Homotopy units in A-infinity algebras

arXiv:1111.2723v4 [math.AT] 30 Apr 2014 HOMOTOPY UNITS IN A-INFINITY ALGEBRAS FERNANDO MURO Abstract. We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures. 1. Introduction It is well known that monoids in a monoidal category, a.k.a. algebras, may have at most one unit. Hence, being unital can be regarded as a property, rather than a structure. In other words, the set of unital monoid structures on a given object embeds as a subset of the set of non-unital monoid structures. This fact can be deduced from the following stronger and fancier statement. Proposition 1.1. Given a closed symmetric monoidal category Vwith an initial object, the canonical morphism φV:AssV→uAssVfrom the associative operad to the unital associative operad is an epimorphism in the category Op(V)of nonsymmetric operads in V. The canonical morphism φVmodels the forgetful functor from unital monoids to non-unital monoids. If Vis also a model category, one is often more interested in homotopy algebra structures rather than strict algebra structures. This is because, given a monoid M and a weak equivalence ϕ:X∼ →Min V, there need not be a monoid structure on Xcompatible with ϕ, but there is always a compatible homotopy monoid structure on X, at least if Xis fibrant and cofibrant. Homotopy (unital) associative algebras are known as (unital) A-infinity algebras. They are formally defined as algebras over cofibrant resolutions of the operads AssVand uAssV. If V= Top is the category of topological spaces, there are nice resolutions of these operads given by associahedra [Sta63] and unital associahedra [MT14]. The cellular homology of (unital) associahedra yield resolutions for V= Ch(k) the category of chain complexes over a commutative ring k. The strongest possible homotopical generalization of Proposition 1.1 is the following result, which is the main theorem of this paper. Theorem 1.2. Let Vbe a simplicial or complicial closed symmetric monoidal model category. Assume that Vsatisfies the monoid axiom and the strong unit axiom. Suppose further that Vis cofibrantly generated and has sets of generating (trivial) cofibrations with presentable sources. Then the morphism φV:AssV→ uAssVis a homotopy epimorphism in Op(V). 1991 Mathematics Subject Classification. 18D50, 18G55. Key words and phrases. Operad, A-infinity algebra, unit, model category, mapping space. 1 2 FERNANDO MURO This means that taking derived mapping spaces in the model category Op(V) [Mur11a] out of φV, (φV)∗: MapOp(V)(uAssV,O)−→ MapOp(V)(AssV,O), is essentially an inclusion of connected components for any operad O, i.e. an injection on π0and an isomorphism in all homotopy groups πn,n > 0, with all possible base points. Putting O=EndV(X), the endomorphism operad of an object X in V, we deduce that the homotopical moduli space [Rez96] of unital A-infinity algebra structures on Xembeds as a subset of connected components of the homotopical moduli space of all A-infinity algebra structures on X. In [Mur11b] we go beyond, showing that if Xis perfect then φVinduces an affine Zariski open immersion of geometric moduli spaces in many homotopical algebraic geometry contexts, including derived, complicial, and brave new algebraic geometry. A friendly characterization of the image of the injective map π0(φV)∗when O=EndV(X) is an endomorphism operad is possible for many V’s thanks to results of Lyubashenko–Manzyuk and Lurie, see Remarks 5.11 and 6.5. Let us comment on the hypotheses of Theorem 1.2. A symmetric monoidal model category V[SS00] is simplicial if it is equipped with a symmetric monoidal Quillen adjunction from the category of simplicial sets, Set∆op F//W. G oo The upper arrow will always be the left adjoint in this kind of diagram. Similarly, Vis complicial if it is equipped with a symmetric monoidal Quillen adjunction Ch(k)F//W. G oo The strong unit axiom, introduced in [Mur14, Definition A.9], says that tensoring with a cofibrant replacement ˜ Iof the tensor unit Ipreserves all weak equivalences. This obviously holds if Iis cofibrant, but it is also true in many other cases of interest, such as diagram spectra with the positive stable model structure [MMSS01]. The rest of the hypotheses are needed to have a model structure on Op(V) with fibrations and weak equivalences defined as in V, see [Mur11a, Theorem 1.1]. The paper is structured as follows. Section 2 studies homotopy epimorphisms in arbitrary model categories. In Section 3 we recall what we need about operads and their homotopy theory. Sections 4 and 5 contain the proof of Theorem 1.2 for two special categories V: groupoids and unbounded complexes over a commutative ring. In the last section, Section 6, we deduce the main theorem from these two specific cases. We assume the reader familiarity with category theory and abstract homotopy theory. Some standard references are [Mac98, Hov99, Hir03]. For monoidal categories, functors, and adjunctions, we refer to [AM10, Chapter 3]. Acknowledgements. A previous version of this paper only contained Theorem 5.1, with a substantially more complicated proof. Lecturing about this result at the Homotopical Algebra Summer Day in Barcelona 2012, I realised of the possibility of simplifying the proof, as it is given in Section 5. The simplification needs the results in [Mur14], which are of independent interest. I’m grateful to the organizers of that Summer Day, Imma G´alvez and Javier Guti´errez, for providing such an inspiring HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 3 environment. The results in [Mur14] also alowed me to extend Theorem 5.1 to a wide class of model categories, see Theorem 1.2. I wished to do this since I obtained the first proof of Theorem 5.1, and I’m grateful to Joe Hirsh for encouraging me to do so during the Summer Day. I 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. Homotopy epimorphisms Recall that a morphism f:X→Yin a category Cis an epimorphism if C(f, Z): C(Y, Z)→C(X, Z) is an injective map for any object Zin C. The following characterization of epimorphisms is well known and easy to check. Proposition 2.1. Let f:X→Ybe a morphism in a category C. Assume the push-out Xf// f  push Y i2  Yi1 //Y∪XY exists. The following statements are equivalent: (1) fis an epimorphism. (2) i1is an isomorphism. (3) i2is an isomorphism. (4) The codiagonal ∇= (1Y,1Y): Y∪XY→Yis an isomorphism. If they hold, then i1=i2=∇−1. The strongest homotopy invariant property which generalizes the notion of injective map is the following one. Definition 2.2. A map g:K→Lbetween simplicial sets is a homotopy monomorphism if it gives rise to an injection on connected components, π0(g): π0(K)֒→π0(L), and isomorphisms on homotopy groups for all possible base points x∈K0, πn(g): πn(K, x)∼ = −→ πn(L, g(x)), n ≥1. There are other obvious characterizations of homotopy monomorphisms of simplicial sets. Lemma 2.3. Given a map g:K→Lbetween simplicial sets, the following statements are equivalent: (1) gis a homotopy monomorphism. (2) gcorestricts to a weak equivalence between Kand a subset of connected components of L. (3) For any x∈K0, the homotopy fiber of gat g(x)is contractible. (4) The homotopy fibers of gare empty or contractible. 