transferredobjectswasformulatedin[ ].Thepresentarticlecontributestothe tions,anunconditionalandelaborateanswerproducingexplicitformulasforthe algebrastructureon ( ) andandinwhichsenseisit can structurebetransferredto ( ) andsecondly,whatisitsexplicitformi

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Tomus 53 (2017), 267–312

ON THE HOMOTOPY TRANSFER OF A_∞ STRUCTURES

Jakub Kopřiva

Dedicated to the memory of Martin Doubek

Abstract. The present article is devoted to the study of transfers forA∞

structures, their maps and homotopies, as developed in [7]. In particular, we supply the proofs of claims formulated therein and provide their extension by comparing them with the former approach based on the homological perturbation lemma.

1. Introduction

The notion of strongly homotopy associative orA_∞ algebras is a generalization of the concept of differential graded algebras. These algebras were introduced by J. Stasheff with the aim of a characterization of (de)looping and bar construction in the category of topological spaces. Since then they found many applications ranging from algebraic topology and operads to quantum theories in theoretical physics.

We consider the following situation: let (V, ∂V) and (W, ∂W) be two chain complexes of modules, and f: (V, ∂V)→(W, ∂W) andg: (W, ∂W)→(V, ∂V) two mappings of chain complexes such thatgf is homotopic to the identity map onV and (V, ∂_V) is equipped withA_∞ algebra structure. Then a natural question arises

−canA_∞ structure be transferred to (W, ∂_W) and secondly, what is its explicit form in terms of A_∞ algebra structure on (V, ∂_V) and and in which sense is it unique?

While the existence of a transfer follows from general model structure considera- tions, an unconditional and elaborate answer producing explicit formulas for the transferred objects was formulated in [7]. The present article contributes to the problem of transfer ofA_∞structures. Its modest aim is to supply detailed proofs of many claims omitted in the original article [7], thereby facilitating complete subtle proofs to a reader interested in this topic. This exposition also extends the results of the aforementioned article in several ways, and sheds a light on its relationship with the homological perturbation lemma.

2010Mathematics Subject Classification: primary 18D10; secondary 55S99.

Key words and phrases:A∞structures, transfer, homological perturbation lemma.

Received March 11, 2017, revised September 2017. Editor M. Čadek.

DOI: 10.5817/AM2017-5-267

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The content of our article goes as follows. In the Section 2 we recall a well-known correspondence between A_∞algebras and codifferentials on reduced tensor coalgebras. This allows us to simplify the proofs in Section 3 considerably. The Section 3 is devoted to the problem of homotopy transfer of A_∞ algebras. We first derive the formulas introduced in [7], and then give their self-contained proofs. Here we achieve a substantial simplification of all proofs due to the reduction of sign factors.

We also comment on another remark in [7], namely, the relationship between the homological perturbation lemma and homotopy transfer ofA_∞ algebras. We prove that on certain assumptions the explicit formulas in [7] do coincide with those coming from the homological perturbation lemma.

We shall work in the category ofZ-graded modules over an arbitrary commutative unital ringR, and their gradedR-homomorphisms.

We first briefly recall the concepts ofA∞algebra,A∞morphism ofA∞algebras andA∞homotopy ofA∞ morphisms, cf. [7], [4].

Definition 1.1. Let (V, ∂V) be a chain complex of modules indexed by Z, i.e.

(V, ∂V) is aZ-graded modulesV =L∞

i=−∞Viwith∂V(Vi)⊂Vi−1and∂V◦∂V = 0.

Let µn: V^⊗n → V be a collection of linear mappings of degree n−2 (n ≥ 2), satisfying

∂Vµn−

n

X

i=1

(−1)ⁿµn 1^⊗i−1V ⊗∂V ⊗1^⊗n−iV

= X

A(n)

(−1)^i(`+1)+nµk 1^⊗i−1V ⊗µ`⊗1^⊗k−iV

(1)

for all n ≥ 2 and A(n) = {k, ` ∈ N | k+` = n+ 1, k,` ≥ 2,1≤ i ≤k}. The structure (V, ∂_V, µ₂, µ₃, . . .) is calledA_∞ algebra.

Throughout the article, we use the Koszul sign convention. This means that for U,V a W graded modules andf:U →V,g:U →V,h:V →W andi:V →W linear maps of degrees|f|,|g|,|h|and|i|, respectively, holds

(h⊗i)(f ⊗g) = (−1)^|f||i|hf⊗ig .

