An infinite loop space machine for infinity-operads

An infinite lo op space mac hine for ∞ -op erads Gijs Heuts Abstract This pap er describes a consequ ence of the more genera l results of [4] whic h is of indep end ent interes t. W e construct a functor from the cate- gory of dend roidal sets, which mo dels the theory of ∞ -op erads, into the category of E ∞ -spaces. Applying Ma y’s infi nite lo op space machine for E ∞ -spaces th en gives an infinite loop space machine for ∞ -op erads. W e sho w th at our m achine ex hibits the homotopy theory of E ∞ -spaces as a localization of the homotopy theory of ∞ -op erads. Con ten ts 1 Prerequisites 2 2 Left fibrations and the co v arian t mo del structure 5 3 The straight eni ng functor 7 4 The co v ariant mo del structure on dSets 9 In tro duction In this pa per we intro duce a construction which asso cia tes an E ∞ -space to any dendroidal set X . F rom this one ca n then get a n infinite lo op spa ce. This construction follo ws from the mo re genera l results of [4], which treats the theory of algebras ov er ∞ -o per ads v alued in either spaces or ∞ -categor ies. Since the topic of an infinite lo op space machine for ∞ -ope rads seems of independent int er e s t, this pa per is a summary o f the re sults r elating to this machine. Most pro ofs howev er a re not given here, but o nly in the mo re lengthy pap er [4]. The idea is as follows. W e will first fix a cofibr ant resolution o f the ter minal ob ject in dSets (which is s imply the nerve o f the commutativ e op erad), which we call E ∞ . The induced adjunction dSets /E ∞ / / dSets o o is a Q uillen equiv a lence [3]. W e then apply the so-called st ra ightening functor , which go es from dSets /E ∞ to the catego ry of E ∞ -spaces. One can then a pply May’s infinite lo op space machine [8] to o btain an infinite lo o p space. The 1 straightening functor we co nstruct g eneralizes Lurie’s straig ht ening functor for simplicial sets [7]. W e will describ e a mo del structure on dSe ts /E ∞ , which we call the c ovariant mo del structur e , for which the stra ig ht ening functor in fact induces a Quillen equiv ale nce b etw een dSets /E ∞ and the catego r y of E ∞ -spaces endow ed with the p r o jective mo del structure. W e then show that the cov ariant mo del structure is a lo c alization of the us ual Cisinski-Mo erdijk mo del structure on dSets /E ∞ constructed in [3 ]. The cov aria n t mo del structure can in fact b e regarded a s a gener alization o f the usual K an-Quillen mo del structur e o n simplicial sets to the ca tegory of dendroidal sets. The pla n of this pap er is as follows: • Section 1 con tains a brief discussio n of necessary prerequisites concern- ing dendroida l sets. This discussion is by no mea ns comprehe ns ive, but contains p ointers to references which a r e. • Section 2 will discuss left fibrations of dendroidal sets. F or any dendroida l set S we will define the cov aria n t mo del str ucture on the slice category dSets /S . • Section 3 discusses the construction of the s traightening functor, which sends dendroidal sets over S to a lgebras ov er the simplicial op erad hc τ d ( S ) asso ciated to S . In this pa per we will only need the sp ecial ca se S = E ∞ . • Section 4 discuss e s our infinite lo op space machine and contains further discussion of the cov ar iant mo del structur e. In par ticular, we show that the cov ariant mo del structure on dSets is a left Bousfield lo calizatio n of the Cisins ki-Mo erdijk mo del structure. Ac knowle dgemen ts I would like to tha nk Urs Sc hreib er and Dav e Carchedi for some illuminating conv ersatio ns. Many thanks are due to Iek e Mo erdijk, who introduced me to dendroidal sets. 1 Prerequisites Throughout this text the word op erad will alw ays mean a symmetric c olour e d op er ad . An op erad P in a given closed