4 FERNANDO MURO When we say that a simplicial set is contractible we mean that it is weakly equivalent to a point. The usual terminology is ‘weakly contractible’ but we prefer to shorten it. There are also less obvious characterizations along the lines of the dual of Proposition 2.1. Proposition 2.4. Let g:K։Lbe a Kan fibration between Kan complexes. Consider the pull-back square K×LKp2//// p1  pull K g  Kg////L The following statements are equivalent: (1) gis a homotopy monomorphism. (2) p1is a weak equivalence. (3) p2is a weak equivalence. (4) The diagonal ∆ = 1K 1K:K→K×LKis a weak equivalence. If they hold, then p1=p2= ∆−1in the homotopy category of simplicial sets. Proof. Since pj∆ = 1K,j= 1,2, the equivalences (2) ⇔(3) ⇔(4) and the final statement are clear. In order to show (1) ⇔(2), notice that parallel arrows in the square of the statement have essentially the same fibers. More precisely, if Fxdenotes the fiber of p1over a base point x∈K0, then Fxis isomorphic to the fiber of gover g(x). The map π0(p1) is surjective since p1∆ = 1K. Therefore, by the long exact sequence in homotopy groups, Fxis contractible for any x∈K0if and only if p1is a weak equivalence.  This proposition is actually useful to characterize when an arbitrary morphism gof simplicial sets is a homotopy monomorphism, since this property is homotopy invariant, so we can replace gby a weakly equivalent morphism which is a Kan fibration between Kan complexes. We now define homotopy epimorphisms in model categories via mapping spaces and homotopy monomorphisms of simplicial sets. This definition is dual to the notion of homotopy monomorphism in [To¨e07]. Definition 2.5. A morphism f:X→Yin a model category Mis said to be a homotopy epimorphism if for any object Zin M, the induced morphism on derived mapping spaces, f∗= MapM(f, Z): MapM(Y, Z)−→ MapM(X, Z), is a homotopy monomorphism of simplicial sets. Remark 2.6.This definition is compatible with Definition 2.2, i.e. a morphism g:K→Lof simplicial sets is a homotopy monomorphism in the sense of Definition 2.2 if and only if it is a homotopy epimorphism in the opposite of the model category of simplicial sets in the sense of Definition 2.5. This follows from Proposition 2.4. The construction of derived mapping spaces we have in mind is the simplicial set MapM(X, Z) = M(˜ X, Z•), HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 5 where ˜ Xis a cofibrant resolution of Xand Z•is a simplicial resolution of Z. In particular, f∗= MapM(f, Z) = M(˜ f, Z•), where ˜ f:˜ X→˜ Yis a lifting of fto cofibrant resolutions of Xand Y, ˜ X˜ f// ∼  ˜ Y ∼  Xf //Y It is usual to require cofibrant resolutions ˜ X∼ →Xto be trivial fibrations from a cofibrant object. However, for us it is enough to have a weak equivalence with cofibrant source. Notice that f:X→Ybeing a homotopy epimorphism only depends on the image of fin the homotopy category Ho M. Actually, it only depends on the isomorphism class of fin Ho M. The following result characterizes homotopy epimorphisms along the lines of Proposition 2.1. The dual caracterization of homotopy monomorphisms was noticed in [To¨e07]. Proposition 2.7. Let f:X֌Ybe a cofibration between cofibrant objects in a model category M. Consider the push-out square X// f//  f  push Y  i2  Y//i1 //Y∪XY The following statements are equivalent: (1) fis a homotopy epimorphism. (2) i1is a weak equivalence. (3) i2is a weak equivalence. (4) The codiagonal ∇is a weak equivalence. If they hold, then i1=i2=∇−1in Ho M. Proof. Since ∇ij= 1Y,j= 1,2, the equivalences (2) ⇔(3) ⇔(4) and the final statement are clear. If we apply MapM(−, Z) = M(−, Z•) to the push-out in the statement, we obtain a pull-back of simplicial sets consisting of Kan fibrations between Kan complexes, M(X, Z•)oooo f∗ OOOO f∗pull M(Y, Z•) OOOO i∗ 2 M(Y, Z•)ooooi∗ 1 M(Y∪XY, Z•) Hence (1) ⇔(2) follows from Proposition 2.4 and the fact that i∗ 1is a weak equivalence of simplicial sets for all objects Zin Mif and only if i1is a weak equivalence in M. 6 FERNANDO MURO Remark 2.8.Proposition 2.7 is actually useful to check whether any morphism in Mis a homotopy epimorphism. A morphism f:X→Yin Mis a homotopy epimorphism if and only if a cofibrant resolution ˜ f:˜ X֌˜ Yof fis. Such a cofibrant resolution is a cofibration between cofibrant objects fitting into a commutative diagram ˜ X// ˜ f// ∼  ˜ Y ∼  Xf //Y The statements (2), (3) and (4) in Proposition 2.7 only depend on the isomorphism class of the commutative square ˜ X// ˜ f//  ˜ f  push ˜ Y  i2  ˜ Y//i1 //˜ Y∪˜ X˜ Y in the homotopy category Ho(M) of commutative squares in M. The isomorphism class of this square only depends on the isomorphism class of f:X→Y in Ho M. Actually, it can be constructed using the derivator DMof M, which consists of all homotopy categories of diagrams in Mwith the shape of a finite direct category, such as ,• → •, or • ← • → •, see [Cis10]. A category is finite and direct if its nerve has finitely-many non-degenerate simplices. Let Cat be the category of categories and functors and Dirf ⊂Cat the full subcategory of finite direct categories. The derivator DMis the 2-functor, DM: Dirfop −→ Cat, I7→ Ho(MIop ). Apparently, there is a problem here with the size of Cat. Morphism ‘sets’ in Cat may be proper classes. Nevertheless, there is really no trouble, since morphism sets in Dirf are honest sets, so the derivator DMis insensitive to the problems of Cat. Let D: Dirfop →Cat be an abstract derivator, more precisely, a right derivator satisfying [Cis10, Der 5], i.e. [Mal07, Der 5]. If edenotes the category with only one object and one morphism (the identity), one can give a definition of homotopy epimorphism in D(e) along the lines of (2), (3) and (4) above, extending the notion of homotopy epimorphism in DM(e) = Ho M. Homotopy epimorphisms are preserved by cocontinuous morphisms of right derivators, in particular by equivalences of derivators. This observation yields a quick justification for the following corollary. The first part also follows easily from the elementary properties of mapping spaces. Corollary 2.9. Let F:M⇄N:Gbe a Quillen adjunction between model categories and let LF: Ho M⇄Ho N:RGbe the derived adjoint pair between homotopy categories. The functor LFpreserves homotopy epimorphisms. Moreover, if F⊣Gis a Quillen equivalence then RGalso preserves homotopy epimorphisms. Furthermore, if LFreflects isomorphisms then it also reflects homotopy epimorphisms. HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 7 Remark 2.10.This is a continuation of the previous remark. Let X ¯ f ֌˜ Y∼ →Y be a factorization of finto a cofibration