimo dule, LR - S -bimo dule, and LR - LS -bimo dule struct ur es that exten d the given R - S - bimo dule stru ctur e. (v) If f : M 1 − → M 2 is a morphism of left R -m o dules, wher e R is a monoid in E , then Lf : LM 1 − → LM 2 is a morphism of left R -mo dules and a morphism of left L R -mo dules. The analo gous statements ar e true for right R -mo dules and for R - S -bimo dules, wher e S is another monoid. Pr o of. This follows from Theorem 3.4 using the coloured op era ds o f Subsections 2.1 and 2.2, by cho o sing a suitable ideal in each case. In part (i), pick the o p- erads A ss and C om, viewed as colour e d op er ads w ith one colour, together with the ideal J = ∅ in ea ch ca s e. In par t (ii), pic k the co lo ured ope r ad M or A ss of Subsection 2.2, toge ther with the idea l J = ∅ . (Note that f − → Lf is a commuta- tive diagr am of mo rphisms of monoids, and therefo r e LR 1 and L R 2 are eq uipp ed with the monoid structur e given by (i).) In pa rt (iii), use ﬁrst the c oloured op- erad L Mo d A ss of Subsection 2.1 with the ideal J = { r } in order to endow LM with a left R -mo dule structure, a nd choo se the ideal J = ∅ to endow L M with an LR - mo dule structure extending the prev io us R -mo dule structure. Similarly for right mo dules. F o r bimo dules, in part (iv ), use the coloured oper ad B Mod A ss with each of the ideals J = { r, s } , J = { r } , J = { s } , and J = ∅ , in o rder to endow LM with an R - S -bimo dule structure , a n R - LS -bimo dule structure, an LR - S -bimo dule structure, and an LR - LS -bimo dule structure, resp ectively . In part (v), use the coloured op erad M or Q describ ed in Subsection 2.2 for Q = L Mo d A ss , with the ideal J = { (0 , r ) , (1 , r ) } in o rder to infer that Lf is a morphism of R -mo dules, and J = ∅ in order to infer that Lf is a morphism of LR -mo dules . Similarly with Q = R Mo d A ss for right R -mo dules a nd Q = B Mo d A ss for R - S -bimodules . As in (ii), the mo dule or bimodule structures on LM 1 and LM 2 are those given b y (iii) or (iv), since f − → Lf is a commutativ e diagram of morphisms.  12 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT Corollary 3 .6. L et ( L, η ) b e a close d lo c alization on a close d symmetric monoidal c ate gory E . If P is a C -c olour e d op er ad in E , then the C -c olour e d c ol le ction LP deﬁne d as ( LP )( c 1 , . . . , c n ; c ) = L ( P ( c 1 , . . . , c n ; c )) has a unique C -c olo ur e d op er ad structu re such that t he map P − → LP induc e d by η is a morphism of C -c olour e d op er ads. Pr o of. F or ea ch set C , there is a co loured o p erad whose algebra s are pr e c isely the C -co lo ured o pe rads in E ; for a descr iption, s ee [BM07, Examples 1 .5 .6 and 1.5 .7]. The s tatement follows by applying Theo r em 3.4 to this colo ur ed op er ad, with the empt y ideal.  As we next explain, Corollar y 3.5 implies a num ber of known res ults ab out preserv ation o f certain structures under lo ca lizations. Subsections 3.2 a nd 3 .3 re- fer to the homotopy categories o f spa c es and spe c tra, r e s p e ctively , in which the corres p o nding results are weak forms of the stronger results describ ed in Section 6. 3.1. Discrete rings and mo dules. In the category A b of a b elian groups, giv en a homomorphism f : A − → B , an ab elian group G is called f -lo c al if it is ortho gonal to f , tha t is, if A b( f , G ) : A b( B , G ) − → A b( A, G ) is a bijection. A homomorphis m is called an f -e quivalenc e if it is o r thogonal to all f - lo cal groups. B y gener al results ab out lo cally presentable categories (see [AR94, Theorem 1.39]), there is a lo c alization functor ( L f , η ) for every f o n the category of ab elian groups, called f -lo c alizatio n , such that η G : G − → L f G is an f - equiv a le nc e int o an f -lo cal gro up for all G . Thu s L f is a clo s ed lo caliza tion if we endow the category of ab elian groups with the closed symmetric monoidal struc tur e g iven by the tensor pr o duct ov er Z and the canonica l enrichmen t of A b ov er itself. Hence, we infer fr o m Corollar y 3.5 the following obser v ation made in [Ca s00, Theorems 3.8 and 3.9 ], wher e by a ring we mean an asso c ia tive ring R with a unit morphism Z − → R (so the zero r ing is not excluded): Prop ositi on 3.7 . In t he c ate gory of ab elian gr oups, every f -lo c alizatio n pr eserves the classes of rings, c ommutative rings, left or right mo dules over a ring, and bimo dules over rings.  3.2. H -spaces. Let Ho b e the homotopy category of k -space s with the Quillen mo del str ucture (as in [Qui67, I I.3]), or the homotopy ca tegory of simplicial s ets with the Kan mo del structur e . Ea ch of these (equiv alent ) categories is clos e d sym- metric monoidal with the cor resp onding derived pro duct a s tensor pro duct, the one-p oint space as unit, and the derived mapping space map( − , − ) as internal hom; cf. [Hov99, Theorem 4.3 .2]. A monoid in Ho is a homotopy a sso ciative H -space, i.e., a space X together with a multiplication map X × X − → X that is asso ciative up to homotopy and with a homoto py unit . F o r ev ery ma p f b etw een spaces there is an f -lo c alization functor on Ho (see [F a r96] or [Hir03, 4.1 .1]), which is closed by construction. Hence, the follo wing fact is deduced from Co rollary 3.5: Prop ositi on 3.8 . Every f -lo c alization o n sp ac es pr eserves the classes of homotopy asso ciative H -sp ac es and homotopy c ommutative H -sp ac es.  LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 13 3.3. Hom otop y ring s p ectra and homotopy mo dule sp ectra. Let Ho s be the stable homotopy ca teg ory of Adams–Boardman, which is clos ed symmetric monoidal w ith the derived smas h pr o duct as tensor pro duct, the spher e sp ectrum as unit, and the derived function spectr um F ( − , − ) as in ternal hom. A monoid in Ho s is a ho motopy r ing sp ectrum and a mo dule over a monoid is a homotopy mo dule sp ectrum. Hence, the following res ult, which extends [CG05, Theorem 4.2], is a consequence of Co rollary 3.5: Prop ositi on 3.9. If ( L, η ) is a close d lo c alization on sp e ctr a and R is a homo- topy ring sp e ctrum, then LR admits a unique homotopy ring str u ctur e such that η R : R − → LR is a homotopy ring map. If M is a homotopy R -mo dule, then LM admits a u nique homotopy R -mo dule s tructur e such that η M : M − → LM is a ho- motopy R -mo dule map, and LM admits a unique homotopy LR - mo dule structur e extending the R -mo dule structur e.  Note that, in par ticula r, if M is a homoto py R - mo dule and LR ≃ 0, then we deduce that LM ≃ 0 a s well. Examples of closed lo c alizations o n Ho s are stable f -lo caliza tions when f is a wedge of maps { Σ k g } for all k ∈ Z and some map g ; see [CG05, Theor em 2.7]. Homologica l lo calizations a re of this kind. A lo calizatio n on Ho s is closed if a nd only if it commutes with susp ension, a nd this is equiv alent to the prop e rty of pr eserving coﬁbre sequences (see also the re - marks made in Subse c tion 6 .2 b elow). Thus it is importa nt to distinguish closed lo calizations from other localiz a tions on Ho s that do not preserve coﬁbr e sequences, such as P ostniko v sections. Indeed, Pro p o sition 3.9 do es not hold if L is a Postnik ov section and R = K ( n ); see (6.5) b elow for details. 4. H omotopical lo caliza tion functors When one w orks with mo del categor ies, orthogona lit y b etw een maps and o b jects is more conv eniently discussed in terms o f homotopy function c omplexes . This is a stronger notion than or thogonality deﬁned in terms o f homotopy classes of maps, and distinct from orthog onality deﬁned in terms o f an internal hom (if av ailable), in general. A homotopy function c omplex in a model category M is a functorial c hoice, for every tw o ob jects X and Y in M , o f a ﬁbra nt simplicial set map( X , Y ) whos e homotopy t yp e is the s ame as the diagonal of the bisimplicial set M ( X ∗ , Y ∗ ) where X ∗ − → X is a cosimplicia l res o lution of X and Y − → Y ∗ is a simplicial r esolution of Y , a s deﬁned, e.g., in [Hir03, 16.1]. Th us, the homotopy type of map( X , Y ) do e s not change if we replace X or Y by weakly equiv alent ob jects, and π 0 map( X , Y ) is in natural bijectiv e corresp ondence with the set [ X , Y ] of morphisms from X to Y in Ho( M ). F or mor e details, see [Hir03, Theorem 17.7.2]. The existence o f homotopy function complexes in every mo del categor y is proved in [Hov99, 5.4] and [Hir03, 17.3]. Recall that a simpli cial c ate gory is a categor y C eq uipped with an enr ichmen t, a tensor and a cotensor o v er the ca tegory of simplicial sets. Thus, there are functors Map( − , X ) : C op − → sSets; − ⊠ X : sSe ts − → C ; X ( − ) : sSets op − → C , for every o b ject X of C , satisfying certain co mpatibility r elations. See [GJ99] o r [Hir03, § 9 .1] for details. Among these, for every tw o ob jects X and Y of C a nd 14 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT every simplicial set K , ther e are natura l bijections (4.1) C ( K ⊠ X , Y ) ∼ = sSets( K, Map( X , Y )) ∼ = C ( X, Y K ) . A simplicial mo del c ate gory is a mo del category M that is als o a simplicial category and sa tisﬁes Quillen’s SM7 axiom: If f : X − → Y is a co ﬁbr ation in M and g : U − → V is a ﬁbration in M , then the induced map Map( Y , U ) − → Map( Y , V ) × Map( X,V ) Map( X , U ) is a ﬁbration of simplicial sets that is trivia l if f o r g is trivial. If M is a simplicial mo del categor y , then map( X , Y ) = Map ( QX , F Y ) deﬁnes a homotopy function complex, where Map( − , − ) deno tes the simplicial enrichment , Q is a functor ial coﬁbra nt replacemen t and F is a functoria l ﬁbrant replacemen t. Now let M b e any model ca tegory with a c hoice of homotopy function complexes denoted by ma p( − , − ). W e will also as s ume that M has functoria l factoriza tions, as in [Hov99] and [Hir03]. A morphism f : X − → Y and an ob ject Z a re called simplicia l ly ortho gonal if the induced map (4.2) map( f , Z ) : map( Y , Z ) − → map( X , Z ) is a weak equiv a le nce of simplicial sets. This form of or thogonality is used in the following deﬁnitio n. Deﬁnition 4.1. A homotopic al lo c aliza tion on a mo del category M with homotop y function complexes map( − , − ) is a functor L : M − → M that preser ves weak e q uiv a- lences and ta kes ﬁbrant v alues, together w ith a natura l transforma tion η : I d M − → L such tha t, for every ob ject X , the following hold: (i) L η X : LX − → L L X is a weak equiv alence; (ii) η LX and Lη X are equal in the homotopy category Ho( M ); (iii) η X : X − → LX is a co ﬁbration such that the map map( η X , LY ) : map( LX , LY ) − → map( X , L Y ) is a weak equiv a lence of simplicial s ets for all Y . The co nditio n that L takes ﬁbrant v alues and the condition that η X is a coﬁ- bration for all X are technical, yet useful in pra ctice. None o f the t w o imp oses a restriction on the deﬁnition, since, if L do es not take ﬁbr ant v a lues, then we may replace it b y F L , where F is a ﬁbra nt r eplacement functor, a nd we may also decomp ose η X functorially in to a coﬁbration follow ed by a trivial ﬁbra tion for a ll X , X ξ X − → K X ν X − → LX . Then K b e c omes a functor a nd ξ : Id M − → K a natur al tra nsformation for which (i), (ii) and (iii) hold. (The condition ξ K X ≃ K ξ X is sa tisﬁed since η LX ≃ Lη X and ν LX ◦ K ν X = Lν X ◦ ν K X , as ν : K − → L is also a natur al transfor ma tion.) Every homoto pica l lo calization b ecomes just an idemp otent functor when we pass to the homo topy category Ho( M ), since π 0 map( X , Y ) ∼ = [ X , Y ]. If ( L, η ) is a homotopical loca lization, then the ﬁbra nt ob jects of M weakly equiv alen t to LX for some X are ca lled L -lo c al , and the maps f : X − → Y such tha t Lf : LX − → LY is a weak equiv alence are called L -e quivalenc es . In addition to o r thogonality in Ho( M ), L -lo cal ob jects and L -equiv alences are simplicially or thogonal as deﬁned in (4.2), and in fact a ﬁbra nt ob ject is L -lo cal if a nd only if it is s implicially orthogo nal to all L -equiv alences, while a map is an L -eq uiv alence if a nd only if it is simplicially orthogo nal to all L -lo cal ob jects. LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 15 If ( L, η ) is a homotopica l lo ca liz ation on simplicial sets or k -spaces, then either all nonempt y L -lo c al spaces are w eakly equiv alent to a p oint, or all L -equiv alences are bijectiv e on connected comp onents. The following arg ument to prove this claim is well known. If ther e is an L -equiv alence that is no t bijective on connected com- po nents, th en it has a r etract of the form S 0 − → ∗ or ∗ − → S 0 (bes ide s the trivial case ∅ − → ∗ ). Since every r etract of an L -eq uiv alence is an L -eq uiv alence, it fol- lows that, if X is L -lo cal, then X has the s ame homotopy type as X × X , which implies that X is weakly contractible (or empty). Because of this o bs erv a tion, we will assume thr o ughout that η induces an iso morphism π 0 ( X ) ∼ = π 0 ( LX ) for any homotopical lo calizatio n L and all spaces X . Suﬃcien t conditions for a loca lization on Ho( M ) in o rder that it b e induced by a homotopical lo caliz a tion on M were g iven in [CC06]. Most lo calizations encountered in pra ctice, including all f -lo caliz a tions in the sense of [F ar9 6], are homotopical lo calizations. In fact, if M is a left prop er, coﬁbrantly gener ated, loca lly pr esentable simplicial mo del category without empty hom-s e ts , and one as sumes the v alidit y o f V o pˇ enk a’s principle from set theory , then every lo c alization on Ho( M ) co mes from an f -loc a lization on M for so me map f ; see [CC06, Theor em 2.3]. 5. Mod el structures on ca tegories of o perads Before pres entin g our main results, we still need to recall from [B M03] and [BM07] the terminology and basic prop erties of a model structure for the category of coloured op erads ov er a ﬁxed set of colo ur s. A monoida l mo del c ate gory E is a closed symmetric monoidal category with a mo del str uctur e that satisﬁes the pushout- pr o duct axio m (s e e [Hov99, § 4 .2], [SS00]): If f : X − → Y and g : U − → V are coﬁbrations in E , then the induced ma p ( X ⊗ V ) a X ⊗ U ( Y ⊗ U ) − → Y ⊗ V is a coﬁbra tio n that is trivial if f o r g is tr ivial. W e will also ass ume that the unit I o f E is coﬁbrant. Using the a djunction b etw een ⊗ and Hom E ( − , − ), one obtains the following equiv a lent fo r mulation of the pushout-pro duct axiom: If f : X − → Y is a coﬁbra tio n in E and g : U − → V is a ﬁbra tio n in E , then the induced map Hom E ( Y , U ) − → Hom E ( Y , V ) × Hom E ( X,V ) Hom E ( X, U ) is a ﬁbration in E that is trivial if either f o r g is trivial. Let E be a monoidal mo del catego r y . If E is coﬁbrantly g enerated, then, as explained in [BM03, § 3] and [BM07, § 3], the ca tegory of C - coloured collections in E admits a mo de l str ucture in which a morphis m K − → L is a weak equiv alence (resp. a ﬁbratio n) if and only if for each tuple of colour s ( c 1 , . . . , c n ; c ) the map K ( c 1 , . . . , c n ; c ) − → L ( c 1 , . . . , c n ; c ) is a weak equiv a lence (resp. a ﬁbra tio n) in E . This mo del structure c a n b e trans- ferred along the fr e e-forgetful adjunction (5.1) F : Coll C ( E ) ⇆ Ope r C ( E ) : U to provide a mo del structure on the category of C -co loured op er a ds, under suitable assumptions on the category E , including still the assumption that E be coﬁbrantly generated; see [BM03, Theorem 3.2] and [BM07, Theor em 2.1 and Example 1.5.7]. 16 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT The monoidal mo del ca tegories of k -s paces (with the Q uillen mo del structure ) and simplicial sets satisfy these assumptions. Thus, in a ny of these ca tegories , a morphism of C -coloured op erads f : P − → Q is a w eak eq uiv alence (resp. a ﬁbration) if a nd only if U f is a weak equiv a lence (resp. a ﬁbra tion) of C -colour e d collections. Co ﬁbr ations are deﬁned by the left lifting pro p erty with r e s p e ct to the trivial ﬁbratio ns. A C -coloured op erad P is coﬁbra nt if the unique morphism I C − → P is a coﬁbration, wher e I C is the initial C -co loured op er ad deﬁned by I C ( c ; c ) = I for a ll c , and zero otherwise. The W -construction of Boardman–V ogt for C -coloure d op erads (see [BV73], [V o g03], [BM06], [BM07, § 3]) provides a coﬁbran t replac ement for C -colo ured o p e r - ads P who s e under lying C -colo ured collection (p ointed b y the unit) is Σ-coﬁbrant; that is, the unit ma p I − → P ( c ; c ) is a coﬁbra tion for a ll c a nd P ( c 1 , . . . , c n ; c ) is coﬁbra nt as a Σ c 1 ,...,c n -space for all ( c 1 , . . . , c n ; c ), wher e Σ c 1 ,...,c n denotes the subgroup of Σ n leaving ( c 1 , . . . , c n ) inv ariant; see [BM0 7, Theorem 3.5]. This was implicit in [BV73] fo r topolog ic al op erads, and further dev elop ed in [V o g03] for the category of k - spaces with the Strø m mo del structure . F r om now on w e will only consider categor ies of coloured op erads admitting the mo del structure transferre d along (5.1). Under this ass umption, for ev ery function α : C − → D , the adjunction α ! : Op er C ( E ) ⇄ Op er D ( E ) : α ∗ given by (2.1) is a Quillen pair, since α ∗ preserves ﬁbrations and w eak equiv alences . Given a C -co loured operad P , a c oﬁbr ant r esolut ion of P is a trivial ﬁbration of C -co lo ured op era ds P ∞ − → P where P ∞ is c oﬁbrant. (F or notational conv