Homotopy theory of nonsymmetric operads

Algebraic & Geometric Topology 11 (2011) 1541–1599 1541 Homotopy theory of nonsymmetric operads FERNANDO MURO We endow categories of nonsymmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched nonsymmetric monoidal model categories. 18D50, 55U35; 18D10, 18D35, 18D20 1 Introduction Operads are well-known devices encoding the laws of algebras defined by multilinear operations and relations, eg there are operads Ass , Com and Lie whose algebras are associative, commutative and Lie algebras, respectively. Morphisms of operads codify relations between different kinds of algebras, eg there are morphisms Lie!Ass!Com telling us that any commutative algebra is an associative algebra, and that commutators in an associative algebra yield a Lie algebra. There are two kinds of operads: symmetric and nonsymmetric operads. Symmetric operads are needed whenever it is necessary to permute variables in order to describe the laws of the corresponding algebras, eg Com and Lie . Nonsymmetric operads are specially useful to deal with algebras in nonsymmetric monoidal categories, eg given a commutative ring k and a set S which is not a singleton, the category of k –modules with object set S , which are collections of k –modules indexed by SS , MD fM.x;y/gx;y2S, has a nonsymmetric tensor product, .M˝SN/.x;y/DM z2S M.z;y/˝kN.x;z/; whose associative algebras, ie algebras over the operad Ass , are k –linear categories with object set S. Any object M in a symmetric monoidal category V , such as the category of k – modules, has an endomorphism symmetric operad EndV.M/ in V such that, if O is another symmetric operad in V , the set of O –algebra structures on M is the set of symmetric operad morphisms O!EndV.M/ . If M belongs to a nonsymmetric Published: 26 May 2011 DOI: 10.2140/agt.2011.11.1541 1542 Fernando Muro monoidal category C enriched over V , such as the category of k –modules with object set S , then there is a nonsymmetric operad EndC.M/ in V such that the set of algebra structures on M over a nonsymmetric operad O in V is the set of nonsymmetric operad morphisms O!EndC.M/. When the underlying symmetric monoidal category V carries homotopical information, eg if we replace k –modules with differential graded k –modules, one is often more interested in a space of O –algebra structures on M rather than a plain set. Such a space can be constructed by using the powerful machinery developed by Dwyer and Kan [9;7;8] provided we can place the operads O and EndC.M/ in an appropriate model category of operads. Model categories of operads were first considered by Hinich in the differential graded context [13;12], and by Berger and Moerdijk in a more general setting [4]. They dealt with symmetric operads and showed that restrictive hypotheses are necessary to endow the category of all operads with an appropriate model category structure, eg when k is a Q –algebra or when the symmetric monoidal structure in V is cartesian closed and there is a symmetric monoidal fibrant replacement functor. Motivated by our interest in spaces of differential graded category structures, we consider the nonsymmetric case, which surprisingly enough does not need any restrictive hypotheses, just usual hypotheses for model categories with a monoidal structure; see Schwede and Shipley [20]. Theorem 1.1 Let V be a cofibrantly generated closed symmetric monoidal model category. Assume that V satisfies the monoid axiom. Moreover, suppose that there are sets of generating cofibrations and generating trivial cofibrations in V with presentable sources. Then the category Op.V/ of nonsymmetric operads in V is a cofibrantly generated model category such that a morphism fWO!P in Op.V/ is a weak equivalence (resp. fibration) if and only if f .n/WO.n/!P.n/ is a weak equivalence (resp. fibration) in V for all n0 . Moreover, if V is right proper then so is Op.V/ . Furthermore, if Vis combinatorial then Op.V/is also combinatorial. This theorem can be applied to all examples in [20] (see also the references therein): (1) Complexes of modules over a commutative ring k with the usual tensor product of complexes. (2) Simplicial k–modules with the levelwise tensor product ˝k. (3) Modules over a finite-dimensional Hopf algebra R over a field k with the tensor product over k, eg RDkG the group-ring of a finite group G. (4) Symmetric spectra with their smash product, and more generally modules over a commutative ring spectrum. Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1543 (5) –spaces with Lydakis’ smash product. (6) Simplicial functors with their smash product. (7) S–modules with their smash product. In particular, Theorem 1.1 will also be useful to study spaces of spectral category structures. Recall from Adámek and Rosický [1, Definition 1.13 (2)]that an object X of V is presentable if there exists a cardinal  such that the representable functor V.X;/ commutes with  –filtered colimits in V . Presentable objects are also called small or compact in some references. All objects are presentable in many categories of interest, eg in all combinatorial model categories. Actually, up to set theoretical principles any cofibrantly generated model category is Quillen equivalent to a combinatorial model category; see Raptis [19]. Categories of algebras over symmetric operads do not always have a model structure with fibrations and weak equivalences defined in the underlying category. Sufficient conditions can be found in Berger and Moerdijk [4]. In the framework of nonsymmetric operads they do. When both algebras and operads live in the same ambient symmetric monoidal model category V , satisfying the monoid axiom, this has been recently proved by J E Harper [11, Theorem 1.2]. We here extend this result to algebras in a monoidal model category C satisfying the monoid axiom and appropriately enriched over V . This is necessary, for instance, to construct model categories of enriched categories, of enriched A1 –categories, or of any other categorified algebraic structure; see Section 10. Theorem 1.2 Let V and C be cofibrantly generated biclosed monoidal model categories. Suppose V is symmetric and C has a V –algebra structure given by a strong braided monoidal functor zWV!Z.C/ to the center of C such that the composite functor Vz ! Z.C/forget ! C is a left Quillen functor. Moreover, assume that V and C satisfy the monoid axiom (see Definition 6.1 and Definition 9.1). Furthermore, suppose that C has sets of generating cofibrations and generating trivial cofibrations with presentable source. Let O be a nonsymmetric operad in V . The category AlgC.O/ of O –algebras in C is a cofibrantly generated model category such that an O –algebra morphism gWA!B is a weak equivalence (resp. fibration) if and only if g is a weak equivalence (resp. fibration) in C . Moreover, if C is right proper then so is AlgC.O/ . Furthermore, if C is combinatorial then AlgC.O/is also combinatorial. Algebraic & Geometric Topology, Volume 11 (2011) 1544 Fernando Muro The notion of monoidal model category in [20, Definition 3.1]makes sense with no modification in the nonsymmetric context; see Definition 4.2. Any operad morphism WO!Pinduces a change of operad functor WAlgC.P/! AlgC.O/ by restricting the action of P to O along  . This functor is the identity on underlying objects in C , hence it preserves fibrations and weak equivalences. Moreover, the functor has a left adjoint , therefore we have a Quillen adjunction (1) AlgC.O/ //AlgC.P/:  oo The following result establishes conditions so that this is a Quillen equivalence if  is a weak equivalence of operads. These conditions are the nonsymmetric analogues of those considered in [4] for symmetric operads. Theorem 1.3 In the conditions of the previous theorem, assume further that C is left proper. Let WO!P be a weak equivalence between operads in V such that for all n0 the objects O.n/ and P.n/ are cofibrant in V . Then Equation (1) is a Quillen equivalence, in particular the derived adjoint pair is an equivalence between the homotopy categories of algebras: Ho AlgC.O/ L//Ho AlgC.P/:  oo This result will be useful to show that in many examples the homotopy theory of enriched categories coincides with the homotopy theory of A1 –categories, eg when the underlying symmetric monoidal category V is any of the categories in the examples (1)–(6) listed above; see Section 10. When C is a simplicial model category and the simplicial structure is compatible with zWV!Z.C/ in a suitable way, the derived equivalence of homotopy categories in Theorem 1.3 was obtained by Batanin in [2, Section 2]using totally different methods closer to categorical algebra than to homotopy theory. The paper is structured as follows. Sections 2,5and 6deal with operads and Sections 7, 8and 9deal with algebras in a rather parallel way: we recall the basics on these algebraic structures, we give very detailed constructions of some pushouts which are the main ingredients for the proofs of our main theorems, and then we proceed with the proofs. The other sections are auxiliary. Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1545 Acknowledgements The author wishes to thank Michael Batanin, Clemens Berger, Benoit Fresse, Javier J Gutiérrez, Ieke Moerdijk, Andy Tonks and Bruno Vallette for conversations related to the contents of this paper, in particular for providing very interesting references. FM was partially supported by the Spanish Ministry of Education and Science under the MEC-FEDER grants MTM2007-63277 and MTM2010-15831, by the Government of Catalonia under the grant SGR-119-2009 and by the Andalusian Ministry of Economy, Innovation and Science under the grant FQM-5713. Notation Throughout this paper V and C will denote complete and cocomplete biclosed monoidal categories (see Kelly [17, 1.5]) with tensor product X˝Y and unit objects IV and IC , respectively. We drop the subscript when it is clear from the context. The category V will be symmetric and internal morphism objects in V will be denoted by Hom.X;Y/. We will add homotopical hypotheses when needed. 2 Operads In this section we recall the well-known notion of nonsymmetric operad. Definition 2.1 The category VN of sequences of objects VD fV.n/gn0 in V is the product of countably many copies of V . It has a right-closed nonsymmetric monoidal structure given by the composition product UıV(compare [2, Definition 1.2]) .UıV/.m/Da n0a n P iD1 piDm U.n/˝V.p1/˝˝V.pn/: The unit object is Iı: Iı.n/D(Ithe unit of ˝in V;if nD1; 0the initial object of V;if n¤1: Remark 2.2 The fact that ı is nonsymmetric is obvious from the very definition. One can easily check by writing down explicitly the formulas of .UıV/ıW and Uı.VıW/ how the symmetry constraint of ˝ is used to define the associativity constraint of ı. The right adjoint of  ı Vis the functor Homı.V;/defined by Homı.V;W/.n/DY p1;:::;pn0 Hom.V.p1/˝˝V.pn/; W.p1CCpn//; in particular  ı V preserves all colimits. On the contrary, the functor Uı  does not preserve all colimits, but it does preserve filtered colimits. Algebraic & Geometric Topology, Volume 11 (2011) 1546 Fernando Muro Remark 2.3 If V is a model category then the product category VN is also a model category with fibrations, cofibrations and weak equivalences defined coordinatewise [15, Example 1.1.6]. Moreover, if V is cofibrantly generated (resp. combinatorial) then VNis also cofibrantly generated (resp. combinatorial). Indeed, let I be a set of generating cofibrations and J a set of generating trivial cofibrations in V . For any n0 , let snWV!VN be the left adjoint of the projection onto the n–th factor, which is defined by .sn.V//.m/D(Vif mDn; 0the initial object, if m¤n: Given a set S of morphisms in V we consider the following set of morphisms in VN : SND[ n0 sn.S/: The sets IN and JN are sets of generating cofibrations and generating trivial cofibrations in VN, respectively. Definition 2.4 Anonsymmetric operad O in V is a monoid in the monoidal category of sequences VNwith the composition product ı. Remark 2.5 The previous condensed definition of an operad O can be unraveled by noticing that the multiplication WOıO!O consists of a series of multiplication morphisms, 1in,pi0, nIp1;:::;pnWO.n/˝O.p1/˝˝O.pn/! O.p1CCpn/: The associativity condition amounts to saying that the following diagram is always commutative: O.n/˝ n O iD1O.pi/˝ pi O jD1 O.qij / O.n/˝ n O iD1 O.pi/˝ n O iD1 pi O jD1 O.qij / O.n/˝ n O iD1 Opi X jD1 qij  On X iD1 pi˝ n O iD1 pi O jD1 O.qij / On X iD1 pi X jD1 qij  Šass. and sym.  id˝ n N iD1  55 ˝id ))    ;; Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1547 Here the order of tensor factors in Nn iD1Npi jD1O.qij / is determined by the lexicographic order of the pair .i;j/ . Moreover, the unit is just a morphism uWI!O.1/ such that the following morphisms are (compositions of) unit constraints in V: I˝O.n/u˝id //O.1/˝O.n/1In//O.n/; O.n/˝I˝nid˝u˝n //O.n/˝O.1/˝nnI1;:::;1 //O.n/: Remark 2.6 The multiplication morphisms in the previous remark are determined by the following morphisms, 1im,n0, ıiWO.m/˝O.n/! O.mCn1/; defined as O.m/˝O.n/Š .left and right unit/1 // ıi  O.m/˝I˝.i1/˝O.n/˝I˝.mi/ id˝u˝.i1/˝id˝u˝.mi/  O.mCn1/O.m/˝O.1/˝.i1/˝O.n/˝O.1/˝.mi/: mI1;i1 ::::::;1;n;1;mi :::::: ;1 oo An operad can actually be defined as a collection of morphisms ıi as above together with a unit morphism uWI!O.1/ such that, for 1im , the following diagrams commute: (1) If 1j<i: .O.l/˝O.m// ˝O.n/ .O.l/˝O.n// ˝O.m/ O.lCm1/˝O.n/ O.lCn1/˝O.m/ O.lCmCn2/ ıi˝id ıj˝id ıj ıiCn1 Šass. and sym.  ?? ## $$ :: (2) If ij<mCi: .O.l/˝O.m// ˝O.n/ O.l/˝.O.m/˝O.n// O.lCm1/˝O.n/ O.l/˝O.mCn1/ O.lCmCn2/ ıi˝id id˝ıjiC1 ıj ıi Šass.  ?? ## $$ 99 Algebraic & Geometric Topology, Volume 11 (2011) 1548 Fernando Muro These relations are illustrated by the trees in Figure 10 below. Moreover, for all 1in the following composite morphisms must be unit constraints in V: (3) I˝O.n/u˝id //O.1/˝O.n/ı1//O.n/; (4) O.n/˝Iid˝u//O.n/˝O.1/ıi//O.n/: 3 Trees The combinatorics of operads is that of trees with additional structure. In this section we recall some facts about trees that we need in order to prove our main theorems. We also give a different characterization of operads in terms of trees. Definition 3.1 Aplanted tree is a contractible finite 1 –dimensional simplicial complex T with set of vertices V.T/ , a nonempty set of edges E.T/ , and a distinguished vertex r.T/2V.T/ of degree 1 , called root. Recall that the degree of v2V.T/ is the number of edges containing v. Nevertheless, we will mostly use the number zvD.degree of v/ 1: The level of a vertex v2V.T/ is the distance to the root, level.v/ Dd.v; r.T// , with respect to the usual metric d such that the distance between two adjacent vertices fv; wg 2 E.T/is d.v; w/ D1. The height ht.T/of a planted tree Tis ht.T/Dmax v2V.T/level.v/: Definition 3.2 Aplanted planar tree is a planted tree T together with a total order  in V.T/, called planar order, such that: If level.v/ < level.w/ then v < w. If fv1; v2g;fw1; w2g 2 E.T/are edges with level.v1/Dlevel.w1/Dlevel.v2/1Dlevel.w2/1; and v1< w1, then v2< w2. Given eD fv; wg 2 E.T/ with v < w we say that e is an incoming edge of v and the outgoing edge of w(there is only one if w¤r.T/and none otherwise). There is another useful order in V.T/ that we call the path order  . Given v2V.T/ , consider the shortest path from r.T/ to v and let r.T/Dv0; : : : ; vnDv be the vertices within this path in order of appearance. We associate with v the word v0   vn in V.T/ . The path order in V.T/ is the order induced by the lexicographic order of words in V.T/with respect to . Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1549 jTj D  v0Dr.T/  v1  v2v3  v4 v5 v6  v7 v8 v9 Figure 1: The geometric realization of a planted planar tree T with vertices ordered by the subscript Remark 3.3 Notice that the path order restricted to level sets fv2V.T/Ilevel.v/ Dng;n0; coincides always with the planar order . The words associated to the vertices of the planted planar tree in Figure 1 are given in the following tables: vertex word v0v0 v1v0v1 v2v0v1v2 v3v0v1v3 v4v0v1v3v4 vertex word v5v0v1v3v5 v6v0v1v3v6 v7v0v1v3v4v7 v8v0v1v3v4v8 v9v0v1v3v4v9 Hence the path order in V.T/is v0v1v2v3v4v7v8v9v5v6. Definition 3.4 Aplanted planar tree with leaves is a planted planar tree T together with a fixed set of degree 1 vertices L.T/ , called leaves, different from the root, r.T/…L.T/ . An inner vertex is a vertex which is neither a leaf nor the root. The set of inner vertices will be denoted by I.T/and V.T/D fr.T/g t I.T/tL.T/: We denote by kTk the open subspace of the geometric realization of T obtained by removing the root and the leaves (see Figure 2): kTkDjTj n .fr.T/g t L.T//: Abusing of terminology, we say that an edge is the root or a leaf if it contains the root or a leaf vertex, respectively. The rest of edges are called inner edges. Algebraic & Geometric Topology, Volume 11 (2011) 1556 Fernando Muro Examples of planted planar trees with leaves illustrating relations (1) and (2) in Remark 2.6 are depicted in Figure 10.       Figure 10: The planted planar trees with leaves illustrating the associativity relations .C3ı2C4/ı1C5D.C3ı1C5/ı6C4 in Remark 2.6 (1) and .C3ı2C4/ı3C5DC3ı2.C4ı2C5/in Remark 2.6 (2), respectively 4 The monoidal category of morphisms The category Mor.C/ of morphisms in C can be regarded as the category of functors 2!C , where 2 is the category with two objects, 0 and 1 , and only one nonidentity morphism 0!1 , ie it is the poset f0<1g . A morphism fWU!V in C is identified with the functor fW2!Cdefined by f .0/DU,f .1/DVand f .0!1/Df. The category Mor.C/ carries a biclosed monoidal structure given by the ˇ product of morphisms fˇg: U˝X V ˝X U˝YU˝YS U˝X V˝X V˝Y f˝idX// push idU˝g  // idV˝g  f˝idY00 fˇg )) This monoidal structure is symmetric provided ˝ is. If 0 denotes the initial object of C, the functor C! Mor.C/; X7! .0!X/; is strong (symmetric) monoidal. We regard C as a full subcategory of Mor.C/ through this functor. Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1557 Notice that pushouts in C are a special kind of morphism in Mor.C/ . The following lemma asserts that the ˇproduct preserves pushouts in C. Lemma 4.1 Given two pushout diagrams in C,iD1;2, Ui fi// gi  push Vi g0 i  Xif0 i //Yi the following diagram in Cis also a pushout: U1˝V2S U1˝U2 V1˝U2 X1˝Y2S X1˝X2 Y1˝X2 V1˝V2 Y1˝Y2 push f1ˇf2// g0 1˝g0 2  f0 1ˇf0 2 // g1˝g0 2S g1˝g2 g0 1˝g2  This lemma follows straightforwardly from the very definition of ˇ together with the fact that ˝is biclosed, and hence it preserves colimits in both variables. Definition 4.2 The category C is a monoidal model category if it is endowed with a model structure satisfying the pushout product axiom:  Let f and g be cofibrations in C . The morphism fˇg is also a cofibration. If in addition for gis a weak equivalence, then so is fˇg. This axiom was considered by Schwede and Shipley [20, Definition 3.1]for C symmetric, but it also makes sense in the nonsymmetric case. Actually, following the terminology of Meyer [18] and Batanin [2, Definition 2.2], which work in a nonsymmetric context, the first half of the pushout product axiom says that all cofibrations in Care closed. Remark 4.3 The pushout product axiom implies that the tensor product of cofibrant objects is cofibrant. Moreover, if X is a cofibrant object and f is a (trivial) cofibration in C then X˝f and f˝X are (trivial) cofibrations. In particular, by Ken Brown’s lemma [15, Lemma 1.1.12], for X cofibrant the functors X˝  and  ˝ X preserve weak equivalences between cofibrant objects. Furthermore, if f and g are (trivial) cofibrations with cofibrant source, then so is fˇg. Algebraic & Geometric Topology, Volume 11 (2011) 1558 Fernando Muro Lemma 4.4 Let C be a left proper monoidal model category. Consider two commutative squares in Mor.C/ where the rows are cofibrations and the columns are weak equivalences between cofibrant objects, iD1;2, Ui// fi// gi  Vi g0 i   Xi// f0 i //Yi Then in the following diagram the rows are also cofibrations and the columns are weak equivalences between cofibrant objects: U1˝V2S U1˝U2 V1˝U2 X1˝Y2S X1˝X2 Y1˝X2 V1˝V2 Y1˝Y2 // f1ˇf2// g0 1˝g0 2   // f0 1ˇf0 2 // g1˝g0 2S g1˝g2 g0 1˝g2  Proof Looking at Definition 4.2 and the remark afterwards we notice that it is only left to check that the left column is a weak equivalence. This follows easily from the gluing property in left proper model categories [14, Proposition 13.5.4]. Given morphisms fiWUi!Vi in C , 1in , the target of f1ˇ    ˇ fn is the iterated tensor product of the targets V1˝    ˝ Vn . This object is the colimit of the diagram f1˝˝fnW2n! C; since 2n has a final object .1;n : : :; 1/ . The source of f1ˇ    ˇ fn is the colimit of the restriction of this diagram to the full subcategory of 2n obtained by removing the final object. For simplicity, we denote it by s.f1ˇˇfn/: f1ˇˇfnWs.f1ˇˇfn/! V1˝˝Vn: The universal property of s.f1ˇˇfn/in Crefers to canonical morphisms iWV1˝˝Vi1˝Ui˝ViC1˝˝Vn! s.f1ˇˇfn/; 1in; with .f1ˇˇfn/iDid˝.i1/˝fi˝id˝.ni/. Any collection of morphisms giWV1˝˝Vi1˝Ui˝ViC1˝˝Vn! X;1in; Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1559 such that the following squares commute, 1i<jn, V1˝˝Ui˝˝Uj˝˝Vn id˝˝fi˝˝id // id˝˝fj˝˝id  V1˝˝Vi˝˝Uj˝˝Vn gj  V1˝˝Ui˝˝Vj˝˝Vngi //X induces a unique morphism gWs.f1ˇ    ˇ fn/!X such that giDgi , 1in . Compare the paragraph preceding [2, Lemma 2.2]. 