Similarly for u1,u2∈U of degree|u1| and|u2|, respectively, holds (f ⊗g)(u1⊗u2) = (−1)^|u¹^||g|f(u1)⊗g(u2).

Definition 1.2. Let (V, ∂V, µ2, . . .) and (W, ∂W, ν2, . . .) be A_∞ algebras. Then the set{fn:V^⊗n →W,|fn|=n−1}_n≥1is called A_∞ morphism if

∂Wfn+X

B(n)

(−1)^ϑ(r¹^,...,r^k⁾νk(fr₁⊗ · · · ⊗fr_k)

=f1µn−

n

X

i=1

(−1)ⁿfn 1^⊗i−1V ⊗∂V ⊗1^⊗n−iV

−X

A(n)

(−1)^i(`+1)+nfk 1^⊗i−1V ⊗µ`⊗1^⊗k−iV

(2)

(3)

holds for alln≥1 withB(n) ={k, r1, . . . , r_k ∈N|k≥2, r₁, . . . , r_k≥1, r₁+· · ·+ r_k =n} andϑ(r₁, . . . , r_k) =P

1≤i<j≤kr_i(r_j+ 1).

Morphisms ofA∞ algebras can be composed: for (U, ∂U, %2, . . .), (V, ∂V, µ2, . . .) and (W, ∂W, ν2, . . .)A∞ algebras, {fn:U^⊗n →V}n≥1 and {gn:V^⊗n →W}n≥1

A_∞morphisms, their composition{(gf)n:U^⊗n→W}n≥1 is defined as (gf)n =g1fn+X

B(n)

(−1)^ϑ(r¹^,...,r^k⁾gk(fr₁⊗ · · · ⊗fr_k). (3)

Definition 1.3. Let{fn:V^⊗n →W}_n≥1and{gn:V^⊗n→W}_n≥1be morphisms betweenA_∞ algebras (V, ∂V, µ2, . . .) and (W, ∂W, ν2, . . .). The set of linear mappings {hn:V^⊗n→W,|hn|=n}_n≥1 is anA_∞ homotopy betweenA_∞morphisms {fn:V^⊗n →W}_n≥1 and{gn:V^⊗n →W}_n≥1 provided

fn−gn=h1µn−

n

X

i=1

(−1)ⁿhn 1^⊗i−1V ⊗∂V ⊗1^⊗n−iV

−X

A(n)

(−1)^i(`+1)+nh_k 1^⊗i−1V ⊗µ_`⊗1^⊗k−iV

+δ_Wh_n

+X

B(n)

X

1≤i≤k

(−1)^ϑ(r¹^,...,r^k⁾νk

×(fr1⊗ · · · ⊗fri−1⊗hri⊗gri+1⊗ · · · ⊗gr_k), (4)

is true for alln≥1 withB(n) ={k, r1, . . . , rk∈N|k≥2, r1, . . . , rk≥1, r1+· · ·+ rk =n}.

2. Reduced tensor coalgebras

In the present section we introduce a bijective correspondence between A_∞ algebras and codifferentials on reduced tensor coalgebras, cf. [4]. We retain the notationV =L∞

i=−∞Vi forZ-graded modules as well as A(n) ={k, `∈N|k+`=n+ 1, k,`≥2,1≤i≤k}, (A)

B(n) ={k, r1, . . . , rk ∈N|k≥2, r1, . . . , rk≥1, r1+· · ·+rk =n}

(B)

for n∈N, andA(1) =A(2) =B(1) =∅. We use a few natural variations on this notation, e.g. A⁰(n) ={k⁰, `⁰ ∈N|k⁰+`⁰=n+ 1, k⁰,`⁰ ≥2,1≤i⁰≤k⁰}.

2.1. Codiferentials on tensor coalgebras.

Definition 2.1. LetT V =L∞

n=1V^⊗n, where the elements in V^⊗i have degree (or homogeneity)i, and let the mappingC:T V →T V ⊗T V be defined in such a way thatC:v7→0 forv∈V^⊗1=V and

C:v₁⊗ · · · ⊗v_n 7→

n−1

X

i=1

(v₁⊗ · · · ⊗v_i)⊗(v_i+1⊗ · · · ⊗v_n), (5)

for n ≥ 2 and v1, . . . , vn ∈ V. The pair (T V, C) is called the reduced tensor coalgebra.

