# An Alternative to Homotopy Transfer for $A_{\infty}$-Algebras

C. A. Cremonini  Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-
Maximilians-Universität München, Theresienstr. 37, D-80333 Munich, Germany
<EMAIL_ADDRESS>and  V. E. Marotta  Università di Trieste,
Dipartimento di Matematica e Geoscienze, Via A. Valerio 12/1, 34127 Trieste,
Italy<EMAIL_ADDRESS>To the memory of Roberto “Rob”
Catenacci, a friend.

###### Abstract.

In this work, we propose a novel approach to the homotopy transfer procedure
starting from a set of homotopy data such that the first differential complex
is a differential graded module over the second one. We show that the module
structure may be used to induce an $A_{\infty}$-algebra on the second
differential complex, constructed in a similar fashion to the homotopy
transfer $A_{\infty}$-algebra. We prove that, under certain conditions, the
$A_{\infty}$-algebras obtained with this procedure are quasi-isomorphic to the
homotopy transfer one. On the other hand, when the side conditions do not
hold, we find that there are cases where the existence of an
$A_{\infty}$-quasi-isomorphism with the homotopy transfer $A_{\infty}$-algebra
is obstructed. In other words, we obtain a new $A_{\infty}$-algebra on the
second complex, inequivalent to the homotopy transfer one. Lastly, we prove
that these $A_{\infty}$-algebras are not infinitesimal Hochschild deformations
of the homotopy transfer $A_{\infty}$-algebra.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 $A_{\infty}$-Algebras
    3. 1.3 Homotopy Transfer of Algebraic Data
    4. 1.4 Outline and Results
    5. 1.5 Acknowledgements
  2. 2 Homotopy Data and Algebraic Structures
    1. 2.1 Homotopy Data with Module Structure
    2. 2.2 Homotopy Transfer and Side Conditions
    3. 2.3 Weakening the Side Conditions
  3. 3 $A_{\infty}$-Algebras from Module Structures
    1. 3.1 Homotopy Induced Products from Module Structures
    2. 3.2 Homotopy Module-Induction and (Higher) Associativity
    3. 3.3 $A_{\infty}$-Algebras from Homotopy Module-Induction with Side Conditions
    4. 3.4 General Construction of Homotopy Module-Induced $A_{\infty}$-Algebras
  4. 4 Homotopy Transfer, Homotopy Module-Induction and Deformations
    1. 4.1 Homotopy Transfer and Homotopy Module-Induction
    2. 4.2 Massey Products
    3. 4.3 Infinitesimal Hochschild Deformations and Homotopy Module-Induction

##  1\. Introduction

$\infty$-algebras were introduced by Stasheff [Sta63], under the name
_strongly homotopy algebras_ , to describe “group-like” topological space.
Soon after their first appearance, they found a vast realm of applications,
both in mathematics and physics. For instance, they represent a cornerstone in
the study of formality of Kähler manifolds [Del+75, Mer99] and deformation
quantisation of Poisson manifolds [Kon03], whereas in physics they appear to
be the underlying structure describing quantum field theory [Jur+19, Jur+20,
Ćir+20] and string field theory, see [Wit86, Zwi98, EKS14, CG20, Dou+20] and
references therein.

###  1.1. Motivation

In this paper, we aim at constructing an $A_{\infty}$-algebra by considering
two quasi-isomorphic differential complexes such that (a subring of) the first
one admits a bimodule structure on the second one. In this way, we may induce
an $A_{\infty}$-algebra on the second complex by a technique that is
reminiscent of the one used for the Homotopy Transfer Theorem. The main
difference resides in the type of algebraic data that is used to induce an
$A_{\infty}$-algebra on the second complex. In order to motivate why this
specific setting became relevant to us, let us discuss the setting of Open
Superstring Field Theory as in [EKS14]. In that context, the authors consider
three complexes, the BRST complexes of string fields with picture numbers
$0,\,-1$ and $-2$, respectively. For the sake of simplicity, consider the
picture numbers as labels for the three complexes. These three complexes are
quasi-isomorphic, where the quasi-isomorphisms are called _Picture Changing
Operators_ , and are bimodules over the picture-number 0 complex, which is
also the only one naturally equipped with an algebraic structure. The authors
used these structures to define on the picture $-1$ complex a 2-product
(interpreted as a vertex of interaction between three string fields) which
turned out to be non-associative, thus spoiling the gauge-invariance of the
action. The gauge-invariance is recovered by adding higher products (i.e.,
higher vertices of interactions) that encoded the structure of an
$A_{\infty}$-algebra. Notably, the construction of this $A_{\infty}$-algebra
does not rely on the transfer of algebraic data between complexes. We analyse
here a similar setting and study under which conditions one can leverage on
module structures to induce algebraic structures from one complex to another.
In particular, this construction will heavily rely on a novel type of
interplay between algebraic and homotopy data.

Supermanifolds and the quasi-isomorphic complexes of differential and integral
forms [Man97, Noj23, Cre23] give another relevant case at the heart of our
motivations. Integral forms are defined as a complex of modules over
differential forms but are not equipped with a ring structure. Thus the quest
for the induction of algebraic structures on this complex. By using the quasi-
isomorphisms and the dga-algebra structure on differential forms one can rely
on the usual Homotopy Transfer Theorem to transfer an $A_{\infty}$-algebra on
integral forms. Nonetheless, the very rich structure we just described
suggests alternative constructions of $A_{\infty}$-algebras coming from the
module structure. We partly explore this topic here and leave the full
discussion to the upcoming work [CM], where we shall use this technology to
introduce intersection theory on supermanifolds.

We shall briefly discuss here the basic setting of the $A_{\infty}$-category,
for more details see [Kel01, LV14].

###  1.2. $A_{\infty}$-Algebras

Let $\Bbbk$ be a field and let $A$ be a $\mathbb{Z}$-graded vector space over
$\Bbbk$,

$A\coloneqq\bigoplus_{i\in\mathbb{Z}}A_{i},$ (1)

where the $A_{i}$ are $\Bbbk$-vector spaces for every $i$.

######  Definition 1 ($A_{\infty}$-Algebra).

An ${A}_{\infty}$-algebra is a pair $(A,\\{m_{n}\\}_{n\in\mathbb{N}_{+}})$,
where $A$ is a $\mathbb{Z}$-graded $\Bbbk$-vector space and
$\\{m_{n}\\}_{n\in\mathbb{N}_{+}}$ is a collection of homogeneous
$\Bbbk$-linear maps of degree $2-n$,

$m_{n}\colon A^{\otimes n}\longrightarrow A,$ (2)

such that they satisfy the following condition for every $n\in\mathbb{N}_{+}$:

$\sum_{n=r+s+t}(-1)^{r+st}m_{r+t+1}(\mathds{1}^{\otimes r}\otimes
m_{s}\otimes\mathds{1}^{\otimes t})=0.$ (3)

The maps $m_{n}$ are called $n$-products. We will denote it by $(A,m_{n}).$

This definition encodes the fact that all of the $n$-products satisfy higher
associativity relations _up to homotopy_. For more details, see [Sta63, Kel01,
Val14, LV14] and references therein.

Note that the maps $\\{m_{n}\\}_{n\in\mathbb{N}_{+}}$ in the definition of the
$A_{\infty}$-algebra belong to the $\mathsf{Hom}$-space of $\Bbbk$-multilinear
maps $\mathsf{Hom}(A^{\otimes n},A)$ for any $n\geq 0$. The space
$\mathsf{Hom}(A^{\otimes n},A)$ is a chain complex $\mathsf{Hom}(A^{\otimes
n},A)\coloneqq\bigoplus_{k\geq 0}\mathsf{Hom}_{k}(A^{\otimes n},A),$ where
$\mathsf{Hom}_{k}(A^{\otimes n},A)$ is the space of multilinear maps $f\colon
A^{\otimes n}\rightarrow A$ of degree $k$, with the differential

$\displaystyle\partial_{A}\colon\mathsf{Hom}(A^{\otimes
n},A)\longrightarrow\mathsf{Hom}(A^{\otimes n},A)$ (4)
$\displaystyle\partial_{A}\varphi\coloneqq[\mathrm{d},\varphi]=\mathrm{d}\circ\varphi-(-1)^{|\varphi|}\varphi\circ\mathrm{d}_{A^{\otimes
n}}.$ (5)

for any $\varphi\in\mathsf{Hom}(A^{\otimes n},A),$ where the differential
$\mathrm{d}_{A}$ is extended to a map $\mathrm{d}_{A^{\otimes n}}\colon
A^{\otimes n}\rightarrow A^{\otimes n}$ by $\mathrm{d}_{A^{\otimes
n}}\coloneqq\sum_{n-1=s+t}\mathds{1}^{\otimes
s}\otimes\mathrm{d}\otimes\mathds{1}^{\otimes t}$.

Lastly, let us recall the notion of morphism between $A_{\infty}$-algebras. An
$A_{\infty}$_-morphism between_ the $A_{\infty}$-algebras $(A,m_{n})$ and
$(A^{\prime},m^{\prime}_{n})$ is a family of maps $f_{n}\colon A^{\otimes
n}\to A^{\prime},$ for all $n\geq 1$ of degree $1-n$ satisfying the relations

$\displaystyle\sum_{r+s+t=n}(-1)^{r+st}f_{r+t+1}\left(\mathds{1}_{A}^{\otimes
r}\otimes m_{s}\otimes\mathds{1}_{A}^{\otimes
t}\right)=\sum_{l=1}^{n}(-1)^{u}m^{\prime}_{l}(f_{i_{1}}\otimes
f_{i_{2}}\otimes...\otimes f_{i_{l}})\,,$ (6)

where $i_{1}+i_{2}+...+i_{n}=n$ and
$u=(l-1)(i_{1}-1)+(l-2)(i_{2}-1)+\ldots+(i_{l-1}-1)$.

###  1.3. Homotopy Transfer of Algebraic Data

The homotopy transfer of algebraic data represents the main source of
inspiration for this work. We briefly discuss it here and refer to [Kad80,
Val14, LV14] for more details on this construction.

Consider the homotopy data

${\left(A^{\bullet},\mathrm{d}_{A}\right)}$${\left(B^{\bullet},\mathrm{d}_{B}\right)}$$\scriptstyle{h_{A}}$$\scriptstyle{Y}$$\scriptstyle{Z}$
(7)

with $h_{A}$ degree $-1$ map satisfying

$\displaystyle\mathds{1}_{A}-Z\circ Y=\mathrm{d}_{A}\circ
h_{A}+h_{A}\circ\mathrm{d}_{A}\,,$ (8)

together with the algebraic data given by an associative product $\wedge$ on
$A^{\bullet}$ making $A^{\bullet}$ a differential graded associative algebra.
By using the quasi-isomorphism $Z,Y$, the product on $A^{\bullet}$ can be
transported to $B^{\bullet}$ by defining

$\displaystyle m_{2}^{\text{ht}}\coloneqq Y\circ\wedge\circ Z^{\otimes
2}\colon B^{\otimes 2}\to B\,.$ (9)

It can be shown that, in general, $m_{2}^{\text{ht}}$ is not associative, in
other words, this transport procedure does not preserve the associativity of
the product. Nonetheless, it can easily be seen that the associativity of
$m_{2}^{\text{ht}}$ is violated in a controlled way, i.e.

$\displaystyle m_{2}^{\text{ht}}(m_{2}^{\text{ht}}(b\otimes b^{\prime})\otimes
b^{\prime\prime})-m_{2}^{\text{ht}}(b\otimes
m_{2}^{\text{ht}}(b^{\prime}\otimes
b^{\prime\prime}))=\partial_{B}m_{3}^{\text{ht}}(b\otimes b^{\prime}\otimes
b^{\prime\prime})\,,$ (10)

for all $b,b^{\prime},b^{\prime\prime}\in B,$ where

$\displaystyle
m_{3}^{\text{ht}}\coloneqq-Y\circ\wedge\circ\bigl{(}(h_{A}\circ\wedge\circ
Z^{\otimes 2})\otimes
Z\bigr{)}+Y\circ\wedge\circ\bigl{(}Z\otimes(h_{A}\circ\wedge\circ Z^{\otimes
2})\bigr{)}$ (11)

and $\partial_{B}$ is defined as in Equation (5). This construction can be
iterated and it is possible to show that $m_{2}^{\text{ht}}$ and
$m_{3}^{\text{ht}}$ are associative up to an exact term with respect to
$\partial_{B}.$ This leads to stating the following remarkable result, proved
in [Kad80] for $B=\mathsf{H}_{\bullet}(A)$.

######  Theorem 1 (Homotopy Transfer Theorem).

Consider the homotopy data introduced above and let $B^{\bullet}$ be endowed
with the (higher) products $m_{n}^{\text{ht}},$ for all $n\in\mathbb{N}_{+},$
obtained as described above. Then $(B^{\bullet},m_{n}^{\text{ht}})$ forms an
$A_{\infty}$-algebra.

This construction is one of our main guides in order to build a procedure
inducing an $A_{\infty}$-algebra structure on the differential complex
$(B^{\bullet},\mathrm{d}_{B})$ starting from a different type of algebraic
data on $A^{\bullet}$.

###  1.4. Outline and Results

The outline of the paper is as follows: in Section 2 we discuss the basic
assumptions that will be the main framework of this paper. In Section 3 we
show the existence of new $A_{\infty}$-algebras coming from an analogue of the
homotopy transfer procedure performed under new assumptions. In Section 4 we
discuss how this novel procedure relates to homotopy transfer.

Section 2 begins with the definition of the main object of this work, the
homotopy data given by two quasi-isomorphic cochain complexes
$(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$ with quasi-isomorphisms $Y,Z$
such that $A$ is a differential graded associative algebra, $B$ is a
differential graded $\mathrm{Im}(Z)$-bimodule and the quasi-isomorphisms $Y,Z$
are $\mathrm{Im}(Z)$-bimodule homomorphisms, where $\mathrm{Im}(Z)$ is assumed
to be a subring of $A$. We call this _Homotopy Data with Module Structure_ ,
see Definition 2. As an example, we show that the de Rham complex on a (split)
supermanifold and its quasi-isomorphic complex of integral forms can be
endowed with homotopy operators such that they represent an example of
Homotopy Data with Module Structure. We thus proceed to show that the above
homotopy data satisfying at least one of the so-called side conditions only
admit an associative algebra on $B$ coming from the homotopy transfer of the
algebra structure on $A,$ i.e. we have the following

Proposition 2. Let $(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$ be homotopy
data with module structure as in Definition 2, endowed with the homotopy
transfer $A_{\infty}$-algebra $(B,m_{n}^{\text{ht}})$. Moreover, suppose that
$h_{A}$ satisfies the left-side condition. Then the homotopy transfer
$A_{\infty}$-algebra $(B,m_{n}^{\text{ht}})$ reduces to an associative
algebra, i.e., $m_{n}^{\text{ht}}=0$ for $n>2.$

This represents one of our motivations for the investigation of an alternative
homotopy transfer procedure, so that we may obtain an $A_{\infty}$-algebra on
$B$ admitting higher terms even when the aforementioned side condition holds.

Inspired by [Mar01], we explore under which circumstances we can preserve the
right side condition under a change of homotopy data removing the obstruction
to the existence of a strong homotopy equivalence. Moreover, we show that if
the quasi-isomorphisms satisfy $Z\circ Y\circ Z=Y,$ there always exists a
redefinition of the homotopy operator $h_{B}$ implementing the right side
condition with the same quasi-isomorphisms. We stress that if the right side
condition $Z\circ h_{B}=0$ holds, then we have that the quasi-isomorphisms
satisfy $Z\circ Y\circ Z=Z.$ Furthermore, if the right side condition holds,
then the obstruction to the existence of a strong homotopy equivalence can
always be removed.