followed by a weak equivalence. If Mis left proper the gluing lemma holds, see [Hir03, Proposition 13.5.4]. Therefore, the previous push-out square is isomorphic to X// ¯ f// f  push ˜ Y i2  Y//i1 //Y∪X˜ Y in Ho(M). In particular, fis a homotopy epimorphism if and only if this i1is a weak equivalence. The gluing lemma also holds in cofibration categories [Bau89, II.1.2 (b)]. Hence, the same is true if X,Yand ˜ Ybelong to a full subcategory of Mwhich is a cofibration category with cofibrations and weak equvialences defined as in M. 3. Operads All operads considered in this paper are non-symmetric. Definition 3.1. Let Vbe a symmetric monoidal category with tensor product ⊗ and tensor unit I. An operad Oin Vis a sequence O={O(n)}n≥0of objects in Vequipped with an identity, idO:I→ O(1), and composition laws, 1 ≤i≤p,q≥0, ◦i:O(p)⊗ O(q)−→ O(p+q−1), satisfying certain associativity and unit equations, see [Mur11a, Remark 2.6]. We refer to O(n) as the arity ncomponent of O. Amorphism of operads f:O → P is a sequence of morphisms f(n): O(n)→ P(n) in V,n≥0, compatible with the identities and composition laws in the obvious way. We usually drop the arity from the notation f(n) in order to simplify. We denote by Op(V) the category of operads in V. Remark 3.2.If V= Set is the category of sets, the identity is simply an element idO∈ O(1) and the associativity and unit equations are: (1) (a◦ib)◦jc= (a◦jc)◦i+q−1bif 1 ≤j < i and c∈ O(q). (2) (a◦ib)◦jc=a◦i(b◦j−i+1 c) if b∈ O(p) and i≤j < p +i. (3) idO◦1a=a. (4) a◦iidO=a. The same happens if V= Top is the category of topological spaces or the category Mod(k) of modules over a commutative ring k. If V= Mod(k)Zis the category of Z-graded k-modules then idOmust be in degree 0, idO∈ O(1)0, and (1) must be replaced with (1′) (a◦ib)◦jc= (−1)|b||c|(a◦jc)◦i+q−1bif 1 ≤j < i and c∈ O(q). This reflects the use of the Koszul sign rule in the definition of the symmetry constraint for the tensor product in Mod(k)Z. 8 FERNANDO MURO Furthermore, if V= Ch(k) is the category of differential graded k-modules the identity must be a cycle, d(idO) = 0, and the differential must behave as a derivation with respect to all composition laws, d(a◦ib) = d(a)◦ib+ (−1)|a|a◦id(b). In this paper differentials have degree |d|=−1, i.e. we consider chain complexes. The category V= Grd of groupoids with the cartesian symmetric monoidal structure behaves essentially as Set. The identity idOis an object of the groupoid O(1). Remark 3.3.Operads can be alternatively (and are usually) described in terms of multiplication morphisms,n≥1, p1,...pn≥0, O(n)⊗ O(p1)⊗ · · · ⊗ O(pn)−→ O(p1+···+pn), defined by iterating composition laws, e.g. if Vis any of the categories in the previous remark, this morphism is given by (a, b1,...,bn)7→ a(b1,...,bn) = (···((a◦1b1)◦p1+1 b2)◦p1+p2+1 ···)◦p1+···+pn−1+1 bn. This iterated composition can be expressed in many different ways, for instance, if V= Set or Mod(k), a(b1,...,bn) = (···((a◦nbn)◦n−1bn−1)◦n−2···)◦1b1. If V= Mod(k)Zor Ch(k) this formula would be true up to a sign determined by the Koszul rule. The multiplication morphisms together with the identity and certain associativity and unit equations yield an equivalent definition of operad, see [Mur11a, Remark 2.5]. If Vis the category of sets or k-modules these equations are: a(b1(c11,...,c1p1),......,bn(cn1,...,cnpn)) =a(b1,...,bn)(c11,...,c1p1,......,cn1,...,cnpn), idO(a) = a, a(idO,...,idO) = a. If Vis the category of graded modules we must alter the first equation with a sign, according to the Koszul rule. In the differential graded case, in addition, the differential must behave like a derivation with respect to the multiplication morphisms, i.e. d(a(b1,...,bn)) = d(a)(b1,...,bn) + n X i=1 (−1) |a|+ i−1 P j=1 |bj| a(b1,...,d(bi),...,bn). Example 3.4.The unital associative operad uAssVin V, whose algebras are unital monoids, is given by uAssV(n) = Ifor all n≥0. The identity of this operad iduAssV:I→uAssV(1) is simply the identity morphism in I, and all composition laws are given by the unit isomorphism I⊗I∼ =I, wich is part of the symmetric monoidal structure of V. Example 3.5.Suppose Vis closed and has an initial object ∅. The associative operad AssVin V, whose algebras are non-unital monoids, is given by AssV(n) = I HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 9 for all n≥1 and AssV(0) = ∅. The operad structure is determined by the fact that the sequence of morphisms φV:AssV−→ uAssV given by the identity in Iin all positive arities, φV(n) = idI,n≥1, is a morphism of operads. Remark 3.6.Suppose Vis cartesian closed and has an initial object ∅. If ⊗=×is the cartesian product then Iis the final object in Vand uAssVis the final operad, i.e. the final object in Op(V). Hence, φVis the only possible map. Moreover, AssV is the final object of the full subcategory of operads which are ∅in arity 0. This happens when Vis Set, Top, or Grd. Remark 3.7.If Vis cocomplete, then so is Op(V). In this case, the forgetful functor from Op(V) to the category VNof sequences V={V(n)}n≥0of objects in Vhas a left adjoint, the free operad functor, VNF//Op(V). forget oo We sometimes write F=FV. This adjunction is monadic, i.e. Op(V) is the category of algebras over the free operad monad. These facts allow the construction of operads by presentations. A presentatation of an operad Oconsists of describing Oas the coequalizer of two parallel arrows between free operads, F(U)////F(V)//O. The free operad functor Fcan be explicitly described in terms of planted planar trees with leaves, see [Mur11a, §3 and §5]. Planting a tree consists of choosing a degree 1 vertex, called root. The degree of a vertex is the number of adjacent edges. The planar structure is given by an order in the set of vertices which indicates how to draw them from left to right. The leaves are specified degree 1 vertices different from the root. They can be distinguished in pictures since we do not draw them. We do not draw the root either, but there is no confusion since the root is placed at the bottom. Vertices are distrubuted in ascending layers according to the distance to the root. We call inner vertices those which are drawn, i.e. the vertices which are neither leaves nor the root. A cork is an inner vertex of degree 1. An inner edge is an edge which is not adjacent to the root or to a leaf. These notions are better illustrated with a picture, b b b b b This is a planted planar tree with four leaves and five inner vertices, including two corks. There are four inner edges. Sometimes, abusing language, we also call leaf or root to the adjacent edge, which is what we really depict. From now on, in the whole paper, whenever we talk about trees we mean planted planar trees with leaves. 