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Definition 2.2. A linear mappingδ:T V →T V of degree−1 is called coderivation ifC◦δ= (δ⊗1+1⊗δ)◦C. Moreover, ifδsatisfiesδ◦δ= 0, it is called codifferential.

Remark 2.3. We notice thatC is coassociative, (1⊗C)◦C= (C⊗1)◦C. For all v ∈ T V holds C(v) = 0 if and only if v is of homogeneity 1. For all maps ϕ:V^⊗n →T W,n ≥1, holds C_{T W} ◦ϕ= 0 if and only if ϕ(V^⊗n)⊆W. For all v=v1⊗. . .⊗vn∈T V andw=w1⊗. . .⊗wm∈T V, we have

C(v⊗w) =

n−1

X

i=1

(v_1,i)⊗(v_i+1,n⊗w) + (v)⊗(w) +

m−1

X

i=1

(v⊗w_1,i)⊗(w_i+1,m), withv_i,j = v_i⊗. . .⊗v_j, i ≤j, i,j ∈ {1, . . . , n}, and analogously for w_i,j. This little calculation expresses a fact thatT V is a bialgebra which is, as a conilpotent coalgebra, cogenerated byV.

Lemma 2.4. Let E : T V → T W be a linear mapping for which there exist {en:V^⊗n→W}_n≥1 withE|_V^⊗n=en+P

B(n)er₁⊗. . .⊗er_k, andB(n)given in (B). Then

C_{T W} ◦E|_V⊗n=

n−1

X

i=1

(E|_V⊗i)⊗(E|_V⊗n−i). (6)

Proof. Obviously, we can writeE|V^⊗n=en+Pn−1

i=1 ei⊗E|V^⊗n−i. The proof is by induction onn: the claim holds forn= 1 and we assume it is true foll all natural numbers less thann. Then

C_{T W} ◦E|_V⊗n =C_{T W} ◦ e_n+

n−1

X

i=1

e_i⊗E|_V⊗n−i

=C_{T W} ◦ⁿ⁻¹X

i=1

e_i⊗E|_V⊗n−i

=

n−1

X

i=1

(e_i)⊗(E|V^⊗n−i) +

n−1

X

i=1 n−1−i

X

j=1

(e_i⊗E|V^⊗j)⊗(E|V^⊗n−i−j)

=

n−1

X

i=1

(ei)⊗(E|_V⊗n−i) +

n−1

X

`=2

`−1

X

j=1

(ej⊗E|_V⊗`−j)⊗(E|_V⊗n−`)

= (e1)⊗(E|V^⊗n−1) +

n−1

X

`=2

e`+

`−1

X

j=1

ej⊗E|_V^⊗`−j

⊗(E|_V^⊗n−`), and the proof follows by induction hypothesis from E|_V⊗` = e_` +P`−1

i=1e_i ⊗

E|_V⊗`−i.

Theorem 2.5. Let E: T V → T W and G: T V → T W be linear mappings for which there exist linear mappings {e_n :V^⊗n→W}_n≥1,{g_n:V^⊗n→W}_n≥1 such that E|_V⊗n =e_n+P

B(n)e_r₁⊗ · · · ⊗e_r_k and G|_V⊗n=g_n+P

B(n)g_r₁⊗. . .⊗g_r_k

(5)

with B(n) given in (B). Given a linear mapping F : T V →T W, the following conditions are equivalent:

(1) C_{T W} ◦F = (E⊗F+F⊗G)◦C_{T V},

(2) there exist linear mappings {f_n:V^⊗n→W}_n≥1 such that F|_V^⊗n =fn+X

B(n)

X

1≤i≤k

er₁⊗ · · · ⊗eri−1⊗fr_i⊗gr_i+1⊗ · · · ⊗gr_k. Proof. (2)⇒(1): We haveF|V^⊗n =fn+Pn−1

i=1 E|V^⊗i⊗fn−i+Pn−1

i=1 fi⊗G|V^⊗n−i+ Pn−1

i=1

Pn−i−1

j=1 E|_V⊗j⊗fi⊗G|_V⊗n−i−j for alln≥1. We now verify (1) by expanding both sides:

(E⊗F+F⊗G)◦C_{T V}|_V^⊗n = (E⊗F+F⊗G)◦

n−1

X

i=1

1^⊗n−iV

⊗ 1^⊗iV

=

n−1

X

i=1

(E|_V^⊗n−i)⊗(F|_V^⊗i) + (F|_V^⊗n−i)⊗(G|_V^⊗i) , and by Lemma 2.4 we get

C_{T W} ◦ⁿ⁻¹X

i=1

E|V^⊗i⊗f_n−i

=

n−1

X

i=1

(E|_V^⊗n−i)⊗(fi) +

n−1

X

i=1 n−1−i

X

j=1

(E|_V^⊗n−i−j)⊗(E|_V^⊗j ⊗fi),

C_{T W} ◦ⁿ⁻¹X

i=1

fi⊗G|_V^⊗n−i

=

n−1

X

i=1

(fi)⊗(G|_V^⊗n−i) +

n−1

X

i=1 n−1−i

X

j=1

(fi⊗G|_V^⊗j)⊗(G|_V^⊗n−i−j),

C_{T W} ◦ⁿ⁻¹X

i=1 n−i−1

X

j=1

E|_V⊗j ⊗fi⊗G|_V⊗n−i−j

=

n−1

X

i=1 n−i−1

X

j=1

(E|V^⊗n−i−j)⊗(f_i⊗G|V^⊗j) +

n−1

X

i=1 n−i−1

X

j=1

(E|V^⊗j⊗f_i)⊗(G|V^⊗n−i−j)

+

n−1

X

i=1 n−i−1

X

j=1 j−1

X

k=1

(E|_V⊗n−i−j−k)⊗(E|_V⊗j ⊗f_i⊗G|_V⊗k)

+

n−1

X

i=1 n−i−1

X

j=1 j−1

X

k=1

(E|_V⊗j ⊗f_i⊗G|_V⊗k)⊗(G|_V⊗n−i−j−k).

(6)

The summation in the variablesi+j andi+j+k, respectively, yields C_{T W}◦ⁿ⁻¹X

i=1

E|_V⊗i⊗f_n−i

=

n−1

X

i=1

(E|V^⊗n−i)⊗(fi) +

n−1

X

`=2

`−1

X

j=1

(E|_V^⊗n−`)⊗(E|_V^⊗`−j ⊗fj),

C_{T W} ◦ⁿ⁻¹X

i=1

f_i⊗G|V^⊗n−i

=

n−1

X

i=1

(fi)⊗(G|_V⊗n−i) +

n−1

X

`=2

`−1

X

j=1

(fj⊗G|_V⊗`−j)⊗(G|_V⊗n−`),

C_{T W}◦ⁿ⁻¹X

i=1 n−i−1

X

j=1

E|_V⊗j ⊗f_i⊗G|_V⊗n−i−j

=

n−1

X

`=2

`−1

X

j=1

(E|_V⊗n−`)⊗(fj⊗G|_V⊗`−j)

+

n−1

X

`=2

`−1

X

j=1

(E|V^⊗`−j⊗f_j)⊗(G|V^⊗n−`)

+

n−1

X

`=3 m−1

X

j=1 j−1

X

i=1

(E|_V⊗n−`)⊗(E|_V⊗`−j ⊗f_i⊗G|_V⊗j−i)

+

n−1

X

`=3 m−1

X

j=1 j−1

X

i=1

C_{T W}◦F|_V^⊗n=

n−1

X

i=1

B(m)

P

1≤i≤ker₁⊗· · ·⊗eri−1⊗fr_i⊗gr_i+1⊗· · ·⊗gr_kforn > m≥1, that

C_{T W} ◦ⁿ⁻¹X

i=1

E|V^⊗i⊗f_n−i+

n−1

X

i=1

f_i⊗G|V^⊗n−i+

n−1

X

i=1 n−i−1

X

j=1

E|V^⊗j⊗f_i⊗G|V^⊗n−i−j

=

n−1

X

i=1

(E|V^⊗n−i)⊗(F|V^⊗i) + (F|V^⊗n−i)⊗(G|V^⊗i) .

