Homotopy theory of non-symmetric operads

HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS FERNANDO MURO Abstract. W e endo w cate gories of non-symmetric op erads with natural mo del structures. W e wo rk with no restriction on our operads and only assume the usual h ypotheses for mo del cate gories with a symmetric monoidal struc- ture. W e also study cat egories of algebras o ver th ese op erads in enrich ed non-symmetric monoidal model categories. Contents 1. Int ro duction 1 2. Op erads 4 3. T rees 7 4. The monoidal category of morphisms 14 5. The relev an t operad push-out 17 6. Pro of of Theorem 1.1 41 7. Algebras 42 8. The relev an t algebra push-out 45 9. Pro of of Theorems 1.3 and 1.5 48 10. An application to enriched categ ories and A ∞ -categor ies 51 References 53 1. In troduction Op erads are w ell-known devices e nco ding the la ws of algebras deﬁned b y m ul- tilinear op erations and relations, e.g . there are op er ads Ass , C om and Li e whose algebras a re ass o ciative, commutativ e and Lie alg ebras, resp ectively . Mor phisms of op erads co dify r elations b et ween diﬀeren t kinds of alg ebras, e.g. there are mor- phisms Lie → Ass → Com telling us that any co mm utative alg ebra is an asso ciative alegbra, and that comm utators in an asso ciative alg ebra yield a Lie a lgebra. There are tw o kinds of op erads: sy mmetric and non-sy mmetric op erads. Sym- metric oper ads are needed whenev er it is necessary to permute v ariables in order to describ e the la ws of the corresp onding a lgebras, e.g. Co m and Lie . Non-symmetric op erads ar e sp ecially useful to deal with algebr as in non-sy mmetr ic monoida l cat- egories, e .g. given a co mm utative ring k and a s e t S w hich is not a singleton, the 2010 M athematics Subje ct Classiﬁc ation. 18D50, 55U35, 18D10, 18D35, 18D20 . Key wor ds and phr ases. op er ad, algebra, model category , enriched A ∞ -category . The author was partially supp orted by the Spanish M inistry of Education and Science under the MEC-FEDER grants MTM 2007-63277 and MTM 2010-15831, and by the Go ve rnment of Catalonia under the gr an t SGR-119-2009. 1 2 FERNANDO M URO category o f k -mo dules w ith ob ject set S , whic h ar e collections of k - mo dules indexed by S × S , M = { M ( x, y ) } x,y ∈ S , has a non-symmetric tensor product, ( M ⊗ S N )( x, y ) = M z ∈ S M ( z , y ) ⊗ k N ( x, z ) , whose ass o ciative algebra s, i.e. algebr as ov er the op era d Ass , a re k -linea r categ ories with ob ject set S . An y ob ject M in a symmetric monoidal category V , such as the categ ory of k -mo dules, has an endomo rphism symmetric oper ad End V ( M ) in V suc h that, if O is another symmetric operad in V , the s e t of O - algebra structures on M is the set of symmetric op erad morphis ms O → En d V ( M ). If M b elongs to a non-symmetric monoidal categ ory C enriched over V , s uch a s the catego r y of k - mo dules with ob ject set S , then there is a no n-symmetric o p e r ad End C ( M ) in V such that the set of algebr a structures on M ov er a non-symmetric o pe rad O in V is the set of non-symmetric oper ad morphisms O → End C ( M ) (Deﬁnition 7.1). When the underlying symmetric monoidal ca tegory V carries homotopical in- formation, e.g. if we r eplace k -mo dules with diﬀerential gr aded k - mo dules, one is often more in terested in a s pace of O -a lgebra structures on M ra ther than a plain set. Suc h a space ca n b e cons tr ucted by using the p ow erful machinery developed by Dwyer and Kan [DK80c, D K80 a, DK80b] provided w e ca n place the op erads O and End C ( M ) in an appropriate model categ ory of opera ds . Mo del ca teg ories of op er ads were ﬁr s t cosidered by Hinich in the diﬀer ent ial graded co nt ext [Hin97, Hin0 3], and by Berg er and Mo er dijk in a more gener al setting [BM0 3]. They dealt with s ymmetric o p erads a nd showed that restrictive hypotheses are necess a ry to endow the categ ory o f all op er ads with an appro pri- ate mo del catego ry structure, e.g. when k is a Q -algebr a or when the symmetric monoidal structure in V is ca rtesian closed a nd there is a symmetric mo noidal ﬁbrant r e pla cement functor. Motiv ated b y our interest in s paces of diﬀerential g raded catego ry structur es, we conside r the non-s ymmetric case, whic h surpris ing ly enough does not need any restrictive hypothesys, just usual hypotheses for model categor ies with a monoidal structure [SS00]. Theorem 1.1. L et V b e a c oﬁbr antly gener ate d close d symmetric monoidal mo del c ate gory. Assume that V satisﬁes the monoid axiom. Mor e ove r, supp ose that ther e ar e sets of gener ating c oﬁbr a tions and gener ating trivial c oﬁbr ations in V with pr esentable sourc es. Then the c ate gory Op( V ) of non-symmetric op er ads in V is a c oﬁbr ant ly gener ate d mo del c ate gory su ch that a m orphism f : O → P in O p( V ) is a we ak e quivalenc e (r esp. ﬁbr ation) if and only if f ( n ) : O ( n ) → P ( n ) is a we ak e quivale nc e (r esp. ﬁbr ation) in V for al l n ≥ 0 . Mor e over, if V is right pr op er then so is Op( V ) . F u rthermor e, if V is c ombinatorial then Op( V ) is also c ombinatorial. This theor em can b e applied to all examples in [SS00 ], see also the refer ences therein: (1) Complexes o f mo dules ov er a c ommut ative ring k with the usual tensor pro duct of complexes. (2) Simplicial k -mo dules with the lev elwise tensor pro duct ⊗ k . (3) Mo dules ov er a ﬁnite-dimensional Ho pf algebra R ov er a ﬁe ld k with the tensor pro duct ov er k , e.g. R = k G the gro up-ring of a ﬁnite g r oup G . HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 3 (4) Symmetric sp ectr a with their smash pro duct, and more generally modules ov er a co mmu tative ring spectrum. (5) Γ-spa ces with Lydakis’ smash product. (6) Simplicial functors with their smas h product. (7) S -mo dules with their smas h pro duct. In particula r, Theorem 1.1 will also be useful to study spaces of sp ectra l categor y structures. R emark 1.2 . Recall from [AR94, D eﬁnition 1.13 (2)] that an ob ject X of V is pr e- sentable if there exis ts a cardina l λ such that the representable functor V ( X, − ) commutes with λ -ﬁlter ed co limits in V . Presentable ob jects a re also calle d smal l or c omp act in some refer e nc e s. All ob jects are presentable in many ca tegories of int erest, e.g. in all combinatorial mo del categor ies. Actually , up to set theoreti- cal principles any coﬁbr antly genera ted mo del categ ory is Q uillen equiv alent to a combinatorial mo del categ ory [Rap09]. Categorie s o f algebras over symmetric o p erads do not alwa ys hav e a mo del struc- ture with ﬁbratio ns and weak equiv alences deﬁned in the underlying category . Suf- ﬁcient conditions ca n b e found in [B M03]. In the framework of non-symmetric op erads they do. When b oth alge br as and op er ads live in the same ambien t sym- metric monoida l mo del categ ory V , s a tisfying the monoid axiom, this has b een recently pro ved b y J. E. Harp er [Har1 0, Theorem 1 .2]. W e here extend this result to alg ebras in a monoidal mo del c ategory C sa tisfying the monoid ax io m and ap- propriately enriched ov er V . This is nece s sary , for instance, to construct mo del categorie s of enrie ched categories, of enric hed A ∞ -categor ies, o r of any other cate- goriﬁed algebraic structure, see Section 10. Theorem 1. 3. L et V and C b e c oﬁbr antly gener ate d bi close d monoidal mo del c ate- gories. Supp ose V is symmetric and C has a V - algebr a st ructur e given by a str ong br aid e d monoidal functor to t he c enter of C , z : V → Z ( C ) , such that the c omp osite functor V z − → Z ( C ) − → C is a le ft Quil len fun ctor. Mor e ov er, assume that V and C satisfy the monoid axio m (se e D eﬁ nitions 6.1 and 9.1). F urthermor e, supp ose that C has sets of gener ating c oﬁbr ations and gener ating tr ivial c oﬁbr ations with pr esentable sour c e. Le t O b e a n on-symmetric op er ad in V . The c ate go ry Alg C ( O ) of O -algebr as in C is a c oﬁbr antly gener ate d mo del c ate go ry such t hat an O -algebr a morphism g : A → B is a we ak e quivalenc e (r esp. ﬁbr ation) if and only if g is a we ak e quival enc e (r esp. ﬁbr ation) in C . Mor e over, if C is right pr op er then so is Alg C ( O ) . F urthermor e, if C is c ombinatorial then Alg C ( O ) is also c ombinatoria l. The no tion of monoidal mo del catego ry in [SS00 , Deﬁnition 3.1] makes sense with no modiﬁcatio n in the non-symmetric con text, see Deﬁnition 4.2. An y op era d mo rphism φ : O → P induces a c hange of oper ad functor φ ∗ : Alg C ( P ) − → Alg C ( O ) by res tr icting the action of P to O along φ . This functor is the identit y on under- lying ob jects in C , hence it pres erves ﬁbrations and w eak equiv alences. Moreover, 4 FERNANDO M URO the functor φ ∗ has a left adjoint φ ∗ , therefore w e hav e a Q uillen adjunction, (1.4) Alg C ( O ) φ ∗ / / Alg C ( P ) . φ ∗ o o The following result esta blishes co nditions so that this is a Quillen equiv a lence if φ is a w eak equiv alence of opera ds. These conditions are the non-symmetric analog ues of those considered in [BM03] for symmetric opera ds. Theorem 1.5. In the c onditions of the pr evio us the or em, assume further that C is left pr op er. L et φ : O → P b e a we ak e quivale nc e b etwe en op er ads in V s u ch that for al l n ≥ 0 the obje cts O ( n ) and P ( n ) ar e c oﬁbr ant in V . Then (1.4) is a Quil len e quivalenc e, in p articular the de rive d ad joint p air is an e quiva lenc e b et we en the homotopy c ate go ries of algebr as, Ho Alg C ( O ) L φ ∗ / / Ho Alg C ( P ) . φ ∗ o o This result will be use ful to show that in many examples the homotopy theory of enriched catego ries co incides with the ho motopy theor y of A ∞ -categor ies, e.g. when the underlying symmetric monoida l categ ory V is any of the c ategories in the examples (1)–(6) listed ab ove, see Section 10. Ac kno wledgements. The author wishes to thank Michael Batanin, Clemens Ber- ger, Benoit F r esse, Javier J. Guti´ errez, Ieke Mo er dijk, Andy T o nks a nd Bruno V allette for conversations related to the conten ts of this pap er , in particula r fo r providing very in teresting references. Notation. Throug hout this paper V and C will denote complete and co complete biclosed monoida l categories [Kel0 5, 1 .5] with tens or pro duct X ⊗ Y and unit ob ject I V and I C , r esp ectively . W e dr op the subscript when it is clear from the context. The ca tegory V will b e s y mmetric and internal morphism ob jects in V will b e denoted b y Hom( X , Y ). W e will add homotopical hypotheses when needed. 2. Operads In this section w e recall the w ell-known no tion of non-symmetric op erad. Deﬁnition 2.1 . The category V N of se q uenc es o f ob jects V = { V ( n ) } n ≥ 0 in V is the pro duct o f co un tably ma ny copies of V . It ha s a r ight-closed no n-symmetric monoidal structure giv en b y the c omp osition pr o duct U ◦ V , ( U ◦ V )( m ) = a n ≥ 0 a n P i =1 p i = m U ( n ) ⊗ V ( p 1 ) ⊗ · · · ⊗ V ( p n ) . The unit ob ject is I ◦ , I ◦ ( n ) =  I , the unit of ⊗ in V , if n = 1 ; 0 , the initial ob ject of V , if n 6 = 1 . R emark 2 .2 . The fact that ◦ is no n-symmetric is ob vious from the v ery deﬁnition. One can eas ily chec k by wr iting down explicitly the formulas o f ( U ◦ V ) ◦ W and HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 5 U ◦ ( V ◦ W ) how the s ymmetry co nstraint of ⊗ is used to deﬁne the asso cia tivit y constraint of ◦ . The right a djoint of − ◦ V is the functor Hom ◦ ( V , − ) deﬁned by Hom ◦ ( V , W )( n ) = Y p 1 ,...,p n ≥ 0 Hom( V ( p 1 ) ⊗ · · · ⊗ V ( p n ) , W ( p 1 + · · · + p n )) , in particular − ◦ V pr eserves a ll colimits. On the contrary , the functor U ◦ − do es not preserve all colimits, but it do es preserve ﬁlter ed colimits. R emark 2.3 . If V is a mo del catego ry then the pro duct ca teg ory V N is also a mo del category with ﬁbrations, coﬁbrations and weak equiv alences deﬁned co or- dinatewise [Hov99 , Ex ample 1.1.6]. Mor e over, if V is coﬁbrantly gener ated (re s p. combinatorial) then V N is also coﬁbrantly gener a ted (resp. combinatorial). Indeed, let I b e a set of g enerating coﬁbrations and J a s et of gener ating tr ivial coﬁbrations in V . F or any n ≥ 0, let s n : V → V N be the left adjoint of the pro jection on to the n th factor, which is deﬁned by ( s n ( V ))( m ) =  V , if m = n ; 0 , the initial ob ject, if m 6 = n. Given a set S o f morphisms in V we cons ider the following set of morphisms in V N , S N = [ n ≥ 0 s n ( S ) . The sets I N and J N are sets o f genera ting coﬁbrations a nd generating tr ivial coﬁ- brations in V N , respec tively . Deﬁnition 2. 4. A non-symmetric op er ad O in V is a monoid in the monoida l category of sequences V N with the compo sition pro duct ◦ . R emark 2.5 . The previo us condensed deﬁnition of an op era d O can b e unrav eled by noticing that the multiplication µ : O ◦ O → O consists of a series of m ultiplication morphisms, 1 ≤ i ≤ n , p i ≥ 0, µ n ; p 1 ,...,p n : O ( n ) ⊗ O ( p 1 ) ⊗ · · · ⊗ O ( p n ) − → O ( p 1 + · · · + p n ) . The asso ciativity condition amounts to say that the following diagra m is alwa ys commutativ e, O ( n ) ⊗ n N i =1 O ( p i ) ⊗ p i N j =1 O ( q ij ) !  