We also discuss a weaker notion of the side conditions given by $Y\circ Z\circ
h_{B}=0$ and $h_{B}\circ Y\circ Z=0,$ called _Weak Side Conditions_. We show
that if the quasi-isomorphisms satisfy the condition $(Z\circ Y)^{2}=Z\circ
Y,$ then the homotopy operator can always be redefined in such a way that the
weak side conditions hold. The weak side conditions will appear later on in
the construction of an $A_{\infty}$-algebra from the Homotopy Module-
Induction.

Section 3 is the core of this work and contains the main results concerning
the homotopy transfer product induced by a new set of algebraic data. In
particular, in Subsection 3.1, we define the main object of our study the
Homotopy Module-Induction procedure as in

Definition 5. Let $(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$ be quasi-
isomorphic chain complexes endowed with the homotopy data with bimodule
structure of Definition 2. Then we define the _Homotopy Module-Induced Product
(HMIP)_

$\displaystyle m_{2}(b\otimes b^{\prime})\coloneqq k_{1}Z(b)\rhd
b^{\prime}+k_{2}\,h_{B}(b)\lhd Z(b^{\prime})\,,$ (12)

for all $b,b^{\prime}\in B$ and $k_{1},k_{2}\in\Bbbk$.

Our analysis starts off by showing that the product $m_{2}$ differs from the
homotopy transfer product $m_{2}^{\text{ht}}$ by $\partial_{B}\tilde{m_{2}},$
for some $\tilde{m}_{2}\in\mathsf{Hom}_{-1}(B^{\otimes 2},B)$ and is
associative up to homotopy, i.e., $\mathsf{Ass}=\partial_{B}m_{3},$ for some
$m_{3}\in\mathsf{Hom}(B^{\otimes 3},B),$ see Subsection 3.2. We thus check the
first higher associativity relation involving the 3-product $m_{3}$ and
$m_{2}$ and find out that it is verified in two non-trivial cases: when the
coefficients $k_{1},\,k_{2}$ appearing in the definition of $m_{2}$ are such
that $k_{1}=-k_{2},$ or when the homotopy data satisfies the weak side
conditions, see Lemma 10.

Before completing the discussion of these two cases, we show that we recover
an $A_{\infty}$-algebra from the Homotopy Module-Induction when the side
conditions hold, i.e., we have the following

Theorem 11. Let $(A,\mathrm{d}_{A}),(B,\mathrm{d}_{B})$ be homotopy data with
module structure as in Definition 2 such that the homotopy $h_{B}$ satisfies
the side conditions of Definition 3. Then the Homotopy Module-Induction gives
an $A_{\infty}$-algebra $(B,m_{n})$ with $m_{1}=\mathrm{d}_{B},$ $m_{2}$ given
by Equation (78), and higher products

$\displaystyle m_{n}\coloneqq(m_{n-1}\circ m_{2})\circ h_{B}\ ,\qquad\text{for
}n\geq 3\,.$ (13)

Thus, we can move on to showing that the Homotopy Module-Induction gives an
$A_{\infty}$-algebra also in the weaker assumptions $k_{1}=-k_{2}$ or homotopy
data satisfying the weak side conditions.

Theorem 12. Let $(A,\mathrm{d}_{A}),(B,\mathrm{d}_{B})$ be homotopy data with
module structure as in Definition 2 and assume that either of the following
conditions hold

  1. (1)

the homotopy $h_{B}$ satisfies the weak side conditions;

  2. (2)

the coefficients appearing in the definition of $m_{2}$ are such that
$k_{1}=-k_{2}.$

Then the Homotopy Module-Induction gives the $A_{\infty}$-algebras $(B,m_{n})$
with $m_{1}\coloneqq\mathrm{d}_{B},$ $m_{2}$ as in Definition 5 and higher
products

$\displaystyle m_{n}\coloneqq$ $\displaystyle(-1)^{n-1}\left(m_{2}\circ
m_{n-1}\right)\circ_{fb}h_{B}+\left(m_{n-1}\circ m_{2}\right)\circ_{fb}h_{B}$
(14) $\displaystyle=$
$\displaystyle\tilde{m}_{2}\circ_{Z}m_{n-1}+m_{n-1}\circ_{h_{B}}\tilde{m}_{2}\,,$
(15)

where for $n=3$ we consider only the second equality.

Section 4 concentrates on the description of the relation between the homotopy
transfer $A_{\infty}$-algebra and the Homotopy Module-Induced one. In
particular, we can find a strict quasi-isomorphims between them when the side
conditions hold, as in

Proposition 13. Let Theorem 11 hold. Then the $A_{\infty}$-algebra $(B,m_{n})$
is strictly quasi-isomorphic to the dga algebra $(A,\mathrm{d}_{A},\wedge)$
seen as an $A_{\infty}$-algebra if and only if $k_{1}\neq-k_{2}$. The quasi-
isomorphism reads $f_{1}\coloneqq(k_{1}+k_{2})Z$ and $f_{i}\coloneqq 0,$ for
all $i\neq 1$.

This yields that, in this case, the homotopy transfer $A_{\infty}$-algebra and
the Homotopy Module-Induced one are (strictly) quasi-isomorphic. Moreover, the
$A_{\infty}$-Massey products they induce on the cohomology $\mathsf{H}(B)$ are
isomorphic.

When Theorem 12 holds in the case of hypothesis 2, there is an obstruction to
the existence of such quasi-isomorphism, as shown in

Proposition 15. Let Theorem 12 hold in the case of hypothesis 2 and assume
that the product $\wedge$ on $A$ induces a non-trivial product on
$\mathsf{H}(A)$, i.e.,
$0\neq[\wedge]\in\mathsf{H}\left(\mathsf{Hom}\left(A^{\otimes
2},A\right)\right)$. Then the cohomology class of $\wedge$ represents an
obstruction to the existence of a quasi-isomorphism
$f_{\bullet}:(B,m_{n})^{\otimes\bullet}\to(B,m_{n}^{\text{ht}})$.

We also address the problem of determining whether the Homotopy Module-Induced
$A_{\infty}$-algebra is an infinitesimal deformation of the standard homotopy
transfer $A_{\infty}$-algebra, which is stated in the following

Proposition 18. Assume that Theorem (11) holds and let $(B,m_{n}^{\text{ht}})$
and $(B,m_{n})$ be the $A_{\infty}$-algebras induced by the homotopy transfer
and the Homotopy Module-Induction, respectively. Then $(B,m_{n})$ is not an
infinitesimal deformation of $(B,m_{n}^{\text{ht}}).$

A relevant example of this result is given by the case of an associative
Homotopy Module-Induced Product.

###  1.5. Acknowledgements

We thank Branislav Jurčo, Simone Noja, Ruggero Noris, Ingmar Saberi, Ivo Sachs
and Richard Szabo for helpful discussions. The work of C.A.C. and V.E.M. was
supported in part by the GACR Grant EXPRO 19-28268X. The work of C.A.C. is
supported by the Excellence Cluster Origins of the DFG under Germany’s
Excellence Strategy EXC-2094 390783311. The work of V.E.M. is supported by
PNRR MUR projects PE0000023-NQSTI.

##  2\. Homotopy Data and Algebraic Structures

In this Section, we introduce the notion of homotopy data endowed with a
bimodule structure and discuss the interplay between these data and the side
conditions. For a concise introduction to homotopy data and homotopy transfer
see [Kel01], [Cra04], [Val14] and references therein.

###  2.1. Homotopy Data with Module Structure

The cornerstone of this work is the following set of homotopy data. Here and
throughout the paper every (graded) vector space is defined over a field
$\Bbbk$ of characteristic zero, which will always be omitted from our
notation.

######  Definition 2 (Homotopy data with bimodule structure).

Let $(A^{\bullet},\mathrm{d}_{A})$ and $(B^{\bullet},\mathrm{d}_{B})$ be two
quasi-isomorphic cochain complexes together with two degree $-1$ maps
(henceforth homotopies) $h_{A}\colon A^{\bullet}\to A^{\bullet-1}$,
$h_{B}\colon B^{\bullet}\to B^{\bullet-1}$, i.e.,

${\left(A^{\bullet},\mathrm{d}_{A}\right)}$${\left(B^{\bullet},\mathrm{d}_{B}\right)}$$\scriptstyle{h_{A}}$$\scriptstyle{Y}$$\scriptstyle{Z}$$\scriptstyle{h_{B}}$
(16)

such that the following conditions hold

$\displaystyle\mathds{1}_{A}-Z\circ Y$
$\displaystyle=\left[\mathrm{d}_{A},h_{A}\right]\equiv\mathrm{d}_{A}\circ
h_{A}+h_{A}\circ\mathrm{d}_{A}=\partial_{A}h_{A}\,,$ (17)
$\displaystyle\mathds{1}_{B}-Y\circ Z$
$\displaystyle=\left[\mathrm{d}_{B},h_{B}\right]\equiv\mathrm{d}_{B}\circ
h_{B}+h_{B}\circ\mathrm{d}_{B}=\partial_{B}h_{B}\,.$ (18)

For the sake of notation, in the following, we will denote the complexes by
$A^{\bullet}\equiv A$ and $B^{\bullet}\equiv B$. Let $A$ and $B$ be endowed
also with the following structures:

  1. (1)

$(A,\wedge,\mathrm{d}_{A})$ is a differential graded associative algebra,
i.e., $A$ is equipped with a $\Bbbk$-bilinear map

$\displaystyle\wedge\colon A\otimes A\rightarrow$ $\displaystyle\,A$ (19)
$\displaystyle a\otimes a^{\prime}\mapsto$ $\displaystyle\,a\wedge a^{\prime}\
,$ (20)

and $\mathrm{d}_{A}$ satisfies the graded Leibniz rule with respect to
$\wedge$:

$\displaystyle\mathrm{d}_{A}\left(a\wedge
a^{\prime}\right)=\left(\mathrm{d}_{A}a\right)\wedge
a^{\prime}+(-1)^{|a|}a\wedge\left(\mathrm{d}_{A}a^{\prime}\right)\ ,$ (21)

where $|a|$ denotes the parity of an homogeneous element $a\in A$;

  2. (2)

$\mbox{Im}(Z)$ is a subring of $A$ and $(B,\mathrm{d}_{B})$ is a dg-bimodule
over $\mbox{Im}(Z)$, i.e., there is a left action of $\mathrm{Im}(Z)$ on $B$
denoted by

$\displaystyle\rhd\,\colon\mathrm{Im}(Z)\otimes B$ $\displaystyle\to\,B$ (22)
$\displaystyle Z(b)\otimes b^{\prime}$ $\displaystyle\mapsto\,Z(b)\rhd
b^{\prime}\,,$ (23)

and a right action of $\mathrm{Im}(Z)$ on $B$ denoted by

$\displaystyle\lhd\,\colon B\otimes\mathrm{Im}(Z)$ $\displaystyle\to B$ (24)
$\displaystyle b\otimes Z(b^{\prime})$ $\displaystyle\to
b\,\lhd\,Z(b^{\prime})\,,$ (25)

such that the following properties hold:

$\displaystyle Z(b)\rhd\left(Z(b^{\prime})\rhd b^{\prime\prime}\right)$
$\displaystyle=\left(Z(b)\wedge Z(b^{\prime})\right)\rhd b^{\prime\prime}\ ,$
(26) $\displaystyle\left(b\lhd Z(b^{\prime})\right)\lhd Z(b^{\prime\prime})$
$\displaystyle=b\lhd\left(Z(b^{\prime})\wedge Z(b^{\prime\prime})\right)\,,$
(27)

for all $b,\,b^{\prime},\,b^{\prime\prime}\in B$, and

$\displaystyle\mathrm{d}_{B}\left(Z(b)\rhd b^{\prime}\right)$
$\displaystyle=Z\left(\mathrm{d}_{B}b\right)\rhd
b^{\prime}+\left(-1\right)^{|b|}Z(b)\rhd\left(\mathrm{d}_{B}b^{\prime}\right)\,,$
(28) $\displaystyle\mathrm{d}_{B}\left(b\lhd Z(b^{\prime})\right)$
$\displaystyle=\mathrm{d}_{B}b\lhd Z(b^{\prime})+(-1)^{|b|}\lhd
Z(\mathrm{d}_{B}b^{\prime})\,,$ (29)

for all $b,\,b^{\prime}\in B$ homogeneous elements;

  3. (3)

The maps $Y\colon A\rightarrow B$ and $Z\colon B\rightarrow A$ are
$\mbox{Im}(Z)$-bimodule homomorphisms, i.e.,

$\displaystyle Y(Z(b)\wedge a)=Z(b)\rhd Y(a)\,,\qquad$ $\displaystyle
Z(Z(b)\rhd b^{\prime})=Z(b)\wedge Z(b^{\prime})\,,$ (30) $\displaystyle
Y(a\wedge Z(b))=Y(a)\lhd Z(b)\,,\qquad$ $\displaystyle Z(b\lhd
Z(b^{\prime}))=Z(b)\wedge Z(b^{\prime})\,,$ (31)

for all $a\in A,\ b,b^{\prime}\in B$.

The cochain complexes $A,B$ endowed with the above structures are called
_homotopy data with bimodule structure_.

######  Remark 1.

It follows from Property (3) that

$Y(Z(b)\wedge Z(b^{\prime}))=Z(b)\rhd(Y\circ Z)(b^{\prime})=(Y\circ Z(b))\lhd
Z(b^{\prime}),$ (32)

for every $b,b^{\prime}\in B$, and that

$\displaystyle Z(b)\rhd b^{\prime}-b\lhd Z(b^{\prime})\in\ker(Z)\,,$ (33)

for all $b,\,b^{\prime}\in B.$

If $\wedge$ is graded-commutative, then the left $\mathrm{Im}(Z)$-action and
the right $\mathrm{Im}(Z)$-action must agree on $\mathrm{Im}(Y)$ up to the
sign coming from the graded-commutativity of $\wedge.$

We present here one of the motivating examples of the construction described
in this paper. We will limit ourselves to describing the homotopy data with
module structure as in Definition (2); a full analysis with the construction
of the $A_{\infty}$ algebra with the methods described in this paper is
deferred to a future work [CM] where it will be linked to intersection theory
on supermanifolds. The core of this construction relies on the fact that
supermanifolds are naturally equipped with two complexes of forms:
differential forms (or _superforms_), which are a generalisation of usual
differential forms on ordinary manifolds, and _integral forms_ , which define
sections which are integrable on odd-codimension-0 submanifolds. The former
comes naturally with the structure of an algebra, and the latter comes as a
module over superforms, but no algebraic structure is given. We will not
linger on definitions regarding supermanifolds, but rather just mention the
basics which are needed for the present purposes. We refer the reader to
[Man97, Noj23] for a detailed introduction to supermanifolds.

######  Example 1 (Integral Forms on Split Supermanifolds).

Recall that a supermanifold $\mathcal{SM}$ of dimension $(m|n)$ is a locally
decomposable, locally ringed space
$\left(\left|\mathcal{SM}\right|,\mathcal{O}_{\mathcal{SM}}\right),$ where
$\left|\mathcal{SM}\right|$ is a topological space,
$\mathcal{O}_{\mathcal{SM}}$ is the structure sheaf of $\mathbb{Z}_{2}$-graded
commutative rings and there exists a locally free sheaf $\mathcal{F}$ of rank
$n$ such that, for any open subset
$\mathcal{U}\subset\left|\mathcal{SM}\right|,$
$\mathcal{O}_{\mathcal{SM}}\rvert_{\mathcal{U}}\cong\bigwedge^{\bullet}\mathcal{F}^{*}\rvert_{\mathcal{U}},$
as sheaves of $\mathbb{Z}_{2}$-graded rings. We call
$\mathcal{SM}_{red}=\left(\left|\mathcal{SM}\right|,\mathcal{O}_{\mathcal{SM}_{red}}\right)$,
the _reduced manifold_ , where
$\mathcal{O}_{\mathcal{SM}_{red}}\coloneqq\mathcal{O}_{\mathcal{SM}}/\mathcal{J}$
is the sheaf of functions on $|\mathcal{SM}_{red}|$, with $\mathcal{J}$ the
ideal sheaf generated by nilpotent elements. We denote by
$i^{*}\colon\mathcal{O}_{\mathcal{SM}}\to\mathcal{O}_{\mathcal{SM}_{red}}$ the
surjective sheaf morphism corresponding to the embedding of locally ringed
spaces $i\colon\mathcal{SM}_{red}\hookrightarrow\mathcal{SM}$ and its
extension to forms.

