Homotopy theory of non-symmetric operads, II: Change of base category and left properness
Abstract
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.
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msp Algebraic & Geometric Topology 14 (2014) 229–281 Homotopy theory of non-symmetric operads, II: Change of base category and left properness FERNANDO MURO We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads. 18D50, 55U35; 18G55 1 Introduction In this paper, we continue the study of the homotopy theory of non-symmetric operads started in [11]. We deal with two new topics: change of base category and left properness. Given a Quillen equivalence FWVWWG between symmetric monoidal model categories, we wish to have a Quillen equivalence Op.V/Op.W/ between their operad categories. The conditions so that FaG induces an adjoint pair on operads are too strong. They are not satisfied in many cases of interest, such as the Dold–Kan equivalence Mod.k/op Ch.k/0 between simplicial modules over a commutative ring k and non-negative chain complexes; see Schwede and Shipley [15]. They are neither satisfied in one step of the four-step zig-zag of Quillen equivalences between unbounded chain complexes Ch.k/ and modules Mod.Hk/ over the Eilenberg–MacLane symmetric ring spectrum Hk; see Shipley [16]. Schwede and Shipley found in [15] sufficient conditions to obtain a Quillen equivalence Mon.V/Mon.W/ between monoid categories. We construct, under the same hypotheses, a Quillen equivalence between operads. Theorem 1.1 Let FWVWWG be a weak symmetric monoidal Quillen equivalence between cofibrantly generated closed symmetric monoidal model categories with cofibrant tensor units. Suppose V and W satisfy the monoid axiom and have sets Published: 9 January 2014 DOI: 10.2140/agt.2014.14.229
230 Fernando Muro of generating (trivial) cofibrations with presentable sources. Then, there is a Quillen equivalence between their model categories of operads Op.V/Foper //Op.W/: G oo Here, operads have the model structure in [11, Theorem 1.1], with weak equivalences and fibrations defined as in the underlying category. Although operads are monoids in the category VN of sequences of objects in V with respect to the composition product, the results of Schwede and Shipley do not apply, since this monoidal category is not left closed. If FaG is simply a weak symmetric monoidal Quillen pair, not necessarily an equivalence, there is still a Quillen pair Foper aG defined as in Theorem 1.1; see Proposition 4.1. The functor Foper is not in general given by F levelwise, but their derived functors coincide. Proposition 1.2 Let FWVWWG be a weak symmetric monoidal Quillen adjunction with V and W as in Theorem 1.1. For any operad O in V , there are natural isomorphisms in Ho W,n0, LF.O.n// ŠLFoper.O/.n/: As a consequence of Theorem 1.1, we obtain Quillen equivalences between operads over the aforementioned Quillen equivalent symmetric monoidal model categories. Corollary 1.3 There is a Quillen equivalence Op.Mod.k/op /Op.Ch.k/0/. Corollary 1.4 The operad categories Op.Ch.k// and Op.Mod.Hk// are Quillen equivalent. It is reasonable to wonder whether two operads corresponding under the Quillen equivalence of Theorem 1.1 have Quillen equivalent categories of algebras. The following result answers this question. Theorem 1.5 In the situation of Theorem 1.1: (1) If O is a cofibrant operad in V , there is a Quillen equivalence between model categories of algebras, AlgV.O/ FO//AlgW.Foper.O//: G oo Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 231 (2) If P is a fibrant operad in W such that P.n/ and G.P.n// are cofibrant for all n0, then there is a Quillen equivalence, AlgV.G.P// FP //AlgW.P/: G oo Here, algebras have the model structure in [11, Theorem 1.2], with weak equivalences and fibrations defined as in the underlying category. We will actually prove a more general version of Theorem 1.5 where algebras may live in a different category than the operad; see Theorems 7.2,7.6 and 7.8. Again, if FaG is simply a weak symmetric monoidal Quillen pair, there are still Quillen pairs FOaG and FPaG as in Theorem 1.5; see Propositions 7.1 and 7.5. Despite that the functors FO and FP need not coincide with F on underlying objects, their derived functors agree. Proposition 1.6 Let FWVWWG be a weak symmetric monoidal Quillen adjunction with V and W as in Theorem 1.1. Consider a cofibrant operad O in V and a fibrant operad P in W such that P.n/ and G.P.n// are cofibrant for all n0 . For any O–algebra Athere is a natural isomorphism in Ho W, LF.A/ŠLFO.A/: If FaG is in addition a Quillen equivalence, then for any P –algebra B there is a natural isomorphism in Ho W, LF.B/ŠLFP.B/: We apply the previous results to show homotopy invariance of the associative operad AssV and the unital associative operad uAssV . Algebras over these operads are non-unital monoids and unital monoids, respectively. We actually prove homotopy invariance of the natural map VWAssV!uAssV modelling the forgetful functor from unital to non-unital monoids. Theorem 1.7 Let FWVWWG be a weak symmetric monoidal Quillen adjunction with Vand Was in Theorem 1.1. Consider the derived adjoint pair, Ho Op.V/ LFoper //Ho Op.W/: RG oo There are isomorphisms in Ho Op.W/, LFoper.AssV/ŠAssW;LFoper.uAssV/ŠuAssW; Algebraic & Geometric Topology, Volume 14 (2014)
232 Fernando Muro such that the following square commutes: LFoper.AssV/LFoper.V/ // Š LFoper.uAssV/ Š AssWW //uAssW For weak symmetric monoidal Quillen equivalences, we deduce the following result by adjunction. Corollary 1.8 In the conditions of Theorem 1.1 there are isomorphisms in Ho Op.V/ , AssVŠRG.AssW/; uAssVŠRG.uAssW/; such that the following square commutes: AssVV // Š uAssV Š RG.AssW/RG.W/ //RG.uAssW/ These results have implications for categories of non-unital monoids Monnu.V/D AlgV.AssV/and unital monoids Mon.V/DAlgV.uAssV/. Proposition 1.9 In the situation of Theorem 1.1, we have Quillen equivalences between unital and non-unital monoid categories: Mon.V///Mon.W/; G ooMonnu.V///Monnu.W/: G oo The Quillen equivalence on the left was obtained in [15]. The Quillen equivalence on the right is new but could also have been obtained by the same methods. If we consider algebras over these operads in the category GraphS.V/ of V –graphs with a fixed object set S (see [11, Section 10]), we obtain a similar result for nonunital V –enriched categories Catnu S.V/DAlgGraphS.V/.AssV/ and unital V –enriched categories CatS.V/DAlgGraphS.V/.uAssV/with fixed object set S. Proposition 1.10 In the situation of Theorem 1.1, we have Quillen equivalences between categories of unital and non-unital V –enriched categories with fixed object set S: CatS.V///CatS.W/; G ooCatnu S.V///Catnu S.W/: G oo Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 233 Recall that a model category is left proper if the push-out of a weak equivalence f along a cofibration gis always a weak equivalence f0: X// g // f push Z f0 Y// g0//Y[XZ In general, this property is only satisfied if f is a trivial cofibration or if X and Y are cofibrant. In particular, any model category whose objects are all