Resolution of coloured operads and rectification of homotopy algebras

We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their categories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, thereby giving a uniform approach to algebraic structures up to homotopy over coloured operads. The Comparison Theorem implies that such structures can be rectified. Many algebraic stuctures are parametrised by operads, and deformations of such structures are then controlled by suitable resolutions of operads. Early examples in topology are the W-construction of Boardman and Vogt [7], and the explicit A∞and E∞operads of Stasheff [30], May [24] and Boardman-Vogt [7]. For operads in chain complexes, one often uses cobar-bar resolutions (cf. GinzburgKapranov [15], Kontsevich-Soibelman [20]), or the smaller Koszul resolutions (cf. Fresse [14]). A coherent theory of such resolutions is provided by a Quillen homotopy theory of operads, as established by Hinich [16] for operads in chain complexes, by Rezk [26] for simplicial operads, and by Spitzweck [29] and the authors [3] for operads in general monoidal model categories. When such a homotopy theory is available, one can define for any operad P the notion of P -algebra up to homotopy or homotopy P -algebra in a homotopy invariant way as an algebra over a cofibrant resolution of P . Indeed, all important instances of homotopy P -algebras occurring in the literature are of this form. For many operads P , it can moreover be proved that homotopy P -algebras, defined in this way, can be rectified, in the sense of being weakly homotopy equivalent to strict P -algebras. One of the first instances of this phenomenon is the well known fact that in topology, any A∞-space is weakly homotopy equivalent to a topological monoid (Stasheff [30], Boardman-Vogt [7]). The model theoretic framework of [3] provides a general rectification result which includes many of the known cases. In categories where the tensor product is the cartesian product, there is another approach to algebraic structures, based on Lawvere’s notion of algebraic theory. 1991 Mathematics Subject Classification. Primary 18D50; Secondary 18G55, 55U35. c ©0000 (copyright holder) 1 2 CLEMENS BERGER AND IEKE MOERDIJK For such an algebraic theory T , this gives rise to a somewhat different notion of homotopy T -algebra (not based on a resolution of T ), for which a rectification result has been proved in the context of simplicial sets by Badzioch [1] and Bergner [6]. One purpose in this paper is to prove a multi-sorted version of such a rectification theorem, based on the notion of coloured operad. This concept goes back to Boardman and Vogt [7], and also occurs in homotopy theory under the name multicategory (cf. e.g. Elmendorf-Mandell [12]). The precise definition and some typical examples of coloured operads and their algebras will be reviewed in Section 1 of this paper. These examples include bimodules over monoids, enriched (e.g. topological, simplicial, or differential graded) categories, diagrams on a fixed such enriched category, morphisms between algebras over a given operad P , and many more. In particular, operads themselves are also algebras for a suitable coloured operad. We will discuss how the homotopy theory of operads developed in [3, 4] extends to coloured operads. The results will be formulated and proved in a general monoidal model category, so as to provide a uniform treatment of operads and their algebras in a variety of contexts, such as spaces, simplicial sets, chain complexes, spectra, and so on. We will prove that under certain conditions, the category of P -algebras for a coloured operad P in a monoidal model category carries a Quillen model category structure (cf. Theorem 2.1). In Section 4, we will prove a general Comparison Theorem (Theorem 4.1), which provides sufficient conditions for a weak