As a consequence of the fact that the structure sheaf
$\mathcal{O}_{\mathcal{SM}}$ has both even and odd sections, the de Rham
complex of a supermanifold is unbounded from above:

${0}$${\mathcal{O}_{\mathcal{SM}}}$${\mathsf{\Omega}^{(1)}\left(\mathcal{SM}\right)}$${\ldots}$${\mathsf{\Omega}^{(m)}\left(\mathcal{SM}\right)}$${\ldots}$$\scriptstyle{\mathrm{d}_{dR}}$$\scriptstyle{\mathrm{d}_{dR}}$$\scriptstyle{\mathrm{d}_{dR}}$$\scriptstyle{\mathrm{d}_{dR}}$

.

In particular, the complex does not contain the space of integrable sections
(i.e., a space of “top forms” analogous to the Determinant for an ordinary
manifold). Nonetheless, on supermanifolds, a second complex can be introduced,
the complex of _integral forms_ :

${\ldots}$${\mathsf{\Sigma}^{(0)}\left(\mathcal{SM}\right)}$${\ldots}$${\mathsf{\Sigma}^{(m-1)}\left(\mathcal{SM}\right)}$${\mathsf{\Sigma}^{(m)}\left(\mathcal{SM}\right)}$${0}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

,

where we denoted
$\mathsf{\Sigma}^{(\bullet)}\left(\mathcal{SM}\right)\equiv\mathpzc{B}er\left(\mathcal{SM}\right)\otimes_{\mathcal{O}_{\mathcal{SM}}}S^{m-\bullet}\Pi\mathcal{T}\mathcal{SM}$,
i.e., integral forms are defined by tensoring the Berezinian sheaf (the super-
analogous of the Determinant sheaf for ordinary manifolds) with (parity-
changed) polyvector fields and $\delta$ is the nilpotent operator defined in
[Man97]. It has been proven that the two complexes are isomorphic in
cohomology, see, e.g., [Noj23]. We shall show that these two complexes fit the
following homotopy data

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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{69.33336pt}{-3.14583pt}\pgfsys@curveto{88.65192pt}{2.03056pt}{88.65192pt}{-14.34999pt}{69.71968pt}{-9.27711pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.96593}{0.25882}{-0.25882}{-0.96593}{69.52652pt}{-9.22536pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{86.17503pt}{-7.90689pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{$\scriptstyle{h_{\mathsf{\Sigma}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (34)

Explicit realisations of these quasi-isomorphisms are well-known in the
Physics community as _Picture Changing Operators_ since [FMS86], were formally
interpreted in a geometric context in [Bel97] and translated in the language
of (co)chain maps in [Cre23]. We will focus only on _split_
supermanifolds111For the sake of clarity, one could think about smooth
supermanifolds, as all smooth supermanifolds are split (see [Bat79]). (i.e.,
those supermanifolds for which the structure sheaf is globally isomorphic to
the reduced structure sheaf times a Grassmann algebra) since in these cases it
is possible to find an easy global expression for the map
$Z\colon\mathsf{\Sigma}\left(\mathcal{SM}\right)\to\mathsf{\Omega}\left(\mathcal{SM}\right)$,
which is surjective on $\mathsf{\Omega}\left(\mathcal{SM}_{red}\right)$. We
will address the case of non-split (and non-projected) supermanifolds in [CM].
Integral forms are naturally equipped with an
$\mathsf{\Omega}(\mathcal{SM})$-module structure, where the action is simply
defined by the pairing forms-vector fields. Given an open subset
$\mathcal{U}\subseteq\left|\mathcal{SM}\right|$ with local coordinates
$\left\\{x^{i},\theta^{\alpha}\right\\}$ (the former coordinates are even, the
latter are odd), the maps $Y$ and $Z$ are defined as follows. Let
$\omega\in\mathsf{\Omega}^{(p)}(\mathcal{SM})$ and consider its local
expression

$\displaystyle\omega\rvert_{\mathcal{U}}=\sum_{i_{1}\ldots
i_{p}=1}^{m}\omega_{i_{1}\ldots
i_{p}}\left(x\right)\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{p}}+\bar{\omega}\,,$
(35)

where $\bar{\omega}\in\ker(i^{*}),$ thus we define

$\displaystyle Y^{(p)}$
$\displaystyle\colon\mathsf{\Omega}^{(p)}\left(\mathcal{U}\right)\to\mathsf{\Sigma}^{(p)}\left(\mathcal{U}\right)$
(36) $\displaystyle\omega\rvert_{\mathcal{U}}\mapsto
Y(\omega\rvert_{\mathcal{U}})$ $\displaystyle\coloneqq\sum_{i_{1}\ldots
i_{p}=1}^{m}\omega_{i_{1}\ldots
i_{p}}\left(x\right)\mathrm{d}x^{i_{1}}\wedge\ldots\wedge\mathrm{d}x^{i_{p}}\rhd\prod_{\alpha=1}^{n}\theta^{\alpha}\mathcal{D}\otimes\bigwedge_{i=1}^{m}\pi\partial_{x^{i}},$
(37)

where
$\rhd\colon\mathsf{\Omega}(\mathcal{SM})\otimes\mathsf{\Sigma}(\mathcal{SM})\to\mathsf{\Sigma}(\mathcal{SM})$
denotes the action of $\mathsf{\Omega}(\mathcal{SM})$ on
$\mathsf{\Sigma}(\mathcal{SM}),$ $\mathcal{D}$ is the local generator of
sections of the Berezinian sheaf (see [Man97, Noj23]), and $\pi$ is the parity
changing operator. Similarly, let
$\mu\in\mathsf{\Sigma}^{(m-p)}(\mathcal{SM})$ and consider

$\displaystyle\mu\rvert_{\mathcal{U}}=\sum_{i_{1},\ldots,i_{p}=1}^{m}\mu_{i_{1}\ldots
i_{p}}\left(x,\theta\right)\mathcal{D}\otimes\bigwedge_{s=1}^{p}\pi\partial_{x^{s}}+\bar{\mu}\,,$
(38)

where $\bar{\mu}$ is the component of the integral form containing at least
one vector field $\pi\partial_{\theta},$ and define

$\displaystyle Z^{(m-p)}$
$\displaystyle\colon\mathsf{\Sigma}^{(m-p)}\left(\mathcal{U}\right)\to\mathsf{\Omega}^{(m-p)}\left(\mathcal{U}\right)$
$\displaystyle\mu\rvert_{\mathcal{U}}\mapsto Z(\mu\rvert_{\mathcal{U}})$
$\displaystyle\coloneqq\sum_{i_{1},\ldots,i_{p}=1}^{m}\mu_{i_{1}\ldots
i_{p}}^{\mathrm{top}}\left(x\right)\iota_{\partial_{x^{i_{1}}}}\ldots\iota_{\partial_{x^{i_{p}}}}\left(\mathrm{d}x^{1}\wedge\ldots\wedge\mathrm{d}x^{m}\right)\,,$
(39)

where $\iota$ is the contraction by a vector field. The global well-
definiteness of these maps is shown in [Cre23]). The fact that
$\mathrm{Im}(Z)=\mathsf{\Omega}\left(\mathcal{SM}_{red}\right)\subset\mathsf{\Omega}\left(\mathcal{SM}\right)$
allows us to easily find candidates for the homotopy operators such that
Equations (17) and (18) hold given by (up to coefficients which we neglect for
the moment)

$\displaystyle h_{\mathsf{\Omega}}\coloneqq\iota\chi\ \ ,\ \
h_{\mathsf{\Sigma}}\coloneqq\bullet\wedge\pi\chi\ \ ,\ \
\chi=\sum_{\alpha}\theta^{\alpha}\frac{\partial}{\partial\theta^{\alpha}}\ ,$
(40)

that is, $h_{\mathsf{\Omega}}$ is the contraction along the _odd Euler vector
field_ , while $h_{\mathsf{\Sigma}}$ is the (symmetrised) multiplication by
the parity-changed field.
$\mathrm{Im}(Z)=\mathsf{\Omega}\left(\mathcal{SM}_{red}\right)$ also allows to
prove that condition (3) of Definition (2) is satisfied, so that Equation (34)
defines an example of homotopy data with module structure. It can easily be
shown that the quasi-isomorphism $Y,Z$ and the homotopy operators
$h_{\mathsf{\Omega}},h_{\mathsf{\Sigma}}$ satisfy the following identities

$\displaystyle Z\circ Y\circ Z$ $\displaystyle=Z\,,\qquad
h_{\mathsf{\Omega}}\circ Z=0=Z\circ h_{\mathsf{\Sigma}}\,,$ (41)
$\displaystyle Y\circ Z\circ Y$ $\displaystyle=Y\,,\qquad
h_{\mathsf{\Sigma}}\circ Y=0=Y\circ h_{\mathsf{\Omega}}\,.$ (42)

We will discuss the results of this paper for the homotopy data of integral
forms on supermanifolds in the upcoming work [CM]. Note that this may be
particularly relevant since the complex of integral forms does not have any
natural algebraic structure.

The last properties of the homotopy data defined by integral forms on
supermanifolds set the stage for the introduction of the so-called side
conditions for the homotopy operator of quasi-isomorphic cochain complexes.

###  2.2. Homotopy Transfer and Side Conditions

Let us discuss the interplay between homotopy transfer and the so-called side
conditions. Recall from [Mar01, Cra04] the following

######  Definition 3 (Side Conditions).

Let $A,B$ be quasi-isomorphic cochain complexes by quasi-isomorphisms $Y,Z$
together with the homotopy $h_{B}.$ This homotopy $h_{B}$ is said to satisfy
the _side conditions_ if the following hold:

$\displaystyle h_{B}\circ Y=0\,,\quad Z\circ h_{B}=0\,,\quad h_{B}^{2}=0\,.$
(43)

An analogous definition holds for the homotopy $h_{A},$ i.e., $h_{A}$
satisfies the side conditions if

$\displaystyle Y\circ h_{A}=0\,,\quad h_{A}\circ Z=0\,,\quad h_{A}^{2}=0\,.$
(44)

In the following, we will be mainly concerned with the implementation of some
side conditions only; in particular, we will use the _right-side condition_
for $h_{A},h_{B}$

$\displaystyle Z\circ h_{B}=0\,,\quad\,Y\circ h_{A}=0\,,$ (45)

and the _left-side condition_ for $h_{A},h_{B}$

$\displaystyle h_{A}\circ Z=0\,,\quad\,h_{B}\circ Y=0\,.$ (46)

Recall that, as discussed in [Mar01, Mar06, Cra04], the side conditions can
always be implemented for a deformation retract,222As discussed in [Cra04],
the deformation retract condition given by $Z\circ Y=1$. In order to change
the homotopy operator $h_{B}$ such that the side conditions still hold, it
suffices to have $Z\circ Y\circ Z=Z$ and $Y\circ Z\circ Y=Y$, as described
below. via an appropriate modification of the homotopy operator.

We point out that the possibility of enforcing the left/right-side conditions,
together with the module structure, greatly simplifies the usual homotopy
transfer procedure. In particular, it gives an associative algebra on $B$.

######  Proposition 2.

Let $(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$ be homotopy data with module
structure as in Definition 2, endowed with the homotopy transfer
$A_{\infty}$-algebra $(B,m_{n}^{\text{ht}})$. Moreover, suppose that $h_{A}$
satisfies the left-side condition as in Definition 3. Then the homotopy
transfer $A_{\infty}$-algebra $(B,m_{n}^{\text{ht}})$ reduces to an
associative algebra, i.e., $m_{n}^{\text{ht}}=0$ for $n>2.$

###### Proof.

In order to discuss the main argument of the proof, let us consider the case
of the 3-product $m_{3}^{\text{ht}}$ (see, for instance, [Val14]):

$\displaystyle m_{3}^{\text{ht}}(b\otimes b^{\prime}\otimes b^{\prime\prime})$
$\displaystyle\coloneqq Y\bigl{(}Z(b)\wedge h_{A}\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)-h_{A}\left(Z(b)\wedge Z(b^{\prime})\right)\wedge
Z(b^{\prime\prime})\bigr{)}$ (47)
$\displaystyle=Y\bigl{(}Z(b)\wedge(h_{A}\circ Z)\left(Z(b^{\prime})\rhd
b^{\prime\prime}\right)-(h_{A}\circ Z)\left(Z(b)\rhd b^{\prime}\right)\wedge
Z(b^{\prime\prime})\bigr{)}=0\ ,$ (48)

where we have used the assumption that $Z$ is an $\mathrm{Im}(Z)$-module
homomorphism $\wedge\circ\left(Z\otimes
Z\right)=Z\circ\rhd\circ\left(Z\otimes\mathds{1}\right)$ and $h_{A}\circ Z=0$.

The above argument may be generalised as follows. Any planar diagram with $n$
entries, of degree $n-2$ (i.e., with an insertion of the homotopy $h_{A}$ in
any internal branch) will be trivially zero, because the homomorphism
assumption of $Z$ will always allow moving one of the operators $Z$ acting on
the entries to an internal leg, thus leading to $h_{A}\circ Z,$ see Equation
(64). ∎

Let us discuss under which assumptions some of the side conditions can be
implemented on quasi-isomorphic cochain complexes. We will focus on the
implementation of the _right-side condition_ (45), because of its relation to
the obstruction to lifting a homotopy equivalence to a _strong homotopy
equivalence_ , i.e. an extension of the homotopy equivalence given by
Equations (17) and (18) to (higher) $2n$-degree maps $Y^{(2n)}$, $Z^{(2n)}$
and (higher) $2n+1$-degree homotopies $h_{A}^{(2n+1)},$ $h_{B}^{(2n+1)},$ for
all $n\geq 1,$ see [Mar01].

In order to explore this relation, let us discuss how the obstruction to the
existence of a strong homotopy equivalence can be understood. Note that
Equation (17) yields

$\displaystyle\mathrm{d}_{A}\circ Z\circ h_{B}+Z\circ
h_{B}\circ\mathrm{d}_{B}=\mathrm{d}_{A}\circ h_{A}\circ Z+h_{A}\circ
Z\circ\mathrm{d}_{B}\,.$ (49)

We can define the usual differential on the space of maps
$\partial_{BA}\colon\mathsf{Hom}\left(B,A\right)\to\mathsf{Hom}\left(B,A\right)$
and recast Equation (49) as

$\displaystyle\partial_{BA}\left(Z\circ
h_{B}\right)=\partial_{BA}\left(h_{A}\circ Z\right)\ \implies\
\partial_{BA}\left(Z\circ h_{B}-h_{A}\circ Z\right)=0\ .$ (50)

The cohomology class

$\displaystyle[Z\circ h_{B}-h_{A}\circ Z]\in\mathsf{H}(\mathsf{Hom}(B,A))$
(51)

is the obstruction, described in [Mar01], to lifting a homotopy equivalence to
a strong homotopy equivalence333Note that an analogous argument can be made by
considering the combination $Y\circ h_{A}-h_{B}\circ Y\in\mathsf{Hom}(A,B)$..
We may interpret Equation (51) as an obstruction to simultaneously
implementing the left and right-side conditions for $h_{A}$ and $h_{B}$ (up to
homotopy). In [Mar01] it is shown that given a homotopy data, it is always
possible to define another one by modifying either the homotopy $h_{A}$ or the
homotopy $h_{B}$ so that the obstruction is removed, i.e., $Z\circ
h_{B}-h_{A}\circ Z$ becomes a $\partial_{BA}$-exact element. In our case, we
aim to remove the obstruction (51) while preserving the right-side condition
given by Equation (45); to do so, we recall the modification of the homotopy
operator described in [Mar01] and show the following property, analogous to
the characterisation given in [Cra04].