enience, we also say that P ∞ is a coﬁbra nt resolution o f P .) Througho ut w e deno te by A ∞ an arbitrar y but ﬁxed coﬁbra nt res o lution of A ss, and b y E ∞ a coﬁbrant resolution of C o m. (It is common practice to denote by A ∞ any non-symmetric o p er ad that is weakly equiv alen t to A ss, and b y E ∞ any op er ad that is weakly equiv alent to C om; he r e we assume them coﬁbra nt for simplicit y in the statement of our results.) W e consider tw o impo rtant special ca ses where change of colour s plays a role. 5.1. P ∞ -mo dul es. Let P b e a ny (one-co loured) op er ad, a nd let P ∞ − → P b e a coﬁbrant resolution. As explained in Subsection 2.1, M o d P is a C -coloured opera d with C = { r, m } . Let α denote the inc lus ion of { r } into C . If ( M o d P ) ∞ − → M o d P is a coﬁbra nt resolution of M o d P , then, since α ∗ preserves trivial ﬁbrations, α ∗ ( M o d P ) ∞ − → α ∗ M o d P = P is a trivial ﬁbra tion. Hence there is a lifting (unique up to homo to py) (5.2) α ∗ ( M o d P ) ∞   P ∞ 9 9 / / P. If a pair ( R, M ) is a ( M o d P ) ∞ -algebra , then R = α ∗ ( R, M ) is an alg ebra ov er α ∗ ( M o d P ) ∞ by (2.4), a nd hence a P ∞ -algebra via (5 .2). Although the second comp onent M need not be a mo dule, w e call it a P ∞ -mo dule ov er R . LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 17 5.2. P ∞ -maps. Let P b e a C -coloured op erad where C is any set of c olours, and choose a coﬁbra nt resolution ϕ : P ∞ − → P . Let M or P be as deﬁned in Subsec- tion 2 .2, with D = { 0 , 1 } × C . F or i ∈ { 0 , 1 } , let α i : C − → D b e the functions deﬁned as α 0 ( c ) = (0 , c ) and α 1 ( c ) = (1 , c ). Thus ( α i ) ∗ M or P = P for b o th i = 0 and i = 1 . If Φ : ( M or P ) ∞ − → M or P is a coﬁbra nt resolution of M or P , then, as in (5.2), there a re morphisms (in fact, weak equiv alences) of C -colour ed opera ds (5.3) ( α i ) ∗ ( M or P ) ∞ ( α i ) ∗ Φ   P ∞ λ i 8 8 ϕ / / P for i = 0 and i = 1, unique up to homotopy , render ing the tria ngle commutativ e. Therefore, by (2.5), an algebra X o v er ( M or P ) ∞ gives rise to a pair of P ∞ -algebra s ( X 0 , X 1 ) with additional structure link ing them, which is w eaker than a morphism of P ∞ -algebra s. Speciﬁca lly , since the unit I of E is c o ﬁbrant, we may choose, for each c ∈ C , a lifting (5.4) ( M or P ) ∞ ((0 , c ); (1 , c ))   I u c / / 7 7 M or P ((0 , c ); (1 , c )) where u c is the map considered in (2.6). The lifting is not unique, but it is unique up to homotopy . Hence, the c o mp o sites X (0 , c ) − → ( M or P ) ∞ ((0 , c ); (1 , c )) ⊗ X (0 , c ) − → X (1 , c ) yield together a homotopy cla ss of maps X 0 − → X 1 . Each o f these will b e called a P ∞ -map . This generalizes the notion of A ∞ -map discussed in [BV73, I.3] and [MSS02, 2.9]. A lifting similar to (5.4) in the topo logical cas e was c o nsidered by Sch w¨ anzl and V ogt in the co ntext of [SV88]. Note that there is als o a lifting (5.5) M or P ∞   ( M or P ) ∞ Φ / / Ψ 9 9 M or P , since the vertical ar row is a trivial ﬁbration of C -co loured o pe rads a nd ( M o r P ) ∞ is coﬁbrant. Hence, every morphism of P ∞ -algebra s admits a P ∞ -map structure. F o r later use, we remark that (5.2) and (5 .5) yield, for i = 0 a nd i = 1, (5.6) ϕ ◦ (( α i ) ∗ Ψ) ◦ λ i = (( α i ) ∗ Φ) ◦ λ i = ϕ. 18 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT 6. P r eser v a tion of structures in monoidal model ca tegories In Section 3 we saw that monoids (including homotopy asso ciative H -spaces and homotopy ring spectra ), mo dules ov er monoids, and morphisms b etw een these, are preserved by clo sed lo calization functors in the unsta ble homotopy categ ory Ho or in the stable homoto py categ ory Ho s . Y et, the categ o ries o f simplicia l sets (or k -spaces ) and symmetric sp ectr a a ls o admit mono idal mo del structures that allow one to deﬁne monoids and modules within the model categories themselv es. In the rest of the article, we study the pres erv a tion of such strict structures in monoidal mo del categorie s , by viewing them as alg e bras over coloure d op era ds and us ing suitable rectiﬁcation functor s. Thu s, from now on, we restrict ours elves to coloure d op erads in simplicial sets (or k -spaces with the Quillen mo del str ucture) acting on simplicial (or top ologica l) monoidal mo del categ ories. A monoida l mo del categ o ry M is called simplic ial if it is also a simplicial mo del categor y , and the simplicial action co mm utes with the monoidal pro duct, i.e., there are na tur al coherent isomorphisms K ⊠ ( X ⊗ Y ) ∼ = ( K ⊠ X ) ⊗ Y where K is any simplicial set and X , Y are ob jects of M . The same deﬁnition applies to the top o lo gical case. While all simplicial sets ar e co ﬁbrant, this is not so for k -spaces . Therefore, coﬁbrancy assumptions will be nee de d at certain places. A re medy would b e to use k -space s with the Strøm structure. Ho wev er, this mo del structure is not known to be co ﬁbr antly genera ted; hence , it do es not ﬁt into the framework describ ed in the preceding section. While it is still p o ssible to talk o f coﬁbr ant o per ads and coﬁbrant algebr as in this setting (see [V og03]) a nd our results remain v alid with the sa me pro o fs, to avoid w orking in tw o diﬀerent settings we stick to the Quillen mo del structure whenever k -spaces ar e considered. F o r the sake of cla r ity , we will emphasize notationally the distinction b etw een the monoidal mo del ca tegory E in which our coloured op erads take v alues and the monoidal mo del categ ory M on which they act. Thus, if P is a C -coloured op erad in the catego r y E of simplicial sets (or k -spaces) and M is a simplicial (res p. top o - logical) mo no idal mo del category , then a P -algebr a X = ( X ( c )) c ∈ C is deﬁned as an ob ject o f M C equipp e d with a morphism of C -co loured op era ds P − → End( X ) in E , where E nd( X ) is now deﬁned as (6.1) End( X )( c 1 , . . . , c n ; c ) = Map( X ( c 1 ) ⊗ · · · ⊗ X ( c n ) , X ( c )) , and Map( − , − ) denotes the simplicial (resp. top o lo gical) enrichmen t of M . This is consistent with the previous deﬁnitions, s inc e the map sSets( P ( c 1 , . . . , c n ; c ) , Map( X ( c 1 ) ⊗ · · · ⊗ X ( c n ) , X ( c )))   M ( P ( c 1 , . . . , c n ; c ) ⊠ ( X ( c 1 ) ⊗ · · · ⊗ X ( c n )) , X ( c )) is bijective b y the adjunction (4.1). Simplicial sets or k - spaces Ho m P ( X , Y )( c 1 , . . . , c n ; c ) and End P ( X )( c 1 , . . . , c n ; c ) are deﬁned analo gously as in (6.1) and (1.3), for every C - coloured op erad P . LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 19 W e say that tw o P -a lgebra structures γ , γ ′ : P − → E nd P ( X ) on an ob ject X of M C coincide up to homotop y if γ ≃ γ ′ in the mo de l catego ry of C -coloured op e rads. (Homotopic means left a nd right homotopic.) The following is the main theore m of this ar ticle: Theorem 6.1. L et ( L, η ) b e a homo topic al lo c alization on a s implicial or top olo gic al monoidal mo del c ate gory M . L et P b e a c oﬁbr ant C - c olour e d op er ad in simplicial sets or k -sp ac es, wher e C is any set, and c onsider the extension of ( L, η ) over M C away fr om an ide al J ⊆ C r elative to P . L et X b e a P -algebr a such that X ( c ) is c oﬁbr ant in M for every c ∈ C . Supp ose that ( η X ) c 1 ⊗ · · · ⊗ ( η X ) c n is an L -e quivale nc e whenever P ( c 1 , . . . , c n ; c ) is nonempty. Then L X admits a homotopy unique P -algebr a structur e such that η X is a map of P -algebr as. Pr o of. W e ﬁrst chec k that the mor phism of C -co loured collectio ns End P ( L X ) − → Hom P ( X , L X ) induced by η X is a triv ial ﬁbra tio n, i.e., a trivial ﬁbration o f simplicia l s ets or k -spaces for e very ( c 1 , . . . , c n ; c ). Since the v alue of b o th these C -colo ured co lle c- tions on ( c 1 , . . . , c n ; c ) is the empty set whenever P ( c 1 , . . . , c n ; c ) is the empty set, we may exclude these cases fro m the arg ument. Now, for all ( c 1 , . . . , c n ; c ) such that P ( c 1 , . . . , c n ; c ) is nonempty , the ma p ( η X ) c 1 ⊗ · · · ⊗ ( η X ) c n : X ( c 1 ) ⊗ · · · ⊗ X ( c n ) − → L c 1 X ( c 1 ) ⊗ · · · ⊗ L c n X ( c n ) is an L - e q uiv alence by assumption. It is also a coﬁbra tion, since, in any monoidal mo del categor y , the tenso r pro duct of tw o coﬁbrations with coﬁbra nt domains is a coﬁbration. Hence , the map Map( L c 1 X ( c 1 ) ⊗ · · · ⊗ L c n X ( c n ) , L c X ( c ))   Map( X ( c 1 ) ⊗ · · · ⊗ X ( c n ) , L c X ( c )) is a trivial ﬁbration. I ndee d, if c ∈ J then c i ∈ J for all i (since J is an ideal) and ther efore the ma p is the identit y; and if c 6∈ J , then it is a weak equiv a lence since L c X ( c ) = LX ( c ) is L -lo ca l, and it is a ﬁbr ation by Quillen’s axiom SM7. The P -alg ebra structure