equivalence between coloured operads to induce a Quillen equivalence between the corresponding categories of algebras. This Comparison Theorem plays a central role in the applications. Important instances of such weak equivalences between operads are provided by coloured versions of various types of resolutions mentioned above. We will present one such resolution in detail; it is an extension of the Boardman-Vogt construction [7]. Our construction applies to an arbitrary coloured operad P in a monoidal model category E possessing a suitable interval H, and provides a functorial cofibrant resolution W(H,P )→ P under some mild conditions on P (cf. Theorem 3.5). The notion of homotopy P -algebra is then captured by that of a W(H,P )-algebra. For example, we obtain in this way explicit definitions of notions like “module up to homotopy over an A∞-algebra”, “operad up to homotopy”, etc. If the conditions of the Comparison Theorem are satisfied, we deduce as a corollary that there is a Quillen equivalence (W (H,P )−algebras) ∼ (P−algebras). (1) This equivalence states in particular that every homotopy P -algebra is weakly homotopy equivalent to a true P -algebra. In the rest of the paper, we elaborate some important special cases. The first one is the rectification of homotopy coherent diagrams of spaces over a topological category, going back to Vogt [31], Segal [28], and others. In fact, for an arbitrary monoidal model category E , a rectification result for E-valued homotopy coherent diagrams over an E-enriched category will be seen to be a special case of an equivalence of type (1). Other examples we consider include modules over A∞-algebras, and weak maps between homotopy P -algebras. Acknowledgements: This paper has had a slow incarnation, as the main results were already presented at the homotopy conference in Arolla in August 2004. A large part of the actual writing was done while the second author was appointed RESOLUTION OF COLOURED OPERADS 3 to a visiting professorship at Nice in the Spring of 2005, and he wishes to express again his gratitude to the Laboratoire Dieudonné for its hospitality and support. We would also like to thank John Harper for pointing out a mistake in an earlier version of this paper. 1. Basic definitions and examples Let E be a cocomplete symmetric monoidal category. We will assume that E is closed, and write HomE(X,Y ) or Y X for the object of E representing the internal hom. The closedness of E implies that the tensor product ⊗ of E preserves colimits in each variable. The unit of E will be denoted by I. The symmetric group on n letters will be denoted by Σn. For any finite group Γ, the category of objects in E equipped with a right Γ-action will be denoted by E. It is again a cocomplete closed symmetric monoidal category. 1.1. C-coloured operads. Let C be a set. We will refer to the elements of C as “colours”. A C-coloured operad P is given by the following data: (i) for each n ≥ 0, and each (n+ 1)-tuple (c1, . . . , cn; c) of colours, an object P (c1, . . . , cn; c) in E ; (ii) for each colour c, a unit 1c : I → P (c; c); (iii) for each (n+ 1)-tuple (c1, . . . , cn; c) of colours and n other colour-tuples (d1,1, . . . , d1,k1), . . . , (dn,1, . . . , dn,kn), of lengths k1, . . . , kn respectively, a composition product P (c1, . . . , cn; c)⊗ P (d1,1, . . . , d1,k1 ; c1)⊗ · · · ⊗ P (dn,1, . . . , dn,kn ; cn) γ −→ P (d1,1, . . . , dn,kn ; c); (iv) for each σ ∈ Σn, a map σ : P (c1, . . . , cn; c)→ P (cσ(1), . . . , cσ(n); c). The object P (c1, . . . , cn; c) represents operations, taking n inputs of colours c1, . . . , cn respectively, and producing an output of colour c; this will be made precise by the definition of a coloured endomorphism-operad below. The four data of a coloured operad are required to satisfy several axioms, which are the obvious analogues of the axioms for ordinary symmetric operads: first, the maps in (iv) define a right action by the symmetric group Σn, in the sense that στ = (τσ) for any σ, τ ∈ Σn; secondly, each 1c is a 2-sided unit for the composition product γ; and finally, this composition product γ is associative and Σn-equivariant in some natural sense. With the obvious morphisms, the C-coloured operads in E form a category, denoted OperC(E). 