symmetric monoida l category E with tensor unit I can b e describ ed by a set of colo urs C a nd for any tuple of colour s ( c 1 , . . . , c n , c ) an o b ject P ( c 1 , . . . , c n ; c ) of E , which is to b e thought of as the ob ject of op era tions of P with inputs c 1 , . . . , c n and output c . F urthermore, for c ∈ C , we should hav e a u nit I − → P ( c ; c ) and we should hav e c omp ositions P ( c 1 , . . . , c n ; c ) ⊗ P ( d 1 1 , . . . , d j 1 1 ; c 1 ) ⊗ . . . ⊗ P ( d 1 n , . . . , d j n n ; c n ) − → P ( d 1 1 , . . . , d j n n ; c ) 2 Finally , p ermutations σ ∈ Σ n should act by transforma tions σ ∗ : P ( c 1 , . . . , c n ; c ) − → P ( c σ (1) , . . . , c σ ( n ) ; c ) All of these data a re required to satisfy v arious well-known as s o ciativity , unit and equiv ar iance axioms. The cas e of in teres t in this pap er is E = sSets , the category of simplicial sets, in which c ase we obtain the definition of a simplicial op er ad . W e will deno te the catego ry o f simplicial op erads by s O p er . An y symmetric monoidal simplicia l ca tegory C gives rise to a simplicial op erad C by C ( c 1 , . . . , c n ; c ) := C ( c 1 ⊗ · · · ⊗ c n , c ) Given a simplicial op era d P , an a lgebra over P in C is then simply a morphism of o per ads P − → C W e briefly r eview the bas ics of dendroidal s ets, o f which a n extensive treatment can b e found in [9] and [10], and fix our notation. The ca teg ory Ω is defined to b e the category of finite ro oted trees. These are trees equipp ed with a distinguished outer vertex called the output and a (p ossibly e mpty) set o f outer vertices not containing the output ca lled inputs . When drawing trees , w e will alwa ys omit out- and input vertices from the picture. Recall that each s uc h ro o ted tree T defines a Sets -op erad Ω( T ), the free op erad genera ted by T , which is colour ed by the edge s o f T . A mo rphism o f trees S − → T is defined to be a morphism of op erads Ω( S ) − → Ω( T ). The categor y of dendro idal sets is defined to b e the category of presheaves on Ω : dSets := Sets Ω op The dendroidal set r epresented by a tree T will b e denoted by Ω[ T ]. W e will de- note the set of T -dendrices of a dendr oidal set X by X T . There is an em b edding of the simplex categor y int o the category of finite ro oted tr ees, denoted i : ∆ − → Ω defined by sending [ n ] to the linea r tree L n with n + 1 edges and n inner vertices. By left Ka n e x tension this induces an a djunction i ! : sSets / / dSets : i ∗ o o As is the ca s e in the simplex category , any morphism in Ω may b e factorized int o fac e and de gener acy m aps . Relations betw een these maps extending the well-kno wn rela tio ns in the simplex catego ry are des crib ed in [9]. An inner fac e of T contracting an inner edge e will b e denoted ∂ e Ω[ T ], an outer fa ce chopping off a corolla with vertex v is denoted ∂ v Ω[ T ]. W e let F ( T ) denote the set o f faces of T . F o r an inner edg e we hav e the inner horn Λ e [ T ] := [ φ ∈ F ( T ) \ ∂ e Ω[ T ] φ and similar ly the outer horn Λ v [ T ] := [ φ ∈ F ( T ) \ ∂ v Ω[ T ] φ 3 W e will call a tree with one vertex and n leav es an n -c or ol la . The tree with no vertices and only a single edge will b e denoted by η . W e will often blur the distinction b etw een Ω[ η ] a nd η and wr ite η for the former , or η c if w e want to be explicit ab out the fact that the unique edge of Ω[ η ] has co lour c . Note that we have an isomorphism dSets /η ≃ sSe ts A dendro idal set X is said to b e an ∞ -op er ad if it has the extens ion prop erty with resp ect to a ll inner ho rn inclusions of tr ees. If X is an ∞ -op era d then the simplicial set i ∗ ( X ) is an ∞ -categ ory and a 1-c o rolla of X is called an e quivalenc e if the induced 1-simplex of i ∗ ( X ) is an equiv ale nce . The functor Ω − → Op e r Sets : T 