Dendroidal sets as models for homotopy operads

DENDR OID AL SETS AS MODELS F OR HOMOTOPY OPERADS DENIS-CHARLES CISINSKI AND IEKE MOERDIJK Abstract. The homotop y theory of ∞ -op erads is deﬁned by exten ding Joy a l’s homotop y theory of ∞ -categories to the category of dendroidal sets. W e prov e that the category of dendroidal sets is endo w ed with a mo del category s truct ure whose ﬁbrant ob jects are the ∞ -operads (i.e. dendroidal inner Kan complexes). This extends the theory of ∞ -categories in the sense tha t the Jo y al mo del category structure on simpli ci al sets whose ﬁbrant ob j ect s are the ∞ -categories is reco v ered from the model category structure on dendroidal set s by si mply slicing ov er the p oin t. Contents Int ro duction 1 1. Dendroidal sets 4 2. Statement of main results 6 3. Construction of an abstr act mo del catego ry for ∞ -ope r ads 9 4. The join op eration on trees 13 5. Subdivis ion of cylinders 21 6. ∞ -op erads as ﬁbrant ob jects 26 Appendices 34 A. Grafting order s onto trees 34 B. Another sub division of cylinders 38 Erra tum 43 References 44 Introduction The notion of dendroidal set is an extension of that of simplicial set, suitable for deﬁning and s tudying nerv es of (co lo ured) o perads in the same wa y as nerves of ca tegories feature in the theory of simplicia l sets. It w as in tro duced by one of the authors and I . W eiss in [MW07]. As explained in that pap er, the catego ry dSet of dendroidal sets ca rries a symmetric monoidal structure, which is closely related to the Bo ardman-V ogt tensor pro duct for op e rads [BV73]. There is also a co rresp onding internal Hom of dendroida l sets . The category of dendroidal sets 2000 Mathema tics Subje c t Classiﬁc ation. 55P48, 55U10, 55U40, 18D10, 18D50, 18G30. Key wor ds and phr ases. Inner Kan complex, operad, ∞ -oper ad, dendroidal set, ∞ -category , quasi-category , simplicial set. 1 2 D.-C. CISINSKI AND I. MOERDIJK extends the categ ory sSet of simplicial se ts , in the precise s ense that there a re adjoint functors (left adjoint on the left) i ! : sSet ⇄ dSet : i ∗ with go od prop erties. In particula r, the functor i ! is strong monoidal and fully faithful, and identiﬁes sSet with the slice category dSet /η , where η is the unit of the monoidal s tr ucture on dSet . (In fa c t, this adjunction is an o pen embedding of top oses.) Using these adjoint functors i ! and i ∗ , we can sa y more precisely ho w v ario us constructions and r esults from the theory o f simplicial sets extend to that of den- droidal se ts. F or example, the nerve functor N : Cat − → sSet a nd its left a djoin t, which we denote b y τ , naturally extend to a pair o f a djoin t functors τ d : dSet ⇄ Ope rad : N d which plays a cen tral role in our work. The goa l of this pa per is to lay the foundations for a ho motop y theor y of den- droidal se ts and “ ∞ -op erads” (or “op erads-up-to -homotopy”, or “quasi-op erads” ) which extends the simplicia l theory o f ∞ - categories (or quasi-catego ries) which has recently b een developed by Joyal, Lurie and others. Our main result is the exis tence of a Quillen closed mo del struc tur e on the catego r y of dendroidal sets, ha ving the following pro perties: 1. This Quillen mo del struc tur e on dSet is symmetric monoida l 1 (in the sense of [Hov99]) and left proper ; 2. The ﬁbrant ob jects of this mo del structure are precisely the ∞ -o p erads. 3. The induced mo del structure on the slice category dSet /η is pr ecisely the Joy al model structure o n simplicial sets [JT07, Lur06]. The existence of such a model structure was suggested in [MW07 ]. The ∞ - opera ds refered to in 2. ar e the dendroidal analog ues of the ∞ -categories forming the ﬁbrant ob jects in the Joy al mo del structure. They are the dendroidal sets s atisfying a lifting condition analogo us to the weak Kan condition of Boa rdman-V ogt, and were int ro duced in [MW07, MW09] under the name “(dendro idal) inner Kan complexes”. The dendro idal nerve of every op erad is s uc h an ∞ -op e r ad; conv ersely , intu itively sp eaking, ∞ -o perads are op erads in which the comp osition of op erations is only deﬁned up to homotopy , in a wa y which is ass ociative up to homoto p y . F or exa mple, the homo to p y co heren t nerve of a symmetric monoidal topolog ic al ca tegory is a n ∞ -op erad. The theory of ∞ -o perads contains the theory o f ∞ -categories, a s well as the theor y o f symmetric mono idal ∞ -categ ories and of oper ads in them. The theory of ∞ -op erads is also likely to b e of use in studying the no tion of ∞ -categor y enriched in a symmetric monoidal ∞ - category (e.g. th e v arious notions of A ∞ - categorie s, dg categor ies, weak n -categor ies). The pro of of o ur main theorem is based o n three sources: Firs t of a ll, we use the general metho ds of constructing model structures on presheaf categ ories developed in [Cis06] (we only use the ﬁrst chapter and Section 8.1 of that b oo k, which a re both elementary). Secondly , we use so me fundamental prop erties o f dendro ida l inner Kan complexes prov ed in [MW09]. And ﬁna lly , we use some impo rtan t notions and 1 This mo del catego r y structure i s not monoidal; see the Err atum at t he end of these notes, where we explain that this does not aﬀect the main r esults of this paper nor of its sequels. W e hav e c hosen not to mo dify the presen t articl e to keep it as close as possible to the published version. DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 3 results from Joyal’s s eminal pa p er [Joy02 ]: namely , the theory o f join op erations and the no tions of left o r right ﬁbration of s implicial sets. Apart from these so urces, our pro of is entirely self-contained. In particular, we do not use the Joyal mo del structure in our pro of, but ins tea d deduce th is model structure as a coro llary , as expressed in 3. ab o ve. It is known that there are several (Quillen) equiv alent models for ∞ -categor ie s: one is given by a left Bousﬁeld lo calisa tion of the Reedy mo del structure on simpli- cial s paces and has as its ﬁbrant ob jects Rezk ’s co mplete Sega l spaces; another is given by a Dwyer-Kan style mo del structure on top ological catego ries established by Bergner , in whic h all o b jects are ﬁbrant. The equiv a lence of these appr oaches is extensively discussed in Lurie’s b o ok [Lur06]; see also [Ber 07, JT07]. I t is nat- ural to ask whether analo gous mo dels exist for ∞ -o perads . In tw o subse q uen t pap ers [CM13a, CM13b], w e will show that this is indeed the c ase. W e will prov e there that the mo del structure on dendr oidal sets describ ed ab ov e is equiv alent to a mo del structure on topolo gical op erads in whic h a ll ob jects are ﬁbr an t, as w ell as to a mo del structure on dendroida l spaces whose ﬁbr an t ob jects a re “dendroidal complete Segal s paces”. The mo dels for ∞ -categories just mentioned as well as the equiv a lences b et ween them will again emer g e simply b y slicing ov er suitable unit ob jects of the resp ective monoidal structures. T ogether these mo del categor ies ﬁt int o a r o w of Quillen eq uiv ale nces s Operad ∼ / / dSet ∼ / / dSpaces s C at ∼ / / O O sSet ∼ / / O O sSpaces O O in whic h the v ertical arrows are (homo top y) full embeddings. This pap er is o rganized a s follows. In the ﬁrst section, we rec a ll the basics ab out dendroidal sets. In Section 2, w e state the main results of this paper : the existence of a mo del category struc tur e on the category o f dendroida l sets whos e ﬁbrant o b jects are the ∞ -op erads, as well as its main prop erties. In Se c tion 3, we construct this mo del structure thr ough rather formal arguments. At this stage, it is clear, b y constructio n, that the ﬁbrant ob jects a re ∞ -oper ads, but the conv er se is not obvious. Sections 4 and 5 pro vide the to ols to prove that any ∞ -op erad is ﬁbrant, following the arg umen ts which are known to hold in the case of simplicial sets for the theor y of ∞ -catego ries. More precisely , in Section 4, we develop a dendroidal analog of Joy al’s join op erations , and pr o ve a ge ner alization of a theorem of Joy a l which ensures a right lifting property for inner Kan ﬁbrations with resp e ct to certain non-inner ho rns, under an additional hypothesis o f weak in vertibility of some 1- c ells. In Section 5, we construct and ex a mine a sub division of c y linders of trees in terms of dendroidal horns. At last, in Sectio n 6, we prov e that any ∞ -op erad is ﬁbran t, and s tudy some o f the go o d prop erties of ﬁbratio ns b et ween ∞ -op erads. This is done by proving an intermediate r esult which is imp ortant by itself: a mo rphism of diagr a ms in an ∞ - opera d is weakly invertible if and only if it is lo cally (i.e. ob ject wise) weakly in vertible (this is where Sections 4 a nd 5 ha ve their roles to play). W e also added tw o appendices , whic h are indep endent of the res t of this paper . In Appendix A, we study the join o peratio ns on leav es (while in Section 4, w e studied join op erations on ro ots), and in App endix B, we study another sub division of 4 D.-C. CISINSKI AND I. MOERDIJK cylinders of trees. In fa c t, thes e appendices can b e used to provide a nother pro of of our main results: Section 6 migh t have b een written using Appendices A and B instead of Sections 4 a nd 5 res pectively , without any changes (except, sometimes, replacing the ev aluatio n by 1 b y the ev aluation by 0, whenev e r necessa ry). How ever, these appendices are not formal consequences of the rest ot these notes, a nd it will be useful to hav e this kind of results av aila ble for further w o rk o n the sub ject. 1. Dendroid al sets 1.1. Recall from [MW07] the category of trees Ω. The ob jects o f Ω are non-empty non-planar trees with a desig nated ro ot, and giv en tw o trees T a nd T ′ , a map from T to T ′ is a mor phism of the corre s ponding op erads whic h, in these notes, we will denote by T and T ′ again. Hence, by deﬁnition, the c ategory of trees is a full sub c ategory of the category of o pera ds. Recall that the category dSet of dendr oidal sets is deﬁned as the categ o ry of presheaves of s ets o n the category of trees Ω. Given a tree T , we denote b y Ω[ T ] the dendroidal set represented by T . Let 0 be the tree with only o ne edge, and set η = Ω[0 ]. Then the categ o ry Ω /η ident iﬁes ca nonically with the category ∆ of simplices, so that the catego r y dSet /η is canonically eq uiv alent to the category sSet o f simplicia l sets. The corr esponding functor (1.1.1) i : ∆ − → Ω , [ n ] 7− → i [ n ] = n is fully faithful and its image is a sieve in Ω. This functor i induces an adjunction (1.1.2) i ! : sSet ⇄ dSet : i ∗ (where i ! is the left Kan extension of i ). Under the identiﬁcation sSet = dSet /η , the functor i ! is simply the forgetful functor from dSet /η to dSet . The functor i ! is fully faithful and makes sS et into an open subtopos of dSet . In other w o rds, if there is a map of dendroidal sets X − → Y with Y a simplicial set, then X has to be a simplicial set as well. W e a lso recall the pairs of adjoint functor s (1.1.3) τ : sSet ⇄ Cat : N and τ d : dSet ⇄ Operad : N d where N and N d denote the nerve functor s from the categor y of categ ories to the category o f simplicial s ets and from the category of (symmetric coloured) oper ads to the category of dendroidal sets. The category of op erads is endow ed with a closed symmetric monoidal structure: the tensor pro duct is deﬁned as the Boardman-V ogt tenso r pro duct; see [MW07, Section 5]. This deﬁnes canonically a unique clo sed symmetric monoidal structure on the catego ry of dendroidal sets suc h that the functor τ d is symmetric mo no idal, and such that, for tw o tr ees T and S , we have Ω[ T ] ⊗ Ω[ S ] = N d ( T ⊗ B V S ) , where T ⊗ B V S is the Bo ardman-V o gt tensor pro duct of o perads. W e will denote int e rnal Hom ob jects by Hom ( A, X ) o r by X A . Note that the functor i ! : sSet − → dSet is a symmetr ic monoida l functor, if we consider sSet with its clos e d c a rtesian monoidal structure. The functor i ∗ turns the categor y of dendroidal sets into a simplicial categor y; given tw o dendroidal sets A and X , w e will write hom ( A, X ) for i ∗ ( Hom ( A, X )), the simplicial set of maps from A to X . DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 5 1.2. W e r ecall her e from [MW07] the diﬀerent kinds of faces of trees in Ω. Let T b e a tree. If e is a n inner edge of T , w e will denote b y T /e the tree obtained from T by contracting e . W e then hav e a canonical inclusion (1.2.1) ∂ e : T /e − → T . A map of type (1.2.1) is called an inner fac e of T . If v is a vertex of T , with the pro perty tha t all but one of the edg es inciden t to v are outer, we will denote b y T /v the tre e obta ined from T by removing the vertex v and all the outer edges incident to it. W e then ha ve a canonical inclusion (1.2.2) ∂ v : T /v − → T . A map of type (1.2.2) is called an outer fac e of T . A map of type (1.2.1) or (1 .2 .2) will be called a n el ementary fac e of T . W e de ﬁne ∂ Ω[ T ] as the union in dSet of all the imag es of elemen ta ry fac e maps Ω[ T /x ] − → Ω[ T ]. W e thus hav e, b y deﬁnition, an inclusion (1.2.3) ∂ Ω[ T ] − → Ω[ T ] . Maps of sha pe (1.2 .3) a re called b oundary inclusions . The image of a face map ∂ x will sometimes b e denoted b y ∂ x ( T ) for short. W e will c a ll fac es the maps of Ω which are obtained, up to an isomor phism, as comp ositions of elementary faces . It can b e chec ked tha t faces ar e exa ctly the monomorphisms in Ω; see [MW07, Lemma 3.1]. 