(1)⇒(2): The proof is again by induction. For allv∈V holdsC_{T W}◦F(v) = 0, which givesF(V)⊂W and so there exists a linear mappingf₁:V →W such that

(7)

F|V =f₁. Assume now the claim of the implication is true for all natural numbers less thann, i.e.F|V^⊗m=f_m+P

B(m)

P

1≤i≤ke_r₁⊗· · ·⊗eri−1⊗fri⊗gri+1⊗· · ·⊗grk, for n > m ≥ 1. The proof of the previous implication claims for F|_V⊗m = fm+P

B(m),ri>0er₁⊗ · · · ⊗er_i−1⊗fr_i⊗gr_i+1⊗ · · · ⊗gr_k withn > m≥1, that C_{T W} ◦F|V^⊗n=

n−1

X

i=1

(E|V^⊗n−i)⊗(F|V^⊗i) + (F|V^⊗n−i)⊗(G|V^⊗i)

=C_{T W}◦ⁿ⁻¹X

i=1

E|_V^⊗i⊗f_n−i+

n−1

X

i=1

fi⊗G|_V^⊗n−i+

n−1

X

i=1 n−i−1

X

j=1

E|_V^⊗j⊗fi⊗G|_V^⊗n−i−j .

BecauseC_{T W}is linear,F|V^⊗ndiffers fromPn−1

i=1 E|V^⊗i⊗fn−i+Pn−1

i=1 fi⊗G|V^⊗n−i+ Pn−1

i=1

Pn−i−1

j=1 E|V^⊗j⊗f_i⊗G|V^⊗n−i−j by a linear mapf_n:V^⊗n→W. This means F|_V^⊗n is of the required form and the proof is complete.

Theorem 2.6. A linear mapping δ: T V → T V of degree −1 fulfills C ◦δ = (δ⊗1V +1V ⊗δ)◦C if and only if there exist a set of maps{δn:V^⊗n→V}n≥1

of degree −1such that δ|V =δ1 and for n≥2holds δ|V^⊗n=δn+Pn

i=11^⊗i−1V ⊗ δ₁⊗1^⊗n−i_V +P

A(n)1^⊗i−1_V ⊗δ_`⊗1^⊗k−i_V , whereA(n)is given by (A).

Proof. In Theorem 2.5 we takeE=G=1V, wheree1=g1=1V anden=gn= 0

forn≥2.

Lemma 2.7. Letδ:T V →T V be a linear map of degree−1such thatδ|V =δ1and forn≥2holdsδ|_V⊗n=δn+Pn

i=11^⊗i−1V ⊗δ1⊗1^⊗n−iV +P

A(n)1^⊗i−1V ⊗δ`⊗1^⊗k−iV . Then the following conditions are equivalent:

(1) δ◦δ= 0,

(2) δ1◦δ1= 0 and for alln≥2 we have (7) δ1(δn) +

n

X

i=1

δn 1^⊗i−1V ⊗δ1⊗1^⊗n−iV

+X

A(n)

δk 1^⊗i−1V ⊗δ`⊗1^⊗k−iV

= 0,

whereA(n) is given by (A).

Proof. (1)⇒(2): The proof goes by induction. By assumption we have for v∈V δ(δ1(v)) = 0,soδ1:V →V impliesδ1(δ1(v)) = 0. Now assume (7) is true for all natural numbers less thann. Then

δ²|_V⊗n=δ1(δn) +

n

X

i=1

δ|_V⊗n 1^⊗i−1V ⊗δ1⊗1^⊗n−iV

+X

A(n)

δ|_V⊗k 1^⊗i−1V ⊗δ`⊗1^⊗k−iV

.

(8)

Schematically, this means δ²|_V⊗n=δ1(δn) +

n

X

i=1

δn(1^⊗i−1_V ⊗δ1⊗1^⊗n−i_V )

+X

A(n)

δk(1^⊗i−1V ⊗δ`⊗1^⊗k−iV )+X

1^⊗aV ⊗δb+d+1(1^⊗bV ⊗δc⊗1^⊗dV )⊗1^⊗eV

+X

1^⊗a_V ⊗δ_b⊗1^⊗c_V ⊗δ_d⊗1^⊗e_V −X

1^⊗a_V ⊗δ_b⊗1^⊗c_V ⊗δ_d⊗1^⊗e_V , where the last row is a consequence of the Koszul sign convention:

1^⊗a_V ⊗δ_b⊗1^⊗c+1+e_V 1^⊗a+b+c_V ⊗δ_d⊗1^⊗e_V

=1^⊗a_V ⊗δ_b⊗1^⊗c_V ⊗δ_d⊗1^⊗e_V , 1^⊗a+1+cV ⊗δd⊗1^⊗eV

1^⊗aV ⊗δb⊗1^⊗c+d+eV

= (−1)^|δ^b^||δ^d^|1^⊗aV ⊗δb⊗1^⊗cV ⊗δd⊗1^⊗eV

with|δn|=−1 for all n∈N. The term P1^⊗aV ⊗δb+d+1 1^⊗bV ⊗δc⊗1^⊗dV

⊗1^⊗eV

can be written as

1^⊗a_V ⊗δb+d+1 1^⊗b_V ⊗δc⊗1^⊗d_V

⊗1^⊗e_V = 1^⊗a_V ⊗δb+d+1⊗1^⊗e_V 1^⊗a+b_V ⊗δc⊗1^⊗d+e_V . We havea+b+c+d+e=n, choose arbitrarya,e≥0, 1≤a+e < nand sum over all b,c, dsuch that 0 ≤b, d≤n−a−eand 1≤c ≤n−a−esuch that b+c+d=n−e−a:

X

b,c,d

δb+d+1 1^⊗bV ⊗δc⊗1^⊗dV

=δ1(δn⁰) +

n⁰

X

i=1

δn⁰ 1^⊗i−1V ⊗δ1⊗1^⊗nV ⁰⁻ⁱ

+ X

A(n⁰)

δk 1^⊗i−1V ⊗δ`⊗1^⊗k−iV

, wheren⁰=n−a−e. By induction hypothesis, the last display is equal to 0, and we have

X1^⊗aV ⊗δb+d+1 1^⊗bV ⊗δc⊗1^⊗dV

⊗1^⊗eV =X

a,e

1^⊗aV ⊗ X

b,c,d

δb+d+1 1^⊗bV ⊗δc⊗1^⊗dV

⊗1^⊗eV =X

a,e

1^⊗aV ⊗0 ⊗1^⊗eV = 0. Consequently, (7) is true fornand δ₁(δ_n) +

n

X

i=1

δ_n 1^⊗i−1_V ⊗δ₁⊗1^⊗n−i_V

+X

A(n)

δ_k 1^⊗i−1_V ⊗δ_`⊗1^⊗k−i_V

=δ²|_V⊗n= 0. (2)⇒(1): The second implication can be easily deduced from the first one.

2.2. Morphisms and homotopies.

Definition 2.8. Letδ^V be a codifferential on (T V, C) andδ^W be a codifferential on (T W, C). A linear mappingF: T V, C, δ^V

→ T W, C, δ^W

of degree 0 is called morphism provided C_{T W} ◦F = (F⊗F)◦C_{T V} andδ^W ◦F =F◦δ^V.

(9)

Lemma 2.9. Let F: T V, δ^V

→ T W, δ^W

be a linear map of degree 0. Then the following claims are equivalent:

(1) C_{T W} ◦F = (F⊗F)◦C_{T V},

(2) there is a set of linear mappings {fn :V^⊗n →W}_n≥1 of degree0such that F|_V^⊗n=fn+P

B(n)fr₁⊗. . .⊗fr_k, with B(n)given in (B).

Proof. (2)⇒(1): A consequence of Lemma 2.4.

(1)⇒(2) The proof goes by induction. Forv∈V we haveC(v) = 0, which implies 0 = (F⊗F)◦C_{T V} =C_{T W} ◦F and so F(v)∈W.

Assuming the claim is true for all natural numbers less thann, (F⊗F)◦C|_V⊗n= (F⊗F)◦

n−1

X

i=1

1^⊗i

⊗ 1^⊗n−i

=

n−1

X

i=1

(F|_V⊗i)⊗(F|_V⊗n−i) and by induction hypothesisF|_V⊗m =fm+P

B(m)fr₁⊗ · · · ⊗fr_kfor alln > m≥1.

Lemma 2.4 gives

n−1

X

i=1

(F|V^⊗i)⊗(F|V^⊗n−i) =C_{T W} ◦ⁿ⁻¹X

i=1

fi⊗F|V^⊗n−i

and becauseC_{T W} is linear,F|_V^⊗n differs fromPn−1

i=1 fi⊗F|_V^⊗n−i by a linear map fn: V^⊗n→W. Then F|_V⊗n is of the required form and the proof is complete.