7− → Ω( T ) defines, b y left K an extension, an adjunction τ d : dSets / / Op er Sets : N d o o The r ight a djo in t N d is c a lled the dendr oidal nerve . Recall that the categ ory Op er Sets carries a tensor pro duct ⊗ B V called the Bo ar dman-V o gt t ensor pr o d- uct . F or t wo representables Ω[ S ] , Ω[ T ] ∈ dSets their tensor pro duct is defined by Ω[ S ] ⊗ Ω[ T ] := N d (Ω( S ) ⊗ B V Ω( T )) W e extend this definition by colimits to all of dSets . By g eneral ar guments the functor − ⊗ X ha s a r ight adjoin t Hom dSets ( X, − ), making dSets into a closed symmetric mono idal categ o ry . The ca tegory of de ndr oidal sets is close ly related to the category s O p er of simplicial op era ds. The B o ardman-V ogt W -constr uction with res pect to the int er v al ∆ 1 ∈ s Sets (see [2] and [9] for a deta iled description) yields a functor Ω − → sOp er : T 7− → W (Ω( T )) By left Ka n e x tension this g ives an a djunction hc τ d : dSets / / sOp er : hc N d o o The right adjoint hc N d is called the homotopy c oher ent dendr oidal nerve . Let us describ e the s implicial oper ad W (Ω( T )) explicitly . Given colours c 1 , . . . , c n , c of T , we describ e the space of op erations W (Ω( T ))( c 1 , . . . , c n ; c ). Supp ose there exists a subtree S of T s uch that the leaves of S a re c 1 , . . . , c n and its ro ot is c . If it exists, such an S is unique. Define W (Ω( T ))( c 1 , . . . , c n ; c ) := (∆ 1 ) I ( S ) where I ( S ) de no tes the set o f inner edges of the tree S . If this set is e mpty the right-hand side is understo o d to b e the p oint ∆ 0 . If there ex ists no S matching the descr iption ab ov e, we let this space of oper ations b e e mpt y . Co mpos ition is defined by grafting trees, a ssigning length 1 to newly ar ising inner edges (i.e. the edg es along which the gr afting o ccurs). W e will without explicit mention us e ba sic fac ts from the theory o f mo del cat- egories, which can for example b e found in [5] and [6]. 4 A mono morphism f : X − → Y of dendroidal sets is s aid to b e normal if, for any tree T ∈ Ω and any α ∈ Y T which is no t in the image of f , the stabilizer Aut( T ) α is tr ivial. A dendroidal s et X is called normal if the unique map ∅ − → X is normal. The no rmal monomo rphisms are the weakly s aturated class generated by the b oundary inclusions of trees ∂ Ω[ T ] ⊆ Ω[ T ]. By Quillen’s small o b ject ar gument a n y map o f dendr oidal se ts may b e factore d as a no rmal monomorphism follow ed by a morphism having the right lifting pro pe r t y with resp ect to all nor mal monomor phis ms, which we r efer to as a trivial fibr ation . In particular, factoring a map ∅ − → X in this wa y , we obtain a norma l dendroidal set X ( n ) which we call a normalization of X . An e a sy to prov e and very useful fact is that any dendroidal set admitting a ma p to a normal dendroidal s e t is itself no r mal. A map f : X − → Y of dendro ida l sets is c a lled an op er adic e quivalenc e if there exists nor malizations X ( n ) and Y ( n ) and a map f ( n ) : X ( n ) − → Y ( n ) making the obvious diag ram commute such that for any ∞ -op erad Z the induced map i ∗ Hom dSets ( Y ( n ) , Z ) − → i ∗ Hom dSets ( X ( n ) , Z ) is a categor ical equiv a le nce o f simplicial sets, i.e. an equiv alence in the J oy al mo del structure. The fo llowing was established by Cisinski a nd Mo erdijk in [3]: Theorem 1.1 . Ther e exists a c ombinatorial mo del stru ctur e on the c ate gory of dendr oidal sets in which the c ofibr ations ar e the normal monomorphisms and the we ak e quivalenc es ar e the op er adic e quivalenc es. The fi br ant obje cts of this mo del struct ur e ar e pr e cisely the ∞ -op er ads. By slicing over η we obtain a mo del structur e on sSets which c oincides with the Joyal mo del st ructur e. W e will refer to this mo del structure as the Cisinski-Mo er dijk mo del structur e . 