cofibrant is left proper. The model category of operads Op.V/ is not left proper, even if V is. Nevertheless, we here show the following result along this line. Theorem 1.11 Let V be a cofibrantly generated closed symmetric monoidal model category satisfying the monoid axiom with sets of generating (trivial) cofibrations with presentable sources. Consider a push-out diagram in Op.V/as follows: O// // ' push Q '0 P// 0//P[OQ If O.n/and P.n/are cofibrant in Vfor all n0, then '0is a weak equivalence. If V has a cofibrant tensor unit, the last condition is satisfied by cofibrant operads, but also by many non-cofibrant operads of interest, such as the associative operad AssV and the unital associative operad uAssV . We take advantage of this result in [12], where we show that the natural map VWAssV!uAssV considered above is a homotopy epimorphism for a wide class of base categories V. We would like to stress that Theorem 1.11 does not require left properness for V . This theorem yields left properness for operads in the following special case. Corollary 1.12 Let V be as in Theorem 1.11. Assume all objects in V are cofibrant. Then Op.V/is left proper. This result applies, for instance, to the category Setop of simplicial sets and to the category Ch.k/of unbounded chain complexes over a field k. We have similar results for algebras. Algebraic & Geometric Topology, Volume 14 (2014)
234 Fernando Muro Theorem 1.13 Let V be as in Theorem 1.11. Consider an operad O in V such that O.n/ is cofibrant for all n0 , and a push-out diagram in the category of O –algebras as follows: A// // ' push C '0 B// 0//B[AC If the underlying objects of A and B are cofibrant in V then '0 is a weak equivalence. Corollary 1.14 Let V be as in Theorem 1.11. Assume all objects in V are cofibrant. Then, AlgV.O/is left proper for any operad Oin V. These results will also be proved for algebras living in a different category than the operad; see Theorem 8.1 and Corollary 8.2. The reader must have noticed the cofibrancy hypotheses on tensor units in many results above. In the appendices, we show how to get rid of them, still imposing some (weaker) hypotheses. The tensor unit is not cofibrant in some examples of interest, such as symmetric spectra and other diagram spectra with the positive stable model structure (Mandell, May, Schwede and Shipley [10]). Positive stable model structures are very important, actually unavoidable in brave new algebraic geometry, where they must be used in order to have transferred model structures on commutative monoids (Toën and Vezzosi [17]). In [13] we construct moduli spaces of algebras over an operad in brave new algebraic geometry. This is why we need to avoid cofibrancy hypotheses on tensor units. In Appendix A we isolate a class of objects, that we call pseudo-cofibrant, which share many properties with cofibrant objects. The tensor unit is always pseudo-cofibrant, and the components O.n/,n0, of a cofibrant operad Oare also pseudo-cofibrant. Pseudo-cofibrant objects share even more properties with cofibrant objects when the strong unit axiom is satisfied. This axiom holds, for instance, whenever cofibrant objects are flat, ie, if tensoring with a cofibrant object preserves weak equivalences. This is a rather common property, satisfied by symmetric spectra with the positive model structure. Appendix B analyzes a relevant subclass of pseudo-cofibrant objects: the I –cofibrant objects, ie, those objects Xadmitting a cofibration IXfrom the tensor unit. Weak monoidal Quillen adjunctions must satisfy two axioms in order to extend the previous results: the pseudo-cofibrant and the I –cofibrant axioms, introduced in Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 235 Appendix B. These axioms mitigate the fact that left Quillen functors need not preserve pseudo-cofibrant objects and weak equivalences between them. The three new axioms are obviously satisfied when tensor units are cofibrant. They are also satisfied in many other examples related to spectra where tensor units are not cofibrant. The statements of our main results with these weaker hypotheses are in Appendices C,Dand E. The paper is structured as follows. We first prove change of base category and left properness results for operads and then for algebras. The homotopy invariance of the (unital) associative operad comes later, just before the appendices. We intercalate three sections with background on monoidal model categories, operads, and algebras over operads. We assume the reader is familiar with category theory and abstract homotopy theory. Some standard references are Mac Lane [9], Hovey [7] and Hirschhorn [6]. For monoidal categories, functors, and adjunctions, we refer to Aguiar and Mahajan [1, Chapter 3]. Acknowledgements The author is grateful to David White for Lemma A.11, suggested as an answer to a question of the author on MathOverflow. The author was partially supported by the Andalusian Ministry of Economy, Innovation and Science under the grant FQM-5713, by the Spanish Ministry of Education and Science under the MEC-FEDER grant MTM2010-15831 and by the Government of Catalonia under the grant SGR-119-2009. 2 Monoidal model categories We here recall the compatibility conditions between monoidal and model structures introduced in Hovey [7], Schwede and Shipley [14] and Schwede and Shipley [15]. See also [11] for the non-symmetric version of the monoid axiom. The tensor product and tensor unit of a monoidal category C will be denoted by ˝ and I , respectively. We will sometimes write ˝C and IC if we wish to distinguish between different monoidal categories. Initial objects will be denoted by ¿. Definition 2.1 Given a category C , the category of morphisms C2 is the category of functors 2!C , where 2 is the category with two objects, 0 and 1 , and only one Algebraic & Geometric Topology, Volume 14 (2014)
236 Fernando Muro non-identity morphism 0!1 , ie, it is the poset f0<1g . A morphism fWU!V in C is identified with the functor fW2!C defined by f .0/DU , f .1/DV and f .0!1/Df. If C is a cocomplete biclosed (symmetric) monoidal category, then C2 is biclosed (symmetric) monoidal with respect to the push-out product of morphisms. Given morphisms fWU!V and gWX!Y in C , the push-out product fˇg is defined by the following diagram: (2-1) U˝X push f˝X // U˝g V˝X xg V˝g U˝Y x f // f˝Y00 U˝Y[U˝XV˝X fˇg ((V˝Y Notice that fˇ.¿!X/Df˝X and .¿!X/ˇfDX˝f . The unit object in C2is ¿!I. Remark 2.2 Given morphisms fiWUi!Vi in C , 1in , the iterated push-out product f1ˇˇfncan be constructed as follows. Consider the diagram f1˝˝fnW2n! C: Then f1ˇˇfnis the morphism colim 2nnf.1;n :::;1/g f1˝˝fn! colim 2nf1˝˝fnDV1˝˝Vn induced by the inclusion of the full subcategory 2nn f.1;n : : : ; 1/g 2n obtained by removing the final object of 2n, which is the n–dimensional cube category. Definition 2.3 Amonoidal model category is a biclosed monoidal category C endowed with a model structure such that the following axioms hold: (1) Push-out product axiom If f and g are cofibrations then fˇg is also a cofibration. Moreover, if in addition f or g is a trivial cofibration then fˇg is a trivial cofibration. (2) Unit axiom If X is a cofibrant object and qWz I I is a cofibrant resolution of the tensor unit, then X˝qand q˝Xare weak equivalences. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 237 These two axioms imply that the homotopy category Ho C is biclosed monoidal; see [7]. In order to have transferred model structures on monoids, operads, algebras over operads, etc we need to impose the following extra axiom (see [14;11]): (3) Monoid axiom Relative K0–cell complexes are weak equivalences, where K0D(f1ˇˇfnW n1;¿¤S f1;:::;ng; fiis a trivial cofibration if i2S; fiW¿!Xifor some object Xiin Cif i…S:) This axiom is usually not incorporated in the definition of monoidal model category but we do include it, since we will always need it. We will also assume that all monoidal model categories are cofibrantly generated. Moreover, we will suppose that there is a set of generating cofibrations I and a set of generating trivial cofibrations J with presentable sources. Asymmetric monoidal model category V is a monoidal model category as above whose underlying monoidal category is symmetric. In this case, (3) can be replaced with: .30/Monoid axiom Relative K–cell complexes are weak equivalences, where KD ff˝XWfis a trivial cofibration and Xis an object in Vg: Remark 2.4 If the unit axiom holds for a certain cofibrant resolution of I then it holds for any cofibrant resolution of I. It is obviously satisfied if Iis cofibrant. In many examples, tensoring with a cofibrant object preserves weak equivalences. This is a usual condition to ensure that a weak equivalence of monoids induces a Quillen equivalence between their module categories; see [14]. This condition also implies the unit axiom. The push-out product axiom has many immediate consequences that the reader may work out, eg cofibrant objects are closed under tensor products, cofibrations between cofibrant objects are closed under push-out products, weak equivalences between cofibrant objects are closed under tensor products, etc. The following result is another straightforward consequence of the push-out product axiom. Lemma 2.5 Given cofibrations between cofibrant objects f1; : : : ; fn in a monoidal model category C, the diagram f1˝˝fnW2n! Cis Reedy cofibrant in C2n. Algebraic & Geometric Topology, Volume 14 (2014)
244 Fernando Muro Proposition 4.2 Let FWVWWG be a weak symmetric monoidal Quillen adjunction. Suppose that the tensor units of V and W are cofibrant. If O is cofibrant in Op.V/ then Ois a weak equivalence in WN. Notice that Proposition 1.2 is a corollary of this result. In the proof of Proposition 4.2, we need the following technical result. Lemma 4.3 Let FWCDWG be a weak monoidal Quillen adjunction. Consider cofibrations with cofibrant source and target f1; : : : ; fn and cofibrant objects X1;:::;Xm in C. The morphism in D2 Fn K iD1 fi˝ m O jD1 Xj! n K iD1 F.fi/˝ m O jD1 F.Xj/ defined by the comultiplication of the colax monoidal functor F is a weak equivalence in D2. Proof Denote fiWUiVi , 1in . The target of the morphism in the statement is the iterated comultiplication Fn O iD1 Vi˝ m O jD1 Xj! n O iD1 F.Vi/˝ m O jD1 F.Xj/; which is a weak equivalence by Lemma 2.7. The difficult part of this proof is to show that the source is also a weak equivalence. The diagram n O iD1 fi˝ m O jD1 XjW2n! C is Reedy cofibrant in C2nby Lemma 2.5. In particular, Fn O iD1 fi˝ m O jD1 XjW2n! D is Reedy cofibrant in D2n by [6, Proposition 15.4.1(1)], since left Quillen functors preserve cofibrant objects. Moreover, n O iD1 F.fi/˝ m O jD1 F.Xj/W2n! D Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 245 is also Reedy cofibrant in D2n by Lemma 2.5. The restrictions to 2nn f.1;n : : : ; 1/g are also Reedy cofibrant; see [6, Lemma 15.3.7. (1)]. Let WFn O iD1 fi˝ m O jD1 XjH) n O iD1 F.fi/˝ m O jD1 F.Xj/W2n! D be the natural transformation defined pointwise by the iterated comultiplication weak equivalence in Lemma 2.7. The categories 2n and 2nn f.1;n : : : ; 1/g are direct in the sense of [7, Definition 5.1.1]. Therefore, colim2nnf.1;n :::;1/g is a weak equivalence by [7, Corollary 5.1.6], since left Quillen functors preserve weak equivalences between cofibrant objects. This finishes the proof, since colim2nnf.1;n :::;1/g is the source of the morphism in the statement; compare Remark 2.2. Remark 4.4 Lemma 4.3 is obviously also true if we reorder the tensor and push-out product factors, ie, we may have .X1˝f1ˇf2/ˇ.X2˝f3˝X3˝X4/ˇf4ˇf5: Proof of Proposition 4.2 If ODIıV is the initial operad in V then Foper.IıV/DIıW is the initial operad in W , since Foper is a left adjoint. Moreover, the morphism IıVWF.IıV/!IıW is the counit F.IV/!IW of F in arity 1 . This counit is a weak equivalence by Definition 2.6(2), since IV is cofibrant. In arities n¤1 , IıV is the identity morphism on the initial object. Hence IıVis a weak equivalence. Consider a push-out in Op.V/, FV.U/ push // FV.f / // g FV.V/ g0 O// f0//P with f a cofibration in VN and O a cofibrant operad such that O is a weak equivalence. We are going to show that Pis also a weak equivalence in WN. Let Oxg Uf V be the diagram adjoint to the left upper corner. We can suppose without loss of generality that U.n/ is cofibrant for all n0 . This happens, for instance, if U is the underlying Algebraic & Geometric Topology, Volume 14 (2014)
246 Fernando Muro sequence of a cofibrant operad, such as O ; see Corollary 3.8. If this condition did not hold, we could replace fwith z fin the following push-out diagram in VN, U// f // xg push V O// z f //O[UV Indeed, if xg0WV!Pis adjoint to g0then FV.O/ push // FV.z f / // adjoint to 1O FV.O[UV/ adjoint to .1O;xg0/ O// f0//P is also a push-out in Op.V/. Recall from Remark 3.3 the decomposition of the morphism of sequences underlying f0 as a transfinite (countable) composition. This transfinite composition consists of cofibrations by Lemma 3.5. Since F is a left Quillen functor, F.f 0/ is the transfinite composition of F.O/DF.P0/////: : : ////F.Pt1/// F.'t/ //F.Pt/////: : : ; where F.'t.n//,n0, is a push-out along the cofibration (4-2) a T FK v2Ie.T/ f .zv/ ˝O w2Io.T/ O.zw/: The cofibrant operad Foper.P/fits into the following push-out diagram: FW.F.U// FW.F.f // // Foper.g/ FW.F.V// Foper.g0/ Foper.O/Foper.f 0/ //Foper.P/ Hence, the morphism of sequences underlying Foper.f 0/ is the transfinite composition of Foper.O/Dz P0////: : : ////z Pt1// z't//z Pt////: : : ; Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 247 where z't.n/,n0, is a push-out along the cofibration (4-3) a TK v2Ie.T/ F.f .zv// ˝O w2Io.T/ Foper.O.zw//: The morphism Pis the colimit of the diagram: F.O/ O F.P0///// 0 O ////F.Pt1/// F.'t/ // t1 O F.Pt///// t O Foper.O/z P0//// ////z Pt1// z't //z Pt//// The morphism t O.n/ , n0 , is obtained from the previous one by taking push-out of horizontal arrows in the following diagram: F.Pt1/.n/ t1 O.n/ oo//(4-2) // z Pt1.n/ oo//(4-3) // The commutative square on the right is a weak equivalence (4-2) ! (4-3) in W2 defined as follows. It is a coproduct `T of weak equivalences between cofibrant objects. Each factor of the coproduct decomposes as: FK v2Ie.T/ f .zv/ ˝O w2Io.T/ O.zw/ K v2Ie.T/ F.f .zv// ˝O w2Io.T/ F.O.zw// K v2Ie.T/ F.f .zv// ˝O w2Io.T/ Foper.O/. zw/ Here, the first arrow is the weak equivalence in Lemma 4.3. The second weak equivalence is K v2Ie.T/ F.f .zv// ˝O w2Io.T/ O.zw/: Algebraic & Geometric Topology, Volume 14 (2014)