on L X is now obtained simila rly a s in Le mma 3 .2, a s follows. Consider the C -colour ed o pe r ad End P ( η X ), obtained as the following pullback of C -co lo ured collections: (6.2) End P ( η X ) ρ / / τ   End P ( L X )   End P ( X ) / / Hom P ( X , L X ) . The morphism τ is a trivial ﬁbration s ince it is a pullba ck o f a tr ivial ﬁbration, and the colour ed oper ad P is coﬁbra nt b y h ypothesis . Hence ther e is a lifting End P ( η X )   P / / : : End P ( X ) 20 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT where P − → E nd P ( X ) is the given P - a lgebra structure of X . Now, co mp o sing this lifting with the upp er mor phis m ρ in (6.2) gives a P -a lgebra struc tur e on L X such that η X is a map o f P -alg e br as, as claimed. F o r the uniqueness, suppo se that we hav e tw o P -a lgebra structure s on L X , which we denote b y γ , γ ′ : P − → End P ( L X ), and ass ume further that η X is a map of P - algebra s fo r each of them, meaning that γ and γ ′ factor thro ugh End P ( η X ). Thu s, let δ, δ ′ : P − → End P ( η X ) b e such that γ = ρ ◦ δ and γ ′ = ρ ◦ δ ′ , where ρ is the upp er mor phism in (6 .2). Since τ ◦ δ = τ ◦ δ ′ and τ is a trivial ﬁbration, it follows that δ and δ ′ are le ft homotopic. Since P is coﬁbra nt and End P ( L X ) is ﬁbrant, w e obtain that, in fact, γ ≃ γ ′ ; see [Hir0 3, 7.4.8].  This result also holds if the C - coloured op er ad P is non- symmetric. In this case, we can replace it by its symmetric version Σ P , since bo th yield the sa me class of algebras (see Remar k 1.1). Moreov er, Theorem 6.1 is als o true for top ologica l C -coloured op er ads without the ass umption that they admit a mo del structure (e.g., if one uses the Strøm mo del category s tructure on k - spaces). F or this, one has to a ssume that the C -coloured op erad P given in the statement of Theorem 6 .1 is “coﬁbr ant” in the sense that it has the left lifting pr op erty with resp ect to mo r phisms of C -coloured op era ds yielding trivial ﬁbra tions of spaces a t each tuple of colours . The assumption that P is coﬁbrant as a C -colo ur ed op era d is essential in the pro of of Theo rem 6.1. In or der to o btain a similar res ult for ar bitrary colo ur ed op erads, one needs that the mo noidal mo del c ategory M allo ws r e ctiﬁcation of algebras over resolutions of coloured o p e rads. According to [EM06], this holds when M is the categ ory of symmetric s pec tra. W e will use this fact in Section 7 to extend Theorem 6.1 in the cas e of sp e c tra. 6.1. A ∞ -spaces and E ∞ -spaces. Let us s p ec ia lize to the model catego ry o f sim- plicial sets acting on itself. Let P b e any C -co loured op era d in simplicial sets and choose a coﬁbr a nt resolution P ∞ − → P . If ( L, η ) is any homotopical lo ca lization, then the pro duct of any tw o L -equiv a lences is an L -equiv a lence by the argument used in (3.2). Hence, we ca n a pply Theo rem 6.1. Therefore, if X = ( X ( c )) c ∈ C is any P ∞ -algebra , then L X is again a P ∞ -algebra a nd η X : X − → L X is a map of P ∞ -algebra s. The same sta tements a re true in the categ ory of k -spa ces, although in this case we need suitable co ﬁbrancy ass umptions on the spaces X ( c ) for c ∈ C for the v alidit y of the ar gument, if the Quillen mo del structure is used. In particular , this r esult applies to the o p erads A s s a nd C om, yielding the fol- lowing result. Recall tha t an A ∞ -sp ac e is an algebra over an a rbitrary but ﬁxed coﬁbrant r esolution of A ss, a nd an E ∞ -sp ac e is a n algebra ov er a coﬁbrant r esolu- tion o f C om. Analogously , as deﬁned in Subsection 5.2, b y an A ∞ -map we mean an alg ebra ov er a co ﬁbrant resolution of M or A ss , and by an E ∞ -map we mean an algebra ov er a coﬁbrant r esolution of M o r C om . Corollary 6.2 . L et ( L, η ) b e a homotopic al lo c alization on the c ate gory of simplicial sets or k -sp ac es. If X is a c oﬁbr ant A ∞ -sp ac e, then LX has a homotop y u nique A ∞ -sp ac e structur e such that η X : X − → LX is a map of A ∞ -sp ac es. Mor e ove r, if g is an A ∞ -map b et we en c oﬁbr ant A ∞ -sp ac es, t hen L g is also an A ∞ -map. The same statements ar e true for E ∞ -sp ac es and E ∞ -maps.  As explained in the In tro duction, the following is a co nsequence of Corollar y 6.2, using the fact that every lo op space is an A ∞ -space, a nd, con v ersely , every A ∞ -space LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 21 X for whic h the monoid of co nnected comp onents π 0 ( X ) is a group is w eakly equiv alent to a lo o p space, na mely Ω B X , wher e B denotes the cla ssifying space functor; cf. [Sta63], [BV73, Theorem 1 .26], [May74 ]. Corollary 6 .3. I f ( L, η ) is a homotopic al lo c alization on the c ate gory of simplicial sets or k - s p ac es, and X is a lo op sp ac e, then LX is n atur al ly we akly e quivale nt to a lo op sp ac e and t he lo c alization map η X : X − → LX is natur al ly we akly e qu ivalent to a lo op map. Mor e over, if g : X − → Y is a lo op map b etwe en lo op sp ac es, then Lg is natur al ly we akly e quivalent to a lo op map. Pr o of. Let Q b e a coﬁbrant repla cement functor, so that QX − → X is a trivial ﬁbration and QX is coﬁbr ant. Her e X is an A ∞ -space and, b y homotopy inv ariance, QX is also an A ∞ -space (see [BV73, Theo rem 4.58 ], [BM03, Theorem 3 .5.b]). Therefore, by Cor ollary 6.2, LQX is a n A ∞ -space and η QX is a map of A ∞ -spaces. Since π 0 ( LQX ) ∼ = π 0 ( X ) is a group, we may a pply the classifying s pace functor, hence obtaining a co mmutative dia gram (6.3) X η X   QX ≃ o o η QX   ≃ / / Ω B Q X Ω B η QX   LX LQX ≃ o o ≃ / / Ω B L QX . T o prove the third claim, vie w g as an A ∞ -map. Then Qg is also an A ∞ -map, and, b y Cor ollary 6.2, LQg is an A ∞ -map, hence natur ally weakly equiv alent (as a functor on g ) to a lo op map b etw ee n lo op spa c es.  Essentially the same result was obtained in [Bou94, Theo rem 3.1] for nulliﬁca- tions a nd in [F ar96, Lemma A.3] for f -lo calizatio ns, using Segal’s theory o f loo p spaces. As we next s how, their delo oping of L f Ω c o incides, up to ho motopy , with the one given b y (6.3). Prop ositi on 6.4. L et f b e any map. Then L f Ω Y ≃ Ω L Σ f Y for al l Y . Pr o of. It follows from (6.3) that L f Ω Y ≃ Ω F Y , where F is a functor , namely F = B L f Q Ω. Note that there is a natural transformatio n ζ : B Q Ω − → F and there is a lso a natural is omorphism ξ : B Q Ω − → Id on the homo topy categ ory of connected spa c e s. It follows that, if λ = ζ ◦ ξ − 1 , then ( F, λ ) is a lo calization o n this category . On one hand, a connected space Y is F -lo cal if and only if Ω Y is L f -lo cal. On the o ther ha nd, Ω Y is L f -lo cal if and o nly if it is simplicially orthog onal to f , and this happ ens if and only if Y is simplicially or thogonal to Σ f . Hence, F and L Σ f are lo calizatio ns o n the same ca tegory with the sa me class of lo cal o b jects, from which it follows that there is a homo topy equiv alence F Y ≃ L Σ f Y under Y , for all connected spa ces Y . If Y is no t connected, then we take the basep o int co mpo nent Y 0 and hav e L f Ω Y = L f Ω Y 0 ≃ Ω L Σ f Y 0 = Ω( L Σ f Y ) 0 = Ω L Σ f Y , hence completing the pro of.  The preserv ation of inﬁnite lo op spaces and inﬁnite loo p maps under ho mo topical lo calizations follows either iterativ ely o r by rep eating the ab ove ar guments with E ∞ instead of A ∞ . 22 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT 6.2. A ∞ -structures and E ∞ -structures on sp e ctra. Now let M be the cate- gory of symmetric sp ectra over simplicial se ts. In order to ha ndle commutative ring s p ectr a and their mo dules conv enien tly , we endow M with the p ositive stable mo del structure, which w as discussed in [MMSS01], [Sch01], or [Shi04]. Thus, weak equiv alence s in M ar e the usual stable weak equiv alences (as deﬁned in [HSS00]), and p o sitive coﬁbr ations a re stable coﬁbr ations as in [HSS00] with the a dditional assumption that they are isomorphis ms in level zer o. Positive ﬁbr ations are deﬁned by the right lifting prop erty with resp ect to the trivia l p ositive co ﬁbr ations. By [Shi04, Pro po sition 3.1], the catego ry of symmetric spectr a ov er simplicial se ts with the p ositive sta ble mo del structure is a coﬁbrantly genera ted, pro p er, monoidal mo del categor y , a nd so is the categor y of R -mo dules for every ring sp ectrum R . A spectrum X is c a lled c onne ctive if it is ( − 1)-connected, i.e., π k ( X ) = 0 for k < 0. If map( − , − ) is any homotop y function complex in M , then π n map( X , Y ) ∼ = [Σ n X , Y ] ∼ = π n F ( X , Y ) for a ll sp ectra X , Y and n ≥ 0, where F ( − , − ) denotes the derived function sp ec- trum; cf. [Hov99, Lemma 6.1 .2]. In other words, the