######  Proposition 3.

Let $A,B$ be homotopy data with quasi-isomorphisms $Y,Z$ and suppose that
$Z\circ Y\circ Z=Z.$ Then there always exists a homotopy operator
$h_{B}^{\prime}$ such that $Z\circ h^{\prime}_{B}=0.$ Conversely, let the
homotopy $h_{B}$ satisfy Equation (45), then $Z=Z\circ Y\circ Z$.

###### Proof.

We show that such $h_{B}^{\prime}$ always exists by defining (see [Cra04,
Remark 2.3 (i)])

$\displaystyle h_{B}^{\prime}\coloneqq(\partial_{B}h_{B})\circ h_{B}\,.$ (52)

Then $h_{B}^{\prime}$ satisfies $\mathds{1}_{B}-Y\circ
Z=\partial_{B}h_{B}^{\prime}$ because of the Leibniz rule for $\partial_{B}$
and $Y\circ Z$ is idempotent because of $Z=Z\circ Y\circ Z$.

It follows that

$\displaystyle Z\circ h_{B}^{\prime}=Z\circ(\mathds{1}_{B}-Y\circ Z)\circ
h_{B}=0\,.$ (53)

Consider now the case where the homotopy $h_{B}$ satisfies Equation (45). By
acting with $\partial_{A}$ on Equation (45), we obtain

$\displaystyle 0=\partial_{A}\left(Z\circ
h_{B}\right)=Z\circ\partial_{B}h_{B}=Z-Z\circ Y\circ Z\ .$ (54)

∎

######  Remark 2.

Note that an analogous result holds for the right side condition: if $Y=Y\circ
Z\circ Y$, the given homotopy $h_{B}$ can always be modified in order to
obtain the condition $h_{B}^{\prime}\circ Y=0$. In particular, one can always
define the homotopy $h_{B}^{\prime}=h_{B}\circ\partial_{B}h_{B}$ which
satisfies $h_{B}^{\prime}\circ Y=0$ and defines a set of homotopy data
together with the quasi-isomorphisms $Y,Z$ between the cochain complexes
$(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$.

######  Proposition 4.

Let $(A,\mathrm{d}_{A}),$ $(B,\mathrm{d}_{B})$ be quasi-isomorphic cochain
complexes with quasi-isomorphisms $Y\in\mathsf{Hom}(A,B),$
$Z\in\mathsf{Hom}(B,A)$ and homotopies $h_{A}\in\mathsf{Hom}_{-1}(A,A),$
$h_{B}\in\mathsf{Hom}_{-1}(B,B)$. Suppose that the right-side condition (45)
holds. Then there always exists a new homotopy data still satisfying (45) such
that the obstruction (51) is removed.

###### Proof.

The homotopy $h_{A}$ can be modified preserving the side condition (45) and
removing the obstruction (51), i.e.,

$\displaystyle h_{A}\mapsto h_{A}^{\prime}=h_{A}-Z\circ Y\circ
h_{A}=\partial_{A}h_{A}\circ h_{A}\ .$ (55)

That gives

$\displaystyle Z\circ h_{B}-h_{A}^{\prime}\circ Z=$ $\displaystyle-
h_{A}^{\prime}\circ Z=-h_{A}\circ Z+Z\circ Y\circ h_{A}\circ Z$ (56)
$\displaystyle=$ $\displaystyle-\partial_{A}h_{A}\circ h_{A}\circ Z$ (57)
$\displaystyle=$ $\displaystyle-h_{A}\circ\partial_{A}h_{A}\circ
Z-\partial_{BA}\left(h_{A}\circ h_{A}\circ Z\right)$ (58) $\displaystyle=$
$\displaystyle-\partial_{BA}\left(h_{A}\circ h_{A}\circ Z\right)\,,$ (59)

where we used $\left(\partial_{A}h_{A}\right)\circ
Z=\partial_{BA}\left(h_{A}\circ Z\right)=0$, as follows from Equation (50).
Note that the homotopy $h^{\prime}_{A}$ satisfies Equation (17) because of
Proposition 3. ∎

######  Remarkl 3.

Note that our case of interest shows some peculiarities that do not arise in
[Mar01]. In particular, by modifying the homotopy $h_{B}$, the obstruction is
removed, but the condition (45) would be lost. This can be seen by considering
the redefinition

$\displaystyle h_{B}\mapsto h_{B}^{\prime}=h_{B}-Y\circ Z\circ h_{B}+Y\circ
h_{A}\circ Z=h_{B}+Y\circ h_{A}\circ Z\,,$ (60)

which yields

$\displaystyle Z\circ h_{B}^{\prime}=Z\circ Y\circ h_{A}\circ Z=h_{A}\circ
Z-\partial_{BA}\left(h_{A}\circ h_{A}\circ Z\right)\neq 0\,.$ (61)

Note that this redefinition preserves the right-side condition if all the side
conditions of Definition 3 hold, but, in general, it is impossible to modify
the homotopy $h_{B}$ while preserving the right-side condition (45) and
removing the obstruction (51) at the same time. This is due to the fact that
even if the side condition (45) is achieved, the obstruction (51) will be
completely determined by the (eventual) cohomology class of $h_{A}\circ Z,$
which is non-trivial.

Note that a crucial condition in order to redefine the homotopy $h_{B}$ is
given by $Z=Z\circ Y\circ Z$. This condition may be characterised as follows:

######  Lemma 5.

Let $A,B$ be homotopy data with quasi-isomorphisms $Y,Z$. Then

$\displaystyle Z-Z\circ Y\circ
Z=0\Longrightarrow\mathrm{Im}(Z)\cap\ker(Y)=\\{0\\}\ ,$ (62)

and, analogously, $Y-Y\circ Z\circ
Y=0\Longrightarrow\mathrm{Im}(Y)\cap\ker(Z)=\\{0\\}.$ Moreover, if we assume
$Y\circ Z=\left(Y\circ Z\right)^{2}$ and $Z\circ Y=\left(Z\circ Y\right)^{2}$
the converse holds as well.

###### Proof.

$(\Rightarrow)$ : By assumption, we have that $Z\circ Y=\mathds{1}_{A}$ on
$\mathrm{Im}(Z)$, and let $a\in\mathrm{Im}(Z)\cap\ker(Y)$. We thus have

$\displaystyle a=\mathds{1}_{A}a=\left(Z\circ Y\right)(a)=0\ \implies\ a=0\ .$
(63)

The proof for the second part follows the same steps as the one presented
above.

$(\Leftarrow)$ : Since $Y\circ Z$ is a projector, it follows that
$\left(Z-Z\circ Y\circ Z\right)\left(b\right)\in\ker(Y),$ for all $b\in B.$ On
the other hand, $\left(Z-Z\circ Y\circ
Z\right)\left(b\right)=\left(Z\circ\left(1-Y\circ
Z\right)\right)\left(b\right)\in\mathrm{Im}(Z),$ for all $b\in B$. Since
$\mathrm{Im}(Z)\cap\ker(Y)=\\{0\\}$, the statement holds. ∎

Lastly, we observe that the module structure and the assumption $Z=Z\circ
Y\circ Z$ simplify the homotopy transfer procedure as follows:

######  Proposition 6.

Let $(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$ be homotopy data with module
structure as in Definition 2, endowed with the homotopy transfer
$A_{\infty}$-algebra $(B,m_{n}^{\text{ht}})$ as in Subsection 1.3. Suppose
that $Z=Z\circ Y\circ Z.$ Then $\partial_{B}m_{n}^{\text{ht}}=0,$ for all
$n\geq 1.$

###### Proof.

The proof will be carried on by induction. Note that
$\partial_{B}m_{2}^{\text{ht}}=0,$ which follows from the Leibniz rule for the
associative dga $(A,\mathrm{d}_{A},\wedge).$ Furthermore, the higher homotopy
transferred products are given by the recursive formula

$\displaystyle
m_{n+1}^{\text{ht}}(\mathds{1}_{B}^{\otimes(n+1)})\coloneqq\sum_{s=0}^{n-1}\pm
m_{n}^{\text{ht}}\bigl{(}\mathds{1}_{B}^{\otimes
s}\otimes(h_{A}\circ\wedge\circ Z^{\otimes
2})\otimes\mathds{1}_{B}^{\otimes(n-s-1)}\bigr{)}\,,$ (64)

for $n\geq 2.$ Assume, by induction, that $\partial_{B}m_{n}^{\text{ht}}=0.$
Thus, we have

$\displaystyle\partial_{B}m_{n+1}^{\text{ht}}(\mathds{1}^{\otimes(n+1)})=$
$\displaystyle\sum_{s=0}^{n-1}\pm(\partial_{B}m_{n}^{\text{ht}})\bigl{(}\mathds{1}_{B}^{\otimes
s}\otimes(h_{A}\circ\wedge\circ Z^{\otimes
2})\otimes\mathds{1}_{B}^{\otimes(n-s-1)}\bigr{)}$ (65)
$\displaystyle\pm\sum_{s=0}^{n-1}\pm
m_{n}^{\text{ht}}\bigl{(}\mathds{1}_{B}^{\otimes
s}\otimes\partial_{A}(h_{A}\circ\wedge\circ Z^{\otimes
2})\otimes\mathds{1}_{B}^{\otimes(n-s-1)}\bigr{)}\,,$ (66)

where the first term on the RHS of Equation (66) vanishes because of the
induction hypothesis. The second term on the RHS of Equation (66) vanishes as
well, as

$\displaystyle\partial_{A}(h_{A}\circ\wedge\circ Z^{\otimes
2})=\partial_{A}(h_{A}\circ
Z\circ\rhd\circ(Z\otimes\mathds{1}_{B}))=(\mathds{1}_{A}-Z\circ Y)\circ
Z\circ\rhd\circ(Z\otimes\mathds{1}_{B})=0\,,$ (67)

since $Z\circ Y\circ Z=Z$. Note that the signs in Equation (64) do not play
any role in the proof, since every summand on the RHS of Equation (66)
vanishes separately. ∎

In the following section, we will discuss a novel type of homotopy transfer
based on the module structure of Definition 2 involving only the homotopy
$h_{B}$ on $B.$

###  2.3. Weakening the Side Conditions

In this Subsection, we discuss a weaker version of the side conditions, which
we refer to as _weak side conditions_. They will be used in the next section
to construct $A_{\infty}$-algebras which can be introduced in a slightly
broader regime than those where the SC hold.

######  Definition 4 (Weak Side Conditions).

Let $A,B$ be quasi-isomorphic cochain complexes by quasi-isomorphisms $Y,Z$
together with the homotopy $h_{B}.$ This homotopy $h_{B}$ is said to satisfy
the _Weak Side Conditions (WSC)_ if the following hold:

$\displaystyle Y\circ Z\circ h_{B}=0\,,\qquad h_{B}\circ Y\circ Z=0\ .$ (68)

In the previous paragraphs, we have characterised the side conditions in terms
of properties of the maps $Y$ and $Z$ in Proposition 3; an analogous result
holds here:

######  Proposition 7.

Let $A,B$ be homotopy data with quasi-isomorphisms $Y,Z$ and suppose that
$\left(Z\circ Y\right)^{2}=Z\circ Y$, i.e., $Z\circ Y$ is a projector. Then
there always exists a homotopy operator $h_{B}^{\prime}$ such that

$\displaystyle Y\circ Z\circ h^{\prime}_{B}=0\,,\qquad h^{\prime}_{B}\circ
Y\circ Z=0\ .$ (69)

Conversely, let the homotopy $h_{B}$ satisfy the conditions (68), then $Z\circ
Y=\left(Z\circ Y\right)^{2}$.

###### Proof.

The construction of $h_{B}^{\prime}$ follows the very same prescription given
in [Cra04, Remark 2.3 (i)], for the side conditions:

$\displaystyle h_{B}^{\prime}\coloneqq(\partial_{B}h_{B})\circ
h_{B}\circ(\partial_{B}h_{B})\,.$ (70)

Then $h_{B}^{\prime}$ satisfies $\mathds{1}_{B}-Y\circ
Z=\partial_{B}h_{B}^{\prime}$ because of the Leibniz rule for $\partial_{B}$
and $Y\circ Z$ is idempotent by assumption.

It follows that

$\displaystyle Y\circ Z\circ h_{B}^{\prime}$ $\displaystyle=Y\circ
Z\circ(\mathds{1}_{B}-Y\circ Z)\circ h_{B}(\mathds{1}_{B}-Y\circ Z)=0\,,$ (71)
$\displaystyle h^{\prime}_{B}\circ Y\circ Z$
$\displaystyle=(\mathds{1}_{B}-Y\circ Z)\circ h_{B}(\mathds{1}_{B}-Y\circ
Z)\circ Y\circ Z=0\,.$ (72)

Consider now the case where the homotopy $h_{B}$ satisfies the conditions
(68). By acting with $\partial_{B}$ on either of the conditions (68), we
obtain

$\displaystyle 0=\partial_{B}\left(Y\circ Z\circ h_{B}\right)=Y\circ
Z\circ\partial_{B}h_{B}=Y\circ Z-Y\circ Z\circ Y\circ Z\ ,$ (73)

thus $Y\circ Z$ is idempotent. ∎

######  Remark 4.

Note that the idempotency of $Y\circ Z$ allows to define a homotopy operator
$h^{\prime}_{B}$ that satisfies $(h_{B}^{\prime})^{2}=0$ as well. In
particular, by following the prescription given in [Cra04], we can introduce

$\displaystyle h^{\prime}_{B}\coloneqq h_{B}\circ\mathrm{d}_{B}\circ h_{B}\ .$
(74)

If $h^{\prime}_{B}$ satisfies the conditions (68), we have

$\displaystyle(h_{B}^{\prime})^{2}$
$\displaystyle=h_{B}\circ\mathrm{d}_{B}\circ h_{B}^{2}\circ\mathrm{d}_{B}\circ
h_{B}$ (75) $\displaystyle=h_{B}^{2}\circ\mathrm{d}_{B}\circ
h_{B}-h_{B}^{2}\circ\mathrm{d}_{B}\circ h_{B}\circ\mathrm{d}_{B}\circ
h_{B}-h_{B}\circ Y\circ Z\circ h_{B}\circ\mathrm{d}_{B}\circ h_{B}$ (76)
$\displaystyle=h_{B}^{2}\circ\mathrm{d}_{B}\circ
h_{B}-h_{B}^{2}\circ\mathrm{d}_{B}\circ h_{B}+h_{B}^{3}\circ Y\circ Z\circ
d_{B}\circ h_{B}+h_{B}^{3}\circ\mathrm{d}_{B}^{2}\circ h_{B}=0\ ,$ (77)

where we used Equations (18) and (68). We will discuss in more depth the weak
side conditions in [CM].

In the next section, the condition $(h_{B}^{\prime})^{2}=0$ will not be needed
in order to construct an $A_{\infty}$-algebra out of the homotopy data with
bimodule structure, thus we will not assume it.

##  3\. $A_{\infty}$-Algebras from Module Structures

We present in this Section a new homotopy transfer construction that only
relies on the homotopy and algebraic data of Definition 2. In particular, we
show how the homotopy data with module structure induces a product which is
associative up to homotopy. This procedure presents some analogies with the
homotopy transfer of an associative algebra. We shall discuss later on how
these constructions are related.

###  3.1. Homotopy Induced Products from Module Structures

Starting from the homotopy data with bimodule structure, we can use the
$\mathrm{Im}(Z)$-bimodule actions and the quasi-isomorphism $Z\colon A\to B$
to define a product on $B$ inspired by the construction given in [EKS14,
CG20].

