Note that every strictly commutative algebraA is anEn-algebra for any 16n6∞, and that every symmetric A-bimodule is a representation of the En-algebra A

ITERATED BAR COMPLEXES AND E_n-HOMOLOGY WITH COEFFICIENTS

BENOIT FRESSE AND STEPHANIE ZIEGENHAGEN

Abstract. The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized toEn-algebras, and that one can calculate En- homology with trivial coefficients via this iterated bar construction. We extend this result to En-homology andEn-cohomology of a commutative algebraAwith coefficients in a symmetric A-bimodule.

Introduction

Boardman-Vogt [BV73] and May [Ma72] showed in the early 70’s thatn-fold loop spaces are equivalent to algebras over the little n-cubes operad (at least, up to group completion issues, or provided that we restrict ourselves to connected spaces). In this paper, we deal with E_n- operads in chain complexes, which are defined as operads quasi-isomorphic to the chain operad of littlen-cubes, for any 16n6∞. The category ofEn-algebras, algebras over anEn-operad, can be thought of as an algebraic version of the category ofn-fold loop spaces. If we choose a Σ∗-cofibrantEn-operad, letEn, then we can define homological invariants specifically suited to E_n-algebras, namely theE_n-homology andE_n-cohomology of anE_n-algebraAwith coefficients in a representation M of A. These invariants are defined via derived Quillen functors. Note that every strictly commutative algebraA is anE_n-algebra for any 16n6∞, and that every symmetric A-bimodule is a representation of the E_n-algebra A. The homology of E_n-algebras can also be regarded as a particular case of the factorization homology theory (see [Fra13]), and, in the case of commutative algebras, agrees with Pirashvili’s higher Hochschild homology theory (see [Pir00]). In the casen=∞, we retrieve the Γ-homology theory of [RW02].

In [Fre11] the first author also proved that the classical iterated bar complex of augmented commutative algebras Bⁿ(A) = B ◦ · · · ◦ B(A), such as defined by Eilenberg and MacLane (see [EM53]), computes the E_n-homology of A with coefficients in the commutative unital ground ring M = k. The purpose of this paper is to extend this construction to the non- trivial coefficient case in order to get an explicit chain complex computing the En-homology and the E_n-cohomology of commutative algebras with coefficients. In short, we define a chain complex (M ⊗Bⁿ(A), ∂θ) by adding a twisting differential ∂θ :M ⊗Bⁿ(A) →M ⊗Bⁿ(A) to the tensor product of the Eilenberg-MacLane iterated bar complex Bⁿ(A) with any symmetric A-bimodule of coefficients M (see Definition 1.2). We then prove that the chain complex B∗^[n](A, M) = (M⊗Σ⁻ⁿBⁿ(A), ∂_θ), where we use ann-fold desuspension Σ⁻ⁿto mark an extra degree shift, computes theE_n-homology ofA with coefficients in M (see Theorem 1.3). Thus, we get

H_∗^En(A, M) =H∗(M⊗Σ⁻ⁿBⁿ(A), ∂_θ).

In the cohomology case, our result reads

H_E^∗_n(A, M) =H∗(Hom_k(Σ⁻ⁿBⁿ(A), M), ∂θ),

2000Mathematics Subject Classification. 55P48, 57T30, 16E40.

Key words and phrases. En-homology, Iterated bar construction, En-operad, Module over an operad, Hochschild homology.

The first author is supported in part by grant ANR-11-BS01-002 “HOGT” and by the Labex ANR-11-LABX- 0007-01 “CEMPI”. The second author gratefully acknowledges support by the DFG.

1

for a twisted cochain complex B_[n]^∗ (A, M) = (Hom_k(Σ⁻ⁿBⁿ(A), M), ∂θ) which we obtain by replacing the tensor product of the homological constructionM⊗Bⁿ(A) by a differential graded module of homomorphisms Hom_k(Bⁿ(A), M). These results hold for all 16n6∞, including n=∞for a suitable infinite bar complex Σ^−∞B^∞(A) which we define as a colimit of the finitely iterated bar constructions Σ⁻ⁿBⁿ(A),n= 1,2, . . ..

Let us emphasize that we use the plain Eilenberg-MacLane iterated bar complexes in our approach. The chain complexes which we define in this paper therefore give a small algebraic counterpart of the constructions of factorization homology theory [Fra13], which are based on higher category theory models, and of the constructions of Pirashvili’s higher Hochschild homology theory [Pir00], which is defined in terms of functors on simplicial sets. But we do not use this analogy further in this paper. To get our result, we mainly rely on the theory of modules over operads [Fre09] and on the methods of [Fre11] which we extend to appropriately address the non-trivial coefficient setting. The small explicit complexes for calculating En-homology and -cohomology we exhibit in this article have been employed by the second author in [Z] to show thatEn-homology and -cohomology can be interpreted as functor homology.

Plan. We define the chain complexes B∗^[n] and B_[n]^∗ and we state the main theorems in the first section of the paper. We explain the correspondence between these chain complexes and modules over operads in the second section, and we prove that these chain complexes effectively compute theEn-homology and theEn-cohomology of commutative algebras in the third section of the paper.

Conventions. We fix a unital commutative ringk. We work in the categorydg-mod of lowerZ- graded differential gradedk-modules, with the differential lowering the degrees. For a differential graded module M and m∈M we denote the corresponding element in the suspension ΣM by sm. For an associative nonunital differential graded algebra B we denote by B+ the unital algebra obtained by adjoining a unit to B.

Forea finite set, we denote by|e|its arity. Fors>0 we sets={1, . . . , s}and [s] ={0, . . . , s}.

By k < e >we denote the freek-module with basis e.