simplicial set map( X , Y ) has the same homoto py groups (with an y c hoice of a basep o int) as the connective cov e r F c ( X, Y ) o f the spectrum F ( X , Y ). Hence, if L is a homoto pic a l loca lization on M , then a map f : X − → Y is an L -equiv alence if and only if (6.4) F c ( f , Z ) : F c ( Y , Z ) − → F c ( X, Z ) is a weak equiv alence of sp ectra for every L -lo cal sp ectrum Z . As a cons e quence of this fact, the smash pr o duct of two L - e quivalenc es ne e d not b e an L -e quivalenc e , but it is so if any one of the suﬃcient co nditions s tated in Theorem 6.5 is satisﬁed; cf. [CG05]. W e s ay that the fun ctor L c ommutes with su sp ension if L Σ X ≃ Σ LX for all X . Note that, by (6.4), if f is a ny L - equiv alence , then so is Σ f . Therefor e, for every sp ectrum X , the map Σ η X : Σ X − → Σ LX is a n L -equiv alence. F or X coﬁbrant, this yields a ma p (in fact, an L -equiv alence) g X : Σ LX − → L Σ X , unique up to homotopy , such that g X ◦ Σ η X ≃ η Σ X . It is natural to s ay that L commut es with susp ensio n if g X is a weak equiv alence for all (coﬁbrant) X . How ev er, this is equiv a lent to the co ndition that Σ LX b e weakly equiv a lent to an L -lo cal spec tr um for all X , and hence to the conditio n that L Σ X ≃ Σ LX fo r all X . If L comm utes with s usp ension and Z is L -lo cal, then (a ﬁbrant replacement of ) Σ n Z is a lso L -lo cal, not only for n ≤ 0, but also for n > 0 . F rom this fact it follows that a map f is an L - equiv alenc e if and only if F ( f , Z ) is a weak equiv alence for every L -loca l spectrum Z . Hence, the condition that L c o mmut es with susp ension holds if and only if L is close d on Ho( M ) in the sens e of Section 3 ab ov e. Theorem 6.5. L et ( L, η ) b e a homotopic al lo c alization on symmetric sp e ctr a. L et f 1 : X 1 − → Y 1 and f 2 : X 2 − → Y 2 b e L -e quival enc es. Supp ose t hat any one of t he fol lowing c onditions is satisﬁe d: (i) L c ommutes with su s p ension. (ii) X 1 and Y 2 ar e c onne ctive. (iii) f 1 is a we ak e quival enc e b etwe en c onne ctive sp e ctr a. Then the derive d s m ash pr o duct f 1 ∧ f 2 : X 1 ∧ X 2 − → Y 1 ∧ Y 2 is an L -e quivalenc e. LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 23 Pr o of. If L commutes with suspensio n, then L is closed on Ho( M ) and therefore we may use the sa me argument as in (3.2). Now as sume that the sp ectra X 1 and Y 2 are connective. The follo wing argument is due to Bousﬁeld [Bou9 9]. One ﬁrst prov es that X 1 ∧ f 2 is an L - equiv ale nce as follows. If Z is any L -loca l spectrum, then F c ( X 1 ∧ f 2 , Z ) ≃ F c ( X 1 , F ( f 2 , Z )) ≃ F c ( X 1 , F c ( f 2 , Z )) since X 1 is connective. Here F c ( f 2 , Z ) is a weak equiv alence and this implies that F c ( X 1 ∧ f 2 , Z ) is a lso a weak e q uiv alence. Then the same metho d proves that f 1 ∧ Y 2 is a n L -equiv a lence. Finally , f 1 ∧ f 2 = ( f 1 ∧ Y 2 ) ◦ ( X 1 ∧ f 2 ), and the argument is complete. O f co urse, the same argument is v alid if, instead, X 2 and Y 1 are connective. Mo r eov er, if f 1 is a weak equiv alence, then we o nly need that X 1 be connective, since f 1 ∧ Y 2 is in this case a weak equiv alence, and similarly if the indices are exchanged.  W e emphasize that this apparently ad ho c r esult is crucial in the proo f o f Co r ol- lary 6.6, where connectivit y co nditions are imposed in the case when L do es not commute with susp ension. These co nnectivity conditions a re justiﬁed b y the result that we hav e just sho wn, a nd their necessity will b e demo nstrated with count erex- amples at the end of this s ection. Let us reca ll that an A ∞ -ring is an algebra over a coﬁbrant r esolution of A ss (whic h need therefore not be a strict ring, although it is weakly equiv alen t to one). An A ∞ -map of A ∞ -rings is a n alg ebra ov er a coﬁbrant resolutio n of M or A ss (whic h, as expla ined in Subsection 5.2, is a weaker notion than a morphism of A ∞ -rings). If ( R, M ) is an algebra over a coﬁbrant res olution of L Mo d A ss , then M is ca lled a left A ∞ -mo dule ov er R , as in Subsectio n 5.1. Accor dingly , an A ∞ -map of left A ∞ -mo dules is a n alg ebra over a coﬁbrant resolutio n of M or P where P = L Mo d A ss . The same terminolog y is used with E ∞ . Note that, if the v alue of a C -coloured o p erad P on a given tuple of c olours ( c 1 , . . . , c n ; c ) is the empt y set, and P ∞ − → P is a coﬁbrant resolution, then the v alue of P ∞ on ( c 1 , . . . , c n ; c ) is also the e mpty set, since P ∞ ( c 1 , . . . , c n ; c ) − → P ( c 1 , . . . , c n ; c ) is a weak e quiv alence. This ensures that, if J is an ideal r elative to P , then J is a lso an ideal relative to P ∞ . This fact is imp o r tant for the v alidity of the next r esult. Corollary 6.6. L et ( L, η ) b e a homotopic al lo c alization on symmetric sp e ctr a that c ommut es with susp ension. L et M b e a left A ∞ -mo dule over an A ∞ -ring R , and assume that b oth R and M ar e c oﬁbr ant as sp e ctr a. Then the fol lowing hold: (i) L R has a homotopy u nique A ∞ -ring structu r e such that η R : R − → LR is a morphism of A ∞ -rings. (ii) L M has a homotopy u nique left A ∞ -mo dule st ructur e over R such that η M : M − → LM is a m orphism of A ∞ -mo dules. (iii) L M admits a homotopy unique left A ∞ -mo dule structu r e over L R extending the A ∞ -mo dule structure ove r R . (iv) If f : R − → T is an A ∞ -map of c oﬁbr ant A ∞ -rings, then Lf admits a homotopy unique c omp atible A ∞ -map structu re . (v) If g : M − → N is an A ∞ -map of c oﬁbr ant left A ∞ -mo dules over R , then Lg admits a homotopy unique c omp atible struct ur e of an A ∞ -map of left A ∞ -mo dules over R , and also over LR . 24 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT Similar s tatements a re true for rig ht mo dules and bimo dules, and the same results hold for E ∞ -rings and their mo dules. If L do es not commute with susp ensio n, then the same statements hold b y as- suming that R and LR are c o nnective in (i); that R is connective in (ii); tha t R and LR ar e connective, and at least one of M and LM is connective in (iii); that R , T , L R , and LT ar e connective in (iv); that R is connective for the ﬁrs t claim in (v), and that R , LR , M o r LM , and N or LN are connective for the second claim in (v). Pr o of. In part (i), use a co ﬁbrant resolution of the op era d A ss. If L commu tes with susp ens io n, then the re s ult follows from Theo rem 6.1, since every ﬁnite smash pro duct o f L -equiv ale nces is an L -equiv ale nce. If L does not commute with sus- pens ion, then we need to prov e that η R ∧ · · · ∧ η R is a n L -equiv ale nc e for an y ﬁnite nu m be r of factor s . By Theo rem 6.5, this follows fro m the fact that R and LR are connective. In part (ii), use a coﬁbra nt resolution o f the no n-symmetric C -colour ed op- erad L Mo d A ss with C = { r, m } describ ed in Subse c tion 2.1, and choose the ideal J = { r } . Thus, ( R, M ) is an algebr a over this C -c oloured op erad. In order to prov e that R ∧ · · · ∧ R ∧ η M is an L -eq uiv alence (where R app ears an ar bitrary num b er of times, while η M app ears pre c isely once), we only need that R be connective. T o prov e (iii), use again a c o ﬁbrant resolution of L Mo d A ss with C = { r, m } , and choose the ideal J = ∅ . Her e we need that η R ∧ · · · ∧ η R be an L -equiv alenc e for any nu m be r of factors, which is the cas e if either L comm utes with susp ensio n or R and LR are connective, and we also need that η R ∧ · · · ∧ η R ∧ η M be a n L - equiv ale nce for any n um ber of factors, where η M app ears precise ly o nc e . This is the ca se if either L co mm utes with susp ensio n, o r R and LR and at leas t o ne of M a nd LM are connective. F o r par t (iv), use a co ﬁbrant resolution of the colour ed op erad M or A ss with J = ∅ . W e need tha t η R ∧ · · · ∧ η R ∧ η T ∧ · · · ∧ η T be an L -equiv a lence for any nu m be r of factors, which happens if either L commut es with suspensio n or R , LR , T , and L T a r e connective. Similarly , in part (v) use a coﬁbrant reso lution of the colour ed op erad M or Q with Q = L Mo d A ss . In order to infer that Lg is an A ∞ -map o f A ∞ -mo dules o ver R , choose the ideal J = { (0 , r ) , (1 , r ) } . In the ca se when L does not comm ute with susp ension, it is eno ugh to a ssume that R be co nnec tive. If we wish to infer that Lg is an A ∞ -map of A ∞ -mo dules ov er LR , then we hav e to choose J = ∅ , a nd, if L do es not co mmu te with susp ension, we need to add the as sumption that LR be connective and a t least one of M and LM be connective and furthermor e at least one of N and LN b e connective, by the same rea son as in par t (iii).  