######  Definition 5 (Homotopy Module-Induced Product).

Let $(A,\mathrm{d}_{A})$ and $(B,\mathrm{d}_{B})$ be quasi-isomorphic chain
complexes endowed with the homotopy data with bimodule structure of Definition
2. Then we define the _Homotopy Module-Induced Product (HMIP)_

$\displaystyle m_{2}\colon B^{\otimes 2}\longrightarrow B$

by

$\displaystyle m_{2}(b\otimes b^{\prime})\coloneqq k_{1}Z(b)\rhd
b^{\prime}+k_{2}\,b\lhd Z(b^{\prime})\ ,$ (78)

for all $b,b^{\prime}\in B$ and $k_{1},k_{2}\in\Bbbk$. We call this procedure
_Homotopy Module-Induction (HMI)_.

The first thing we want to investigate is if there is any relation between the
product (78) and the product that one could transfer on $B$ via the usual
homotopy transfer, i.e.,

$m_{2}^{\text{ht}}\left(b\otimes b^{\prime}\right)\coloneqq Y\left(Z(b)\wedge
Z(b^{\prime})\right)\,,$ (79)

for all $b,b^{\prime}\in B$.

Let us consider the graded vector space
$\mathsf{Hom}\left(B^{\otimes\bullet},B\right)$ and equip it with the
differential $\partial_{B}$ defined as in Equation (5).

In order to discuss the equivalence up to homotopy of the Homotopy Module-
Induced Product $m_{2}$ and the standard homotopy transfer
$m_{2}^{\text{ht}},$ we define the map
$\tilde{m}_{2}\in\mathsf{Hom}\left(B^{\otimes 2},B\right)$ as

$\displaystyle\tilde{m}_{2}\left(b\otimes b^{\prime}\right)\coloneqq
k_{1}\left(-1\right)^{|b|}Z\left(b\right)\rhd
h_{B}(b^{\prime})+k_{2}\,h_{B}(b)\lhd Z(b^{\prime})\,,$ (80)

for all $b,b^{\prime}\in B.$ The map $\tilde{m}_{2}$ is the key to
understanding how the Homotopy Module-Induced Product $m_{2}$ and the standard
homotopy transfer product $m_{2}^{\text{ht}}$ are equivalent up to homotopy.

######  Lemma 8.

In the assumptions of Definition 2, the Homotopy Module-Induced Product
satisfies

$\displaystyle\partial_{B}\tilde{m}_{2}=m_{2}-(k_{1}+k_{2})\,m_{2}^{\text{ht}}\,,$
(81)

i.e., if $k_{1}+k_{2}\neq 0,$ the Homotopy Module-Induced Product $m_{2}$ and
the (rescaled) homotopy transfer $m_{2}^{\text{ht}}$ are equivalent up to
homotopy.

###### Proof.

Let us consider the map $\tilde{m}_{2}$ given by Equation (80) and compute its
differential by applying $\partial_{B}$. We obtain

$\displaystyle\partial_{B}\tilde{m}_{2}\left(b\otimes
b^{\prime}\right)=m_{2}\left(b\otimes
b^{\prime}\right)-k_{1}Z\left(b\right)\rhd(Y\circ
Z)(b^{\prime})-k_{2}\left(Y\circ Z\right)\left(b\right)\lhd
Z\left(b^{\prime}\right)\,,$ (82)

for all $b,b^{\prime}\in B.$ Then Equation (81) follows straightforwardly from
Equation (82) by using that $Y$ is an $\mathrm{Im}(Z)$-module homomorphism. ∎

######  Remark 5.

If $k_{1}+k_{2}=0$ it simply follows that the product $m_{2}$ is exact. This
in particular means that if we restrict on the cohomology
$\mathsf{H}\left(B\right)$ the product is identically zero; in other words,
the restriction to cohomology trivialises the ring structure.

###  3.2. Homotopy Module-Induction and (Higher) Associativity

The assumptions of Definition 2 represent the cornerstone of the Homotopy
Module-Induction and we shall see that they are crucial to the associativity
up to homotopy of the induced product $m_{2}$. To this, recall that the
_Associator_ for any product $m_{2}\in\mathsf{Hom}(B^{\otimes 2},B)$ is a
trilinear map $\mathsf{Ass}\in\mathsf{Hom}(B^{\otimes 3},B)$ given by

$\displaystyle\mathsf{Ass}(b\otimes b^{\prime}\otimes
b^{\prime\prime})\coloneqq m_{2}\left(m_{2}\left(b\otimes
b^{\prime}\right)\otimes b^{\prime\prime}\right)-m_{2}\left(b\otimes
m_{2}\left(b^{\prime}\otimes b^{\prime\prime}\right)\right)\,,$ (83)

for all $b,b^{\prime},b^{\prime\prime}\in B.$

######  Proposition 9.

Let $(A,\mathrm{d}_{A}),(B,\mathrm{d}_{B})$ be homotopy data with module
structure as in Definition 2. Then the Homotopy Module-Induced Product
$m_{2}\in\mathsf{Hom}(B^{\otimes 2},B)$ is associative up to homotopy, i.e.
there exists a map $m_{3}\in\mathsf{Hom}(B^{\otimes 3},B)$ such that

$\displaystyle\mathsf{Ass}=\partial_{B}\,m_{3}\,,$ (84)

where $\partial_{B}$ is defined by Equation (5).

###### Proof.

The associator for the Homotopy Module-Induced Product $m_{2}$ in the
assumptions of Definition 2 is given by

$\displaystyle\mathsf{Ass}(b\otimes b^{\prime}\otimes
b^{\prime\prime})=k_{1}k_{2}\left(Z(b)\wedge Z(b^{\prime})\right)\rhd
b^{\prime\prime}-k_{1}k_{2}\,b\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)\,,$ (85)

for any $b,b^{\prime},b^{\prime\prime}\in B$.

In order to construct a map $m_{3}\in\mathsf{Hom}(B^{\otimes 3},B)$ that
satisfies Equation (84), we use the following prescription. We obtain the
3-product $m_{3}$ by inserting the homotopy $h_{B}$ on the free external
branches of $\mathsf{Ass},$ i.e. the external branches that do not have any
input in $Z,$ such as the entries $b^{\prime\prime}$ in the first term and $b$
in the second term of Equation (85), respectively. We will discuss this
prescription in its full generality later on. Hence we have

$\displaystyle m_{3}\left(b\otimes b^{\prime}\otimes b^{\prime\prime}\right)=$
$\displaystyle k_{1}k_{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}(b^{\prime\prime})$ (86) $\displaystyle-
k_{1}k_{2}h_{B}(b)\lhd\left(Z(b^{\prime})\wedge Z(b^{\prime\prime})\right)\,,$
(87)

for all $b,b^{\prime},b^{\prime\prime}\in B$. Note that $m_{3}$ given by
Equation (86) is not unique, since one can always add a closed term with
respect to $\partial_{B}.$ Its differential becomes

$\displaystyle\partial_{B}m_{3}(b\otimes b^{\prime}\otimes
b^{\prime\prime})=k_{1}k_{2}\left(Z(b)\wedge Z(b^{\prime})\right)\rhd
b^{\prime\prime}-k_{1}k_{2}b\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)\,,$ (88)

for all $b,b^{\prime},b^{\prime\prime}\in B.$ Therefore, by comparing
Equations (88) and (85) we see that Equation (84) is verified. ∎

######  Remark 6.

We mentioned above that the product $m_{3}$ is constructed by suitably
inserting the homotopy $h_{B}$ on a free branch of each term of
$\mathsf{Ass}.$ Let us discuss what happens when a different choice, i.e. a
different insertion, is made. For instance, consider

$\displaystyle m_{3}^{(1,cl)}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\right)\coloneqq\left((Z\circ h_{B})(b)\wedge
Z(b^{\prime})-(-1)^{|b|}Z(b)\wedge\left(Z\circ
h_{B}\right)(b^{\prime})\right)\rhd b^{\prime\prime}\,.$ (89)

This term is $\partial_{B}$-closed, because of the compatibility with the maps
$Y$ and $Z$ with the $\mathrm{Im}(Z)$-module action. In this sense, we follow
the prescription used in the proof and we will not consider contributions to
$m_{3}$ given by closed terms.

######  Remark 7.

Let us briefly comment on the relevance of the Property 3 in Definition 2. Let
us drop assumption 3, then the associator of $m_{2}$ becomes

$\displaystyle\mathsf{Ass}(b\otimes b^{\prime}\otimes b^{\prime\prime})=$
$\displaystyle k_{1}^{2}Z\left(Z\left(b\right)\rhd b^{\prime}\right)\rhd
b^{\prime\prime}+k_{1}k_{2}Z\left(b\lhd Z\left(b^{\prime}\right)\right)\rhd
b^{\prime\prime}$ (90) $\displaystyle+k_{2}^{2}b\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)-k_{1}^{2}\left(Z(b)\wedge Z(b^{\prime})\right)\rhd
b^{\prime\prime}$ (91) $\displaystyle-k_{1}k_{2}b\lhd
Z\left(Z\left(b^{\prime}\right)\rhd b^{\prime\prime}\right)-k_{2}^{2}b\lhd
Z\left(b^{\prime}\lhd Z\left(b^{\prime\prime}\right)\right)\ ,$ (92)

for all $b,b^{\prime},b^{\prime\prime}\in B.$

By following the same argument as in the proof of Proposition 9, we define a
map $m_{3}$ by

$\displaystyle m_{3}(b\otimes b^{\prime}\otimes b^{\prime\prime})=$
$\displaystyle(-1)^{|b|+|b^{\prime}|}k_{1}^{2}Z\left(Z\left(b\right)\rhd
b^{\prime}\right)\rhd h_{B}(b^{\prime\prime})$ (93)
$\displaystyle+(-1)^{|b|+|b^{\prime}|}k_{1}k_{2}Z\left(b\lhd
Z\left(b^{\prime}\right)\right)\rhd h_{B}(b^{\prime\prime})$ (94)
$\displaystyle+k_{2}^{2}h_{B}(b)\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)-(-1)^{|b|+|b^{\prime}|}k_{1}^{2}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}(b^{\prime\prime})$ (95) $\displaystyle-
k_{1}k_{2}h_{B}(b)\lhd Z\left(Z\left(b^{\prime}\right)\rhd
b^{\prime\prime}\right)-k_{2}^{2}h_{B}(b)\lhd Z\left(b^{\prime}\lhd
Z\left(b^{\prime\prime}\right)\right)\ ,$ (96)

for all $b,b^{\prime},b^{\prime\prime}\in B.$

By applying the differential $\partial_{B}$ to Equation (94), we obtain

$\displaystyle\partial_{B}m_{3}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\right)=$ $\displaystyle m_{2}\left(m_{2}\left(b\otimes
b^{\prime}\right)\otimes b^{\prime\prime}\right)-m_{2}\left(b\otimes
m_{2}\left(b^{\prime}\otimes b^{\prime\prime}\right)\right)$ (97)
$\displaystyle-k_{1}^{2}Z\left(Z\left(b\right)\rhd
b^{\prime}\right)\rhd(Y\circ Z)(b^{\prime\prime})-k_{1}k_{2}Z\left(b\lhd
Z\left(b^{\prime}\right)\right)\rhd(Y\circ Z)(b^{\prime\prime})$ (98)
$\displaystyle-k_{2}^{2}(Y\circ Z)(b)\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)+k_{1}^{2}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd(Y\circ Z)(b^{\prime\prime})$ (99)
$\displaystyle+k_{1}k_{2}(Y\circ Z)(b)\lhd Z\left(Z\left(b^{\prime}\right)\rhd
b^{\prime\prime}\right)+k_{2}^{2}(Y\circ Z)(b)\lhd Z\left(b^{\prime}\lhd
Z\left(b^{\prime\prime}\right)\right)\ ,$ (100)

that is, $\partial_{B}m_{3}$ is given by the associator of $m_{2}$ plus six
extra terms. It can be verified that it is not possible to modify the
definition of the 3-product with other terms constructed with different
insertions of the homotopy $h_{B}$ (or the homotopy $h_{A}$) that compensate
for these extra terms. This means that, in general, an induced product defined
as in Equation (78) will not be associative up to homotopy by constructing the
map $m_{3}$ with the prescription discussed in the proof of Proposition 9.

We can now proceed with the construction of the first higher associator. In
particular, we study the _Pentagonator_ , denoted by $\mathsf{Pen},$ of the
first two products coming from the Homotopy Module-Induction, which is defined
by

$\displaystyle\mathsf{Pen}(b\otimes b^{\prime}\otimes b^{\prime\prime}\otimes
b^{\prime\prime\prime})\coloneqq$ $\displaystyle
m_{2}\left(m_{3}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\right)\otimes
b^{\prime\prime\prime}\right)+\left(-1\right)^{|b|}m_{2}\left(b\otimes
m_{3}\left(b^{\prime}\otimes b^{\prime\prime}\otimes
b^{\prime\prime\prime}\right)\right)$ (101) $\displaystyle-
m_{3}\left(m_{2}\left(b\otimes b^{\prime}\right)\otimes
b^{\prime\prime}\otimes b^{\prime\prime\prime}\right)+m_{3}\left(b\otimes
m_{2}\left(b^{\prime}\otimes b^{\prime\prime}\right)\otimes
b^{\prime\prime\prime}\right)$ $\displaystyle-m_{3}\left(b\otimes
b^{\prime}\otimes m_{2}\left(b^{\prime\prime}\otimes
b^{\prime\prime\prime}\right)\right)\,,$ (102)

for all $b,b^{\prime},b^{\prime\prime},b^{\prime\prime\prime}\in B.$ By using
Equations (78), (86) and Definition 2, we obtain

$\displaystyle\mathsf{Pen}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\otimes b^{\prime\prime\prime}\right)=$ $\displaystyle
k_{1}k_{2}^{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}(b^{\prime\prime})\lhd
Z(b^{\prime\prime\prime})$ (103) $\displaystyle-
k_{1}k_{2}^{2}\,h_{B}(b)\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\wedge Z(b^{\prime\prime\prime})\right)$ (104)
$\displaystyle+k_{1}^{2}k_{2}\left(-1\right)^{|b|+|b^{\prime}|+|b^{\prime\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\wedge Z(b^{\prime\prime})\right)\rhd
h_{B}(b^{\prime\prime\prime})$ (105) $\displaystyle-
k_{1}^{2}k_{2}\left(-1\right)^{|b|}Z(b)\rhd
h_{B}(b^{\prime})\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (106)
$\displaystyle+k_{1}^{2}k_{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\wedge\left(Z\circ h_{B}\right)(b^{\prime\prime})\right)\rhd
b^{\prime\prime\prime}$ (107) $\displaystyle-k_{1}^{2}k_{2}\left((Z\circ
h_{B})(b)\wedge Z(b^{\prime})\wedge Z(b^{\prime\prime})\right)\rhd
b^{\prime\prime\prime}$ (108)
$\displaystyle+k_{1}k_{2}^{2}\left(-1\right)^{|b|+|b^{\prime}|+|b^{\prime\prime}|}b\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\wedge(Z\circ h_{B})(b^{\prime\prime\prime})\right)$ (109)
$\displaystyle-k_{1}k_{2}^{2}\left(-1\right)^{|b|}b\lhd\left((Z\circ
h_{B})(b^{\prime})\wedge Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$
$\displaystyle+k_{1}^{2}k_{2}\,h_{B}\left(Z(b)\rhd
b^{\prime}\right)\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (110)
$\displaystyle+k_{1}k_{2}^{2}\,h_{B}\left(b\lhd
Z(b^{\prime})\right)\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (111) $\displaystyle-
k_{1}^{2}k_{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}\left(Z(b^{\prime\prime})\rhd
b^{\prime\prime\prime}\right)$ (112) $\displaystyle-
k_{1}k_{2}^{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}\left(b^{\prime\prime}\lhd
Z(b^{\prime\prime\prime})\right)\ ,$ (113)

for all $b,b^{\prime},b^{\prime\prime},b^{\prime\prime\prime}\in B.$ The
product $m_{4}\in\mathsf{Hom}\left(B^{\otimes 4},B\right)$ can be constructed,
out of the expression of the pentagonator, by following the same prescription
discussed in the proof of Proposition 9: whenever there is a “naked” external
branch, i.e., an external branch without insertions of $Z$ or $h_{B}$, apply
the homotopy operator $h_{B}$444Of course, one should also fix the sign
depending on which inputs and how many homotopy operators have been crossed
when applying $h_{B}$.. If a term presents no naked branches, it will be
neglected. In particular, out of the twelve terms of Equation (103), we can
construct the product $m_{4}$ as