We denote the symmetric operad encoding associative algebras in differential graded k- modules byAsand the symmetric operad encoding commutative algebras in differential graded k-modules byCom. We choose to work with the differential graded version of the Barratt-Eccles operad E. Recall that E is a Σ∗-cofibrant E∞-operad which admits a filtration E1 ⊂ E2 ⊂

· · · ⊂ E_n ⊂ E_n+1 ⊂ · · · ⊂ E, such that E_n is a Σ∗-cofibrant E_n-operad, see [Be97]. We also set E∞ = E for n = ∞. Note that E admits a morphism E → Com, hence by restriction of structure we can view every differential graded commutative algebra as an En-algebra. We mostly deal with non-unital algebras in what follows. We therefore assume E(0) = 0 when we deal with the Barratt-Eccles operad E. We similarly set As(0) = 0 for the associative operad As, and Com(0) = 0 for the commutative operad Com.

For the necessary background on symmetric sequences, operads, algebras over an operad and right modules over an operad we refer the reader to [Fre09]. We denote the plethysm for symmetric sequences by ◦. We will frequently switch between considering Σ∗-modules (with the symmetric groups acting on the right) and contravariant functors defined on the category Bij of finite sets and bijections as explained for example in [Fre11, 0.2].

Let K◦ P be the free right P-module over the operad P generated by the Σ∗-module K.

Given a morphism f: K → R of Σ∗-modules with target a right P-module, we denote by

∂f:K◦ P →R the induced morphism of right P-modules.

1. The results We briefly explain our results in this section.

SinceE_nis Σ∗-cofibrant, the category ofE_n-algebras forms a semi-model category (see [Fre09, ch.12]). In particular there is a notion of cofibrant replacement for En-algebras, and we can

2

define their homology and cohomology as derived functors

H_E^∗_n(A;M) =H∗(Der_En(Q_A, M)) and H_∗^En(A;M) =H∗(M ⊗_UEn(QA)Ω¹_En(Q_A)) for a cofibrant replacementQA of the En-algebra Aand a representation M of A. Here for an operad P and a P-algebra A, a representation M of A can be thought of as generalizing the notion of anA-module. ByUP(A) we denote the enveloping algebra ofA. Representations ofA correspond to left modules overUP(A). We denote by DerP(A;M) the differential graded mod- ule ofP-derivations from Ato M, and Ω¹_P(A) is the associated module of K¨ahler differentials, satisfying

DerP(A, M)∼= Hom_UP(A)(Ω¹_P(A), M),

where Hom_UP(A)(−,−) denotes the hom-object enriched in differential graded modules of the category of differential graded left modules overUP(A).

ForP =Com we retrieve the usual notions of symmetric bimodules, derivations and K¨ahler differentials. The universal enveloping algebra is given by

UCom(A) =A₊.

All of these objects are appropriately natural, in particular they are natural in P. Hence any symmetric bimodule over a commutative algebra A is also a representation of A as an E_n-algebra. We refer the reader to [Fre09, ch.4, ch.10, ch.13] for further background on these constructions.

In [Fre11] the first author proves that the iterated bar construction can be extended to E_n- algebras, and that up to a suspension this complex calculates theEn-homology of a givenEn- algebra with coefficients in k. In this article, we prove a corresponding result forE_n-homology of commutative algebras with coefficients in a symmetric bimodule.

The bar construction is usually defined for augmented unital associative algebras, but we prefer to deal with nonunital algebras. To retrieve the usual construction, we simply use that a nonunital algebraA is equivalent to the augmentation ideal of the algebraA+ which we define by adding a unit to A.

The bar construction B(A) = (T^c(ΣA), ∂s) of a differential graded associative algebra A is given by the reduced tensor coalgebra T^c(−) on the suspension of A together with a twisting differential ∂_s which is determined by the multiplication of A. If A is commutative, then the bar construction B(A) also becomes a commutative differential graded algebra with the shuffle product as product, and we can iteratively defineBⁿ(A). We review this construction in more detail in the next section.

The iterated bar construction Bⁿ(A) is spanned by planar fully grown n-level trees whose leaves are labeled by a tensor in A. For 16 n < ∞, we formally define a planar fully grown n-level treet (ann-level tree for short) as a sequence of order-preserving surjections

t= [r_n] ^{fn //}· · · ^{f2 //}[r₁].

The elements in [r_i] are the vertices at leveliof the tree. The elements of [r_n] are the leaves of t. The labeled trees which we consider in the definition of the bar construction are pairs (t, π), wheretis ann-level tree andπ =a0⊗ · · · ⊗arn ∈A^⊗[rn]is a tensor whose factors are indexed by the leaves [r_n]. We adopt the notationt(a₀, . . . , a_rn) for the element of the n-fold bar complex defined by such a pair (t, π).

In what follows, we also consider trees equipped with a labeling by a finite sete. In this case, the labeling is a bijection

σ:e−^∼=→[rn].

We refer the reader to [Fre11] and [FrApp] for a description of the differential onBⁿ(A). Be- fore we define the complex calculatingE_n-homology we need to introduce a notation concerning trees.

3

Definition 1.1. Lett= [r_n] ^{fn //}· · · ^{f2 //}[r₁] be an n-level tree. Fors∈[r_n] such that sis not the only element in the corresponding 1-fiber oft containingswe let t\sbe the tree

t\s= [r_n−1] ^f

n0 //[rn−1] ^{fn−1 //}· · · ^{f2 //}[r₁] with

f_n⁰(x) =

(fn(x), x < s, f_n(x+ 1), x>s.

To a tree (t, σ) labeled by a finite set e, we also associate the tree (t\s, σ⁰_σ−1(s)) equipped with the labeling σ⁰_σ−1(s):e\ {σ⁻¹(s)} →[rn−1] such that:

σ_σ⁰−1(s)(e) = (

σ(e), σ(e)< s, σ(e) + 1, σ(e)> s.

Definition 1.2. Let A be a differential graded commutative algebra. We define a twisting differential ∂θ by setting

∂_θ(u⊗t(a₀, . . . , a_rn)) = X

06l6rn−1,

|fn⁻¹(l)|>1, x=minfn⁻¹(l)

(−1)^{sn,x−1+|ax|(|a0|+···+|ax−1|)}ua_x⊗(t\x)(a₀, . . . ,aˆ_x, . . . , a_rn)

+ X

06l6rn−1

|f_n⁻¹(l)|>1, y=maxfn⁻¹(l)

(−1)^{sn,y+|ay|(|u|+|a0|+···+|ay−1|)}a_yu⊗(t\y)(a₀, . . . ,aˆ_y, . . . , a_rn)

for a tree t = [rn] ^{fn //}· · · ^{f2 //}[r1] labeled by a0, . . . , arn ∈ A and u ∈ A+. We omit the definition of the signs sn,a at this point, the definition will be given in 2.4.