16 FERNANDO MURO Proof. The ‘underlying set’ functor V(I,−): V→Set is part of a lax-lax symmetric monoidal adjoint pair Set −⊗I //V. V(I,−) oo The left adjoint sends a set Sto the coproduct of copies of the tensor unit indexed by this set S⊗I=∐s∈SI. This adjoint pair induces an adjoint pair between categories of operads Op(Set) −⊗I //Op(V). V(I,−) oo Notice that AssSet ⊗I=AssV,uAssSet ⊗I=uAssV, and φSet ⊗I=φV. Hence, this proposition follows from the previous one, since left adjoints preserve epimorphisms.  Proposition 3.12 is true even if Vdoes not have coproducts. The proof of Proposition 3.11 can be translated into diagrams in order to check this general case, Propositon 1.1. If Vhas a suitable model structure, compatible with the monoidal structure, then the category of operads Op(V) carries an induced model structure. Theorem 3.13 ([Mur11a, Theorem 1.1]).Let Vbe a cofibrantly generated closed symmetric monoidal model category. Assume that Vsatisfies the monoid axiom. Moreover, suppose that there are sets of generating cofibrations Iand generating trivial cofibrations Jin Vwith presentable sources. Then the category Op(V)of operads in Vis a cofibrantly generated model category such that a morphism f:O → Pin Op(V)is a weak equivalence (resp. fibration) if and only if f(n): O(n)→ P(n) is a weak equivalence (resp. fibration) in Vfor all n≥0. We refer the reader to [Hov99, §4] and [SS00] for the theory of symmetric monoidal model categories. All categories Vin this paper will satisfy the assumptions in this theorem, and this will be the only model structure considered on Op(V). Remark 3.14.Let us describe sets of generating (trivial) cofibrations in Op(V). The model structure in the previous theorem is transferred along the free operad adjunction in Remark 3.7. The category of sequences VNis endowed with the product model structure. Recall that given an object Xin Vand n≥0, we denote by X[n] the sequence consisting of Xconcentrated in arity nand the initial object ∅elsewhere. Given a morphism f:X→Yin Vwe denote by f[n]: X[n]→Y[n] the morphism of sequences defined by fin arity nand the identity in ∅elsewhere. For any set Sof morphisms in V, we write SN=[ n≥0 {f[n] ; f∈S}. The sets INand JNare sets of generating cofibrations and generating trivial cofibrations in VN, respectively. Hence, F(IN) and F(JN) are sets of generating cofibrations and generating trivial cofibrations in Op(V). HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 17 Definition 3.15. Let Vbe a symmetric monoidal model category satisfying the hypotheses of Theorem 3.13. An A-infinity operad AV ∞in Op(V) is a cofibrant resolution of AssV, AV ∞ ∼ −→ AssV. Similarly, a unital A-infinity operad uAV ∞is a cofibrant resolution of uAssVin Op(V), uAV ∞ ∼ −→ uAssV. Au-infinity associative operad u∞AVis the middle term of a factorization of φV as a cofibration ¯ φV ∞followed by a weak equivalence, AssV¯ φV ∞ ֌u∞AV∼ −→ uAssV. Remark 3.16.For each specific V, we may choose a cofibrant resolution of φV, φV ∞:AV ∞֌uAV ∞. If Vsatisfies the strong unit axiom [Mur14, Definition A.9], e.g. if the tensor unit is cofibrant, then we can define a u-infinity associative operad as the following push-out, AV ∞// φV ∞// ∼  push uAV ∞ ∼  AssV// ¯ φV ∞//u∞AV Here, the right vertical map is a weak equivalence by [Mur14, Corollary C.3 and Theorem C.7]. Hence, the map u∞AV→uAssVinduced by the universal property of the push-out is a weak equivalence by the 2-out-of-3 axiom. Definition 3.17. Au-infinity unital associative operad u∞uAVis an operad fitting into a push-out square as follows, AssV// ¯ φV ∞// φV  push u∞AV ψV  uAssV// ϕV //u∞uAV We will prove Theorem 1.2 using the following lemma. Lemma 3.18. Let Vbe a symmetric monoidal model category as in Theorem 3.13. Assume further that Vsatisfies the strong unit axiom. The morphism φV:AssV→uAssVis a homotopy epimorphism in Op(V)if and only if ϕVis a weak equivalence. Proof. The operads AssVand uAssVbelong to the full subcategory Oppc(V)⊂ Op(V) spanned by the operads Owhose components O(n) are pseudo-cofibrant for all n≥0. Recall from [Mur14, Definition A.1] that an object Xin Vis pseudocofibrant if the functor X⊗ − preserves cofibrations. The tensor unit Iand the initial object ∅are obviously pseudo-cofibrant. The category Oppc(V) inherits from Op(V) the structure of a cofibration category, see [Mur14, Proposition C.8]. The operads u∞AVand u∞uAVare also in Oppc(V), see [Mur14, Corollary C.2]. Hence, this lemma follows from Remark 2.10.  18 FERNANDO MURO The following lemma is useful to check that some symmetric monoidal categories carry a compatible model structure. Lemma 3.19. Let F:V⇄W:Gbe a lax-lax symmetric monoidal adjunction between symmetric monoidal categories. Suppose that Vis a cofibrantly generated model category satisfying the push-out product axiom in [SS00, Definition 3.1]. Assume further that Wpossesses a transferred model structure along this adjunction, in the sense of [Hir03, Theorem 11.3.2]. Then Walso satisfies the push-out product axiom. Proof. Let Iand Jbe sets of generating (trivial) cofibrations of V. Then F(I) and F(J) are sets of generating (trivial) cofibrations of W. Denote by f⊙g:U⊗Y[ U⊗X V⊗X−→ V⊗Y the push-out product of two morphisms f:U→Vand g:X→Y. In order to check the push-out product axiom for W, it is enough to prove that the sets F(I)⊙F(I) and F(I)⊙F(J) consist of cofibrations and trivial cofibrations in W, respectively, compare [Hov99, Corollary 4.2.5]. The monoidal functor Fis strong, see [AM10, Proposition 3.96]. It also preserves push-outs, since it is a left adjoint. Hence, Fpreserves push-out products. In particular, F(I)⊙F(I) = F(I⊙I), F(I)⊙F(J) = F(I⊙J). These sets consist of cofibrations and trivial cofibrations, respectively, since V satisfies the push-out product axiom and Fpreserves (trivial) cofibrations.  4. Main theorem for operads of groupoids Let Gpd be the model category of small groupoids. Morphisms are functors and weak equivalences are equivalences of categories. A cofibration is a functor which is injective on objects. Fibrations are functors satisfying the isomorphism lifting property. Recall that ϕ:G→Hhas the isomorphism lifting property if for any object xin Gand any isomorphism f:ϕ(x)→yin Hthere exists an isomorphism f′:x→x′in Gwith ϕ(f′) = f, in particular ϕ(x′) = y. Trivial fibrations have a simple characterization. Lemma 4.1. A trivial fibration in Gpd is a fully-faithful functor surjective on objects. It is enough to notice that an equivalence of categories satisfies the isomorphism lifting property if and only if it is surjective on objects. Proposition 4.2. The category Gpd with the cartesian product is a combinatorial closed symmetric monoidal model category satisfying the monoid axiom where all objects are cofibrant. Proof. The model structure on Gpd is transferred along the following adjoint pair Set∆op Π1//Grd . Ner oo Here, Ner is the nerve functor and Π1is the fundamental groupoid functor. We regard Set∆op as a symmetric monoidal model category with the usual model structure and the cartesian product monoidal structure, see [Hov99, Proposition 4.2.8]. HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 19 Since Set∆op is cofibrantly generated, then so is Gpd. Moreover, Gpd is locally presentable (this is an elementary fact from category theory). Hence, Gpd is a combinatorial model category. The functor Π1is known to preserve products. Therefore, the push-out product axiom for Grd follows from Lemma 3.19. All objects are cofibrant by the very definition of cofibration. Hence, the monoid axiom follows from the push-out product axiom, see [SS00, Remark 3.4].  