248 Fernando Muro All these objects are indeed cofibrant, since the cofibration f has cofibrant source and the sequences F.O/and Foper.O/are cofibrant by Corollary 3.8. The cube lemma [7, Lemma 5.2.6]shows that if t1 O.n/ is a weak equivalence then t O.n/ is also a weak equivalence. Since 0 ODO is a weak equivalence by hypothesis, we deduce that t O is a weak equivalence for all t0 . Therefore, the colimit PDcolimtt Ois a weak equivalence by [6, Proposition 15.10.12(1)]. Now, a standard inductive argument shows that O is a weak equivalence for any FV.IN/ –cell complex O , and hence for any cofibrant operad O by the usual retract argument. Since O is a natural morphism of sequences, in order for the inductive argument to work we should check that any F.IN/ –cell complex is a transfinite composition of cofibrations of sequences with cofibrant starting sequence. The starting sequence is the initial operad, which is cofibrant because tensor units are cofibrant, and bonding morphisms are cofibrations of sequences by Corollary 3.6. Proof of Theorem 1.1 Let 'WFoper.O/!P be a morphism in Op.W/ with fibrant target such that O is cofibrant in Op.V/ . We must show that ' is a weak equivalence if and only if its adjoint '0WO!G.P/is a weak equivalence. Consider the following morphisms of sequences in W: F.O/O ! Foper.O/' ! P: Here O is a weak equivalence by Proposition 4.2. Recall that a morphism of operads is a weak equivalence (resp. fibration) if and only if the underlying morphism of sequences if a weak equivalence (resp. fibration). Hence ' is a weak equivalence if and only if 'O is a weak equivalence. Moreover, the components of P are fibrant. Furthermore, the components of O are cofibrant by Corollary 3.8. Therefore, since FaG is a Quillen equivalence, 'O is a weak equivalence if and only if its adjoint along FaG , ie, the morphism of sequences underlying '0WO!G.P/ , is a weak equivalence. Hence we are done. 5 Left properness for operads In this section we prove Theorem 1.11 and deduce a gluing lemma for weak equivalences in Op.V/. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 249 Proof of Theorem 1.11 Assume first that fits into a push-out diagram in Op.V/, F.U/ push // F.f / // g F.V/ g0 O// //Q; where f is a cofibration in VN . In this case, 0 fits into the following push-out diagram: F.U/ push // F.f / // 'g F.V/ '0g0 P// 0//P[OQ We can suppose that U.n/ is cofibrant for all n0 ; compare the proof of Proposition 4.2. Recall from Remark 3.3 that the morphisms of sequences underlying and 0 decompose as transfinite compositions of ODQ0 Qt1 t Qt ;PDR0 Rt1 0 t Rt ; where t.n/,n0, is a push-out along (5-1) a TK v2Ie.T/ f .zv/ ˝O w2Io.T/ O.zw/; and 0 t.n/,n0, is a push-out along (5-2) a TK v2Ie.T/ f .zv/ ˝O w2Io.T/ P.zw/: All these morphisms are cofibrations by Lemma 3.5. The morphism of sequences underlying '0is the colimit of: O ' Q0//// '0 ////Qt1// t// 't1 Qt//// 't PR0//// ////Rt1// 0 t //Rt//// Algebraic & Geometric Topology, Volume 14 (2014)
250 Fernando Muro Here, the morphism 't.n/ , n0 , is obtained from the previous one by taking push-out of horizontal arrows in the following diagram: Qt1.n/ 't1.n/ oo//(5-1) // Rt1.n/ oo//(5-2) // The square on the right is the following weak equivalence (5-1) ! (5-2) in V2: a TK v2Ie.T/ f .zv/ ˝O w2Io.T/ '. zw/: This is indeed a weak equivalence, since each factor of the coproduct is a weak equivalence with cofibrant source and target. Here we use that the cofibration f has cofibrant source and that the underlying sequences of O and P are assumed to be cofibrant. The cube lemma [7, Lemma 5.2.6]shows that if 't1.n/ is a weak equivalence then 't.n/ is also a weak equivalence. Since '0D' is a weak equivalence by hypothesis, we deduce that 't is a weak equivalence for all t0 . Therefore, the colimit '0Dcolimt't is a weak equivalence by [6, Proposition 15.10.12(1)]. Now, an inductive argument as in the proof of Proposition 4.2 shows that Theorem 1.11 is true if is a relative F.IN/ –cell complex. We then deduce that Theorem 1.11 is true for any cofibration by the usual retract argument. Theorem 1.11 is an important ingredient to show that operads with underlying cofibrant sequence satisfy the axioms of a cofibration category in the sense of Baues [3, I.1.1]. Proposition 5.1 Let V be a symmetric monoidal model category. The full subcategory Opc.V/Op.V/ spanned by the operads O such that O.n/ is cofibrant for all n0 is a cofibration category. Proof Axiom (C1) follows since Opc.V/ is a full subcategory of a model category. Given a push-out diagram in Op.V/, O// // ' push Q '0 P// 0//P[OQ; Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 251 if O and P are in Opc.V/ then so are Q and P[OQ by Corollary 3.7. Hence (C2)(a) follows since any push-out of a trivial cofibration in a model category is a trivial cofibration, and (C2)(b) follows from Theorem 1.11. Axioms (C3) and (C4) also follow easily from Corollary 3.7 and well known properties of model categories. The following result holds in any cofibration category; see [3, II.1.2 (b)]. Corollary 5.2 (Gluing lemma) Let V be a symmetric monoidal model category. Consider a commutative diagram in Opc.V/as follows: P O oo// Q P0O0 oo//Q0 If at least one of the morphisms in each row is a cofibration, then the induced morphism P[OQ!P0[O0Q0is a weak equivalence. 6 Algebras We here recall some facts about algebras over operads. See [11] for details. Definition 6.1 Let V be a closed symmetric monoidal category with coproducts and C a biclosed V –algebra with coproducts. An operad O in V gives rise to a monad FOWC!Cgiven by FO.X/Da n0 z.O.n// ˝X˝n: The monad operations F2 O)FO and 1C)FO are defined by the lax monoidal structure of z (see Definition 2.8) and by the multiplications and the unit of the operad O. An O–algebra in Cis an algebra over this monad. We denote AlgC.O/the category of O–algebras in C. We also denote by FO the free O –algebra functor, left adjoint to the forgetful functor AlgC.O/!C: C FO//AlgC.O/ forget oo Algebraic & Geometric Topology, Volume 14 (2014)