2 Left fibrations and the co v arian t mo del struc- ture In this sectio n we will define left fibr ations o f dendroidal sets, which are a gen- eralization of left fibrations o f simplicial sets. In the s ame way that left fibra- tions with co domain a fixed simplicial set S mo del functors from S into the ∞ -categor y of s paces, our left fibrations of dendro idal s ets with co domain a dendroidal set S mo del S -algebr as in the ∞ -categ o ry of spa ces. Definition 2.0.1. Let p : X − → S b e a map of dendro idal sets. Then p is a left fibr ation if the following co nditions are satisfied: • p is an inner fibr ation • F o r any c orolla σ of S having inputs { s 1 , . . . , s n } (note that the set of inputs co uld b e empt y) and colors { x 1 , . . . , x n } of X satisfying p ( x i ) = s i for 1 ≤ i ≤ n , there exists a co rolla ξ of X with inputs { x 1 , . . . , x n } such that p ( ξ ) = σ • F o r a ny tree T with at least tw o vertices and a n y lea f vertex v of T , there 5 exists a lift in any diagr am of the form Λ v [ T ] / /   X p   Ω[ T ] / / = = z z z z S Remark 2.0.1. 1. If p : X − → S is a left fibra tion, then the induced map i ∗ p : i ∗ X − → i ∗ S is a le ft fibratio n of simplicia l sets, i.e. it has the right lifting prop erty with resp ect to horn inclusio ns Λ n i − → ∆ n for 0 ≤ i < n . In particular, if i ∗ S is a Kan co mplex, then i ∗ X is a Kan co mplex as well by a fundamental result o f Joy al (see P rop osition 1.2 .5.1 of [7]). Also, if q : K − → L is a left fibration o f simplicial sets, then i ! ( q ) is a left fibration o f dendro idal sets. Our goal in this section is to describ e a mo del structure on the categ ory dSets /S in which the fibrant ob jects are precise ly the left fibra tions with co domain S . W e will first endow dSets / S with the str ucture of a s implicial catego ry . Definition 2.0.2. Given maps X − → S and Y − → S , we define the simplicial set Ma p S ( X, Y ) as fo llows: Map S ( X, Y ) n := dSets /S ( X ⊗ i ! (∆ n ) , Y ) where the map X ⊗ i ! (∆ n ) − → S is o btained by comp osing the pro jection map X ⊗ i ! (∆ n ) − → X with the ma p X − → S . Note that Map S ( X, Y ) satisfies the following universal pro per t y: for any sim- plicial set K there is a n isomor phism sSets ( K, Map S ( X, Y )) ≃ dSets /S ( X ⊗ i ! ( K ) , Y ) This is omorphism is natural in K . The mapping ob jects Map S ( X, Y ) make dSe ts /S in to a simpicia l categ ory . Also recall tha t dSets / S is tensored a nd cotens ored over sSe ts , a fact we already used in the definition a bove. Definition 2.0.3. W e will ca ll a ma p in dSets /S a c ovariant c ofibr ation if its underlying map of dendroida l sets is a cofibra tion. W e will call a map f : X − → Y in dSets / S a c ovariant e quivalenc e if for any left fibration Z − → S and norma lizations X ( n ) and Y ( n ) of X resp. Y the induced map Map S ( Y ( n ) , Z ) − → Map S ( X ( n ) , Z ) is a weak homotopy equiv alence of simplicia l se ts . Remark 2. 0.3.1. The condition that there ex ist norma liz ations X ( n ) and Y ( n ) of X resp. Y such that the induced map Map S ( Y ( n ) , Z ) − → Map S ( X ( n ) , Z ) is a weak homotopy equiv alence is equiv alent to the condition that for an y choice of normaliza tions X ( n ) and Y ( n ) the stated map is a weak homotopy equiv alence. 6 The following result can b e found in Section 6 .2 of [4]. Theorem 2. 