The main result of this section is the following theorem. Theorem 4.3. The morphism φGpd :AssGpd →uAssGpd in Example 3.5 is a homotopy epimorphism in Op(Gpd). This theorem follows from Lemma 3.18 above and Lemma 4.15 below. Remark 4.4.The sets of generating (trivial) cofibrations of Gpd obtained by taking fundamental groupoids on the usual sets of generating (trivial) cofibrations of Set∆op are too big. We can alternatively take I={∅֌e, i:{e, e′}֌E, p:Z֌e}, J ={j:e∼ ֌E}. Here eis the final groupoid, which consists of only one object and one morphism (the identity), Zis the groupoid with one object with automorphism group Z, and Eis the groupoid with two isomorphic objects, e∼ =e′, with trivial automorphism groups. The functor iis the inclusion of the discrete subgroupoid formed by the two objects, and jis the inclusion of an object. Recall that a groupoid Gis discrete if the only morphisms in Gare the indentities. Indeed, a functor satisfying the right lifting property with respect to ∅֌e,i, or p, is a functor surjective on objects, full, or faithful, respectively. Remark 4.5.Limits are easier than colimits in the category of groupoids, at least easier than non-filtered colimits. However, colimits behave well on objects, in the sense that the set of objects of the colimit of a diagram of groupoids is the colimit of the diagram of object sets. This follows from the fact that the ‘set of objects’ functor from groupoids to sets has a right adjoint, Grd Ob //Set . contractible oo The right adjoint, called ‘contractible groupoid’ functor, sends the empty set to the empty groupoid, and any non-empty set Sto the contractible groupoid with object set S. Recall that a groupoid Gis contractible if it is equivalent to e, i.e. if it has a non-empty set of objects and there exists a unique isomorphism between any to objects of G. Hence, morphisms into contractible groupoids are usually denoted by simply indicating the source, the target, and the map between object sets. The ‘contractible groupoid’ and the ‘set of objects’ functors preserve products, hence they induce an adjoint pair on operads, Op(Grd) Ob //Op(Set). contractible oo In particular, the ‘set of objects’ functor also preserves colimits at the level of operads. This fact is used in the proof of the following lemma. 20 FERNANDO MURO In this section we consider free operads of sets and free operads of groupoids. For the sake of simplicity, we omit the subscript from the free operad functor for sets F=FSet, but not for groupoids FGrd, in order to avoid confusion. These free operad functors and the object set functor commute, Ob FGrd =FOb, since the latter preserves cartesian products, compare Remark 3.7. Lemma 4.6. A morphism is a cofibration in Op(Grd) if and only if it is a retract of a morphism f:O → P such that Ob(f): Ob(O)→Ob(P) = Ob(O)∐ F(V)is an inclusion of a factor of a binary coproduct such that the other factor is a free operad in Op(Set). Proof. Any relative FGrd(IN)-cell complex is as fin the statement. Indeed, on the one hand, the functors i, p ∈Iare the identity on objects, hence a push-out along FGrd(i[n]) or FGrd(p[n]) is the identity on objects. On the other hand, a push-out along FGrd(∅֌e[n]) adds freely a new object in arity n. Hence, the ‘only if’ part follows. The converse is also true, i.e. any morphism as fis a relative FGrd(IN)-cell complex, but this is complicated to show directly. In order to prove the ‘if’ part, it is easier to check that fin the statement satisfies the left lifting property with respect to trivial fibrations. We therefore consider a commutative diagram of solid arrows in Op(Grd) as follows, Og// f  Q ∼q  Ph // l ?? ⑦ ⑦ ⑦ ⑦R Here, qis a trivial fibration. In order to obtain a lifting l, we first consider the diagram of objects Ob(O)Ob(g)// Ob(f)  Ob(Q) Ob(q)  Ob(O)∐ F(V)Ob(h) // l′77 ♣ ♣ ♣ ♣ ♣ ♣Ob(R) Here, Ob(q) is levelwise surjective by Lemma 4.1. Hence, it is easy to obtain a lifting l′in Op(Set). Define l′as Ob(g) on the first factor. On the second factor, we choose preimages of the objects h(V(n)) along q(n), n≥0, and extend to a morphism from the free operad F(V). Finally, since q(n) is fully faithful, there is a unique functor l(n): P(n)→ Q(n) given by l′(n) on objects and such that q(n)l(n) = h(n), n≥0. One can easily check that the sequence of functors {l(n)}n≥0is an operad morphism lwhich also satisfies lf =g. Corollary 4.7. An operad Oin Op(Grd) is cofibrant if and only if the operad of object sets Ob(O)is a retract of a free operad in Op(Set). We can now easily define a u-infinity associative operad of groupoids. Definition 4.8. The u-infinity associative operad u∞AGrd in Op(Grd) is the levelwise contractible operad with operad of objects Ob(u∞AGrd) = AssSet ∐ F({u}[0]). Lemma 4.9. The operad u∞AGrd in the previous definition is indeed a u-infinity associative operad in the sense of Definition 3.15. The morphism ¯ φGrd ∞:AssGrd ֌ u∞AGrd is given on objects by the inclusion of the first factor of the coproduct. HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 21 Proof. The morphism ¯ φGrd ∞is a cofibration by Lemma 4.6. Moreover, the unique morphism u∞AGrd →uAssGrd is a weak equivalence since u∞AGrd is levelwise contractible. The composition of these morphisms is φGrd since uAssGrd is final in Op(Grd).  Remark 4.10.We now describe the operad of objects of u∞AGrd following Remarks 3.9 and 3.10. The set Ob(u∞AGrd)(n), for n > 1, can be identified with the set of corollas with nleaves, at least two branches, and possibly corks, e.g. µ4(id, u, u, id,id) = b b b ∈Ob(u∞AGrd(3)), Abranch of a tree is an edge adjacent to a leaf or a cork. For n= 0,1, in addition to the corollas with nleaves, at least two branches, and possibly corks, we have u= b ∈Ob(u∞AGrd(0)),id = ∈Ob(u∞AGrd(1)). By requiring at least two branches we are explicitly excluding the following two corollas, b , b b . The composition laws (between trees different from id = |) are given by grafting and then contracting the newly created inner edge, b b ◦2 b b b b b b b b b b b b , µ2(id, u, id) ◦2µ4(id, u, u, id,id) = µ6(id, u, id, u, u, id,id), except when the grafted tree is u. In that case, the new inner edge is not contracted, µ2(id, u, id) ◦2u= b b ◦2 b = b b b =µ2(id, u, u). Compare Example 3.10. Lemma 4.11. The operad u∞AGrd is generated by the objects µand uand by the isomorphisms µ(u, id) ∼ =id and µ(id, u)∼ =id. Proof. The previous remark shows that any object in u∞AGrd can be obtained from µand u. We must show that the unique existing isomorphism between any two objects can be obtained from µ(u, id) ∼ =id and µ(id, u)∼ =id. It is enough to prove that we can get all morphisms with target µn−1, the corolla with nleaves and no corks, n≥2, all morphisms with target id = |, and all morphisms with target u. 22 FERNANDO MURO Starting with an object of positive arity represented by a corolla as in the previous remark, the isomorphisms in the statement, that we can respectively denote b b λ −→ , b b ρ −→ , allow to delete one cork at a time, ending up with a corolla with no corks or with id = |, depending on the arity, e.g. b b b b b ρ // b b b λ // b . In arity 0, we can use the isomorphism λ◦1u=ρ◦1u: b b b −→ b , to reduce the number of corks, ending up with u. Hence, we are done.  