If L do es not commute with susp ension, then the co ndition that R b e connec tive cannot be dro pp ed in pa rt (i). Indeed, the n th Postniko v sectio n functor P n is a homotopical lo ca lization for all n , a nd, if R is nonconnective, then P − 1 R do es not admit a ring s pe c trum s tructure —not even up to homotopy— since the comp osite of the unit map ν : S − → P − 1 R with the multiplication map S ∧ P − 1 R − → P − 1 R ∧ P − 1 R − → P − 1 R has to b e a ho motopy equiv alence, yet ν is null since π 0 ( P − 1 R ) = 0. Similarly , in pa rt (ii), w e need that R b e connective, since otherwis e the Post- niko v sections of R need not b e homo topy mo dules ov er R . The following ex a mple LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 25 is a simpler version of [CG05, Ex ample 4 .4]. Let K ( n ) b e the Mo rav a K -theory sp ectrum for a ny pr ime p and n ≥ 1, and let i b e a n y integer. If P i K ( n ) were a homotopy mo dule s pe ctrum over K ( n ), then the compos ite of the unit map o f K ( n ) with the structure map of P i K ( n ) (6.5) S ∧ P i K ( n ) − → K ( n ) ∧ P i K ( n ) − → P i K ( n ) would b e a ho motopy equiv alence. Howev er, K ( n ) ∧ H Z /p ≃ 0 while P i K ( n ) has nonzero mo d p homo logy; see [Rud98, p. 545]. In part (iii), we nee d in addition that either M or LM b e connective; otherwise a counterexample can be dis play ed as follows. Let R b e the in tegral Eilenberg–Ma c Lane sp ectrum H Z a nd let L b e localiza tion with resp ect to the map f : S − → S Q , where S Q denotes a rational Mo ore spectr um, and the map f is induced by the inclusion Z ֒ → Q . Then LR ≃ H Q . How ev er, if M = Σ − 1 H Z , then M is L -lo cal, yet it is not an H Q -mo dule. Inciden tally , this example shows that the condition that either M or LM b e co nnective w as also necessary in [CG05, Theor em 4.5]. 7. Rectifica tion resul ts f or spectra Let M b e, as ab ov e, the category of symmetric sp ectr a ov er simplicia l sets with the p ositive stable mo del s tructure. According to [E M06, Theorem 1.3 ], for every set C and every C -coloured o per ad P in simplicial sets, there is a mo del structure on the category of P -algebr as in M in which a map of P -a lgebras X − → Y is a weak equiv alence (resp. a ﬁbr a tion) if and only if, for eac h c ∈ C , the map X ( c ) − → Y ( c ) is a weak equiv a lence (resp. a po sitive ﬁbration) of symmetr ic sp ectra. If P = A ss or P = C om, then the corre s po nding mo del structures coincide with the mo del structures used in categorie s of ring sp e c tra by other authors, e.g. in [Shi04]. If P is a C -colo ured op era d in s implicia l sets and ϕ : P ∞ − → P is a coﬁbrant resolution, then it follows from [EM0 6, Theorem 1.4] that the adjoint pair (7.1) ϕ ! : Alg P ∞ ( M ) ⇄ Alg P ( M ) : ϕ ∗ , where ϕ ∗ assigns to each P -alg ebra the P ∞ -algebra structure g iven b y comp osing with ϕ , and ϕ ! is its left a djoint , deﬁnes a Quillen eq uiv alence. Co ns equently , if X is a P ∞ -algebra , and Q is a coﬁbra nt replac e ment functor on the mo del categ ory of P ∞ -algebra s, while F is a ﬁbrant replacement functor on the mo del ca tegory o f P -alg ebras, then the unit map Q X − → ϕ ∗ F ϕ ! Q X is a weak equiv alence; see [Hov99, Corollar y 1 .3.16]. T hus, ϕ ! Q X is a functoria l r e ctiﬁc ation of X for each P ∞ -algebra X . Indeed, ϕ ! Q X is a P -alg ebra and its comp onent at c is w eakly equiv alen t to X ( c ) in M for a ll c ∈ C . Dually , if X is a P -alg ebra, then the co unit map ϕ ! Qϕ ∗ F X − → F X is a weak equiv alence of P - algebra s. Since weak equiv alences o f P - algebra s are deﬁned com- po nent wis e , ϕ ∗ preserves them. This implies that, if X is a P -a lgebra, then (7.2) ϕ ! Qϕ ∗ X ≃ ϕ ! Qϕ ∗ F X ≃ F X ≃ X as P -algebr as (thus, r ectifying a P ∞ -algebra which is in fac t a P -a lg ebra yields a weakly e q uiv alent P -algebr a). This will b e re le v ant in the rest of the article. W e lab el the following statemen ts for subsequent reference. 26 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT Lemma 7.1. L et X and Y b e P -algebr as in M , wher e P is a C - c olour e d op era d. L et ϕ : P ∞ − → P b e a c oﬁbr ant r esolution. If ϕ ∗ X and ϕ ∗ Y ar e we akly e quivalent as P ∞ -algebr as, then X and Y ar e we akly e quivalent as P -algebr as. Pr o of. Let Q be a coﬁbr a nt re pla cement functor on P ∞ -algebra s. Since ϕ ! pre- serves weak equiv alences b etw een c o ﬁbrant ob jects, w e hav e ϕ ! Qϕ ∗ X ≃ ϕ ! Qϕ ∗ Y as P - a lgebras , a nd it follows from (7.2) that X ≃ Y , a s claimed.  Lemma 7.2. If P is a c oﬁbr ant C -c olour e d op er ad in simplic ial sets and X is a c oﬁbr ant P -algebr a in M , then X ( c ) is c oﬁbr ant for al l c ∈ C . Pr o of. If C ha s only one colo ur, this follows from [BM03, Prop os itio n 4.3] and [BM03, Corolla r y 5.5 ]. The extension to several colours fo llows fro m the argument used in the pr o of of [BM0 7, Theor em 4.1].  In the catego r y of ar rows of M C we co nsider the model structur e whose weak equiv alence s and ﬁbrations are comp onent wise. Thus, tw o vertical ar rows f and f ′ are weakly equiv alent if there is a zig- zag of commutativ e squares X f   ≃ / / X 0 f 0   X 1 f 1   ≃ / / ≃ o o · · · X n f n   ≃ / / ≃ o o X ′ f ′   Y ≃ / / Y 0 Y 1 ≃ / / ≃ o o · · · Y n ≃ / / ≃ o o Y ′ whose horizo nt al arrows a re w eak equiv a lences at each colour. W e say tha t tw o functors F and F ′ from any given categ ory to a mo del cate- gory are natur al ly we akly e quiva lent if there is a zig-zag of natura l transformations betw een F and F ′ that a re weak equiv alences at e very o b ject. F or an ob ject X , w e will say that F X and F ′ X are natur ally weakly eq uiv alent if F and F ′ are clea r from the context and naturally weakly equiv ale nt . The following result is inferred from Theorem 6.1 and will yield the main results in this section a s sp e cial cases . T o g rasp its sig niﬁca nce, note that no coﬁbrancy assumption is made on the colo ured oper ad P . Theorem 7.3. L et ( L, η ) b e a homotopic al lo c alizatio n on the mo del c ate gory M of symmet ric sp e ctr a. L et P b e a C - c olour e d op er ad in simplicial s et s, wher e C is any set, and c onsider the extension of ( L, η ) over M C away fr om an ide al J ⊆ C r elative to P . L et X b e a P -algebr a such that X ( c ) is c oﬁbr ant for e ach c ∈ C , and let η X : X − → L X b e the lo c alization map. Supp ose that ( η X ) c 1 ∧ · · · ∧ ( η X ) c n is an L -e quivalenc e whenever P ( c 1 , . . . , c n ; c ) is n onempty. Then t her e is a map ξ X : D X − → T X of P -algebr as, dep ending functorial ly on X , su ch that: (i) X and D X ar e natu ra l ly we akly e qu ivalent as P -algebr as; (ii) L X and T X ar e n atu r ally we akly e quivalent as P ∞ -algebr as; (iii) η X and ξ X ar e n atur al ly we akly e quivalent as ( M or P ) ∞ -algebr as. Pr o of. Let ϕ : P ∞ − → P b e a coﬁbr ant r esolution of P , and let ( ϕ ! , ϕ ∗ ) be the cor re- sp onding Quillen equiv alence b etw e e n the ca tegories o f P ∞ -algebra s and P -alg ebras, as in (7.1). W e view X as a P ∞ -algebra via ϕ ∗ . Let Φ : ( M or P ) ∞ − → M or P be a coﬁbrant r esolution of M o r P , and let Φ ! : Alg ( M or P ) ∞ ( M ) ⇄ Alg M or P ( M ) : Φ ∗ be the cor resp onding adjoint pair. LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 27 Since P ∞ ( c 1 , . . . , c n ; c ) is nonempty prec is ely when P ( c 1 , . . . , c n ; c ) is nonempt y , it follows from Theor em 6.1 that L X a dmits a P ∞ -algebra structur e such that η X is a map of P ∞ -algebra s. Thus η X is an algebra ov er M or P ∞ , and it is als o an algebra ov er ( M or P ) ∞ using (5.5). If Q denotes a coﬁbra nt replacement functor on ( M or P ) ∞ -algebra s and F denotes a ﬁbrant replacement functor on M or P -algebra s, then there ar e w eak equiv alences of ( M or P ) ∞ -algebra s (7.3) η X Qη X ≃ o o ≃ / / Φ ∗ F Φ ! Qη X . Hence, η X is weakly equiv alent to Φ ∗ ξ X as a ( M or P ) ∞ -algebra , w her e ξ X = F Φ ! Qη X . Note that ξ X depe nds functorially on X . Hence , if we denote by D X the domain of ξ X and by T X its targ et, then D and T are endofunctors in the catego ry of P -alg ebras. F o r i ∈ { 0 , 1 } , let α i denote the inclusion o f C into { 0 , 1 } × C as α i ( c ) = ( i , c ), and choose a lifting λ i as in (5.3 ), (7.4) ( α i ) ∗ ( M or P ) ∞ ( α i ) ∗ Φ   P ∞ λ i 8 8 ϕ / / P. Now, if we apply ( λ i ) ∗ ( α i ) ∗ to (7.3), we obta in weak equiv a lences of P ∞ -algebra s. Let us choo s e ﬁrst i = 0. O n one hand, using (7.4), ( λ 0 ) ∗ ( α 0 ) ∗ Φ ∗ ξ X = ( λ 0 ) ∗ (( α 0 ) ∗ Φ) ∗ D X = ϕ ∗ D X . On the other ha nd, it follows from (5.6) that ( λ 0 ) ∗ ( α 0 ) ∗ η X = ϕ ∗ X . Therefore, Lemma 7.1 implies that X ≃ D X as P -alg e bras, and the ar gument given in the proo f o f Lemma 7.1 pr e serves na turality . Secondly , for i = 1 we obtain similarly weak equiv alences of P ∞ -algebra s L X ( λ 1 ) ∗ ( α 1 ) ∗ Qη X ≃ o o ≃ / / ϕ ∗ T X , as claimed.  