5 The relevant operad pushout The forgetful functor from operads to sequences Op.V/!VN has a left adjoint FWVN!Op.V/ , the free operad functor, explicitly constructed for example in [3, Appendix B]. An alternative construction in terms of trees is as follows (see [2, Section 3]): F.V/.n/Da TO v2I.T/ V.zv/; where T runs over a set of isomorphism classes of trees with n leaves in PPTL . The product ıi,1im, F.V/.m/˝F.V/.n/D` T0N u2I.T0/ V.zu/˝` TN v2I.T/ V.zv/card L.T0/Dm card L.T/Dn Š` T0;TN u2I.T0/ V.zu/˝N v2I.T/ V.zv/ F.V/.mCn1/D` T00 N w2I.T00/ V.zw/ card L.T00/DmCn1 ıi  sends the factor corresponding to the trees T and T0 in the source to the factor of T00 DT0ıiTin the target: I.T0ıiT/DI.T0/tI.T/; O u2I.T0/ V.zu/˝O v2I.T/ V.zv/ DO w2I.T0ıiT/ V.zw/: The unit uWI!F.V/.1/ is the inclusion of the factor of the coproduct corresponding to the tree with one leaf a no inner vertex, ie the unit of the grafting operation. Algebraic & Geometric Topology, Volume 11 (2011) 1560 Fernando Muro The unit of the adjunction V!F.V/ in VN is given by the morphisms in V , n0 , V.n/ inclusion of the factor corresponding to Cn //F.V/.n/: Given an operad O with associated operadic functor L.O/ , if we denote pTWT!Cn the morphism in PPTL collapsing all inner edges of a tree T with n leaves, then the counit F.O/!Ois defined by the following morphisms, n0, F.O/.n/D` TN v2I.T/ O.zv/ D` T L.O/.T/L.O/.Cn/DO.n/: .L.O/.pT//T// An analogous construction for symmetric operads was considered by Ginzburg and Kapranov in [10, 2.1]. In this section we give an explicit construction of the pushout of two morphisms in Op.V/as follows: (2) OF.U/ g oo F.f / //F.V/: Consider the adjoint diagram in VN: OU xg oo f //V: The pushout of Equation (2) is an operad P together with morphisms f0WO!P in Op.V/ and xg0WV!P in VN such that f0xgD xg0f in VN . Moreover, given an operad P0 and morphisms f00WO!P0 in Op.V/ and xg00WV!P0 in VN with f00 xgD xg00f in VN , there is a unique morphism hWP!P0 in Op.V/ such that f00 Dhf0 and xg00 Dhxg0in VN. Given a planted planar tree with leaves Twe denote Ve.T/D fv2V.T/Ilevel.v/ is eveng;Vo.T/DV.T/nVe.T/; Ie.T/DI.T/\Ve.T/; Io.T/DI.T/\Vo.T/ (see Figure 11). From now on, we will only consider one tree in each isomorphism class of objects in PPTL. The idea behind our construction of the pushout of Equation (2) is as follows. For any planted planar tree with leaves concentrated in even levels, such as T in Figure 11, we replace any inner even (resp. odd) vertex v with the piece of V (resp. O ) in degree zv , Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1561 jTj D  ı  v ııı  ı   kTk D ı v ııı   ı Figure 11: For a planted planar tree with leaves T , on the left (resp. right) we denote ı the vertices in Vo.T/ (resp. Io.T/ ) and  the vertices in Ve.T/ (resp. in Ie.T/). and transform adjacency relations into tensor products. ı  ı ı ı   ı O.2/ V.3/ O.3/O.0/O.2/ V.0/V.1/ O.0/ ˝ ˝ ˝ ˝ ˝ ˝ ˝ In order to simplify the exposition of this intuitive idea, let us allow ourselves to talk about elements of this object in V . We want to attach to O the product of these elements in a coherent way. More precisely, if T has n leaves, we attach these elements to O.n/ . For this, we must proceed by induction on the number of inner even vertices and require that, for any even inner vertex v , the image of the morphism induced by f .zv/ , 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/ ˝ ˝ ˝ ˝ ˝ ˝ ˝ Algebraic & Geometric Topology, Volume 11 (2011) 1562 Fernando Muro is attached according to the attachment of the tree T0 with less even inner vertices obtained from Tby contracting the edges surrounding v: (3) kTk D ı  ı ı ı   ı _ _ _ _  M        M_ _ __ contraction morphism in PPTL // _[VSRSV[_ı   ı D kT0k O.2/ U.3/ O.3/O.0/O.2/ V.0/V.1/ O.0/ ˝ ˝ ˝ ˝ ˝ ˝ ˝ // O.2/ O.3/ O.3/O.0/O.2/ V.0/V.1/ O.0/ 9 > > > = > > > ; ˝ ˝ ˝ ˝ ˝ ˝ ˝ idO.2/ı23I3;0;2 composition in O according to the structure of Tin a neighbourhood of v //O.6/ V.0/V.1/ O.0/ ˝ ˝ ˝ This inductive construction is carried out in the following lemma. In order to state it we need to introduce some terminology. The star of a vertex v2V.T/ is the subtree St.v/T formed by the edges containing v , and the link Lk.v/ V.T/ consists of the vertices adjacent to v ; see Figure 12. When jTj D  ı  v ııı  ı   St.v/ D ı v ııı Lk.v/ D ı ııı Figure 12: The star and the link of the vertex vof the tree Tin Figure 11 Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1563 the star is formed by inner edges, the natural projection pT St.v/WT! T=St.v/ is a morphism in PPTL ; see Equation (3). This is the case if v2Ie.T/ and L.T/ Ve.T/. Moreover, in this case pT St.v/ induces identifications Ie.T/n fvg D Ie.T=St.v//; Io.T/nLk.v/ DIo.T=St.v// n fŒSt.v/g: Furthermore, we will also consider the extended star St.v/ T , which is the planted planar tree with leaves whose inner part is St.v/ , the root edge is the outgoing edge of the minimum vertex u2Lk.v/ , the leaves are the incoming edges of the vertices in Lk.v/ except from fu; vg , and the planar order is the restriction of the planar order in T; see Figure 13. Notice that St.v/=St.v/ DCrv, where (4) rvDz ŒSt.v/ Dcard L.St.v// D zu1CX w2Lk.v/nfug zwDX w2Lk.v/ zw1: The inductive construction of the pushout of Equation (2) is the in following scaring kSt.v/k D ı u v ııı Figure 13: The extended star of the vertex v of the planted planar tree with leaves Tin Figure 11 and Figure 12 lemma, whose statement is actually more complicated than its proof. For the sake of simplicity, from now on we use the same notation for an operad and for its associated operadic functor. Lemma 5.1 There is a sequence of morphisms in VN, ODP0 '1 ! P1!    ! Pt1 't ! Pt!    ; such that, for all n0 , the morphism 't.n/WPt1.n/!Pt.n/ is the pushout of the following coproduct of morphisms indexed by the set of planted trees with n leaves Algebraic & Geometric Topology, Volume 11 (2011) 1564 Fernando Muro concentrated in even levels and t inner even vertices, ie card L.T/Dn , L.T/Ve.T/ , and card Ie.T/Dt, (5) a TK v2Ie.T/ f .zv/ ˝O w2Io.T/ O.zw/; along the unique morphism (6) . T t/TWa T sK v2Ie.T/ f .zv/˝O w2Io.T/ O.zw/ ! Pt1.n/ such that, given u2Ie.T/, for tD1the morphism T 1is U.zu/˝N w2Io.T/ O.zw/ O.zu/˝N w2Io.T/ O.zw/ DO.T/O.n/; T 1 ,, xg.zu/˝id 33O.pT/ 44 and for t>1the composite morphism U.zu/˝N v2Ie.T/nfug V.zv/ ˝N w2Io.T/ O.zw/ s.J v2Ie.T/ f .zv// ˝N w2Io.T/ O.zw/ Pt1.n/ u˝id // T t// coincides with the following composition that we call T t;u: U.zu/˝N v2Ie.T/nfug V.zv/ ˝N w2Io.T/ O.zw/ O.zu/˝N v2Ie.T/nfug V.zv/ ˝N w2Io.T/ O.zw/ N v2Ie.T/nfug V.zv/ ˝N w2Io.T/nLk.u/ O.zw/ ˝O.St.u// N v2Ie.T=St.u// V.zv/ ˝N w2Io.T=St.u//nfŒSt.u/g O.zw/ ˝O.ru/ Pt1.n/ xg.zu/˝id  Šsymmetry  id˝O.pSt.u//  x T=St.u/ t1  Here .x T0 t1/T0 denotes the pushout of . T0 t1/T0 , ie Equation (6) for t1 , along Equation (5). Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1565 Proof The proof is by induction on t0 . Notice that there is nothing to check for tD0;1 . Let t>1 and assume everything works up to t1 . By the universal property of the source of an iterated ˇ product, described in Section 4, we only have to check the following compatibility condition: given two different vertices u;u02Ie.T/, the following square commutes: (a) U.zu/˝U.z u0/˝N v2Ie.T/nfu;u0g V.zv/ ˝N w2Io.T/ O.zw/ V.zu/˝U.z u0/˝N v2Ie.T/nfu;u0g V.zv/ ˝N w2Io.T/ O.zw/ U.zu/˝V.z u0/˝N v2Ie.T/nfu;u0g V.zv/ ˝N w2Io.T/ O.zw/ Pt1.n/ f .zu/˝id ;; id˝f . z u0/˝id $$ T t;u0 && T t;u :: Here, for simplicity, we omit some symmetry isomorphisms in V. Denote St.u;u0/DSt.u/[St.u0/ and Lk.u;u0/DLk.u/[Lk.u0/ . Suppose that d.u;u0/ > 2 . Then St.u/\St.u0/D∅ (see Figure 14), and moreover t>2 . By induction hypothesis, in this case both compositions coincide with (7) U.zu/˝U.z u0/˝N v2Ie.T/nfu;u0g V.zv/ ˝N w2Io.T/ O.zw/ O.zu/˝O.z u0/˝N v2Ie.T/nfu;u0g V.zv/ ˝N w2Io.T/ O.zw/ N v2Ie.T/nfu;u0g V.zv/ ˝N w2Io.T/nLk.u;u0/ O.zw/ ˝O.St.u// ˝O.St.u0// N v2Ie.T=St.u;u0// V.zv/ ˝N w2Io.T=St.u;u0//nfŒSt.u/;ŒSt.u0/g O.zw/ ˝O.ru/˝O.ru0/ Pt2.n/ Pt1.n/ xg.zu/˝xg.z u0/˝id  Šsymmetry  id˝O.pSt.u//˝O.pSt.u0//  x T=St.u;u0/ t2  't1.n/  See Figure 14 and Figure 15. Algebraic & Geometric Topology, Volume 11 (2011) 1572 Fernando Muro kTk D ı x x ı uı ı  kT0k D ıu0   ı Figure 20: For the trees T and T0 and iD2 we depict u , u0 and two possible choices of x , one with fx;ug 2 E.T/ and the other one with fx;ug … E.T/. Suppose fx;ug … E.T/ . Then u…Lk.x/ . Using the definition of ds;t i.T;T0/ in the statement of this lemma and the definition of x .TıiT0/=e sCt in Lemma 5.1 we deduce