1.2. P -algebras and coloured endomorphism-operads. For a C-coloured operad P , a P -algebra is a family A = (A(c))c∈C of objects of E , together with maps αc1,...,cn;c : P (c1, . . . , cn; c)⊗A(c1)⊗ · · · ⊗A(cn)→ A(c) 4 CLEMENS BERGER AND IEKE MOERDIJK satisfying obvious axioms for associativity, unit and equivariance. For example, for each σ ∈ Σn, the diagram P (c1, . . . , cn; c)⊗A(c1)⊗ · · · ⊗A(cn) αc1,...,cn;c A(c) αcσ(1),...,cσ(n);c P (cσ(1), . . . , cσ(n); c)⊗A(cσ(1))⊗ · · · ⊗A(cσ(n)) σ∗ ⊗ σ−1 ∗ ? commutes, where σ∗ denotes the left action on tensor products induced by σ and the symmetry of E . We will denote such an algebra by (A,α), or simply by A. Equivalently, a P -algebra (A,α) can also be defined as a map of coloured operads α : P −→ End(A) with values in the endomorphism-operad End(A) of the family (A(c))c∈C . This coloured operad is defined by setting End(A)(c1, . . . , cn; c) = HomE(A(c1)⊗ · · · ⊗A(cn), A(c)) where the composition products (resp. the Σn-actions) are induced by substitution (resp. permutation) of the tensor factors. A map of P -algebras f : A → B is a family (fc : A(c) → B(c))c∈C of maps in E , respecting the algebra structures in the sense that each diagram of the form P (c1, . . . , cn; c)⊗A(c1)⊗ · · · ⊗A(cn) A(c) P (c1, . . . , cn; c)⊗B(c1)⊗ · · · ⊗B(cn) id⊗ fc1 ⊗ · · · ⊗ fcn ? B(c) fc ? commutes. This defines a category of P -algebras, denoted AlgE(P ). Exactly as in the uncoloured case, a map of C-coloured operads α : P → Q induces adjoint functors α! : AlgE(P ) AlgE(Q) : α ∗. Remark 1.3. Suppose the set C of colours is equipped with a linear ordering ≤. If c1 ≤ · · · ≤ cn are elements of C, write Σc1...cn ⊆ Σn for the subgroup of permutations σ for which cσ(1) ≤ · · · ≤ cσ(n) (so in particular (cσ(1), . . . , cσ(n)) is the same n-tuple as (c1, . . . , cn)). Then a C-coloured operad P is completely determined by the objects P (c1, . . . , cn; c) for c1 ≤ · · · ≤ cn, and Σc1...cn acts from the right on this object. In other words, we can view a C-coloured operad as an object in ∏ c1≤···≤cn,c Ec1...cn , equipped with units I → P (c; c) and suitably equivariant and associative composition maps. Similarly, a P -algebra structure on a family A = {A(c)}c∈C is completely determined by (Σc1...cn -equivariant) action maps P (c1, . . . , cn; c)⊗Σc1...cn (A(c1)⊗ · · · ⊗A(cn))→ A(c), for any c and any ordered sequence c1 ≤ · · · ≤ cn. It will often be convenient to work with this “smaller” representation of a coloured operad P and its algebras. RESOLUTION OF COLOURED OPERADS 5 Remark 1.4. If the set C of colours is singleton, a C-coloured operad P is the same as a classical (symmetric) operad, and P -algebra has its classical meaning. We will speak of uncoloured operads if we want to emphasize that there is just one colour. There is an obvious notion of non-symmetric coloured operad, obtained from 1.1 by deleting all references to the symmetric group actions. If we speak of a “Ccoloured operad”, we will always mean one in the sense of Definition 1.1, although we will sometimes speak of “symmetric C-coloured operads”, for emphasis. Exactly as for classical operads, every non-symmetric C-coloured operad has an induced symmetric C-coloured operad which defines the same category of algebras. 1.5. Examples of algebras over coloured operads. 