For example, the element represented by the decorated 2-level tree

u ⊗

@

@@ S

SS a₀ a₁ a₂

A AA

a₃ a₄

is mapped by∂_θ to

−ua₀ ⊗

@

@@ A

AA a1 a2

A AA

a3 a4

− (−1)^{|a3|(|a0|+|a1|+|a2|)}ua₃ ⊗

@

@@ S

SS

a0 a1 a2 a4

+ (−1)^{|a2|(|u|+|a0|+|a1|)}a₂u ⊗

@

@@ A

A A

a₀ a₁

A AA

a₃ a₄

− (−1)^{|a4|(|u|+|a0|+···+|a3|)}a₄u ⊗

@

@@ S

SS

a₀ a₁ a₂ a₃

For a symmetric A-bimodule M, we set:

B∗^[n](A, M) =M⊗_A+(A+⊗Σ⁻ⁿBⁿ(A), ∂_θ)

4

and

B_[n]^∗ (A, M) = Hom_A+((A₊⊗Σ⁻ⁿBⁿ(A), ∂_θ), M).

In this construction, we just use that the algebraUCom(A) =A₊ is identified with a symmetric bimodule over itself.

This article is dedicated to proving the following results.

Theorem 1.3. Let 16n6∞. Let A be a commutative differential graded algebra and let M be a symmetric A-bimodule in dg-mod. Provided that A is cofibrant in dg-mod, we have the identities

H_∗^En(A;M) =H∗(B∗^[n](A, M)) and H_E^∗_n(A;M) =H∗(B_[n]^∗ (A, M)).

We complete the proof of this theorem in Section 3.

2. From iterated bar complexes to modules over operads

Reminder on the trivial coefficient case. The (unreduced) bar construction was originally defined by Eilenberg-MacLane for augmented algebras in dg-mod in [EM53, II.7]. We recall from [Fre11, 1.6] the definition of the nonunital reduced bar construction in the context of right modules over an operad, which we will be working in.

Let P be an operad in differential graded modules and let R be an As-algebra in right P- modules. The bar construction associated to R is a twisted right P-module such that BR = (T^c(ΣR), ∂_s), where we use the tensor product of rightP-modules to form the tensor coalgebra

T^c(ΣR) =M

l>1

(ΣR)^⊗l.

For a finite setethe object (ΣR)^⊗l(e) is spanned by tensorssr1⊗· · ·⊗sr_l∈ΣR(e1)⊗· · ·⊗ΣR(e_l) such thate=e₁t · · · te_l. The twist∂_s is defined by

∂_s(sr₁⊗ · · · ⊗sr_l) =

l−1

X

i=1

(−1)^{i+|r1|+···+|ri|}sr₁⊗ · · · ⊗sγ(id₂;r_i, r_i+1)⊗ · · · ⊗sr_l where id2 ∈ As(2) andγ refers to the action ofAson R.

If R is commutative, i.e.if the action ofAs on R factors through Com, then BR is a Com- algebra in right P-modules. Forsr₁⊗ · · · ⊗sr_p∈ΣR(e₁)⊗ · · · ⊗ΣR(e_p), sr_p+1⊗ · · · ⊗sr_p+q∈ ΣR(e_p+1)⊗ · · · ⊗ΣR(e_p+q) the multiplication is given by the shuffle product

sh(sr₁⊗ · · · ⊗sr_p, sr_p+1⊗ · · · ⊗sr_p+q) = X

σ∈sh(p,q)

(−1)sr_σ⁻¹₍₁₎⊗ · · · ⊗sr_σ⁻¹_(p+q), where sh(p, q) ⊂ Σp+q denotes the set of shuffles of {1, . . . , p} with {p+ 1, . . . , p+q}. The sign is the graded signature of the shuffle σ, i.e. it is determined by picking up a factor (−1)^{(|ri|+1)(|rj|+1)} whenever i < j and σ(i)> σ(j).

In particular, if R is commutative we can iterate the construction and define an n-fold bar complexBⁿ(R). Applying this toCom itself, regarded as aCom-algebra in rightCom-modules, we define the commutative algebra B_Comⁿ in right Com-modules by

B_Comⁿ :=Bⁿ(Com).

According to [Fre11, 2.7, 2.8] the iterated bar module B_Comⁿ is a quasifree right Com-module B_Comⁿ = (Tⁿ◦ Com, ∂_γ) with Tⁿ = (T^cΣ)ⁿ(I) a free Σ∗-module, which we will discuss in more detail later. By [Fre11, 2.5] it is possible to lift∂_γ to a twisting differential∂:Tⁿ◦E →Tⁿ◦E and setB_Eⁿ = (Tⁿ◦E, ∂). For an E-algebraA we call

B_Eⁿ(A) =B_Eⁿ◦_EA

then-fold bar complex ofA. Using thatEis equipped with a cell structure indexed by complete graphs the first author proves in [Fre11, 5.4] that∂ restricts to ∂:Tⁿ◦En→Tⁿ◦En. Hence

5

setting B_Eⁿ_n = (Tⁿ◦En, ∂) we can define ann-fold bar complexB_Eⁿ_n(A) =Bⁿ_En◦_En Afor any E_n-algebra A. IfA is commutative, i.e.if A is an E-algebra such that the action of E on A factors as the standard trivial fibration E → Com followed by an action of Com on A, then B_Eⁿ_n(A) =B_Eⁿ(A) =B_Comⁿ (A) is the usual bar construction.

For any E-algebra A in dg-mod or in right modules over an operad there is a suspension morphismσ: ΣA→BE(A), induced by the inclusioni: ΣI →T^cΣI. Hence we can set

Σ^−∞B^∞(A) = colim_nΣ⁻ⁿBⁿ_E(A).