1.3. A monomorphism of dendroidal sets X − → Y is normal if for any tree T , any non deg enerate dendrex y ∈ Y ( T ) which do es not b elong to the image of X ( T ) has a tr ivial stabilizer Aut( T ) y ⊂ Aut( T ). A dendro idal set X is normal if the map ∅ − → X is normal. F or instance, for any tree T , the dendroida l set Ω[ T ] is no rmal. Prop osition 1.4. The class of n ormal monomorphisms is stable by pushouts, tr ansﬁnite c omp ositions and r etra ct s . F urt hermor e, this is the smal lest class of maps in dSet which is clo se d under pushouts and tr anﬁnite c omp ositions, and which c ontains the b oundary inclusions ∂ Ω[ T ] − → Ω[ T ] , T ∈ Ω . Pr o of. This follows fr om [Cis06, Prop osition 8.1.35].  Prop osition 1.5. A monomorphi sm of dendr oidal sets X − → Y is normal if and only if for any tr e e T , the action of Aut( T ) on Y ( T ) − X ( T ) is fr e e. Pr o of. It is eas ily seen that the cla ss of monomorphisms whic h satisfy the abov e prop ert y is stable by pusho uts and transﬁnite comp ositions, and contains the bo undary inclusions ∂ Ω[ T ] − → Ω[ T ]. It thus follows from the preceding prop o- sition that any normal monomorphism has this prop erty . But it is also obvious that an y monomorphism with this pro perty is normal.  Corollary 1. 6 . A dendr oidal set X is normal if and only if for any tre e T , the action of the gr oup Aut( T ) on X ( T ) is fr e e. Corollary 1.7. Given any map of dendr oidal sets X − → Y , if Y is normal, then X is normal. Corollary 1. 8 . Any monomorphism i : A − → B with B n ormal is a normal monomorphism. 6 D.-C. CISINSKI AND I. MOERDIJK Prop osition 1.9. L et A − → B and X − → Y b e two n ormal monomorphi sms. The induc e d map A ⊗ Y ∐ A ⊗ X B ⊗ X − → B ⊗ Y is a normal monomorphism. Pr o of. As the class of normal mo nomorhisms is genera ted b y the b oundary inclu- sions, it is suﬃcient to check this prop ert y in th is case; see e.g. [Hov99 , Lemma 4.2.4]. Consider now t wo trees S and T . W e ha ve to show that the map ∂ Ω[ S ] ⊗ Ω[ T ] ∐ ∂ Ω[ S ] ⊗ ∂ Ω[ T ] Ω[ S ] ⊗ ∂ Ω[ T ] − → Ω[ S ] ⊗ Ω[ T ] is a normal monomor phism. But as Ω[ S ] ⊗ Ω[ T ] is the dendro idal nerve of the Boardman- V ogt tenso r pro duct of S and T , which is Σ- fr ee, it is a normal dendroidal set. Hence we are reduced to prov e that the a bov e map is a monomorphism. This latter prop erty is equiv alent to the fact that the commutativ e square ∂ Ω[ S ] ⊗ ∂ Ω[ T ] / /   ∂ Ω[ S ] ⊗ Ω[ T ]   Ω[ S ] ⊗ ∂ Ω[ T ] / / Ω[ S ] ⊗ Ω[ T ] is a pullback squar e in which any map is a mono mo rphism. As th e nerve functor preserves pullbacks, this reduces to the following prop ert y: for any elementary faces S/x − → S and T /y − → T the c o mm utative square S/x ⊗ B V T /y / /   S/x ⊗ B V T   S ⊗ B V T /y / / S ⊗ B V T is a pullback s quare of mono morphisms in the categ ory of op erads. T his is an elementary co ns equence of the deﬁnitions in volved.  1.10. Under the assum tions of Prop osition 1.9, w e s hall write A ⊗ Y ∪ B ⊗ X instead of A ⊗ Y ∐ A ⊗ X B ⊗ X . 2. St a tement of main resul ts In this section, we state the main results of this pap er. 2.1. Recall from [MW09 , Section 5] the notion of inner horn . Given an inner edge e in a tree T , we get an inclusion (2.1.1) Λ e [ T ] − → Ω[ T ] , where Λ e [ T ] is obta ine d as the union o f all the images of element ary fac e maps which are distinct from the face ∂ e : T /e − → T . The maps of shape (2.1.1) are called inner horn inclusions. A map of dendr oidal sets is called an inner ano dyne ex tension if it b elongs to the smalles t class of maps which is stable by pushouts, transﬁnite composition and retracts, and which con tains the inner hor n inc lus ions. A map of dendroida l sets is ca lled an inner Kan ﬁbr ation if it ha s the r igh t lifting prop ert y with resp ect to the class of inner ano dyne extensions (or, equiv ale ntly , to the set of inner horn inclusions). DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 7 A dendroidal se t X is an inner Kan c omplex if the map from X to the terminal dendroidal set is a n inner K an ﬁbration. W e will also call inner Kan complexes ∞ - op er ads . F or ex ample, for an y o perad P , the dendroidal set N d ( P ) is an ∞ - opera d; see [MW09 , Pr opos ition 5.3]. In particular, for any tree T , the dendroidal set Ω[ T ] is an ∞ -op erad. F or a simplicial set K , its image by i ! is an ∞ -op erad if and only if K is an ∞ -c ate gory (i.e. K is a quasi-categ o ry in the se nse of [J o y0 2 ]). A map of de ndr oidal sets will b e called a trivial ﬁbr ation if it has the right lifting prop ert y with respec t to normal monomorphisms. Note that the small o b ject argument implies that we can factor any map of dendroidal se ts into a no rmal monomorphism fo llo wed by a triv ial ﬁbration (resp. int o a n inner ano dyne extension follo wed b y an inner Kan ﬁbra tion). R emark 2.2 . A morphism betw een normal dendroidal sets is a trivial ﬁbration if and only if it has the rig ht lifting prop erty with respect t o monomo rphisms: this follows immediately from Corollaries 1.7 and 1.8. 2.3. Recall the na iv e mo de l structur e on the category o f op erads [W ei07]: the weak equiv a lences are the equiv alences of op erads, i.e. the maps f : P − → Q which are fully faithf ul and essent ia lly surjective: for any n + 1- uple of ob jects ( a 1 , . . . , a n , a ) in P , f induces a bijection P ( a 1 , . . . , a n ; a ) − → Q ( f ( a 1 ) , . . . , f ( a n ); f ( a )) , and any o b ject of Q is iso mo rphic to the image of some ob ject in P . The ﬁbr a tions are op er adic ﬁbr ations , i.e. the maps f : P − → Q such that, given any isomorphism β : b 0 − → b 1 in Q , and any ob ject a 1 in P such that f ( a 1 ) = b 1 , there exists a n isomorphism α : a 0 − → a 1 in P , such that f ( α ) = β . This mo de l str ucture is clo s ely related with the naive mo del structure on Cat (for w hich the weak equiv alences a re the equiv a le nces of categories). In fact, the latter can b e recovered from the one on op erads by slicing ov er the unit op erad (whic h is also the termina l ca tegory). The ﬁbrations of the naive mo del structur e on Cat will b e called the c ate goric al ﬁbr ations . Theorem 2.4. The c ate gory of d endr oidal set s is endowe d with a mo del c ate gory structur e for which the c oﬁbr ations ar e t he normal monomorphisms, t he ﬁbr ant obje cts ar e the ∞ -op er ads, and the ﬁbr ations b etwe en ﬁbr ant obje cts ar e the inner Kan ﬁbr ations b etwe en ∞ -op er ads whose image by τ d is an op er adic ﬁbr ation. The class of we ak e quivalenc es is the smal lest class of maps of dendr oidal sets W which satisﬁes the fol lowing t hr e e pr op erties. (a) ( ‘ 2 out 3 pr op erty’) In any c ommu tative triangle, if t wo maps ar e in W , then so is the thir d. (b) Any inner ano dyne exten sion is in W . (c) Any trivial ﬁbr ation b etwe en ∞ - op er ads is in W . Pr o of. This follows fr om Propo sition 3 .12, Theorem 6.10, and Corollary 6.11.  Corollary 2.5. The adjunction τ d : dSet ⇄ Operad : N d is a Quil len p air. Mor e- over, the two fu n ctors τ d and N d b oth pr eserve we ak e quivalenc es. In p articular, a morphism o f op er ads is an e quivalenc e of op er ads if and only if its dendr oidal nerve is a we ak e quivalenc e. Pr o of. See 6.17.  8 D.-C. CISINSKI AND I. MOERDIJK Prop osition 2. 6. The mo del c ate gory structu r e of The or em 2.4 has the fol lowing additiona l pr op erties: (a) it is left pr op er; (b) it is c oﬁbr ant ly gener ate d (it is even c ombinatorial); (c) it is symmet r ic monoidal. Pr o of. See Prop ositions 3.12 and 3.17.  Corollary 2.7. F or any normal dendr oidal set A and any ∞ -op er ad X , the set of maps [ A, X ] = Hom Ho ( dSet ) ( A, X ) is c anonic al ly identiﬁe d wi th the set of isomor- phism classes of obje cts in the c ate gory τ hom ( A, X ) . Pr o of. See Prop osition 6.20.  Corollary 2. 8. L et f : X − → Y b e a morphism of ∞ -op er ads. The fol lowing c onditions ar e e quivalent. (a) The map f : X − → Y is a we ak e qu iva lenc e. (b) F or any normal dendr oidal set A , the map τ d Hom ( A, X ) − → τ d Hom ( A, Y ) is an e quivalenc e of op er ads. (c) F or any normal dendr oidal set A , the m ap τ hom ( A, X ) − → τ hom ( A, Y ) is an e quivalenc e of c ate gories. Pr o of. Remem be r that, by deﬁnition (and any ∞ - opera d b eing ﬁbrant), the map f is a weak equiv alence if and only if, for a n y normal dendr o idal set A , the induced map [ A, X ] − → [ A, Y ] is bijective. This corollary is thus a direc t c o nsequence of Cor ollaries 2.5 and 2 .7 and of the fact the mo del category structure on dSet is mo no idal.  Corollary 2. 9. L et u : A − → B b e a morph ism of normal dendr oidal sets. The fol lowing c onditions ar e e quivalent. (a) The map u : A − → B is a we ak e quivalenc e. (b) F or any ∞ -op er ad X , the map τ d Hom ( B , X ) − → τ d Hom ( A, X ) is an e quivalenc e of op er ads. (c) F or any ∞ -op er ad X , the map τ hom ( B , X ) − → τ hom ( A, X ) is an e quivalenc e of c ate gories. Pr o of. The ﬁbra n t ob jects of dSet are ex actly the ∞ -op erads. Hence, the map u : A − → B is a weak equiv alence if and only if, for any ∞ -op erad X , the map [ B , X ] − → [ A, X ] is bijectiv e. W e co nclude the pr oof using the same arguments as in the proo f of Corollar y 2.8.  DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 9 Corollary 2.10 (Joyal) . The c ate gory of simplicia l sets is endowe d with a left pr op er, c oﬁbr antly gener ate d, symm et ric monoidal mo del c ate gory stru ctur e for which the c oﬁbr ations ar e the monomorphisms, the ﬁbr ant obje cts ar e the ∞ - c a- tegories, and t he ﬁbr ations b etwe en ﬁbr ant obje cts ar e the inner Kan ﬁbr ations b etwe en ∞ - c ate gories whose image by τ is a c ate goric al ﬁbr ation. Pr o of. The mo del category structure o n dSet induces a mode l category struc tur e on dSet /η ≃ sSet ; s e e also Rema rk 3.14 for B = η .  R emark 2.11 . Note that, the functor i ! : sSet − → dSet is fully faithful and symmet- ric monoida l. Mor eo ver, for any s implicial sets A and X , w e hav e hom ( i ! ( A ) , i ! ( X )) = X A . W e deduce from this that the induced map Hom Ho ( sSet ) ( A, X ) − → Hom Ho ( dSet ) ( i ! ( A ) , i ! ( X )) is bijective (where Ho ( sSet ) deno tes the homo top y catego ry of the Joy a l model structure, giv en b y Corollary 2.10). As a consequence, w e als o hav e for mally the simplicial analogs of Corolla ries 2.7, 2.8 a nd 2.9. 3. Constr uction of an a bstract model ca tegor y f or ∞ -operads This section is devoted to the construction of a mo del categor y structure on dSet . The construction is r elativ ely formal and uses v er y little of the theory o f dendroida l sets. By deﬁnition, we will have that any ﬁbra n t ob ject of this model ca tegory is an ∞ - opera d. The proo f o f the conv erse (any ∞ -o perad is ﬁbrant) is the ‘ra ison d’ˆ etre’ of the next sections. Prop osition 3. 1. L et A − → B and X − → Y b e a n inner ano dyne ext ension and a n ormal monomorphism r esp e ctively. The induc e d map A ⊗ Y ∪ B ⊗ X − → B ⊗ Y is an inner ano dyne extension. Pr o of. Using [Cis 0 6, Co rollary 1.1.8], w e see that it is suﬃcien t to chec k this pro p- erty when A − → B is an inner horn inclusion and when X − → Y is a b oundary inclusion. This prop osition th us fo llo ws from [MW09, Pr opos ition 9.2 ].  3.2. W e denote b y J the ner v e of the con tra ctible g roupo id with t wo ob jects 0 and 1 (i.e. J is the nerve of the fundament al group oid of ∆[1 ]). W e will write J d = i ! ( J ) for the corr e s ponding dendroidal set. A mo rphism o f dendroidal sets is a J - ano dyne exten s ion if it be lo ngs to the smallest class of maps which con tains the inner a nodyne extensions a nd the maps ∂ Ω[ T ] ⊗ J d ∪ Ω[ T ] ⊗ { e } − → Ω[ T ] ⊗ J d T ∈ Ω , e = 0 , 1 , and whic h is closed under pushouts, tra nsﬁnite c ompositions a nd retr acts. A morphism of dendroidal sets will b e called a J -ﬁbr ation if it has the rig h t lifting prop erty with resp ect to J - anodyne extensio ns. A dendro idal set X is J -ﬁbrant if the m ap f rom X to the terminal dendroida l set is a J -ﬁbr ation. Prop osition 3.3 . L et A − → B and X − → Y b e a J -ano dyne extension and a normal m onomorphism re sp e ctively. The induc e d map A ⊗ Y ∪ B ⊗ X − → B ⊗ Y is a J -ano dyne extension. 10 D.-C. CISINSKI AND I. MOERDIJK Pr o of. Using [Cis06, Co rollary 1.1.8], this follows formally from the deﬁnition a nd from Prop osition 3.1.  3.4. Let B b e a dendroidal set. Denote by An B the class of maps o f dSet /B whose image in dSet is J -ano dyne. F or each dendroidal set X over B , with structura l map a : X − → B , we deﬁne a cylinder of X ov er B X ∐ X ( ∂ 0 X ,∂ 1 X ) / / ( a,a ) % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ J d ⊗ X σ X / / a ′   X a { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ B (3.4.1) in which ∂ e X is the tensor pro duct of { e } − → J d with 1 X , while σ X is the tensor pro duct of J d − → η with 1 X , and a ′ is the compo s ition of 1 J d ⊗ a with the map σ B . These cylinders ov er B deﬁne the no tion J -homotopy over B (or ﬁb erwise J - homotopy ) betw een ma ps in dSet /B . Given tw o dendroidal sets A and X ov er B , we deﬁne [ A, X ] B as the quotien t of the s et Hom dSet /B ( A, X ) b y the equiv alence relation g e nerated b y the relation of J -homotopy ov er B . A morphism A − → A ′ of dendroidal sets ov er B is a B -e quivalenc e if, for any de ndr oidal set X o ver B such that the structural map X − → B is a J -ﬁbratio n, the map [ A ′ , X ] B − → [ A, X ] B is bijectiv e . In the case B is no rmal, any monomorphism over B is no rmal; see Corolla r ies 1.7 and 1 .8. W e see fro m Prop osition 3.3 and from [C is 06, Lemma 1.3.52] that the cla ss An B is a cla s s of ano dyne e x tensions with res p ect to the functor ial cylinder (3.4.1) in the sense of [Cis06, Deﬁnition 1 .3.10]. In other words, the functorial cylinder (3.4.1) and the cla s s An B form a homotopic al structur e on the catego ry dSet /B in the sense of [Cis06, Deﬁnition 1.3.14 ]. As a consequence, a dir ect application of [Cis06, Theorem 1.3.22, Pr opos ition 1.3.36 and Lemma 1.3 .52] leads to the following statement 2 . Prop osition 3. 