1.5.1. Modules over operads. Let P be an (uncoloured) operad, and let C = {a,m} be a 2-element set. There is a C-coloured operad ModP whose algebras are pairs (A,M) = (A(a), A(m)) where A is a P -algebra andM is an A-module: One sets ModP (c1, . . . , cn; c) = P (n) if c = a and all the ci are equal to a also. And one sets ModP (c1, . . . , cn,m) = P (n) when exactly one of the ci is m, and ModP (c1, . . . , cn,m) = 0 in all other cases. The structure maps of ModP are induced by those of P . 1.5.2. (Bi)modules over monoids. Write Ass for the (non-symmetric) operad whose algebras are (unitary associative) monoids in E ; so Ass(n) = I for every n ≥ 0. There is a non-symmetric operad LMod on two colours, a and m, whose algebras are pairs (M,E) where M is a monoid in E acting from the left on an object E of E : LMod(c1, . . . , cn; c) = I if c = a and each ci = a, or if c = m and c1 = · · · = cn−1 = a while cn = m, and LMod(c1, . . . , cn;m) = 0 in all other cases. There are similar operads RMod on two colours and BiMod on three colours, whose algebras are pairs (M,E) where M is a monoid acting from the right on E, respectively triples (M,E,N) where M and N are monoids and E is an M -N bimodule in E . 1.5.3. Morphisms. Let P be an arbitrary (non-coloured) operad. There is a coloured operad P 1 on a set {0, 1} of two colours, whose algebras are triples (A0, A1, f) where A0 and A1 are P -algebras, and f : A0 → A1 is a map of P algebras. Explicitly, P (i1, . . . , in; i) = { P (n) if max(i1, . . . , in) ≤ i; 0 otherwise. The structure maps of P 1 are induced by those of P (for n = 0, we agree that max(i1, . . . , in) = −1). Given a P -algebra on two objects A0 and A1, the objects P (0, . . . , 0; 0) and P (1, . . . , 1; 1) give A0, respectively A1, their P -algebra structure; furthermore, 1 : I → P (1) corresponds to a map α : I → P (0; 1) giving a map of P -algebras f : A0 → A1. This coloured operad has been discussed extensively in the context of chain complexes by Markl [21]. 1.5.4. Categories enriched in E. Let O be a set, and consider the product C = O × O. There is a (non-symmetric) C-coloured operad CatO whose algebras are the E-enriched categories with O as set of objects, and for which the maps between algebras are the functors which act by the identity on objects: One puts CatO((c1, c1), . . . , (cn, c ′ n); (c ′ 0, cn+1)) = I 6 CLEMENS BERGER AND IEKE MOERDIJK whenever ci = ci+1 for i = 0, . . . , n, and zero in all other cases. (In particular, for n = 0 we have CatO(; (c, c)) = I for each c ∈ C, providing the CatO-algebras with the necessary identity arrows.) 1.5.5. Diagrams in E. Let C be a fixed E-enriched category, with set of objects O. There is a non-symmetric O-coloured operad DiagC whose algebras are covariant E-valued diagrams on C: one puts DiagC(o1, . . . , on; o) = { HomC(o1; o) if n = 1; 0 if n > 1. Composition in DiagC is given by composition in C. There is of course a similar operad for contravariant diagrams. 1.5.6. Operads. We will describe a coloured operad SE , whose category of algebras is the category of (uncoloured) operads in E . The set of colours in this case is the set N of natural numbers. In fact, the operad to be defined is a coloured operad S in Sets. Then, the strong symmetric monoidal functor Sets → E X 7→ XE = ∐ x∈X I maps S to a coloured operad SE whose algebras are operads in E . The elements of S(n1, . . . , nk;n) are equivalence classes of triples (T, σ, τ) where T is a planar rooted tree with n input edges and k vertices, σ is a bijection {1, . . . , k} → V (T ) (i.e. the set of vertices of T ) with the property that the vertex σ(i) has valence ni (i.e. ni input edges), and τ is a bijection {1, . . . , n} → in(T ), the set of input edges of T . Two such triples (T, σ, τ), (T ′, σ′, τ ′) represent the same element of S(n1, . . . , nk;n) if there is a (planar) isomorphism φ : T → T ′ with φ ◦ τ = τ ′ and φ ◦ σ = σ′. Any α ∈ Σk induces a map α : S(n1, . . . , nk;n) → S(nα(1), . . . , nα(k);n) sending (the equivalence class