In particular we can define the right E-module Σ^−∞B_E^∞. Since the associated suspension morphism is of the form σ = i◦E this is again a quasifree right E-module, generated by the Σ∗-module Σ^−∞T^∞= colimnΣ⁻ⁿTⁿ. Similar considerations apply to Σ^−∞B_Com^∞ , and we have Σ^−∞B^∞_E(Com) = Σ^−∞B_Com^∞ .

As a quasifree right En-module in nonnegatively graded chain complexesB_Eⁿ_n is a cofibrant right En-module. The augmentation of the desuspended n-fold bar complex : Σ⁻ⁿB_Eⁿ_n → I is defined as the composite Σ⁻ⁿB_Eⁿ_n → Σ⁻ⁿTⁿI → Σ⁻ⁿΣⁿ(I) =I with the first map induced by the operad morphism E_n → Com → I and the second map an iteration of the projection T^c(ΣI)→ ΣI. In [Fre11, 8.21, 9.4] the first author shows that is a quasiisomorphism, hence Σ⁻ⁿBⁿ_E

n is a cofibrant replacement ofI. From this one deduces that, for anE_n-algebraAwhich is cofibrant as a differential gradedk-module, there is an isomorphism

H_∗^En(A;k)∼=H∗(Σ⁻ⁿBⁿ_En(A)) for 16n6∞ (see [Fre11, 8.22, 9.5]).

The twisted module modeling iterated bar complexes with coefficients. We now define the twist∂_θ on M⊗B_Comⁿ (A) which will incorporate the action of the nonunital commutative algebra A on the symmetric A-bimodule M. As we will see the general case follows from the case of universal coefficientsM =UCom(A) =A₊. We first recall the definition of the universal enveloping algebra associated toCom. Note that this is a special case of the notion of enveloping algebras associated to operads, which we will discuss in Proposition 3.7.

Definition 2.1. We denote by UCom the algebra in right Com-modules given by UCom(i) = Com(i+ 1),i.e.UCom(i) =kfor all i>0 with trivial Σi-action, seee.g. [Fre09, 10.2.1]. Equiv- alently,

UCom(e) =k

for all finite sets e. Denote by µ_e the generator of Com(e) = k. Setf₊=f t {+} and denote the generatorµ_f

+ ∈UCom(f) =Com(f t+) by µ^U_f. Then the multiplication in UCom is µ^U_e ·µ^U_f =µ^U_etf,

while the right Com-module structure is given by

µ^U_e ◦_eµ_f =µ^U_(etf)\{e}

fore∈e.

Observe thatA+=UCom◦_ComA. Hence

UCom(A)⊗Bⁿ(A) = (UCom⊗B_Comⁿ )◦ComA and it suffices to define

∂_θ:UCom⊗B_Comⁿ →UCom⊗B_Comⁿ .

Before giving the definition of this twisting differential, we check thatUCom⊗(Tⁿ◦ Com) is a freeUCom-module in right Com-modules, and we examine the generating Σ∗-moduleTⁿ closer.

6

Definition 2.2. Let P be an operad and (U, µU,1U) an associative unital algebra in right P- modules with P-action γ_U:U ◦ P →U. The category _UMod(M_P) of left U-modules in right P-modules consists of rightP-modules (M, γ_M) equipped with a leftU-actionµ_M:U⊗M →M which is a morphism of right P-modules. The morphisms of this category are morphisms of right P-modules which preserve the U-action.

Proposition 2.3. Let M be a Σ∗-module. The free object in _UMod(M_P) generated by M is U ⊗(M ◦ P) with U-action given by

U⊗U⊗(M◦ P) ^{µU⊗(M◦P) //}U⊗(M◦ P) and right P-module structure defined by

(U⊗(M ◦ P))◦ P ^{∼= //}(U◦ P)⊗(M◦ P ◦ P) ^{γU⊗(M◦γP)//}U ⊗(M◦ P) .

In the following, for a map f:M → N from a Σ∗-module M to N ∈ _UMod(M_P) we will denote by

∂f:U⊗(M◦ P)→N

the associated morphism in_UMod(M_P). We will use the same notation for morphisms defined on free right P-modules, which version applies will be clear from the context.

Let 1 6 n < ∞. By definition, we have Tⁿ = (T^cΣ)ⁿ(I) and hence, this object has an expansion such that

Tⁿ(e) = M

e=e1t···te_l

ΣTⁿ⁻¹(e₁)⊗ · · · ⊗ΣTⁿ⁻¹(e_l) and with

T¹(e) = (I^⊗s)(e) = M

σ:e→s

k·σ

for|e|=s. Forn= 1 it is obvious thatTⁿ(e) has generators corresponding to decorated 1-level trees (t = [r1], σ : e → [r1]). For n > 1, let sx1 ⊗ · · · ⊗sxl ∈ ΣTⁿ⁻¹(e1)⊗ · · · ⊗ΣTⁿ⁻¹(el).

Then the corresponding n-level tree is the n-level tree with l vertices in level 1 such that the ith vertex is the root of then−1-level tree defined byxi. Hence elements inTⁿ(e) correspond ton-level trees with leaves decorated bye. Consequently, as ak-moduleTⁿ◦ Com is generated by planar fully grown n-level trees in Tⁿ with leaves labeled by elements in Com.

Definition 2.4. The morphism∂_θ:UCom⊗(Tⁿ◦Com)→UCom⊗(Tⁿ◦Com) in_UComMod(M_Com) is induced by a map

θ:Tⁿ→UCom⊗(Tⁿ◦ Com)

which we define as follows: for (t, σ) ∈ Tⁿ(e) with t = [r_n] ^{fn //}· · · ^{f2 //}[r₁] labeled by σ:e→[rn] we set

θ(t, σ) = X

06l6rn−1,

|f_n⁻¹(l)|>1, x=minfn⁻¹(l)

(−1)^sn,x−1µ^U_{σ−1(x)}⊗(t\x, σ_σ⁰−1(x))

+ X

06l6rn−1

|fn⁻¹(l)|>1, y=maxfn⁻¹(l)

(−1)^sn,yµ^U_{σ−1(y)}⊗(t\y, σ_σ⁰−1(y)).