5. F or any normal dendr oidal set B , the c ate gory dSet /B of den- dr oidal sets over B is endowe d with a left pr op er c oﬁbr antly gener ate d mo del c ate gory structur e fo r which the we ak e quivalenc es ar e the B -e quivalenc es, the c oﬁbr ations ar e the monomorp hisms, and the ﬁbr ant obj e ct s ar e the dendr oidal sets X over B such that the st ructur al map is a J -ﬁbr ation. Mor e over, a morphism b etwe en ﬁbr ant obje cts is a ﬁbr ation in dSet / B if and only if its image in dSet is a J -ﬁ br ation. R emark 3.6 . Any J -a nodyne extension ov er B is a trivial coﬁbration in the model structure of the preceding prop osition; see [Cis06, Pro p osition 1.3.31]. Lemma 3.7. L et p : X − → Y b e a trivial ﬁbr ation b etwe en n ormal dendr oidal sets. Any se ction s : Y − → X is a J - ano dyne extension. Pr o of. This is a particular case of [Cis0 6, Corollary 1.3.35] applied to the ho mo - topical structure deﬁned in 3.4 on dSet / Y .  2 The r esults of [Cis06] are stated for pr eshea v es cate gories, so that, strictly sp eaking, to apply them, we implicitely use the canonical equiv alence of categories b et ween dSet /B and the category of preshea ves on Ω /B . DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 11 3.8. W e ﬁx once and for all a normalization E ∞ of the terminal dendr o idal set: i.e., we choo se a normal dendroidal set E ∞ such that the map from E ∞ to the terminal dendroidal set is a trivia l ﬁbra tion. Lemma 3.9. F or any normal de ndr oidal set X , and any map a : X − → E ∞ , the map ( a, 1 X ) : X − → E ∞ × X is a J -ano dyne extension. Pr o of. This follows immediately fro m Lemma 3.7 because ( a, 1 X ) is a section of the pro jection X × E ∞ − → X , which is a trivial ﬁbration by deﬁnition of E ∞ .  Lemma 3.10. L et i : A − → B b e a m orphism of n ormal dendr oidal s et s, and p : X − → Y a morphism of dendr oidal sets. The map p has the right lifting pr op erty wi t h re sp e ct to i in dSet if and only if, for any m orph ism B − → E ∞ , the map 1 E ∞ × p has the right lifting pr op erty with r esp e ct t o i in dSet /E ∞ . Pr o of. Suppo se that 1 E ∞ × p has the righ t lifting prop erty with resp ect to i , and consider the lifting problem b elo w. A a / / i   X p   B b / / ℓ > > Y As B is normal, there exists a map β : B − → E ∞ . If we write α = β i , w e see immediately that the lif ting pr o blem above is now equiv alent to the lifting problem A ( α,a ) / / i   E ∞ × X 1 E ∞ × p   B ( β ,b ) / / ( β ,ℓ ) ; ; E ∞ × Y and this prov es the lemma.  3.11. Given a no r mal dendr oidal set A and a J -ﬁbrant dendroidal set X , we denote by [ A, X ] the quotient of Hom dSet ( A, X ) b y the equiv ale nc e relation generated by the J -homotopy relation (i.e., with the notations of 3.4, [ A, X ] = [ A, X ] e , where e denotes the terminal dendroidal set). Prop osition 3.12. The c ate gory of dendr oidal sets is endowe d with a left pr op er c oﬁbr antly gener ate d mo del c ate gory in which the c oﬁbr ations ar e the normal mono- morphisms, the ﬁbr ant obj e cts ar e the J -ﬁbr ant dendr oidal sets, and t he ﬁbr ations b etwe en ﬁ br ant obje ct s ar e the J - ﬁbr ations. F urthermor e, given a normal dendr oidal set A and a J -ﬁbr ant dendr oidal set X , we have a c anonic al identiﬁc ation [ A, X ] = Hom Ho ( dSet ) ( A, X ) . Pr o of. Prop osition 3.5 applied to B = E ∞ gives us a mo del ca teg ory structure on dSet /E ∞ . Consider the adjunction p ! : dSet /E ∞ ⇄ dSet : p ∗ , where p ∗ is the functor X 7− → E ∞ × X . It follows obviously from Lemma 3.9 that the functor p ∗ p ! is a left Quillen equiv a lence from the categor y dSet /E ∞ to itself. This implies immediately that the adjunction ( p ! , p ∗ ) sa tisﬁes all the necessa ry hypothesises to deﬁne a mo del structure o n dSet b y transfer; see e.g. [Cra95] 12 D.-C. CISINSKI AND I. MOERDIJK or [Cis06, Pr opos ition 1.4.23 ]. In other words, the categ ory of dendro idal sets is endowed with a co ﬁbran tly genera ted mo del ca tegory structure for which the weak equiv alences (resp. the ﬁbratio ns) a re the maps whose imag e by p ∗ is a weak equiv ale nc e (resp. a ﬁbration) in dSet /E ∞ . The desc r iption of coﬁbrations follows from Prop osition 1.4. W e know that the ﬁbra tions betw een ﬁbr a n t ob jects in dSet / E ∞ are the maps whos e image in dSet is a J -ﬁbration; see Pr opositio n 3.5. The description of ﬁbrant o b jects and o f ﬁbrations b et ween ﬁbrant ob jects in dSet a s J -ﬁbra n t o b jects and J -ﬁbrations is th us a direct conseq uence of Lemma 3.10. The identiﬁcation [ A, X ] = Hom Ho ( dSet ) ( A, X ) is obtained from the g eneral description of the set o f maps fr o m a coﬁbr a n t ob ject to a ﬁbrant ob ject in an abstract mo del catego ry . It r emains to prov e left pr opernes s: this follows fro m the left prop erness of the model category structure of Prop osition 3.5 for B = E ∞ (whic h is obvious, a s any ob ject over E ∞ is coﬁbrant), and from the fa ct that p ∗ preserves coﬁbrations as well as colimits, while it preserves and detects weak equiv a lences.  3.13. The weak equiv a lences of the mo del structure deﬁned in Prop osition 3.12 will be c a lled the we ak op er adic e quivalenc es . Given a dendr oidal set A , a normal ization of A is a trivial ﬁbr a tion A ′ − → A with A ′ normal. F or instance, the pro jection E ∞ × A − → A is a nor ma lization of A (as E ∞ is no r mal, it follows fro m Corolla ry 1.7 that E ∞ × A is no rmal). F or a morphism of dendroidal sets f : A − → B , the following conditions are equiv alent. (a) T he map f is a w ea k opera dic equiv alence. (b) F or a n y commutativ e squa r e A ′ / /   A   B ′ / / B in which the horizo n tal maps a re no rmalizations, and for a n y J -ﬁbrant dendroidal set X , the map [ B ′ , X ] − → [ A ′ , X ] is bijectiv e. (c) There exists a commutativ e square A ′ / /   A   B ′ / / B in which the horizontal maps ar e normaliza tions s uc h that, for a n y J -ﬁbra n t dendroidal set X , the map [ B ′ , X ] − → [ A ′ , X ] is bijectiv e. R emark 3.1 4 . Given a normal J -ﬁbrant dendro idal se t B , the mo del structure induced on dSet / B by t he mode l str uc tur e of Prop osition 3 .12 coincide with the mo del structure of Pro position 3.5 (this f ollows, fo r instance, from the fact these mo del struc tur es have the same coﬁbratio ns and ﬁbra tio ns b et ween ﬁbrant ob jects). R emark 3.15 . The mo del categ ory structure of Pr opos itio n 3.12 is coﬁbrantly gen- erated. The gener ating coﬁbrations are the inclusions of shap e ∂ Ω[ T ] − → Ω[ T ] for any tree T . W e don’t k now any explicit set o f generating trivial coﬁbr a tions. How ever, we know (from the proof of Pro position 3.1 2 ) that there exists a gener- ating set of trivial coﬁbr ations J for the mo del structure o n dSet / E ∞ , suc h that DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 13 p ! ( J ) is a g enerating set of tr ivial coﬁbr ations of dSet . In pa rticular, there exists a generating set o f trivial coﬁbratio ns of dSet which consists of trivial coﬁbrations betw een normal dendroidal sets. Statements a bout trivial co ﬁbrations will o ften b e reduced to statements about J -ano dyne extensions using the following a rgumen t. Prop osition 3. 16. The class of trivial c oﬁbr ations b et we en normal dendr oidal sets is the smal lest class C of monomorphisms b etwe en normal dendr oidal set s which c ontains J -ano dyne extensions, and such that, giv en any monomorphisms b etwe en normal dendr oidal sets A i / / B j / / C , if j and j i ar e in C , so is i . Pr o of. Let i : A − → B b e a monomorphism b et ween nor mal dendroidal sets. As B is normal, we ca n choo s e a map from B to E ∞ . W e can then c ho ose a c omm utative diagram ov er E ∞ A a / / i   A ′ i ′   B b / / B ′ in which a and b are J -ano dyne extens ions, A ′ and B ′ are ﬁbrant in dSet /E ∞ , and i ′ is a mo no morphism: this follows, for instance, from the fact that a n y J - ﬁbrant res o lution functor constructed with the small ob ject ar gumen t applied to the generating set of J -ano dyne extensions preserves monomor phisms; see [Cis06, Prop osition 1.2 .35]. Applying [Cis06, Co rollary 1.3 .35] to the model structure of Prop osition 3.5 for B = E ∞ , w e see that i is a trivial coﬁbration if a nd only if i ′ is a J -a nodyne extension. This proves the prop osition.  Prop osition 3.17. The mo del c ate gory structure on dSet is symmetric monoida l. Pr o of. As we a lready know that no rmal monomorphisms a re well b eha ved with resp ect to the tenso r pro duct (Prop osition 1.9) it just remains to prove that, g iv en a normal mo nomorphism i : A − → B and a trivial coﬁbration j : C − → D , the induced map A ⊗ D ∪ B ⊗ C − → B ⊗ D is a trivial coﬁbr ation. According to [Hov99, Lemma 4.2 .4], we can assume that i is a ge ne r ating coﬁbra tion, and j a genera ting trivial coﬁbration. In particular, w e can a ssume that i and j are monomorphisms b etw een normal dendr oidal sets; see Remark 3.14. It is thus suﬃcient to prove that, given a normal dendro idal set A , the functor X 7− → A ⊗ X preserves trivia l coﬁbr a tions b etw een nor mal dendro idal sets. By Prop osition 3.16, it is even suﬃcient to pr o ve that tensor pro duct by A preserves J -ano dyne extensions, which follows from Prop osition 3.3.  4. The join opera tio n on trees The aim of this s ection is to s tudy a dendroidal ana log of the join oper ations on simplicial sets introduce d b y Joy al in [Joy02]. W e shall pr o ve a generaliza tion of [Joy02 , Theo rem 2.2]; see Theorem 4.2. 14 D.-C. CISINSKI AND I. MOERDIJK 4.1. Let X b e a ∞ -op erad. A 1-simplex of X (i.e. a map ∆[1] − → i ∗ ( X )) will b e called we akly invertible if the corr esponding morphism in the ca teg ory τ ( i ∗ ( X )) is an isomorphism. Note that, for a n y ∞ -o perad X , the category τ ( i ∗ ( X )) is canonica lly iso morphic to the catego ry underlying the o p erad τ d ( X ): this comes from the explicit descrip- tion of τ ( i ∗ ( X ) g iv en by Boardman and V og t (see [Joy02, P r opos ition 1.2]) and from its dendr oidal g eneralization, which describ es τ d ( X ) explicitely; see [MW09, Prop osition 6.1 0 ]. As a co nsequence, weakly inv er tible 1-cells in X can b e de- scrib ed a s the maps i ! ∆[1] = Ω[1] − → X whic h induce inv ertible morphisms in the underlying categor y of the op erad τ d ( X ). Theorem 4.2 . L et T b e a t r e e with at le ast t wo vertic es as wel l as a u nary vertex r at the r o ot, and let p : X − → Y b e an inner Kan ﬁbr ation b etwe en ∞ -op er ads. Then any solid c ommutative squar e of the form Λ r [ T ] f / /   X p   Ω[ T ] g / / h = = Y in which f ( r ) is we akly invertible in X has a diagonal ﬁl ling h . 4.3. In order to prov e this theorem, we will in tro duce join op erations on forests. A for est is a ﬁnite set of tree s (i.e. o f ob jects of Ω). Given a for est T = ( T 1 , . . . , T k ), k > 0, we write T / dSet for the ca tegory of dendroidal sets under the copro duct Ω[ T ] = ∐ k i =1 Ω[ T i ]. The ob jects of T / dSet are thus of shap e ( X , x i ) = ( X, x 1 , . . . , x k ), where X is a dendroidal set, and x i ∈ X ( T i ), fo r 1 6 i 6 k . Morphisms ( X, x i ) − → ( Y , y i ) are maps f : X − → Y such that f ( x i ) = y i for all i , 1 6 i 6 k . Given an integer n > 0, we construct the tree T ⋆ n by joining the trees T 1 , · · · , T k together ov er a ne w v ertex v , and then g r afting the result onto i [ n ] (i.e. ont o [ n ] viewed a s a tree). • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 1 • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 2 ······ • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T k • a 2 a 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ a k ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ( T 1 , . . . , T k ) ⋆ n = • 0 v 1 • n . . . (4.3.1) DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 15 W e ins ist that the forest T mig h t b e empt y: for k = 0, we ha ve • • 0 v 1 ( ) ⋆ n = • n . . . (4.3.2) As each T i , 1 6 i 6 k , embeds canonically into T ⋆ n , we can view Ω[ T ⋆ n ] a s an ob ject of T / dSet . One chec k s that there is a unique functor ∆ − → Ω , [ n ] 7− → T ⋆ n such that the inclusions T i − → T ⋆ n are functorial in T i and such that the ca nonical inclusion i [ n ] − → T ⋆ n is functorial in [ n ]. This deﬁnes a functor (4.3.3) T ⋆ ( − ) : ∆ − → T / dSet . By Kan extension, we o btain a colimit preserving functor which extends (4.3.3): (4.3.4) T ⋆ ( − ) : sSet − → T / dSet . W e have T ⋆ ∆[ n ] = Ω[ T ⋆ n ]. The functor (4.3.4) has a right a djoin t (4.3.5) T \ ( − ) : T / dSet − → sSet . F o r a one tree forest T = ( T ), w e will simply write T ⋆ K = T ⋆ K and T \ X = T \ X for any simplicial set K and any dendroidal set X under Ω[ T ]. Under these conv entions, these op erations extend the join op erations in tr oduced by Joyal in [Joy02 ] in the sense that we have the fo llo wing formulas. i [ n ] ⋆ i ! ( K ) = i ! (∆[ n ] ⋆ K ) i [ n ] \ i ! ( L ) = i ! (∆[ n ] \ L ) Note that the inclus io ns Ω[ n ] − → T ⋆ ∆[ n ] in dSet induce a natural pro jection map (4.3.6) π X : T \ X − → i ∗ ( X ) for any dendroida l set X under T . R emark 4.4 . Note that a n y tree with at least one vertex T is obtained by jo ining a forest with an or dinal, i.e. as T = T ⋆ n for some fores t T and some int eger n > 0. A tree T has at least tw o v ertices and a unary v ertex a t the ro ot (as in the statement of The o rem 4.2) if a nd only if there exists a forest T such that T = T ⋆ 1. 