$\displaystyle m_{4}\left(b\otimes b^{\prime}\otimes b^{\prime\prime}\otimes
b^{\prime\prime\prime}\right)\coloneqq$
$\displaystyle-(-1)^{|b^{\prime\prime}|}k_{1}^{2}k_{2}\left(Z(b)\wedge
Z(b^{\prime})\wedge\left(Z\circ h_{B}\right)(b^{\prime\prime})\right)\rhd
h_{B}(b^{\prime\prime\prime})$ (114)
$\displaystyle+(-1)^{|b|+|b^{\prime}|+|b^{\prime\prime}|}k_{1}^{2}k_{2}\left((Z\circ
h_{B})(b)\wedge Z(b^{\prime})\wedge Z(b^{\prime\prime})\right)\rhd
h_{B}(b^{\prime\prime\prime})$ (115)
$\displaystyle+k_{1}k_{2}^{2}\left(-1\right)^{|b|+|b^{\prime}|+|b^{\prime\prime}|}h_{B}(b)\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\wedge(Z\circ h_{B})(b^{\prime\prime\prime})\right)$ (116)
$\displaystyle-k_{1}k_{2}^{2}\left(-1\right)^{|b|}h_{B}(b)\lhd\left((Z\circ
h_{B})(b^{\prime})\wedge Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ $\displaystyle-
k_{1}^{2}k_{2}\,h_{B}\left(Z(b)\rhd
h_{B}(b^{\prime})\right)\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (117) $\displaystyle-
k_{1}k_{2}^{2}\,h_{B}\left(h_{B}(b)\lhd
Z(b^{\prime})\right)\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (118)
$\displaystyle+k_{1}^{2}k_{2}\left(-1\right)^{|b^{\prime\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}\left(Z(b^{\prime\prime})\rhd
h_{B}(b^{\prime\prime\prime})\right)$ (119)
$\displaystyle+k_{1}k_{2}^{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}\left(h_{B}(b^{\prime\prime})\lhd
Z(b^{\prime\prime\prime})\right)\,,$ (120)

for all $b,b^{\prime},b^{\prime\prime},b^{\prime\prime\prime}\in B,$ where
only eight terms are involved in the construction.

In order to have the map $m_{4}$ satisfying the higher associativity relation
for $\mathsf{Pen},$ we can either constrain the homotopy data or fix a
relation between the coefficients $k_{1}$ and $k_{2}$ appearing in the product
$m_{2},$ Equation (78). This can be stated in the following

######  Lemma 10.

The product $m_{4}$ satisfies the equation

$\displaystyle\mathsf{Pen}=\partial_{B}m_{4}$ (121)

if either of the following conditions hold

  1. (1)

$k_{1}\vee k_{2}=0$; in this case, the map $m_{2}$ becomes associative, the
maps $m_{3}$ and $m_{4}$ are zero, i.e. Equation (121) is trivially satisfied;

  2. (2)

$k_{1}=-k_{2}$; in this case, the non-associative map $m_{2}$ becomes
$\partial_{B}$-exact, as it is shown in Equation (81);

  3. (3)

the homotopy $h_{B}$ satisfies the weak side conditions, see Definition 4.

###### Proof.

Equation (114) satisfies

$\partial_{B}m_{4}=\mathsf{Pen}+\mathfrak{R}_{Y}\ ,$ (122)

where $\mathfrak{R}_{Y}$ is a collection of extra terms depending explicitly
on the map $Y$ given by

$\displaystyle\mathfrak{R}_{Y}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\otimes b^{\prime\prime\prime}\right)=$ $\displaystyle-
k_{1}k_{2}\left(k_{1}+k_{2}\right)Y\left(Z\left(b\right)\wedge
Z\left(b^{\prime}\right)\wedge\left(Z\circ
h_{B}\right)\left(b^{\prime\prime}\right)\wedge
Z\left(b^{\prime\prime\prime}\right)\right)$ (123)
$\displaystyle+k_{1}k_{2}\left(k_{1}+k_{2}\right)Y\left(\left(Z\circ
h_{B}\right)\left(b\right)\wedge Z\left(b^{\prime}\right)\wedge
Z\left(b^{\prime\prime}\right)\wedge
Z\left(b^{\prime\prime\prime}\right)\right)$ (124) $\displaystyle-
k_{1}k_{2}\left(k_{1}+k_{2}\right)Y\left(Z\left(b\right)\wedge
Z\left(b^{\prime}\right)\wedge
Z\left(b^{\prime\prime}\right)\wedge\left(Z\circ
h_{B}\right)\left(b^{\prime\prime\prime}\right)\right)$ (125)
$\displaystyle+k_{1}k_{2}\left(k_{1}+k_{2}\right)Y\left(Z\left(b\right)\wedge\left(Z\circ
h_{B}\right)\left(b^{\prime}\right)\wedge Z\left(b^{\prime\prime}\right)\wedge
Z\left(b^{\prime\prime\prime}\right)\right)$ (126) $\displaystyle-
k_{1}k_{2}\left(k_{1}+k_{2}\right)\left(h_{B}\circ
Y\right)\left(Z\left(b\right)\wedge
Z\left(b^{\prime}\right)\right)\lhd\left(Z\left(b^{\prime\prime}\right)\wedge
Z\left(b^{\prime\prime\prime}\right)\right)$ (127)
$\displaystyle+k_{1}k_{2}\left(k_{1}+k_{2}\right)\left(Z\left(b\right)\wedge
Z\left(b^{\prime}\right)\right)\rhd\left(h_{B}\circ
Y\right)\left(Z\left(b^{\prime\prime}\right)\wedge
Z\left(b^{\prime\prime\prime}\right)\right)\,,$ (128)

for all $b,b^{\prime},b^{\prime\prime},b^{\prime\prime\prime}\in B.$ If any of
the conditions above hold, Equation (126) vanishes. ∎

######  Remark 8.

Note that the weak side conditions of Definition 4 hold true whenever the side
conditions on Definition 3 are satisfied.

In order to show that the Homotopy Module-Induction yields an
$A_{\infty}$-algebra, consider the $A_{\infty}$-relations

$\displaystyle\partial_{B}m_{n}=\mathsf{Ass}_{n}\coloneqq\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i+1}\,,$ (129)

for all $n\in\mathbb{N}_{+},$ where on the RHS we define

$\displaystyle m_{i}\circ
m_{k}\bigl{(}\mathds{1}_{B}^{\otimes(i+k-1)}\bigr{)}\coloneqq\sum_{j=0}^{i-1}(-1)^{j+i+k(i-j-1)}m_{i}\bigl{(}\mathds{1}^{\otimes
j}_{B}\otimes m_{k}(\mathds{1}_{B}^{\otimes
k})\otimes\mathds{1}_{B}^{\otimes(i-j-1)}\bigr{)}\,.$ (130)

Lemma 10 shows under which assumptions we may obtain an $A_{\infty}$-algebra
from the Homotopy Module-Induction;555We will neglect the situation described
in case 1. before exploring the construction under these assumptions, we show
that a simpler $A_{\infty}$-algebra can be obtained when the homotopy data
satisfies the side conditions (see Definition 3), as a special case of Lemma
10, see Remark 8.

###  3.3. $A_{\infty}$-Algebras from Homotopy Module-Induction with Side
Conditions

We shall show that the Homotopy Module-Induction yields an
$A_{\infty}$-algebra when the homotopy data with bimodule structure satisfies
the side conditions.

In particular, by enforcing the left and right side conditions for the
homotopy $h_{B}$, as in Definition 3, we immediately see that the product
$m_{4}$ defined by Equation (114) simplifies to

$\displaystyle m_{4}\left(b\otimes b^{\prime}\otimes b^{\prime\prime}\otimes
b^{\prime\prime\prime}\right)\coloneqq$ $\displaystyle-
k_{1}^{2}k_{2}h_{B}\left(Z(b)\rhd
h_{B}(b^{\prime})\right)\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (131) $\displaystyle-
k_{1}k_{2}^{2}h_{B}\left(h_{B}(b)\lhd
Z(b^{\prime})\right)\lhd\left(Z(b^{\prime\prime})\wedge
Z(b^{\prime\prime\prime})\right)$ (132)
$\displaystyle+k_{1}^{2}k_{2}\left(-1\right)^{|b^{\prime\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}\left(Z(b^{\prime\prime})\rhd
h_{B}(b^{\prime\prime\prime})\right)$ (133)
$\displaystyle+k_{1}k_{2}^{2}\left(-1\right)^{|b|+|b^{\prime}|}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd h_{B}\left(h_{B}(b^{\prime\prime})\lhd
Z(b^{\prime\prime\prime})\right)\ ,$ (134)

for all $b,b^{\prime},b^{\prime\prime},b^{\prime\prime\prime}\in B,$ and that
Equation (126) vanishes with no further assumptions on the coefficients
$k_{1},k_{2},$ i.e. Equation (121) holds and the pentagonator of $m_{3}$ and
$m_{2}$ vanishes up to homotopy.

Note that, as stated in Remark 8, $m_{4}$ given by Equation (131) satisfies
Lemma 10. We can thus show that the following $A_{\infty}$-algebra structure
can be induced on $B$:

######  Theorem 11 (Homotopy Module-Induction with SC).

Let $(A,\mathrm{d}_{A}),(B,\mathrm{d}_{B})$ be homotopy data with module
structure as in Definition 2 such that the homotopy $h_{B}$ satisfies the side
conditions of Definition 3. Then the Homotopy Module-Induction gives an
$A_{\infty}$-algebra $(B,m_{n})$ with $m_{1}=\mathrm{d}_{B},$ $m_{2}$ given by
Equation (78), and higher products

$\displaystyle m_{n}\coloneqq(m_{n-1}\circ m_{2})\circ h_{B}\ ,\qquad\text{for
}n\geq 3\ .$ (135)

######  Remark 9.

Note that the definition of $n$-product given by Equation (135) reflects the
prescription described in the previous paragraphs: to define the $n$-ary
product $m_{n}$, start from the expression of the $n$-agonator
$\mathsf{Ass}_{n}=\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ m_{n-i+1}$ and compose
(with proper signs) the homotopy operator $h_{B}$ on the resulting free
inputs. Because of the properties of the map $Z$ and the side conditions, we
have that the only contributions with a free input come from $m_{n-1}\circ
m_{2}$. Note that terms in $m_{2}\circ m_{n-1}$ could have a free input, but
they vanish because of the side conditions. In particular, the only terms
having a free input are given by the composition of $m_{n-1}$ on the input of
$m_{2}$ presenting $Z$. Since $Z$ is an $\mathrm{Im}(Z)$-module homomorphism,
we can always move it in the resulting expression until, eventually, it is
composed with a homotopy operator $h_{B}$, thus giving zero because of the
side conditions.

###### Proof.

We shall start by considering any $n$-agonator $\mathsf{Ass}_{n}.$ The
building blocks of the $n$-product $m_{n}$ are the terms of $\mathsf{Ass}_{n}$
presenting one free input (or one free external branch, in the language of
tree diagrams), as discussed in Remark 9. There will be $2^{n-1}$ of such
terms for $\mathsf{Ass}_{n},$ with $n\geq 3.$ Note that, from the second
equality in Equation (129), it is impossible to have terms in
$\mathsf{Ass}_{n}$ with more than one free input.

We use these $2^{n-1}$ terms to construct the $n$-product by inserting the
homotopy operator $h_{B}$ in the free input. Because of the right side
condition, the terms appearing in the product $m_{n}$ are $2^{n-2}.$ Note that
all the terms contributing to the $n$-product will present the $h_{B}$
operator on connected branches starting from an upper external one and
descending to the innermost one.

The calculation of $\partial_{B}m_{n},$ where $m_{n}$ is constructed with the
prescription discussed above, satisfies Equation (129). Note that by applying
$\partial_{B}$ to each term of $m_{n},$ we will obtain $n-2$ new terms simply
by replacing each one of the operators $h_{B}$ with $\mathds{1}_{B}$ (each
term might pick up a sign based on how many operators $h_{B}$ are in the
diagram, starting from the innermost one, until reaching the one substituted
by the identity). There will be no other contributions, since $\partial_{B}$
can only act on $h_{B}$ and the terms in which $Y\circ Z$ appear will vanish
because of the side conditions and Assumption 3 in Definition 2.

The constructive argument to prove this Theorem leads to an abstract proof
based on the properties of the $n$-agonators with side conditions. From now on
we assume that all the side conditions on $h_{B}$ hold. We start from

$\displaystyle\partial_{B}\mathsf{Ass}_{n}=0\,,$ (136)

which encodes the following relations

$\displaystyle\partial_{B}\mathsf{Ass}_{n}=$
$\displaystyle\partial_{B}\left(\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i+1}\right)$ (137) $\displaystyle=$
$\displaystyle\sum_{i=2}^{n-1}\left(\left(\sum_{j=2}^{i-1}(-1)^{i+j}m_{j}\circ
m_{i-j+1}\right)\circ m_{n-i+1}\right.$ (138)
$\displaystyle+m_{i}\circ\left(\sum_{j=2}^{n-i}(-1)^{j}m_{j}\circ
m_{n-i-j+2}\right)\Bigg{)}=0\ .$ (139)

We can recast it as

$\displaystyle\sum_{j=2}^{n-2}(-1)^{j}\left(m_{j}\circ m_{n-j}\right)\circ
m_{2}=$
$\displaystyle(-1)^{n}\sum_{i=2}^{n-2}\left[\left(\sum_{j=2}^{i-1}(-1)^{i+j}m_{j}\circ
m_{i-j+1}\right)\circ m_{n-i+1}\right.$ (140)
$\displaystyle+m_{i}\circ\left(\sum_{j=2}^{n-i}(-1)^{j}m_{j}\circ
m_{n-i-j+2}\right)\Bigg{]}\,.$ (141)

In order to prove the Theorem, we will proceed by induction (the first
inductive steps have been verified explicitly in the previous paragraphs). Let
us recall the definition of the higher products given by Equation (135), i.e.,

$\displaystyle m_{n}\coloneqq(-1)^{n-1}\mathsf{Ass}_{n}\circ
h_{B}=\left(m_{n-1}\circ m_{2}\right)\circ h_{B}\ ,$ (142)

for all $n\geq 3$, where, from the first to the second equality, we used
$Z\circ h_{B}=0$ and $h_{B}^{2}=0$. Thus, by acting with the differential
$\partial_{B}$, we have

$\displaystyle\partial_{B}m_{n}=\left(\partial_{B}m_{n-1}\circ
m_{2}\right)\circ h_{B}+(-1)^{n-1}m_{n-1}\circ
m_{2}\circ\left(\mathds{1}_{B}-Y\circ Z\right)\ .$ (143)

The term given by $m_{n-1}\circ m_{2}\circ Y\circ Z$ vanishes because of the
Property 3 of Definition 2, i.e., $Y,\,Z$ are $\mathrm{Im}(Z)$-bimodule
homomorphisms, this allows to obtain the composition $h_{B}\circ Y=0$ with the
first of the nested homotopy operator $h_{B}$ appearing in any $m_{n-1}$, thus
giving zero666This is true for all $n\neq 3$. The case of $\mathsf{Ass}_{3}$
has been verified separately. because of the side conditions. By inductive
hypothesis we have $\partial_{B}m_{i}=\sum_{j=2}^{i-1}m_{j}\circ m_{i-j+1}$,
for all $i\leq n-1$, so that the first term reads, by using Equation (141),

$\displaystyle\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ m_{n-i}\right)\circ
m_{2}\circ h_{B}=$
$\displaystyle(-1)^{n}\sum_{i=2}^{n-2}\left[\left(\sum_{j=2}^{i-1}(-1)^{i+j}m_{j}\circ
m_{i-j+1}\right)\circ m_{n-i+1}\right.$ (144)
$\displaystyle+m_{i}\circ\left(\sum_{j=2}^{n-i}(-1)^{j}m_{j}\circ
m_{n-i-j+2}\right)\Bigg{]}\circ h_{B}\ .$ (145)

The first term on the RHS of Equation (145) vanishes, since $m_{n-i+1}\neq
m_{2},$ as $i\leq n-2$, hence the composition with $h_{B}$ is zero because of
the side conditions.