The sign (−1)^sn,i is determined by counting the edges in the treetfrom bottom to top and from left to right. Then sn,i is the number assigned to the edge connected to the ith leave. These signs can be thought of as originating from the process of switching the map M⊗ΣA→M of degree −1 past the suspensions in the bar construction.

7

A tedious but straightforward calculation yields that θis indeed a twisting cochain:

Lemma 2.5. The mapθdefines a twisting cochain onUCom⊗B_Comⁿ , i.e. we have (∂_θ+UCom⊗

∂γ)² = 0.

The suspension morphism ΣB_Comⁿ →Bⁿ⁺¹_Com commutes with the twists∂θ defined onB_Comⁿ and B_Comⁿ⁺¹, hence we can extend the above definition ton=∞.

Proposition 2.6. The twist ∂_θ extends to a twisting differential on UCom⊗Σ^−∞B_Com^∞ . LetηU:k→UCom(0) denote the unit map ofUCom. We want to lift∂_θ+UCom⊗∂_γ=∂_θ+ηU⊗γ

toUCom⊗B_Eⁿ. To achieve this we mimick the construction that is used in [Fre11, 2.4] to lift∂_γ toB_Eⁿ. The following proposition extends [Fre11, 2.5].

Proposition 2.7. Let R,S be operads equipped with differentials dR and dS and let U be an algebra in right R-modules. Suppose there are maps

R

ν 77

ψ //

S oo ι

such that ψ is a morphism of operads, ι is a chain map and

ψι= id, dRν−νdR= id−ιψ and ψν = 0.

Let K =G⊗Σ∗ be a freeΣ∗-module and

β:G→U ⊗(K◦ S) a twisting cochain which additionally satisfies

d_U⊗(K◦S)β+βd_G= 0 and ∂_ββ = 0 with d_U⊗(K◦S) and d_G denoting the differentials onU ⊗(K◦ S) and G.

If K, U and S are nonnegatively graded there exists a twisting cochain α:K→U ⊗(K◦ R) such that

U ⊗(K◦ R)

U⊗(K◦ψ)

∂α //U⊗(K◦ R)

U⊗(K◦ψ)

U ⊗(K◦ S) ^{∂β //}U⊗(K◦ S).

Proof. Extend the homotopyν:R → R to ˆν^(l):R^⊗l→ R^⊗l by setting ˆ

ν^(l)=

l

X

i=1

(−1)ⁱ⁻¹(ιψ)^⊗i−1⊗ν⊗id^⊗l−i_R and extendιto ˆι^(l):S^⊗l → R^⊗l by setting

ˆ

ι^(l)=ι^⊗l. We haveK◦ R=L

i>0G(i)⊗ R^⊗i sinceK is Σ∗-free, and similarly K◦ S =L

i>0G(i)⊗ S^⊗i. We can then define

˜

ν:U⊗(K◦ R)→U ⊗(K◦ R) and ˜ι:U ⊗(K◦ S)→U⊗(K◦ R) as ˜ν= idU⊗L

l>0G(l)⊗ˆν^(l)and ˜ι= idU⊗L

l>0G(l)⊗ˆι^(l). Note that with this definition, we get:

(U ⊗(K◦ψ))˜ι= id, δ(˜ν) = idU⊗(K◦R)−˜ι(U ⊗(K◦ψ)) and (U⊗(K◦ψ))˜ν = 0.

We defineα:G→U ⊗(K◦ R) by settingα0= ˜ιβ,

αm = X

a+b=m−1

˜

ν∂αaαb and α= X

m>0

αm.

8

Observe thatβ:K→U⊗(K◦ S) lowers the degree inK by at least 1, henceαm lowers the degree inK bym+ 1. SinceK is bounded below,α is well defined. Then form>1

(U⊗(K◦ψ))α_m = (U ⊗(K◦ψ)) X

a+b=m−1

˜

ν∂_αaα_b = 0, while (U ⊗(K◦ψ))α₀ =β,hence

(U ⊗(K◦ψ))α=β

and the diagram above commutes. To show that ∂_α is indeed a twisting differential one shows by induction that

δ(α_m) = X

a+b=m−1

∂_αaα_b,

which then yields the claim forα.

Recall that the Barratt-Eccles operad has an extension to finite sets and bijections given by E(e)_l=k <Bij(e, r)^l+1> .

In our constructions we will need to choose a distinguished element in Bij(e, r) for all e, cor- responding to an ordering of e. We fix a choice of a family (τ_e)e∈Bij such that for e⊂N0 the elementτe corresponds to the canonical order.

Proposition 2.8. [Fre11, 1.4] The trivial fibration from the Barratt-Eccles operad E to the commutative operad Com

ψ:E→ Com, which we explicitly define by

E(e)_l3(σ0, . . . , σ_l)7→

(µe, l= 0, 0, l6= 0,

admits a section ι: Com → E. This section is a morphism of arity-graded differential graded modules defined by

Com(e)3µ_e7→τ_e,

In addition, we have a k-linear homotopy ν:E→E betweenιψ andidE, given by the mapping E(e)l3(σ0, . . . , σl)7→(σ0, . . . , σl, τe),

and such that ψν = 0. (Note that this is not a homotopy retract of operads since ι does not even preserve the action of bijections.)

Hence we can lift ∂_θ+UCom⊗∂_γ as desired:

Proposition 2.9. For 1 6n 6 ∞, there is a map λ: Σ⁻ⁿTⁿ → UCom⊗(Σ⁻ⁿTⁿ◦E) which induces a twisting differential ∂_λ such that

UCom⊗(Σ⁻ⁿTⁿ◦E) ^{∂λ //}

UCom⊗(Σ⁻ⁿTⁿ◦ψ)

UCom⊗(Σ⁻ⁿTⁿ◦E)

UCom⊗(Σ⁻ⁿTⁿ◦ψ)

UCom⊗(Σ⁻ⁿTⁿ◦ Com) ^{∂θ+UCom⊗∂γ//}UCom⊗(Σ⁻ⁿTⁿ◦ Com) commutes.