252 Fernando Muro Remark 6.2 In a more explicit fashion, an O –algebra is an object A in C together with structure morphisms A nWz.O.n//˝A˝n!A , n0 , satisfying certain relations. An O –algebra morphism fWA!B is a morphism in C compatible with the structure morphisms. Remark 6.3 The initial O–algebra is z.O.0// with the following structure: z.O.0// nWz.O.n// ˝z.O.0//˝nmult. of z//z.O.n/˝O.0/˝n/z.nI0;:::;0/ //z.O.0// Remark 6.4 Suppose that the V –algebra C is complete and cocomplete. Then the category of algebras AlgC.O/ in C over an operad O in V is also complete and cocomplete. Limits and filtered colimits in AlgC.O/ are easy, since they are computed in C . Push-outs in AlgC.O/ are very complicated in general, but push-outs along free maps were carefully analyzed in [11, Section 8]: (6-1) FO.U/ push FO.f / // g FO.V/ g0 Af0//B The underlying morphism of f0 in C is a transfinite (actually countable) composition of morphisms ADB0 '1 ! B1! ! Bt1 't ! Bt! ; where 'tWBt1!Btis a push-out along (6-2) a n1a Sf1;:::;ng card.S/Dt z.O.n// ˝K S f˝O f1;:::;ngnS A: Here we are abusing terminology, since the tensor and push-out product factors should be ordered according to the ordering of f1;:::;ng if C is non-symmetric. Nevertheless, we will keep this notation throughout this paper, so as to simplify the exposition. We will not recall the explicit definition of the map from the source of (6 - 2) to Bt1 ; see [11, Lemma 8.1]for details. Remark 6.5 We showed in [11, Theorem 1.2]that if C is a model V –algebra with sets of generating cofibrations I0 and generating trivial cofibrations J0 with presentable sources, then AlgC.O/ has a cofibrantly generated model structure transferred along the free O –algebra adjunction, ie, an O –algebra morphism is a weak equivalence or fibration if and only if the underlying morphism in C is one. Moreover, FO.I0/ and FO.J0/are sets of generating (trivial) cofibrations of AlgC.O/. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 253 The following result is similar to Lemma 3.5. Lemma 6.6 Consider a model V –algebra C . If f is a cofibration in C , O is an operad in V such that z.O.n// is cofibrant for all n0 , and A is an O –algebra with underlying cofibrant object in C, then (6-2) is a cofibration in C. In practical examples, z.O.n// is cofibrant because O.n/is cofibrant, n0. We recall [11, Theorem 1.3]. Theorem 6.7 Let C be a model V –algebra. Consider a weak equivalence 'WO ! P between operads in V such that the objects O.n/ and P.n/ are cofibrant for all n0 . Then the change of coefficients Quillen pair is a Quillen equivalence: AlgC.O/ '//AlgC.P/ ' oo In [11, Theorem 1.3]we further assumed C to be left proper, but this is actually unnecessary since the cube lemma [7, Lemma 5.2.6]applies where we invoked left properness, in the proof of [11, Lemma 9.6]. We should also remark that in that proof we can suppose, without loss of generality, that Y is a cofibrant object in C , applying if necessary the same trick as in the proof of Proposition 4.2 above. This trick can be applied since, under the hypotheses of Theorem 6.7, any cofibrant O –algebra has a cofibrant underlying object in C; see [11, Lemma 9.4]. This kind of result goes back to Berger and Moerdijk [4, Theorem 4.4]. 7 Change of base category for algebras In this section we prove Theorem 1.5 and Proposition 1.6. Actually, we show more general versions of these results. Throughout this section, we place ourselves in the following context. We have a diagram VF// zC W G oo zD C x F//D x G oo where C is a model V –algebra, D is a model W –algebra, the Quillen pairs FaG and x Fax G are colax-lax (symmetric) monoidal adjunctions, and there is a fixed natural transformation (7-1) .X/Wx FzC.X/! zDF.X/; Algebraic & Geometric Topology, Volume 14 (2014)
260 Fernando Muro Proof Let BWx F.B/!x FP.B/ be the adjoint of the unit of x FPax G along x Fax G . It is enough to show that B is a weak equivalence for B a cofibrant G.P/ –algebra in C . Let qWx FP.B/ C be a fibrant replacement in AlgD.P/ . By the two-out-of-three axiom, B is a weak equivalence if and only if qB is a weak equivalence. Since x Fax G is a Quillen equivalence, qB is a weak equivalence if and only if its adjoint along x Fax G is a weak equivalence. The adjoint of qB is a weak equivalence by [7, Proposition 1.3.13(b)], since x FPax Gis a Quillen equivalence. In the special case that VDW and FaG is the identity adjunction, Theorems 7.2 and 7.6 admit the following common generalization. Theorem 7.8 Suppose VDWhas a cofibrant tensor unit, FDGD1V,x Fax Gis a weak monoidal Quillen equivalence, and O is an operad in V with cofibrant underlying sequence. Then AlgC.O/ x FPDx FP //AlgD.O/ x G oo is a Quillen equivalence. Proof Let 'Wz O O be a cofibrant resolution. Consider the following diagram of Quillen pairs: AlgC.z O/ x Fz ODx Fz O // ' OO ' AlgD.z O/ x G oo ' OO ' AlgC.O/ x FODx FO //AlgD.O/ x G oo The right adjoints commute, 'x GDx G' , hence the left adjoints commute up to natural isomorphism. The vertical Quillen pairs are Quillen equivalences by Theorem 6.7, since under our hypotheses any cofibrant operad has an underlying cofibrant sequence; see Corollary 3.8. Moreover, the upper horizontal Quillen pair is a Quillen equivalence by Theorem 7.2. Hence the bottom horizontal Quillen pair is also a Quillen equivalence. We omit the proof of the following corollary, which is very similar to the proofs of Corollaries 7.4 and 7.7. Corollary 7.9 Under the assumptions of Theorem 7.8, for any O –algebra B in C there is a natural isomorphism in Ho D, Lx F.B/ŠLx FO.B/DLx FO.B/: Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 261 8 Left properness for algebras In this section we prove the following stronger version of Theorem 1.13. Theorem 8.1 Let C be a model V –algebra. Consider an operad O in V such that z.O.n// is cofibrant for all n0, and a push-out diagram in AlgC.O/as follows: A// // ' push C '0 B// 0//B[AC If the underlying objects of A and B are cofibrant in C then '0 is a weak equivalence. Proof This proof follows the same pattern as the proof of Theorem 1.11 in Section 5 above. Assume first that fits into a push-out diagram in AlgC.O/ FO.U/ push // FO.f / // g FO.V/ g0 A// //C; where f is a cofibration in C . In this case, 0 fits into the following push-out diagram: FO.U/ push // FO.f / // 'g FO.V/ '0g0 B// 0//B[AC We can suppose that Uis cofibrant; compare the proof of Proposition 4.2. Recall from Remark 6.4 that the morphisms underlying and 0 decompose as transfinite compositions, ADC0 Ct1 t Ct ;BDD0 Dt1 0 t Dt ; where tis a push-out along (8-1) a n1a Sf1;:::;ng card.S/Dt z.O.n// ˝K S f˝O f1;:::;ngnS A; Algebraic & Geometric Topology, Volume 14 (2014)
262 Fernando Muro and 0 tis a push-out along (8-2) a n1a Sf1;:::;ng card.S/Dt z.O.n// ˝K S f˝O f1;:::;ngnS B: All these morphisms are cofibrations by Lemma 6.6. The morphism of sequences underlying '0is the colimit of: A ' C0//// '0 : : : ////Ct1// t// 't1 Ct//// 't : : : B D0////: : : ////Dt1// 0 t //Dt////: : : Here, the morphism 't is obtained from the previous one by taking push-out of the horizontal arrows in the following diagram: Ct1 't1 oo//(8-1) // Dt1 oo//(8-2) // The square on the right is the following weak equivalence (8-1) ! (8-2) in C2, a n1a Sf1;:::;ng card.S/Dt z.O.n// ˝K S f˝O f1;:::;ngnS ': This is indeed a weak equivalence, since each factor of the coproduct is a weak equivalence with cofibrant source and target. The cube lemma [7, Lemma 5.2.6]shows that if 't1 is a weak equivalence then 't is also a weak equivalence. Since '0D' is a weak equivalence by hypothesis, we deduce that 't is a weak equivalence for all t0 . Therefore, the colimit '0Dcolimt't is a weak equivalence by [6, Proposition 15.10.12(1)]. Now, a standard inductive argument shows that the statement is true for any relative FO.I0/ –cell complex , and hence for any cofibration by the usual retract argument. The following corollary generalizes Corollary 1.14. Corollary 8.2 Let C be a model V –algebra. If all objects in C are cofibrant then, for any operad Oin V, the category AlgC.O/is left proper. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 263 The following result can be proved as Proposition 5.1, replacing Corollary 3.7 and Theorem 1.11 with [11, Corollary 9.5]and Theorem 8.1, respectively. Proposition 8.3 Let C be a model V –algebra and O an operad in V such that O.n/ is cofibrant for all n0 . The full subcategory Algc C.O/AlgC.O/ spanned by the O–algebras whose underlying object in Cis cofibrant is a cofibration category. In particular, the gluing lemma also holds for Algc C.O/ in these circumstances; compare Corollary 5.2. 9 Homotopy invariance of the (unital) associative operad Here we prove Theorem 1.7,Corollary 1.8, and Propositions 1.9 and 1.10. Let V be a closed symmetric monoidal category with coproducts. The unital associative operad uAssV is defined by uAssV.n/DI for all n0 , the unit I!uAssV.1/ is the identity morphism in I , and all composition laws are given by the unit isomorphism I˝IŠI . Algebras over this operad are monoids. The associative operad AssV is identical to uAssV except for arity zero, AssV.0/D¿ . The operad structure is determined by the existence of an operad morphism VWAssV!uAssV , which is the identity in positive arities. Algebras over this operad are non-unital monoids. Moreover, the functor .V/ induced on algebras by the morphism V is the usual forgetful functor from unital monoids to non-unital monoids. If V is a symmetric monoidal model category, a (unital) A –infinity operad .u/AV 1 is a cofibrant resolution of .u/AssV in Op.V/ . Here we mean cofibrant resolution in the following weak sense: we have weak equivalences with cofibrant sources, qWuAV 1 ! uAssV;q0WAV 1 ! AssV; we do not ask them to be fibrations. A (unital) A –infinity algebra or (unital) A –infinity monoid is a .u/AV 1–algebra. Let FWVWWG be a colax-lax symmetric monoidal adjunction between closed symmetric monoidal categories with coproducts. The (unital) associative operad is very special since the unit IV!G.IW/ of the lax symmetric monoidal functor G induces morphisms in Op.V/, 'WuAssV! G.uAssW/; '0WAssV! G.AssW/: Algebraic & Geometric Topology, Volume 14 (2014)
264 Fernando Muro Theorem 9.1 Let FWVWWG be a weak symmetric monoidal Quillen adjunction. Suppose Vand Whave cofibrant tensor units. Then the morphisms WFoper.uAV 1/! uAssW; 0WFoper.AV 1/! AssW; adjoint to 'qand '0q0, respectively, are weak equivalences in Op.W/. Proof We use the natural morphism (4-1). For any n0, the composite morphism F.uAV 1.n// uAV 1.n/ ! Foper.uAV 1/.n/ .n/ ! uAssW.n/DIW coincides with F.uAV 1.n// F.q.n// ! F.uAssV.n// DF.IV/counit of F ! uAssW.n/DIW: The counit of F is a weak equivalence by Definition 2.6(2), and F.q.n// is a weak equivalence since q.n/ is a weak equivalence between cofibrant objects; see Corollary 3.8. Moreover, uAV 1.n/ is a weak equivalence by Proposition 4.2. Hence, .n/ is a weak equivalence by the two-out-of-three axiom. One can similarly check that 0.n/is a weak equivalence for n>0. In arity zero, F.AV 1.0// AV 1.0/ ! Foper.AV 1/.0/ 0.0/ ! AssW.0/D¿ coincides with F.AV 1.0// F.q0.0// ! F.AssV.0// DF.¿/D¿DAssW.0/: This last morphism is a weak equivalence since q0.0/ is a weak equivalence between cofibrant objects, and AV 1.0/ is a weak equivalence by Proposition 4.2. Hence, 0.0/ is also a weak equivalence by the two-out-of-three axiom. Theorem 1.7 is a corollary of the previous result, since the following square commutes: AssVV // '0 uAssV ' G.AssV/G.V/ //G.uAssV/ Monoids and A –infinity monoids, unital and non-unital, have Quillen equivalent model categories. In other words, A–infinity monoids can be rectified. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 265 Proposition 9.2 Let C be a model V –algebra. Assume that the tensor unit of V is cofibrant. Then qand q0induce Quillen equivalences AlgC.uAV 1/ q//AlgC.uAssV/; q ooAlgC.AV 1/ q0 //AlgC.AssV/: .q0/ oo This is a corollary of Theorem 6.7. It allows the proof of the following corollary, from which Propositions 1.9 and 1.10 follow. Corollary 9.3 Let us place ourselves in the situation described in the first paragraph of Section 7. Suppose that FaG and x Fax G are weak (symmetric) monoidal Quillen equivalences, and that V and W have cofibrant tensor units. Then, there are Quillen equivalences, AlgC.uAssV///AlgD.uAssW/; x G ooAlgC.AssV///AlgD.AssW/: x G oo Proof Consider the following diagram of Quillen pairs: AlgC.uAV 1/ x FuAV 1// .qV/ OO q V AlgD.Foper.uAW 1// x G oo OO AlgC.uAssV///AlgD.uAssW/ x G oo The bottom x G is the usual functor between categories of monoids induced by a lax monoidal functor. This functor preserves (trivial) fibrations and has a left adjoint for abstract reasons, hence it is a right Quillen functor. The square formed by the right adjoints commutes, so left adjoints commute up to a natural isomorphism. The left vertical pair is a Quillen equivalence by Proposition 9.2. The right vertical pair is a Quillen equivalence by Theorems 9.1 and 6.7. The upper horizontal pair is a Quillen equivalence by Theorem 7.2. Hence, the bottom Quillen pair is also a Quillen equivalence. The proof of the non-unital case is formally the same. Algebraic & Geometric Topology, Volume 14 (2014)
266 Fernando Muro Appendix A: Pseudo-cofibrant objects In the series of appendices that we now start, we develop technical tools to avoid unnecessary cofibrancy hypotheses on tensor units. With this purpose, in this first appendix we isolate a certain property of cofibrant objects. The objects satisfying this property are called pseudo-cofibrant. We show that there are many pseudo-cofibrant objects, including the tensor unit, which are possibly non-cofibrant. Moreover, we prove that pseudo-cofibrant objects share many other features with cofibrant objects. Definition A.1 Apseudo-cofibrant object X in a monoidal model category C is an object such that given a cofibration fWUV , the morphisms f˝X and X˝f are cofibrations. Remark A.2 Some examples of pseudo-cofibrant objects are: (1) Cofibrant objects, by the push-out product axiom. (2) The tensor unit I. (3) Coproducts of pseudo-cofibrant objects. (4) Tensor products of pseudo-cofibrant objects. Tensoring with pseudo-cofibrant objects also preserves trivial cofibrations, by the monoid axiom. Therefore, pseudo-cofibrant objects are those objects X such that ˝ Xand X˝ are left Quillen functors. As it usually happens in cofibrantly generated model categories, it is enough to check the property defining pseudo-cofibrant objects on generating (trivial) cofibrations. Lemma A.3 An object X is pseudo-cofibrant if and only if f˝X and X˝f are cofibrations for any generating (trivial) cofibration f. Proof The “only if” part is trivial. For the “if” part, assume X˝ takes generating (trivial) cofibrations to cofibrations. The monoid axiom shows that it actually takes generating trivial cofibrations to trivial cofibrations. Hence it is a left Quillen functor [7, Lemma 2.1.20]. Similarly for ˝ X . Therefore X is pseudo-cofibrant by the previous remark. The following result is a source of examples of pseudo-cofibrant objects. Lemma A.4 If gWXY is a cofibration and X is pseudo-cofibrant then Y is also pseudo-cofibrant. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 267 Proof Let fWUV be a cofibration. We are going to prove that f˝Y is a cofibration. One can similarly check that Y˝fis also a cofibration. Consider the diagram (2 - 1) for the construction of fˇg . The morphism f˝X is a cofibration since X is pseudo-cofibrant, hence x f is also a cofibration. Moreover, fˇg is a cofibration by the push-out product axiom, so f˝YD.f ˇg/x f is a cofibration. Corollary A.5 Given morphisms Y XZ , if Y is pseudo-cofibrant then so is Y[XZ. Proof Simply recall that the canonical morphism YY[XZis a cofibration. Corollary A.6 If fWUV and gWXY are cofibrations between pseudocofibrant objects, then the cofibration fˇg has pseudo-cofibrant source, and hence target. Proof The source of fˇgis the push-out of U˝Yoo U˝g ooU˝X// f˝X //V˝X: These three objects are pseudo-cofibrant, therefore the previous corollary applies. Pseudo-cofibrant objects become cofibrant if we tensor them with an honestly cofibrant object. Lemma A.7 Given two objects U and X , if one of them is cofibrant and the other one is pseudo-cofibrant then U˝Xis cofibrant. Proof If U is the cofibrant object then .¿ U/˝XD¿ U˝X is a cofibration, ie, U˝Xis cofibrant, and the proof is similar if Xis the cofibrant object. Corollary A.8 If fWUV and gWXY are cofibrations between pseudocofibrant objects and U or X is cofibrant, then the cofibration fˇg has cofibrant source, and hence target. Proof The source of fˇgis the push-out of U˝Yoo U˝g ooU˝X// f˝X //V˝X; and these three objects are cofibrant by the previous lemma. Algebraic & Geometric Topology, Volume 14 (2014)
268 Fernando Muro Definition A.9 The following axiom may or may not be satisfied by a monoidal model category: Strong unit axiom If X is a pseudo-cofibrant object and qWz I I is a cofibrant resolution of the tensor unit then X˝qand q˝Xare weak equivalences. Remark A.10 If this axiom holds for a certain cofibrant resolution of I then it holds for any cofibrant resolution of I . In particular, it is satisfied when I is cofibrant. The following result shows that it also holds in monoidal model categories where tensoring with a cofibrant object preserves weak equivalences. This property is rather common. The proof of the following lemma is due to David White. Lemma A.11 Suppose that, for qWz I I a cofibrant resolution of the tensor unit, the functors z I˝ and ˝ z I preserve weak equivalences between pseudo-cofibrant objects. Then the strong unit axiom holds. Proof Let X be a pseudo-cofibrant object. Take a cofibrant resolution pWz X X . Consider the following commutative diagram: z I˝z X z I˝p // q˝z X z I˝X q˝X z Xp ////X Here q˝z X is a weak equivalence by the unit axiom, and z I˝p is a weak equivalence by hypothesis. Hence q˝X is a weak equivalence by the two-out-of-three axiom. One can similarly check that X˝qis a weak equivalence. The converse is also true, even something stronger holds. Lemma A.12 If the strong unit axiom holds, a morphism between pseudo-cofibrant objects fWU!V is a weak equivalence if and only if f˝z I is a weak equivalence for some cofibrant replacement z I of the tensor unit. The same is true replacing f˝z I with z I˝f. Proof Let qWz I I be a cofibrant resolution of the tensor unit. Consider the following commutative diagram: U˝z If˝z I // U˝q V˝z I V˝q Uf //V Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 269 The vertical morphisms are weak equivalences by the strong unit axiom. Hence, by the two-out-of-three axiom, f is a weak equivalence if and only if f˝z I is. The proof is similar for z I˝f. Lemma A.13 Let C be a monoidal model category satisfying the strong unit axiom. If fWU ! V is a weak equivalence with pseudo-cofibrant source and target and X is a pseudo-cofibrant object, then f˝Xand X˝fare weak equivalences. Proof If U and V are cofibrant the result follows from the monoid axiom and Ken Brown’s Lemma [7, Lemma 1.1.12]. In general, if z I is a cofibrant replacement of the tensor unit, f˝z I is a weak equivalence between cofibrant objects by Lemmas A.7 and A.12. Hence X˝.f ˝z I/Š.X˝f / ˝z I is a weak equivalence, so X˝f is a weak equivalence between pseudo-cofibrant objects again by Lemma A.12; see Remark A.2(4). One can similarly check that f˝Xis a weak equivalence. Corollary A.14 In a monoidal model category satisfying the strong unit axiom, weak equivalences between pseudo-cofibrant objects are closed under tensor products. Lemma A.13 is a typical example of the following situation. We want to show that a certain morphism is a weak equivalence if some objects are pseudo-cofibrant, and the result happens to be known if those objects are honestly cofibrant. The pseudocofibrancy hypotheses and some of the previous results and remarks show that has pseudo-cofibrant source and target. We tensor everything with a cofibrant replacement z I of the tensor unit, to turn pseudo-cofibrant objects into cofibrant objects by Lemma A.7. The “only if” part of Lemma A.12 shows that we are in the same situation as at the beginning, but now objects are cofibrant. Then we apply the known result to conclude that ˝z I or z I˝ is a weak equivalence, and we use the “if” part of Lemma A.12 to deduce that is indeed a weak equivalence. This technique can be used to prove the rest of the results in this section. Therefore we leave proofs as an exercise for the reader and content ourselves with indicating why the corresponding result is true when objects are cofibrant. Lemma A.15 If S is a set and fs are weak equivalences between pseudo-cofibrant objects in a monoidal model category satisfying the strong unit axiom, s2S , then `s2Sfsis a weak equivalence between pseudo-cofibrant objects. The result for cofibrant objects follows from [6, Proposition 7.2.5]and Ken Brown’s Lemma [7, Lemma 1.1.12]. The following result is a pseudo-cofibrant version of the cube lemma [7, Lemma 5.2.6]. Algebraic & Geometric Topology, Volume 14 (2014)