Lemma 2.10. LetF : T V, δ^V

→ T W, δ^W

be a linear map of degree 0 such that F|_V⊗n =f_n+P

B(n)f_r₁⊗. . .⊗f_r_k, with all{f_n:V^⊗n →W}_n≥1 linear of degree 0. Then the following are equivalent:

(1) δ^W ◦F =F◦δ^V, (2) for all n≥1 holds

δ₁^W(f_n) +X

B(n)

δ_k^W(f_r₁⊗ · · · ⊗f_r_k) =f₁ δ_n^V

+

n

X

i=1

fn 1^⊗i−1V ⊗δ₁^V ⊗1^⊗n−iV

+X

A(n)

fk 1^⊗i−1V ⊗δ_`^V ⊗1^⊗k−iV

. (8)

Proof. (1)⇒(2): The proof goes by induction. The restriction toV,δ^W ◦F|V = F ◦δ^V|V, corresponds to δ^W₁ ◦f₁ = f₁◦δ₁^V. We now assume (8) applies to all natural numbers less thann. We expand both sides of (8),

δ^W ◦F|_V^⊗n

=δ^W₁ (f_n) +X

B(n)

X

a,b

f_r₁⊗· · ·⊗f_r_a⊗δ_b^W f_r_a+1⊗· · ·⊗f_r_a+b

⊗fra+b+1⊗· · ·⊗f_r_k,

F◦δ^V|_V⊗n

=f1 δ₁^V

+X

B(n)

X

j,`

fr₁⊗· · ·⊗fri−1⊗fr_i 1^⊗jV ⊗δ_`^V⊗1^⊗rV ⁱ^−j−1

⊗fr_i+1⊗· · ·⊗fr_k

(10)

and compare the terms of same homogeneities. We fix j ≥1 andr₁, . . . , r_j ≥1, r₁+· · ·+r_j < n and 0 ≤ m ≤ j, and focus on terms of the form f_r₁ ?· · · ⊗ fri−1⊗?⊗fr_i⊗ · · · ⊗fr_j, where? is an expression of the formδ^W_? (f?⊗ · · · ⊗f?) or f_? 1^⊗?_V ⊗δ^V_? ⊗1^⊗?_V

.

Terms on the right hand side of the formfr₁⊗ · · · ⊗fr_i−1⊗δ_?^W(f?⊗ · · · ⊗f?)⊗ fr_i⊗ · · · ⊗fr_j correspond to

fr₁⊗ · · · ⊗fr_m⊗

δ₁^W(f_n⁰) + X

B(n⁰)

δ_k^W f_r⁰

1⊗ · · · ⊗f_r⁰

k ⊗fr_m+1⊗ · · · ⊗fr_j, while the terms of the formf_r₁⊗ · · · ⊗f_r_i−1⊗f_? 1^⊗?_V ⊗δ^V_? ⊗1^⊗?_V

⊗f_r_i⊗ · · · ⊗f_r_j correspond to

fr₁⊗ · · · ⊗fr_m⊗ ⊗ f1 δ_n^V0

+

n⁰

X

i=1

fn⁰ 1^⊗i−1_V ⊗δ^V₁ ⊗1^⊗n_V ⁰⁻ⁱ

+ X

A(n⁰)

f_k 1^⊗i−1V ⊗δ_`^V ⊗1^⊗k−iV

⊗ ⊗frm+1⊗ · · · ⊗f_r_j withn⁰ =n−r1+· · ·+rj. Becausen⁰< n, they fulfill the equality (8) and hence are equal. Subtracting from both sides all elements of homogeneity greater than 1, we arrive at

δ^W₁ (fn) + X

B(n)

δ^W_k (fr₁⊗ · · · ⊗fr_k)

=f₁ δ_n^V +

n

X

i=1

f_n 1^⊗i−1_V ⊗δ₁^V ⊗1^⊗n−i_V

+X

A(n)

f_k 1^⊗i−1_V ⊗δ^V_` ⊗1^⊗k−i_V . However, this equality is true by (8) forn.

(2)⇒(1): This implication can be again reduced to the previous one.