of) (T, σ, τ) to (T, σα, τ). The identity element 1n ∈ S(n;n) is represented by the tree tn (the corolla with n leaves) whose inputs are numbered 1, . . . , n from left to right with respect to the planar structure. The composition product is defined as follows: given (T, σ, τ) as above, and k other such (T1, σ1, τ1), . . . , (Tk, σk, τk), with n1, . . . , nk inputs and p1, . . . , pk vertices respectively, one obtains a new planar rooted tree T ′ by replacing the vertex σ(i) in T by the tree Ti, identifying the ni input edges of σ(i) in T with the ni input edges of Ti via the bijection τi (the l-th input edge of σ(i) in the planar order is matched with the input edge τi(l) of Ti). The vertices of the new tree T ′ are numbered in the following order: first the vertices of Tσ(1) in the order given by σ1, then the vertices in Tσ(2) in the order given by σ2, etc. In other words, the map {1, . . . , p1 + · · · + pk} → V (T ′) is given by (σ1 × · · · × σk) ◦ σ(p1, . . . , pk) where σ(p1, . . . , pk) permutes the blocks of size pi. The new tree T ′ still has n input edges, which are ordered as given by τ and the identifications given by the τi. Notice that S(n1;n) = Σn if n1 = n, and S(ni, n) = φ otherwise. More precisely, S(n, n) consists of pairs (tn, τ) where tn is the tree above and τ is a numbering of its inputs. The composition product of S in particular gives a map S(n, n) × S(n, n) → S(n, n) which sends ((tn, τ), (tn, ρ)) to (tn, ρτ), so that S(n;n) is identified with the opposite group of Σn. The S-algebras are exactly the operads in sets. Indeed, given such an operad P , a triple (T, σ, τ) ∈ S acts on (p1, . . . , pk) ∈ P (n1) × · · · × P (nk) by labelling the vertex σ(i) ∈ T by pi, and then using the operad structure of P to compose RESOLUTION OF COLOURED OPERADS 7 (p1, . . . , pk) along the tree T to get an element p ∈ P (n), and then applying the right action by τ to this element; i.e. (T, σ, τ)(p1, . . . , pk) = p · τ. In particular, the action S(n, n)× P (n)→ P (n) encodes the right Σn-action. There is a similar coloured operad S, whose algebras are operads P = (P (n))n≥1 without 0-term. It is the coloured suboperad of S given by considering only vertices of valence ≥ 1. This operad S is in fact induced by a non-symmetric N-coloured operad, cf. Remark 1.4. Indeed, it is sufficient to show that for any planar tree T and any ordering τ of its input edges, there is an ordering σT,τ of its vertices, which is compatible with the composition product just described, and is invariant in the sense that for a planar isomorphism φ : T → T ′, the following relation holds: σT ′,φ◦τ = φ ◦ σT,τ . The easiest way to define σT,τ is to view σT,τ (resp. τ) as linear orderings of the vertices (resp. input edges) of T . So, given a linear ordering of the input edges of T , we have to define a linear ordering of its vertices; we use induction on T . Suppose that T is obtained by grafting p trees on the corolla tp, for short: T = tp(T1, . . . , Tp), with root r; write ≤i for the linear order on Ti corresponding to the linear ordering of its input edges induced by that of T . Also write Ti < Tj if the first input edge (in the ordering) of Ti comes before the first one of Tj . This defines a linear order on the p-element set {T1, . . . , Tp}. For vertices v, w ∈ T , now define v ≤ w ⇐⇒  v = r; or v, w ∈ Ti and v ≤i w; or v ∈ Ti and w ∈ Tj and Ti < Tj . 1.5.7. Coloured operads. Let C be a set of colours. There is a coloured operad SC whose algebras are C-coloured operads. The set of colours of SC is the set of sequences of the form (c1, . . . , cn; c) for all n ≥ 0 (or, more economically, those sequences for which c1 ≤ · · · ≤ cn after having chosen an order on C). We leave a detailed description of SC to the reader. 