1. Ther e exists a mo del structure on dSets /S in which the c ofibr a- tions (r esp. we ak e quivalenc es) ar e the c ovariant c ofibr ations ( re sp. c ovariant we ak e quivalenc es). This mo del struct u r e is c ombinatorial, left pr op er and sim- plicial . In this m o del structure the fibr ant obje cts ar e pr e cisely t he left fibr ations over S . W e will refer to this mo del structure as the c ovariant m o del s t ructur e . It turns out tha t the w eak equiv ale nces b etw een fibr ant o b jects of dSets /S are easily characterized: Prop ositio n 2.2. Su pp ose we ar e given a diagr am X p   @ @ @ @ @ @ @ f / / Y q          S such that p and q ar e left fibr ations. Then t he fol lowing ar e e quivalent: (i) The map f is a c ovariant e quivalenc e (ii) The map f is an op er adic e quivalenc e (iii) F or every c olor s ∈ S t he induc e d map of fib ers f s : X s − → Y s is a we ak homotopy e quivalenc e simplicial sets Remark 2.2.0. 2. The fib ers app earing in condition (iii) are auto matically Kan complexes. One se e s this b y noting that left fibratio ns are stable under pullback and inv oking Remark 2 .0.1.1. 3 The straigh tening functor In this sec tio n we will describ e the relation b et ween the categor y dSets /S a nd the category of algebras ov er the simplicial oper ad hc τ d ( S ) under the a ssumption that S is co fibrant in the Cis inski-Mo erdijk mo del structure, i.e. no rmal. The r esults of Berger a nd Mo erdijk [1][2] in pa rticular yield the following: Theorem 3. 1. If S is normal, so that hc τ d ( S ) is c ofibr ant, ther e exists a left pr op er s implicial mo del st ructur e on the simplicial c ate gory Alg hc τ d ( S ) ( sSets ) of simplicial hc τ d ( S ) -algebr as in which a map of algebr as is a we ak e quivalenc e (r esp. a fibr ation) if and only if it is a p ointwise we ak e quivalenc e (r esp. a p ointwise fibr ation). W e will now de fine the so- called str aightening functor S t S : dSets /S − → Alg hc τ d ( S ) ( sSets ) Note that w e can a lso describ e dSets /S a s a preshea f catego ry; indeed, we hav e dSets /S ≃ Sets ( R Ω S ) op 7 where R Ω S is the category of elements of S . F r om this w e conclude that dSets /S is g enerated under colimits by ob jects of the form Ω[ T ] − → S for T ∈ Ω . Since the straightening functor is supp ose d to b e a left adjoint, it will suffice to constr uct it o n these genera tors a nd then extend its definition by a left K an extension. First, consider the sp ecial ca se where S = Ω[ T ] a nd p is the identit y map of Ω[ T ]. F or a n y color c of T let T /c denote the subtre e of T which consists of c and ‘everything ab ove c ’. Example 3.1. 0 .1. If T is the tree • • • C C C C C { { { { { • • c C C C C C { { { { { then T /c is the tree • • • C C C C C { { { { { c Define the cub e ∆[ T / c ] := (∆ 1 ) col( T /c ) \{ c } where the pro duct on the right is understo o d to b e ∆ 0 if the set o cc ur ring in the exp onent is empty . The hc τ d (Ω[ T ])-a lgebra S t Ω[ T ] (id Ω[ T ] ) is given by S t Ω[ T ] (id Ω[ T ] )( c ) := ∆[ T /c ] The s tructure maps hc τ d (Ω[ T ])( c 1 , . . . , c n ; c ) × S t Ω[ T ] (id Ω[ T ] )( c 1 ) × · · · × S t Ω[ T ] (id Ω[ T ] )( c n )   S t Ω[ T ] (id Ω[ T ] )( c ) are given by gr a fting trees, a ssigning length 1 to the newly ar ising inner edges c 1 , . . . , c n . Now let S b e any dendroidal set a nd p : Ω[ T ] − → S a map. W e g et a map hc τ d ( p ) o f simplicial op erads, which induces an adjunction hc τ d ( p ) ! : Alg hc τ d (Ω[ T ]) ( sSets ) / / Alg hc τ d ( S ) ( sSets ) : hc τ d ( p ) ∗ o o W e set S t S ( p ) := hc τ d ( p ) ! ( S t Ω[ T ] (id Ω[ T ] )) 8 F unctoria lit y in p is given a s follows. Supp ose we are given ma ps Ω[ R ] f / / Ω[ T ] p / / S If f is a face ma p of T , the map S t S ( f ) is describ ed b y the inclusions ∆[ S/c ] ≃ ∆[ S/c ] × { 0 } col( T /f ( c ) ) \ f (col( S/c )) − → ∆[ T /c ] for co lo urs c of R . If f is a degener acy , it is clear how to define S t S ( f ). Having defined the functor S t S on all ma ps o f the for m Ω[ T ] − → S , we take a left Ka n extension of S t S to a ll of dSets /S to obtain a functor S t S : dSets /S − → Alg hc τ d ( S ) ( sSets ) : X 7− → lim − → Ω[ T ] → X S t S (Ω[ T ] → X ) Since S t S preserves colimits, the