The second term on the RHS of Equation (145) has only non-zero term given by
the one with $m_{2}$ composed on the right, i.e., the term with $j=n-i$ in the
sum, whereas all the other terms vanish because of the side conditions.
Equation (145), together with the second term of Equation (143), leads to

$\displaystyle\partial_{B}m_{n}=\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i+1}+(-1)^{n-1}m_{n-1}\circ m_{2}=\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i+1}=\mathsf{Ass}_{n}\,,$ (146)

thus completing the proof. ∎

###  3.4. General Construction of Homotopy Module-Induced
$A_{\infty}$-Algebras

In this subsection, we construct an $A_{\infty}$-algebra via HMI in different
cases than Theorem 11. In particular, we will prove an analogous theorem that
holds both when the weak side conditions are assumed and in the case where we
do not assume any condition on the maps involved, but we fix the coefficients
of the product $m_{2}$ to satisfy $k_{1}=-k_{2}$. These are the relevant cases
2 and 3 of Lemma 10.

Before proceeding with the construction, we describe some notations that will
be used in the following. Given a $n$-ary map
$\varphi\in\mathsf{Hom}\left(B^{\otimes n},B\right)$, we denote

$\displaystyle\varphi\circ h_{B^{\otimes
n}}=\varphi\circ(h_{B}\otimes\mathds{1}_{B}^{\otimes
n-1}+\mathds{1}_{B}\otimes h_{B}\otimes\mathds{1}_{B}^{\otimes
n-2}+\ldots+\mathds{1}_{B}^{\otimes n-1}\otimes h_{B})\equiv\varphi\circ
h_{B}$ (147)

as the $n$-ary map where we inserted once the homotopy operator $h_{B}$ on the
inputs of $\varphi$. We denote

$\displaystyle\varphi\circ_{fb}h_{B}$ (148)

as the $n$-ary map where $h_{B}$ has been composed in the free inputs (“free
branches” in the graphical representation of maps as planar trees) of
$\varphi$ only. For example, given the definitions in Equations (78) and (80)
one sees that

$\displaystyle\tilde{m}_{2}\coloneqq m_{2}\circ_{fb}h_{B}\ .$ (149)

Note that the maps we will use will always have at most one free input. Given
$m$ and $n$-ary maps $\varphi\in\mathsf{Hom}\left(B^{\otimes m},B\right)$ and
$\psi\in\mathsf{Hom}\left(B^{\otimes n},B\right)$, we denote with

$\displaystyle\varphi\circ_{Z}\psi\quad\text{and}\quad\varphi\circ_{h_{B}}\psi$
(150)

the $m+n-1$-ary maps where $\psi$ has only been composed in the inputs with
$Z$, respectively $h_{B}$, of $\varphi$. For example, consider
$\varphi(b\otimes b^{\prime})=\psi(b\otimes b^{\prime})\coloneqq b\lhd
Z(b^{\prime})$, for all $b,b^{\prime}\in B$, we have

$\displaystyle(\varphi\circ_{Z}\psi)(b\otimes b^{\prime}\otimes
b^{\prime\prime})=-b\lhd\left(Z\left(b^{\prime}\lhd
Z(b^{\prime\prime})\right)\right)\ ,$ (151)

for all $b,b^{\prime},b^{\prime\prime}\in B$, where the minus sign comes from
the prescription of Equation (130).

######  Theorem 12 (Homotopy Module-Induced $A_{\infty}$-Algebra).

Let $(A,\mathrm{d}_{A}),(B,\mathrm{d}_{B})$ be homotopy data with module
structure as in Definition 2 and assume that either of the following
conditions hold

  1. (1)

the homotopy $h_{B}$ satisfies the WSC of Definition 4;

  2. (2)

the coefficients appearing in Equation (78) are such that $k_{1}=-k_{2}.$

Then the Homotopy Module-Induction gives the $A_{\infty}$-algebras $(B,m_{n})$
with $m_{1}\coloneqq\mathrm{d}_{B},$ $m_{2}$ is defined by Equation (78) and
higher products777For $n=3$ only the second equality should be considered.

$\displaystyle m_{n}\coloneqq$ $\displaystyle(-1)^{n-1}\left(m_{2}\circ
m_{n-1}\right)\circ_{fb}h_{B}+\left(m_{n-1}\circ m_{2}\right)\circ_{fb}h_{B}$
(152) $\displaystyle=$
$\displaystyle\tilde{m}_{2}\circ_{Z}m_{n-1}+m_{n-1}\circ_{h_{B}}\tilde{m}_{2}\,,$
(153)

where $\tilde{m}_{2}$ is given by Equation (80).

###### Proof.

We follow a similar strategy to the proof of Theorem 11. We start with the
relation

$\displaystyle\partial_{B}\mathsf{Ass}_{n}=0\ ,$ (154)

which yields

$\displaystyle\partial_{B}\mathsf{Ass}_{n}=$
$\displaystyle\partial_{B}\left(\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i+1}\right)$ (155) $\displaystyle=$
$\displaystyle\sum_{i=2}^{n-1}\left(\left(\sum_{j=2}^{i-1}(-1)^{i+j}m_{j}\circ
m_{i-j+1}\right)\circ m_{n-i+1}\right.$ (156)
$\displaystyle+m_{i}\circ\left(\sum_{j=2}^{n-i}(-1)^{j}m_{j}\circ
m_{n-i-j+2}\right)\Bigg{)}=0\,.$ (157)

We recast Equation (155) as

$\displaystyle\sum_{j=2}^{n-2}$ $\displaystyle(-1)^{j}\left(m_{j}\circ
m_{n-j}\right)\circ m_{2}+(-1)^{n-1}m_{2}\circ\sum_{j=2}(-1)^{j}m_{j}\circ
m_{n-i-j+2}=$ (158)
$\displaystyle(-1)^{n}\sum_{i=2}^{n-2}\left[\left(\sum_{j=2}^{i-1}(-1)^{i+j}m_{j}\circ
m_{i-j+1}\right)\circ m_{n-i+1}\right]$ (159)
$\displaystyle+(-1)^{n}\sum_{i=3}^{n-2}\left[m_{i}\circ\left(\sum_{j=2}^{n-i}(-1)^{j}m_{j}\circ
m_{n-i-j+2}\right)\right]\ .$ (160)

We now proceed by induction (the first inductive steps have been proved in the
previous paragraphs) and define (as stated in Footnote 7, for $n=3$ only the
second expression should be considered)

$\displaystyle m_{n}\coloneqq$ $\displaystyle(-1)^{n-1}\left(m_{2}\circ
m_{n-1}\right)\circ_{fb}h_{B}+\left(m_{n-1}\circ m_{2}\right)\circ_{fb}h_{B}$
(161) $\displaystyle=$
$\displaystyle\tilde{m}_{2}\circ_{Z}m_{n-1}+m_{n-1}\circ_{h_{B}}\tilde{m}_{2}\
,$ (162)

where $\circ_{Z}$ and $\circ_{h_{B}}$ denote the composition along an input
with $Z$ or $h_{B}$, respectively, as introduced above. This means we are
enforcing the prescription of inserting a homotopy operator on every term with
a free external input of the associator, neglecting the other terms. By using
the inductive hypothesis $\partial_{B}m_{i}=\sum_{i=2}^{i-1}(-1)^{j}m_{j}\circ
m_{i-j+1}$ and
$\partial_{B}\tilde{m}_{2}=m_{2}-(k_{1}+k_{2})m_{2}^{\text{ht}}$ (see Lemma
8), we can compute its differential as

$\displaystyle\partial_{B}m_{n}=$
$\displaystyle\left(m_{2}-(k_{1}+k_{2})m_{2}^{\text{ht}}\right)\circ_{Z}m_{n-1}-\tilde{m}_{2}\circ_{Z}\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}$ (163) $\displaystyle+\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\circ_{h_{B}}\tilde{m}_{2}+(-1)^{n-1}m_{n-1}\circ_{h_{B}}\left(m_{2}-(k_{1}+k_{2})m_{2}^{\text{ht}}\right)\
.$ (164)

Let us first discuss the terms containing $m_{2}^{\text{ht}}$ in Equation
(163): on the one hand, if condition 2 of the theorem holds, these terms have
vanishing coefficients. On the other hand, if condition 1 of the theorem
holds, by using the Property 3 of the maps $Y$ and $Z$ in Definition 2, the
composition $Y\circ Z$ can be moved until it composes with a homotopy operator
$h_{B}$888This is true for $n>3$. The case $n=3$ has been verified explicitly
in the previous section., thus giving zero because of the weak side conditions
(68). Let us now focus on the second term on the RHS of Equation (163), which
gives (by summing and subtracting the same terms)

$\displaystyle-\tilde{m}_{2}\circ_{Z}\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}=$
$\displaystyle-(-1)^{n-2}\left[m_{2}\circ\left(\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i}\right)\right]\circ_{fb}h_{B}$ (165)
$\displaystyle+(-1)^{n-2}\left[m_{2}\circ_{h_{B}}\left(\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i}\right)\right]\circ_{fb}h_{B}\ .$ (166)

Here there is a slight abuse of notation: when writing
$m_{2}\circ_{h_{B}}\ldots$, we actually mean the composition on the free input
of $m_{2}$, since $m_{2}$ does not contain any $h_{B}$. This notation comes
from the fact that this $m_{2}$ originated from $\tilde{m}_{2}$ which instead
contains $h_{B}$. The second term on the RHS of Equation (165) yields

$\displaystyle(-1)^{n-2}\left[m_{2}\circ_{h_{B}}\left(\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i}\right)\right]\circ_{fb}h_{B}$ $\displaystyle=$ (167)
$\displaystyle\left[(-1)^{n-2}m_{2}\circ_{h_{B}}\left(m_{2}\circ
m_{n-2}+m_{n-2}\circ m_{2}\right)\right]\circ_{fb}h_{B}$
$\displaystyle=m_{2}\circ_{h_{B}}m_{n-1}\ ,$ (168)

where the non-trivial summands are given by $i=2,n-2$ and we used the
definition of $m_{n-1}$. Note that the other terms in the sum vanish because
of the free branch composition. Equation (167), together with the first term
on the RHS of Equation (163), gives

$\displaystyle m_{2}\circ_{Z}m_{n-1}+m_{2}\circ_{h_{B}}m_{n-1}=m_{2}\circ
m_{n-1}\ .$ (169)

A similar argument holds for the third term on the RHS of Equation (163),
leading to

$\displaystyle\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\circ_{h_{B}}\tilde{m}_{2}=$
$\displaystyle\left[\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\circ m_{2}\right]\circ_{fb}h_{B}$ (170)
$\displaystyle-\left[\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\circ_{Z}m_{2}\right]\circ_{fb}h_{B}\,.$ (171)

For the second term on the RHS of Equation 170, we have

$\displaystyle-\left[\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\circ_{Z}m_{2}\right]\circ_{fb}h_{B}$ $\displaystyle=$ (172)
$\displaystyle-\left[\left(m_{2}\circ m_{n-2}+(-1)^{n-2}m_{n-2}\circ
m_{2}\right)\circ_{Z}m_{2}\right]\circ_{fb}h_{B}$
$\displaystyle=(-1)^{n-1}m_{n-1}\circ_{Z}m_{2}\,.$ (173)

Equation (172), together with the last one on the RHS of Equation (163),
yields

$\displaystyle(-1)^{n-1}m_{n-1}\circ_{Z}m_{2}+(-1)^{n-1}m_{n-1}\circ_{h_{B}}m_{2}=(-1)^{n-1}m_{n-1}\circ
m_{2}\,.$ (174)

Let us now consider the remaining terms from Equations (165) and (170), that
is

$\displaystyle(-1)^{n-1}$
$\displaystyle\left[m_{2}\circ\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\right]\circ_{fb}h_{B}+\left[\left(\sum_{i=2}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i}\right)\circ m_{2}\right]\circ_{fb}h_{B}=$ (175)
$\displaystyle(-1)^{n-1}\left[-\sum_{i=2}^{n-2}(-1)^{i}\sum_{j=2}^{i-1}(-1)^{j}\left(m_{j}\circ
m_{i-j+1}\right)\circ m_{n-i+1}\right]$ (176)
$\displaystyle-(-1)^{n-1}\left[\sum_{i=3}^{n-1}m_{i}\circ\sum_{j=2}^{n-i}(-1)^{j}m_{j}\circ
m_{n-i-j+2}\right]\circ_{fb}h_{B}\,,$ (177)

where we used Equation (155). The free branch composition simplifies the
Equation above as follows: the only non-vanishing contributions to the first
term are given by $j=2,i-1,$ whereas the non-vanishing contributions to the
second term are $j=2,n-i$, thus leading to Equation (176) being equal to

$\displaystyle\left[(-1)^{n}\sum_{i=3}^{n-2}(-1)^{i}\left(m_{2}\circ
m_{i-1}+(-1)^{i-1}m_{i-1}\circ m_{2}\right)\circ
m_{n-i+1}\right]\circ_{fb}h_{B}$ (178)
$\displaystyle(-1)^{n}\left[\sum_{i=3}^{n-2}m_{i}\circ\left(m_{2}\circ
m_{n-i}+(-1)^{n-i}m_{n-i}\circ m_{2}\right)\right]\circ_{fb}h_{B}\,.$ (179)

Moreover, we may have a further simplification due to the free branch
composition. In particular, in order to have a free input, the second
composition of the first term can be done on legs with $Z$ only and the first
composition on the second term can be done on legs with $h_{B}$ only999We are
just listing the terms which result in having a free input, i.e. relevant to
the free branch composition.. This leads to

$\displaystyle\left[(-1)^{n}\sum_{i=3}^{n-2}(-1)^{i}\left(m_{2}\circ
m_{i-1}+(-1)^{i-1}m_{i-1}\circ
m_{2}\right)\circ_{Z}m_{n-i+1}\right]\circ_{fb}h_{B}$ (180)
$\displaystyle+(-1)^{n}\left[\sum_{i=3}^{n-2}m_{i}\circ_{h_{B}}\left(m_{2}\circ
m_{n-i}+(-1)^{n-i}m_{n-i}\circ m_{2}\right)\right]\circ_{fb}h_{B}$ (181)
$\displaystyle=\sum_{i=3}^{n-2}(-1)^{i}m_{i}\circ m_{n-i+1}\ ,$ (182)

where we have used Equation (161) and $\circ_{Z}+\circ_{h_{B}}=\circ$ when
composing any two (higher) products constructed with the prescription given
above, see Equation (153). Gathering Equation (169), (174) and (181), we have

$\displaystyle\partial_{B}m_{n}$ $\displaystyle=m_{2}\circ
m_{n-1}+(-1)^{n-1}m_{n-1}\circ m_{2}+\sum_{i=3}^{n-2}(-1)^{i}m_{i}\circ
m_{n-i+1}$ (183) $\displaystyle=\sum_{i=2}^{n-1}(-1)^{i}m_{i}\circ
m_{n-i}=\mathsf{Ass}_{n}\,,$ (184)

that completes the proof. ∎

##  4\. Homotopy Transfer, Homotopy Module-Induction and Deformations

In this section, we explicitly show that the $A_{\infty}$-algebras induced on
$B$ via Homotopy Module-Induction are not ($A_{\infty}$) Hochschild
deformations of the $A_{\infty}$-algebra induced via usual homotopy transfer.
For a detailed description of ($A_{\infty}$) Hochschild cohomology, we refer
to [PS94], where the cyclic cohomology is discussed as well101010The
introduction of an invariant, non-degenerate inner product is useful in many
contexts, e.g., in Physics, in order to write down action functionals..