O ( n ) ⊗ n N i =1 O ( p i )  ⊗ n N i =1 p i N j =1 O ( q ij ) O ( n ) ⊗ n N i =1 O p i P j =1 q ij ! O  n P i =1 p i  ⊗ n N i =1 p i N j =1 O ( q ij ) O n P i =1 p i P j =1 q ij ! ∼ = assoc. and sym.   id ⊗ n N i =1 µ 5 5 k k k k k k k µ ⊗ id ) ) R R R R R R R µ   ? ? ? ? ? ? ? ? ? µ = = z z z z z z z z z z Here the or der of tensor fa c tors in N n i =1 N p i j =1 O ( q ij ) is determined by th e le x ico- graphic order of the pa ir ( i, j ). Moreov er, the unit is just a morphism u : I → O (1) 6 FERNANDO M URO such that the following morphisms are (compositions of ) unit constraints in V , I ⊗ O ( n ) u ⊗ id / / O (1 ) ⊗ O ( n ) µ 1; n / / O ( n ) , O ( n ) ⊗ I ⊗ n id ⊗ u ⊗ n / / O ( n ) ⊗ O (1) ⊗ n µ n ;1 ,..., 1 / / O ( n ) . R emark 2.6 . The m ultiplication morphisms in the pr evious remar k are determined by the following m orphisms , 1 ≤ i ≤ m , n ≥ 0, ◦ i : O ( m ) ⊗ O ( n ) − → O ( m + n − 1 ) , deﬁned as O ( m ) ⊗ O ( n ) ∼ = (left and right unit) − 1   O ( m ) ⊗ I ⊗ ( i − 1) ⊗ O ( n ) ⊗ I ⊗ ( m − i ) id ⊗ u ⊗ ( i − 1) ⊗ id ⊗ u ⊗ ( m − i )   O ( m ) ⊗ O (1) ⊗ ( i − 1) ⊗ O ( n ) ⊗ O (1) ⊗ ( m − i ) µ m ;1 , i − 1 ......, 1 ,n, 1 , m − i ......, 1   O ( m + n − 1 ) An o p e r ad can actually b e de ﬁned as a co llection o f morphis ms ◦ i as a b ov e together with a unit morphism u : I → O (1) suc h that, fo r 1 ≤ i ≤ m , the follo wing diagrams comm ute: (1) If 1 ≤ j < i , ( O ( l ) ⊗ O ( m )) ⊗ O ( n ) ( O ( l ) ⊗ O ( n )) ⊗ O ( m ) O ( l + m − 1) ⊗ O ( n ) O ( l + n − 1) ⊗ O ( m ) O ( l + m + n − 2 ) ◦ i ⊗ id ◦ j ⊗ id ◦ j ◦ i + n − 1 ∼ = assoc. and sym.   7 7 o o o o o o o o o ' ' O O O O O O O O O $ $ I I I I I I I I I I I I I I I I : : u u u u u u u u u u u u u u u u (2) If i ≤ j < m + i , ( O ( l ) ⊗ O ( m )) ⊗ O ( n ) O ( l ) ⊗ ( O ( m ) ⊗ O ( n )) O ( l + m − 1) ⊗ O ( n ) O ( l ) ⊗ O ( m + n − 1) O ( l + m + n − 2 ) ◦ i ⊗ id id ⊗◦ j − i +1 ◦ j ◦ i ∼ = assoc.   7 7 o o o o o o o o o ' ' O O O O O O O O O $ $ I I I I I I I I I I I I I I I I : : u u u u u u u u u u u u u u u u HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 7 These relations ar e illustrated by the trees in Figures 10 and 11 b elow. Moreover, for all 1 ≤ i ≤ n the following compo site morphisms must be unit constraints in V , (3) I ⊗ O ( n ) u ⊗ id / / O (1 ) ⊗ O ( n ) ◦ 1 / / O ( n ) , (4) O ( n ) ⊗ I id ⊗ u / / O ( n ) ⊗ O (1) ◦ i / / O ( n ) . 3. Trees The combinatorics of op erads is that of trees with additional structur e . In this section we rec a ll some facts ab out trees that we nee d in order to pr ov e o ur main theorems. W e a lso giv e a diﬀeren t characterization of op era ds in terms of t rees. Deﬁnition 3.1. A plante d tr e e is a contractible ﬁnite 1-dimensional simplicial com- plex T with set of vertices V ( T ), a no n-empty set o f e dges E ( T ), and a distinguished vertex r ( T ) ∈ V ( T ) of degr ee 1, ca lled r o ot . Re c a ll that the de gr e e of v ∈ V ( T ) is the num b e r of edges containing v . Nev ertheless, we will mo stly us e the following nu mber, e v = (degr e e of v ) − 1 . The level of a v ertex v ∈ V ( T ) is t he distance to the ro ot, lev el( v ) = d ( v, r ( T )), with resp ect to the usual metric d s uch that the distance b etw een tw o adjacent vertices { v , w } ∈ E ( T ) is d ( v , w ) = 1. The height ht( T ) of a pla n ted tree T is ht( T ) = max v ∈ V ( T ) level( v ) . Deﬁnition 3. 2. A plante d planar tr e e is a pla nt ed tree T together with a total order ≤ in V ( T ), ca lled planar or der , such that: • If level( v ) < level( w ) then v < w . • If { v 1 , v 2 } , { w 1 , w 2 } ∈ E ( T ) are edges with level( v 1 ) = level( w 1 ) = level( v 2 ) − 1 = lev el( w 2 ) − 1 , and v 1 < w 1 , then v 2 < w 2 . | T | = • v 0 = r ( T ) • v 1 • v 2 • v 3 • v 4 • v 5 • v 6 • v 7 • v 8 • v 9 ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 9         Figure 1. The geometric realization of a pla nted plana r tree T with vertices o rdered b y the subscript. Given e = { v , w } ∈ E ( T ) with v < w w e say that e is an inc oming e dge o f v and the out going e dge of w (there is only one if w 6 = r ( T ) and none otherwise). 8 FERNANDO M URO There is another useful order in V ( T ) that we call the p ath or der  . Given v ∈ V ( T ), consider the sho rtest path from r ( T ) to v a nd let r ( T ) = v 0 , . . . , v n = v be the vertices within this path in order of app ea r ance. W e as so ciate with v the word v 0 · · · v n in V ( T ). The path order in V ( T ) is the or de r induced by the lexicogra phic order of w ords in V ( T ) with respect to ≤ . R emark 3 .3 . Notice that the path or der  r estricted to lev el sets { v ∈ V ( T ) ; level ( v ) = n } , n ≥ 0 , coincides alwa ys with the planar or der ≤ . The words as s o ciated to the v ertices of the plan ted planar tree in Figure 1 are vertex word v 0 v 0 v 1 v 0 v 1 v 2 v 0 v 1 v 2 v 3 v 0 v 1 v 3 v 4 v 0 v 1 v 3 v 4 vertex word v 5 v 0 v 1 v 3 v 5 v 6 v 0 v 1 v 3 v 6 v 7 v 0 v 1 v 3 v 4 v 7 v 8 v 0 v 1 v 3 v 4 v 8 v 9 v 0 v 1 v 3 v 4 v 9 hence the path order in V ( T ) is v 0 ≺ v 1 ≺ v 2 ≺ v 3 ≺ v 4 ≺ v 7 ≺ v 8 ≺ v 9 ≺ v 5 ≺ v 6 . Deﬁnition 3.4. A plante d planar tr e e with le a ves is a planted planar tree T to- gether with a ﬁx ed se t of degree 1 vertices L ( T ), called le aves , diﬀere n t from the ro ot, r ( T ) / ∈ L ( T ). An inner vertex is a vertex which is neither a leaf nor the ro o t. The set of inner v ertices will be denoted by I ( T ), V ( T ) = { r ( T ) } ⊔ I ( T ) ⊔ L ( T ) . W e denote k T k the op en subspace o f the geometr ic realization of T o btained by removing the ro ot a nd the leav es, s ee Figure 2, k T k = | T | \ ( { r ( T ) } ⊔ L ( T )) . k T k = • v 1 • v 3 • v 4 • v 5 ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 9         Figure 2. The space k T k for the planted plana r tree in Figure 1 with set of leav es L ( T ) = { v 2 , v 6 , v 7 , v 8 , v 9 } . Abusing of terminolog y , we say that an edge is the ro ot o r a le af if it contains the ro ot or a leaf v ertex, respectively . The rest of edges are called inner e dges . Given n ≥ 0, the c o r ol la with n leav es is a planted planar tr ee C n with n + 2 vertices and n leav es, see Figure 3. A morphism o f pla nt ed plana r tre e s w ith leaves is a s implicial map f : T → T ′ such that: HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 9 k C 0 k = • , k C 1 k = • , k C 2 k = • / / / / / / /        , k C 3 k = • / / / / / / /        , k C n k = • n · · · · · · ? ? ? ? ? ? ? ? ?          . Figure 3 . A class of planted planar trees w ith leaves: the co rol- las C n , n ≥ 0. • If v  w ∈ V ( T ) then f ( v )  f ( w ) ∈ V ( T ′ ). • f − 1 ( { r ( T ′ ) } ) = { r ( T ) } . • ca rd L ( T ) = card L ( T ′ ) and f − 1 ( L ( T ′ )) = L ( T ). W e denote PPTL the catego ry of planted pla nar trees with leaves. No tice that this category has no non-trivial automorphism. R emark 3.5 . Any morphism f : T → T ′ is uniquely determined by t he inner edges e = { v , w } ∈ E ( T ) that f cont racts f ( v ) = f ( w ). Moreover, given a planted planar tree w ith leav es T and an inner e dg e e = { v , w } ∈ E ( T ) the quotient tr ee T / e , obtained by contracting e to a vertex [ e ] ∈ V ( T /e ), car r ies a unique s tructure of planted planar tree with leav es s uch t hat the na tural pro jection p T e : T → T /e is a morphism in PPTL , see Figure 4. This morphism induces identiﬁcations V ( T ) \ { v , w } = V ( T /e ) \ { [ e ] } , E ( T ) \ { e } = E ( T /e ) . k T k = • v 1 • v 3 • e v 4 • v 5 • v 1 • [ e ] • v 5 ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          / / / / / / /        ? ? ? ? ? ? ? ? ?                   J J J J J J J J J J J ? ? ? ? ? ? ? ? ? / / / / / / / p T e 5 5 S V Y \ _ b e h k = k T /e k Figure 4. The mor phism p T e : T → T /e in PPTL c ontracting the inner edge e = { v 3 , v 4 } . One can similarly deﬁne a mor phism p T K : T → T /K in PPTL contracting the connected co mpo nent s of a ny sub complex K ⊂ T formed by inner edges, s ee Fig- ure 15 below for a more complicated e xample. 10 FERNANDO M URO Deﬁnition 3.6. Given a plant ed pla nar tree T with n lea ves and n plan ted plana r trees with leaves T 1 , . . . , T n , we denote T ( T 1 , . . . , T n ) the planted planar tree with the s a me ro o t as T , the leaves are the disjoin t unio n o f the leav es of all T i , a nd the space k T ( T 1 , . . . , T n ) k is obtained b y gr afting the ro ot edge of k T i k in the i th leaf edge of k T k with resp ect to the path order  in L ( T ) ⊂ V ( T ), 1 ≤ i ≤ n , see Figure 5. Grafting is asso ciative, i.e. T ( T 1 ( T 1 , 1 , . . . , T 1 ,p 1 ) , . . . . . . , T n ( T n, 1 , . . . , T n,p n )) = ( T ( T 1 , . . . , T n ))( T 1 , 1 , . . . , T 1 ,p 1 , . . . . . . , T n, 1 , . . . , T n,p n ) . The planted pla nar tree U with only one edge and one leaf, k U k = | , is a unit for the grafting opera tion, U ( T ) = T , T ( U, . . . , U ) = T . k T ′ k = • • • • • • • • • ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 9         2 2 2 2 2 2 2        Figure 5. The gra fting T ′ = T ( U, C 0 , C 1 , C 1 ( C 0 ) , C 2 ) for T in Figure 2. The category PPTL splits as the copro duct of the full subcatego ries PPTL ( n ) of trees with n lea ves, PPTL = a n ≥ 0 PPTL ( n ) . Notice that the gr a fting oper ation is functorial in PPTL in the sense of the follow- ing ob vious lemma. Lemma 3.7. The se quenc e { PPTL ( n ) } n ≥ 0 with the gr afting op er a tion and the unit U is an op er ad in the c artesia n close d c ate gory of smal l c ate gories. Lemma 3.8. Al l plante d planar tr e es with le aves c an b e obtaine d by gr afting c or ol las and U . Pr o of. By induction on the heig ht of the planted pla nar tr ee withe leaves T . On the one hand, if ht( T ) = 1 then T = U or C 0 . On the other hand, any T 6 = U, C 0 can be deco mpo sed as T = C n ( T 1 , . . . , T n ) where n + 1 is the degree of the unique level 1 vertex o f T a nd h t( T i ) < ht( T ), 1 ≤ i ≤ n .  HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 11 F or instance, T in Figure 2 is T = C 2 ( U, C 3 ( C 3 , C 0 , U )) = C 2 ◦ 2 (( C 3 ◦ 2 C 0 ) ◦ 1 C 3 ) . Deﬁnition 3.9 . An op er a dic functor with v a lues in V is a functor G : PPTL − → V equipp e d with a unit morphism u : I → G ( C 1 ) and natural isomorphisms G ( T ( T 1 , . . . , T n )) ∼ = G ( T ) ⊗ G ( T 1 ) ⊗ · · · ⊗ G ( T n ) , that w e call gr af ting isomorphisms , such that: • G ( U ) = I . • The following comp osition o f gra fting isomo r phisms is a coherent compo si- tion of asso cia tivit y and symmetry constrain ts in V , G ( T ) ⊗ G ( T 1 ) ⊗ G ( T 1 , 1 ) ⊗ · · · ⊗ G ( T 1 ,p 1 ) ⊗ · · · · · · ⊗ G ( T n ) ⊗ G ( T n, 1 ) ⊗ · · · ⊗ G ( T n,p n ) ∼ = G ( T ) ⊗ G ( T 1 ( T 1 , 1 , . . . , T 1 ,p 1 )) ⊗ · · · · · · ⊗ G ( T n ( T n, 1 , . . . , T n,p n )) ∼ = G ( T ( T 1 ( T 1 , 1 , . . . , T 1 ,p 1 ) , . . . . . . , T n ( T n, 1 , . . . , T n,p n ))) = G (( T ( T 1 , . . . , T n ))( T 1 , 1 , . . . , T 1 ,p 1 , . . . . . . , T n, 1 , . . . , T n,p n )) ∼ = G ( T ( T 1 , . . . , T n )) ⊗ G ( T 1 , 1 ) ⊗ · · · ⊗ G ( T 1 ,p 1 ) ⊗ · · · · · · ⊗ G ( T n, 1 ) ⊗ · · · ⊗ G ( T n,p n ) ∼ = G ( T ) ⊗ G ( T 1 ) ⊗ · · · ⊗ G ( T n ) ⊗ G ( T 1 , 1 ) ⊗ · · · ⊗ G ( T 1 ,p 1 ) ⊗ · · · · · · ⊗ G ( T n, 1 ) ⊗ · · · ⊗ G ( T n,p n ) . • The follo wing gra fting isomorphisms are (co mpo sitions of ) unit constraints in V , I ⊗ G ( T ) = G ( U ) ⊗ G ( T ) ∼ = grafting / / G ( U ( T )) = G ( T ) , G ( T ′ ) ⊗ I ⊗ · · · ⊗ I = G ( T ′ ) ⊗ G ( U ) ⊗ · · · ⊗ G ( U ) grafting ∼ = / / G ( T ( U, . . . , U )) = G ( T ) . • Supp ose T ′ = C 1 ( T ), see Figure 6. Let f : T ′ → T b e the morphism which contracts the incoming edge of the lev el 1 vertex of T ′ . Then the following morphism is the left unit constraint in V , I ⊗ G ( T ) u ⊗ id / / G ( C 1 ) ⊗ G ( T ) ∼ = grafting / / G ( T ′ ) G ( f ) / / G ( T ) . • Supp ose T ′ = T ( C 1 , . . . , C 1 ), see Figur e 7. Let f : T ′ → T be the morphism which contracts all the inner edges a dja cent to the leaf edg es in T ′ . Then the following mo rphism is a compo s ition of righ t unit constraints in V , G ( T ) ⊗ I ⊗ · · · ⊗ I id ⊗ u ⊗···⊗ u / / G ( T ) ⊗ G ( C 1 ) ⊗ · · · ⊗ G ( C 1 ) ∼ = grafting / / G ( T ′ ) G ( f ) / / G ( T ) . A morphism of o pe r adic functors ϕ : G → H is a natural tra ns formation c o m- patible w ith the gr a fting isomorphisms, with the unit mor phism, and such that ϕ ( U ) = id I . The following equiv ale nce b etw een o per ads and op era dic functors was sket ched by Ginzburg and Ka pranov in the symmetric case [GK94, 1.2]. Prop ositi o n 3 .10. Ther e is an e quivalenc e b etwe en the c ate gories of op er ads in V and op er ad ic fun ct ors with values in V . 12 FERNANDO M URO k C 1 ( T ) k = e • • • • • ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 9         Figure 6. The planted planar tree with leaves T ′ = C 1 ( T ) for T as in Figure 2. Here we denote e the inco ming edge of the level 1 vertex o f T ′ . k T ( C 1 , . . . , C 1 ) k = e 1 e 2 e 3 e 4 e 5 • • • • • • • • • ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 9         Figure 7. The planted planar tree with leaves T ′ = T ( C 1 , . . . , C 1 ) for T a s in Fig ur e 2. Here we denote e i the inner edges adjacent to the leaf edges in T ′ . Pr o of. Denote OpF unc( V ) the catego r y of o pe r adic functor s with v alues in V . W e are going to deﬁne adjoin t equiv ale nces Op( V ) L / / OpF unc( V ) . R o o Given an op er adic functor G we set R ( G )( n ) = G ( C n ) , the unit of the op erad R ( G ) is u : I → G ( C 1 ) = R ( G )(1), and multip lications in R ( G ) are deﬁned b y the morphisms f n ; p 1 ,...,p m : C n ( C p 1 , . . . , C p n ) − → C p 1 + ··· + p n HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 13 which contract all inner edges, R ( G )( n ) ⊗ R ( G )( p 1 ) ⊗ · · · ⊗ R ( G )( p n ) µ n ; p 1 ,...,p n   G ( C n ) ⊗ G ( C p 1 ) ⊗ · · · ⊗ G ( C p n ) ∼ = grafting   G ( C n ( C p 1 , . . . , C p n )) G ( f n ; p 1 ,...,p n )   R ( G )( p 1 + · · · + p n ) G ( C p 1 + ··· + p n ) Conv ersely , if O is an op era d then the corr esp onding o pe radic functor L ( O ) is deﬁned on ob jects as L ( O )( T ) = O u ∈ I ( T ) O ( e u ) , see Figure 8. The morphism u : I → O (1) = L ( O )( C 1 ) is the unit of the op erad. k T k = • v 1 • v 3 • v 4 • v 5 ? ? ? ? ? ? ? ? ?          ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 9         O (2 ) O (3 ) O (3 ) O (0) ⊗ ⊗ ⊗ = L ( O )( T ) Figure 8. The o b ject L ( O )( T ) asso ciated to the pla n ted plana r tree with leav es T in Figure 2. Grafting iso morphisms are coherent comp ositions of asso ciativity and symmetry constraints in V . Moreover, let T b e a planted planar tree with leaves and e = { v , w } ∈ E ( T ) an inner edge which is the i th incoming edge o f v . The mo rphism induced b y the natural pro jection p T e : T → T /e in Remark 3.5 is L ( O )( T ) L ( O )( T /e ) O ( e v ) ⊗ O ( e w ) ⊗ N u ∈ I ( T ) \{ v ,w } O ( e u ) O ( f [ e ]) ⊗ N u ∈ I ( T /e ) \{ [ e ] } O ( e u ) ∼ = symmetry / / ∼ = symmetry / / L ( O )( p T e )   ◦ i ⊗ id   Here w e use that f [ e ] = e v + e w − 1 , see Figure 9. 