We now define an operad that we will later show to be a u-infinity unital associative operad of groupoids, see Lemma 4.15 below. Definition 4.12. The operad of groupoids Uis defined as the levelwise contractible operad with objects Ob(U) = uAssSet ∐ F({u′}[0]). Remark 4.13.Following Remarks 3.9 and 4.10, we here describe the operad of sets Ob(U). We can consider the morphism in Op(Grd) ψ:u∞AGrd −→ U given on objects by Ob(ψ) = φSet ∐(iso. u7→ u′): AssSet ∐ F({u}[0]) −→ uAssSet ∐ F({u′}[0]). The morphism ψis an isomorphism in positive arities, that we use as an identification. In arity 0, ψin an inclusion of objects (and hence morphisms). We also identify u∞AGrd(0) with its image in U(0) through ψ(0). The extra object of U(0) is u, which is represented by a trivial corolla with a white cork, as in Example 3.10. Therefore, black cork means u′and white cork means u, µ4(id, u′, u′,id,id) = b b b , u′= b , u = bc . The composition laws are defined in terms of trees as in Remark 4.10 when uis not involved. If uappears, it is almost always given by grafting and then contracting the newly created inner edge, b b b ◦2 bc b b b bc b b b , HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 23 µ4(id, u′, u′,id,id) ◦2u=µ4(id, u′, u′, u, id) = µ3(id, u′, u′,id). There are four exceptions, the two exceptions in Example 3.10 and (µ(id, u′))◦1u b bc b b =u′,(µ(u′,id))◦1u b bcb b =u′. Lemma 4.14. The operad Ufits into the following push-out diagram, FGrd(e[0]) // FGrd(j[0]) ∼// ζ  push FGrd(E[0]) ζ′  uAssGrd //∼ ϕ//U where ϕis given on objects by the inclusion of the first factor of the coproduct, ζis defined by ζ(e) = u, and ζ′is defined by ζ′(e) = uand ζ′(e′) = u′. Proof. Denote by FGrd(e[0]) // FGrd(j[0]) ∼// ζ  push FGrd(E[0]) ¯ ζ  uAssGrd //∼ ¯//P the push-out in Op(Grd). The square in the statement is clearly commutative, so it induces a unique compatible morphism χ:P → U. We are going to show that χ is an isomorphism. The square in the statement, on objects, is a push-out in Op(Set), F({e}[0]) incl. // Ob(ζ)  push F({e, e′}[0]) F({e}[0]) ∐ F({e′}[0]) Ob(ζ′)=Ob(ζ)∐(iso. e′7→u′)  uAssSet incl. //uAssSet ∐ F({u′}[0]) Hence, χis bijective on objects. Therefore, in order to show that χis an isomorphism it is enough to prove that Pis levelwise contractible. This is obvious. Indeed FGrd(j[0]) is a generating trivial cofibration, so ¯is a trivial cofibration, in particular a weak equivalence, i.e. ¯(n): e=uAssGrd(n)→ P(n) is an equivalence of categories for all n≥0.  Lemma 4.15. Consider the commutative square AssGrd // ¯ φGrd ∞// φGrd  u∞AGrd ψ ∼  uAssGrd //∼ ϕ//U where ϕand ψwere defined in Lemma 4.14 and Remark 4.13, respectively. The morphisms ϕand ψare weak equivalences since their sources and their target are levelwise contractible by definition. We assert that the previous commutative square is a push-out in Op(Grd). 24 FERNANDO MURO Proof. In order to warm up, the reader can easily check that the square in the statement is a push-out on objects. We tackle directly the statement. We are going to prove that the square satisfies the universal property of a push-out. With this purpose, we consider a commutative diagram of solid arrows in Op(Grd), AssGrd // ¯ φGrd ∞// φGrd  u∞AGrd ψ ∼ g  uAssGrd //∼ ϕ// f// U h ## ● ● ● ● ● O where fφGrd =g¯ φGrd ∞. We will show that there exists a unique morphism hcompleting the diagram in a commutative way, i.e. with two new commutative triangles, f=hϕ and g=hψ. The following equation holds in U, b −→ bc = b b −→ !◦1 bc . Therefore, if hexisted, it should satisfy h b −→ bc =g b b −→ !◦1f bc . By Lemma 4.14, there exists a unique hsatisfying this equation and f=hϕ (notice that the equation implies h(u) = f(u)). Hence, it is only left to prove that g=hψ. It is enough to show that this equation holds for the generators in Lemma 4.11. This is obvious for µ, since it comes from AssGrd. For u∈u∞AGrd(0), which is the black cork, h b =g b b !◦1f bc = g b !◦2g b !◦1f bc  = f b !◦1f bc !◦1g b =f  ◦1g b  = idO◦1g b =g b . The generating isomorphisms satisfy the following equations in U, b b λ −→ = b ◦1 b −→ bc , b b ρ −→ = b ◦2 b −→ bc . HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 25 Hence, h(λ) = h b !◦1h b −→ bc  =g b !◦1 g b b −→ !◦1f bc ! = g b !◦1g b b −→ !! ◦1f bc  =g b b −→ b !◦1f bc  = g b !◦2g b b −→ !! ◦1f bc  = f b !◦1f bc !◦1g b b −→ ! =g b b −→ !=g(λ), h(ρ) = h b !◦2h b −→ bc  =g b !◦2 g b b −→ !◦1f bc ! = g b !◦2g b b −→ !! ◦2f bc  =g b b −→ b !◦2f bc  = g b b −→ !◦1g b !! ◦2f bc  =g b b −→ !◦1 f b !◦2f bc ! =g b b −→ !=g(ρ). This concludes the proof.  5. Main theorem for DG-operads Let Ch(k) be the category of DG-modules over a ground commutative ring k. Weak equivalences in Ch(k) are quasi-isomorphisms and fibrations are levelwise surjective maps. In this way, Ch(k) with the usual tensor product becomes a combinatorial closed symmetric monoidal model category with cofibrant tensor unit 32 FERNANDO MURO if (n, m)6= (2,1),(1,1), d(νS n) = (−1)nµ◦1νS n−1unless lm=n +µ◦2νS−1 n−1unless l1= 1 +X 1≤v≤m+1 lv−1<i+v−1<lv−1 (−1)i+v−1νSv∪(S′ v−1) n−1◦iµ; and if m= 1 also d(ν{1} 1) = 0, d(ν{1} 2) = µ◦1ν{1} 1−id, d(ν{2} 2) = µ◦2ν{1} 1−id. Notice that P, whose underlying graded operad is uAssCh(k)∐ F(k· {νS n}n,S), has a linear differential regarded as an operad under uAssCh(k). As a graded operad Q=uAssCh(k)∐ F(k· {σS n, d(σS n)}n,S). Here σS nis the top generator of the copy of Dn+m−1indexed by nand S, so it has degree n+m−1 and arity n−m. The differential of Qis linear too. The morphism of DG-poperads Φ under uAssCh(k)is defined by Φ(σS n) = (−1)l1+1νS+1 n+1 ◦l1u. Notice that Φ is linear. The definition of Ψ is more complicated. Let h:uAssCh(k)∐P1→uAssCh(k)∐Q1 be the degree +1 morphism of uAssCh(k)-modules defined by h(νS n) = σS n, h(id) = 0. Moreover, let p:uAssCh(k)∐P1→uAssCh(k)∐Q1be the degree 0 morphism given by p(id) = id, p(νS n) = 0 if n > 1, p(ν{1} 1) = uif m= 1. We define Ψ as a morphism of graded operads under uAssCh(k)by Ψ(νS n) = dh(νS n) + hd(νS n) + p(νS n) = d(σS