4.5. In o rder to prove Theorem 4.2, we will have also to cons ider some sp eciﬁc maps of forests. F or this purpo se, w e intro duce the follo wing terminolog y . Let T b e a tree. A set A o f edges in T is is called admissible if, for any input edge e of T , and any v ertex v in T , if A con tains a path (branch) from e to v , then A contains all the edg es abov e v . If A is an admissible set of edges in T , we will deﬁne a fo r est ∂ A ( T ), and for each tree S in ∂ A ( T ), a face map S − → T in the category Ω. Roughly sp eaking, one deletes from T all edg e s in A , and deﬁnes ∂ A ( T ) as the resulting co nnected comp onen ts. A forma l deﬁnitio n is b y induction on the cardinality of A . (i) If A is empty , then ∂ A ( T ) = T . 16 D.-C. CISINSKI AND I. MOERDIJK (ii) If A contains the ro ot edge e of T , let T 1 , . . . , T k be the trees obtained from T by deleting e a nd the vertex immediately above it, let A i = T i ∩ A , and deﬁne ∂ A ( T ) as the union of the for ests ∂ A i ( T i ), 1 6 i 6 k . (iii) If A con ta ins an input edge a o f T , it must co n tain all the edges a bov e the vertex v just below a . Let T ( v ) be the tree o btained from T by pruning aw ay v and all the edges ab ov e it. Let A ( v ) = T ( v ) ∩ A , and deﬁne ∂ A ( T ) = ∂ A ( v ) ( T ( v ) ). (iv) If A co n tains an inner edge a of T , let T /a be the tree obtaine d by con- tracting a , and deﬁne ∂ A ( T ) to be ∂ A −{ a } ( T /a ). One can c heck that th e s teps (i)–(iv) can be perfor med in any o rder, so that the forest ∂ A ( T ) is well deﬁned. Each tree S in this for est ∂ A ( T ) is a fa ce of T , hence comes with a canonical map S − → T . Example 4 .6 . The tree • c ❈ ❈ ❈ ❈ d ④ ④ ④ ④ • T = • b ❂ ❂ ❂ ❂ ❂ ❂ e ✁ ✁ ✁ ✁ ✁ ✁ a v has t wo input e dg es c and d . The e dges b and c form a path from c down to v . So any admissible set A whic h contains b a nd c , for example, m ust cont ain d and e a s well. 4.7. This constructio n extends to for ests in the follo wing w ay . Let T = ( T 1 , . . . , T k ) be a forest. An admissible subset of e dges A in T is a k -tuple A = ( A 1 , . . . , A k ), where A i is an admissible set o f edges of T i for 1 6 i 6 k . W e can then deﬁne the forest ∂ A ( T ) as the union of the forests ∂ A i ( T i ). Given an y int e ger n > 0, we ha ve a canonical map (4.7.1) ∂ A ( T ) ⋆ n − → T ⋆ n which is characterized by the fact that, given any tree S in some ∂ A i ( T i ), for 1 6 i 6 k , the diagra m S ⋆ n   / / T i ⋆ n   ∂ A ( T ) ⋆ n / / T ⋆ n (4.7.2) commutes. The map (4.7.1) is a monomorphism of trees in Ω and is natur al in [ n ] (as an ob ject of ∆). Mor e generally , given an inclusion A ⊂ B betw een a dmissible subsets of edges in T , we have ca nonical monomorphisms of trees (4.7.3) ∂ B ( T ) ⋆ n − → ∂ A ( T ) ⋆ n (whic h is just a nother instance of (4 .7.1 ) for the forest ∂ A ( T ) with admissible sub- set of e dg es given by the sets B i ∩ ∂ A i ( T i )). The maps (4.7.3) deﬁne a co n trav a rian t functor from t he se t of a dmissible subsets of edges in T (partially or dered by in- clusion) to Ω. Given an inclusion A ⊂ B of admissible subsets of edges in T , there exists a unique morphism (4.7.4) Ω[ ∂ B ( T )] − → Ω[ ∂ A ( T )] DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 17 such that the following diagr am commutes for any simplicial set K . Ω[ ∂ B ( T )] / /   Ω[ ∂ A ( T )]   ∂ B ( T ) ⋆ K / / ∂ A ( T ) ⋆ K (4.7.5) By adjunction, we a lso ha ve natural morphisms (4.7.6) ∂ B ( T ) \ X − → ∂ A ( T ) \ X for all dendroidal sets X under Ω[ ∂ A ( T )]. Example 4.8 . If a is the ro ot of T , and if T is obtained b y gr afting trees T i with ro ot edges a i onto a corolla , then the map of type (4 .7.3) for A = ∅ and B = { a } is the map ∂ a given by contracting a : • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 1 • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 2 ······ • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T k • a 2 a 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ a k ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ • 0 1 • n . . . ∂ a / / • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 1 • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 2 ······ • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T k • a 2 a 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ a k ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ • a 0 • n . . . 4.9. W e will now study an elementary com binatorial situation which w e will ha ve to consider t wice to prov e Theore m 4.2: in the proof o f Prop osition 4.11 and in the pro of of 4.15.2. Consider a tree T . Assume that T = T ⋆ n fo r a non-e mpty fores t T = ( T 1 , . . . , T k ) and an ordinal [ n ], n > 0. Let i , 0 6 i < n , be an integer, a nd { A 1 , . . . , A s } , s > 1, a ﬁnite family of admissible subsets of edges in T . Deﬁne C ⊂ D ⊂ Ω[ T ] by C =  s [ r =1 ∂ A r ( T ) ⋆ Λ i [ n ]  ∪ Ω[ n ] and D = s [ r =1 ∂ A r ( T ) ⋆ ∆[ n ] , where Ω[ n ] is considere d as a sub complex of Ω[ T ] through the cano nical map. Lemma 4.10. Under the assumptions of 4.9 , t he map C − → D is an inner ano dyne extension. Pr o of. If T is the empty for est, w e must hav e s = 1 and A 1 = ∅ , so that D = Ω[ T ], and C = Λ i [ T ] is an inner horn. F rom now on, we will assume that T is non- empt y . Given a fore s t T ′ , the n umber of edges in T ′ is simply deﬁned as the sum of the nu mber of edges in eac h of the trees which o ccur in T ′ . F or each int eger p > 0, write F p for the set of faces F which b elong to D but not to C , and whic h ar e of shap e F = Ω[ ∂ A ( T ) ⋆ n ] for an admissible subset of e dg es A in T , such that ∂ A ( T ) has exactly p edges. Deﬁne a ﬁltratio n C = C 0 ⊂ C 1 ⊂ . . . ⊂ C p ⊂ . . . ⊂ D 18 D.-C. CISINSKI AND I. MOERDIJK by C p = C p − 1 ∪ [ F ∈ F p F , p > 1 . W e hav e D = C p for p big enough, and it is suﬃcient to prov e that the inclusions C p − 1 − → C p are inner ano dyne for p > 1. If F and F ′ are in F p , then F ∩ F ′ is in C p − 1 . Moreover, if F = Ω[ ∂ A ( T ) ⋆ n ] for an admissible subset o f edg es A , then w e hav e F ∩ C p − 1 = Λ i [ ∂ A ( T ) ⋆ n ] , which is an inner horn. Hence we can describ e the inclusion C p − 1 − → C p as a ﬁnite comp osition of pushouts b y inner horn inclusions o f shape F ∩ C p − 1 − → F for F ∈ F p .  Prop osition 4.11. Le t T = ( T 1 , . . . , T k ) b e a for est, and n > 1 , 0 6 i < n , b e inte gers. The inclusion ( T ⋆ Λ i [ n ]) ∪ Ω[ n ] − → T ⋆ ∆[ n ] is an inn er ano dyne extension. Pr o of. This is a particular case of the pr eceding le mma .  4.12. Remember from [Joy02] that a morphism of s implicia l sets is called a left (resp. right ) ﬁbr ation if it has the right lifting pro perty with res pect to inclusions of shap e Λ i [ n ] − → ∆[ n ] for n > 1 and 0 6 i < n (resp. 0 < i 6 n ). A mo rphism b et ween ∞ - categories X − → Y is c onservative if the induced func- tor τ ( X ) − → τ ( Y ) is conser v a tive (whic h can be refor m ulated by saying that a 1-simplex of X is w eakly invertible if and o nly if its imag e in Y is weakly in vert- ible). F or insta nce, by v ir tue of [Joy02 , Pr o position 2.7 ], a n y left (res p. right) ﬁbration b et ween ∞ -categories is conserv ative. Prop osition 4.13. L et p : X − → Y b e an inner Kan ﬁbr ation o f dendr oidal sets under a for est T . The map T \ X − → T \ Y × i ∗ ( Y ) i ∗ ( X ) is a left ﬁbr ation. In p articular, for any ∞ -op er ad X under a for est T , the map T \ X − → i ∗ ( X ) is a left ﬁbr ation. Pr o of. This follows immediately from Pr opos ition 4.1 1 by a standard adjunction argument.  Corollary 4. 14. F or any ∞ -op er ad X and any for est T over X , the simplicial set T \ X is an ∞ -c ate gory. Similarly, for any inner Kan ﬁbr ation b etwe en ∞ -op er ads X − → Y and any for est T over X , the simplicial set T \ Y × i ∗ ( Y ) i ∗ ( X ) is an ∞ -c ate gory. Pr o of. If X is an ∞ -op erad, then i ∗ ( X ) is clearly an ∞ - c ategory . Since the pro jec- tion T \ X − → i ∗ ( X ) is a left ﬁbration, this implies this corollar y .  As a warm up to pro ve Theorem 4.2, w e shall consider a particular case. Lemma 4.15. The or em 4.2 is true if T = ( ) ⋆ 1 (wher e ( ) denotes the empty for est) . DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 19 Pr o of. In this case, T is a tree of shap e 1 • r 0 • v and Λ r [ T ] is the union of the tw o faces 1 • v and 1 • r 0 In other w o rds, w e get Λ r [ T ] = ( ) ⋆ Λ 1 [1] ∪ Ω[1]. Th us , a lifting problem of shap e Λ r [ T ] f / /   X p   Ω[ T ] g / / h = = Y is equiv a len t to a lifting pro blem o f shape { 1 } ˜ f / /   ( ) \ X ϕ   ∆[1] ˜ g / / ˜ h 7 7 ( ) \ Y × i ∗ ( Y ) i ∗ ( X ) By v ir tue of Pr opositio n 4.13, the map ϕ is a left ﬁbration, a nd, as left ﬁbrations are stable b y pullback and by comp osition, so is the pro jection of ( ) \ Y × i ∗ ( Y ) i ∗ ( X ) to i ∗ ( X ). The image of ˜ g by the latter is nothing but f ( r ), and, as we know that left ﬁbrations b etw een ∞ -categ ories ar e conserv ative (see [Joy02, Pro position 2.7]), the 1- c ell ˜ g is quasi- in vertible in ( ) \ Y × i ∗ ( Y ) i ∗ ( X ). W e conclude the pro of using [Joy02 , P ropo sitions 2 .4 and 2.7].  Pr o of of The or em 4.2. Let T be a tree with at lea st tw o vertices and a unary vertex r at the ro ot. There is a forest T = ( T 1 , · · · , T k ), k > 0, such that T = T ⋆ 1. By virtue of Lemma 4 .1 5, we may assume that T is not the empty forest, or, equiv a len tly , that k > 1. W e will write T ′ = T ⋆ 0. The trees T and T ′ can be represented a s follo ws. T = • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 1 • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 2 ······ • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T k • a 2 a 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ a k ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ • 0 1 r T ′ = • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 1 • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T 2 ······ • ❈ ❈ ❈ ❈ ④ ④ ④ ④ T k • a 2 a 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ a k ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ 0 Given a dendro ida l set X , a map Λ r [ T ] − → X corr esponds to a c ompatible family of maps of simplicial sets { 1 } = ∆[0] − → ∂ A ( T ′ ) \ X , 20 D.-C. CISINSKI AND I. MOERDIJK indexed by the non-empty admiss ible subset of edges A in T ′ . This family corre- sp onds to a map ∆[0] − → lim ← − A ∂ A ( T ′ ) \ X . By separating the case A = { 0 } (the ro ot edge of T ′ ) from the others, the map Λ r [ T ] − → X corr esponds to a commut ative s quare of shape ∆[0] / / ∂ 0   T \ X   ∆[1] / / lim ← − ∂ ¯ B ( T ) \ X in which the limit lim ← − ∂ ¯ B ( T ) \ X is ov er the non-empt y admissible subsets of edges B in T ′ with 0 / ∈ B , and ¯ B = ( B ∩ T 1 , . . . , B ∩ T k ). Consider fro m no w on an inner Ka n ﬁbration b et ween ∞ -op erads p : X − → Y . Lifting problems of shap e Λ r [ T ] f / /   X p   Ω[ T ] g / / h = = Y corres p ond to lifting problems ∆[0] ˜ f / / ∂ 0   P ϕ   ∆[1] ˜ g / / ˜ h > > Q where P = T \ X a nd Q = U × W V , with U = lim ← − ∂ ¯ B ( T ) \ X , V = T \ Y , W = lim ← − ∂ ¯ B ( T ) \ Y . Exactly like in the pro of of Prop osition 4.15, it now suﬃces to prov e the following three statements: (i) the map ϕ : P − → Q is a left ﬁbra tion; (ii) the simplicial set Q is an ∞ -ca tegory; (iii) if f ( r ) is w eak ly in vertible in X , then the 1-cell ˜ g is weakly in vertible in i ∗ ( Q ). Note that, as left ﬁbrations a r e conser v ative and are stable by pullback a nd com- po sition, statemen ts (ii) and (iii) will follow fro m the following tw o ass ertions: (iv) the map V − → W is a left ﬁbration; (v) the map U − → i ∗ ( X ) is a left ﬁbra tion. But (iv) is a pa rticular ca s e of (i): just r eplace p b y the map from Y to the ter minal dendroidal set. It thus remains to prov e (i) and (v). 4.15.1. Pr o of of (i) . DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 21 F o r 0 6 i < n , a lifting problem of the form Λ i [ n ] / /   P ϕ   ∆[ n ] / / = = Q corres p ond to a lifting problem of the form Λ i [ T ⋆ n ] / /   X p   Ω[ T ⋆ n ] / / ; ; Y As Λ i [ T ⋆ n ] is an inner horn, (i) thus follows from the fact p is a n inner Kan ﬁbration. 4.15.2. Pr o of of (v) . F o r 0 6 i < n , a lifting problem of the form Λ i [ n ] / /   U   ∆[ n ] / / ; ; i ∗ ( X ) corres p onds to a lifting problem of the form C / /   X D > > where the inclusio n C − → D can be describ ed as follows. The dendroida l set D is the union of all the faces ∂ x ( T ⋆ n ) g iv en b y cont racting a n inner edge or a ro ot edge in one of the trees T i , or by deleting a top v ertex in the tree T ⋆ n . The dendr oidal set C is the union of the image o f Ω[ n ] − → Ω[ T ⋆ n ] a nd all the ‘co dimension 2’ faces of Ω[ T ⋆ n ] of shap e ∂ j ∂ x ( T ⋆ n ), wher e ∂ x is as ab o ve, a nd 0 6 j 6 n is distinct from i . It is now suﬃcient to chec k that the inclusion C − → D is an inner anody ne extension, which follows from a straightforward application of Lemma 4.1 0 .  5. Subdivision of cylind ers 5.1. Let S b e a tree with a t leas t one vertex, and cons ider the tenso r pro duct Ω[ S ] ⊗ ∆[1]. It has a sub ob ject A 0 = ∂ Ω[ S ] ⊗ ∆[1] ∪ Ω[ S ] ⊗ { 1 } where { 1 } − → ∆[1 ] is ∂ 0 : ∆[0] − → ∆[1]. In this section, w e will prov e the following result. Theorem 5.2 . Ther e exists a ﬁ lt r ation of Ω [ S ] ⊗ ∆[1] of the form A 0 ⊂ A 1 ⊂ · · · ⊂ A N − 1 ⊂ A N = Ω[ S ] ⊗ ∆[1] , wher e: 22 D.-C. CISINSKI AND I. MOERDIJK (i) the inclu s io n A i − → A i +1 is inner ano dyne for 0 6 i < N − 1 ; (ii) the inclu s io n A N − 1 − → A N ﬁts into a pushout of the form Λ r [ T ] / /   A N − 1   Ω[ T ] / / A N for a t r e e T with at le ast two vertic es and a unary vertex r at t he r o ot; (iii) the m ap ∆[1] − → Λ r [ T ] − → A N − 1 ⊂ Ω[ S ] ⊗ ∆[1] c orr esp onding to the vertex r in (ii) c oincides with the inclusion { e S } ⊗ ∆[1 ] − → Ω[ S ] ⊗ ∆[1] wher e e S is the e dge at the r o ot of the tr e e S . 