that, in this case, the left hand side of (a) is the following composite morphism (see Figure 21): (9) U.zx/˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/ O.z w0/ O.zx/˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/ O.z w0/ O.St.x// ˝O.zu/˝O.z u0/ ˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/n.Lk.x/[fug/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/nfu0g O.z w0/ O.rx/˝O.zuCz u01/ ˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/n.Lk.x/[fug/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/nfu0g O.z w0/ N v2Ie...T=St.x//ıiT0/=e/ V.zv/ ˝N w2Io...T=St.x//ıiT0/=e/ O.zw/ PsCt1.mCn1/ PsCt.mCn1/ xg.zx/˝id  Šsymmetry  O.pSt.x//˝ık˝id  Šsymmetry  x ..T=St.x//ıiT0/=e sCt1 'sCt.mCn1/  Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1573 Moreover, by induction, since .s1;t/ < .s;t/ one can easily check that this is also the right hand side of (a). ı  x ı u u0 e ı ı ı   ı  kTı2T0k pTı2T0 St.x/te // _YURQRUY_ı  ı Œeı ıŒSt.x/   ı k..T=St.x// ı2T0/=ek O.2/ V.3/ O.1/ O.3/ O.0/O.2/ V.0/V.1/ O.0/ U.0/ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ xg.0/22 O.2/ V.3/ O.1/ O.3/ O.0/O.2/ V.0/V.1/ O.0/ O.0/ _   :__   : _K l  ! KK l  ! K ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ı1  O.pSt.x//D ı2 OO O.2/ V.3/ O.3/O.0/O.1/ V.0/V.1/ O.0/ ˝ ˝˝˝ ˝ ˝ ˝ Figure 21: An illustration of Equation (9) for T and T0 as in Figure 20 in case fx;ug … E.T/ Suppose now that fx;ug 2 E.T/ . Then u2Lk.x/ . Assume that e is the l –th leaf of St.x/ . We denote T00 the planted planar tree with leaves T00 DSt.x/ılCz u0 . The inner part of T00 is identified with the subtree T000 TıiT0 formed by adjoining the edge e to St.x/ ; see Figure 22. Using the definition of ds;t i.T;T0/ in the statement, kTı2T0k ı x ı u u0 e ı ı ı   ı  kT00k ı x ı u u0 e ı ı ı k.Tı2T0/=T000k ı   ı  ŒT000 Figure 22: For the choice of x in Figure 20 with fx;ug 2 E.T/ we here depict T00 . The subtree T000 is indicated with double lines. Algebraic & Geometric Topology, Volume 11 (2011) 1574 Fernando Muro the definition of x .TıiT0/=e sCt in Lemma 5.1, and relation (2) in Remark 2.6 for O , we deduce that, in this case, the left hand side of (a) is the following composite morphism (see Figure 23): (10) U.zx/˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/ O.z w0/ O.zx/˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/ O.z w0/ O.T00/˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/nLk.x/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/nfu0g O.z w0/ O.rxCz u01/˝N v2Ie.T/nfxg V.zv/ ˝N w2Io.T/nLk.x/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/nfu0g O.z w0/ N v2Ie..TıiT0/=T000/ V.zv/ ˝N w2Io..TıiT0/=T000/ O.zw/ PsCt1.mCn1/ PsCt.mCn1/ xg.zx/˝id  Šsymmetry  O.pT00 /˝id  Šsymmetry  x .TıiT0/=T000 sCt1  'sCt.mCn1/  Moreover, by induction one can easily check that this is also the right hand side of (a), hence we are done with this proof. Let Pbe the sequence defined as P.n/Dcolim t0Pt.n/: By the previous lemma, the morphisms cs;t i.m;n/ induce composition laws in the colimit: (11) ıiWP.m/˝P.n/! P.mCn1/; 1im;n0: Consider the morphism (12) IO.1/DP0.1/colim t0Pt.1/DP.1/: canonical // u// Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1575 O.2/ U.3/ O.1/ O.3/ O.0/O.2/ V.0/V.1/ O.0/ V.0/ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ xg.3/ // O.2/ O.3/ O.1/ O.3/ O.0/O.2/ V.0/V.1/ O.0/ V.0/ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ ˝ O.pT00 / //         ________ ___    ______  O.6/ V.0/V.1/ O.0/ V.0/ ˝˝ ˝ ˝ Figure 23: An illustration of Equation (10) for T and T0 as in Figure 20 in case fx;ug 2 E.T/(see Figure 22) Proposition 5.3 The sequence P , the unit in Equation (12) and the composition laws in Equation (11) define an operad. Proof We must check that relations (1)–(4) in Remark 2.6 hold for P . Each of these relations for P can be derived from the corresponding relation for O . As relations (1) and (2) are very similar to each other, just as (3) and (4), we here check (2) and (3). In order to prove relation (2) for P it is enough to check that the following two morphisms Pr.l/˝Ps.m/˝Pt.n/!PrCsCt.lCmCn2/coincide: crCs;t j.lCm1;n/.cr;s i.l;m/˝idPt.n//Dcr;sCt i.l;mCn1/.idPr.l/˝cs;t jiC1.m;n//: We check this by induction on .r;s;t/2N3 with respect to the graded lexicographic order. For rDsDtD0 this is just relation (2) for the operad O . If we assume that the relation holds up to the predecessor of .r;s;t/ , then by using the universal property of the pushout definition of Pr.m/˝Ps.n/˝Pt.p/ arising from Lemmas 4.1 and 5.1, we only have to check that, with the notation of Lemma 5.2, given planted planar trees with leaves concentrated in even levels T;T0;T00 with card L.T/Dl , card L.T0/Dm , card L.T00/Dn, card Ie.T/Dr, card Ie.T0/Ds, and card Ie.T00/Dt, then (a) crCs;t j.lCm1;n/.dr;s i.T;T0/˝x T00 t/ Dcr;sCt i.l;mCn1/.x T r˝ds;t jiC1.T0;T00//: Let u2Io.T/ be the inner vertex of the i –th leaf edge of T , u0 12Io.T0/ the unique level 1 vertex of T0 , u0 22Io.T0/ the inner vertex of the .jiC1/ –st leaf edge of T0 , and u00 2Io.T00/ the unique level 1 vertex of T00 . Suppose that the i –th leaf edge Algebraic & Geometric Topology, Volume 11 (2011) 1576 Fernando Muro of T is the k1 –st incoming edge of u , and that the .jiC1/ –st leaf edge of T0 is the k2 –th incoming edge of u0 2 . The most complicated case is when u0 1Du0 2 , and even this case is easy, although somewhat tedious. kTk ı  ı uı ı  kT0k ıu0 1.Du0 2if jD2/ u0 2if jD3   ı kT00k ıu00   Figure 24: For the planted planar trees with leaves T , T0 and T00 we depict u,u0 1,u0 2and u00 for iD2and jD2;3. Assume u0 1Du0 2 and denote this vertex simply by u0 . Notice that .TıiT0/ıjT00 D Tıi.T0ıjiC1T00/ ; compare the second tree in Figure 10. Let K.TıiT0/ıjT00 be the subtree with V.K/D fu;u0;u00g and E.K/D ffu;u0g;fu0;u00gg ; see Figure 25. Then by Lemma 5.2 and relation (2) for O , both sides of (a) coincide with the following: N v2Ie.T/ V.zv/ ˝N w2Io.T/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/ O.z w0/˝N v002Ie.T00/ V.z v00/˝N w002Io.T00/ O.z w00/ O.zu/˝O.z u0/˝O.z u00/˝N v2Ie.T/[Ie.T0/[Ie.T00/ V.zv/ ˝N w2.Io.T/[Io.T0/[Io.T00//nfu;u0;u00g O.zw/ O.zuCz u0Cz u00 2/˝N v2Ie..TıiT0/ıjT00/=K V.zv/ ˝N w2Io...TıiT0/ıjT00/=K/nfŒKg O.zw/ PsCt.mCnCp2/ Šsymmetry  .ık1.id ˝ ık2// ˝id  x ..TıiT0/ıjT00/=K rCsCt  Assume now that u0 1¤u0 2 . In this case it is not even necessary to use any of the relations in Remark 2.6 for O . Actually, by Lemma 5.2, if K.TıiT0/ıjT00 is the (disjoint) union of the edges e1D fu;u0 1g and e2D fu0 2;u00g ; see Figure 25, then both Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1577 k.Tı2T0/ı2T00k ı  ı u u00 u0 1Du0 2 ı ı ı   ı ı    k.Tı2T0/ı3T00k ı  ı u u0 1 u0 2 u00 ı ı ı   ı ı    Figure 25: Here we depict the subtree K.TıiT0/ıjT00 in double lines for the trees in Figure 24,iD2and jD2;3. sides of (a) coincide with the following: N v2Ie.T/ V.zv/ ˝N w2Io.T/ O.zw/ ˝N v02Ie.T0/ V.z v0/˝N w02Io.T0/ O.z w0/˝N v002Ie.T00/ V.z v00/˝N w002Io.T00/ O.z w00/ O.zu/˝O.z u0 1/˝O.z u0 2/˝O.z u00/˝N v2Ie.T/[Ie.T0/[Ie.T00/ V.zv/ ˝N w2.Io.T/[Io.T0/[Io.T00//nfu;u0 1;u0 2;u00g O.zw/ O.zuCz u0 11/˝O.z u0 2Cz u00 1/˝N v2Ie..TıiT0/ıjT00/=K V.zv/ ˝N w2Io...TıiT0/ıjT00/=K/nfŒe1;Œe2g O.zw/ PsCt.mCnCp2/ Šsymmetry  ık1˝ık2˝id  x ..TıiT0/ıjT00/=K rCsCt  Relation (3) is a consequence of the fact that the following composite morphism is a right unit constraint in V: Pr.l/˝Iid˝u//Pr.l/˝O.1/DPr.l/˝P0.1/cr;0 i.l;1/ //Pr.l/: This follows by induction on r . For rD0 this is just relation (3) for