14 FERNANDO M URO L ( O )( T ) = O (2 ) O (3 ) O (3 ) O (0)  3 ; B B I O   3 ; B B I O ⊗ ⊗ ⊗ ◦ 1 / / L ( O )( p T e ) / / L ( O )( T /e ) = O (2 ) O (5 ) O (0 ) ⊗ ⊗ Figure 9 . The mor phism L ( O )( p T e ) for T and e = { v 3 , v 4 } as in Figure 4, see also Figure 8. The unit na tur al tra nsformation O → R L ( O ) is the iden tit y morphism, and the counit ε : L R ( G ) → G is deﬁned by grafting isomorphisms, ε ( T ) : LR ( G )( T ) = N u ∈ I ( T ) G ( C e u ) G ( T ) . ∼ = grafting / / Here w e use that any plant ed planar tree with leaves T can b e o btained by g rafting appropria tely the cor ollas C e u , u ∈ I ( T ), compare the previous le mma .  Examples of plant ed pla na r tr e e s with lea ves illustrating relations (1) and (2) in Remark 2.6 are depicted in Figures 10 and 11 , resp e c tively . • • • N N N N N N N N N N N N N p p p p p p p p p p p p 9 9 9 9 9 9 9 9 , , , , , , ,                2 2 2 2 2 2 2 % % % % % %              Figure 10 . The planted planar tree with leaves illustrating the asso ciativity rela tio n ( C l ◦ i C m ) ◦ j C n = ( C l ◦ j C n ) ◦ i + n − 1 C m in Remark 2.6 (1) for l = 3, m = 4, n = 5, a nd j = 1 < i = 2. 4. The monoidal ca tegor y of morphisms The c ate gory Mor( C ) of morphisms in C can b e rega rded as the category o f functors 2 → C , where 2 is the catego ry with tw o o b jects, 0 and 1, and only one non-identit y morphism 0 → 1 , i.e. it is the p ose t { 0 < 1 } . A morphism f : U → V in C is ident iﬁed with the functor f : 2 → C deﬁned by f (0) = U , f (1) = V and f (0 → 1) = f . HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 15 • • • ? ? ? ? ? ? ? ? ?          9 9 9 9 9 9 9 , , , , , ,              2 2 2 2 2 2 2 % % % % % %              Figure 11 . The planted planar tree with leaves illustrating the asso ciativity relatio n ( C l ◦ i C m ) ◦ j C n = C l ◦ i ( C m ◦ j − i +1 C n ) in Remark 2.6 (2) for l = 3, m = 4, n = 5, and i = 2 ≤ j = 3 < m + i = 6. The category Mo r( C ) carr ies a biclosed monoidal str ucture given b y the ⊙ pro d- uct of morphisms f ⊙ g , U ⊗ X V ⊗ X U ⊗ Y U ⊗ Y S U ⊗ X V ⊗ X V ⊗ Y f ⊗ i d X / / push id U ⊗ g     / / id V ⊗ g   f ⊗ i d Y / / f ⊙ g ) ) R R R R R R R R R R R R R R R This monoidal structur e is symmetric provided ⊗ is. If 0 denotes the initial ob ject of C , the functor C − → Mor( C ) , X 7→ (0 → X ) , is s trong (symmetric) mo no idal. W e regar d C as a full sub categ ory of Mor( C ) through this functor. Notice tha t push-outs in C are a spec ia l kind of morphism in Mor( C ). The following lemma as serts that the ⊙ product preserves push-outs in C . Lemma 4.1 . Given two push-out diagr ams in C , i = 1 , 2 , U i f i / / g i   push V i g ′ i   X i f ′ i / / Y i 16 FERNANDO M URO the fol lowing diagr a m in C is also a push-out, U 1 ⊗ V 2 S U 1 ⊗ U 2 V 1 ⊗ U 2 X 1 ⊗ Y 2 S X 1 ⊗ X 2 Y 1 ⊗ X 2 V 1 ⊗ V 2 Y 1 ⊗ Y 2 push f 1 ⊙ f 2 / / g ′ 1 ⊗ g ′ 2   f ′ 1 ⊙ f ′ 2 / / g 1 ⊗ g ′ 2 S g 1 ⊗ g 2 g ′ 1 ⊗ g 2   This lemma follows straightforwardly from the very deﬁnition of ⊙ together with the fact that ⊗ is biclosed, and hence it pr eserves co limits in both v ariables. Deﬁnition 4.2. The category C is a monoidal mo del c ate go ry if it is endow ed with a model s tr ucture satisfying the push-out pr o d uct axiom : • Let f a nd g be coﬁbrations in C . The morphism f ⊙ g is a lso a coﬁbr ation. If in addition f or g is a weak e quiv alence, th en so is f ⊙ g . This a xiom was consider ed in [SS00, Deﬁnition 3.1] for C symmetric, but it a lso makes sense in the non-symmetric case. R emark 4.3 . The push-out pro duct axiom implies that the tensor pro duct of co ﬁ- brant ob jects is coﬁbra nt . Mo reov er, if X is a coﬁbr ant ob ject and f is a (trivia l) coﬁbration in C then X ⊗ f and f ⊗ X are (trivial) coﬁbrations. In particular , by Ken Brown’s lemma [Hov99, Lemma 1.1 .1 2], for X coﬁbra nt the functors X ⊗ − and − ⊗ X pr e s erve w eak equiv a lences betw een coﬁbrant ob jects. F urthermore, if f and g a re (trivial) coﬁbrations with coﬁbrant sour ce, then so is f ⊙ g . Lemma 4 .4. L et C b e a left pr op er monoidal m o del c ate go ry. Consider t wo c om- mutative squ ar es in Mo r( C ) wher e the r ows ar e c oﬁbr a tions and the c olumns ar e we ak e quivale nc es b etwe en c o ﬁbr ant obje cts, i = 1 , 2 , U i / / f i / / g i ∼   V i g ′ i ∼   X i / / f ′ i / / Y i Then in the fol lowing diagr am the r ows ar e also c oﬁbr a tions and the c olumns ar e we ak e quivale nc es b etwe en c o ﬁbr ant obje cts, U 1 ⊗ V 2 S U 1 ⊗ U 2 V 1 ⊗ U 2 X 1 ⊗ Y 2 S X 1 ⊗ X 2 Y 1 ⊗ X 2 V 1 ⊗ V 2 Y 1 ⊗ Y 2 push / / f 1 ⊙ f 2 / / g ′ 1 ⊗ g ′ 2 ∼   / / f ′ 1 ⊙ f ′ 2 / / g 1 ⊗ g ′ 2 S g 1 ⊗ g 2 g ′ 1 ⊗ g 2 ∼   Pr o of. Lo ok ing at Deﬁnition 4.2 and the remark a fterwards we notice that it is only left to chec k that the left column is a weak equiv alence. T his follows eas ily from the gluing pr o p erty in left proper model catego ries [Hir03, Prop o sition 13.5.4].  HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 17 Given morphisms f i : U i → V i in C , 1 ≤ i ≤ n , the targe t of f 1 ⊙ · · · ⊙ f n is the iterated tensor pro duct of the tar gets V 1 ⊗ · · · ⊗ V n . This ob ject is the colimit of the diagram f 1 ⊗ · · · ⊗ f n : 2 n − → C , since 2 n has a ﬁnal ob ject (1 , n . . ., 1). The source of f 1 ⊙ · · · ⊙ f n is the colimit of the restr ic tio n of this diagr am to the full sub categ ory of 2 n obtained by removing the ﬁnal ob ject. F or simplicity , w e denote it b y s ( f 1 ⊙ · · · ⊙ f n ), f 1 ⊙ · · · ⊙ f n : s ( f 1 ⊙ · · · ⊙ f n ) − → V 1 ⊗ · · · ⊗ V n . The univ ersal prop erty o f s ( f 1 ⊙ · · · ⊙ f n ) in C re fers to canonical morphisms κ i : V 1 ⊗ · · · ⊗ V i − 1 ⊗ U i ⊗ V i +1 ⊗ · · · ⊗ V n − → s ( f 1 ⊙ · · · ⊙ f n ) , 1 ≤ i ≤ n, with ( f 1 ⊙ · · · ⊙ f n ) κ i = id ⊗ ( i − 1) ⊗ f i ⊗ id ⊗ ( n − i ) . Any collection of morphisms g i : V 1 ⊗ · · · ⊗ V i − 1 ⊗ U i ⊗ V i +1 ⊗ · · · ⊗ V n − → X , 1 ≤ i ≤ n, such that the following squares c o mmut e, 1 ≤ i < j ≤ n , V 1 ⊗ · · · ⊗ U i ⊗ · · · ⊗ U j ⊗ · · · ⊗ V n id ⊗·· ·⊗ f i ⊗···⊗ id / / id ⊗···⊗ f j ⊗···⊗ id   V 1 ⊗ · · · ⊗ V i ⊗ · · · ⊗ U j ⊗ · · · ⊗ V n g j   V 1 ⊗ · · · ⊗ U i ⊗ · · · ⊗ V j ⊗ · · · ⊗ V n g i / / X induces a unique morphism g : s ( f 1 ⊙ · · · ⊙ f n ) → X such that g i = g κ i , 1 ≤ i ≤ n . 5. The relev ant operad push-out The forg etful functor fr om op er ads to s equences Op( V ) → V N has a left adjoint F : V N → Op( V ), the fr e e op er ad functor, explicitly constr ucted for example in [BJT97, Appendix B]. An alter native construction using trees is as follows, F ( V )( n ) = a T O v ∈ I ( T ) V ( e v ) , where T runs over a set of isomo rphism classes of tr e es with n leaves in PPTL . The product ◦ i , 1 ≤ i ≤ m , F ( V )( m ) ⊗ F ( V )( n ) = ` T ′ N u ∈ I ( T ′ ) V ( e u ) ⊗ ` T N v ∈ I ( T ) V ( e v ) card L ( T ′ ) = m card L ( T ) = n ∼ = ` T ′ ,T N u ∈ I ( T ′ ) V ( e u ) ⊗ N v ∈ I ( T ) V ( e v ) F ( V )( m + n − 1) = ` T ′′ N w ∈ I ( T ′′ ) V ( e w ) card L ( T ′′ ) = m + n − 1 ◦ i   sends the factor corr e sp onding to the trees T a nd T ′ in the source to the factor of T ′′ = T ′ ◦ i T in the tar get, I ( T ′ ◦ i T ) = I ( T ′ ) ⊔ I ( T ) , O u ∈ I ( T ′ ) V ( e u ) ⊗ O v ∈ I ( T ) V ( e v ) = O w ∈ I ( T ′ ◦ i T ) V ( e w ) . 18 FERNANDO M URO The unit u : I → F ( V )(1) is the inclusio n of the facto r of the copr o duct cor resp o ding to the tree with one leaf a no inner vertex, i.e. the unit of the grafting op eration. The unit of the a djunction V → F ( V ) in V N is given b y the following mor phis ms in V , n ≥ 0, V ( n ) inclusion of the f actor corresp ondin g to C n / / F ( V )( n ) . Given a n op era d O with asso ciated o p er adic functor L ( O ), if we denote p T : T → C n the mo r phism in PPTL collapsing all inner edges of a tree T with n leav es, then the counit F ( O ) → O is deﬁned by the following morphisms, n ≥ 0, F ( O )( n ) = ` T N v ∈ I ( T ) O ( e v ) = ` T L ( O )( T ) L ( O )( C n ) = O ( n ) . ( L ( O )( p T )) T / / An analog ous construction for symmetric op erads was considered by Ginzburg and Kapranov in [GK9 4, 2.1]. In this sec tio n we give an explicit construction of the push- o ut of a diagram in Op( V ) a s follows, (5.1) F ( U ) g   F ( f ) / / F ( V ) O Consider the adjoint diag ram in V N , U ¯ g   f / / V O The push-out of (5.1) is a n op er a d P together w ith morphisms f ′ : O → P in Op( V ) and ¯ g ′ : V → P in V N such that f ′ ¯ g = ¯ g ′ f in V N . Mor eov er, g iven an op er ad P ′ and mo r phisms f ′′ : O → P ′ in Op( V ) and ¯ g ′′ : V → P ′ in V N with f ′′ ¯ g = ¯ g ′′ f in V N , there is a unique morphism h : P → P ′ in O p( V ) such that f ′′ = hf ′ and ¯ g ′′ = h ¯ g ′ in V N . Given a planted planar tree with lea ves T we denote V e ( T ) = { v ∈ V ( T ) ; level( v ) is even } , V o ( T ) = V ( T ) \ V e ( T ) , I e ( T ) = I ( T ) ∩ V e ( T ) , I o ( T ) = I ( T ) ∩ V o ( T ) , see Figure 1 2. F ro m now on, we will o nly co nsider o ne tree in each iso mo rphism class of ob jects in PPT L . The idea behind our construction of the push- o ut of (5.1) is as fo llows. F or an y planted planar tree with leaves concen trated in even levels, such as T in Figure 12, we replace a ny inner even (resp. o dd) vertex v with the piece of V (resp. O ) in degree e v , and transform adjacency relations in to tensor pro ducts. HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 19 | T | = • ◦ • • v ◦ ◦ ◦ • • • ◦ • • ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         9 9 9 9 9 9 9 9         2 2 2 2 2 2 2        k T k = ◦ • v ◦ ◦ ◦ • • ◦ ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         9 9 9 9 9 9 9 9         2 2 2 2 2 2 2        Figure 12. F o r a planted planar tr ee with leav es T , on the left (resp. rig ht) we denote ◦ the vertices in V o ( T ) (res p. I o ( T )) and • the vertices in V e ( T ) (resp. in I e ( T )). H OM OT OP Y T H E OR Y OF N ON - S Y M M E T R I C OP E R A D S 1 7 i n , t h e r e i s a u n i q u e m or p h i s m i n O p ( ) s u c h t h at ′ ′ h f an d ′ ′ i n G i v e n a p l an t e d p l an ar t r e e w i t h l e a v e s w e d e n ot e ) = ) ; l e v e l ( ) i s e v e n , V ) = ) = , I ) = s e e F i gu r e e o 9. F r om n o w on , w e w i l l on l y c on s i d e r on e t r e e i n e ac h i s om or p h i s m c l as s of ob j e c t s i n P P TL • • ◦ ◦ ◦ • • • • • ◦ ◦ ◦ • • F i g u r e 9 . F or a p l an t e d p l an ar t r e e w i t h l e a v e s , on t h e l e f t ( r e s p . r i gh t ) w e d e n ot e t h e v e r t i c e s i n ) ( r e s p . ) ) an d t h e v e r t i c e s i n ) ( r e s p . i n ) ) . eo T h e i d e a b e h i n d ou r c on s t r u c t i on of t h e p u s h - ou t of ( p o 5. 1) i s as f ol l o w s . F or an y p l an t e d p l an ar t r e e w i t h l e a v e s c on c e n t r at e d i n e v e n d e gr e e s , s u c h as i n F i gu r e e o 9, w e r e p l ac e an y i n n e r e v e n ( r e s p . o d d ) v e r t e x w i t h t h e p i e c e of ( r e s p . ) i n d e gr e e v al ) , an d t r an s f or m ad j ac e n c y r e l at i on s i n t e n s or p r o d u c t s . ◦ • ◦ ◦ ◦ • • ◦  O (2) V (3) O (3) O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ I n or d e r t o s i m p l i f y t h e e x p os i t i on of t h i s i n t u i t i v e i d e a, l e t u s al l o w ou r s e l v e s t o t al k ab ou t e l e m e n t s of t h i s ob j e c t i n . W e w an t t o at t ac h t o t h e p r o d u c t of t h e s e e l e m e n t s i n a c oh e r e n t w a y . M or e p r e c i s e l y , i f h as l e a v e s , w e at t ac h t h e s e e l e m e n t s t o ) . F or t h i s , w e m u s t p r o c e e d b y i n d u c t i on on t h e n u m b e r of In o r der to simplify the expo sition of this intuit ive idea, let us allow ourselves to talk ab out elements of this ob ject in V . W e want to attach to O the pro duct of these elements in a c o herent way . More pr ecisely , if T has n leav es, we attach these elements to O ( n ). F or this, we m ust proc e ed by indu ction on the num ber of inner even vertices and require that, for any even inner vertex v , the image of the morphism induced b y f ( e v ), 20 FERNANDO M URO 1 8 F E R N A N D O M U R O i n n e r e v e n v e r t i c e s an d r e q u i r e t h at , f or an y e v e n i n n e r v e r t e x , t h e i m age of t h e m or p h i s m i n d u c e d b y ( v al ) ) , O (2) U (3) O (3) O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ f (3)   O (2) V (3) O (3) O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ i s at t ac h e d ac c or d i n g t o t h e at t ac h m e n t of t h e t r e e w i t h l e s s e v e n i n n e r v e r t i c e s ob t ai n e d f r om b y c on t r ac t i n g t h e e d ge s s u r r ou n d i n g ◦ ◦ ◦ • • c o n t r a c t i o n m o r p h i s m i n P P T L • • ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ ( 3 ) ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ i d ( 2 ) 3 ; 3 c o m p o s i t i o n i n a c c o r d i n g t o t h e s t r u c t u r e o f i n a n e i g h - b ou rh o o d of ( 6) ( 0) ( 1) ( 0) ⊗ ⊗ T h i s i n d u c t i v e c on s t r u c t i on i s c ar r i e d ou t i n t h e f ol l o w i n g l e m m a. I n or d e r t o s t at e i t w e n e e d t o i n t r o d u c e s om e t e r m i n ol ogy . T h e s t a r of a v e r t e x ) i s t h e s u b t r e e S t ( f or m e d b y t h e e d ge s c on t ai n i n g , an d t h e l i n k Lk ( ) c on s i s t s of t h e v e r t i c e s ad j ac e n t t o , s e e F i gu r e s t a r l i n k 10. W h e n t h e s t ar i s f or m e d b y i n n e r e d ge s , t h e n at u r al p r o j e c t i on S t( − → T / S t ( is attac hed acco r ding to the a ttachmen t of the tree T ′ with less even inner v ertices obtained from T by con tracting the edges surrounding v , (5.2) 1 8 F E R N A N D O M U R O i n n e r e v e n v e r t i c e s an d r e q u i r e t h at , f or an y e v e n i n n e r v e r t e x , t h e i m age of t h e m or p h i s m i n d u c e d b y ( v al ) ) , ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ ( 3 ) ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ i s at t ac h e d ac c or d i n g t o t h e at t ac h m e n t of t h e t r e e w i t h l e s s e v e n i n n e r v e r t i c e s ob t ai n e d f r om b y c on t r ac t i n g t h e e d ge