n) + hd(νS n) +uif (n, m) = (1,1). Let us check that Ψ is compatible with differentials. Notice that Ψ is linear by definition and the formula Ψ = dh +hd +p holds on the uAssCh(k)-module generators id and νS nof uAssCh(k)∐ P1, hence it holds after (co)restricting to linear parts, i.e. when we evaluate each side at an element in uAssCh(k)∐ P1we obtain an equality in uAssCh(k)∐ Q1. Then, Ψd(νS n) = dhd(νS n) + hd2(νS n) + pd(νS n) = dhd(νS n), dΨ(νS n) = d2h(νS n) + dhd(νS n) + dp(νS n) = dhd(νS n). Here we use that pd(νS n) = 0 = dp(νS n) for any νS n. Indeed, p(νS n) = 0 if (n, m)6= (1,1) and dp(ν{1} 1) = d(u) = 0, so dp(νS n) = 0 in any case. Moreover, as pointed out HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 33 above, if νS n6=ν{1} 1, ν{1} 2, ν{2} 2each summand in d(νS n) contains a certain νS′ n′6=ν{1} 1, so p(νS′ n′) = 0 and pd(νS) = 0. Furthermore, pd(ν{1} 1) = p(0) = 0, pd(ν{1} 2) = p(µ◦1ν{1} 1−id) = µ◦1p(ν{1} 1)−id = µ◦1u−id = id −id = 0, and similarly pd(ν{2} 2) = 0. We finally prove that ΦΨ = 1P. It is enough to check the equation on generators νS nsince Φ and Ψ are morphisms of DG-operads under uAssCh(k). We start with the general case (n, m)6= (1,1),(2,1): ΦΨ(νS n) = Φ(d(σS n) + hd(νS n)) =dΦ(σS n) + Φhd(νS n) = (−1)l1+1d(νS+1 n+1 ◦l1u) + (a) unless lm=n z}| { (−1)n+l1+1µ◦1(νS+1 n◦l1u) + (b) unless l1=1 z}| { (−1)l1µ◦2(νS n◦l1−1u) +X 0<i<l1−1 (ci) z}| { (−1)i+l1(νS n◦l1−1u)◦iµ +X 1<v≤m+1 lv−1<i+v−1<lv−1 (di) z}| { (−1)i+v+l1(ν(Sv+1)∪S′ v n◦l1u)◦iµ, (−1)l1+1d(νS+1 n+1 ◦l1u) = −(a) unless lm=n z}| { (−1)n+l1(µ◦1νS+1 n)◦l1u+ (b′) z}| { (−1)l1+1(µ◦2νS n)◦l1u +X 0<i<l1 −(ci) z}| { (−1)i+l1+1(νS n◦iµ)◦l1u +X 1<v≤m+1 lv−1+1<i+v−1<lv (d′ i) z}| { (−1)i+v+l1(ν(Sv+1)∪S′ v n◦iµ)◦l1u . The summands (a) and −(a) either do not occur (if lm=n) or cancel. Moreover, (di) = −(d′ i+1), so all the (di) and the (d′ i) cancel. If l1= 1, there are no (ci), (b) does not occur, and (b′) = (µ◦2νS n)◦1u= (µ◦1u)◦1νS n= id ◦1νS n=νS n. If l1>1 then (b) = −(b′) and all the (ci) cancel except for the last one: −(cl1−1) = (νS n◦l1−1µ)◦l1u=νS n◦l1−1(µ◦2u) = νS n◦l1−1id = νS n. Therefore ΦΨ(νS n) = νS n. 34 FERNANDO MURO Let us finally check the special cases (n, m) = (1,1),(2,1): ΦΨ(ν{1} 1) = Φ(d(σ{1} 1) + hd(ν{1} 1) + u) =dΦ(σ{1} 1) + u =d(ν{2} 2◦1u) + u = (µ◦2ν{1} 1−id) ◦1u+u = (µ◦1u)◦1ν{1} 1−u+u = id ◦1ν{1} 1 =ν{1} 1, ΦΨ(ν{1} 2) = Φ(d(σ{1} 2) + hd(ν{1} 2)) =dΦ(σ{1} 2) + Φhd(ν{1} 2) =d(ν{2} 3◦1u) + Φh(µ◦1ν{1} 1−id) = (−µ◦1ν{2} 2+µ◦2ν{1} 2)◦1u+µ◦1(ν{2} 2◦1u) = (µ◦1u)◦1ν{1} 2 =ν{1} 2, ΦΨ(ν{2} 2) = Φ(d(σ{2} 2) + hd(ν{2} 2)) =dΦ(σ{2} 2) + Φhd(ν{2} 2) =−d(ν{3} 3◦2u) + Φh(µ◦2ν{1} 1−id) =−(µ◦2ν{2} 2−ν{2} 2◦1µ)◦2u+µ◦2(ν{2} 2◦1u) =ν{2} 2◦1(µ◦2u) =ν{2} 2.  Corollary 5.10. The morphism ϕCh(k)in Remark 5.6 is a trivial cofibration. Proof. By Remark 5.8 and Lemma 5.9, ϕCh(k)is a transfinite (countable) composition of trivial cofibrations. Hence ϕCh(k)is itself a trivial cofibration.  Remark 5.11.There are friendly characterizations of the image of the injective map π0(φCh(k))∗:π0MapOp(Ch(k))(uAssCh(k),O)֒→π0MapOp(Ch(k))(AssCh(k),O) when O=EndCh(k)(X) is the endomorphism operad of a cofibrant complex X. Since all complexes are fibrant, these sets are the sets of homotopy classes of maps from the (unital) A-infinity DG-operad to EndCh(k)(X), i.e. homotopy classes of (unital) A-infinity structures on X. Recall from Remark 5.5 the description of the (unital) A-fininity DG-operad (u)ACh(k) ∞and the morphism φCh(k) ∞:ACh(k) ∞֌uACh(k) ∞. An A-infinity structure on Xis given by graded morphisms mn:X⊗n→Xof degree n−2 satisfying certain equations. A unital A-infinity structure is similarly defined by graded morphisms mS n:X⊗n−|S|→Xof degree n−2 + |S|. The underlying A-infinity structure is HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 35 given by the morphisms mn=m∅ n:X⊗n→X. A unital A-infinity structure is strict if mS n= 0 for S6=∅and n > 1. Following the terminology in [Lur12, Definition 5.2.3], we say that an A-infinity structure is quasi-unital if there exists a cycle v∈X0,d(v) = 0, such that the chain maps m2(v, −), m2(−, v): X→Xare chain homotopic to the identity. The underlying A-infinity structure of a unital A-infinity structure is quasi-unital. We can take v=m{1} 1∈X0and the chain homotopies m{1} 2, m{2} 2:X→X. If kis a field we can suppose that Xhas trivial differential. Then quasi-unital means that m2has a unit. It is well known that such a quasi-unital A-infinity structure is quasi-isomorphic, in the A-infinity sense, to a strictly unital A-infinity structure, via a quasi-isomorphism whose linear term is the identity [LH03, §3.2.1]. One can check that two A-infinity structures which are quasi-isomorphic in this way represent the same element in π0MapOp(Ch(k))(AssCh(k),EndCh(k)(X)). Therefore, the image of π0(φCh(k))∗is formed exactly by the homotopy classes of quasi-unital A-infinity structures. A deeper result of Lyubashenko and Manzyuk [LM08, Theorem 3.7] shows that this statement is actually true over any commutative ring k. We do not think that a similar result holds for any target operad O. Our opinion is based in the following facts. The generalization of a quasi-unital A-infinity structure can be straightforwadly defined as follows. A morphism ξ:ACh(k) ∞→ O is quasi-unital if there exists a cycle v∈ O(0)0,d(v) = 0, such that the following three homology classes coincide [idO] = [ξ(µ2)◦1v] = [ξ(µ2)◦2v]∈H0(O(1)). If ξextends to uACh(k) ∞by a morphism ¯ ξ:uACh(k) ∞→ O then ξis quasi-unital. We can take v=¯ ξ(µ{1} 1) since for j= 1,2, d(¯ ξ(µ{j} 2)) = ¯ ξ(d(µ{j} 2)) = ¯ ξ(µ∅ 2◦jµ{1} 1−iduACh(k) ∞) = ξ(µ2)◦j¯ ξ(µ{1} 1)−idO. Actually, it is enough that ξextends to the suboperad P ⊂ uACh(k) ∞spanned by µ∅ n, n≥2, µ{1} 1,µ{1} 2, and µ{2} 2. In particular, the inclusion ACh(k) ∞⊂ P is quasi-unital. The image of the injective map π0(φCh(k))∗consists of homotoy classes with a quasi-unital representative. Suppose that, conversely, all homotoy classes with a quasi-unital representative where in the image for any target operad O. Taking O=Pwe would obtain a morphism ¯ ξ:uACh(k) ∞→ P whose restriction to ACh(k) ∞ would be homotopic to the inclusion. Below we show that the composition of ¯ ξfollowed by the inclusion P ⊂ uACh(k) ∞would be a homotopy automorphism of uACh(k) ∞, so uACh(k) ∞would be a homotopy retract of P. We think this is very unlikely to happen since Pseems too small in terms of both size and coherence for the quasi-unit. Nevertheless, we have been unable to reach a contradiction. Proposition 5.12. Any endomorphism of uACh(k) ∞is a homotopy automorphism. Proof. The monoid of homotopy classes of maps from uACh(k) ∞to itself coincides with the endomorphism monoid of uAssCh(k), since the homology operad of uACh(k) ∞ is uAssCh(k), which is concentrated in degree 0. Recall the presentation of uAssCh(k) in Example 3.10. Any morphism ϕ:uAssCh(k)→uAssCh(k)is determined by the image of the generators, which by arity and degree reasons must be of the form ϕ(µ) = α·µ, ϕ(u) = β·u, α, β ∈k. 36 FERNANDO MURO The relation id = µ◦2uimplies id = ϕ(id) =ϕ(µ◦2u) =ϕ(µ)◦2ϕ(u) = (α·µ)◦2(β·u) =α·β·(µ◦2u) =α·β·id. Therefore α·β= 1, i.e. α∈k×and β=α−1. One can conversely check that for any α∈k×,µ7→ αµ and u7→ α−1udefine an automorphism of uAssCh(k). Hence, π0MapOp(Ch(k))(uACh(k) ∞,uACh(k) ∞) = AutOp(Ch(k))(uAssCh(k))∼ =k×.  