O . Assume this holds up to r1 . By Lemma 5.1 and the induction hypothesis, we only have to check that the morphism cr;0 i.l;1/.x T r˝u/ coincides with the composition of the right unit isomorphism and x T r. By Lemma 5.2, cr;0 i.l;1/.x T r˝idO.1//Ddr;0 i.T;C1/: Algebraic & Geometric Topology, Volume 11 (2011) 1578 Fernando Muro Let u02Io.C1/ be now the unique inner vertex of C1 , and eD fu;u0g 2 E.TıiC1/ . In this case .TıiC1/=eDT . Moreover, by the definition of dr;0 i.T;C1/ in the statement of Lemma 5.2 and by relation (3) for O , the morphism dr;0 i.T;C1/.id ˝u/ is the composition of the right unit isomorphism and x T r, hence we are done. Consider the morphisms of sequences f0WO!Pand xg0WV!Pdefined as: f0.n/WO.n/DP0.n/colim t0Pt.n/DP.n/; canonical // V.n/˝I˝.nC1/ŠV.n/ V.n/˝O.1/˝.nC1/P1.n/colim t0Pt.n/DP.n/ id˝u˝.nC1/ x C1.Cn.C1;:::;C1// 1//canonical // xg0.n/ '' kC1.C5.C1;:::;C1//k D ı v  ıı ıı ı Figure 26: The planted planar tree with leaves in even levels C1.Cn.C1;:::;C1// for nD5 Theorem 5.4 The morphism f0WO!P is an operad morphism. Moreover, if g0WF.V/!P is the operad morphism adjoint to xg0 , then the following diagram is a pushout in Op.V/: F.U/ g  F.f / //F.V/ g0  Of0//P Proof The morphism f0 is an operad morphism by the very definition of the operad structure in P , since c0;0 iD ıi is the structure morphism of O and the unit of P is the composition of the unit of O and f0 ; see Lemma 5.2 and Equation (12). Moreover, Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1579 the square U.l/ xg.l/  f .l/ //V.l/ xg0.l/  O.l/f0.l/ //P.l/ commutes for all l0 . In fact, the following diagram commutes by some trivial facts, including the very definition of P1.l/in Lemma 5.1: U.l/ Š.right unit/1  xg.l/ vv f .l/ //V.l/ Š .right unit/1  O.l/ Š .right unit/1 (( id ++ U.l/˝I˝.lC1/ xg.l/˝id˝.lC1/ I  id˝u˝.lC1/ (( f .l/˝id.lC1/ I //V.l/˝I˝.lC1/ id˝u˝.lC1/  O.l/˝I˝.lC1/ idO.l/˝u˝.lC1/  right unit && U.l/˝O.1/˝.lC1/ xg.l/˝id˝.lC1/ vvf .l/˝id.lC1/ O.1/ 88 C1.Cl.C1;:::;C1// 1 mm V.l/˝O.1/˝.lC1/ x C1.Cl.C1;:::;C1// 1  O.l/˝O.1/˝.lC1/ O.pC1.Cl.C1;:::;C1///  O.l/ '1.l/ ?? P1.l/ Also, its outer (commutative) square U.l/ xg.l/  f .l/ //V.l/ Š.right unit/1  V.l/˝I˝.lC1/ id˝u˝.lC1/  V.l/˝O.1/˝.lC1/ x C1.Cl.C1;:::;C1// 1  O.l/'1.l/ //P1.l/ composed with the canonical morphism P1.l/!colimr0Pr.l/DP.l/ yields the former square. Algebraic & Geometric Topology, Volume 11 (2011) 1580 Fernando Muro Suppose we are given an operad P0 and morphisms f00WO!P0 in Op.V/ and xg00WV!P0in VNsuch that the square (a) U.l/ xg.l/  f .l/ //V.l/ xg00.l/  O.l/f00.l/ //P0.l/ commutes for all l0 . We must show that there is a unique morphism hWP!P0 in Op.V/such that f00 Dhf0and xg00 Dhxg0in VN. We define morphisms hr.l/WPr.l/! P0 by induction on r0 as follows. We set h0.l/Df00.l/ . Assume we have defined up to hr1.l/ . Then we define hr.l/ so that hr.l/'r.l/Dhr1.l/ and, for any planted planar tree T with l leaves concentrated in even levels and r inner vertices in even levels: (b) hr.l/x T rDP0.pT/. O v2Ie.T/ xg00.zv/ ˝O w2Io.T/ f00.zw//: The morphism hr.l/ is well defined by the universal property of the pushout definition of Pr.l/in Lemma 5.1 since, given u2Ie.T/, P0.pT/O v2Ie.T/ xg00.zv/ ˝O w2Io.T/ f00.zw/.f .zu/˝id/ DP0.pT/O v2Ie.T/nfug xg00.zv/ ˝O w2Io.T/[fug f00.zw/.id ˝ xg.zu// DP0.pT=St.u//O v2Ie.T=St.u// xg00.zv/ ˝O w2Io.T=St.u// f00.zw/.id ˝O.pSt.u///.id ˝ xg.zu// Dhr1.l/x T=St.u/ r1.id ˝O.pSt.u///.id ˝ xg.zu// Dhr1.l/ T r;u: Here, in the first equation we use the commutativity of (a), in the second equation we use the fact that f00 is an operad morphism, and in the third equation we use the induction hypothesis. The fourth equation follows from the very definition of T r;u in the statement of Lemma 5.1. For simplicity, in these equations we have omitted some symmetry isomorphisms in V. Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1581 We have checked that the morphisms hr.l/ induce a morphism of sequences hWP!P0 in the colimit. It is clear that hf0Df00 since h0Df00 , in particular h preserves units. Moreover, hg0Dg00 since for x TDC1.Cl.C1;:::;C1// (see Figure 26), if v2Ie.x T/ is the unique inner vertex in even levels, then h1.l/x x T 1.idV.l/˝u˝.lC1//DP0.px T/.xg00.l/˝f00.1/˝.lC1//.idV.l/˝u˝.lC1// DP0.px T/.xg00.l/˝u˝.lC1// D xg00.l/: Here, in the first equation we use (b), in the second equation we use that f00 is an operad morphism and therefore it preserves units, and in the third equation we use relations (3) and (4) in Remark 2.6 for the operad P0. In order to check that his indeed an operad morphism, we show that hrCs.lCm1/cr;s i.l;m/Dhr.l/ıihs.m/: We proceed by induction on .r;s/2N2 with respect to the graded lexicographic order. This is obvious for rDsD0 , since f00 is an operad morphism. If the equation holds up to the predecessor of .r;s/ then by induction hypothesis we only have to check that the following equation holds: hrCs.lCm1/cr;s i.l;m/.x T r˝x T0 s/D.hr.l/x T r/ıi.hs.m/x T0 s/; for T0 a planted planar tree with m leaves concentrated in even levels and s inner vertices in even levels. Let u2Io.T/ be the inner vertex of the i –th leaf edge of T , u02Io.T0/ the unique level 1 vertex of T0 , and eD fu;u0g 2 E.TıiT0/ . Suppose that the i–th leaf edge of Tis the k–th incoming edge of u. Then, hrCs.lCm1/cr;s i.l;m/.x T r˝x T0 s/ DhrCs.lCm1/dr;s i.T;T0/ DhrCs.lCm1/x .TıiT0/=e sCt.ık˝id/ DP0.p.TıiT0/=e/O v2Ie..TıiT0/=e/ xg00.zv/ ˝O w2Io..TıiT0/=e/ f00.zw/.ık˝id/ DP0.p.TıiT0/=e/.ık˝id/O v2Ie.T/[Ie.T0/ xg00.zv/ ˝O w2Io.T/[Io.T0/ f00.zw/ Algebraic & Geometric Topology, Volume 11 (2011) 1588 Fernando Muro The action of Oon FO.Y/, z.O.n// ˝FO.Y/˝nDz.O.n// ˝ n O iD1a pi0 z.O.pi// ˝Y˝pi Ša p1;:::;pn0 z.O.n// ˝ n O iD1z.O.pi// ˝Y˝pi Ša p1;:::;pn0 z.O.n// ˝z.O.p1// ˝˝z.O.pn// ˝Y˝Pn iD1pi Ša p1;:::;pn0 z.O.n/˝O.p1/˝    O.pn//˝Y˝Pn iD1pi FO.Y/Da p0 z.O.p// ˝Y˝p; n  is defined as the morphism which sends the factor .p1;:::;pn/2Nn in the source to the factor pDp1CCpn2N in the target via z.nIp1;:::;pn/˝id , n1 . For nD0 , the morphism 0Wz.O.0// !FO.Y/ is the inclusion of the factor pD0 of the coproduct. The unit of the adjunction is the following composite morphism in C: YIC˝Yz.IV/˝Y z.O.1// ˝YFO.Y/: Š .left unit/1 //Š unit˝id // z.u/˝id // inclusion of the factor pD1 // Moreover, given an O –algebra A , the counit of the adjunction is defined by the multiplication morphisms in Equation (13): .p/p0WFO.A/! A: In this section we give an explicit construction of the pushout of two morphisms in AlgC.O/as follows: (14) AFO.Y/ g oo FO.f / //FO.Z/: Consider the adjoint diagram in C: A Y xg oo f //Z: The pushout of Equation (14) is an O –algebra B together with morphisms f0WA!B in AlgC.O/ and xg0WZ!B in C such that f0xgD xg0f in C . Moreover, given an Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1589 O –algebra B0 and morphisms f00WA!B0 in AlgC.O/ and xg00WZ!P0 in C with f00 xgD xg00f in C , there is a unique morphism hWB!B0 in AlgC.O/ such that f00 Dhf0and xg00 Dhxg0in C. The following lemma allows an inductive definition of the pushout of Equation (14) as an object in C . We omit proofs in this section since the results are simpler analogs of those in Section 5, and the proofs follow very much the same steps. Lemma 8.1 There is a sequence in C, ADB0 '1 ! B1!    ! Bt1 't ! Bt!    ; where the morphism 'tis the pushout of (15) a n1a Sf1;:::;ng card.S/Dt z.O.n// ˝kS 1ˇˇkS n;kS iD(fi2S; 0!A i …S; along the unique morphism (16) . n;S t/n;SWa n1a Sf1;:::;ng card.S/Dt z.O.n// ˝s.kS 1ˇˇkS n/! Bt1; such that for tD1and 1in, n;fig 1Dn.idz.O.n// ˝id˝.i1/˝ xg˝id˝.ni//; and for t>1and i2S, n;S t.idz.O.n// ˝i/Dx n;Snfig t1.idz.O.n// ˝id˝.i1/˝ xg˝id˝.ni//: Here .x n;S0 t1/n;S0 denotes the pushout of . n;S0 t1/n;S0 , ie Equation (16) for t1 , along Equation (15). We now endow BDcolim t0Bt with an O–algebra structure. Lemma 8.2 There are unique morphisms in C, ct1;:::;tn nWz.O.n// ˝Bt1˝˝Btn! Bt1CCtn;n1;ti0; Algebraic & Geometric Topology, Volume 11 (2011) 1590 Fernando Muro such that c0;n :::;0 nDnWz.O.n// ˝A˝n! A; and, with the convention x pi;Si 0Dpi , if Si f1;:::;pig is a subset of cardinality card SiDti,1in, then ct1;:::;ti;:::;tn n.id˝.i1/˝'ti˝id˝.ni//D't1CCtnct1;:::;ti1;:::;tn n; ct1;:::;tn n.x p1;S1 t1˝˝x pn;Sn tn/ Dx p1CCpn;Sn iD1.SiC.p1CCpi1// t1CCtnz.nIp1;:::;pn/˝id˝Pn iD1pi: Here SCpD fiCpIi2Sg and  is the multiplication of the operad O . For simplicity, in these equations we have omitted some obvious structure isomorphisms of V,Cand z. We define f0WADB0! colim t0BtDB as the canonical morphism to the colimit. Moreover, for n1we define B nWz.O.n// ˝B˝n! B as the colimit of the morphisms ct1;:::;tn n in the previous lemma, ti0 , and for nD0 , B 0Wz.O.0// A 0 ! Af0 ! B: Furthermore, we define xg0WZ!Bas the composite morphism ZIC˝Zz.IV/˝Z z.O.1// ˝ZB1B: Š .left unit/1 //Š unit˝id // z.u/˝id // x 1;f1g 1// projection to the colimit // Theorem 8.3 The morphisms B n , n0 , define an O –algebra structure on B , f0WA!B is an O –algebra morphism, and if g0WFO.Z/!B is the adjoint of xg0WZ!B, then the following square is a pushout in AlgC.O/: FO.Y/ g  FO.f / //FO.Z/ g0  Af0 //B Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1591 9 Proofs of Theorem 1.2 and Theorem 1.3 Suppose that we are in the same conditions as in the two previous sections. Assume also that V and C are monoidal model categories (see Definition 4.2) and that the composite functor Vz ! Z.C/forget ! C is a left Quillen functor [15, Definition 1.3.1]. We will need a nonsymmetric version of the monoid axiom in Definition 6.1. Definition 9.1 The monoid axiom for Csays that, for K0D ff1ˇˇfnIn1;S f1;:::;ngis a subset with card S1; fiis a trivial cofibration if i2S; fiW0!Xifor some object Xiin Cif i…Sg; all relative K0–cell complexes are weak equivalences. Notice that, as a consequence of the pushout product axiom, this is indeed equivalent to the monoid axiom in Definition 6.1 when C is symmetric. In any case, if all objects in C are cofibrant then the monoid axiom is a consequence of the pushout product axiom. Suppose from now on that C satisfies the monoid axiom and is cofibrantly generated with sets of generating cofibrations and generating trivial cofibrations I and J , respectively, with presentable sources. Proposition 9.2 Consider a pushout diagram in AlgC.O/as follows. FO.Y/ g  FO.f / // push FO.Z/ g0  Af0 //B (1) If f is a trivial cofibration in C , then the underlying morphism f0WA!B in Cis a relative K0–cell complex, where K0is the class in Definition 9.1. (2) Suppose A is cofibrant in C , f is a cofibration in C , and O.n/ is cofibrant in V , n0 . Then the morphism f0WA!B is a cofibration in C , in particular B is cofibrant in C. Algebraic & Geometric Topology, Volume 11 (2011) 1592 Fernando Muro Proof In case (1), the morphism Equation (15) in Lemma 8.1 is in K0 , hence (1) follows from Theorem 8.3. In case (2), since z is a left Quillen functor, the objects z.O.n// are cofibrant in C . Therefore, by the pushout product axiom (Definition 4.2) the morphism Equation (15) is a cofibration in C . Furthermore, by Theorem 8.3 the morphism f0WA!B is a transfinite composition of cofibrations in C , hence a cofibration in C itself [14, Proposition 10.3.4]. As an immediate consequence of (1) here and the monoid axiom, we obtain the following. Corollary 9.3 A morphism in C underlying a relative FO.J/ –cell complex in AlgC.O/ is a weak equivalence in C. Now we are ready to prove Theorem 1.2. Proof of Theorem 1.2 Using the explicit description of the free operad adjunction at the beginning of Section 8, it is easy to see that O –algebras are the same thing as algebras over the monad associated to the free O –algebra adjunction; compare [2, Proposition 1.3]. Moreover, this monad preserves filtered colimits (see again the explicit construction), therefore the category AlgC.O/ is complete and cocomplete [5, Proposition 4.3.6]. Furthermore, the forgetful functor AlgC.O/!C also preserves filtered colimits [5, Proposition 4.3.2], in particular, since FO is a left adjoint and sources of morphisms in I and J are presentable in C , then sources of morphisms in FO.I/and FO.J/are presentable in AlgC.O/. We can apply [20, Lemma 2.3]in order to prove the existence of the claimed model structure in AlgC.O/ . The smallness condition has already been checked, and condition (1) of [20, Lemma 2.3]has been established in Corollary 9.3. The statement about right properness is obvious since fibrations and weak equivalences in AlgC.O/are detected by the forgetful functor AlgC.O/!C, and this functor is a right adjoint, so it preserves all limits, in particular pullbacks. If C is combinatorial then AlgC.O/ is locally presentable by [1, 2.3 (1) and the Theorem in 2.78], hence it is combinatorial. Lemma 9.4 Suppose that O is an operad in V with O.n/ cofibrant for all n0 . Then any cofibrant O–algebra is also cofibrant as an object in C. Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1593 Proof Cofibrant O –algebras are retracts of FO.I/ –cell complexes, and cofibrant objects in C are closed under retracts, so it is enough to check that FO.I/ –cell complexes are cofibrant in C . The initial O –algebra in C (see Remark 7.2) is cofibrant in C , since O.0/ is cofibrant in V and z is a left Quillen functor. Using Proposition 9.2 (2), an induction argument proves that any FO.I/–cell complex is cofibrant in C. Corollary 9.5 Let O be an operad in V with O.n/ cofibrant for all n0 . Then, the forgetful functor AlgC.O/!Cpreserves cofibrations with cofibrant source. Proof This is an immediate consequence of Lemma 9.4 and Proposition 9.2 (2), since cofibrations in AlgC.O/ are retracts of relative FO.I/ –cell complexes, the forgetful functor preserves filtered colimits, and cofibrations in C are closed under transfinite compositions and retracts. Lemma 9.6 Under the hypotheses of Theorem 1.3, suppose that we have a pushout diagram in AlgC.O/, FO.Y/ g  // FO.f / // push FO.Z/ g0  A// f0 //B where f is a cofibration in C and A is a cofibrant O –algebra. If the unit of the adjunction evaluated at A is a weak equivalence AWA !A , then it is also a weak equivalence when evaluated at B,BWB !B. Proof Since  is left adjoint to  , which is the identity on the underlying object in C , there is a natural isomorphism FOŠFP that we regard as an identification, and the morphism .f 0/fits into the following pushout diagram in AlgC.P/: FP.Y/ .g/  // FP.f / // push FP.Z/ .g0/  A// .f 0/ //B The O –algebra A is cofibrant and  is a left Quillen functor, therefore A is a cofibrant P –algebra, in particular, both A and A are cofibrant in C by Lemma 9.4. Notice that the underlying object of Aand Ain Cis the same. Algebraic & Geometric Topology, Volume 11 (2011) 1594 Fernando Muro Let us call CDB . By Lemma 8.1, the morphism in C underlying B is the colimit in t2N of an inductively constructed diagram of cofibrant objects in C , t>0 , (a)    Bt1// 'B t// t1  Bt   t      Ct1// 'C t//Ct   such that B0DA , C0DA , 0DA , the morphism t is the pushout of the