adjoint functor theor em provides us with a right adjoint to the stra ightening functor. W e call this rig ht adjoint the un- str aightening functor a nd denote it U n S . O ne of the main results of [4] is the following: Theorem 3.2. L et S b e a normal dendr oidal set. Then the adjunction S t S : dSets /S / / Alg hc τ d ( S ) ( sSets ) : U n S o o is a Quil len e qu ivalenc e. 4 The co v arian t mo del struc tur e on dSets In the s pecia l case that P is the terminal ob ject ∗ of dSe ts , which is isomor phic to N d ( C omm ), Theorem 2 .1 g ives us a simplicial mo del str uctur e on dSets itself. In this se ction we will denote the categ ory of dendr oidal sets equipped with this mo del s tructure by dSets cov in order to avoid confusion with the usual Cisinski-Mo erdijk model structure. The norma lization ν : E ∞ − → ∗ induces a Quillen equiv ale nc e ν ! : ( dSets /E ∞ ) cov / / dSets cov : ν ∗ o o The s traightening functor and Theorem 3.2 give us a Quillen equiv a lence S t E ∞ : ( dSets /E ∞ ) cov / / Alg hc τ d ( E ∞ ) ( sSets ) : U n E ∞ o o One can then compo se with May’s infinite lo o p space mac hine for E ∞ -spaces [8] to obtain a functor a ssigning an infinite lo op spa c e to a dendroida l set. Remark 4 . 0.0.3. I n [4] we showed how to define symmetric monoidal ∞ - categorie s as a certain type of fibration over E ∞ . Therefore our co nstruction in particula r provides a n infinite lo op space machine for symmetric monoidal ∞ -categor ies. Our g oal for the re ma inder of this section is to inv estiga te dSets cov a little more closely and show tha t it is in fact a lo calization of dSets equipped with the Cisins ki-Mo erdijk mo del structure. 9 Prop ositio n 4.1. The fibr ant obje cts of dSets cov ar e pr e cisely the dendr oidal sets X satisfying the fol lowing: (i) L et n ≥ 0 and let c 1 , . . . , c n denote the c olors of the le aves of C n . Then X has al l fil lers of the form ` n i =1 η c i / /   X Ω[ C n ] ; ; x x x x x (ii) If T is a tr e e with at le ast 2 vertic es and φ is any fac e of T ap art fr om a p ossible fac e chopping of t he r o ot, then any map Λ φ [ T ] − → X c an b e extende d to Ω[ T ] , i.e. X has al l horn fil lers of the form Λ φ [ T ] / /   X Ω[ T ] = = { { { { { (iii) In c ase the r o ot vertex v of T is unary, X has horn fil lers of the form Λ v [ T ] / /   X Ω[ T ] = = z z z z In p articular, if X is fibr ant the simplicial set i ∗ ( X ) is a Kan c omplex. Pr o of. B y definition, fibrancy is equiv alent to ( i ) and ( ii ), so we nee d to show that a fibrant o b ject of dSe ts cov satisfies ( iii ). Let α b e a unar y c orolla of X . F or any n ≥ 2 there exists a lift in any diag ram as follows: i ! (∆ 1 ) { 0 , 1 }   α " " D D D D D D D D D i ! (Λ n 0 )   / / X i ! (∆ n ) < < y y y y y By a fundamental lemma of Joyal (see P rop osition 1.2 .4.3 of [7]) we see that α is an equiv alence in X . But then Theo rem 4.2 of [3] tells us that X has pr op erty (iii) if the ro o t cor olla of T is mapped to α . Since α was arbitrar y , the result follows.  Definition 4.1.1 . W e call a dendroidal set X sa tisfying the conditions o f the previous pro po sition a dendr oidal Kan c omplex . 10 Remark 4. 1.1.1. If X is iso mo rphic to i ! ( K ) for a simplicial s e t K , then X is a dendroidal K an co mplex if and only if K is a Ka n complex in the usual sense. Kan complexes in the ca tegory of s implicial sets c a n be thought of as mo delling ∞ -group oids. Similarly , our previo us zigza g of Quillen equiv alences b etw een dSets cov and the categor y of E ∞ -spaces allows us to think of dendroidal Kan complexes a s symmetric monoidal ∞ -gro up oids. Remark 4.1. 