s s u r r ou n d i n g T = ◦ •                       ◦ ◦ ◦ • • ◦ contraction morphism in PPTL ◦ • • ◦ = T ′ ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ ( 3 ) ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ i d ( 2 ) 3 ; 3 c o m p o s i t i o n i n a c c o r d i n g t o t h e s t r u c t u r e o f i n a n e i g h - b ou rh o o d of ( 6) ( 0) ( 1) ( 0) ⊗ ⊗ T h i s i n d u c t i v e c on s t r u c t i on i s c ar r i e d ou t i n t h e f ol l o w i n g l e m m a. I n or d e r t o s t at e i t w e n e e d t o i n t r o d u c e s om e t e r m i n ol ogy . T h e s t a r of a v e r t e x ) i s t h e s u b t r e e S t ( f or m e d b y t h e e d ge s c on t ai n i n g , an d t h e l i n k Lk ( ) c on s i s t s of t h e v e r t i c e s ad j ac e n t t o , s e e F i gu r e s t a r l i n k 10. W h e n t h e s t ar i s f or m e d b y i n n e r e d ge s , t h e n at u r al p r o j e c t i on S t( − → T / S t ( 1 8 F E R N A N D O M U R O i n n e r e v e n v e r t i c e s an d r e q u i r e t h at , f or an y e v e n i n n e r v e r t e x , t h e i m age of t h e m or p h i s m i n d u c e d b y ( v al ) ) , ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ ( 3 ) ( 2) ( 3) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ ⊗ i s at t ac h e d ac c or d i n g t o t h e at t ac h m e n t of t h e t r e e w i t h l e s s e v e n i n n e r v e r t i c e s ob t ai n e d f r om b y c on t r ac t i n g t h e e d ge s s u r r ou n d i n g ◦ ◦ ◦ • • c o n t r a c t i o n m o r p h i s m i n P P T L • • O (2) U (3) O (3) O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ¯ g (3)   O (2) O (3) O (3) O (0) O (2) V (0) V (1) O (0)              ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ id O (2) ◦ 2 µ 3;3 , 0 , 2 compos ition in O accor ding to the structure of T in a neigh- bourho o d of v   O (6) V (0) V (1) O (0) ⊗ ⊗ ⊗ T h i s i n d u c t i v e c on s t r u c t i on i s c ar r i e d ou t i n t h e f ol l o w i n g l e m m a. I n or d e r t o s t at e i t w e n e e d t o i n t r o d u c e s om e t e r m i n ol ogy . T h e s t a r of a v e r t e x ) i s t h e s u b t r e e S t ( f or m e d b y t h e e d ge s c on t ai n i n g , an d t h e l i n k Lk ( ) c on s i s t s of t h e v e r t i c e s ad j ac e n t t o , s e e F i gu r e s t a r l i n k 10. W h e n t h e s t ar i s f or m e d b y i n n e r e d ge s , t h e n at u r al p r o j e c t i on S t( − → T / S t ( This inductive construction is c a rried out in the following lemma. In or der to state it w e need to in tro duce some terminology . The st ar of a vertex v ∈ V ( T ) is the subtre e St( v ) ⊂ T formed by the edges containing v , a nd the link Lk( v ) ⊂ V ( T ) consists of the vertices adjacen t to v , see Figure 13. When the star is formed by inner edg es, the natural pro jection p T St( v ) : T − → T / St( v ) HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 21 | T | = • ◦ • • v ◦ ◦ ◦ • • • ◦ • • ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         9 9 9 9 9 9 9 9         2 2 2 2 2 2 2        St( v ) = ◦ • u ◦ ◦ ◦         ? ? ? ? ? ? ? ?         Lk( v ) = ◦ ◦ ◦ ◦ Figure 13. The star and the link of the vertex v of the tree T in Figure 12. is a morphism in PPTL , see (5.2). This is the case if v ∈ I e ( T ) and L ( T ) ⊂ V e ( T ). Moreov er, in this cas e p T St( v ) induces identiﬁcations I e ( T ) \ { v } = I e ( T / St( v )) , I o ( T ) \ Lk( v ) = I o ( T / St( v )) \ { [St( v )] } . F urthermore, we will a lso c o nsider the extende d st ar St( v ) ⊂ T , which is the planted planar tree with le aves whose inner part is St( v ), the r o ot edg e is the outgoing edge of the minim um v ertex u ∈ Lk ( v ), the leav es ar e the incoming edges of the v ertices in Lk( v ) except from { u, v } , a nd the pla nar o rder is the restriction of the planar order in T , see Figur e 14. Notice that St( v ) / St( v ) = C r v , where (5.3) r v = ^ [St( v )] = card L ( St( v )) = e u − 1 + X w ∈ Lk( v ) \{ u } e w =   X w ∈ Lk( v ) e w   − 1 . k St( v ) k = ◦ u • v ◦ ◦ ◦ ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         9 9 9 9 9 9 9        2 2 2 2 2 2 2        Figure 14 . The ex tended star of the vertex v of the planted planar tree with leav es T in Figures 12 and 13. 22 FERNANDO M URO The inductive constr uction o f the push-out of (5.1) is the in following scaring lemma, whose statemen t is actually mor e complicated than its pro o f. F o r the sak e of simplicity , from now o n we use the same notation for an o pe rad and for its asso ciated op eradic functor. Lemma 5.4 . Ther e is a se quenc e of morphisms in V N , O = P 0 ϕ 1 − → P 1 → · · · → P t − 1 ϕ t − → P t → · · · , such t hat, for al l n ≥ 0 , the morphism ϕ t ( n ) : P t − 1 ( n ) → P t ( n ) is the push-out of the fol lowing c op r o d uct of morphisms indexe d by the set of plante d t re es with n le ave s c onc ent r ate d in even levels and t inner even vertic es, i.e. card L ( T ) = n , L ( T ) ⊂ V e ( T ) , and c ard I e ( T ) = t , (5.5) a T K v ∈ I e ( T ) f ( e v ) ⊗ O w ∈ I o ( T ) O ( e w ) , along the unique morphism (5.6) ( ψ T t ) T : a T s   K v ∈ I e ( T ) f ( e v )   ⊗ O w ∈ I o ( T ) O ( e w ) − → P t − 1 ( n ) such that, given u ∈ I e ( T ) , for t = 1 t he morphism ψ T 1 is U ( e u ) ⊗ N w ∈ I o ( T ) O ( e w ) O ( e u ) ⊗ N w ∈ I o ( T ) O ( e w ) = O ( T ) O ( n ) , ψ T 1 * * ¯ g ( e u ) ⊗ id 6 6 O ( p T ) 9 9 and for t > 1 the c omp osite morphism U ( e u ) ⊗ N v ∈ I e ( T ) \{ u } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) s ( J v ∈ I e ( T ) f ( e v )) ⊗ N w ∈ I o ( T ) O ( e w ) P t − 1 ( n ) κ u ⊗ id / / ψ T t / / HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 23 c oinci des with the fol lowing c omp osition, that we c al l ψ T t,u , U ( e u ) ⊗ N v ∈ I e ( T ) \{ u } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) O ( e u ) ⊗ N v ∈ I e ( T ) \{ u } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) N v ∈ I e ( T ) \{ u } V ( e v ) ⊗ N w ∈ I o ( T ) \ Lk ( u ) O ( e w ) ⊗ O ( St( u )) N v ∈ I e ( T / S t( u )) V ( e v ) ⊗ N w ∈ I o ( T / S t( u )) \{ [St( u )] } O ( e w ) ⊗ O ( r u ) P t − 1 ( n ) ¯ g ( e u ) ⊗ id   ∼ = symmetry   id ⊗O ( p St( u ) )   ¯ ψ T / St( u ) t − 1   Her e ( ¯ ψ T ′ t − 1 ) T ′ denotes the push-out of ( ψ T ′ t − 1 ) T ′ , i.e. (5.6) for t − 1 , along (5.5) . Pr o of. The pro of is by induction on t ≥ 0. No tice that there is nothing to chec k for t = 0 , 1. Let t > 1 and ass ume everything works up to t − 1 . By the univ ersa l prop erty of the sour ce of an iterated ⊙ pro duct, describ ed in Section 4, we only hav e to check the following compatibility condition: given tw o diﬀere n t vertices u, u ′ ∈ I e ( T ), the following square comm utes, (a) U ( e u ) ⊗ U ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) V ( e u ) ⊗ U ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) U ( e u ) ⊗ V ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) P t − 1 ( n ) f ( e u ) ⊗ id < < y y y y y y y y y y y id ⊗ f ( e u ′ ) ⊗ id " " E E E E E E E E E E E ψ T t,u ′ $ $ J J J J J J J J J J J J J ψ T t,u ; ; x x x x x x x x x x x x x Here, for simplicit y , we omit so me symmetry isomorphisms in V . Denote St( u, u ′ ) = St( u ) ∪ St( u ′ ) and Lk( u, u ′ ) = Lk( u ) ∪ Lk( u ′ ). Supp os e that d ( u, u ′ ) > 2. Then St ( u ) ∩ St( u ′ ) = ∅ , see Figure 15. Moreover, in this case t > 2. 24 FERNANDO M URO By induction h yp otesis, in this case both comp os itions coincide with (5.7) U ( e u ) ⊗ U ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) O ( e u ) ⊗ O ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) \ Lk ( u,u ′ ) O ( e w ) ⊗ O ( St( u )) ⊗ O (St( u ′ )) N v ∈ I e ( T / S t( u,u ′ )) V ( e v ) ⊗ N w ∈ I o ( T / S t( u,u ′ )) \{ [St( u )] , [St( u ′ )] } O ( e w ) ⊗ O ( r u ) ⊗ O ( r u ′ ) P t − 2 ( n ) P t − 1 ( n ) ¯ g ( e u ) ⊗ ¯ g ( e u ′ ) ⊗ id   ∼ = symmetry   id ⊗O ( p St( u ) ) ⊗O ( p St( u ′ ) )   ¯ ψ T / St( u,u ′ ) t − 2   ϕ t − 1 ( n )   See Figures 15 and 16. k T k ◦ • u ′ ◦ ◦ ◦ • u • ◦ • ◦ ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9         2 2 2 2 2 2 2               p T St( u,u ′ ) ? ? _ V Q O O S Z g  k T / St( u, u ′ ) k ◦ • [St( u ′ )] ◦ ◦ ◦ [St( u )] • ◦ ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?                 2 2 2 2 2 2 2 Figure 15. A plan ted planar tr ee T with leav es in ev en levels and t wo even inner vertices u and u ′ with d ( u, u ′ ) > 2. The discon- nected sub co mplex St( u, u ′ ) ⊂ T is in double lines. W e illustra te the morphism p T St( u,u ′ ) . HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 25 O (2 ) V (3) O (3 ) O (0) O (2) U (0) V (1) O (0 ) U (1) O (0 ) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ¯ g (1)   ¯ g (0)   O (2 ) V (3) O (3 ) O (0) O (2 ) O (0 ) V (1) O (0 ) O (1 ) O (0 ) _ n   _ _  . C T          . C T _ _ _ n   { ( 2 ; ; F Q { { ( 2 ; ; F Q ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ O ( p St( u ′ ) )   O ( p St( u ) ) O O O (2 ) V (3) O (2 ) O (0) O (1) V (1) O (0 ) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ Figure 16. A sketch of (5.7) for the planted pla nar tr ee T with leav es in even levels and the tw o even inner vertices u and u ′ in Figure 15. • ◦ • u ′ u • ◦ • u ′ u u ′′         ? ? ? ? ? ? ? ? Figure 1 7 . The only tw o p o ssible r e lative pos itions of u and u ′ , u < u ′ , within the plan ted planar tree with leaves T if d ( u, u ′ ) = 2. Suppo se no w that d ( u, u ′ ) = 2. Then the subco mplex St( u, u ′ ) ⊂ T is connected. Both factors shar e the unique vertex which is one step awa y from bo th u a nd u ′ , see Figure 17. Let T ′ ⊂ T b e in this ca s e the plan ted planar tree with leav es whose inner part is St( u, u ′ ), the ro ot edge is the outgoing edge of the minimun v ertex u ′′ ∈ Lk( u, u ′ ), the leav es are the incoming edges of the vertices in Lk( u, u ′ ) not containing u or u ′ , and the plana r o rder is the r estriction of the planar order in T . This plan ted planar tree has m leav es, wher e m = r u + r u ′ − 1 , when the rela tive po sition of u and u ′ is as in the ﬁr st diagr am of Figure 17, s ee also Figure 18. If the relative p o sition is as in the second diag ram of Figure 17, then m = f u ′′ + r u + r u ′ − 2 , see Figure 19. 26 FERNANDO M URO ◦ • u ◦ ◦ ◦ • u ′ k T k • ◦ ? ? ? ? ? ? ? ?                 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?                 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9         2 2 2 2 2 2 2        ◦ u ′′ • u ◦ ◦ ◦ • u ′ k T ′ k ? ? ? ? ? ? ? ?                 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?                 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9        2 2 2 2 2 2 2        ◦ [St( u,u ′ )] k T / St( u, u ′ ) k • ◦ ? ? ? ? ? ? ? ?         v v v v v v v v v v o o o o o o o o o o o o o Figure 18. An ex a mple of the plan ted plana r tree with leav es T ′ for the r elative p os ition of the vertices u and u ′ as in the ﬁrst digram of Figure 17. The sub complex St( u, u ′ ) ⊂ T is in do uble lines. W e also depict T / St( u, u ′ ). In this ca s e, by inductio n hypothes is, the t wo p ossible comp ositions in the square (a) coincide with the follo wing morphism, see Figure 20 for an illustration, (5.8) U ( e u ) ⊗ U ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) O ( e u ) ⊗ O ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) N v ∈ I e ( T ) \{ u, u ′ } V ( e v ) ⊗ N w ∈ I o ( T ) \ Lk ( u,u ′ ) O ( e w ) ⊗ O ( T ′ ) N v ∈ I e ( T / S t( u,u ′ )) V ( e v ) ⊗ N w ∈ I o ( T / S t( u,u ′ )) \{ [St( u,u ′ )] } O ( e w ) ⊗ O ( m ) P t − 2 ( n ) P t − 1 ( n ) ¯ g ( e u ) ⊗ ¯ g ( e u ′ ) ⊗ id   ∼ = symmetry   id ⊗O ( p T ′ )   ¯ ψ T / St( u,u ′ ) t − 2 or the identity if t =2   ϕ t − 1 ( n )    HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 27 k T k = ◦ • ◦ ◦ ◦ • u • u ′ ◦ ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9                 2 2 2 2 2 2 2        k T ′ k = ◦ u ′′ • u • u ′ ◦ ? ? ? ? ? ? ? ? 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9                 Figure 19. An ex a mple of the plan ted plana r tree with leav es T ′ for the r elative p osition of the vertices u and u ′ as in the second digram of Figure 17. The sub complex St( u, u ′ ) ⊂ T is in do uble lines. O (2 ) U (3) O (3 ) O (0) O (2) U (0) V (1) O (0 ) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ¯ g (3) / / ¯ g (0)   O (2 ) O (3 ) O (3 ) O (0) O (2) O (0 ) V (1) O (0 ) T T T T T T T T T T T T T % % % % % % % % % % % % % { { { { { { { { { { { { { ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ O ( p T ′ ) / / O (4 ) V (1) O (0 ) ⊗ ⊗ Figure 20. An illustration of (5.8) for T , u and u ′ as in Figure 18. In the following lemma, we inductively construct an o per ad str uc tur e o n the colimit of the sequence deﬁned in the former. Roughly spea king, we need to deﬁne the multiplications ◦ i of elemen ts attached to O through plan ted plana r trees with leav es c o ncentrated in even levels T and T ′ , where i is less than or equal to the nu mber of leav es o f T . Co nsider for instance 28 FERNANDO M URO 2 4 F E R N A N D O M U R O I n t h i s c as e , b y i n d u c t i on h y p ot h e s i s , t h e t w o p os s i b l e c om p os i t i on s i n t h e s q u ar e ( a) c oi n c i d e w i t h ( v al ) ) ( v al ) ) \ { u , u ( v al ) ) ( v al ) ) ( v al ) ) ( v al ) ) \ { u , u ( v al ) ) ( v al ) ) \ { u , u ( v al ) ) ( L k ( L k ( ) ) ( v al ) ) T / S t( u , u ) ) ( v al ) ) T / S t( u , u ) ) \ { [ S t ( u , u ) ] ( v al ) ) ( v a l ) ) ( v a l ) ) i d r e o r d e r i n g i d T / St ( u, u o r t h e i d e n t i t y i f =2 I n t h e f ol l o w i n g l e m m a, w e i n d u c t i v e l y c on s t r u c t an op e r ad s t r u c t u r e on t h e c ol i m i t of t h e s e q u e n c e d e ﬁ n e d i n t h e f or m e r l e m m a. R ou gh l y s p e ak i n g, w e n e e d t o d e ﬁ n e t h e m u l t i p l i c at i on s of e l e m e n t s at t ac h e d t o t h r ou gh p l an t e d p l an ar t r e e s w i t h l e a v e s c on c e n t r at e d i n e v e n l e v e l s an d , w h e r e i s l e s s or e q u al t h an t h e n u m b e r of l e a v e s of . C on s i d e r f or i n s t an c e ! T ! = ◦ • ◦ ◦ ◦ ! T ′ ! = ◦ • • ◦ In this case, in or der to deﬁne ◦ 2 we take T ◦ 2 T ′ and the following asso ciated tens or pro duct of ob jects in V and O , H OM OT OP Y T H E OR Y OF N ON - S Y M M E T R I C OP E R A D S 2 5 I n t h i s c as e , i n or d e r t o d e ﬁ n e w e t ak e an d t h e f ol l o w i n g as s o c i at e d t e n s or p r o d u c t of ob j e c t s i n an d T ◦ 2 T ′ = ◦ • ◦ ◦ ◦ ◦ • • ◦ e  O (2) V (3) O (1) O (3) O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ Not i c e t h at t h i s ob j e c t i n i s j u s t t h e t e n s or p r o d u c t of t h e ob j e c t s as s o c i at e d t o an d . T h e n w e c on t r ac t t h e r o ot e d ge of , w h i c h i s i d e n t i ﬁ e d w i t h t h e s e c on d l e af of , an d w e ge t a p l an t e d p l an ar t r e e w i t h l e a v e s i n e v e n v e r t i c e s ( /e /e ◦ ◦ ◦ • • T h i s c an b e al ge b r ai c al l y m i m i c k e d on t h e as s o c i at e d t e n s or p r o d u c t b y m e an s of m u l t i p l i c at i on i n ac c or d i n g t o t h e l o c al s t r u c t u r e of i n a n e i gh b ou r h o o d of , e . g. i s t h e s e c on d l e af of b u t i t i s t h e ﬁ r s t ( an d t h e on l y ) on e at t ac h e d t o Notice that this ob ject in V is just the tensor pro duct of the ob jects as so ciated to T and T ′ . Then we contract the ro ot edge e of T ′ , which is identiﬁed with the second leaf of T , and we get a planted planar tree with leav es in ev en lev els ( T ◦ 2 T ′ ) /e . H OM OT OP Y T H E OR Y OF N ON - S Y M M E T R I C OP E R A D S 2 5 I n t h i s c as e , i n or d e r t o d e ﬁ n e w e t ak e an d t h e f ol l o w i n g as s o c i at e d t e n s or p r o d u c t of ob j e c t s i n an d ◦ ◦ • • ( 2) ( 3) ( 1) ( 3) ( 0) ( 2) ( 0) ( 1) ( 0) ⊗ ⊗ ⊗ ⊗ Not i c e t h at t h i s ob j e c t i n i s j u s t t h e t e n s or p r o d u c t of t h e ob j e c t s as s o c i at e d t o an d . T h e n w e c on t r ac t t h e r o ot e d ge of , w h i c h i s i d e n t i ﬁ e d w i t h t h e s e c on d l e af of , an d w e ge t a p l an t e d p l an ar t r e e w i t h l e a v e s i n e v e n v e r t i c e s ( /e ( T ◦ 2 T ′ ) /e = ◦ • ◦ ◦ ◦ • • ◦ T h i s c an b e al ge b r ai c al l y m i m i c k e d on t h e as s o c i at e d t e n s or p r o d u c t b y m e an s of m u l t i p l i c at i on i n ac c or d i n g t o t h e l o c al s t r u c t u r e of i n a n e i gh b ou r h o o d of , e . g. i s t h e s e c on d l e af of b u t i t i s t h e ﬁ r s t ( an d t h e on l y ) on e at t ac h e d t o This can b e a lgebraica lly mimic ked on the asso ciated tensor pro duct by means of m ultiplication in O a ccording to the lo c al structure of T ◦ 2 T ′ in a neighbourho o d HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 29 of e , e.g. e is the second leaf of T but it is the ﬁrst (and the only) one attac hed to its inner v ertex in T , 2 6 F E R N A N D O M U R O i t s i n n e r v e r t e x i n O (2) V (3) O (1) O (3)    O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ◦ 1   O (2) V (3) O (3) O (0) O (2) V (0) V (1) O (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ W e c an d e ﬁ n e t h e m u l t i p l i c at i on of e l e m e n t s as s o c i at e d t o an d v i a t h i s m or p h i s m an d t h e at t ac h i n g of e l e m e n t s as s o c i at e d t o ( /e . W e n o w f or m al i z e t h i s i d e a. pond L e m m a 5. 5. T h er e a r e u n i qu e m o r p h i s m s i n m, n , s , t s , t m, n ) : − → 1) s u ch t h a t − → 1) i s t h e o p er a d c o m p o s i t i o n l a w , s , t m, n ) ( i d ) = 1) , t m, n s , t m, n ) ( i d ) ) = 1) s , t m, n a n d gi ven p l a n t e d p l a n a r t r e es a n d w i t h l e a ves c o n c en t r a t e d i n even d e gr e es w i t h c ar d ) = c ar d ) = c ar d ) = , a n d c ar d ) = , i f i s t h e u n i qu e l evel ver t ex , b el o n gs t o t h e t h l e a f e d ge ( w i t h r es p e ct t o t h e p a t h o r d er ) , t h e t h l e a f e d ge o c cu p i es t h e t h p l a c e a m o n g a l l i n c o m m i n g e d ges o f , a n d u , u , t h en t h e t h e m o r p h i s m s , t m, n ) ( W e can deﬁne the ◦ 2 m ultiplication o f elements as so ciated to T and T ′ via this morphism a nd the attaching of elements asso cia ted to ( T ◦ 2 T ′ ) /e . W e now forma liz e this idea. Lemma 5.9 . Ther e ar e u nique morphisms in V , n, s, t ≥ 0 , 1 ≤ i ≤ m , c s,t i ( m, n ) : P s ( m ) ⊗ P t ( n ) − → P s + t ( m + n − 1) , such that c 0 , 0 i = ◦ i : O ( m ) ⊗ O ( n ) − → O ( m + n − 1 ) is t he op er a d c omp osition law, c s,t i ( m, n )( ϕ s ( m ) ⊗ id ) = ϕ s + t ( m + n − 1) c s − 1 ,t i ( m, n ) , c s,t i ( m, n )(id ⊗ ϕ t ( n )) = ϕ s + t ( m + n − 1) c s,t − 1 i ( m, n ) , and given plante d planar tr e es T and T ′ with le aves c onc entr ate d in even levels, card L ( T ) = m , car d L ( T ′ ) = n , card I e ( T ) = s , and car d I e ( T ′ ) = t , if u ′ ∈ I o ( T ′ ) is the u n ique level 1 vertex, u ∈ I o ( T ) b elongs t o the i th le af e dge (with r esp e ct to the p ath or der), the i th le af e dge o c cupies the k th plac e among al l inc omming e dges of u , and e = { u , u ′ } ∈ E ( T ◦ i T ′ ) , then the the morphism c s,t i ( m, n )( ¯ ψ T s ⊗ ¯ ψ T ′ t ) 30 FERNANDO M URO c oinci des with the fol lowing morphism, that we c all d s,t i ( T , T ′ ) , N v ∈ I e ( T ) V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) O ( e u ) ⊗ O ( e u ′ ) ⊗ N v ∈ I e ( T ) ∪ I e ( T ′ ) V ( e v ) ⊗ N w ∈ ( I o ( T ) \{ u } ) ∪ ( I o ( T ′ ) \{ u ′ } ) O ( e w ) O ( e u + e u ′ − 1 | {z } = f [ e ] ) ⊗ N v ∈ I e (( T ◦ i T ′ ) /e ) V ( e v ) ⊗ N w ∈ I o (( T ◦ i T ′ ) /e ) \{ [ e ] } O ( e w ) P s + t ( m + n − 1) ∼ = symmetry   ◦ k ⊗ id   ¯ ψ ( T ◦ i T ′ ) /e s + t   Her e we u s e the c onvention that ¯ ψ T 0 = id O ( m ) and ¯ ψ T ′ 0 = id O ( n ) . Pr o of. The map c s,t i ( m, n ) is deﬁned from c s − 1 ,t i ( m, n ), c s,t − 1 i ( m, n ) and d s,t i ( T , T ′ ) by using the universal property of the push-o ut deﬁnitio n of P s ( m ) ⊗ P t ( n ) arising from Lemmas 4.1 a nd 5 .4, b y induction on ( s, t ) ∈ N × N , N = { 0 , 1 , 2 . . . } , with resp ect to the graded lexicographic order, ( s, t ) ≤ ( s ′ , t ′ ) ⇔  either s + t < s ′ + t ′ , or s + t = s ′ + t ′ and s ≤ s ′ . There is nothing to chec k for the ﬁrs t three elements (0 , 0), (0 , 1 ), (1 , 0). Assume that everything holds up to the predece ssor of ( s, t ) with s + t > 1 . W e hav e to show that, for a n y x ∈ I e ( T ) and x ′ ∈ I e ( T ′ ), the follo wing compatibilit y conditions hold: d s,t i ( T , T ′ )( f ( e x ) ⊗ id ) = ϕ s + t ( m + n − 1) c s − 1 ,t i ( m, n )( ψ T s,x ⊗ ¯ ψ T ′ t ) , (a) d s,t i ( T , T ′ )( f ( e x ′ ) ⊗ id ) = ϕ s + t ( m + n − 1) c s,t − 1 i ( m, n )( ¯ ψ T s ⊗ ψ T ′ t,x ′ ) . (b) Since (a) and (b) are very similar to each other, we here just chec k (a). W e must distinguish t wo ca ses: { x, u } ∈ E ( T ) and { x, u } / ∈ E ( T ), see Figure 21. Suppo se { x, u } / ∈ E ( T ). Then u / ∈ Lk( x ). Using the deﬁnition of d s,t i ( T , T ′ ) in the statement of this le mma and the deﬁnition of ¯ ψ ( T ◦ i T ′ ) /e s + t in L emma 5.4 w e de duce that, in this case, the left hand side o f (a) is the following comp os ite morphism, HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 31 k T k = ◦ • x x ◦ u ◦ ◦ • ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         2 2 2 2 2 2 2        k T ′ k = ◦ u ′ • • ◦ 9 9 9 9 9 9 9 9         Figure 21. F or the trees T a nd T ′ and i = 2 we depict u , u ′ and t wo p o ssible choices of x , one with { x, u } ∈ E ( T ) a nd the o ther one with { x, u } / ∈ E ( T ). see Figure 22, (5.10) U ( e x ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) O ( e x ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) O ( St( x )) ⊗ O ( e u ) ⊗ O ( e u ′ ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) \ ( Lk ( x ) ∪{ u } ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) \{ u ′ } O ( f w ′ ) O ( r x ) ⊗ O ( e u + e u ′ − 1) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) \ ( Lk ( x ) ∪{ u } ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) \{ u ′ } O ( f w ′ ) N v ∈ I e ((( T / St( x )) ◦ i T ′ ) /e ) V ( e v ) ⊗ N w ∈ I o ((( T / St( x )) ◦ i T ′ ) /e ) O ( e w ) P s + t − 1 ( m + n − 1) P s + t ( m + n − 1) ¯ g ( e x ) ⊗ id   ∼ = symmetry   O ( p St( x ) ) ⊗◦ k ⊗ id   ∼ = symmetry   ¯ ψ (( T / St( x )) ◦ i T ′ ) /e s + t − 1   ϕ s + t ( m + n − 1)   32 FERNANDO M URO ◦ • x ◦ u u ′ e ◦ ◦ ◦ • • ◦ • k T ◦ 2 T ′ k ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         2 2 2 2 2 2 2        9 9 9 9 9 9 9 9         p T ◦ 2 T ′ St( x ) ⊔ e / / _ X V U T S S S S S T U V X _ ◦ • ◦ [ e ] ◦ ◦ [St( x )] • • ◦ k (( T / St( x )) ◦ 2 T ′ ) /e k ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         2 2 2 2 2 2 2 9 9 9 9 9 9 9 9         O (2 ) V (3) O (1 ) O (3 ) O (0 ) O (2) V (0) V (1 ) O (0 ) U (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ¯ g (0) 2 2 O (2 ) V (3) O (1 ) O (3 ) O (0 ) O (2) V (0) V (1 ) O (0 ) O (0 ) _   7 _ _   7 _ K n }     K K n }     K ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ◦ 1   O ( p St( x ) ) = ◦ 2 O O O (2 ) V (3) O (3 ) O (0) O (1) V (0) V (1) O (0 ) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ Figure 22. An illustr ation of (5.10) for T a nd T ′ as in Figure 21 in case { x, u } / ∈ E ( T ). Moreov er, b y induction, sinc e ( s − 1 , t ) < ( s, t ) one can easily chec k that this is als o the right hand side of (a). Suppo se no w that { x, u } ∈ E ( T ). Then u ∈ Lk( x ). Assume tha t e is the l th leaf of St( x ). W e denote T ′′ the pla nt ed planar tree with leaves T ′′ = St ( x ) ◦ l C e u ′ . The inner pa rt of T ′′ is iden tiﬁed with the subtree T ′′′ ⊂ T ◦ i T ′ formed b y adjoining the edge e to St( x ), see Figure 23. Using the deﬁnition of d s,t i ( T , T ′ ) in the statemen t, the deﬁnition of ¯ ψ ( T ◦ i T ′ ) /e s + t in Lemma 5.4, and rela tion (2) in Remark 2.6 for O , we deduce that, in this ca se, the left hand side o f (a) is the following comp osite HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 33 k T ◦ 2 T ′ k ◦ • x ◦ u u ′ e ◦ ◦ ◦ • • ◦ • ? ? ? ? ? ? ? ?                 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?                 2 2 2 2 2 2 2        9 9 9 9 9 9 9 9         k T ′′ k ◦ • x ◦ u u ′ e ◦ ◦ ◦ ? ? ? ? ? ? ? ?                 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?                 2 2 2 2 2 2 2        9 9 9 9 9 9 9 9         k ( T ◦ 2 T ′ ) /T ′′′ k ◦ • • ◦ • [ T ′′′ ] ? ? ? ? ? ? ? ?         t t t t t t t t t t t / / / / / / /        Figure 23. F or the choice of x in Figure 21 with { x, u } ∈ E ( T ) we here depict T ′′ . The subtree T ′′′ is indicated with double lines. morphism, see Figure 24, (5.11) U ( e x ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) O ( e x ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) O ( T ′′ ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) \ Lk ( x ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) \{ u ′ } O ( f w ′ ) O ( r x + e u ′ − 1 ) ⊗ N v ∈ I e ( T ) \{ x } V ( e v ) ⊗ N w ∈ I o ( T ) \ Lk ( x ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) \{ u ′ } O ( f w ′ ) N v ∈ I e (( T ◦ i T ′ ) /T ′′′ ) V ( e v ) ⊗ N w ∈ I o (( T ◦ i T ′ ) /T ′′′ ) O ( e w ) P s + t − 1 ( m + n − 1) P s + t ( m + n − 1) ¯ g ( e x ) ⊗ id   ∼ = symmetry   O ( p T ′′ ) ⊗ id   ∼ = symmetry   ¯ ψ ( T ◦ i T ′ ) /T ′′′ s + t − 1   ϕ s + t ( m + n − 1)   34 FERNANDO M URO O (2 ) U (3) O (1 ) O (3 ) O (0 ) O (2) V (0) V (1 ) O (0 ) V (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ¯ g (3) / / O (2 ) O (3 ) O (1 ) O (3 ) O (0 ) O (2) V (0) V (1 ) O (0 ) V (0) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ O ( p T ′′ ) / /           _ _ _ _ _ _ _ _ _ _ _ _    _ _ _ _ _ _         O (6 ) V (0) V (1) O (0 ) V (0) ⊗ ⊗ ⊗ ⊗ Figure 24. An illustr ation of (5.11) for T a nd T ′ as in Figure 21 in case { x, u } ∈ E ( T ), see Figure 23. Moreov er, b y induction one can eas ily check that this is also the rig ht hand side of (a), hence w e are done with this proo f.  Let P b e the seque nce deﬁned as P ( n ) = co lim t ≥ 0 P t ( n ) . By the previous lemma, the morphisms c s,t i ( m, n ) induce comp os itio n laws in the colimit, (5.12) ◦ i : P ( m ) ⊗ P ( n ) − → P ( m + n − 1 ) , 1 ≤ i ≤ m, n ≥ 0 . Consider the morphism (5.13) I O (1 ) = P 0 (1) colim t ≥ 0 P t (1) = P (1) . canonical / / u / / Prop ositi o n 5.14. The se quenc e P , t he unit (5.13) and the c omp ositio n laws (5.12) deﬁne an op er ad. Pr o of. W e must chec k that rela tio ns (1)–(4) in Remark 2.6 hold for P . Each of these rela tions for P ca n b e derived from the corresp o nding relation for O . As relations (1) and (2) ar e very similar to each other , just a s (3) and (4), w e here chec k (2) and (3). In order to prove relation (2) for P it is enough to chec k that the follo wing t wo morphisms P r ( l ) ⊗ P s ( m ) ⊗ P t ( n ) → P r + s + t ( l + m + n − 2) coincide, c r + s,t j ( l + m − 1 , n )( c r,s i ( l, m ) ⊗ id P t ( n ) ) = c r,s + t i ( l, m + n − 1)(id P r ( l ) ⊗ c s,t j − i +1 ( m, n )) . W e chec k this by induction on ( r, s, t ) ∈ N 3 with resp ect to the graded lexico - graphic order. F or r = s = t = 0 this is just re la tion (2) for the op era d O . If we assume that the relation holds up to the predecess or of ( r , s, t ), then b y us ing the universal prop erty of the push- out deﬁnit ion of P r ( m ) ⊗ P s ( n ) ⊗ P t ( p ) a rising from HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 35 k T k ◦ • ◦ u ◦ ◦ • ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         2 2 2 2 2 2 2        k T ′ k ◦ u ′ 1 (= u ′ 2 if j = 2) u ′ 2 if j = 3 • • ◦ 9 9 9 9 9 9 9 9         k T ′′ k ◦ u ′′ • • 9 9 9 9 9 9 9 9         Figure 25. F or the planted pla na r trees with leaves T , T ′ and T ′′ we depict u , u ′ 1 , u ′ 2 and u ′′ for i = 2 and j = 2 , 3. Lemmas 4.1 and 5.4, w e only ha ve to chec k that, with the no ta tion of Lemma 5.9, given planted pla nar trees with leav es concentrated in even levels T , T ′ , T ′′ with card L ( T ) = l , card L ( T ′ ) = m , ca r d L ( T ′′ ) = n , ca rd I e ( T ) = r , card I e ( T ′ ) = s , and card I e ( T ′′ ) = t , then (a) c r + s,t j ( l + m − 1 , n )( d r,s i ( T , T ′ ) ⊗ ¯ ψ T ′′ t ) = c r,s + t i ( l, m + n − 1)( ¯ ψ T r ⊗ d s,t j − i +1 ( T ′ , T ′′ )) . Let u ∈ I o ( T ) b e the inner vertex of the i th leaf edge of T , u ′ 1 ∈ I o ( T ′ ) the unique level 1 vertex of T ′ , u ′ 2 ∈ I o ( T ′ ) the inner vertex of the ( j − i + 1) th leaf edge of T ′ , and u ′′ ∈ I o ( T ′′ ) the unique level 1 v ertex of T ′′ . Suppo se that the i th leaf edge of T is the k th 1 incomming edge of u , and that the ( j − i + 1) th leaf edge o f T ′ is the k th 2 incomming edge o f u ′ 2 . The most complicated case is when u ′ 1 = u ′ 2 , and ev en this case is easy , although somewhat tedious. Assume u ′ 1 = u ′ 2 and denote this vertex simply by u ′ . Notice that ( T ◦ i T ′ ) ◦ j T ′′ = T ◦ i ( T ′ ◦ j − i +1 T ′′ ), compa r e Figure 11. Let K ⊂ ( T ◦ i T ′ ) ◦ j T ′′ be the subtree with V ( K ) = { u , u ′ , u ′′ } and E ( K ) = { { u, u ′ } , { u ′ , u ′′ }} , see Figure 2 6. Then by 36 FERNANDO M URO k ( T ◦ 2 T ′ ) ◦ 2 T ′′ k ◦ • ◦ u u ′′ u ′ 1 = u ′ 2 ◦ ◦ ◦ • • ◦ ◦ • • • ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         2 2 2 2 2 2 2        9 9 9 9 9 9 9 9         , , , , , ,       k ( T ◦ 2 T ′ ) ◦ 3 T ′′ k ◦ • ◦ u u ′ 1 u ′ 2 u ′′ ◦ ◦ ◦ • • ◦ ◦ • • • ? ? ? ? ? ? ? ?         ? ? ? ? ? ? ? ?         2 2 2 2 2 2 2        9 9 9 9 9 9 9 9         2 2 2 2 2 2       Figure 26. Here we depict the subtree K ⊂ ( T ◦ i T ′ ) ◦ j T ′′ in double lines for the trees in Figure 25, i = 2 and j = 2 , 3. Lemma 5.9 and relation (2) for O , b oth sides of (a) coincide with N v ∈ I e ( T ) V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) ⊗ N v ′′ ∈ I e ( T ′′ ) V ( f v ′′ ) ⊗ N w ′′ ∈ I o ( T ′′ ) O ( f w ′′ ) O ( e u ) ⊗ O ( e u ′ ) ⊗ O ( f u ′′ ) ⊗ N v ∈ I e ( T ) ∪ I e ( T ′ ) ∪ I e ( T ′′ ) V ( e v ) ⊗ N w ∈ ( I o ( T ) ∪ I o ( T ′ ) ∪ I o ( T ′′ )) \{ u,u ′ ,u ′′ } O ( e w ) O ( e u + e u ′ + f u ′′ − 2) ⊗ N v ∈ I e (( T ◦ i T ′ ) ◦ j T ′′ ) /K V ( e v ) ⊗ N w ∈ I o ((( T ◦ i T ′ ) ◦ j T ′′ ) /K ) \{ [ K ] } O ( e w ) P s + t ( m + n + p − 2) ∼ = symmetry   ( ◦ k 1 (id ⊗◦ k 2 )) ⊗ id   ¯ ψ (( T ◦ i T ′ ) ◦ j T ′′ ) /K r + s + t   Assume now that u ′ 1 6 = u ′ 2 . In this case it is not even necessar y to us e a ny of the relations in Remark 2.6 for O . Actually , by Lemma 5.9, if K ⊂ ( T ◦ i T ′ ) ◦ j T ′′ is the (disjoint) union of the edges e 1 = { u, u ′ 1 } and e 2 = { u ′ 2 , u ′′ } , see Figur e 26, HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 37 then bo th side s of (a) coincide with N v ∈ I e ( T ) V ( e v ) ⊗ N w ∈ I o ( T ) O ( e w ) ⊗ N v ′ ∈ I e ( T ′ ) V ( e v ′ ) ⊗ N w ′ ∈ I o ( T ′ ) O ( f w ′ ) ⊗ N v ′′ ∈ I e ( T ′′ ) V ( f v ′′ ) ⊗ N w ′′ ∈ I o ( T ′′ ) O ( f w ′′ ) O ( e u ) ⊗ O ( f u ′ 1 ) ⊗ O ( f u ′ 2 ) ⊗ O ( f u ′′ ) ⊗ N v ∈ I e ( T ) ∪ I e ( T ′ ) ∪ I e ( T ′′ ) V ( e v ) ⊗ N w ∈ ( I o ( T ) ∪ I o ( T ′ ) ∪ I o ( T ′′ )) \{ u,u ′ 1 ,u ′ 2 ,u ′′ } O ( e w ) O ( e u + f u ′ 1 − 1) ⊗ O ( f u ′ 2 + f u ′′ − 1) ⊗ N v ∈ I e (( T ◦ i T ′ ) ◦ j T ′′ ) /K V ( e v ) ⊗ N w ∈ I o ((( T ◦ i T ′ ) ◦ j T ′′ ) /K ) \{ [ e 1 ] , [ e 2 ] } O ( e w ) P s + t ( m + n + p − 2 ) ∼ = symmetry   ◦ k 1 ⊗◦ k 2 ⊗ id   ¯ ψ (( T ◦ i T ′ ) ◦ j T ′′ ) /K r + s + t   Relation (3) is a consequence of the fact tha t the following compo site mor phism is a righ t unit constraint in V , P r ( l ) ⊗ I id ⊗ u / / P r ( l ) ⊗ O (1) = P r ( l ) ⊗ P 0 (1) c r, 0 i ( l, 1) / / P r ( l ) . This follows by inductio n o n r . F or r = 0 this is just relation (3) for O . Assume this holds up to r − 1. By Lemma 5 .4 and the induction hypo thesis, we only ha ve to chec k that the morphism c r, 0 i ( l, 1)( ¯ ψ T r ⊗ u ) coincides with the composition o f the right unit iso mo rphism and ¯ ψ T r . By Lemma 5 .9, c r, 0 i ( l, 1)( ¯ ψ T r ⊗ id O (1) ) = d r, 0 i ( T , C 1 ) . Let u ′ ∈ I o ( C 1 ) be now the unique inner v ertex of C 1 , and e = { u, u ′ } ∈ E ( T ◦ i C 1 ). In this case ( T ◦ i C 1 ) /e = T . More over, by the very deﬁnition o f d r, 0 i ( T , C 1 ) in the statement of Lemma 5.9 a nd b y relation (3) for O , the morphism d r, 0 i ( T , C 1 )(id ⊗ u ) is the compos ition of the right unit isomor phis m and ¯ ψ T r , hence w e are done.  Consider the morphisms of sequences f ′ : O → P and ¯ g ′ : V → P deﬁned as f ′ ( n ) : O ( n ) = P 0 ( n ) colim t ≥ 0 P t ( n ) = P ( n ) , canonical / / V ( n ) ⊗ I ⊗ ( n +1) ∼ = V ( n ) V ( n ) ⊗ O (1) ⊗ ( n +1) P 1 ( n ) colim t ≥ 0 P t ( n ) = P ( n ) id ⊗ u ⊗ ( n +1)   ¯ ψ C 1 ( C n ( C 1 ,...,C 1 )) 1 / / canonical / / ¯ g ′ ( n ) ' ' Theorem 5.15 . The morphism f ′ : O → P is an op er ad morphism. Mor e over, if g ′ : F ( V ) → P is t he op er ad morphism adjoint to ¯ g ′ , then the fol lowing diagr a m is 38 FERNANDO M URO k C 1 ( C 5 ( C 1 , . . . , C 1 )) k = ◦ v • ◦ ◦ ◦ ◦ ◦ / / / / / ? ? ? ? ? ?            Figure 27. The planted plana r tr ee with leaves in even levels C 1 ( C n ( C 1 , . . . , C 1 )) for n = 5. a push-out in Op( V ) , F ( U ) g   F ( f ) / / F ( V ) g ′   O f ′ / / P Pr o of. The mor phism f ′ is an op er ad morphism b y the very deﬁnition of the op era d structure in P , since c 0 , 0 i = ◦ i is the structure morphism of O and the unit of P is the compositio n o f the unit of O and f ′ , see Lemma 5 .9 and (5 .13). Mor eov er, the square U ( l ) ¯ g ( l )   f ( l ) / / V ( l ) ¯ g ′ ( l )   O ( l ) f ′ ( l ) / / P ( l ) commutes for all l ≥ 0. In fact, the following dia g ram commutes by some trivial facts, including the v ery deﬁnition of P 1 ( l ) in Lemma 5.4, U ( l ) ∼ = (right unit) − 1   ¯ g ( l ) v v m m m m m m m m m m m m m m m m f ( l ) / / V ( l ) ∼ = (right unit) − 1   O ( l ) ∼ = (right unit) − 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q id , , U ( l ) ⊗ I ⊗ ( l +1) ¯ g ( l ) ⊗ id ⊗ ( l +1) I   id ⊗ u ⊗ ( l +1) ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q f ( l ) ⊗ i d ( l +1) I / / V ( l ) ⊗ I ⊗ ( l +1) id ⊗ u ⊗ ( l +1)   O ( l ) ⊗ I ⊗ ( l +1) id O ( l ) ⊗ u ⊗ ( l +1)   right unit & & U ( l ) ⊗ O (1) ⊗ ( l +1) ¯ g ( l ) ⊗ id ⊗ ( l +1) v v m m m m m m m m m m m m m f ( l ) ⊗ i d ( l +1) O (1) 9 9 ψ C 1 ( C l ( C 1 ,...,C 1 )) 1 l l V ( l ) ⊗ O (1) ⊗ ( l +1) ¯ ψ C 1 ( C l ( C 1 ,...,C 1 )) 1   O ( l ) ⊗ O (1) ⊗ ( l +1) O ( p C 1 ( C l ( C 1 ,...,C 1 )) )   O ( l ) ϕ 1 ( l ) @ @ P 1 ( l ) HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 39 and its outer (comm utative) squa re U ( l ) ¯ g ( l )   f ( l ) / / V ( l ) ∼ = (right unit) − 1   V ( l ) ⊗ I ⊗ ( l +1) id ⊗ u ⊗ ( l +1)   V ( l ) ⊗ O (1) ⊗ ( l +1) ¯ ψ C 1 ( C l ( C 1 ,...,C 1 )) 1   O ( l ) ϕ 1 ( l ) / / P 1 ( l ) comp osed with the cano nical morphism P 1 ( l ) → colim r ≥ 0 P r ( l ) = P ( l ) yields the former square. Suppo se we are g iven an op erad P ′ and mor phisms f ′′ : O → P ′ in O p( V ) and ¯ g ′′ : V → P ′ in V N such that the squar e (a) U ( l ) ¯ g ( l )   f ( l ) / / V ( l ) ¯ g ′′ ( l )   O ( l ) f ′′ ( l ) / / P ′ ( l ) commutes for all l ≥ 0. W e m ust show that there is a unique morphism h : P → P ′ in Op( V ) such that f ′′ = hf ′ and ¯ g ′′ = h ¯ g ′ in V N . W e deﬁne morphisms h r ( l ) : P r ( l ) − → P ′ by induction on r ≥ 0 a s follows. W e set h 0 ( l ) = f ′′ ( l ). Assume we have deﬁned up to h r − 1 ( l ). Then we deﬁne h r ( l ) s o that h r ( l ) ϕ r ( l ) = h r − 1 ( l ) a nd, for any plan ted planar tree T with l leav es concen trated in ev en lev els and r inner v ertices in even levels, (b) h r ( l ) ¯ ψ T r = P ′ ( p T )( O v ∈ I e ( T ) ¯ g ′′ ( e v ) ⊗ O w ∈ I o ( T ) f ′′ ( e w )) . The morphism h r ( l ) is well deﬁned b y the univ ersal proper ty of th e push-out deﬁ- nition of P r ( l ) in Lemma 5.4 since, given u ∈ I e ( T ), P ′ ( p T )( O v ∈ I e ( T ) ¯ g ′′ ( e v ) ⊗ O w ∈ I o ( T ) f ′′ ( e w ))( f ( e u ) ⊗ id ) = P ′ ( p T )( O v ∈ I e ( T ) \{ u } ¯ g ′′ ( e v ) ⊗ O w ∈ I o ( T ) ∪ { u } f ′′ ( e w ))(id ⊗ ¯ g ( e u )) = P ′ ( p T / S t( u ) )( O v ∈ I e ( T / S t( u )) ¯ g ′′ ( e v ) ⊗ O w ∈ I o ( T / S t( u )) f ′′ ( e w ))(id ⊗ O ( p St( u ) ))(id ⊗ ¯ g ( e u )) = h r − 1 ( l ) ¯ ψ T / S t( u ) r − 1 (id ⊗ O ( p St( u ) ))(id ⊗ ¯ g ( e u )) = h r − 1 ( l ) ψ T r,u . 40 FERNANDO M URO Here, in the ﬁrst eq ua tion w e use the co mm utativity of (a), in the second eq uation we use the fact that f ′′ is an op erad morphism, and in the third equatio n we use the induction hypothesis . The fourth equa tio n fo llows from the very deﬁnition of ψ T r,u in the statement of Lemma 5.4. F or simplicity , in these equations we have omitted some symmetry isomorphisms in V . W e hav e chec k ed that the morphisms h r ( l ) induce a mor phism of sequences h : P → P ′ in the colimit. It is clear that h f ′ = f ′′ since h 0 = f ′′ , in particu- lar h preserves units. Moreover, hg ′ = g ′′ since fo r ¯ T = C 1 ( C l ( C 1 , . . . , C 1 )), see Figure 27, if v ∈ I e ( ¯ T ) is the unique inner vertex in ev en lev els, then h 1 ( l ) ¯ ψ ¯ T 1 (id V ( l ) ⊗ u ⊗ ( l +1) ) = P ′ ( p ¯ T )( ¯ g ′′ ( l ) ⊗ f ′′ (1) ⊗ ( l +1) )(id V ( l ) ⊗ u ⊗ ( l +1) ) = P ′ ( p ¯ T )( ¯ g ′′ ( l ) ⊗ u ⊗ ( l +1) ) = ¯ g ′′ ( l ) . Here, in the ﬁrst equation we use (b), in the second equatio n w e use that f ′′ is an op erad morphism and therefo r e it pres e r ves units, and in the third equation we use relations (3) and (4) in Remark 2.6 fo r the op er ad P ′ . In order to c heck that h is indeed an op era d mo rphism, w e show that h r + s ( l + m − 1 ) c r,s i ( l, m ) = h r ( l ) ◦ i h s ( m ) W e pro ceed by induction on ( r, s ) ∈ N 2 with res pec t to the graded lex icogra phic order. This is obvious for r = s = 0, s inc e f ′′ is a n o p erad mo rphism. If the equation holds up to the predecessor o f ( r, s ) then b y induction h yp othesis we only hav e to chec k that the following equa tion holds, h r + s ( l + m − 1) c r,s i ( l, m )( ¯ ψ T r ⊗ ¯ ψ T ′ s ) = ( h r ( l ) ¯ ψ T r ) ◦ i ( h s ( m ) ¯ ψ T ′ s ) , for T ′ a planted planar tree with m le aves concent rated in even levels and s inner vertices in ev en levels. Let u ∈ I o ( T ) be the inner vertex of the i th leaf edge of T , u ′ ∈ I o ( T ′ ) the unique level 1 v ertex of T ′ , and e = { u, u ′ } ∈ E ( T ◦ i T ′ ). Supp ose that the i th leaf edge of T is the k th incomming edge of u . Then, h r + s ( l + m − 1 ) c r,s i ( l, m )( ¯ ψ T r ⊗ ¯ ψ T ′ s ) = h r + s ( l + m − 1 ) d r,s i ( T , T ′ ) = h r + s ( l + m − 1 ) ¯ ψ ( T ◦ i T ′ ) /e s + t ( ◦ k ⊗ id ) = P ′ ( p ( T ◦ i T ′ ) /e )( O v ∈ I e (( T ◦ i T ′ ) /e ) ¯ g ′′ ( e v ) ⊗ O w ∈ I o (( T ◦ i T ′ ) /e ) f ′′ ( e w ))( ◦ k ⊗ id ) = P ′ ( p ( T ◦ i T ′ ) /e )( ◦ k ⊗ id )( O v ∈ I e ( T ) ∪ I e ( T ′ ) ¯ g ′′ ( e v ) ⊗ O w ∈ I o ( T ) ∪ I o ( T ′ ) f ′′ ( e w )) = ( P ′ ( p T ) ◦ i P ′ ( p T ′ ))( O v ∈ I e ( T ) ∪ I e ( T ′ ) ¯ g ′′ ( e v ) ⊗ O w ∈ I o ( T ) ∪ I o ( T ′ ) f ′′ ( e w )) = ( h r ( l ) ¯ ψ T r ) ◦ i ( h s ( m ) ¯ ψ T ′ s ) . Here ◦ k denotes either ◦ k : O ( e u ) ⊗ O ( e u ′ ) → O ( f [ e ]) or ◦ k : P ′ ( e u ) ⊗ P ′ ( e u ′ ) → P ′ ( f [ e ]). Moreov er, in the ﬁr st equatio n we use the inductive deﬁnition of c r,s i ( l, m ) in Lemma 5.9, in the s econd eq ua tion we use the deﬁnition of d r,s i ( T , T ′ ) a lso in Lemma 5.9, in the third equation we use (b), in the fourth equa tio n we use that f ′′ is a n o pe r ad mor phism, in the ﬁfth equation we use the construction of o p eradic HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 41 functors fro m op erads in Prop osition 3 .1 0, and in the ﬁnal equation we use (b) again. F urthermor e , for simplicity we hav e omitted some symmetry isomo r phisms in V in t hese equa tio ns. The uniqueness of h follows from the fa ct that the morphism ¯ ψ T r deﬁned in Lemma 5.4 is related to the opera dic functor of P by the follo wing equation, ¯ ψ T r = P ( p T )( O v ∈ I e ( T ) ¯ g ′ ( e v ) ⊗ O w ∈ I o ( T ) f ′ ( e w )) . Therefore, if h ′ : P → P ′ is an opera d morphism sa tisfying h ′ f ′ = f ′′ and h ′ ¯ g ′ = ¯ g ′′ , and if we denote h ′ r ( l ) the compo sition o f h ′ ( l ) with the cano nical morphism to the colimit P r ( l ) → P ( l ), then the morphisms h ′ r ( l ) must satisfy h ′ 0 ( l ) = f ′′ ( l ), a nd also (b) after repla cing h r ( m ) with h ′ r ( m ), there fore h ′ = h b y the univ ersal prop erty of the push-outs P r ( l ) and the colimit P .  6. P roof of Theorem 1.1 Assume in this s e c tion that V is also a coﬁbrantly ge ner ated monoida l mo del category (see Deﬁnition 4.2) satisfying the monoid axiom [SS00, Deﬁnition 3.3]. In order to explain what this means, let us recall so me terminology from [Ho v99]. Given an ordinal λ , a directed diagra m X : λ → V is c ontinuous if for an y limit ordinal α < λ , the canonical morphism colim i<α X i − → X α is an isomorphism. The natura l morphism from the ﬁrst o b ject to the colimit X 0 − → colim i<λ X i is said to b e the t r ansﬁnite c omp osition o f the morphisms in the diagram. W e here do not exclude the possibility that λ be ﬁnite. Given a class of morphisms K in V , a r elative K - c el l c omplex is a transﬁnite comp osition o f mor phisms X : λ → V such that for any i < λ with i + 1 < λ the morphism X i → X i +1 ﬁts into a push-out diag ram as follows, where the top horizontal a rrow is in K , A   push in K / / B   X i / / X i +1 A plain K -c el l c omplex is a r elative K -cell complex with X 0 = 0 the initial ob ject of V . If I and J are sets of generating coﬁbr ations and generating trivial coﬁbr a tions in V , resp ectively , then the coﬁbrations in V are exa ctly the r etracts o f rela tive I -c ell complexes, and the trivia l coﬁbr ations are t he r etracts of relativ e J - cell com- plexes. In particular the coﬁbran t ob jects in V are the retracts of I -cell complexes. So far, nothing of this needs either the monoida l structure of V or its mode l c a te- gory structure. Deﬁnition 6.1 . The monoid axiom for V s ays that, for K = { f ⊗ X ; f is a trivial coﬁbration and X is a n ob ject in V } , all relative K -cell complexes are w eak equiv alences. 42 FERNANDO M URO Prop ositi o n 6 .2. Consider a push-out diagr a m in Op( V ) as fol lows, F ( U ) g   F ( f ) / / push F ( V ) g ′   O f ′ / / P If f is a trivial c oﬁbr ation t hen f ′ ( n ) : O ( n ) → P ( n ) is a r elative K - c el l c omplex, n ≥ 0 , wher e K is the class in t he pr evious deﬁnition. Pr o of. By the push-out pro duct axiom (Deﬁnition 4 .2), the mo r phism (5.5) in Lemma 5.4 is the tensor pro duct of a trivial coﬁbra tion and an ob ject in V , i.e. (5.5) ∈ K . Ther efore, by Theor em 5.15, f ′ ( n ) is a relative K -cell complex.  