5.3. The pro of of Theorem 5 .2 is in fact v er y similar to that of [MW09, Prop osition 9.2], stated here as Prop osition 3 .1. W e recall from lo c. cit. that, for an y tw o trees S a nd T , one can write Ω[ S ] ⊗ Ω[ T ] = N [ i =1 Ω[ T i ] , where Ω[ T i ] − → Ω[ S ] ⊗ Ω[ T ] are ‘percolatio n sc he mes ’. Drawing vertices o f S as white, and those of T as blac k, these per colation schemes a re partia lly ordere d in a natural wa y , starting with the tree obtained by stacking a c o p y of the black tree T o n top of each input edge of the white tree S , and e nding with the tree obtained by stacking copies of S on top of T . The intermediate tre e s are o btained by letting the blac k vertices of T p ercolate thr ough the white tree S , b y successive ‘mov es’ of the form • ❈ ❈ ❈ ❈ ④ ④ ④ ④ ··· ··· • ❈ ❈ ❈ ❈ ④ ④ ④ ④ ··· ④ ④ ④ ④ ◦ s n ⊗ w ④ ④ ④ ④ ④ s 1 ⊗ w ❈ ❈ ❈ ❈ ❈ v ⊗ t = ⇒ ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ ··· ··· ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ ··· ··· ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ ··· • v ⊗ t 1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ v ⊗ t n ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ s ⊗ w with ◦ s n ④ ④ ④ ④ s 1 ❈ ❈ ❈ ❈ ··· s in S , and • t m ④ ④ ④ ④ t 1 ❈ ❈ ❈ ❈ ··· t in T . In the spec ia l case wher e T = [1], the ﬁltration referr ed to in Theorem 5.2 is given by A i = A 0 ∪ Ω[ T 1 ] ∪ · · · ∪ Ω[ T i ] , where T 1 , . . . , T N is any linear o rder o n the p ercolation schemes extending the natural partial order. R emark 5.4 . F o r any tr ee S with at least one v e r tex, a nd ro ot edge named e S ( e for ‘exit’), the last tre e T N in the partial or der o f perco la tion sc hemes for Ω[ S ] ⊗ ∆[1] DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 23 is of shap e ◦ ④ ④ ④ ④ ❈ ❈ ❈ ❈ S T N = • r ( e S , 0) ( e S , 1) It alwa ys has a unique predecesso r T N − 1 of the form ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ S 1 ··· ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ S n ④ ④ ④ ④ T N − 1 = • ( s 1 , 0) ··· • ( s n , 0) ◦ ( s n , 1) ✄ ✄ ✄ ✄ ✄ ✄ ( s 1 , 1) ❀ ❀ ❀ ❀ ❀ ❀ ( e S , 1) where S is of the form ( S 1 , · · · , S n ) ⋆ [0]. ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ S 1 ··· ◦ ❈ ❈ ❈ ❈ ④ ④ ④ ④ S n ④ ④ ④ ④ S = ◦ s n ✁ ✁ ✁ ✁ ✁ ✁ s 1 ❂ ❂ ❂ ❂ ❂ ❂ e S This observ ation already enables us to g e t Pr o of of p arts (ii) and (iii) of The or em 5.2. Consider all the faces of T N . F or such a face F − → T N , there ar e three pos s ibilities; (a) it misses an S -colour en tirely (i.e. there is an edg e s in S so that neither ( s, 0 ) nor ( s, 1) ar e in F , so that Ω[ F ] facto r s thro ugh ∂ Ω[ S ] ⊗ ∆[1]; (b) F is given b y contracting the edge ( e S , 0 ), in which cas e Ω[ F ] factors through Ω[ T N − 1 ] (since the face F then coincides with the face of T N − 1 obtained by con tra cting ( s 1 , 1 ) , . . . , ( s n , 1 )); (c) F is given by chopping oﬀ the edge ( e S , 1 ) and the bla c k vertex ab ov e it, i.e. Ω[ F ] = Ω[ S ] ⊗ { 0 } . This face canno t factor thro ugh A 0 , nor throug h any of the earlier pe rcolation s chemes since none of these has a n edge coloured ( e S , 0 ). Thu s, Ω[ T N ] ∩ A N − 1 = Λ r [ T N ], where r denotes the black v ertex as pictured ab o ve. This shows tha t Λ r [ T ] / /   A N − 1   Ω[ T ] / / A N is a pushout, ex actly a s stated in part (ii) of Theo rem 5.2. Moreov er, the statement of part (iii) of Theorem 5.2 is obvious from the construction.  5.5. The pro of of part (i) of Theorem 5 .2 is mo r e involv ed, but it is completely analogo us to the pro of of [MW09 , Pr opos ition 9.2]. The diﬀere nce with the situation in lo c. cit. is tha t, no w, w e ar e dealing with an inclusion of the form ∂ Ω[ S ] ⊗ Ω[ T ] ∪ Ω[ S ] ⊗ Λ e [ T ] − → Ω[ S ] ⊗ Ω[ T ] , 24 D.-C. CISINSKI AND I. MOERDIJK where e is an outer edge of T = i [1], whereas in lo c. cit. , we dealt with Ω[ S ] ⊗ ∂ Ω[ T ] ∪ Λ e [ S ] ⊗ Ω[ T ] − → Ω[ S ] ⊗ Ω[ T ] , where e is a n inner edg e of S . T his forces us to lo ok at diﬀere nt ‘s pines’ and ‘characteristic edges’ compa red to the ones in lo c. cit. (notice als o in this connection that although the tenso r pr oduct is s y mmetric, the partial or de r on the percola tion schemes is r ev er sed). The following le mma w as also used (implicitly) in [MW09]. Lemma 5.6. L et T i and T j b e two distinct p er c olation schemes for Ω[ S ] ⊗ ∆[1] . Then Ω[ T i ] ∩ Ω[ T j ] ⊂ ∪ k Ω[ T k ] as sub obje ct s of Ω[ S ] ⊗ ∆[1] , wher e the u nion r anges over a l l the p er c olation schemes T k which pr e c e de b oth T i and T j in the p artial or der. Pr o of. Let F b e a c ommon face of Ω[ T i ] and Ω [ T j ]. If T j 6 T i in the partial order, there is nothing to prove. Otherwise, w e will g iv e an algorithm for replacing T j by successively earlier percola tion schemes, T j = T j 0 > T j 1 > T j 2 > · · · each having F as a face, and even tually preceding T i in the par tial order. As a ﬁrst step, T j is obtained from an earlier p ercolation scheme T j ′ by c hanging • ··· ( s 1 , 0) • ( s n , 0) ◦ ( s n , 1) t t t t t t ( s 1 , 1) ❏ ❏ ❏ ❏ ❏ ❏ ( s, 1) in T j ′ int o ◦ ✉ ✉ ✉ ✉ ✉ ■ ■ ■ ■ ■ ··· • ( s, 0) ( s, 1) in T j If F is a lso a fa c e of T j ′ , we ‘push up the black vertices’ b y replacing T j by T j 1 = T j ′ . If not, then the colour ( s, 0) must o ccur in F , hence in T j as w ell as in T i . So the o ccurrence of ( s, 0) in T j is not the re ason that T j  T i , and we put T j 1 = T j . T r eating all blac k vertices in this w ay , we can push them up if they o ccur below black v er tices in T i , un til we even tually reach a p ercola tion sc heme T j n 6 T j , still having F a s a face, for whic h T j n 6 T i .  5.7. W e r eturn to the proof of Theo rem 5.2. Cons ide r the inclusion (5.7.1) A k − → A k +1 = A k ∪ Ω[ T k +1 ] , for k + 1 < N . The per colation scheme T k +1 will have a t le ast one bla c k v ertex. Consider all the black v er tices in T k +1 , and the cor respo nding faces o f T k +1 which are formed b y paths from these black v ertices to the ro ot of T k +1 : • ( s, 0) ◦ ( s, 1) ... ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ β = ◦ ❀ ❀ ❀ ❀ ❀ ... q q q q q q q q ◦ . . . ❀ ❀ ❀ ❀ ❀ ... q q q q q q q q ( e S , 1) (5.7.2) DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 25 The face β is the minimal external face which cont ains the given black vertex as well as the root edg e. W e call a face β of T k +1 of this form a spine in T k +1 . Notice that the v er tex just a b ov e ( e S , 1 ) is indeed white, as in the picture, b e cause k + 1 < N . Notice also that t he outer face of β given by c hopping oﬀ this vertex misse s the colour e S , hence b elongs to ∂ Ω[ S ] ⊗ ∆[1] ⊂ A 0 . F urther more, the outer face of β given b y chopping oﬀ its black top vertex b elongs to Ω[ S ] ⊗ { 1 } ⊂ A 0 . Finally , all the inner faces of β miss an S -colo r, hence factor thr ough ∂ Ω[ S ] ⊗ ∆[1], except po ssibly the one given by con tr a cting the edge ( s, 1 ) near the top. How ever, if this last face ∂ ( s, 1) ( β ) of β belongs to A k , then some earlie r T i , 1 6 i 6 k , contains the edge ( s, 0), hence all of β . T hus, either Ω[ β ] is con tained in A k , or we can adjoin it by an inner ano dyne extension Λ ( s, 1) [ β ] / /   A k   Ω[ β ] / / A k ∪ Ω[ β ] . (5.7.3) Such a spine β is an ex ample of an initial se gment of T k +1 . Rec a ll from [MW09] that a face R − → T k +1 is called a n initial se gment if it is obtained b y successively chopping oﬀ to p v ertices. Our strategy will b e to adjoin more initial segments of T k +1 to A k , starting with the spines. T o this end, we ne e d the following deﬁnition and lemma fro m [MW09], in which we use the notation m ( R ) ⊂ Ω[ T k +1 ] for the image of the map Ω[ R ] − → Ω[ T k +1 ] given by an initial segment R . Deﬁnition 5 .8 ([MW09]) . Let R , Q 1 , . . . , Q p be initial seg ments of T k +1 , and let B = m ( Q 1 ) ∪ · · · ∪ m ( Q p ). Supp ose that, for every to p face F of R , we hav e m ( F ) ⊂ A k ∪ B . In this situation, an inner edge ξ o f R is ca lle d char acteristic with res p ect to Q 1 , . . . , Q p if, for any inner face F of R , if m ( F / ξ ) is contained in A k ∪ B , then so is m ( F ) (where F /ξ − → F is the face obtained by contracting ξ ). Example 5.9 . In any spine β as in picture (5.7.2), the edge ξ = ( s, 1) is character is tic with resp ect to any family of initia l s egmen ts. Example 5.10 . More generally , supp ose R is an initial segment of T k +1 given by a spine β expa nded b y one (or mor e) white v ertices , sa y • ( s, 0) ◦ ( s, 1) ④ ④ ④ ④ ◦ ④ ④ ④ ④ R = ◦ ( s ′ , 1) ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ◦ . . . ❂ ❂ ❂ ❂ ❂ ( e S , 1) Then ξ = ( s, 1 ) is a gain c ha racteristic with resp ect to any family Q 1 , . . . , Q p . In- deed, if R/ ξ is a face of a n initial seg men t Q i , then so is R itself; see [MW09, Remark 9.6 (iv)]. And if R/ξ is a face of T j for a p ercolation sc he me T j , then T j 26 D.-C. CISINSKI AND I. MOERDIJK either contains R , or loo ks like ◦ ( s, 0) ④ ④ ④ ④ • ( s ′ , 0) ◦ ④ ④ ④ ④ ◦ ( s ′ , 1) ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ◦ . . . ❈ ❈ ❈ ❈ ( e S , 1) But, by Lemma 5.6, we can assume T j comes b efore T k +1 in the partial order, so this is imp ossible. Finally , if Ω[ R /ξ ] − → Ω[ S ] ⊗ ∆[1] factors through A 0 , then R/ξ misses an S - c olour, and hence so does R . 5.11. T he pro of of Theo rem 5.2 (i) is base d on a repeated us e of argumen ts lik e the preceding one in Example 5.10. W e q uo te the following le mma on character istic edges from [MW09]. Lemma 5.12 ([MW09 , Le mma 9.7]) . L et R , Q 1 , . . . , Q p b e initial se gments of T k +1 . L et B = m ( Q 1 ) ∪ · · · ∪ m ( Q p ) , and supp ose e ach top fac e of R has the pr op erty that m ( F ) is c ontaine d in A k ∪ B . If R p ossesses a char acteristic e dge with re sp e ct t o Q 1 , . . . , Q p , t hen the inclusion A k ∪ B − → A k ∪ B ∪ m ( R ) of su b obje cts of Ω[ S ] ⊗ ∆[1] is inn er ano dyne. Lemma 5 .13. L et R , Q 1 , . . . , Q p b e initial se gments of T k +1 , satisfying c ondition (i) in Deﬁn ition 5.8, and let β b e a spine in R . Then the e dge ξ = ( s, 1) imme diately b elow t he black vertex on t he spine is a char acteristic e dge for R . Hint for a pr o of. This is prov ed exactly as Example 5.10; cf. also [MW09, Lemma 9.8].  5.14. Using the characteristic edg e s fro m Lemma 5.13, one ca n now co p y the pro of of [MW09, Lemma 9.9], repeated b elow as Lemma 5.15, verbatim. This proof is b y induction on l , and describes a precise stra tegy for adjoining more and/ or large r initial segments o f T k +1 to A k . Lemma 5.15. Fix l 6 0 , and let Q 1 , . . . , Q p b e a family of initial se gments in T k +1 , e ach c ontaining at le ast one spine, and at most l spines (so, ne c essarily, p = 0 if l = 0 ). L et R 1 , . . . , R q b e initial se gments which e ach c ontain exactly l + 1 spines. Th en the inclusion A k − → A k ∪ B ∪ C is inner ano dyne, wher e B = m ( Q 1 ) ∪ · · · ∪ m ( Q p ) and C = m ( R 1 ) ∪ · · · ∪ m ( R q ) . 5.16. T his strategy terminates when one arrives at the n umber l o f al l spines in T k +1 . Indeed, for this l and p = 0, q = 1, Lemma 5.1 5 states for R 1 = T k +1 that A k − → A k +1 is inner ano dyne, as asserted in Theor em 5.2 (i). This completes the pro of of Theorem 5.2. 6. ∞ -operads as fibrant objects 6.1. The aim o f this section is to c haracter ize ∞ -o perads a s the ﬁbrant ob jects of the mo del categor y structure on the the categor y o f dendroidal sets given b y Prop osition 3.12. This c ha r acterization is s tated in Theorem 6.10 b e lo w. DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 27 Given an ∞ -category X , we denote b y k ( X ) the maximal Kan complex co n tained in X ; see [Joy02 , Corollary 1.5]. Recall that, given t wo dendroidal sets A and X , w e write hom ( A, X ) = i ∗ Hom ( A, X ) . Note that, b y virtue of Propo s ition 3.1, if X is a n ∞ -op e rad, and if A is nor mal, then Hom ( A, X ) is an ∞ -op erad, so that hom ( A, X ) is an ∞ -ca teg ory . F o r an ∞ -op erad X and a simplicial set K , we will write X ( K ) for the sub complex of Hom ( i ! ( K ) , X ) which consists of dendrices a : Ω[ T ] × i ! ( K ) − → X such that, fo r a n y 0-ce ll u in T , the induced ma p a u : K − → i ∗ ( X ) factors thro ugh k ( i ∗ ( X )) (i.e. all the 1 -cells in the image o f a u are w ea kly in vertible in i ∗ ( X )). F o r an ∞ - opera d X a nd a normal dendroidal set A , we will write k ( A, X ) for the sub complex of hom ( A, X ) which cons is ts of maps u : A ⊗ i ! (∆[ n ]) − → X such that, fo r a ll v ertices a of A (i.e. ma ps a : η − → A ), the induced map u a : ∆[ n ] − → i ∗ ( X ) factors thro ugh k ( i ∗ ( X )). So, by deﬁnition, for a ny nor mal dendroidal set A , any simplicial set K , and any ∞ - o pera d X , there is a natural bijection: (6.1.1) Hom sSet ( K, k ( A, X )) ≃ Hom dSet ( A, X ( K ) ) . R emark 6.2 . The simplicia l set k ( A, X ) is by deﬁnition the ∞ -categor y of ob ject wise weakly inv ertible 1-cells in hom ( A, X ). W e can reformulate the deﬁnition of k ( A, X ) as follows (still with A normal and X an ∞ -op erad). Deﬁne (6.2.1) Ob A = a A 0 η . W e hav e a unique mono mo rphism i : Ob A − → A which is the identit y on 0-cells. As A is normal, i is a normal monomo rphism. W e also hav e (6.2.2) k (Ob A, X ) = k ( hom (Ob A, X )) = Y A 0 k ( i ∗ X ) , and k ( A, X ) ﬁts by deﬁnition in the following pullba c k squar e. k ( A, X ) / /   hom ( A, X )   k ( hom (Ob A, X )) / / hom (Ob A, X ) (6.2.3) In par ticular, the pro jection of k ( A, X ) o n k (Ob A, X ) is an inner Kan ﬁbration, and a s the latter is a Ka n complex, this shows tha t k ( A, X ) is a n ∞ -categor y . One of the key res ults of this sectio n asserts that k ( A, X ) is a Kan complex as well, which can be reformulated b y saying that the inclusion k ( hom ( A, X )) ⊂ k ( A, X ) is in fact an equa lit y . In other w ords, a map in the ∞ -categor y hom ( A, X ) is w e akly inv ertible if and only it is ob ject wise w eak ly in vertible; see Corollary 6.8. 