1.2. There is a ca nonical iso morphism of categor ies dSets /η ≃ sSe ts Under this isomorphis m the mo del str ucture on dSets /η induced by the cov ar i- ant mo del str uc tur e co incides with the usual K an-Quillen mo del structure on sSets . Indeed, the co fibr ations are the mo nomorphisms and the fibrant ob jects are pr ecisely the Ka n complex es. Therefore the cov ar iant mo del structure on dSets can be regar ded a s a generaliza tion o f the Ka n-Quillen mo del s tructure to dendroida l sets. Prop ositio n 4.2. If dSets is e quipp e d with the Cisinksi-Mo er dijk mo del stru c- tur e, then the identity functor id : dSets − → dSets cov is a left Bousfield lo c alization. Pr o of. The tw o mo del structur e s under consider ation have the same co fibr a- tions, so it r emains to prove tha t a ny op era dic equiv alence is a cov ar iant equiv- alence. This will follow immediately fro m the fa c t that the Cisinski-Mo er dijk mo del structure has more fibrant ob jects than the cov ariant mo del s tr ucture. Indeed, let f : X − → Y b e an op era dic equiv alence and let X ( n ) and Y ( n ) be normalizatio ns of X and Y resp ectively . Let Z be any dendroida l Ka n complex. Then it is in pa rticular an ∞ -op era d, s o that the map Map( Y ( n ) , Z ) − → Map( X ( n ) , Z ) is a w eak homoto p y equiv alence by a s sumption. This prov es f is a cov aria nt equiv ale nce .  Remark 4.2. 0.3. In fact, the same argument will prove s omething stronge r ; if S is an ∞ -o per ad, the cov aria n t mo del structure on dSets /S is a left Bousfield lo calization of the model structure on this slice category induced by the Cisinski- Mo erdijk mo del structure. It is stra ightf or ward to describ e the left Bous field lo caliza tio n of Prop ositio n 4.2 as a lo calization with resp ect to a family of cofibrations with cofibr ant domain, which w e will do now. Definition 4. 2 .1. W e define the s et of gener ating left ano dynes to b e the set consisting of the following maps: • F o r any tree T with a t least tw o vertices and any lea f vertex v of T , the map Λ v [ T ] − → Ω[ T ] 11 • F o r any n ≥ 0, the map n a i =1 η c i − → Ω[ C n ] where { c 1 , . . . , c n } is the set o f leaves of C n The weakly satura ted clas s gener a ted by the g enerating left ano dynes is ca lled the class of left ano dyne morphisms, although we will not hav e to us e that class in this pap er. Prop ositio n 4.3. The left Bousfi eld lo c alization of the Cisinski-Mo er dijk mo del structur e with r esp e ct to the set of gener ating left ano dynes is the c ovariant mo del structur e. Pr o of. This follows easily from the fac t that the fibra n t ob jects in the cov ar iant mo del struc tur e are exactly the fibrants in the Cisinski-Mo er dijk mo del structure which are lo cal with r e spec t to generating left a no dynes.  12 References [1] C. Be rger and I. Mo erdijk. Axio matic homoto p y theo r y for op era ds. Com- ment. Math. Helv. , 7 8 (4), 200 3. [2] C. Berger a nd I. Mo erdijk. Resolutio n of coloured op e rads and rectification of homotopy algebr as. Contemp. Math. , (431 ):31–58 , 200 7. [3] D.-C. Cisinski and I. Mo erdijk. Dendroida l se ts as mo dels for homotopy op erads. 2009. arXiv:0 902.19 54. [4] G.S.K.S. Heuts. Algebra s ov er infinity-oper ads. 20 11. a rXiv:1110 .1776. [5] P . Hirschhorn. Mo del Cate gories and Their Lo c alizations . Number 99 in Mathematical Surveys and Monographs . American Mathematical So ciety , Providence, R.I., 200 3 . [6] M. Hovey . Mo del Cate gories . Amer ic an Mathematical So ciety , Providence, R.I., 1 998. [7] J. Lur ie. H igher T op os The ory . Pr inceton Universit y Press, 2 009. [8] J.P . May . The Ge ometry of Iter ate d L o op Sp ac es , volume 271 of L e ctu r e Notes in Math. Springer-V erlag , 197 2. [9] I. Mo er dijk and B. T o¨ en. Simplicial Metho ds for Op er ads and Algebr aic Ge ometry . Birkh¨ auser, 2 010. [10] I. Mo erdijk and I. W eiss. On inner Kan complexes in the category o f dendroidal sets. A dv. in Math. , 22 1 (2):343–3 89, 2009. 13