Consider the a s so ciated sets o f gener ating coﬁbr a tions and genera ting trivial coﬁbrations in V N , I N and J N , resp ectively , see Rema rk 2.3. Corollary 6.3 . If V satisﬁes t he monoid axiom, then a morphism in V N under- lying a r elative F ( J N ) -c el l c omplex in Op( V ) is a we ak e quivalenc e in V N . Now we are ready to pro ve Theorem 1.1. Pr o of of The o r em 1.1. It is ea sy to see that op eradic functors a re the same as alge- bras ov er the monad asso ciated to fr ee op erad adjunction in Sectio n 5. Therefore, using the equiv alence betw een o p er ads a nd op eradic functors in Prop o sition 3.1 0, one can easily show that the natural compar ison functor from ope r ads to alg ebras ov er the the free op era d mo nad is an equiv alence of categ ories. Mo r eov er, this monad preserves ﬁltered colimits, see the explicit construction in Section 5, there- fore the category Op( V ) is complete a nd co complete [Bor94, Pr op osition 4.3.6 ]. F urthermore, the forg etful functor Op( V ) → V N also pr eserves ﬁltered c olimits [Bor94, P rop ositio n 4.3.2 ], in particular, since F is a left a djoint and so urces of morphisms in I and J are presentable in V , then sour ces of morphisms in F ( I N ) and F ( J N ) are presentable in Op( V ). W e can apply [SS00, Lemma 2 .3] in or der to prov e the existence of the claimed mo del structure in Op( V ). The smallness co nditio n has already b een c heck ed, and condition (1) of [SS00, Lemma 2.3] has been established in Corolla ry 6 .3. Recall that a mo del category is right pr o p er if the pull- ba ck of a weak equiv alence along a ﬁbration is again a weak equiv alence [Hir03, Deﬁnition 13.1.1 (2)]. The statement about right prop erness is obvious since ﬁbra tions and w eak eq uiv alences in O p( V ) are detected by the for g etful functor Op( V ) → V N , and this fu nctor is a right adjoint, so it preserves all limits, in particular pull-backs. Recall also that a mo del categ ory is combinatorial if it is coﬁbr antly generated and lo cally pres entable. If V is co mb inatoria l then Op( V ) is lo cally pr esentable b y [AR94, 2.3 (1) and the Theorem in 2 .7 8], hence it is com binatoria l.  7. A l gebras Assume we ha ve a strong br aided monoidal functor V → Z ( C ), where Z ( C ) is the cent er of C , deﬁned in [JS91]. Suc h a functor consists of an ordinary functor z : V − → C , HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 43 together with natural isomorphisms, m ultiplication : z ( X ) ⊗ z ( X ′ ) − → z ( X ⊗ X ′ ) , unit : I C − → z ( I V ) , ζ ( X , Y ) : z ( X ) ⊗ Y − → Y ⊗ z ( X ) , such that the m ultiplication and the unit sa tisfy w ell-known cohere nc e laws [Bor9 4, Deﬁnition 6.4.1] and the following three diagr ams o f isomorphisms comm ute, z ( X ) ⊗ z ( X ′ ) ζ ( X ,z ( X ′ )) / / mult.   z ( X ′ ) ⊗ z ( X ) mult.   z ( X ⊗ X ′ ) z (sym.) / / z ( X ′ ⊗ X ) I C ⊗ Y z ( I V ) ⊗ Y Y Y ⊗ z ( I V ) Y ⊗ I C Y ⊗ unit left un it w w o o o o o o o o unit ⊗ Y ' ' O O O O O O (right unit) − 1   0 0 0 0 0 0 0 ζ ( I V ,Y )          / / z ( X ) ⊗ ( z ( X ′ ) ⊗ Y ) ( z ( X ) ⊗ z ( X ′ )) ⊗ Y z ( X ) ⊗ ( Y ⊗ z ( X ′ )) z ( X ⊗ X ′ ) ⊗ Y ( z ( X ) ⊗ Y ) ⊗ z ( X ′ ) Y ⊗ z ( X ⊗ X ′ ) ( Y ⊗ z ( X )) ⊗ z ( X ′ ) Y ⊗ ( z ( X ) ⊗ z ( X ′ )) z ( X ) ⊗ ζ ( X ′ ,Y )           mult. ⊗ Y   7 7 7 7 7 7 7 7 ζ ( X ,Y ) ⊗ z ( X ′ )   7 7 7 7 7 7 7 7 ζ ( X ⊗ X ′ ,Y )   Y ⊗ mult. C C         o o assoc. assoc.   assoc. / / Moreov er, suppose that the functor z ( − ) ⊗ Y : V → C has a right adjoint, Hom C ( Y , − ) : C − → V . W e will use the evaluatio n morphism , ev aluation : z (Hom C ( Y , Z )) ⊗ Y − → Z, which is the adjoint of the identit y in Hom C ( Y , Z ). Deﬁnition 7.1. The endomorphism op er ad o f an ob ject Y in C is the non- symmetric op erad End C ( Y ) in V with End C ( Y )( n ) = Hom C ( Y ⊗ n · · · ⊗ Y , Y ) . The unit u : I V − → End C ( Y )(1) is the adjoin t of z ( I V ) ⊗ Y unit − 1 ⊗ Y / / I C ⊗ Y left un it / / Y . The compos ition la ws, 1 ≤ i ≤ m , n ≥ 0, ◦ i : End C ( Y )( m ) ⊗ E nd C ( Y )( n ) − → End C ( Y )( m + n − 1) 44 FERNANDO M URO are the adjoin ts of z (Hom C ( Y ⊗ m , Y ) ⊗ Hom C ( Y ⊗ n , Y )) ⊗ Y ⊗ ( m + n − 1) mult. − 1 ⊗ id ∼ =   z (Hom C ( Y ⊗ m , Y )) ⊗ z (Hom C ( Y ⊗ n , Y )) ⊗ Y ⊗ ( m + n − 1) ∼ = id ⊗ ζ (Hom C ( Y ⊗ n ,Y ) ,Y ⊗ ( i − 1) ) ⊗ id   z (Hom C ( Y ⊗ m , Y )) ⊗ Y ⊗ ( i − 1) ⊗ z (Hom C ( Y ⊗ n , Y )) ⊗ Y ⊗ n ⊗ Y ⊗ ( m − i ) id ⊗ evaluation ⊗ id   z (Hom C ( Y ⊗ m , Y )) ⊗ Y ⊗ ( i − 1) ⊗ Y ⊗ Y ⊗ ( m − i ) z (Hom C ( Y ⊗ m , Y )) ⊗ Y ⊗ m ev aluation   Y Here w e hav e omitted some obvious ass o ciativity isomor phisms in C . Given a non-symmetric op era d O in V , an O - algebr a in C is an ob ject Y in C together with an oper ad morphism O → End C ( Y ). Equiv alen tly , an O - algebra structure on Y is given by morphisms in C , n ≥ 0, (7.2) ν n : z ( O ( n )) ⊗ Y ⊗ n − → Y , such that the following diagrams comm ute, 1 ≤ i ≤ m , n ≥ 0, z ( I V ) ⊗ Y z ( u ) ⊗ id / / O O unit ⊗ id ∼ = z ( O (1)) ⊗ Y ν 1   I C ⊗ Y left un it ∼ = / / Y z ( O ( m )) ⊗ z ( O ( n )) ⊗ Y ⊗ ( m + n − 1) mult. ⊗ id ∼ = & & N N N N N N N N N N N id ⊗ ζ ( O ( n ) , Y ⊗ ( i − 1) ) ⊗ id ∼ = u u j j j j j j j j j j j j j j j z ( O ( m )) ⊗ Y ⊗ ( i − 1) ⊗ z ( O ( n )) ⊗ Y ⊗ n ⊗ Y ⊗ ( m − i ) id ⊗ ν n ⊗ id   z ( O ( m ) ⊗ O ( n )) ⊗ Y ⊗ ( m + n − 1) z ( ◦ i ) ⊗ id   z ( O ( m )) ⊗ Y ⊗ m ν m * * T T T T T T T T T T T T T T T T T T z ( O ( m + n − 1)) ⊗ Y ⊗ ( m + n − 1) ν m + n − 1 w w o o o o o o o o o o o o Y HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 45 An O -algebr a morphism f : Y → Z is a morphism in C such that the following squares commut e, n ≥ 0 , z ( O ( n )) ⊗ Y ⊗ n ν Y n / / id ⊗ f ⊗ n   Y f   z ( O ( n )) ⊗ Z ⊗ n ν Z n / / Z The category of O -algebr as in C will b e denoted b y Alg C ( O ). R emark 7 .3 . The initial O -alg ebra in C is z ( O (0)) with structure morphisms ν n : z ( O ( n )) ⊗ z ( O (0)) ⊗ n z ( O ( n ) ⊗ O (0) ⊗ n ) z ( O (0)) . mult. ∼ = / / z ( µ n ;0 , n ..., 0 ) / / Here we use the conven tion µ 0; ∅ = id O (0) . If A is an O - algebra , the structur e morphism ν A 0 : z ( O (0)) → A is the unique morphism of O -a lg ebras z ( O (0)) → A . The ﬁnal O -algebr a in C is the ﬁnal ob ject of C endow ed with the only p ossible O - algebra structure. 8. The relev ant al gebra push-out Assume we are in the same circumstances a s in the previous sec tio n. Let O b e a non-symmetric op er ad in V . The functor Alg C ( O ) → C forgetting the O -a lgebra structure has a left adjoin t F O : C − → Alg C ( O ) , the fr e e O -algebr a functor, explicitly deﬁned as F O ( Y ) = a p ≥ 0 z ( O ( p )) ⊗ Y ⊗ p . The action of O on F O ( Y ), z ( O ( n )) ⊗ F O ( Y ) ⊗ n = z ( O ( n )) ⊗ n O i =1   a p i ≥ 0 z ( O ( p i )) ⊗ Y ⊗ p i   ∼ = a p 1 ,...,p n ≥ 0 z ( O ( n )) ⊗ n O i =1  z ( O ( p i )) ⊗ Y ⊗ p i  ∼ = a p 1 ,...,p n ≥ 0 z ( O ( n )) ⊗ z ( O ( p 1 )) ⊗ · · · ⊗ z ( O ( p n )) ⊗ Y ⊗ P n i =1 p i ∼ = a p 1 ,...,p n ≥ 0 z ( O ( n ) ⊗ O ( p 1 ) ⊗ · · · O ( p n )) ⊗ Y ⊗ P n i =1 p i F O ( Y ) = a p ≥ 0 z ( O ( p )) ⊗ Y ⊗ p , ν n   46 FERNANDO M URO is deﬁned as the morphism which sends the facto r ( p 1 , . . . , p n ) ∈ N n in the so urce to the factor p = p 1 + · · · + p n ∈ N in the target via z ( µ n ; p 1 ,...,p n ) ⊗ id , n ≥ 1. F or n = 0, the mor phism ν 0 : z ( O (0)) → F O ( Y ) is the inclusion of the fa ctor p = 0 o f the copro duct. The unit of the adjunction is the follo wing comp osite morphism in C , Y I C ⊗ Y z ( I V ) ⊗ Y z ( O (1)) ⊗ Y F O ( Y ) . ∼ = (left unit) − 1 / / ∼ = unit ⊗ id / / z ( u ) ⊗ id / / inclusion of the fac tor p =1 / / Moreov er, given a n O -alge br a A , the counit of the adjunction is deﬁned by the m ultiplication morphisms in (7.2), ( ν p ) p ≥ 0 : F O ( A ) − → A. In this sec tio n we give an explicit construction of the push- o ut of a diagram in Alg C ( O ) as follows, (8.1) F O ( Y ) g   F O ( f ) / / F O ( Z ) A Consider the adjoint diag ram in C , Y ¯ g   f / / Z A The push-out of (8.1) is an O - a lgebra B together with morphisms f ′ : A → B in Alg C ( O ) and ¯ g ′ : Z → B in C such that f ′ ¯ g = ¯ g ′ f in C . Moreov er, given an O - algebra B ′ and mor phisms f ′′ : A → B ′ in Alg C ( O ) and ¯ g ′′ : Z → P ′ in C with f ′′ ¯ g = ¯ g ′′ f in C , there is a unique morphism h : B → B ′ in Alg C ( O ) such that f ′′ = hf ′ and ¯ g ′′ = h ¯ g ′ in C . The following lemma a llows an inductive de ﬁnitio n of the push- out (8.1) as an ob ject in C . W e omit proo fs in this section s inc e the results are s impler analogs o f those in Section 5, and the proofs follow v ery m uc h the same steps. Lemma 8.2 . Ther e is a se quenc e in C , A = B 0 ϕ 1 − → B 1 → · · · → B t − 1 ϕ t − → B t → · · · , wher e the morphism ϕ t is the push-out of (8.3) a n ≥ 1 a S ⊂{ 1 ,...,n } card( S )= t z ( O ( n )) ⊗ k S 1 ⊙ · · · ⊙ k S n ; k S i =  f , i ∈ S ; 0 → A, i / ∈ S ; along the unique morphism, (8.4) ( ψ n,S t ) n,S : a n ≥ 1 a S ⊂{ 1 ,...,n } card( S )= t z ( O ( n )) ⊗ s ( k S 1 ⊙ · · · ⊙ k S n ) − → B t − 1 , HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 47 such that for t = 1 and 1 ≤ i ≤ n , ψ n, { i } 1 = ν n (id z ( O ( n )) ⊗ id ⊗ ( i − 1) ⊗ ¯ g ⊗ id ⊗ ( n − i ) ) , and for t > 1 and i ∈ S , ψ n,S t (id z ( O ( n )) ⊗ κ i ) = ¯ ψ n,S \{ i } t − 1 (id z ( O ( n )) ⊗ id ⊗ ( i − 1) ⊗ ¯ g ⊗ id ⊗ ( n − i ) ) . Her e ( ¯ ψ n,S ′ t − 1 ) n,S ′ denotes the push-out of ( ψ n,S ′ t − 1 ) n,S ′ , i.e. (8.4) for t − 1 , along (8.3) . W e now endow B = co lim t ≥ 0 B t with an O -alg ebra structure. Lemma 8.5 . Ther e ar e u nique morphisms in C , c t 1 ,...,t n n : z ( O ( n )) ⊗ B t 1 ⊗ · · · ⊗ B t n − → B t 1 + ··· + t n , n ≥ 1 , t i ≥ 0 , such that c 0 , n ..., 0 n = ν n : z ( O ( n )) ⊗ A ⊗ n − → A, and, with t he c onvention ¯ ψ p i ,S i 0 = ν p i , if S i ⊂ { 1 , . . . , p i } is a su bset of c ar dinali ty card S i = t i , 1 ≤ i ≤ n , then c t 1 ,...,t i ,...,t n n (id ⊗ ( i − 1) ⊗ ϕ t i ⊗ id ⊗ ( n − i ) ) = ϕ t 1 + ··· + t n c t 1 ,...,t i − 1 ,...,t n n , c t 1 ,...,t n n ( ¯ ψ p 1 ,S 1 t 1 ⊗ · · · ⊗ ¯ ψ p n ,S n t n ) = ¯ ψ p 1 + ··· + p n , S n i =1 ( S i +( p 1 + ··· + p i − 1 )) t 1 + ··· + t n  z ( µ n ; p 1 ,...,p n ) ⊗ id ⊗ P n i =1 p i  . Her e S + p = { i + p ; i ∈ S } and µ is the m ultiplic a tion of the op er ad O . F or simplicity, in t hese e quations we have omitte d some obvious structur e isomorph isms of V , C and z . W e deﬁne f ′ : A = B 0 − → colim t ≥ 0 B t = B as the canonical morphism to th e colimit. Mor eov er, for n ≥ 1 w e deﬁne ν B n : z ( O ( n )) ⊗ B ⊗ n − → B as the colimit of the morphisms c t 1 ,...,t n n in the prev ious lemma, t i ≥ 0, and for n = 0, ν B 0 : z ( O (0)) ν A 0 − → A f ′ − → B . F urthermore, w e deﬁne ¯ g ′ : Z → B a s the following comp osite morphism Z I C ⊗ Z z ( I V ) ⊗ Z z ( O (1)) ⊗ Z B 1 B . ∼ = (left un it) − 1 / / ∼ = unit ⊗ id / / z ( u ) ⊗ id / / ¯ ψ 1 , { 1 } 1 / / pro jection to the coli mit / / Theorem 8.6. The morphisms ν B n , n ≥ 0 , deﬁne an O - algebr a struct ur e on B , f ′ : A → B is an O -algebr a morphism, and if g ′ : F O ( Z ) → B is the adjoi nt of ¯ g ′ : Z → B , then the fol lowing squar e is a push-out in Alg C ( O ) , F O ( Y ) g   F O ( f ) / / F O ( Z ) g ′   A f ′ / / B 48 FERNANDO M URO 9. P roof of Theorems 1. 3 and 1 .5 Suppo se that we a re in the same conditions as in the tw o previous sections. Assume also that V and C are monoidal model categor ies (see Deﬁnition 4.2) and that the composite functor V z − → Z ( C ) forget − → C is a left Q uillen functor [Hov99 , Deﬁnition 1.3.1]. W e will need a non-symmetric version o f the monoid axiom in Deﬁnition 6.1. Deﬁnition 9.1 . The monoid axiom for C says that, f or K ′ = { f 1 ⊙ · · · ⊙ f n ; n ≥ 1 , S ⊂ { 1 , . . . , n } is a subset with ca r d S ≥ 1 , f i is a trivial coﬁbration if i ∈ S, f i : 0 → X i for some ob ject X i in C if i / ∈ S } , all relative K ′ -cell complexes are weak equiv alences. Notice that, as a cons e quence of the push-out pro duct axiom, this is indeed equiv alen t to the monoid axiom in Deﬁnition 6.1 when C is symmetric. In any case, if all ob jects in C are coﬁbra nt then the mono id ax io m is a consequence of the push-out pro duct axiom. Suppo se fro m now on that C satisﬁes the monoid axio m and is coﬁbr a ntly gene r - ated with sets of generating coﬁbra tio ns and gener ating trivia l coﬁbratio ns I and J , resp ectively , with presentable sour ces. Prop ositi o n 9 .2. Consider a push-out diagr a m in Alg C ( O ) as fol lows. F O ( Y ) g   F O ( f ) / / push F O ( Z ) g ′   A f ′ / / B (1) If f is a trivial c oﬁbr ation in C , then the underlying morphism f ′ : A → B in C is a re lative K ′ -c el l c omple x, wher e K ′ is t he class in Deﬁnition 9.1. (2) Supp ose A is c oﬁbr ant in C , f is a c oﬁbr atio n in C , and O ( n ) is c oﬁbr ant in V , n ≥ 0 . Then the morphism f ′ : A → B is a c oﬁbr atio n in C , in p articular B is c oﬁbr ant in C . Pr o of. In case (1), the mo rphism (8.3) in Lemma 8 .2 is in K ′ , hence (1) follows from Theorem 8.6. In case (2), since z is a le ft Quillen functor , the o b jects z ( O ( n )) a re coﬁbrant in C . Therefore, by the push-o ut pro duct a xiom (Deﬁnition 4 .2) the mo rphism (8.3) is a coﬁbration in C . F urthermore, b y Theorem 8.6 the morphism f ′ : A → B is a tr a nsﬁnite comp osition of coﬁbratio ns in C , hence a coﬁbra tion in C itse lf [Hir03, Prop osition 10.3.4].  As an immediate consequence of (1) here and the monoid axiom, we obtain the following result. Corollary 9 . 