R emark 7.4 . If the assumption that X ( c ) is co ﬁbrant for all c is not satisﬁed, then L X need not b e a P ∞ -algebra and η X need not be a n algebra ov er ( M or P ) ∞ . In fac t, Theorem 7.3 still holds, altho ug h w e can only deduce that η X and ξ X are naturally weakly equiv ale nt as a rrows in M C , and X ≃ D X a s P -alg ebras. T o pr ov e this, pic k a functoria l coﬁbrant repla cement of X as a P -algebr a, X ′ − → X . By Lemma 7.2, each X ′ ( c ) is then coﬁbra nt . Ther efore the arg ument pro ceeds for X ′ in the sa me way a s ab ov e, and we reach the conclusion that η X ′ is natura lly weakly equiv alent a s an algebr a over ( M or P ) ∞ to a ma p ξ X : D X − → T X of P -alg ebras, still dep ending functorially o n X , where D X is weakly equiv a lent to X ′ (and hence to X ) a s a P - a lgebra. Since η X and η X ′ are natura lly weakly equiv alent as arrows in M C , we ha v e completed the ar g ument. In summary , L is weakly equiv alent in M C to a functor T that sends P -algebr a s to P -a lgebras (where P is any coloure d op er a d, not nece s sarily c o ﬁbrant). How ev er, T is not coagumented, that is, there is no natura l map X − → T X in general. 28 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT Theorem 7.3 s p ec ia lizes to the following co nclusive results. First we state the preserv ation of (strict) ring sp ectra: Theorem 7. 5. L et ( L, η ) b e a homotop ic al lo c alization on symmetric sp e ctr a. If R is a ring sp e ctrum, then η R : R − → LR is natura l ly we akly e quivalent, as a map of sp e ctr a, to a ring morphism ξ R : D R − → T R , pr ovide d that L c ommutes with susp ension or R and L R ar e c onne ctive. Mor e over, D R ≃ R as ring sp e ctr a, and, if R is c ommutative, then D R and T R c an b e chosen to b e c ommutative. Pr o of. This follows fro m Theorem 7 .3 by choosing P = A ss and P = C om, with J empty in each case.  An analogous res ult ho lds for R -mo dules, as stated b elow. Here another subtlety arises since, at a ﬁrst attempt, lo calizing an R -mo dule will yield an R ′ -mo dule whe r e R ′ is weakly equiv alent to R , although they ar e in principle distinct. This diﬃculty is surmounted b y means of the following remar k s. If R is any ring spectr um, we endo w the category of left R -mo dules with the mo del structure of [HSS00 , Corolla ry 5.4.2]; that is, weak equiv a le nces are R -mo dule morphisms that are weak equiv alences of the underly ing spectr a, and ﬁbrations are R -mo dule morphisms that a re p ositive ﬁbra tions of the under lying sp ectra . This is coherent with the mo del s tructure that we ar e considering on the categor y of L Mod A ss -algebra s, b y asso ciating each R -mo dule M with the pair ( R , M ). If R is commutativ e, we also consider the analogous mo del structure on the category of R -alg ebras, as given by [HSS00, Cor ollary 5.4.3]. If ρ : R − → R ′ is a morphism of ring sp ectra, then res triction sends ev ery left R ′ -mo dule M to the left R -mo dule ρ ∗ M (which is the same sp ectrum M with the module structure g iven b y compo sition with ρ ), and induction sends every left R -mo dule N to the left R ′ -mo dule R ′ ∧ R N , where R acts on R ′ via ρ . It then follows tha t, if ρ is a weak equiv alence of r ing sp ectra , then the mo del ca tegories of left R -mo dules and left R ′ -mo dules ar e Quillen equiv alen t via induction and restriction, by [HSS00, Theorem 5.4.5 ]. More g enerally , the following holds: Lemma 7.6. L et R and R ′ b e we akly e quivalent ring sp e ctr a. Then every left R ′ -mo dule M is natura l ly we akly e quivalent as a sp e ctrum to the R -mo dule R ∧ QR Q ′′ ( F R ′ ∧ R ′ Q ′ M ) , wher e Q is a c oﬁbr ant re plac ement funct or and F is a ﬁbr ant r epla c ement functor on ring sp e ctr a, while Q ′ is a c oﬁbr ant r epl ac ement functor on left R ′ -mo dules and Q ′′ is a c oﬁbr ant r eplac ement functor on left F R ′ -mo dules. Pr o of. If R and R ′ are weakly equiv alent as ring sp ectra, there are ring morphisms R ← − QR − → F R ′ ← − R ′ that are weak equiv alences, s ince QR is co ﬁbrant and F R ′ is ﬁbrant. Using these morphisms, we may view F R ′ as a r ight R ′ -mo dule and R as a rig ht QR - mo dule. By [HSS00, L emma 5.4.4 ], sma shing with a co ﬁbr ant left mo dule co nv erts w eak equiv alence s of right modules in to weak equiv ale nce s of spectra . Hence, the zig- zag of weak equiv alences M ← − Q ′ M − → F R ′ ∧ R ′ Q ′ M ← − Q ′′ ( F R ′ ∧ R ′ Q ′ M ) − → R ∧ QR Q ′′ ( F R ′ ∧ R ′ Q ′ M ) prov es o ur claim.  LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 29 If R and R ′ are c o mmut ative, then inductio n and r estriction also yield a Quillen equiv alence betw een the mo del ca tegories of R - algebra s and R ′ -algebra s. Theorem 7.7. L et ( L, η ) b e a homotopic al lo c alization on symmetric sp e ctr a. L et R b e a ring sp e ctrum and M a left R -mo dule. Supp ose either that L c ommutes with sus p ension or t hat R is c onne ctive. Then η M : M − → LM is n atur al ly we akly e quivalent t o a morphism ξ M : D M − → T M of left R -mo dules wher e D M ≃ M as R -mo dules. Pr o of. Cho os e P = L Mod A ss and consider the P - algebra X = ( R, M ) —which depe nds functoria lly o n M — and the idea l J = { r } . W e may as s ume that X is ﬁbrant as a P -a lgebra (other w is e, use a ﬁbrant replacement and L emma 7.6). Now Theorem 7.3 implies that η X : X − → L X is naturally weakly equiv a lent to a map of P - algebra s ξ X : D X − → T X which depends funct orially on X (hence on M ) and where, in addition, D X ≃ X a s P -a lgebras. By co mpo sing ξ X , if necessar y , with a coﬁbrant replacement of D X as a P -alg e bra, we may assume tha t D X is coﬁbrant. Let us denote D X = ( R ′ , M ′ ) and T X = ( R ′′ , M ′′ ), and let µ : M ′ − → M ′′ be the morphism of R ′ -mo dules induced by ξ X on the second v ariable, which is w eakly equiv alent to η M : M − → LM as a ma p of sp ectra. Now a c hange of rings is required. Since D X is coﬁbrant and X is ﬁbrant, there is a weak eq uiv alence of P -alg ebras f : D X − → X . If w e consider the inclusio n α : { r } − → { r, m } , then R = α ∗ X a nd R ′ = α ∗ D X , a nd, since α ∗ P = A ss, we can infer that the res triction of f to the ﬁrst comp onent, ρ : R ′ − → R , is a weak equiv alence of rings. In this situation, by Lemma 7 .6, µ : M ′ − → M ′′ is naturally weakly equiv ale nt to a mo rphism of R -mo dules ξ M : D M − → T M , where D M ≃ R ∧ R ′ M ′ and T M ≃ R ∧ R ′ M ′′ . Hence, η M is naturally weakly equiv a lent to a morphism of R -mo dules, a s claimed. In or der to compar e D M with M , note that, s inc e f : D X − → X is a map o f P -alg ebras, its second comp onent can b e viewed as a morphism of R ′ -mo dules (7.5) M ′ − → ρ ∗ M , which is also a weak eq uiv alence of sp ectra , hence a w eak eq uiv alence of R ′ -mo dules. Here M is ﬁbr ant, and from the fact tha t D X is coﬁbr a nt it follows tha t M ′ is coﬁbrant as an R ′ -mo dule (since it has the left lifting prop erty with resp ect to all trivial ﬁbra tions of R ′ -mo dules). Since induction and res triction set up a Quillen equiv alence , the adjoint map of (7.5), R ∧ R ′ M ′ − → M , is a w eak equiv alence of R -mo dules. This sho ws that D M ≃ M as R -mo dules.  Although this result was stated fo r left R - mo dules, it also holds of c ourse for right R - mo dules or R - S -bimo dules , either by r e p ea ting the a rgument us ing the appropria te colo ured ope r ads, or by r e pla cing the ring sp ectr um R by R op and R ∧ S op , resp ectively . Theorem 7.8. L et ( L , η ) b e a homotopic al lo c alizatio n on symmetric sp e ctr a and f : R − → S a morphism of ring sp e ctr a. If either L c ommutes with su sp ension or R , LR , S , and LS ar e c onne ctive, then η f : f − → Lf is natur al ly we akly e quivalent to a map D f − → T f of ring morphisms, wher e D f ≃ f as su ch. Henc e, Lf is natur al ly we akly e qu ivalent to a morphism of ring sp e ctr a. Pr o of. This follows from Theore m 7 .3 by c hoosing P = M or A ss and J = ∅ .  