6. Transference In this section we prove the main result of this paper, Theorem 1.2. The following proposition is a direct consequence of [Mur14, Proposition 4.1, Corollary C.5, and Theorem E.2] and Corollary 2.9. It allows to transfer our previous main results to a wide class of symmetric monoidal model categories. Proposition 6.1. Let F:V⇄W:Gbe a weak symmetric monoidal Quillen adjunction between symmetric monoidal model categories as in Theorem 3.13. Suppose that Vand Wsatisfy the strong unit axiom and F⊣Gsatisfies the pseudocofibrant axiom and the I-cofibrant axiom. The following statements hold: (1) If φV:AssV→uAssVis a homotopy epimorphism in Op(V)then the map φW:AssW→uAssWis a homotopy epimorphism in Op(W). (2) Suppose in addition that LF: Ho V→Ho Wreflects isomorphisms, e.g. if F⊣Gis a Quillen equivalence. In this case the converse of (1) also holds. The strong unit axiom, the pseudo-cofibrant axiom, and the I-cofibrant axiom were introduced in [Mur14, Definitions A.9 and B.6]. The strong unit axiom holds in all symmetric monoidal model categories with cofibrant unit. The pseudo-cofibrant axiom and the I-cofibrant axiom hold in all Quillen pairs F⊣Gwhere the source of Fhas a cofibrant tensor unit. We now prove the main theorem for the category Ch(k)≥0of non-negative chain complexes. Weak equivalences and the monoidal structure are defined as in Ch(k), fibrations are chain maps which are surjective in positive degrees, compare [SS03, §4.1]. Theorem 6.2. The morphism φCh(k)≥0:AssCh(k)≥0→uAssCh(k)≥0in Example 3.5 is a homotopy epimorphism in Op(Ch(k)≥0). Proof. Consider the symmetric monoidal adjoint pair Ch(k)≥0 inclusion //Ch(k). t≥0 oo The right adjoint t≥0is the truncation functor. It is defined by the fact that the counit t≥0(X)→Xis the identity in positive degrees and the inclusion Ker[d:X0→X−1]⊂X0in degree 0. Clearly, t≥0preserves weak equivalences HOMOTOPY UNITS IN A-INFINITY ALGEBRAS 37 and fibrations, so this is a Quillen pair. The inclusion functor reflects weak equivalences, hence its left derived functor Ho Ch(k)≥0→Ho Ch(k) reflects isomorphisms. The categories Ch(k)≥0and Ch(k) have cofibrant tensor units. These observations show that our adjunction satisfies the assumptions of Proposition 6.1 (2). Therefore the result follows from Theorem 5.1.  The following theorem is the main result for simplicial k-modules with the symmetric monoidal model structure considered in [SS03, §4.1]. Theorem 6.3. The morphism φMod(k)∆op :AssMod(k)∆op →uAssMod(k)∆op in Example 3.5 is a homotopy epimorphism in Op(Mod(k)∆op ). Proof. This follows from Theorem 6.2 and Proposition 6.1 (1) applied to the Dold– Kan equivalence Ch(k)≥0⇄Mod(k)∆op , which is a weak symmetric monoidal Quillen equivalence, see [SS03, §4.2]. Here, both categories have cofibrant tensor unit.  We can now prove the main result for simplicial sets, with the usual cartesian symmetric monoidal model structure, see [Hov99, Proposition 4.2.8]. Theorem 6.4. The morphism φSet∆op :AssSet∆op →uAssSet∆op in Example 3.5 is a homotopy epimorphism in Op(Set∆op ). Proof. We have to show that ϕSet∆op in Definition 3.17 is a weak equivalence, see Lemma 3.18, or equivalently, that u∞uASet∆op (n) is contractible for all n≥0. Consider the following two Quillen pairs, Set∆op Π1//Grd, Ner ooSet∆op Z·− //Mod(Z)∆op . forget oo The first Quillen pair was already considered in the proof of Proposition 4.2. It is a symmetric monoidal Quillen pair in the sense of [Hov99]. The second Quillen pair, induced by the free abelian group functor, is also symmetric monoidal. These four functors happen to preserve weak equivalences, so they coincide with their derived functors. These adjoint pairs induce Quillen pairs between operad categories, see [Mur14, Proposition 4.1]. Applying Π1and Z·− to the push-out square in Definition 3.17 for V= Set∆op , we obtain push-out diagrams in Op(Grd) and Op(Mod(Z)∆op ), respectively, AssGrd φGrd  Π1AssSet∆op // Π1¯ φSet∆op ∞// Π1φSet∆op  push Π1u∞ASet∆op Π1ψSet∆op  uAssGrd Π1uAssSet∆op // Π1ϕSet∆op //Π1u∞uASet∆op AssMod(Z)∆op φMod(Z)∆op  Z·AssSet∆op // Z·¯ φSet∆op ∞// Z·φSet∆op  push Z·u∞ASet∆op Z·ψSet∆op  uAssMod(Z)∆op Z·uAssSet∆op // Z·ϕSet∆op //Z·u∞uASet∆op 38 FERNANDO MURO Here, Π1¯ φSet∆op ∞and Z·¯ φSet∆op ∞are models for ¯ φGrd ∞and ¯ φMod(Z)∆op ∞, respectively, see [Mur14, Theorem 1.7]. Hence, Π1ϕSet∆op and Z·ϕSet∆op are weak equivalences by Theorems 4.3 and 6.3 and Lemma 3.18. In particular, u∞uASet∆op (n) is simply connected and has the homology of a point for all n≥0, therefore it is contractible.  Let us finally prove our main theorem. Proof of Theorem 1.2. It is enough to check that we can apply Proposition 6.1 (1) to the structure symmetric monoidal Quillen pair F⊣Gof the simplicial or complicial monoidal model category V. We are assuming that Vsatisfies the strong unit axiom. The categories Set∆op and Ch(k) have cofibrant tensor units. Hence, they satisfy the strong unit axiom and F⊣Gsatisfies the pseudo-cofibrant axiom and the I-cofibrant axiom.  Remark 6.5.Once Theorem 1.2 is proved, it is reasonable to wonder whether there is a friendly identification of the image of π0(φV)∗as in Remark 5.11 for any V satisfying the hypotheses of that theorem and O=EndV(X) the endomorphism operad of a fibrant-cofibrant object X. Suppose for simplicity that the tensor unit Iis cofibrant. In this case, it is possible to define quasi-unital A-infinity algebras as follows. Let us consider a cofibrant resolution AV ∞ ∼ ։AssVwhich is a trivial fibration. Since Iis cofibrant, the trivial fibration AV ∞(2) ∼ ։AssV(2) = Iis a retraction which admits a section ˜g:I→AV ∞(2). Given an A-infinity structure on X, we define m2:X⊗X→Xas the composite X⊗X∼ =I⊗X⊗X˜g⊗id −→ AV ∞(2) ⊗X⊗X−→ X, where the last morphism is part of the A-infinity structure. We say that an A- infinity structure is quasi-unital if there exists a morphism v:I→Xsuch that the maps m2(v⊗X), m2(X⊗v): X→Xare homotopic to the identity. It looks like if [Lur12, Theorem 5.2.3.5] implied a positive answer for all V satisfying also the hypotheses in [Lur12, Theorem 4.1.4.4], e.g. chain complexes and simplicial sets, but not topological spaces. However, [Lur12, Theorem 5.2.3.5] is not about moduli spaces of algebra structures, but about (generalizations of) Dwyer– Kan simplicial localizations of categories of algebras. The connection between these spaces was established by Rezk [Rez96] for symmetric operads and V= Set∆op or Mod(k)∆op . 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