9

Reminder on the complete graph operad. As in [Fre11] we will use thatEis equipped with a cell structure indexed by complete graphs to prove that ∂_λ restricts to UCom⊗(Σ⁻ⁿTⁿ◦E_n) for 16n <∞. In this subsection we revisit the relevant definitions and results concerning this cell structure. The complete graph operad and its relation to En-operads has been discussed by Berger in [Be96]. We assume that n <∞ in this and the next subsection.

Definition 2.10. Let e be a finite set with r elements. A complete graph κ = (σ, µ) on e consists of an ordering σ:e → {1, . . . , r} together with a symmetric matrix µ= (µef)e,f∈e of elements µ_ef ∈N0 with all diagonal entries 0. We think of σ as a globally coherent orientation of the edges and ofµef as the weight of the edge connectingeand f.

Example 2.11. The complete graph

-

4 0

@

@ I

g 3 f

e

on the set{e, f, g} corresponds to

σ:{e, f, g} → {1,2,3}, σ(e) = 3, σ(f) = 2, σ(g) = 1 and µ_ef = 0, µ_eg = 4, µ_{f g} = 3.

Forσ as above and e, f ∈elet σ_ef = id∈Σ₂ ifσ(e)< σ(f). Otherwise, defineσ_ef to be the transposition in Σ2.

Definition 2.12. The set of complete graphs on eis partially ordered if we set (σ, µ)6(σ⁰, µ⁰)

whenever for all e, f ∈ e either µ_ef < µ⁰_ef or (σ_ef, µ_ef) = (σ_ef⁰ , µ⁰_ef). The poset of complete graphs on eis denoted byK(e).

Proposition 2.13. [Be96] The collection K = (K(e))_e of complete graphs forms an operad in posets: A bijectionω:e→e⁰ acts by relabeling the vertices. The partial composition

◦_e:K(e)× K(f)→ K(etf \ {e})

is given by substituting the vertex e in a complete graph (σ, µ) ∈ K(e) by the complete graph (τ, ν)∈ K(f), i.e.by inserting (τ, ν) at the position ofe, orienting the edges betweeng∈e\ {e}

and g⁰ ∈f like the edge between g and e and giving them the weightµge.

Definition 2.14. [Fre11, 3.4,3.8] Let (P(e))_e be a collection of functorsK(e)→dg-mod. Such a collection is called a K-operad if for all finite sets e and all bijections ω: e⁰ → e there is a natural transformation P(e)→ P(e⁰) with components

P_κ→ P_κ.ω

forκ∈ K(e), as well as partial composition products given by natural transformations

◦_e:P(e)⊗ P(f)→ P(e\ {e} tf) fore∈ewith components

◦_e:P_κ⊗ P_κ⁰ → P_κ◦eκ⁰,

satisfying suitable associativity, unitality and equivariance conditions. A morphism P → P⁰ consists of natural transformations with components P_κ → P_κ⁰ commuting with composition and the action of bijections.

Similarly, a rightK-module R over a K-operad consists of collections (R(e))_e together with natural transformationsRe→R_e⁰ for all bijectionsω:e⁰→ewith components

Rκ →Rκ.ω,

10

as well as natural transformations

◦_e:P(e)⊗R(f)→R(e\ {e} tf) with components

◦_e:R_κ⊗ P_κ⁰ →R_κ◦eκ⁰

fore∈esatisfying again suitable associativity, unitality and equivariance conditions.

Proposition 2.15. [Fre11, 3.4,3.8] There is an adjunction colim :O_dg-mod oo ^//_KO: const

between the category O_dg-mod of operads in differential graded k-modules and the category KO of K-operads defined as follows: For a given K-operad P let

(colimP)(e) = colim_κ∈K(e)P_κ, while for an ordinary operad Q we set

const(Q)_κ =Q(e)

for κ∈ K(e). We say that an operadQ has aK-structure ifQ= colimP. A similar adjunction exists between the category of right K-modules over a K-operad P and the category of right modules overcolimP, and we call right modules of the formcolimR rightcolimP-modules with K-structure.

Example 2.16. [Fre11, 3.5] Besides considering constant K-operads the main example we are interested in is the Barratt-Eccles operadE. Concretely, for a finite set ewithr elements and κ= (σ, µ)∈ K(e) an element (ω₀, . . . , ωl)∈Bij(e, r) is in E_κ if for all e, f ∈ethe sequence

((ω0)ef, . . . ,(ωl)ef)

has either less thanµ_ef variations or has exactly µ_ef variations and (ω_l)_ef =σ_ef. Observe that in

E

ν 77

ψ //

Com, oo ι

the map ψrespects the K-structures onE and Com and that forκ= (τ_e, µ) we have ι(Com_κ)⊂Eκ and ν(Eκ)⊂Eκ.

Definition 2.17. Let K_n(r) be the poset of complete graphsκ = (σ, µ) such thatµ_ij 6n−1 for all pairs i, j∈r. This defines a filtration

K₁ ⊂ · · · ⊂ K_n⊂ K_n+1⊂ · · · ⊂ K= colim_nK_n of K by suboperads.

Remark 2.18. ConsideringK-structures allows more control over the operads in question. A closer look at the K-structure of E yields that colimκ∈KnEκ =En: If x= (ω0, . . . , ωl) ∈Σ^l+1_r is in E_n, then ((ω₀)_ij, . . . ,(ω_l)_ij) has at mostn−1 variations for alli, j∈r. Hencex∈E_κ for κ= (ω_l, µ) withµ_ef =n−1 for alli, j∈r. This will allow us to show that the differentials we are interested in restrict toEn.