276 Fernando Muro Theorem 1.11 is also valid if we replace the cofibrancy hypotheses with pseudocofibrancy hypotheses, provided the strong unit axiom holds. Theorem C.7 Let V be a symmetric monoidal model category satisfying the strong unit axiom. Consider a push-out diagram in Op.V/as follows: O// // ' push Q '0 P// 0//P[OQ If O.n/ and P.n/ are pseudo-cofibrant in V for all n0 , then '0 is a weak equivalence. The following proposition is essentially a corollary of the previous theorem. Proposition C.8 Let V be a symmetric monoidal model category satisfying the strong unit axiom. The full subcategory Oppc.V/Op.V/ spanned by the operads O such that O.n/is pseudo-cofibrant for all n0is a cofibration category. In particular, the gluing lemma also holds for Oppc.V/ in these circumstances; compare Corollary 5.2. Appendix D: Generalized results for algebras In this appendix, we do for algebras what we have just done for operads in the previous appendix. The first result is a version of Lemma 6.6. Lemma D.1 Let C be a model V –algebra satisfying the strong unit axiom. If f is a cofibration in C , O is an operad in V such that z.O.n// is pseudo-cofibrant for all n0 , and A is an O –algebra with underlying pseudo-cofibrant object in C , then (6 - 2) is a cofibration in C. In practical examples, z.O.n// is pseudo-cofibrant because O.n/ is ( I –)cofibrant, n0 . Recall from Remark B.7 that the left Quillen functor zWV!C satisfies the pseudo-cofibrant axiom since it is strong. We derive straightforward generalizations of [11, Lemma 9.4 and Corollary 9.5]. Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 277 Corollary D.2 Let C be a model V –algebra satisfying the strong unit axiom. Consider an operad O in V such that z.O.n// is pseudo-cofibrant for all n0 . If WAB is a cofibration in AlgC.O/ and A is pseudo-cofibrant in C then is also a cofibration in C. Corollary D.3 Let C be a model V –algebra satisfying the strong unit axiom. Consider an operad O in V such that z.O.n// is pseudo-cofibrant for all n0 . Then, any cofibrant O –algebra in C has an underlying pseudo-cofibrant object. Moreover, if in addition O.0/ is cofibrant then any cofibrant O –algebra in C has an underlying cofibrant object. The following theorem generalizes [11, Theorem 1.3]and Theorem 6.7 above. Theorem D.4 Let C be a model V –algebra satisfying the strong unit axiom. Assume zWV!C satisfies the I –cofibrant axiom. Consider a weak equivalence WO ! P between operads in V such that the objects O.n/ and P.n/ are ( I –)cofibrant for all n0. Then the change of coefficients Quillen pair AlgC.O/ //AlgC.P/ oo is a Quillen equivalence. Now, let us place ourselves in the situation described in the first paragraph of Section 7. Assume further that V , W , C and D satisfy the strong unit axiom, and zC , zD , F and x Fsatisfy the pseudo-cofibrant axiom and the I–cofibrant axiom. Lemma D.5 The natural morphism .X/ in (7 - 1) is a weak equivalence if X is (I–)cofibrant. Proposition 7.3 extends as follows. Proposition D.6 Suppose x Fax G is a weak monoidal Quillen adjunction and O is a cofibrant operad in V . Then if A is a cofibrant O –algebra in C , the natural morphism Ain (7-2) is a weak equivalence in D. Corollary D.7 Let x Fax G and O be as in the previous proposition. For any O –algebra Ain Cthere is a natural isomorphism in Ho D, Lx F.A/ŠLx FO.A/: Proposition D.6 allows us to prove the following generalization of Theorem 7.2. Algebraic & Geometric Topology, Volume 14 (2014)
278 Fernando Muro Theorem D.8 Under the conditions of the previous proposition, suppose further that x Fax Gis a Quillen equivalence. Then the Quillen pair AlgC.O/ x FO//AlgD.Foper.O// x G oo in Proposition 7.1 is a Quillen equivalence. Combining Theorems D.4 and D.8 we can easily check that the following extension of Theorem 7.6 holds. Theorem D.9 Suppose FaG and x Fax G are weak (symmetric) monoidal Quillen equivalences, P is a fibrant operad in W , and P.n/ and G.P.n// are ( I –)cofibrant for all n0. Then the Quillen pair in Proposition 7.5 is a Quillen equivalence: AlgC.G.P// x FP //AlgD.P/ x G oo Corollary D.10 In the situation of the previous theorem, for any G.P/ –algebra B in Cthere is a natural isomorphism in Ho D,Lx F.B/ŠLx FP.B/. We can similarly check the following extension of Theorem 7.8. Theorem D.11 Suppose VDW , FDGD1V , x Fax G is a weak monoidal Quillen equivalence, and O is an operad in V such that O.n/ is ( I –)cofibrant for all n0 . Then AlgC.O/ x FPDx FP //AlgD.O/ x G oo is a Quillen equivalence. Corollary D.12 Under the assumptions of the previous theorem, for any O –algebra Bin Cthere is a natural isomorphism in Ho D,Lx F.B/ŠLx FO.B/DLx FO.B/. The following theorem generalizes Theorem 1.11. Theorem D.13 Let C be a model V –algebra satisfying the strong unit axiom. Consider an operad O in V such that z.O.n// is pseudo-cofibrant for all n0 , and a push-out diagram in AlgC.O/as follows: A// // ' push C '0 B// 0//B[AC Algebraic & Geometric Topology, Volume 14 (2014)
Homotopy theory of non-symmetric operads, II 279 If the underlying objects of A and B are pseudo-cofibrant in C then '0 is a weak equivalence. The following proposition is essentially a corollary of the previous theorem. Proposition D.14 Let C be a model V –algebra satisfying the strong unit axiom and O an operad in V such that z.O.n// is pseudo-cofibrant for all n0 . The full subcategory Algpc C.O/AlgC.O/ spanned by the O –algebras whose underlying object in C is pseudo-cofibrant is a cofibration category. In particular, the gluing lemma also holds for Algpc C.O/in these circumstances. Appendix E: Generalized results for A–infinity algebras We finish this paper by indicating how the results in Section 9 extend when tensor units are not cofibrant. Notice that the (unital) associative operad consists of ( I –)cofibrant objects in all arities. Theorem E.1 Let FWVWWG be a weak symmetric monoidal Quillen adjunction between symmetric monoidal model categories satisfying the strong unit axiom. Assume FaG satisfies the pseudo-cofibrant axiom and the I –cofibrant axiom. Then the morphisms WFoper.uAV 1/! uAssW; 0WFoper.AV 1/! AssW; adjoint to 'qand '0q0, respectively, are weak equivalences in Op.W/. In particular, Theorem 1.7 and Corollary 1.8 hold under these weaker hypotheses. Theorem E.2 Under the hypotheses of the previous theorem, there are isomorphisms in Ho Op.W/, LFoper.AssV/ŠAssW;LFoper.uAssV/ŠuAssW; such that the following square commutes: LFoper.AssV/LFoper.V/ // Š LFoper.uAssV/ Š AssWW //uAssW Algebraic & Geometric Topology, Volume 14 (2014)
280 Fernando Muro Corollary E.3 Under the hypotheses of the previous theorem, if in addition FaG is a Quillen equivalence, then there are isomorphisms in Ho Op.V/, AssVŠRG.AssW/; uAssVŠRG.uAssW/; such that the following square commutes: AssVV // Š uAssV Š RG.AssW/RG.W/ //RG.uAssW/ The following result is a consequence of Theorem D.4. Proposition E.4 Let C be a model V –algebra. Suppose V and C satisfy the strong unit axiom. Assume further that zWV!C satisfies the pseudo-cofibrant axiom and the I–cofibrant axiom. Then, qand q0induce Quillen equivalences AlgC.uAV 1/ q//AlgC.uAssV/; q ooAlgC.AV 1/ q0 //AlgC.AssV/: .q0/ oo Let us place ourselves in the situation described in the first paragraph of Section 7. Assume further that V , W , C and D satisfy the strong unit axiom, and zC , zD , F and x Fsatisfy the pseudo-cofibrant axiom and the I–cofibrant axiom. Corollary E.5 Suppose FaG and x Fax G are weak (symmetric) monoidal Quillen equivalences. Then there are Quillen equivalences AlgC.uAssV///AlgD.uAssW/; x G ooAlgC.AssV///AlgD.AssW/: x G oo This shows that Corollaries 1.9 and 1.10 are true provided V and W satisfy the strong unit axiom and Fsatisfies the pseudo-cofibrant axiom and the I–cofibrant axiom. References [1] M Aguiar , S Mahajan ,Monoidal functors, species and Hopf algebras, CRM Monograph Series 29, Amer. Math. Soc. (2010) MR2724388 [2] M A Batanin ,Homotopy coherent category theory and A1 –structures in monoidal categories, J. Pure Appl. Algebra 123 (1998) 67–103 MR1492896 Algebraic & Geometric Topology, Volume 14 (2014)
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