28 D.-C. CISINSKI AND I. MOERDIJK 6.3. Before stating the next theorem, we recall that, fo r a mor phism b et ween ∞ - categorie s f : X − → Y , the induced map τ ( f ) : τ ( X ) − → τ ( Y ) is a catego rical ﬁbration if and only if the map ev 1 : X (∆[1]) − → Y (∆[1]) × Y X induced b y ev aluating at 1 (i.e. by the inclus io n { 1 } − → ∆[1]) has the right lifting prop ert y with respec t to ∂ ∆[0] − → ∆[0]; see [Joy02, Prop osition 2.4]. Theorem 6. 4. L et p : X − → Y b e an inner Kan ﬁbr ation b etwe en ∞ - op er ads. The map ev 1 : X (∆[1]) − → Y (∆[1]) × Y X has the righ t lifting pr op erty with r esp e ct to inclusions ∂ Ω[ S ] − → Ω[ S ] for any tr e e S wi th at le ast one vertex. Conse quently, the functor τ i ∗ ( p ) is a c ate goric al ﬁbr ation if and only if the evaluation at 1 map X (∆[1]) − → Y (∆[1]) × Y X is a trivial ﬁbr ation of dendr oidal sets. Pr o of. Consider a tree S with at least one v e r tex and a solid comm utative squar e ∂ Ω[ S ] f / /   X (∆[1])   Ω[ S ] g / / h 8 8 Y (∆[1]) × Y X W e want to pr o ve the existence o f a diagonal ﬁlling h . This corre s ponds by adjunc- tion to a ﬁlling ˜ h in the following co mm utative square ∂ Ω[ S ] ⊗ ∆[1] ∪ Ω[ S ] ⊗ { 1 } ˜ f / /   X   Ω[ S ] ⊗ ∆[1] ˜ g / / ˜ h 6 6 Y (as the inclusion of ∂ Ω[ S ] in Ω[ S ] is bijective on ob jects, and as the restriction of ˜ h to ∂ Ω[ S ] ⊗ ∆[1] ∪ Ω[ S ] ⊗ { 1 } co inc ide s with ˜ f , the map Ω[ S ] − → X ∆[1] corres p onding to a ﬁlling ˜ h will automatically factor through X (∆[1]) ). Consider the ﬁltration ∂ Ω[ S ] ⊗ ∆[1] ∪ Ω[ S ] ⊗ { 1 } = A 0 ⊂ A 1 ⊂ · · · ⊂ A N − 1 ⊂ A N = Ω[ S ] ⊗ ∆[1] given by Theorem 5.2. As the map X − → Y is an inner Ka n ﬁbra tion, using Theorem 5.2 (i), it is suﬃcient to ﬁnd a ﬁlling in a solid commutativ e diagram of shap e A N − 1 f ′ / /   X   Ω[ S ] ⊗ ∆[1] ˜ g / / ˜ h : : Y in which the r estriction of f ′ to ∂ Ω[ S ] ⊗ ∆[1] ∪ Ω[ S ] ⊗ { 1 } coincides with ˜ f . By virtue of Theo rem 5.2 (ii), it is even suﬃcient to ﬁnd a ﬁlling k in a solid comm utative DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 29 diagram of shap e Λ r [ T ] a / /   X   Ω[ T ] b / / k = = Y in which T is a tree with unar y vertex r at the ro ot, and a is the restriction o f f ′ to Λ r [ T ] ⊂ A N − 1 . F urther mo re, b y Theorem 5.2 (iii), we may assume that a ( r ) is weakly inv ertible in i ∗ ( X ). Thus, the existence o f the ﬁlling k is ensured by Theorem 4.2. The last asser tion of the theorem follows from 6 .3.  Lemma 6.5. Any left ﬁbr ation b etwe en Kan c omplexes is a Kan ﬁbr ation. Pr o of. This follows fr om [Jo y0 2 , Theorem 2.2 and Prop osition 2.7].  Lemma 6.6. A morphi sm of simplicial sets X − → Y is a left (r esp. right) ﬁbr ation if and only it has t he right lifting pr op erty with r esp e ct to maps of shap e ∂ ∆[ n ] × ∆[1] ∪ ∆[ n ] × { e } − → ∆[ n ] × ∆[1] for e = 1 (r esp. for e = 0 ) and n > 0 . Pr o of. The map ∂ ∆[ n ] × ∆[1] ∪ ∆[ n ] × { 0 } − → ∆[ n ] × ∆[1] is obtained as a ﬁnite comp osition of pushouts of horns o f s hape Λ k [ n + 1] − → ∆[ n + 1] with 0 6 k < n + 1; see (the dual version of ) [GZ6 7, Chapter IV, 2.1.1]. Conv erse ly , the inclusion map Λ k [ n ] − → ∆[ n ], 0 6 k < n , is a retr act of the ma p Λ k [ n ] × ∆[1] ∪ ∆[ n ] × { 0 } − → ∆[ n ] × ∆[1]; see [GZ67, Chapter IV, 2.1 .3]. W e deduce easily fr om this that a mor phism of simplicial s ets X − → Y is a right ﬁbration if and only if the ev aluatio n at 0 map X ∆[1] − → Y ∆[1] × Y X is a triv ial ﬁbration (i.e. has the right lifting prop erty with resp ect to monomor phisms). The case of left ﬁbrations follows b y duality .  Prop osition 6. 7. L et p : X − → Y b e a n inn er K an ﬁbr ation b et we en ∞ -op er ads. If τ i ∗ ( p ) is a c ate goric al ﬁbr ation, then, for any monomorphism b et we en normal dendr oidal sets A − → B , t he map k ( B , X ) − → k ( B , Y ) × k ( A,Y ) k ( A, X ) is a Kan ﬁbr ation b et we en Kan c omplexes. Pr o of. The functor i ! : sSet − → dSet b eing symmetric monoidal a nd prese r ving inner ano dyne extensions, Prop osition 3.1 implies that the map hom ( B , X ) − → hom ( B , Y ) × hom ( A ,Y ) hom ( A, X ) is an inner Kan ﬁbration b et ween ∞ -categor ie s. This implies the map k ( B , X ) − → k ( B , Y ) × k ( A,Y ) k ( A, X ) is an inner Kan ﬁbration b et ween ∞ -categor ie s. 30 D.-C. CISINSKI AND I. MOERDIJK W e claim that this map has the right lifting prop erty with resp ect to the inclusion { 1 } − → ∆[1]. Using the iden tiﬁcation (6.1.1), w e see that lifting problems of shap e { 1 } / /   k ( B , X )   ∆[1] / / 6 6 k ( B , Y ) × k ( A,Y ) k ( A, X ) (6.7.1) corres p ond to lifting problems of shap e A / /   X (∆[1])   B / / 9 9 Y (∆[1]) × Y X (6.7.2) so that our claim follows from Theorem 6.4. More generally , the ma p k ( B , X ) − → k ( B , Y ) × k ( A,Y ) k ( A, X ) has the right lifting prop erty with resp ect to maps of shap e ∂ ∆[ n ] × ∆[1] ∪ ∆[ n ] × { 1 } − → ∆[ n ] × ∆[1] , n > 0 . (6.7.3) W e hav e just check ed it a bov e in the case wher e n = 0, so that it r emains to prov e the case where n > 0. Cons ider a lifting problem of shap e ∂ ∆[ n ] × ∆[1] ∪ ∆[ n ] × { 1 } u / /   k ( B , X )   ∆[ n ] × ∆[1] v / / g 4 4 k ( B , Y ) × k ( A,Y ) k ( A, X ) (6.7.4) This lifting problem gives rise to a lifting problem of shap e ∂ Ω[ n ] ⊗ B ∪ Ω[ n ] ⊗ A / /   X (∆[1])   Ω[ n ] ⊗ B / / h 5 5 Y (∆[1]) × Y X . (6.7.5) The existence of the lifting h is provided aga in by Theorem 6.4. The map h deﬁnes a map l : i ! (∆[ n ] × ∆[1]) ⊗ B − → X . As a conseq uence, it is suﬃcient to chec k that, for every non-degenerate m -simplex δ : ∆[ m ] − → ∆[ n ] × ∆[1], m > 1 , a nd fo r any o b ject b : η − → B , the map ( li ! ( δ )) b : ∆[ m ] − → i ∗ ( X ) factors through k ( i ∗ ( X )). Using the ‘2 out of 3 prop erty’ for weakly in vertible 1-cells in i ∗ ( X ), we can assume that m = 1. But then, a s n > 0 , using aga in the ‘2 o ut of 3 pr oper ty’ for weakly inv ertible 1-cells, we may assume that δ facto rs through ∂ ∆[ n ] × ∆[1], which implies then that ( l ◦ i ! ( δ )) ⊗ 1 B factors throug h the sub complex i ! ( ∂ ∆[ n ] × ∆[1]) ⊗ B : the required pro p erty th us follo ws from the f act that the restr iction of the transp ose of h to the ob ject i ! (∆[ n ] × ∆[1] ∪ ∂ ∆[ n ] × { 1 } ) ⊗ B cor respo nds to the map u in (6.7.4). By virtue of Lemma 6.6, the ma p k ( B , X ) − → k ( B , Y ) × k ( A,Y ) k ( A, X ) is a left ﬁbration, hence, b y [Joy02, Prop osition 2.7], is c onserv a tiv e. By applying [J o y0 2 , Corollar y 1 .4], we de duce , from the ca se wher e A = ∅ a nd Y is the ter minal DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 31 dendroidal set, that k ( B , X ) is a Kan complex for a n y normal dendro idal set B and any ∞ -op erad X . As a n y left ﬁbration betw ee n K an complex es is a Ka n ﬁbration (Lemma 6.5), the maps k ( B , X ) − → k ( A, X ) a re thus K an ﬁbrations betw een K an complexes for any monomo rphisms be tw een no rmal dendroidal sets A − → B and any ∞ -op erad X . As a conseque nce , Kan ﬁbra tions b eing stable by pullback, we see that the ﬁber pr oduct k ( B , Y ) × k ( A,Y ) k ( A, X ) is a Kan complex. Using again Lemma 6.5, we conclude that k ( B , X ) − → k ( B , Y ) × k ( A,Y ) k ( A, X ) is a Kan ﬁbration b et ween Kan complexes.  Corollary 6.8. F or any normal dendr oidal set A and any ∞ -op er ad X , we have k ( hom ( A, X )) = k ( A, X ) . F or any inner Kan ﬁ br ation b etwe en ∞ -op er ads p : X − → Y su ch that τ i ∗ ( p ) is a c ate goric al ﬁbr ation, and for any monomorphism b etwe en normal dendr oidal sets A − → B , we have k ( hom ( B , Y ) × hom ( A ,Y ) hom ( A, X )) = k ( B , Y ) × k ( A,Y ) k ( A, X ) . Pr o of. If A is normal, then, for any op erad X , k ( A, X ) is a Kan complex which contains k ( hom ( A, X )). As k ( hom ( A, X )) is the ma ximal sub Kan complex con- tained in the ∞ -category hom ( A, X ), this prov es the ﬁrst assertio n. The second assertion is prov ed similarly .  Corollary 6.9. L et p : X − → Y b e an inner Kan ﬁbr ation b etwe en ∞ -op er ads. If τ i ∗ ( p ) is a c ate goric al ﬁbr ation, then, for any ano dyne ext ension of simplicial sets K − → L , the map X ( L ) − → Y ( L ) × Y ( K ) X ( K ) is a trivial ﬁbr ation of dendr oidal sets. Pr o of. This follows from Pro position 6.7 and from the natural identiﬁcation (6.1.1).  Theorem 6.10 . A dendr oidal set is J -ﬁbr ant if and only if it is an ∞ -op er ad. An inner K an ﬁbr ation b etwe en ∞ -op er ads p : X − → Y is a J -ﬁ br ation (i.e. a ﬁbr ation for the mo del c ate gory structur e of Pr op osition 3.12) if and only if τ i ∗ ( p ) is a c ate goric al ﬁbr ation. Pr o of. Let p : X − → Y be inner K an ﬁbr ation b et ween ∞ -opera ds. W e ha ve to prov e that, for e = 0 , 1, the anodyne extension { e } − → J induces a trivial ﬁbration of dendroidal sets X J d − → Y J d × Y X if a nd only if τ i ∗ ( p ) is a categorica l ﬁbration. But, for an y ∞ -op erad Z , w e clearly hav e Z J d = Z ( J ) and Z = Z ( { e } ) . Hence, by virtue of Corolla ry 6.9, if τ i ∗ ( p ) is a categoric al ﬁbra tion, then p is a J -ﬁbration. The co n verse is a direct co nsequence of [Joy02 , Corollar y 1.6].  Corollary 6. 11. The class of we ak op er adic e quivalenc es is the smal lest class of maps of dendr oidal sets W which satisﬁes the fol lowi n g thr e e pr op erties. (a) ( ‘ 2 out 3 pr op erty’) In any c ommu tative triangle, if t wo maps ar e in W , then so is the thir d. (b) Any inner ano dyne exten sion is in W . (c) Any trivial ﬁbr ation b etwe en ∞ - op er ads is in W . 32 D.-C. CISINSKI AND I. MOERDIJK Pr o of. Consider a class of maps W satisfying conditions (a), (b) and (c) ab o ve. W e wan t to prov e that any w eak op eradic equiv a le nc e is in W . Let f : A − → B b e a mor phis m of dendroidal sets. Using the small ob ject argument applied to the set o f inner horns, we can see there exists a commutativ e square A a / / f   X p   B b / / Y in which the maps a and b are inner ano dyne extensions, and X a nd Y are ∞ - op erads. It is clear that f is a w eak oper adic equiv alence (resp. is in W ) if and only p has the same prop erty . Hence it is suﬃcient to prove that a n y w eak oper adic equiv a lence b et ween ∞ -o perads is in W . As a n y tr iv ial ﬁbration b e t ween ∞ -o pera ds is in W b y assumption, and a s ∞ -op erads are the ﬁbr an t o b jects o f a mo del ca tegory , this corollar y follows fr om Ken Brown’s Lemma [Hov99, Lemma 1.1.12].  6.12. W e will w r ite C n for the corolla with n + 1 edges, ······ C n = • a 2 a 1 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ a n ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ . a Let X b e an ∞ - opera d. Given an ( n + 1 )- tuple of 0-cells ( x 1 , . . . , x n , x ) in X , the space of ma ps X ( x 1 , . . . , x n ; x ) is obtained by the pullbac k below, in whic h the map p is the map induced b y the inclusion η ∐ · · · ∐ η − → Ω[ C n ] (with n + 1 copies of η , corresp onding to the n + 1 ob jects ( a 1 , . . . , a n , a ) of C n ). X ( x 1 , . . . , x n ; x ) / /   Hom (Ω[ C n ] , X ) p   η ( x 1 ,...,x n ,x ) / / X n +1 Using the identiﬁcation sSet = dSet /η , we shall co nsider X ( x 1 , . . . , x n ; x ) as a simplicial set. Prop osition 6.13. The simplici al set X ( x 1 , . . . , x n ; x ) is a Kan c omplex. Pr o of. The ﬁrst assertion of Coro llary 6 .8 for A = Ω[ C n ] can b e r ein terpr eted b y saying we ha ve the pullback square be low (see Remark 6.2). k ( hom (Ω[ C n ] , X )) / /   Hom (Ω[ C n ] , X )   k ( i ∗ X ) n +1 / / X n +1 DENDROID AL SETS AS MODELS FOR HOMOTOPY OPERADS 33 As the terminal simplicia l set η is cer tainly a Kan co mplex, it thus follows from the construction of X ( x 1 , . . . , x n ; x ) that we have a pullback square X ( x 1 , . . . , x n ; x ) / /   k ( hom (Ω[ C n ] , X ))   η ( x 1 ,...,x n ,x ) / / k ( i ∗ X ) n +1 in which the rig ht v er tical map in this diagra m is a Kan ﬁbr ation (b y Pro position 6.7, applied for A = η ∐ · · · ∐ η and B = Ω[ C n ]). The stabilit y of Kan ﬁbrations by pullback a c hieves the proo f.  Prop osition 6.14. Ther e is a c anonic al bije ction π 0 ( X ( x 1 , . . . , x n ; x )) ≃ τ d ( X )( x 1 , . . . , x n ; x ) . Pr o of. W e will use the explicit desc ription o f τ d ( X ) given by [MW09 , Lemma 6.4 and Prop osition 6.6]. The unit map X − → N d τ d ( X ) induces a map X ( x 1 , . . . , x n ; x ) − → ( N d τ d ( X ))( x 1 , . . . , x n ; x ) . It is easily see n that ( N d τ d ( X ))( x 1 , . . . , x n ; x ) is the discrete simplicial set asso ciated to τ d ( X )( x 1 , . . . , x n ; x ), so tha t w e get a surjective map π 0 ( X ( x 1 , . . . , x n ; x )) − → τ d ( X )( x 1 , . . . , x n ; x ) . Using the explicit description of τ d ( X ) giv en b y [MW09, Lemma 6.4 and Pr o po- sition 6.6], it is now suﬃcient to prov e that, if f and g are t wo 0-simplices of X ( x 1 , . . . , x n ; x ) which ar e homotopic alo ng the edge 0 in the sens e of [MW09, Deﬁnition 6.2 ], then they belong to the same connected comp onent . But then, f and g deﬁne tw o ob jects of τ ( hom (Ω[ C n ] , X )) which are isomor phic, whic h can b e expressed by the ex is tence of a map h : ∆[1 ] − → k ( hom (Ω[ C n ] , X )) which connect f a nd g . Using that k ( hom (Ω[ C n ] , X )) − → i ∗ ( X ) n +1 is a Kan ﬁbration betw een Ka n complexes , w e can see b y a path lifting argument that such a map h is homotopic under ∂ ∆[1] to a map ∆[1 ] − → X ( x 1 , . . . , x n ; x ) which connects f and g .  