30 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT Note that this res ult implies Theorem 7.5 b y sp ecializ ing f to b e the iden tit y map of a r ing sp ectrum R . Theorem 7.9. L et ( L , η ) b e a homotopic al lo c alizatio n on symmetric sp e ctr a and g : M − → N a morphi sm of left R -m o dules, wher e R is any ring sp e ctrum . If L c ommut es with susp ension o r R is c onne ctive, then η g : g − → Lg is n atur al ly we akly e quivalent to a map Dg − → T g of R -mo dule morphisms, wher e D g ≃ g as such. Henc e, Lg is n atur al ly we akly e quivalent to a morphism of R -mo dules. Pr o of. Pick P = M or Q with Q = L Mo d A ss and J = { (0 , r ) , (1 , r ) } . If we denote by X the P -algebr a ( R, M ) − → ( R, N ) that is the identit y on the ﬁr st v a riable and g on the second v a riable, then η X : X − → L X is a commutativ e dia gram ( R, M ) (id ,g )   (id ,η M ) / / ( R, LM ) (id ,Lg )   ( R, N ) (id ,η N ) / / ( R, LN ) . By Theor em 7.3, this is naturally weakly equiv alent to a map ξ X : D X − → T X o f P -alg ebras, whic h w e depict a s a commutativ e diagram ( R ′ , M ′ )   / / ( R ′′ , M ′′ ) ( ρ,ν )   ( T ′ , N ′ ) / / ( T ′′ , N ′′ ) . Here ν : M ′′ − → N ′′ is there fore a morphis m of R ′ -mo dules. F rom the fact that D X ≃ X as P -algebr as it follows, by r estriction o f colours a s in the pro of of Theorem 7.7, tha t R ′ ≃ R as r ings. Thus Lemma 7.6 implies tha t ν is naturally weakly e q uiv alent to a morphism T g of R -mo dules, a nd hence so is Lg .  8. A lgebras over commut a tive ring spectra W e ﬁnally discuss, as another application o f our techniques, the pres erv a tion of R -alg ebras under homotopical lo caliza tions, where R is a co mm utative ring sp ec- trum. F or this, we ﬁrst co ns ider a co nv enien t coloured op erad. In a n ar bitrary closed symmetric monoidal categor y E , choose C = { r , a } and deﬁne a C -coloured op erad A as follows: (8.1) A ( c 1 , . . . , c n ; c ) = ( 0 if c = r and c k = a for some k , I [Σ n ] / ∼ otherwise, where I [Σ n ] denotes , as b efor e, a co pro duct of copies of the unit of E indexed by the symmetric group Σ n , and ∼ is the equiv alence relatio n on Σ n deﬁned in the following way , s imilarly as in [EM06, § 9.3]: σ ∼ σ ′ if a nd o nly if, for a ll i and j such that c i = c j = a , the inequality σ ( i ) < σ ( j ) holds pre c isely when σ ′ ( i ) < σ ′ ( j ) holds. F or exa mple, A ( r , ( n ) . . ., r ; r ) = I and A ( a, ( n ) . . ., a ; a ) = I [Σ n ] . Thu s, an a lg ebra ov er A is a pair ( R, A ) w he r e R is a commutativ e monoid and A is a (non- commutativ e) mono id tog ether with a “c e nt ral” ma p R − → A given by the structure map A ( r ; a ) ⊗ R − → A . The ideals relative to A are C , { r } , and ∅ . LOCALIZA TION OF ALGEBRAS OVER COLOURED OPERADS 31 The comm utativ e algebra s A ov er a commutativ e monoid R a re the algebras over a C -co loured op er ad deﬁned as in (8.1), but replacing I [Σ n ] / ∼ with I . Note that the resulting co loured op erad precis e ly coincides with M o r C om after substituting { r, a } b y { 0 , 1 } . Indeed, a comm utativ e R -alg ebra A is nothing else but a morphism R − → A of co mm utative monoids . W e now choo s e E to b e the c a tegory of simplicial sets, acting on the ca teg ory M of symmetric sp ectr a ov er simplicial sets with the po sitive stable mo del str ucture. Then a n a lgebra ov er A in M is a pair ( R , A ) where R is a co mmutative ring sp ectrum and A is an R -algebr a in the us ual sense. Theorem 8.1. L et ( L, η ) b e a homotopic al lo c alization on symmetric sp e ctr a. L et R b e a c ommutative ring s p e ctrum and let A b e an R -algebr a. Supp ose either that L c ommutes with s u sp ens ion or R , A , and LA ar e c onne ctive. Then η A : A − → LA is natu r al ly we akly e quivalent to a morphism of R - algebr as ξ A : D A − → T A wher e D A ≃ A as R -algebr as. Pr o of. Use the colour ed o pe r ad A desc r ib ed a bove and apply Theorem 7.3 lo calizing aw a y from the ideal J = { r } and with a c hange of rings as a ﬁnal step. The details are the same as those in the pr o of of Theo rem 7.7.  Theorem 8.2. L et ( L, η ) b e a homotopic al lo c alization on symmetric sp e ctr a. L et R b e a c ommutative ring sp e ctru m and let g : A − → B b e a morphism of R -algebr as. Supp ose that L c ommutes with susp ension or R , A , LA , B , and LB ar e c onne ctive. Then η g : g − → Lg is natur al ly we akly e quivalent to a map D g − → T g of R - algebr a morphisms, wher e D g ≃ g as such. Henc e, Lg is natur al ly we akly e quiva lent to a morphism of R -algebr as. Pr o of. F or this, use M or A and lo calize aw ay fro m J = { (0 , r ) , (1 , r ) } .  If we let ( L, η ) b e lo calization with resp ect to a n a rbitrary homology theory , then w e essen tially recov er Theo rem VII I.2.1 in [E KMM97], stating that Bousﬁeld lo calizations pr eserve R -algebr as for ev ery commutativ e ring s p e c trum R . A minor complication comes from the fact that [E K MM97] is written in ter ms of S -mo dules instead of symmetric spectra . A comparison can b e made pre cise as follows. Let Ψ set up a Quillen equiv a lence from the ca tegory of R -alg ebras in symmetric spe ctra ov er simplicial sets to the categ ory of Ψ R -a lgebras in S - mo dules, and let Φ b e its right adjo int; see [Sch01] for further details ab out this adjoint pair. F o r an R -a lgebra A in symmetric sp ectra , we denote b y η A : A − → L E A its E ∗ -lo calizatio n, wher e E ∗ is any homo logy theory . By Theorem 8.1, η A is weakly equiv alent to a morphism ξ A : D E A − → T E A of R -a lgebras . Simila rly , denote by λ Ψ A : Ψ A − → (Ψ A ) E the E ∗ -lo calizatio n map in the ca tegory of S -mo dules , and endow (Ψ A ) E with the Ψ R -alge bra structure of [EKMM97, Theorem VI I I.2.1 ]. Prop ositi on 8.3. L et E ∗ b e any homolo gy the ory. L et R b e a c ommutative ring sp e ctrum and let A b e an R -algebr a in symmetric sp e ctr a. Then the Ψ R -algebr as Ψ( T E A ) and (Ψ A ) E ar e n atur al ly we akly e quivalent. Pr o of. The a djoint functors (Ψ , Φ) preserve and reﬂect E ∗ -equiv a lences and E ∗ -lo cal sp ectra. Therefore, Ψ( ξ A ) is a n E ∗ -equiv a lence and Ψ( T E A ) is E -lo ca l. Hence, we infer from [E KMM97, Theorem VII I.2 .1] that λ Ψ A yields a na tural map (Ψ A ) E − → Ψ( T E A ) of Ψ R -a lgebras which is a weak equiv alence.  32 C. CASA CUBER T A, J. J. GUTI ´ ERREZ, I. MOERDIJK, AND R. M. V OGT In a diﬀerent direction, we deduce that, fo r every c onne ctive commutativ e ring sp ectrum R , ea ch c onn e ctive R -algebr a A has a Postnik ov tow er consisting of R -alg ebras. Our ar gument is giv en b elow. This result was proved by Laz a rev in [Laz0 1, § 8] with diﬀerent metho ds, extending previous results of Basterra and Kriz [Bas99, Theor e m 8 .1]. See also [DS06]. Prop ositi on 8 .4. L et R b e a c onne ctive, c ommutative, c oﬁbr ant ring sp e ctrum, and let A b e a c onne ctive c oﬁbr ant R -algebr a. F or e ach i ≥ 0 ther e ar e R -algebr a morphisms A i +1 τ i   A ν i +1 = = { { { { { { { { ν i / / A i such that t he t riangle c ommutes up to homotopy, ν i induc es isomorphisms on π n for n ≤ i , and π n ( A i ) = 0 for n > i . Pr o of. F or each i ≥ 0, let P i denote lo calization with resp ect to f : Σ i +1 S − → 0 (where S denotes the sphere sp ectrum) in the category of s ymmetric sp ectra, and let η i be the cor resp onding coaug ment ation. F r om Theorem 8 .1 we infer that η i : A − → P i A is weakly eq uiv alent to a mor- phism of R -a lg ebras, which we denote by α i : A ′ i − → A ′′ i . Let A i be a ﬁbrant a nd coﬁbrant replacement of A ′′ i in the mo del catego ry o f R -algebr as. Th us A i ≃ P i A as sp ectra. Since A is weakly equiv a lent to A ′ i as an R -algebra , w e can cons ider the following co mp o site of ar rows in the homotopy category of R -alg ebras: (8.2) A ∼ = / / A ′ i α i / / A ′′ i ∼ = / / A i . Since A is co ﬁbrant, ther e is a morphism of R -alge br as ν i : A − → A i lifting (8.2). Thu s ν i induces isomo rphisms on π n for n ≤ i , since η i do es. Now, s ince η i is a natur al transfor mation, the following diagram comm utes: A η i   ν i +1 / / A i +1 η i   P i A P i ν i +1 / / P i A i +1 . In this diagram, the lower hor izontal arrow is a weak equiv alence of sp ectra , and, b y Theorem 8.2, it is weakly equiv alen t to a morphism β i : B ′ i − → B ′′ i of R - a lgebras , which is therefore a weak equiv a lence of R -alg ebras. Likewise, by Theorem 8.1, the rig ht-hand vertical map is weakly e quiv alent to a mo rphism γ i : C ′ i − → C ′′ i of R -alg ebras, where in addition C ′ i ≃ A i +1 as R -a lgebras . It follows from part (iii) of Theorem 7.3 that each of these rectiﬁcation steps is in fact a weak equiv alence of ( M or A ) ∞ -algebra s. (Here we ha ve used the a ssumption that R is coﬁbrant.) Th us, by res triction o f colour s, they induce weak equiv alences of A ∞ -algebra s o n their doma ins and targets. Hence, ( R, C ′′ i ) and ( R, B ′′ i ) ar e weakly equiv alent as A ∞ -algebra s. By Lemma 7.1, they a r e in fact weakly equiv- alent as A -a lgebras , meaning tha t C ′′ i ≃ B ′′ i as R -algebr a s. Similarly , B ′ i ≃ A ′′ i as R -alg ebras. 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Localization of algebras over coloured operads

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