Descending the twisted module to En-modules. We assume that n <∞ in this subsec- tion. In [Fre11] the first author constructs a twisting differential ∂ on Σ⁻ⁿTⁿ◦E. It is then shown that ∂ descends to Σ⁻ⁿTⁿ◦En by using that Tⁿ can be interpreted as a K-diagram and that hence Σ⁻ⁿTⁿ◦E is a right E-module with K-structure. We aim to prove a similar statement for the twisting differential ∂_λ. To achieve this, we will use the same K-structures, hence we now recall the relevant definition.

For a complete graph κ = (σ, µ) ∈ K(e) and f ⊂ e let κ|_f = (σ⁰, µf×f) be the complete subgraph of κ withf ⊂eas vertex set together with the ordering σ⁰:{1, . . . ,|f|} →f defined by the composite

{1, . . . ,|f|} ^{∼= //}σ⁻¹(f) ^{σ //}f .

11

Proposition 2.19. [Fre11, 4.2] There is a K-diagram associated to Tⁿ defined as follows: For n= 1 and κ = (σ, µ) ∈ K(e) set T_κⁿ=k·σ ⊂T¹(e). For general n and κ= (σ, µ) ∈ K(e) an element

sx1⊗ · · · ⊗sx_l∈ΣTⁿ⁻¹(e₁)⊗ · · · ⊗ΣTⁿ⁻¹(e_l)⊂T^c(ΣTⁿ⁻¹)(e₁t · · · te_l) is in T_κⁿ if the following conditions hold:

(a) For16i6l we have that x_i is an element of T_κ|ⁿ⁻¹

ei .

(b) If e, f ∈ewith µ_ef < n−1 then there exists i such thate, f ∈e_i.

(c) If e, f ∈ewith µ_ef =n−1 and if e∈ei, f ∈ej with i < j thenσ_ef = id.

Remark 2.20. The idea behind this definition is the following (see [FrApp]): Interpreting t ∈ Tⁿ(e) as a tree, the smallest complete graph κ with t ∈ T_κⁿ has vertices ordered like the inputs of t and weights µef such that n−1−µef equals the level on which the paths frome and f to the root first join.

For aK-operadP and κa complete graph let (Tⁿ◦colimP)_κ be generated as ak-module by elements t(p1, . . . , p_l) ∈Tⁿ◦colimP such thatt ∈T_κⁿ0, pi ∈ P_κi with κ⁰(κ1, . . . , κ_l) 6 κ. This makesTⁿ◦colimP a right colimP-module withK-structure.

Lemma 2.21. The twist ∂θ satifies

∂θ((Tⁿ◦ Com)_κ)⊂M

e∈e

UCom({e})⊗(Tⁿ◦ Com)κ|(e\{e})

for κ= (σ, µ)∈ K(e).

Proof. We proceed by induction. Forn= 1 we see that x ∈T_κ¹ if and only ifx corresponds to the 1-level tree tr(e1, . . . , er) with r leaves decorated by e1, . . . , er, where σ⁻¹(i) =ei. If r > 1 the map θsends tr(e1, . . . , er) to

−µ^U_{e

1}⊗tr−1(e2, . . . , er) + (−1)^rµ^U_{er}⊗tr−1(e1, . . . , er−1).

Denote by ∂^(a)_θ the morphism ∂θ defined onT^a◦ Com. Forn >1 observe that

∂⁽ⁿ⁾_θ (sx₁⊗ · · · ⊗sx_r) =X

j

±sx₁⊗ · · · ⊗s∂_θ⁽ⁿ⁻¹⁾(x_j)⊗ · · · ⊗sx_r forsx1⊗ · · · ⊗sxr ∈(ΣTⁿ⁻¹)^⊗r. Letxi ∈T_fⁿ⁻¹

i for all i. By the induction hypothesis

∂_θ(xi)∈ M

e∈fi

UCom({e})⊗(Tⁿ⁻¹◦ Com)_κ|f

i\{e},

which yields the claim.

We already noted thatTⁿis Σ∗-free: There are graded modulesG(e) withTⁿ(e)∼=G(e)⊗Σ_e. The functorG is defined inductively by

Gⁿ=M

i>1

(ΣGⁿ⁻¹)^⊗i forn >0, and G⁰(e) =

(k, |e|= 1, 0, |e| 6= 1.

Intuitively, G associates to a finite set e the set of trees with |e| leaves with a tree t ∈ G(r) having degree equal to the number of its edges. The inclusion Gⁿ(e) → Tⁿ(e) is given by mapping a tree tto the tree with leaves labeled by eaccording to the chosen orderτ_e.

We now deduce from Proposition 2.21 that ∂_λ restricts to UCom⊗(Σ⁻ⁿTⁿ◦En). Note that the Lemmata 2.22, 2.23 and Proposition 2.24 are completely analougous to [Fre04, 5.2], [Fre04, 5.3] and [Fre04, 5.4].

For a complete graph κ denote by Gⁿ_κ the arity-graded k-submodule of Gⁿ generated by elements g∈Gⁿ withg∈T_κⁿ.

12

Lemma 2.22. The mapλ0 satisfies λ0(Gⁿ_κ)⊂ M

e=e⁰te⁰⁰

UCom(e⁰)⊗(Tⁿ◦E)κ|_e00

for κ= (τe, µ)∈ K(e) withe⊂N0.

Proof. We know thatλ₀ = ˜ι(θ+η_U⊗γ) and that θ(T_κⁿ)⊂M

e∈e

UCom({e})⊗(Tⁿ◦ Com)_κ|e\{e}, while according to [Fre11, 4.6]

(ηU ⊗γ)(Gⁿ_κ)⊂UCom(∅)⊗(Tⁿ◦ Com)κ.

But by [Fre11, 4.5] (Tⁿ◦ Com)_κ is spanned by elements t(c₁, . . . , c_l) with t∈ T_κⁿ0, c_i ∈ Com_κi such thatκi is also of the form (τ_e(i), µ⁽ⁱ⁾) for some e⁽ⁱ⁾ ⊂N0. Hence we find that

(Tⁿ◦ι)((Tⁿ◦ Com)_κ)⊂(Tⁿ◦E)κ. Observe thatκ|_e\{e}= (τ_e\{e}, µ⁰). Hence

(UCom⊗(Tⁿ◦ι))(M

e∈e

UCom({e})⊗(Tⁿ◦ Com)_κ|

e\{e})⊂M

e∈e

UCom({e})⊗(Tⁿ◦E)_κ|e\{e}

as well. This proves the claim.