Lemma 6.15. L et X − → Y b e a trivial ﬁbr ation b et we en ∞ -op er ads. Th en, for any ( n + 1) -tuple of 0 -c el ls ( x 1 , . . . , x n , x ) in X , the induc e d map X ( x 1 , . . . , x n ; x ) − → Y ( f ( x 1 ) , . . . , f ( x n ); f ( x )) is a trivial ﬁbr ation of simplicial sets. Pr o of. W e k no w that the map Hom (Ω[ C n ] , X ) − → Hom (Ω[ C n ] , Y ) × Y n +1 X n +1 is a trivial ﬁbration (this follows from Prop osition 1.9 by adjunction). As we hav e a pullback o f s ha pe X ( x 1 , . . . , x n ; x ) / /   Hom (Ω[ C n ] , X )   Y ( f ( x 1 ) , . . . , f ( x n ); f ( x )) / / Hom (Ω[ C n ] , Y ) × Y n +1 X n +1 34 D.-C. CISINSKI AND I. MOERDIJK this prov es the lemma.  Prop osition 6.16. The functor τ d : dSet − → Operad sends we ak op er adic e quiva- lenc es to e quivalenc es of op er ads. Pr o of. W e know that τ d sends inner horn inclusions to isomo rphisms of op erads (this follows from [MW09 , Theor em 6.1] b y the Y oneda Lemma). As τ d preserves colimits, w e deduce that τ d sends inner ano dyne extensions to is omorphisms of op erads. B y virtue of Corollary 6.11, it is thus suﬃcien t to prov e that τ d sends trivial ﬁbrations betw een ∞ -op erads to equiv alence s of op erads. Let f : X − → Y be a trivial ﬁbration betw een ∞ -op erads. By virtue o f Pr opositio n 6.14 and of Lemma 6.15, we see that τ d ( f ) is fully faithful. As f is obviously surjectiv e on 0-cells, τ d ( f ) has to b e an equiv alence of oper ads.  Corollary 6.17 . The adjunction τ d : dSet ⇄ Operad : N d is a Quil len p air. Mor e- over, the two fu n ctors τ d and N d b oth pr eserve we ak e quivalenc es. In p articular, a morphism o f op er ads is an e quivalenc e of op er ads if and only if its dendr oidal nerve is a we ak op er adic e quivalenc e. Pr o of. The functor τ d preserves co ﬁbrations, so th at this is a direct consequence of P r opos ition 6.16. Note that any op erad is ﬁbrant, so that the dendro idal nerve functor N d preserves weak equiv alences. Hence the last assertion comes from the fact N d is fully faithful a nd τ d preserves weak equiv a lences.  R emark 6.18 . Theorem 6 .10 also asserts that the functor τ d preserves ﬁbrations betw een ∞ -op erads. 6.19. If A is a normal dendroidal set, and if X is an ∞ -op erad, we ha ve (6.19.1) Hom Ho ( dSet ) ( A, X ) = [ A, X ] ≃ π 0 ( k ( hom ( A, X ))) . Indeed, J ⊗ A is a cylinder of A , and mor phisms J ⊗ A − → X corres p ond to morphisms J − → k ( hom ( A, X )) , so that this form ula follows from the fact X is J -ﬁbr a n t. The nex t statement is a reformulation of (6 .19.1). Prop osition 6.20. L et A b e a normal dendr oidal set, and X an ∞ - op er ad. The set [ A, X ] = Hom Ho ( dSet ) ( A, X ) c an b e c anonic al ly identiﬁe d with the set of iso- morphism classes of obje cts in the c ate gory τ hom ( A, X ) (which is also the c ate gory underlying τ d ( Hom ( A, X )) ). Pr o of. This pro position is a dir ect applicatio n of the explicit descr iption of the op erad τ d ( Hom ( A, X )) g iv en b y [MW09, Prop osition 6.6] and of Corollar y 6.9.  App en dices A. Grafting orders onto trees The main goa l of the tec hnical sections 4 and 5 was to deduce Theorem 6.10, and from it, Cor ollary 6.9. There is an as y mmetry in this a ppr oach, in that Theorem 35 6.10 was o nly proved for ev aluation at one of the end p oin ts, and the symmetry w a s established in Cor ollary 6.9 by using the theory of left ﬁbr ations between simplicia l sets. In these tw o app endices, we will prov e the ana logs of Theore ms 4.2 and 5.2, from which one can deduce dire c tly the symmetric version of Theore m 6.1 0 (for ev a luation a t 0). These tw o app endices c a n also b e used as an alterna tiv e approach to the results in Section 6. Moreover, they are of in terest b y themselv es, as they form the basis of a theory of rig h t ﬁbrations of dendroidal sets. How ever, since the left-right duality for s implicia l sets does not extend to den- droidal sets, the results of these app endices cannot b e deduced from their analog s prov ed earlier. W e b egin by studying the analog of Theorem 4.2 (see Theor e m A.7 b elow). A.1. Let T b e a tree endo wed with a n input edge (leaf ) e . T = ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ e • ... ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ (A.1.1) Given a n in teger n > 0, w e deﬁne the tree n ⋆ e T a s the tree obtained by joining the n -simplex to the edge e by a new v ertex v . n ⋆ e T = • 0 1 • . . . • n v ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ e • ... ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ (A.1.2) This deﬁnes a unique functor (A.1.3) ∆ − → Ω , [ n ] 7− → n ⋆ e T such that the obvious inclusions i [ n ] − → n ⋆ e T a re functorial. W e thus g et a functor (A.1.4) ( − ) ⋆ e T : ∆ − → T / dSet (where T / dSet deno tes the category of dendroidal sets under Ω[ T ]). By Kan ex- tension, we obta in a colimit preserving functor (A.1.5) ( − ) ⋆ e T : sSet − → T / dSet . W e have ∆[ n ] ⋆ e T = Ω[ n ⋆ e T ]. The functor (A.1.5) has a right adjoin t (A.1.6) ( − ) / e T : T / dSet − → sSet . R emark A.2 (F unctoriality in T ) . W e shall say that a fa c e map R − → T is e - admissible if it do es not factor thr ough the external face map which chops oﬀ e . F o r s uc h a face R − → T , e is also a leaf of R , and there a re na tural maps (A.2.1) n ⋆ e R − → n ⋆ e T . 36 Thu s, we obtain, for each simplicial set K , and eac h dendroidal set X under T (i.e. under Ω[ T ]), natural maps (A.2.2) K ⋆ e R − → K ⋆ e T and (A.2.3) X/ e T − → X / e R . Similarly , the inclusions Ω[ n ] − → ∆[ n ] ⋆ e T induce a pro jection (A.2.4) X/ e T − → i ∗ ( X ) for any dendroida l set X under T . A.3. Let 0 < i 6 n b e integers. Let { R 1 , . . . , R t } , t > 1, be a ﬁnite family of e -admissible faces of T , and deﬁne C ⊂ D ⊂ Ω[ n ⋆ e T ] by C =  t [ s =1 Λ i [ n ] ⋆ e R s  ∪ Ω[ n ] and D = t [ s =1 ∆[ n ] ⋆ e R s , where Ω[ n ] is seen as a sub complex of Ω[ n ⋆ e T ] through the canonical embedding. Lemma A. 4 . U nder the assumptions of A.3, the map C − → D is an inner ano dyne extension. Pr o of. F o r p > 1 , write F p for the set of f aces F of Ω[ n ⋆ e T ] which b elong to D but not to C , and which are of the for m F = Ω[ n ⋆ e R ] for an e - admissible face R of T with exactly p edges. Deﬁne a ﬁltra tion C = C 0 ⊂ C 1 ⊂ . . . ⊂ C p ⊂ . . . ⊂ D by C p = C p − 1 ∪ [ F ∈ F p F , p > 1 . W e hav e D = C p for p big enough, and it is suﬃcient to prov e that the inclusions C p − 1 − → C p are inner ano dyne for p > 1. If F and F ′ are in F p , then F ∩ F ′ is in C p − 1 . Moreover, if F = Ω[ n ⋆ e R ] for an e - a dmissible face R of T , then w e hav e F ∩ C p − 1 = Λ i [ n ⋆ e R ] , which is an inner horn. Hence we can describ e the inclusion C p − 1 − → C p as a ﬁnite comp osition of pushouts b y inner horn inclusions o f shape F ∩ C p − 1 − → F for F ∈ F p .  Prop osition A.5. L et 0 < i 6 n b e inte gers. The inclusion (Λ i [ n ] ⋆ e T ) ∪ Ω[ n ] − → Ω[ n ⋆ e T ] is an inner ano dyne extension. Pr o of. Apply Lemma A.4.  Prop osition A. 6 . F or any inner Kan ﬁbr ation p : X − → Y under T , the mor- phism X/ e T − → Y / e × i ∗ ( Y ) i ∗ ( X ) is a right ﬁbr ation of simplici al sets. In p articular, for any ∞ -op er ad X under T , the map X/ e T − → i ∗ ( X ) is a right ﬁbr ation b et we en ∞ -c ate gories. 37 Pr o of. This follows fr om Propo sition A.5 b y a standa rd adjunction argument.  Theorem A.7. L et S b e a tre e with at le ast t wo vertic es, let v b e a un ary t op vertex in S , and let p : X − → Y b e an inner Kan ﬁbr ation b etwe en ∞ -op er ads. Then any solid c ommut ative squar e of t he form Λ v [ S ] ϕ / /   X p   Ω[ S ] ψ / / h = = Y in which ϕ ( v ) is we akly invertible in X has a diagonal ﬁl ling h . Pr o of. The tree S has to b e of s hape S = 1 ⋆ e T for a tree T with a given leaf e . Under this ide ntiﬁcation, we hav e Λ v [ S ] = Λ 0 [1 ⋆ e T ]. A lifting h in the solid commutativ e square Λ 0 [1 ⋆ e T ] ϕ / /   X p   Ω[1 ⋆ e T ] ψ / / h ; ; Y is th us equiv ale nt to a lifting k in the diagram { 0 } ˜ ϕ / /   P   ∆[1] ˜ ψ / / k > > Q in whic h P = X/ e T and Q = U × W V , with U = lim ← − R X/ e R , V = Y / e T , W = lim ← − R Y / e R , where R ranges ov er all the prop er e -admissible face s of T . As in the pro of of Theorem 4.2, it is now suﬃcien t to prove the three following prop erties: (i) the map P − → Q is a right ﬁbra tion; (ii) Q is an ∞ -categor y . (iii) if ϕ ( x ) is weakly in vertible in X , then so is the 1-cell ˜ ψ in Q . Prop erties (ii) and (iii) will follow from the t wo assertions below: (iv) the map V − → W is a right ﬁbra tion; (v) the map U − → i ∗ ( X ) is a right ﬁbration. As (iv) is a particular case of (i), we are th us reduced to prov e (i) and (v). A.7.1. Pr o of of (i) . A lifting problem of shap e Λ i [ n ] / /   P   ∆[ n ] / / = = Q 0 < i 6 n 38 is equiv a len t to a lifting pro blem o f shape C / /   X p   Ω[ n ⋆ e T ] / / : : Y where C is the union of Λ i [ n ] ⋆ e T with the union of the f aces of Ω[ n ⋆ e T ] which are of the form n ⋆ e S − → n ⋆ e T , wher e S r anges ov er the e -admissible elemen tary faces of T . In o ther w o rds, C = Λ i [ n ⋆ e T ] is a n inner horn, so that the r equired lifting exists, b ecause p is assumed to b e an inner Kan ﬁbration. A.7.2. Pr o of of (v) . A lifting problem of shap e Λ i [ n ] / /   U   ∆[ n ] / / ; ; i ∗ ( X ) 0 < i 6 n is equiv a len t to a lifting pro blem o f shape C / /   X D > > in whic h the inclusion C − → D can be describ ed as follows: C = Ω[ n ] ∪ [ R Λ i [ n ] ⋆ e R ⊂ D = [ R Ω[ n ⋆ e R ] ⊂ Ω[ n ⋆ e T ] , where R r anges ov er the e -admissible elemen ta ry faces of T . It is easily seen that the inclusion C − → D is an inner ano dyne extension by Lemma A.4.  B. Another subdivision of cylinders B.1. W e will refer to the ho rn inclus ions o f shap e Λ x [ S ] − → Ω[ S ], where S is a tree with a unary top vertex x , as end extensions . A comp osition of pushouts of end extensions will b e called an end ano dyne map. The goal of this sectio n is to prov e a dual v er sion of Theorem 5.2, namely: Theorem B.2. L et T b e a tr e e with at le ast one vertex, and c onsider t he su b obje ct B 0 = { 0 } ⊗ Ω[ T ] ∪ ∆[1 ] ⊗ ∂ Ω[ T ] ⊂ ∆[1] ⊗ Ω[ T ] . Ther e exists a ﬁltr ation of ∆[1 ] ⊗ Ω[ T ] of the form B 0 ⊂ B 1 ⊂ . . . ⊂ B N − 1 ⊂ B N = ∆[1] ⊗ Ω[ T ] wher e, for e ach i , 0 6 i < N , the map B i − → B i +1 is either inner ano dyne or end ano dyne. 39 Mor e over, the end ano dyne maps ar e al l push outs of the form Λ v [ S ] / /   B i   Ω[ S ] / / B i +1 with t he fol lowing pr op erties: (i) the t r e e S has at le ast two vertic es, and v is a unary top vertex; (ii) the m ap ∆[1] − → Λ v [ S ] − → B i ⊂ ∆[1] ⊗ Ω[ T ] , c orr esp onding to the vertex v in S , c oincides with an inclusion of shap e ∆[1] ⊗ { t } − → ∆[1] ⊗ Ω[ T ] for some e dge t in T . B.3. As in the pro of of Theorem 5.2, we will follow the con ven tion of [MW09], and write Ω[ S ] ⊗ Ω[ T ] = m [ i =1 Ω[ T i ] , where the union ranges ov er the partia lly or dered set of p ercola tion sc hemes , start- ing with a num b er of copies of T g rafted on top of S , and ending with the reverse grafting. F or S = [1] = ◦ , the ﬁrst tree is of shap e T 1 = • T P P P P P P P P ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ◦ , (B.3.1) and the last one is T m = ◦ ◦ ... ◦ ◦ • T P P P P P P P P ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ . (B.3.2) Let us ﬁx a linear order on the p ercolation sc hemes for ∆[1] ⊗ Ω[ T ] which extends the natura l partial orde r . Such a linear ordering induces a ﬁltra tio n on the tensor pro duct ∆[1] ⊗ Ω[ T ], (B.3.3) C 0 ⊂ C 1 ⊂ . . . ⊂ C m − 1 ⊂ C m = ∆[1 ] ⊗ Ω[ T ] by setting (B.3.4) C 0 = B 0 = { 0 } ⊗ Ω[ T ] ∪ ∆[1 ] ⊗ ∂ Ω[ T ] and C i = B 0 ∪ Ω[ T 1 ] ∪ · · · ∪ Ω[ T i ] . the ﬁltration of Theorem B.2 will b e a reﬁnement of this one. 40 Let us start by considering T 1 . If th e r oot edge of T is ca lle d r , then T 1 lo oks like . . . . . . . . . . . . • ... ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❈ ❈ ❈ ❈ ④ ④ ④ ④ ◦ (0 ,r ) (1 ,r ) (B.3.5) With the exception of the faces ∂ (0 ,r ) ( T 1 ) (which contracts (0 , r )) and ∂ (1 ,r ) ( T 1 ) (whic h chops oﬀ (1 , r ) as well a s the white vertex), an y face F of T 1 misses a co lour of T (by this, w e mean there is an edge a in T such that no edge in F is na med ( i, a )). Hence, Ω[ F ] ⊂ ∆[1] ⊗ ∂ Ω[ T ] for these F . Moreover, ∂ (1 ,r ) ( T 1 ) = { 0 } ⊗ T 1 . So Ω[ T 1 ] ∩ B 0 = Λ (0 ,r ) [ T 1 ], and Λ (0 ,r ) [ T 1 ] / /   B 0   Ω[ T 1 ] / / B 0 ∪ Ω[ T 1 ] (B.3.6) is a pushout. So, if we let B 1 = C 1 , then B 0 − → B 1 is obviously inner anodyne. Suppo se w e have deﬁned a ﬁltr a tion up to some B l (B.3.7) B 0 ⊂ B 1 ⊂ . . . ⊂ B l l > 1 , so that B l = C k for some k , 1 6 k 6 m . W e will extend this ﬁltration as B l ⊂ B l +1 ⊂ . . . ⊂ B l ′ , so that B l ′ = C k +1 . The percola tion scheme T k +1 is obtained from a n earlier one T j by pushing a white vertex in T j one s tep up thro ugh a black vertex x , as in • ... ■ ■ ■ ■ ■ ■ ✉ ✉ ✉ ✉ ✉ ✉ ◦ x in T j = ⇒ ◦ ✼ ✼ ✼ ✼ ◦ ✞ ✞ ✞ ✞ • ▼ ▼ ▼ ▼ ▼ ▼ ▼ q q q q q q q ... x in T k +1 (B.3.8) (w e hav e denoted by x the black vertex in b oth trees, although it would b e more accurate to wr ite x for the r elev ant v e r tex of T , and write 0 ⊗ x and 1 ⊗ x for the corres p onding vertices in T j and T k +1 rep ectiv ely ). The Bo ardman-V o gt relation states that, as subo b jects of ∆[1] ⊗ Ω[ T ], the face of T k +1 obtained by co n tracting all input edges of x coincides with the face of T j obtained by co ntracting the output edge of x in T j . In particular, notice that if x has no input edges at all (i.e. if x is a ‘nullary op eration’ in T ), then T k +1 is a face o f T j , so C k +1 = C k , and we let B l ′ = B l , and there is nothing prov e. There fo re, from now on, we will ass ume that the s et of input e dges of x , denoted input ( x ), is non-empt y , and we pro ceed as follows. Let E be the set of a ll c o lours (edges) e in T for which ◦ (0 ,e ) (1 ,e ) (B.3.9) 41 o ccurs in T k +1 . F o r U ⊂ E , let (B.3.10) T ( U ) k +1 ⊂ T k +1 be the face given b y contracting all the edges (1 , e ) fo r e ∈ E but e not in U . Notice that if U ∩ input ( x ) = ∅ , then Ω [ T ( U ) k +1 ] ⊂ B l by the Boa rdman-V og t relation just men tioned. Therefore, we w ill only consider U with U ∩ input ( x ) 6 = ∅ . W e will successively adjoin Ω[ T ( U ) k +1 ] to B l for larger and larg er such U , un til we reach the case where U = E and T ( U ) k +1 = T k +1 . If U = { e } is a sing leton (with e an input edge o f x ), then the face ∂ (1 ,e ) Ω[ T ( { e } ) k +1 ] is contained in B l as said, while the face ∂ (0 ,e ) Ω[ T ( { e } ) k +1 ] is not (it cannot b elong to an earlier T i , and is o b viously not con tained in B 0 = C 0 ). An y o ther face of T ( { e } ) k +1 misses a colour of T and hence is contained in B 0 . Thus, (B.3.11) Ω[ T ( { e } ) k +1 ] ∩ B l ⊂ Ω[ T ( { e } ) k +1 ] is either an inner horn (if (0 , e ) is an inner edge of T k +1 ) or an end extension (if (1 , e ) is an input edge of T k +1 ). In either cas e, w e can adjoin Ω[ T ( { e } ) k +1 ] by for ming the pushout below. Ω[ T ( { e } ) k +1 ] ∩ B l / /   B l   Ω[ T ( { e } ) k +1 ] / / Ω[ T ( { e } ) k +1 ] ∪ B l (B.3.12) W e successively adjoin Ω[ T ( { e } ) k +1 ] to B l in this wa y for all e in E which are input edges of x in T : if these are e 1 , . . . , e p , let (B.3.13) B l + r = B l ∪ Ω[ T ( { e 1 } ) k +1 ] ∪ · · · ∪ Ω[ T ( { e r } ) k +1 ] . Then, for each r < p , the map B l +1 − → B l + r +1 is inner ano dyne or end ano dyne. In gener al, we pro ceed by induction on U . Cho ose U ⊂ E with U ∩ input ( x ) 6 = ∅ , and a s sume we hav e adjoined Ω[ T ( U ′ ) k +1 ] already , for all U ′ of smaller c ardinalit y than U . W e will write B l ′′ for the last ob ject in the ﬁltration constructed up to tha t po in t. Fix an or der o n the set of elemen ts o f U , a nd write it as (B.3.14) U = { α 1 , . . . , α s } . Consider Ω[ T ( U ) k +1 ]. The tree T ( U ) k +1 has edges (0 , c ) or (1 , c ) for c not in E , and the corres p onding face misses the c o lour c alltog e ther , hence ∂ ( i,c ) Ω[ T ( U ) k +1 ] is con ta ined in B 0 for these c . Next, the tree T ( U ) k +1 has edges colour e d (1 , α i ) for 1 6 i 6 s , and contracting a n y of these gives a face (B.3.15) ∂ (1 ,α i ) Ω[ T ( U ) k +1 ] = Ω[ T ( U −{ α i } ) k +1 ] which is contained in B l ′′ by the inductiv e a ssumption on U . None of the faces given b y con tr acting (if it is inner) or by chopping oﬀ (if it is outer) a n edge (0 , α i ) in T ( U ) k +1 can be contained in B l ′′ , ho wev er. Let A 1 , . . . , A t be all the subsets of the set of these edges { (0 , α 1 ) , . . . , (0 , α s ) } of T ( U ) k +1 which cont ain (0 , α 1 ), and or de r them by some linear o r der extending the 42 inclusion order. So there are t = 2 s − 1 such A i , and we could ﬁx the order to be A 1 = { (0 , α 1 ) } A 2 = { (0 , α 1 ) , (0 , α 2 ) } . . . . . . A s = { (0 , α 1 ) , (0 , α s ) } A s +1 = { (0 , α 1 ) , (0 , α 2 ) , (0 , α 3 ) } . . . . . . A t = { (0 , α 1 ) , . . . , (0 , α s ) } . F o r q = 1 , . . . , t , let T ( U,q ) k +1 be the tree o btained fro m T ( U ) k +1 by contracting or chopping oﬀ all the edges (0 , α i ) not in A q . So (B.3.16) T ( U, 1) k +1 = ∂ (0 ,α s ) ∂ (0 ,α s − 1 ) . . . ∂ (0 ,α 2 ) T ( U ) k +1 , and (B.3.17) T ( U,t ) k +1 = T ( U ) k +1 . W e will succe s siv ely adjoin these Ω[ T ( U,q ) k +1 ] to the ﬁltration, to for m the par t (B.3.18) B l ′′ ⊂ B l ′′ +1 ⊂ . . . ⊂ B l ′′ + t = B l ′′ ∪ Ω[ T ( U ) k +1 ] of the ﬁltration, as (B.3.19) B l ′′ + q = B l ′′ ∪ Ω[ T ( U, 1) k +1 ] ∪ . . . Ω[ T ( U,q ) k +1 ] . W e start with T ( U, 1) k +1 . The only face of Ω[ T ( U, 1) k +1 ] not contained in B l ′′ is the one given by the edge (0 , α 1 ). Thu s (B.3.20) Ω[ T ( U, 1) k +1 ] ∩ B l ′′ ⊂ Ω[ T ( U, 1) k +1 ] is either an inner horn (if (0 , α 1 ) is a n inner edge) or an end extension (if (0 , α 1 ) is an input edge of T ( U ) k +1 , i.e. α 1 is a n input edge of T ). So the pushout B l ′′ − → B l ′′ +1 is either inner ano dyne or end ano dyne. Suppo se we have adjoined Ω[ T ( U,q ′ ) k +1 ] for all 1 6 q ′ < q , so ha ve ar r iv ed at the stage B l ′′ + q − 1 of the ﬁltration. Consider now A q and the co rresp onding dendroida l set Ω[ T ( U,q ) k +1 ]. As befor e, its faces giv en b y edge s colo ured b y ( i, c ) for i = 0 , 1 with c not in E are contained in B 0 , and its fa ces g iv en b y edges coloured (1 , e ) with e ∈ U are contained in Ω[ T ( U ′ ) k +1 ] for a smaller U ′ = U − { e } , hence are contained in B l ′′ . Let us cons ide r t he remaining faces g iv en by the edges (0 , α 1 ) , . . . , (0 , α r ) in A q . If i 6 = 1 , the fa c e of Ω[ T ( U,q ) k +1 ] given b y (0 , α i ) ∈ A q is contained in Ω[ T ( U,q ′ ) k +1 ] for some q ′ < q with A q ′ = A q − { (0 , α i ) } . So the only face that is missing is the one given by (0 , α 1 ), i.e. (B.3.21) Ω[ T ( U,q ) k +1 ] ∩ B l ′′ + q − 1 = Λ (0 ,α 1 ) [ T ( U,q ) k +1 ] . Therefore, the induced pushout B l ′′ + q − 1 − → B l ′′ + q is either inner ano dyne or end ano dyne (depending o n whether α 1 is an external edg e o f T o r not). At the end, when q = t , we hav e adjoined all of Ω[ T ( U ) k +1 ]. 43 This co mpletes the contruction of the se g men t o f the ﬁltratio n for the subset U ⊂ E . As said, we contin ue this construction until we reach the stage U = E , when T ( U ) k +1 = T k +1 , whic h c o mpletes the construction of the segmen t of the ﬁltration from B l un til B l ′ , in terp olating b et ween C k and C k +1 . This completes the description of the ﬁltration. F rom it, the last part of the theo rem is clear. Erra tum Prop osition 1.9 is wr ong as sta ted, and should be mo diﬁed as w e will explain below. This mo diﬁcation do es not a ﬀect an y of the main results of this pape r and its t wo sequels [CM13a, CM13b]: the existence of the model str ucture o n dendro idal sets of the present paper , the equiv alent mo del structures for dendroidal co mplete Segal spaces and for Segal op erads in [CM13a], and the Quillen equiv alence to the mo del category of simplicial operads in [CM1 3b]. How ever, the err o r do es aﬀect all the statements c o ncerning the monoidalit y of the model structures. Recall the inclusion i : ∆ → Ω, and the induced left Quille n functor i ! : sSet → dSet . Let us ca ll a tree (an ob ject of Ω) line ar if it lies in the imag e of i , and a dendroidal set simplicial if it lies in the image of i ! . Also, let us call a tree op en if it has no vertices of v alence zero (i.e. no vertices without input edges). If S → T is a morphism in Ω and T is op en, then S is necessa r ily op en as well. Therefor e, the op en tr ees deﬁne a subo b ject U of the terminal ob ject in dSet . Let us call a dendroidal set op en if the unique ma p to the terminal o b ject factors thro ugh U . These op en dendroidal sets form a full sub category dSet /U of dSet . Mor e generally , we ca ll a nor mal monomorphism X → Y line ar if it is obtained by attaching linear trees (i.e. X → Y lie s in the saturation of ∂ Ω[ T ] → Ω[ T ] for T linea r) and op en if it is obtained by attac hing op en trees. The mo diﬁed version o f Pr opositio n 1.9 sho uld be: Prop osition 1.9. L et A → B and X → Y b e normal monomorp hisms. If o ne of them is line ar or b oth ar e op en, then the induc e d map A ⊗ Y ∪ A ⊗ X B ⊗ X → B ⊗ Y is agai n a normal m onomorp hism (and is op en as wel l in the se c ond c ase). By the usual induction, this follows fr om the following lemma (whose pro of is an elementary but tedious combinatorial a rgument , and we refer the reader to [CMn] for the details), which should be added a t the v e ry end of the ﬁrst section: Lemma 1.11. L et S and T b e tr e es. If one of them is line ar or b oth ar e op en, then the pushout-pr o du ct map ∂ Ω[ S ] ⊗ Ω[ T ] ∪ ∂ Ω[ S ] ⊗ ∂ Ω[ T ] Ω[ S ] ⊗ ∂ Ω[ T ] → Ω[ S ] ⊗ Ω[ T ] is a normal monomorphism (and is op en as wel l in the se c ond c ase). The main r esult of [MW09] (namely Prop osition 9.2 ) is wrong as stated, but its pro of sa ys the following: given t wo trees S and T , as well as an inner edge e of S , if W deno tes the the image of the map Λ e [ S ] ⊗ Ω[ T ] ∪ Λ e [ S ] ⊗ ∂ Ω[ T ] Ω[ S ] ⊗ ∂ Ω[ T ] → Ω[ S ] ⊗ Ω[ T ] , then the inclusion W ⊂ Ω[ S ] ⊗ Ω[ T ] 44 is inner a nodyne. This means that Pr opos ition 3.1 should b e replaced b y the following statement. Prop osition 3.1. L et A → B and X → Y b e normal monomorphisms. I f one of them is line ar or b oth ar e op en, and if one o f them is inner ano dyne, then the induc e d map A ⊗ Y ∪ A ⊗ X B ⊗ X → B ⊗ Y is agai n an inner ano dyne ex tension. Prop osition 3.3 should b e mo diﬁed a ccordingly (replacing, in Pr opositio n 3.1 ab o ve, the expression “inner ano dyne” b y “ J -ano dyne”). Let us call a mo del ca tegory M Joyal simplicia l if it satisﬁes the a xioms for a simplicial model category , but with resp ect to the Joyal mo del structure on simplical sets instead of the classical Kan-Q uillen structure. The mo de l structure on dSet established in this pap er is not mono idal. Instead, Prop osition 3.17 sho uld b e replaced by the follo wing result, whic h fo llows immediately from the propo sitions ab o ve and from the arguments expla ined in the o riginal ‘proo f ’ of Pr op. 3.17 : Prop osition 3.17. The Bo ar dman-V o gt tensor pr o duct t urns the mo del st ructur e on dSet int o a Joyal simplicial mo del structure , and induc es a symmetric monoidal mo del st ructur e on the c ate gory dSet / U of op en dendr oidal sets. In the pro of o f the existence of the mo del s tructure, it is only the “line a r half ” of P ropo sitions 1.9 and 3.1 whic h are used, and this pr oof is una ﬀected. Indeed, Prop ositions 3.1 and 3 .3 are used only in the case where one of the maps is in the image of i ! . W e c lo se this err atum with a list of places where the reference to (co nsequences of ) monoidality should be reform ula ted in accor dance with the previo us tw o prop o- sitions: In the pres en t pap er, these are po in t 1 in the int ro duction, Prop osition 2.6(c), Co rollary 2 .8(b), Coro llary 2.9(b); In the pr oof o f Lemma 6.15 , the ﬁrst sentence should b egin as ‘W e know that the image by the functor i ∗ of the map . . . ’; Also the statement b et ween br ac kets at the very end of Prop osition 6.20 s ho uld b e skipp ed (and the pro of of Prop. 6.20 sho uld refer to Boar dman and V og t’s explicit description of the category asso ciated to a quasi-catego ry , instead of its dendroidal analogue). In [CM13a, CM13b], no use of the alleg e d monoidality of the mo del structure is made, and the neces sary changes all co ncern inessential reference s to the monoida lity: in pa per [CM13a], the places where we recall the monoidality from the present pap er, ar e in the abs tr act and the introduction, in Theorem 1.1 , Remarks 6 .14 and 8.16, and ﬁnally in the pro of of Propp osition 6.11, wher e it is Prop osition 3.1 ab ov e whic h should be used rather than the monoidalit y; and in pap er [CM13b], the mentioning of monoidality is in the intro duction as well as in Prop osition 2.8 and Theorem 5.7. W e repeat that this er ror o nly aﬀects the mono idalit y of the mo del structures , and no ne of the main results about the existence of the mo del structure o n dSet and the Quillen equiv alent model categ ories pr esen ted in [CM13 a, CM13b]. References [Ber07] J. Bergner, Thr e e mo dels f or the homotop y the ory of homo topy the ories , T op ology 46 (2007), 397–436 . 45 [BV73] J. M. Boardman and R. M. V ogt, Homotopy invariant algebr aic struct ur es on top olo gica l sp ac es , Lecture Notes in Math., vol. 347, Springer- V erlag, 1973. [Cis06] D.-C. Cisinski, Les pr´ efaisc e aux c omme mo d ` eles des typ es d’ho motopie , Ast´ eri sque, v ol. 308, Soc. M ath . F rance, 2006. [CM13a] , Dendr oidal Se g al sp ac es and ∞ - op er ads , J. T op ol. 6 (2013), no. 3, 675–704. [CM13b] , Dendr oidal sets and simpl icial op er ads , J. T op ol. 6 (2013), no. 3, 705–756. [CMn] , Note on the t e nsor pr o duct of dendr oidal sets . [Cra95] S. E. Cr ans, Quil len close d mo del structur es for she aves , J. Pure A ppl. Algebra 101 (1995), 35–57. [GZ67] P . Gabriel and M. Zisman, Calculus of fr act ions and homotopy the ory , Ergebnisse der Mathematik, v ol. 35, Spri nger-V er l ag, 1967. [Hov9 9] M. Hov ey , Mo del ca t e gories , Math. surveys and monographs, vol. 63, Amer. M ath . So c., 1999. [Joy 02] A. Joy al, Quasi-ca te gories and K an c omplexes , J. Pure Appl. Algebra 1 75 (2002), no. 1- 3, 207–22 2. [JT07] A. Joy al and M . Ti erney , Quasi-c ate gories vs Sega l sp ac es , Categories in Algebra, Ge- ometry and Ph ysics, Contemp. Math., v ol. 431, Amer . M ath . So c., 2007, pp. 277–326. [Lur06] J. Lur ie, Hig her top os the ory , ar Xiv:math/0608 040, 2006. [MW07] I. Moerdij k and I. W eiss , De ndr oidal sets , Algebraic & Geometric T op ology 7 (2007), 1441–147 0. [MW09] , On inner Kan c omplexes in the c ate g ory of dendr oidal sets , Adv. Math. 221 (2009), no. 2, 343–38 9. [W ei07] I. W eiss, Dendr oidal sets , PhD thesis, Univ er s iteit Utrech t, 2007. LA G A, CNRS (UMR 7539 ), Universit ´ e P aris 13, A venue J ean-Baptiste Cl ´ ement, 9 3430 Vil- let aneuse, France E-mail addr ess : ci sinski@ma th.univ-paris13.fr URL : http://ww w.math.un iv-paris13.fr/~cisinski/ Ma thema tisch Instituut, Universiteit Utrecht, PO.Box 80.010 , 3508 T A Utrecht, The Netherlands E-mail addr ess : mo erdijk@ma th.uu.nl URL : http://ww w.math.uu .nl/people/moerdijk/

Dendroidal sets as models for homotopy operads

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