horizontal lines of the following diagram Bt1 t1   induced by and 0  Equation (16) for O oo// Equation (15) for O // induced by and 0  Ct1 Equation (16) for P oo// Equation (15) for P // and 'B tand 'C tare the natural morphisms to the pushout. The objects O.n/ and P.n/ are cofibrant in V and z is a left Quillen functor, hence z.O.n// and z.P.n// are cofibrant in C , n0 . Moreover, f is a cofibration in C and A and A are cofibrant in C . Therefore Lemma 4.4 shows that the square on the right has weak equivalences in the columns and cofibrations in the rows. In particular, 'B t and 'C t are cofibrations in C and, by the gluing property in left proper model categories [14, Proposition 13.5.4],tis a weak equivalence in C. To conclude, BDcolimt0t is a weak equivalence in C since (a) is a weak equivalence between cofibrant objects in the Reedy model category of directed diagrams in C indexed by N [15, Theorem 5.1.3]and Ken Brown’s lemma [15, Lemma 1.1.12] applies, because colimt0is a left Quillen functor [15, Corollary 5.1.6]. Finally, we are ready to prove Theorem 1.3. Proof of Theorem 1.3 We will use the criterion in [15, Corollary 1.3.16 (c)]to detect Quillen equivalences. The functor  preserves and reflects weak equivalences, since it is the identity on the underlying object in C . Therefore, it is enough to check that the unit of the adjunction AWA!A is a weak equivalence for any cofibrant O–algebra A. Weak equivalences are closed under retracts and cofibrant O –algebras are retracts of FO.I/ –cell complexes, so we can suppose that A is an FO.I/ –cell complex, ADcolimi< Ai. We now proceed by induction on the ordinal . Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1595 For D1 , A is the initial O –algebra; see Remark 7.2. Then A is the initial P –algebra, since  is a left adjoint, and ADz..0//Wz.O.0// !z.P.0// . The morphism .0/ is a weak equivalence between cofibrant objects in V and z is a left Quillen functor, therefore z..0// is also a weak equivalence between cofibrant objects in Cby [15, Lemma 1.1.12]. If D˛C1 and the result is true for ˛ , then it is also true for  by the previous lemma. Suppose now that  is a limit ordinal and that the result is true for all i<  . The functor  preserves colimits, since it is a left adjoint, and  preserves filtered colimits, because it is the identity over C and forgetful functors from algebras to C preserve filtered colimits. In particular ADcolimi< i is a colimit of weak equivalences by induction hypothesis. By Proposition 9.2 (2), an FO.I/ –cell complex is a colimit of a continuous diagram of cofibrations between cofibrant objects in C , and the same is true for FP.I/ –cell complexes. This applies to A and A . Such diagrams are cofibrant objects in Reedy model categories of directed diagrams in C [15, Theorem 5.1.3]. Therefore, A is the colimit of a weak equivalence between cofibrant objects in the Reedy model category of directed diagrams in C indexed by  . Now, Ken Brown’s lemma [15, Lemma 1.1.12]shows that A is a weak equivalence, since colimi< is a left Quillen functor [15, Corollary 5.1.6]. 10 An application to enriched categories and A1–categories In this section we lay the foundations to construct model categories of categorified algebraic structures. This is applied to enriched categories and enriched A1 –categories. Definition 10.1 Given a set S , a V –graph M with object set S is a collection of objects in V indexed by SS , MD fM.x;y/gx;y2S . The category GraphS.V/ of V –graphs with object set S , where morphisms are defined in the obvious way, is biclosed monoidal with tensor product .M˝SN/.x;y/Da z2S M.z;y/˝N.x;z/: The unit object ISis IS.x;y/D(Ithe monoidal unit of V;if xDy; 0the initial object of V;if x¤y: Algebraic & Geometric Topology, Volume 11 (2011) 1596 Fernando Muro This monoidal category is clearly nonsymmetric, unless S is a singleton. The right adjoint of M˝  is the functor HomS l.M;/defined as HomS l.M;P/.x;y/DY z2S Hom.M.y;z/; P.x;z//; and the right adjoint of  ˝ Nis the functor HomS r.N;/defined as HomS r.N;P/.x;y/DY z2S Hom.N.z;x/; P.z;y//: We have a strong braided monoidal functor zWV!GraphS.V/defined as z.A/.x;y/D(Aif xDy; 0if x¤y: Moreover, .z.A/˝SM/.x;y/DA˝M.x;y/; .M˝Sz.A//.x;y/DM.x;y/˝A; and the natural isomorphism .A;M/Wz.A/˝SMŠM˝Sz.A/; is defined as the symmetry isomorphism of Vcoordinatewise. Remark 10.2 If V is a model category, the category GraphS.V/ inherits from V a product model category structure, where fibrations, cofibrations and weak equivalences are defined coordinatewise. If V is cofibrantly generated (resp. combinatorial) then so is GraphS.V/ ; compare Remark 2.3. Moreover, since SS is a set, a V –graph M is presentable provided M.x;y/ is presentable for all x;y2S . In particular, if V has sets of generating cofibrations and generating trivial cofibrations with presentable source, then so does GraphV.S/ . Furthermore, if V is right proper then the product model category GraphV.S/is also right proper. Notice that the composite functor Vz !Z.GraphS.V// !GraphS.V/ preserves fibrations, cofibrations and weak equivalences, and it has a right adjoint defined by M7! Y x2S M.x;x/: This adjoint pair is therefore a Quillen adjunction. Proposition 10.3 If V satisfies the monoid axiom then GraphS.V/ also satisfies the monoid axiom. Algebraic & Geometric Topology, Volume 11 (2011) Homotopy theory of nonsymmetric operads 1597 Proof It is enough to notice, using the symmetry of V and the pushout product axiom in V , that any morphism f1ˇ  ˇfn in the class of morphisms K0 of GraphS.V/ in Definition 9.1 is componentwise a morphism in the class K of V in Definition 6.1. Categories enriched on V with set of objects S are the same as monoids in GraphS.V/ . These monoids are the same as algebras over the nonsymmetric operad AssV in V defined by AssV.n/DI , n0 . All compositions in AssV are unit isomorphisms I˝IŠI and the unit of the operad uWI!AssV.1/ is the identity. This operad is generated by the “elements” in degree 0 and 2 ; the degree 2 “element” represents the composition law, and the degree 0“element” represents the identities. In order to simplify notation, we denote CatS.V/DAlgGraphS.V/.AssV/: An A1 –category enriched on V with set of objects S is an algebra over a cofibrant replacement AssV 1 of AssV , which is a trivial fibration WAssV 1  AssV in Op.V/ with cofibrant source. We simply denote A1–CatS.V/DAlgGraphS.V/.AssV 1/: Combining the previous proposition with Theorem 1.2 we obtain the following corollary, which improves [6, Theorem 3.3]. Corollary 10.4 Let V be a cofibrantly generated closed symmetric monoidal category satisfying the monoid axiom. Suppose that V has sets of generating cofibrations and generating trivial cofibrations with presentable source. Then CatS.V/ is a model category where an enriched functor FWC!D is a weak equivalence (resp. fibration) if F.x;y/WC.x;y/!D.x;y/ is a weak equivalence (resp. fibration) in V for all x;y2S , and similarly for A1–CatS.V/ . Moreover, these model categories are right proper (resp. combinatorial) provided Vis. The following corollary also uses Theorem 1.3. Corollary 10.5 In the conditions of the previous corollary, assume in addition that V is left proper and the monoidal unit IV is cofibrant. Then the pullback functor  from enriched categories to enriched A1 –categories and the strictification functor  in the other direction form a Quillen equivalence A1–CatS.V/ //CatS.V/:  oo Algebraic & Geometric Topology, Volume 11 (2011)