Lemma 2.23. The twist ∂λ satisfies

∂λ(UCom⊗(Tⁿ◦E)κ)⊂M

e⁰⊂e

UCom⊗(Tⁿ◦E)κ|_e0

for all κ∈ K(e) withe⊂N0. Proof. We show by induction that

∂_λm(UCom⊗(Tⁿ◦E)_κ)⊂M

e⁰⊂e

UCom⊗(Tⁿ◦E)_κ|

e0

for all m. Let t(e1, . . . , e_l) ∈ (Tⁿ◦E)κ. Then there are κ⁰ ∈ K(f) and κi ∈ K(f

i) such that t∈T_κⁿ0 and e_i ∈E_κi withκ⁰(κ₁, . . . , κ_l)6κ.

Note that (Tⁿ◦E)(e) = L

i>1Tⁿ(i)⊗_Σi(E^⊗i(e)), hence we can assume that κ⁰ = (idl, µ⁰) and t∈ Gⁿ_κ0. Writing down the definition of ∂_λ0 and using Lemma 2.22 yields that ∂_λ0 maps UCom⊗(Tⁿ◦E)_κ toUCom⊗L

{i1<···<ij}=e⁰⊂l(Tⁿ◦E)_κ⁰_|

e0(κi1,...,κ_ij). Since κ⁰|_e⁰(κ_i1, . . . , κ_ij)6κ|_f

i1t···tf

ij

the claim holds for m= 0. Form >0 recall that

∂λm = X

a+b=m−1

˜

ν∂λa∂λ_b.

The induction hypothesis yields that

∂_λa∂_λb(UCom⊗(Tⁿ◦E)_κ)⊂M

e⁰⊂e

UCom⊗(Tⁿ◦E)_κ|e0. Since

˜

ν(u⊗t(e₁, . . . , e_l)) =X

i

±u⊗t(ιψ(e₁), . . . , ιψ(ei−1), ν(e_i), e_i+1, . . . , e_l)

forξ∈Tⁿand y_r∈E the same reasoning as in Lemma 2.22 together with our assumptions on the interaction between ψ, ι, ν and theK-structure yields the claim.

Proposition 2.24. We have

∂_λ(UCom⊗(Tⁿ◦E_n))⊂UCom⊗(Tⁿ◦E_n).

13

Proof. We need to show that ∂λ(Tⁿ) ⊂ UCom⊗(Tⁿ◦En). Observe that for every r > 0 and ξ∈Tⁿ(r) there is a complete graphκ= (σ, µ) withξ ∈T_κⁿsuch thatµ_ef 6n−1 for all vertices e, f. Hence

∂_λ(Tⁿ(r)) =∂_λ(colim_κ∈Kn(r)T_κⁿ) = colim_κ∈Kn(r)(M

e⁰⊂r

UCom⊗(Tⁿ◦E)_κ|

e0).

But

colim_κ∈Kn(r)(M

e⁰⊂r

UCom⊗(Tⁿ◦E)_κ|

e0)

=UCom⊗M

e⁰⊂r

colim_κ∈Kn(r)(Tⁿ◦E)_κ|

e0

⊂U_Com⊗M

e⁰⊂r

(Tⁿ◦E_n)(e⁰),

sinceκ(κ₁, . . . , κ_l)∈ K_n impliesκ₁, . . . , κ_l ∈ K_n. 3. From modules over operads to Quillen homology

The model category of left modules over an algebra in right modules over an operad.

Let P be an operad and U an algebra in right P-modules. We will define a model structure on UMod(M_P) by applying the standard method to transport cofibrantly generated model categories along adjunctions to the adjunction

F =U ⊗ −:M_{P oo} ^//_UMod(M_P) :V

with V the corresponding forgetful functor. This model structure will allow us to compare UCom⊗Σ⁻ⁿB_Eⁿ_n with a standard resolution for computing En-homology.

We start by determining the F I- and F J-cell complexes, where F I (respectivelyF J) is the set of maps

U⊗(i⊗(Fr◦ P)) :U ⊗(C⊗(Fr◦ P))→U ⊗(D⊗(Fr◦ P))

withi:C →Da generating cofibration (respectively a generating acyclic cofibration) indg-mod and

Fr(l) =

(k[Σr], l=r, 0, l6=r.

Recall that the generating cofibrations in dg-mod are of the form S^d−1 → D^d and the acyclic cofibrations are of the form 0 → D^d, with S_d−1^d−1 = k, S_l^d−1 = 0 otherwise and D^d the acyclic chain complex withD^d_d−1 =D^d_d=kandD_l^dotherwise. Also note that the underlying differential graded module of the direct sum of leftU-modules in rightP-modules is the direct sum of their underlying differential graded modules. Using this yields the following observations.

Proposition 3.1. AnF I-cell attachment in _UMod(M_P) is an inclusion K→(K⊕G, ∂) with G=U ⊗(M◦ P) for a free Σ∗-module M with trivial differential and ∂:G→K. An F J-cell attachment in _UMod(M_P) is an inclusion K →K⊕G⁰ with G⁰ =U ⊗(L

αD^nα⊗(F_r◦ P)).

Corollary 3.2. A relativeF I-cell complex in _UMod(M_P) is an inclusion K→(K⊕(U ⊗(M ◦ P)), ∂)

withM a Σ∗-free Σ∗-module with trivial differential, such thatK⊕(U⊗(M◦ P))is filtered by G_λ, λ < κ for a given ordinal κ, with ∂(G_λ)⊂Gλ−1 and G0 =K. A relative F J-cell complex in _UMod(M_P) is the same as an F J-cell attachment.

Now we are in the position to prove that the adjunction between_UMod(M_P) and the category of rightP-modules gives rise to a model structure onUMod(MP).

14