1.6. Change of colour. If P is a C-coloured operad and α : D → C is a map between sets “of colours”, then P pulls back to a D-coloured operad α∗(P ) in the obvious way, α∗(P )(d1, . . . , dn; d) = P (α(d1), . . . , α(dn);α(d)). This defines a functor α∗ : OperC(E)→ OperD(E). In this way, C-coloured operads for varying sets of colours C together form a fibered category over the category of sets. Objects of this fibered category are pairs (C,P ) where P is a C-coloured operad, and arrows (D,Q)→ (C,P ) are pairs (α, φ) where α : D → C and φ : Q → α∗(P ) is a map of D-coloured operads. Such an arrow induces an adjoint pair (α, φ)! : AlgE(Q) AlgE(P ) : (α, φ) ∗. For instance, if SC and SD are the operads whose algebras are C-coloured operads and D-coloured operads respectively, the map α : D → C induces in this way a 8 CLEMENS BERGER AND IEKE MOERDIJK map of coloured operads SD → SC , and hence an adjoint pair α! : OperD(E) OperC(E) : α∗. 1.6.1. Example. Let P be an uncoloured operad. A graded P -algebra is a sequence (An)n≥0 of objects of E (indexed by n ∈ N), such that the coproduct A = ∐ An has a P -algebra structure, which respects the grading in the sense that the structure map P (k)⊗A⊗k → A maps the summand P (k)⊗Ak1 ⊗ · · · ⊗Ank to the summand An for n = n1 + · · · + nk. Associated to P , there is an N-coloured operad Gr(P ) whose algebras are the graded P -algebras. It is given by Gr(P )(n1, . . . , nk;n) = { P (k) if n1 + · · ·+ nk = n; 0 otherwise. There is an evident map of coloured operads (α, φ) : (N,Gr(P )) −→ (∗, P ) given by the unique map α : N → ∗ and the inclusion φ : Gr(P )(n1, . . . , nk;n) α∗(P )(n1, . . . , nk;n) = P (k). The left adjoint (α, φ)! sends the graded P -algebra (An)n≥0 to the P -algebra ∐ An. 1.6.2. Example. Suppose E is additive. Let PLie be the operad for pre-Lie algebras in E , and let S0 be the N-coloured operad for non-symmetric operads in E . (S0 is defined as S in 1.5.7, but without the τ ’s). There is a map of N-coloured operads (σ; c) : (N,Gr(PLie))→ (N, S0) defined as follows: σ(n) = n + 1, and c sends the pre-Lie operation ◦n,m ∈ Gr(PLie)(n,m;n+m) to the sum of the ◦i-operations in S0(n+1,m+1;n+m+1), where the ◦i-operation is the tree with two vertices of valence n+ 1 and m+ 1 respectively and one internal edge at the i-th entry of the lower vertex. The familiar construction of a pre-Lie algebra out of a non-symmetric operad is now given by the composition (α, φ)! ◦ (σ, c)∗, where (∗,PLie) (α, φ) (N,Gr(PLie)) (σ, c) (N, S0) as above. (There is also a symmetric version of this construction, cf. KapranovManin [18].) The functorial Boardman-Vogt resolution to be discussed in Section 4 will evidently have the property that Gr(W(P )) = W(Gr(P )), and the maps above will induce maps between the corresponding cofibrant resolutions (∗,W(PLie)) (N,W(Gr(PLie))) (N,W(S0)). This shows that the same construction yields for any “operad up to homotopy” a pre-Lie algebra up to homotopy (and hence a W(Lie)-algebra, i.e. an L∞-algebra). 2. Model structure on P -algebras In this section, we assume that our cocomplete symmetric monoidal closed category E is equipped with a compatible Quillen model structure, making it into a so-called monoidal model category. We will always assume that the unit I of E is cofibrant and that the model structure is cofibrantly generated. Recall that under the last assumption, for any finite group G, there is an induced monoidal model structure on the category E of objects with right G-action, for which the forgetful RESOLUTION OF COLOURED OPERADS 9 functor E → E preserves and reflects fibrations and weak equivalences. We refer to [13], [17] for basic facts concerning monoidal model categories and associated equivariant categories like E, see also [4, 2.5]. For a set of colours C and a C-coloured operad P , our purpose is to show that, under suitable conditions, the category AlgE(P ) of P -algebras admits a closed model structure for which the forgetful functor