3. A m orphism in C underlying a r elative F O ( J ) -c el l c omplex in Alg C ( O ) is a we ak e quivalenc e in C . Now we are ready to pro ve Theorem 1.3. HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 49 Pr o of of The o r em 1.3. Using the explicit description of the free op era d adjunction at the b eg inning of Section 8, it is easy to see tha t O -algebr as are the same thing as a lgebras ov er the monad asso c iated to the free O -a lgebra a djunction. Moreover, this monad preserves ﬁltered colimits, s ee again the explicit construc tio n, ther e- fore the categor y Alg C ( O ) is complete and co complete [B or94, P rop ositio n 4 .3.6]. F urthermore, the forgetful functor Alg C ( O ) → C also pr eserves ﬁltered co limits [Bor94, Pro po sition 4.3.2], in particular , s ince F O is a left adjoint and source s of morphisms in I and J are prese ntable in C , then sourc es of morphisms in F O ( I ) and F O ( J ) ar e presen table in Alg C ( O ). W e can apply [SS00, Lemma 2 .3] in or der to prov e the existence of the claimed mo del structure in Alg C ( O ). The smallness condition has alr eady b een check ed, and condition (1) of [SS00, Lemma 2.3] has b een es tablished in Corollary 9.3. The statement ab out r ig ht pro p er ness is obvious since ﬁbratio ns and weak equiv- alences in Alg C ( O ) are detected by the for getful functor Alg C ( O ) → C , and this functor is a right adjoint, so it pr e serves all limits, in particular pull-backs. If C is combinatorial then Alg C ( O ) is lo cally presen table by [AR 94, 2.3 (1) a nd the Theorem in 2.78], hence it is com binatorial.  Lemma 9.4. Supp ose that O is an op er ad in V with O ( n ) c oﬁbr ant for al l n ≥ 0 . Then any c oﬁbr a nt O -algebr a is also c oﬁbr ant as an obje ct in C . Pr o of. Coﬁbr a nt O -algebr as ar e retracts of F O ( I )-cell c omplexes, and coﬁbrant ob jects in C are closed under retr a cts, so it is enough to chec k that F O ( I )-cell complexes are coﬁbra nt in C . The initial O -alg ebra in C (see Rema r k 7 .3) is coﬁbrant in C , since O (0) is co ﬁbrant in V and z is a left Quillen functor. Using Prop ositio n 9.2 (2), an inductio n a rgument pro ves that any F O ( I )-cell co mplex is coﬁbrant in C .  Corollary 9.5 . L et O b e an op era d in V with O ( n ) c oﬁbr ant for al l n ≥ 0 . Then, the for getful fun ctor Alg C ( O ) → C pr eserv es c oﬁbr ations with c oﬁbr ant sour c e. Pr o of. This is an immediate co nsequence of Lemma 9.4 and Prop os ition 9 .2 (2), since co ﬁbrations in Alg C ( O ) are retra cts o f relative F O ( I )-cell c omplexes, the forgetful functor preserves ﬁltered co limits, and coﬁbratio ns in C are closed under transﬁnite comp o sitions and retracts.  Lemma 9.6 . Under the hyp otheses of The or em 1.5, supp ose t hat we have a push- out diagr am in Alg C ( O ) , F O ( Y ) g   / / F O ( f ) / / push F O ( Z ) g ′   A / / f ′ / / B wher e f is a c oﬁbr ation in C and A is a c oﬁbr ant O - algebr a . If the unit of the adjunction evaluate d at A is a we ak e quivalenc e η A : A ∼ → φ ∗ φ ∗ A , then it is also a we ak e quivale nc e when evaluate d at B , η B : B ∼ → φ ∗ φ ∗ B . Pr o of. Since φ ∗ is left adjoint to φ ∗ , which is the iden tit y on the underlying ob ject in C , there is a natural isomorphism φ ∗ F O ∼ = F P that w e r egard as an iden tiﬁcation, 50 FERNANDO M URO and the morphism φ ∗ ( f ′ ) ﬁts in to the following push- out diagram in Alg C ( P ), F P ( Y ) φ ∗ ( g )   / / F P ( f ) / / push F P ( Z ) φ ∗ ( g ′ )   φ ∗ A / / φ ∗ ( f ′ ) / / φ ∗ B The O -algebr a A is c o ﬁbrant and φ ∗ is a left Q uillen functor, therefore φ ∗ A is a coﬁbrant P -algebra , in particular , both A a nd φ ∗ A a re coﬁbra nt in C by Lemma 9.4. Notice that the underlying ob ject of φ ∗ A and φ ∗ φ ∗ A in C is the sa me. Let us call C = φ ∗ φ ∗ B . By Lemma 8.2, the morphism in C underlying η B is the colimit in t ∈ N of an inductively constructed diagra m of coﬁbrant ob jects in C , t > 0, (a) · · · ֌ B t − 1 / / ϕ B t / / η t − 1 ∼   B t ֌ · · · η t ∼   · · · ֌ C t − 1 / / ϕ C t / / C t ֌ · · · such that B 0 = A , C 0 = φ ∗ φ ∗ A , η 0 = η A , the morphism η t is the push- out of the horizontal lines of the following diagram B t − 1 η t − 1 ∼   • induce d by φ and η 0 ∼   (8.4) for O o o / / (8.3) for O / / • induce d by φ and η 0 ∼   C t − 1 • (8.4) for P o o / / (8.3) for P / / • and ϕ B t and ϕ C t are the natural morphisms to the push-out. The o b jects O ( n ) a nd P ( n ) are coﬁbra nt in V and z is a le ft Quillen functor , hence z ( O ( n )) and z ( P ( n )) are coﬁbrant in C , n ≥ 0. More over, f is a coﬁbration in C a nd A a nd φ ∗ φ ∗ A ar e coﬁbrant in C . Therefore Lemma 4.4 shows that the square on the right has weak eq uiv alences in the co lumns a nd coﬁbra tions in the rows. In par ticular, ϕ B t and ϕ C t are coﬁbrations in C and, by the gluing prop er t y in left proper model categories [Hir03, Prop osition 13.5.4 ], η t is a w eak equiv alence in C . T o c o nclude, η B = colim t ≥ 0 η t is a weak equiv alence in C s ince (a) is a weak equiv alence b etw een coﬁbrant o b jects in the Reedy mo del catego ry of directed diagrams in C indexed by N [Hov99, Theorem 5 .1.3] and Ken Brown’s lemma [Hov99 , Lemma 1.1 .12] a pplies, b ecause co lim t ≥ 0 is a left Quillen functor [Hov99, Corollar y 5 .1 .6].  Finally , we a re ready to prov e Theor em 1.5. Pr o of of The o r em 1.5. W e will use the cr iterion in [Hov99, Corolla ry 1 .3 .16 (c)] to detect Quillen equiv alences. The functor φ ∗ preserves and reﬂects w eak equiv a - lences, since it is the iden tit y on the underlying ob ject in C . Therefore, it is enoug h to c heck that the unit of the adjunction η A : A → φ ∗ φ ∗ A is a weak equiv alence for any coﬁbra nt O -alge br a A . HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 51 W eak equiv alences are c losed under retra cts and coﬁbrant O -algebra s are r etracts of F O ( I )-cell complexes, so we can suppo se that A is an F O ( I )-cell complex, A = colim i<λ A i . W e now proc eed b y induction on the ordinal λ . F or λ = 1, A is the initial O -alg ebra, s e e Remark 7.3. Then φ ∗ A is the initial P -algebr a, since φ ∗ is a left adjoint, a nd η A = z ( φ (0)) : z ( O (0 )) → z ( P (0)). The morphism φ (0) is a weak equiv alence b etw een coﬁbra nt ob jects in V a nd z is a left Quillen functor, therefore z ( φ (0)) is also a w eak equiv a lence betw een coﬁbrant ob jects in C by [Ho v99, Lemma 1.1.12]. If λ = α + 1 and the result is tr ue for α , then it is als o true for λ b y the previo us lemma. Suppo se now that λ is a limit ordina l a nd that the r esult is true for all i < λ . The functor φ ∗ preserves co limits, since it is a left adjoint, a nd φ ∗ preserves ﬁltered colimits, b ecause it is the identit y over C and forgetful functors fro m algebra s to C preserve ﬁltered colimits. In particular η A = co lim i<λ η i is a co limit of weak equiv- alences by induction hypothesis. By Pro p o sition 9.2 (2), an F O ( I )-cell complex is a colimit o f a contin uous dia gram of co ﬁbrations betw een coﬁbrant ob jects in C , and the sa me is true for F P ( I )-cell complexes. This applies to A and φ ∗ A . Suc h diagrams are co ﬁbr ant ob jects in Reedy mo del catego ries of directed diagra ms in C [Hov99 , Theorem 5.1.3]. Therefor e, η A is the colimit of a weak equiv a lence betw een coﬁbrant ob jects in the Reedy mo del catego ry o f directed diagr ams in C indexed by λ . Now, Ken Brown’s lemma [Hov99, Lemma 1.1.12] shows that η A is a weak equiv alence, since co lim i<λ is a left Quillen functor [Ho v99, Corollar y 5.1 .6 ].  10. An applica tion to enriched ca te gories and A ∞ -ca tegories In this section we lay the foundatio ns to c o nstruct mo del categ ories of cate- goriﬁed a lgebraic structur es. This is applied to enr iched categories a nd enriched A ∞ -categor ies. Deﬁnition 10. 1 . Given a set S , an S -gr aph M with obje ct set S is a collection of ob jects in V indexed by S × S , M = { M ( x, y ) } x,y ∈ S . The category Graph S ( V ) of V -graphs with ob ject se t S , w her e morphisms are deﬁned in the ob vious way , is biclosed monoidal with tensor product, ( M ⊗ S N )( x, y ) = a z ∈ S M ( z , y ) ⊗ N ( x, z ) . The unit ob ject I S is I S ( x, y ) =  I , the monoidal unit of V , if x = y ; 0 , the initial ob ject of V , if x 6 = y . This mo noidal categor y is c le arly no n-symmetric, unless S is a s ingleton. The right adjoint of M ⊗ − is the functor Hom S l ( M , − ) deﬁned as Hom S l ( M , P )( x, y ) = Y z ∈ S Hom( M ( y , z ) , P ( x, z )) , and the right adjoint of − ⊗ N is the functor Hom S r ( N , − ) deﬁned as Hom S r ( N , P )( x, y ) = Y z ∈ S Hom( N ( z , x ) , P ( z , y )) . 52 FERNANDO M URO W e hav e a strong bra ided monoidal functor z : V → Graph S ( V ) deﬁned as z ( A )( x, y ) =  A, if x = y ; 0 , if x 6 = y . Moreov er, ( z ( A ) ⊗ S M )( x, y ) = A ⊗ M ( x, y ) , ( M ⊗ S z ( A ))( x, y ) = M ( x, y ) ⊗ A, and the natural isomorphism ζ ( A, M ) : z ( A ) ⊗ S M ∼ = M ⊗ S z ( A ) , is deﬁned as the symmetry isomorphism of V co or dinatewise. R emark 10 .2 . If V is a mo del ca teg ory , the c a tegory Gr aph S ( V ) inherits from V a pro duct mo de l category structur e, where ﬁbrations, coﬁbrations and w eak equiv- alences ar e deﬁned co or dinatewise. If V is coﬁbrantly gener ated (resp. co mbin ato- rial) then so is Graph S ( V ), co mpare Remark 2.3. Mor e ov er, since S × S is a set, an S -graph M is presentable pr ovided M ( x, y ) is presentable for all x, y ∈ S . In particular, if V ha s s ets of generating co ﬁbrations and genera ting trivia l co ﬁbr a- tions with pres e ntable source, then so does Graph V ( S ). F urther more, if V is right prop er then the product mo del ca tegory Graph V ( S ) is also right prop er . Notice that the comp os ite functor V z → Z (Graph S ( V )) → Graph S ( V ) preserves ﬁbrations, coﬁbr ations and weak equiv alences, and it ha s a r ight a djoint deﬁned b y M 7→ Y x ∈ S M ( x, x ) . This adjoin t pair is therefore a Quillen adjunction. Prop ositi o n 1 0.3. If V satisﬁes the monoid axiom then Graph S ( V ) also satisﬁes the monoid axiom. Pr o of. It is enought to notice, using the symmetry of V and the push-out product axiom in V , that a ny morphism f 1 ⊙ · · · ⊙ f n in the class of morphisms K ′ of Graph S ( V ) in Deﬁnition 9.1 is comp onent wise a morphism in the class K o f V in Deﬁnition 6.1.  Categorie s enriched o n V with set o f ob jects S ar e the same as mono ids in Graph S ( V ). T he s e monoids are the sa me as algebra s ov er the no n- symmetric o p- erad A ss V in V deﬁned b y As s V ( n ) = I , n ≥ 0. All co mp o sitions in Ass V are unit isomorphisms I ⊗ I ∼ = I and the unit opf the operad u : I → Ass V (1) is the iden tit y . This oper ad is generated by the ‘elements’ in degree 0 a nd 2; the degr ee 2 ‘elemen t’ represents the compositio n law, and the degree 0 ‘elemen t’ represents the identities. In order to simplify notation, w e denote Cat S ( V ) = Alg Graph S ( V ) ( Ass V ) . An A ∞ -categor y e nriched on V with set of ob jects S is a n alg ebra over a co ﬁbrant replacement Ass V ∞ of Ass V , which is a trivial ﬁbration φ : Ass V ∞ ∼ ։ Ass V in O p( V ) with coﬁbrant sour ce. W e simply denote A ∞ -Cat S ( V ) = Alg Graph S ( V ) ( Ass V ∞ ) . Combining the previo us prop osition with Theorem 1.3 we o btain the following corolla r y , which impro ves [Dun01, Theo r em 3.3]. HOMOTOPY THEOR Y OF NON-SYMM ETRIC OPERADS 53 Corollary 1 0.4. L et V b e a c oﬁbr antly gener ate d close d symmetric monoidal c at- e gory satisfying the monoid axiom. Supp ose that V has sets of gener ating c oﬁbr a- tions and gener ating trivial c oﬁbr atio ns with pr esentable sourc e. Then Cat S ( V ) is a mo del c a te gory wher e an enriche d functor F : C → D is a we ak e quivalenc e (r esp. ﬁbr ation) if F ( x, y ) : C ( x, y ) → D ( x, y ) is a we ak e quivalenc e (re sp. ﬁbr ation) i n V for al l x, y ∈ S , and similarly for A ∞ - Cat S ( V ) . Mor e ove r, t hese mo del c ate gories ar e right pr op er (r esp. c ombinatorial ) pr ovid e d V is. The following cor ollary also uses Theorem 1.5. Corollary 10.5. In t he c onditions of t he pr evio us c or ol lary, assume in addition that V is left pr op er and the monoidal unit I V is c oﬁbr ant. Then the pul l-b ack functor φ ∗ fr om enriche d c ate gories to enriche d A ∞ -c ate gories and the st rictiﬁc ation functor φ ∗ in the other dir e ction form a Quil len e quivalenc e, A ∞ - Cat S ( V ) φ ∗ / / Cat S ( V ) . φ ∗ o o In p articular, the derive d adjoi nt p air is an e quivalenc e b etwe en t he homotopy c at- e gori es of enriche d c ate go ries and enriche d A ∞ -c ate gories, Ho A ∞ - Cat S ( V ) L φ ∗ / / Ho Cat S ( V ) . φ ∗ o o References [AR94] J. Ad´ amek and J. Rosick´ y, L o c al ly pr esentable and ac cessible c ate gories , London Math- ematical So ciet y Lect ure Note Series, v ol. 189, Cambridge Universit y Press, Cam bridge, 1994. [BJT97] H. -J. Baues, M . Jibladze, and A. T onks, Cohomolo gy of monoids in monoidal c a te gories , Pro ceedings of the Renaissance Conferences (Providence , RI) (J.-L. Lo da y , J. D. Stasheﬀ, and A. A. V oronov, eds.), Contemp. Math., v ol. 202, Amer. Math. Soc., 19 97, pp. 137– 165. [BM03] C. Berger and I. Moerdijk, Ax iomatic homotopy the ory for op er ads , Comment. Math. Helv. 78 (200 3), no. 4, 805–831. [Bor94] F. B or ceux, Handb o ok of ca te goric al algebr a 2 , Encyc lop edia of Math. and its Applica- tions, no. 51, Cambridge Univ ersity Pr ess, 1994. [DK80a] W. G. Dwyer and D. M . Kan, Calculating simplicial lo c alizations , J. Pure Appl. Algebra 18 (1980), no. 1, 17–35. [DK80b] , F u nction c omplexes in homotop ic al algebr a , T opol ogy 19 (1980), no. 4, 427–44 0. [DK80c] , Simplicial lo c alizations of ca te gories , J. Pure Appl. Algebra 17 (1980), no. 3, 267–284. [Dun01] B. I. Dundas, L o c aliza tion of V -c ate gories , Theory Appl. Categ. 8 (200 1), no. 10, 284– 312. [GK94] V. 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[Kel05] G. M. Kel l y , Basic c onc epts of enriche d c ate gory the ory , Repr. Theory Appl. Categ. (2005), no. 10, vi+137, Reprint of the 1982 original [Cambridge Univ. Press]. [Rap09] G. Raptis, On the c oﬁbr ant gener a tion of mo del c ate gories , J. Homotop y Relat. Struct. 4 (2009 ), no. 1, 245–253. [SS00] S. Sch we de and B . E. Shipley , Algebr a s and mo dules i n monoidal mo del ca te gories , Pro c. London Math. Soc. (3) 80 (2000), no. 2, 491–511. Universidad de Sevilla, F acul t ad de Ma tem ´ aticas, Dep ar t amento de ´ Algebra, A vda. Reina Mercedes s/n, 4 1 012 Sevilla, S p ain Home p age : http://personal .us.es/fmuro E-mail addr ess : fmuro@us.e s

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