Definition 2.11. Let δ^V be a codifferential on (T V, C) and δ^W be a codifferential on (T W, C). Let F: T V, C, δ^V

→ T W, C, δ^W

andG: T V, C, δ^V

→ T W, C, δ^W

be morphisms.F andGare homotopy equivalent provided there exist linear mapsH: T V →T W of degree 1 such thatC_{T W} ◦H = (F⊗H+H⊗G)◦ C_{T V} andF−G=Hδ^V +δ^WH.The mapH is a homotopy betweenF aG.

Remark 2.12. Theorem 2.5 implies that H: T V → T W of degree 1 fulfills C_{T W}◦H = (F⊗H+H⊗G)◦C_{T V} if and only if there is a set of maps{hn:V^⊗n→ W}_n≥1 of degree 1 such thatH|_V⊗n=hn+P

B(n),ri>0fr₁⊗ · · · ⊗fri−1⊗hr_i⊗ gr_i+1⊗ · · · ⊗gr_k.

Theorem 2.13. We retain the assumptions of Definition 2.11, and in addition assume the existence of the set of linear maps{e_n:V^⊗n →W}_n≥1,{g_n:V^⊗n→ W}_n≥1of even degreedsuch thatE|_V⊗n=e_n+P

B(n)e_r₁⊗ · · · ⊗e_r_k andG|_V⊗n=

(11)

g_n+P

B(n)g_r₁⊗ · · · ⊗g_r_k. LetF:T V →T W be a linear mapping for which there exists a set of linear maps{fn:V^⊗n →W}_n≥1 of odd degree d+ 1 fulfilling

F|_V⊗n =f_n+ X

B(n),r_i>0

e_r₁⊗ · · · ⊗e_r_i−1⊗f_r_i⊗g_r_i+1⊗ · · · ⊗g_r_k. Then the following assertions are equivalent:

(1) E−G=F δ^V +δ^WF, (2) en−gn=f1(δ_n^V) +Pn

i=1fn(1^⊗i−1V ⊗δ^V₁ ⊗1^⊗n−iV ) +P

A(n)fk(1^⊗i−1V ⊗δ_`^V⊗ 1^⊗k−i_V ) +δ₁^W(fn) +P

B(n),ri>0δ_k^W(er₁⊗ · · · ⊗er_i−1⊗fr_i⊗gr_i+1⊗ · · · ⊗gr_k) for all n≥1.

Proof. The proof can be done along the same lines as the proofs of Lemma 2.7

and Lemma 2.10.

2.3. Codifferentials and A_∞ algebras.

Definition 2.14. For V graded we define sV in such a way that (sV)_i =V_i−1. The graded modulesV andsV are canonically isomorphic:s:V →sV is a linear map of degree 1 called suspension,ω:sV →V is a linear map of degree−1 called desuspension.

Remark 2.15. We haves^⊗n⊗ω^⊗n= (−1)ⁿ⁽ⁿ⁻¹⁾² by the Koszul sign convention.

Theorem 2.16. The following claims are equivalent:

(1) {µn:V^⊗n→V;|µn|=n−2}_n≥1 isA_∞ structure onV,

(2) The linear mapsδ_n =s◦µ_n◦ω^⊗n are of degree−1, and are the components of a codifferential on T sV in the sense of Theorem 2.6.

Proof. (2)⇒(1):δn =s◦µn◦ω^⊗n are the components of a codifferential, and so we have for alln≥1

δ₁(δ_n) +

n

X

i=1

δ_n 1^⊗i−1_V ⊗δ₁⊗1^⊗n−i_V

+X

A(n)

δ_k 1^⊗i−1_V ⊗δ_`⊗1^⊗k−i_V

= 0. This can be rewritten, by Koszul sign convention, as

δ1(δn) =s◦µ1◦ω◦s◦µn◦ω^⊗n =s◦µ1(µn)◦ω^⊗n,

n

X

i=1

δn 1^⊗i−1V ⊗δ1⊗1^⊗n−iV

=

n

X

i=1

s◦µn◦ω^⊗n 1^⊗i−1V ⊗s◦µ1◦ω⊗1^⊗n−iV

=

n

X

i=1

(−1)ⁿ⁻ⁱs◦µ_n ω^⊗i−1⊗µ₁◦ω⊗ω^⊗n−i

=

n

X

i=1

(−1)ⁿ⁻ⁱ(−1)ⁱ⁻¹s◦µ_n 1^⊗i−1_V ⊗µ₁⊗1^⊗n−i_V

◦ω^⊗n,