###  4.1. Homotopy Transfer and Homotopy Module-Induction

The main motivation to investigate infinitesimal Hochschild deformations of
the Homotopy Module-Induced $A_{\infty}$-Algebra is the following result.

######  Proposition 13.

Let Theorem 11 hold. Then the $A_{\infty}$-algebra $(B,m_{n})$ is strictly
quasi-isomorphic to the dga algebra $(A,\mathrm{d}_{A},\wedge)$ seen as an
$A_{\infty}$-algebra if and only if $k_{1}\neq-k_{2}$. The quasi-isomorphism
reads $f_{1}\coloneqq(k_{1}+k_{2})Z$ and $f_{i}\coloneqq 0,$ for all $i\neq
1$.

###### Proof.

We construct the strict quasi-isomorphism by choosing $f_{1}\colon B\to A$ by
$f_{1}\coloneqq(k_{1}+k_{2})\,Z,$ which is a quasi-isomorphism by definition,
and $f_{i}\coloneqq 0,$ for all $i\neq 1.$

Let us start by showing that the $A_{\infty}$-morphism relation (see Equation
(6)) holds for $n=2.$ This equation becomes

$\displaystyle(k_{1}+k_{2})Z\circ m_{2}=(k_{1}+k_{2})^{2}\wedge\circ(Z\otimes
Z)\,$ (185)

since $\partial_{BA}Z=0.$ This relation is satisfied because of the Property 3
of Definition 2.

All the higher $A_{\infty}$-morphism relations (see again Equation (6)) are
trivially satisfied because $f_{i}=0,$ for all $i=0,$ and the
$A_{\infty}$-algebra $A$ only admits a non-trivial 2-product, i.e. all the
$n$-products with $n>2$ vanish. ∎

######  Corollary 14.

The $A_{\infty}$-algebras $(B,m_{n}),$ obtained in Theorem 11, with
$k_{1}\neq-k_{2}$, and $(B,m^{\text{ht}}_{n})$ are quasi-isomorphic.

###### Proof.

The proof simply follows from the composition of the quasi-isomorphism defined
in Proposition 13 and the quasi-isomorphism between
$(A,\mathrm{d}_{A},\wedge)$ and $(B,m_{n}^{\text{ht}})$. ∎

This says that if the side conditions are verified (and $k_{1}\neq-k_{2}$),
the $A_{\infty}$-algebras $(B,m_{n})$ and $(B,m_{n}^{\text{ht}})$ are
equivalent in the $A_{\infty}$-category. With an eye towards Example 1, this
means that on split supermanifolds the Homotopy-Module-Induction and the
Homotopy-Transfer procedures will give rise to equivalent structures.

Let us now consider Theorem 12. In the case where the weak side conditions are
met, it is not yet clear whether there exists a (non-strict) quasi-isomorphism
between the $A_{\infty}$-algebras $(B,m_{n})$ and $(B,m_{n}^{\text{ht}})$.
This will be the subject of further investigation. On the other hand, one can
see that if condition 2 of Theorem 12 is met, then there is an obstruction to
the equivalence of these $A_{\infty}$-algebras:

######  Proposition 15.

Let Theorem 12 hold in the case of hypothesis 2 and assume that the product
$\wedge$ on $A$ induces a non-trivial product on $\mathsf{H}(A)$, i.e.,
$0\neq[\wedge]\in\mathsf{H}\left(\mathsf{Hom}\left(A^{\otimes
2},A\right)\right)$. Then the cohomology class of $\wedge$ represents an
obstruction to the existence of a quasi-isomorphism
$f_{\bullet}:(B,m_{n})^{\otimes\bullet}\to(B,m_{n}^{\text{ht}})$.

###### Proof.

Suppose an $A_{\infty}$-quasi-isomorphism
$f_{\bullet}\colon(B,m_{n})^{\otimes\bullet}\to(B,m_{n}^{\text{ht}})$ exists,
then the maps $f_{\bullet}$ satisfy Equation (6). Recall that, by Lemma 8,
$m_{2}=\partial_{B}\tilde{m}_{2}$. By plugging this in Equation (6) for $n=2$,
we get

$\displaystyle f_{1}\circ
m_{2}=f_{1}\circ\partial_{B}\tilde{m}_{2}=m_{2}^{\text{ht}}\circ
f_{1}^{\otimes 2}+\partial_{B}f_{2}\ \implies\ m_{2}^{\text{ht}}\circ
f_{1}^{\otimes 2}=\partial_{B}\lambda\ ,$ (186)

where $\lambda\in\mathsf{Hom}\left(B^{\otimes 2},B\right)$ of degree $-1$.
Being $f_{1}$ a quasi-isomorphism, this implies that

$\displaystyle m_{2}^{\text{ht}}=\partial_{B}\sigma\ ,$ (187)

for some $\sigma\in\mathsf{Hom}\left(B^{\otimes 2},B\right)$, which is true if
and only if

$\displaystyle\wedge=\partial_{A}\rho\ ,$ (188)

for some $\rho\in\mathsf{Hom}\left(A^{\otimes 2},A\right)$. Hence $f_{1}$ can
be a quasi-isomorphism if and only if $[\wedge]=0$. ∎

###  4.2. Massey Products

In the previous paragraphs, we have discussed under which assumptions the
$A_{\infty}$-algebras $(B,m_{n})$ and $(B,m_{n}^{\text{ht}})$ are quasi-
isomorphic, as shown in Corollary 14 and Proposition 15. This naturally leads
to considerations regarding the _Massey $A_{\infty}$-algebras_ that are
induced on $\mathsf{H}(B)$ (see, e.g., [Val14]). These algebras are
intrinsically related to intersection theory on manifolds and their
_formality_ (see also, e.g., [Del+75]).

These structures are induced by homotopy transfer of $(B,m_{n})$ and
$(B,m_{n}^{\text{ht}})$, respectively, to $\mathsf{H}(B)$. In particular,
consider the following setup of quasi-isomorphic complexes:

${\left(A,\mathrm{d}_{A}\right)}$${\left(B,\mathrm{d}_{B}\right)}$${\left(\mathsf{H}(B),0\right)\,,}$$\scriptstyle{Y}$$\scriptstyle{Z}$$\scriptstyle{p}$$\scriptstyle{i}$
(189)

where the complexes $A$ and $B$ satisfy the assumptions of Definition (2). The
“pivotal” complex $B$ comes with two homotopies $h_{B}$ and $h^{\prime}_{B}$
related to the two maps $i\circ p$ and $Y\circ Z$ as

$\displaystyle\partial_{B}h_{B}=\mathds{1}_{B}-Y\circ Z\ ,\
\partial_{B}h^{\prime}_{B}=\mathds{1}_{B}-i\circ p\ .$ (190)

In particular, the homotopy operator $h_{B}$ is the one used in the Homotopy
Module-Induction to induce an $A_{\infty}$ algebra on $B$, while the homotopy
operator $h^{\prime}_{B}$ is used in the homotopy transfer construction from
$B$ to $\mathsf{H}(B)$ to induce the Massey products. We will denote the
$A_{\infty}$-algebras transferred from $(B,m_{n})$ and $(B,m_{n}^{\text{ht}})$
to $\mathsf{H}(B)$ by $(\mathsf{H}(B),\mathsf{M}_{n})$ and
$\mathsf{H}(B),\mathsf{M}_{n}^{\text{ht}})$, respectively.

Let us consider the situation where Theorem 11 holds: as a Corollary of
Proposition 13 we have

######  Corollary 16.

The $A_{\infty}$-algebras $(\mathsf{H}(B),\mathsf{M}_{n})$ and
$(\mathsf{H}(B),\mathsf{M}_{n}^{\text{ht}})$ are isomorphic.

On the other hand, consider the situation where Proposition 15 holds, we have

######  Corollary 17.

The $A_{\infty}$-algebras $(\mathsf{H}(B),\mathsf{M}_{n})$ and
$(\mathsf{H}(B),\mathsf{M}_{n}^{\text{ht}})$ are not isomorphic. In
particular, $\mathsf{M}_{2}=0$.

An open question is left regarding the situation where Theorem 12 holds in the
case of weak side conditions. Determining whether
$(\mathsf{H}(B),\mathsf{M}_{n})$ and
$(\mathsf{H}(B),\mathsf{M}_{n}^{\text{ht}})$ are isomorphic appears to be non-
trivial. We defer this task to the later work [CM].

We shall now investigate whether the Homotopy Module-Induction can be seen as
a deformation of the Homotopy transfer $A_{\infty}$-algebra. To this, let us
recall some basic facts about Hochschild cohomology.

###  4.3. Infinitesimal Hochschild Deformations and Homotopy Module-Induction

Consider the complex $B$, equipped with the $A_{\infty}$-structure induced via
homotopy transfer $\left(B,m_{n}^{\text{ht}}\right)$ and define the Hochschild
cochains
$C^{\bullet}\left(B\right)\coloneqq\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right),$
where $\mathsf{T}(B)$ is the tensor space of $B.$ We can use the algebraic
structure on $B$ to define a coboundary operator, which is the extension of
the operator
$\partial_{B}\colon\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right)\to\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right)$,
as 111111The signs are chosen according to the convention of Equations (129)
and (130).

$\displaystyle\mathrm{d}^{\mathsf{H}}\mu\coloneqq\sum_{n=1}^{\infty}\left((-1)^{n-1}m_{n}^{\text{ht}}\circ\mu-(-1)^{\mu}\mu\circ
m_{n}^{\text{ht}}\right)\ ,$ (191)

for a given $\mu\in\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right)$,
called the _Hochschild differential_ , where
$m_{1}^{\text{ht}}=\mathrm{d}_{B}$ and the bracket is the extension to
$\mathsf{Hom}(\mathsf{T}(B),B)$ of the commutator on $\mathsf{Hom}(B,B).$ The
nilpotency of $\mathrm{d}^{\mathsf{H}}$ follows from the
$A_{\infty}$-relations (3). Given an element
$\mu\in\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right)$, we say that it is
an _infinitesimal deformation_ of the $A_{\infty}$-algebra if
$\mathrm{d}^{\mathsf{H}}\mu=0$. If $\mu=\mathrm{d}^{\mathsf{H}}\nu$, for some
$\nu\in\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right)$, we say that the
deformation is _trivial_. It is then clear that non-trivial (infinitesimal)
deformations are given by the cohomology groups determined by
$\mathrm{d}^{\mathsf{H}}$; this cohomology is known as _Hochschild cohomology_
and it is denoted as $\mathsf{H}\mathsf{H}\left(B\right)$.

In the previous sections, we have shown how to induce on the complex $B$
algebraic structures with a procedure leveraging on the
$\mathrm{Im}(Z)$-bimodule structure; in this section, we address whether the
Homotopy Module-Induced $A_{\infty}$-structures are related to the
$A_{\infty}$-algebra obtained from the homotopy transfer procedure. In
particular, we will discuss if the structures obtained in Section 3 are a
Hochschild deformation (possibly trivial) of the usual one. It will turn out
that this is not the case.

######  Proposition 18.

Assume that Theorem (11) holds and let $(B,m_{n}^{\text{ht}})$ and $(B,m_{n})$
be the $A_{\infty}$-algebras induced by the homotopy transfer and the Homotopy
Module-Induction, respectively. Then $(B,m_{n})$ is not an infinitesimal
deformation of $(B,m_{n}^{\text{ht}}).$

###### Proof.

Consider the map

$\displaystyle\mu\coloneqq\sum_{i=2}^{\infty}\left(m_{i}-m_{i}^{\text{ht}}\right)\in\mathsf{Hom}\left(\mathsf{T}\left(B\right),B\right)\
;$ (192)

we need to verify whether $\mathrm{d}^{\mathsf{H}}\mu=0$ holds. We will
proceed by checking the equality in each degree. The first relation to check
reads

$\displaystyle\partial_{B}\left(m_{2}-m_{2}^{\text{ht}}\right)=0\ ,$ (193)

which is true since both $m_{2}$ and $m_{2}^{\text{ht}}$ are Leibniz-
compatible with $\mathrm{d}_{B}$. The second relation reads

$\displaystyle\partial_{B}(m_{3}-m_{3}^{\text{ht}})-m_{2}^{\text{ht}}\circ(m_{2}-m_{2}^{\text{ht}})-(m_{2}-m_{2}^{\text{ht}})\circ
m_{2}^{\text{ht}}=0\,,$ (194)

which can be written as

$\displaystyle\partial_{B}m_{3}-m_{2}^{\text{ht}}\circ m_{2}-m_{2}\circ
m_{2}^{\text{ht}}=0\,,$ (195)

where we used $\partial_{B}m_{3}^{\text{ht}}=0$ and $m_{2}^{\text{ht}}\circ
m_{2}^{\text{ht}}=0,$ see Proposition 6. By using the properties of the map
$Z$, we have

$\displaystyle m_{2}^{\text{ht}}\circ m_{2}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\right)=m_{2}^{\text{ht}}\left(m_{2}\left(b\otimes
b^{\prime}\right)\otimes
b^{\prime\prime}\right)-m_{2}^{\text{ht}}\left(b\otimes
m_{2}\left(b^{\prime}\otimes b^{\prime\prime}\right)\right)=0\ .$ (196)

On the other hand, we can explicitly verify that

$\displaystyle m_{2}\circ m_{2}^{\text{ht}}\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\right)=$ $\displaystyle k_{1}\left(Z(b)\wedge
Z(b^{\prime})\right)\rhd b^{\prime\prime}-k_{2}b\lhd\left(Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)$ (197)
$\displaystyle+(k_{2}-k_{1})Y\left(Z(b)\wedge Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)\ .$ (198)

Considering coefficients $k_{1},\,k_{2}$ such that $k_{1}\neq-1$ and
$k_{2}\neq-1$, Equation (197) is different from $\partial_{B}m_{3}$, see
Equation (88), thus showing that $\mu$ does not correspond to a deformation.

In the case $k_{1}=k_{2}=-1,$ we immediately see from Equations (197) and (88)
that Equation (194) is verified. Thus, the next relation to check reads

$\displaystyle\partial_{B}(m_{4}-m_{4}^{\text{ht}})$ $\displaystyle-
m_{2}^{\text{ht}}\circ(m_{3}-m_{3}^{\text{ht}})+(m_{3}-m_{3}^{\text{ht}})\circ
m_{2}^{\text{ht}}$ (199)
$\displaystyle+m_{3}^{\text{ht}}\circ(m_{2}-m_{2}^{\text{ht}})-(m_{2}-m_{2}^{\text{ht}})\circ
m_{3}^{\text{ht}}=0\ .$ (200)

Note that we can use the $A_{\infty}$-relation for the homotopy transfer with
$n=4$ together with Proposition 6 in order to cancel the terms having
$m_{n}^{\text{ht}}$ only. Moreover, $m_{2}^{\text{ht}}\circ m_{3}=0$ because
of the properties 3 of $Z$, together with $Z\circ h_{B}=0,$ and $m_{3}\circ
m_{2}^{\text{ht}}=0$, because of $h_{B}\circ Y=0$. Thus Equation (199) can be
written as

$\displaystyle\partial_{B}m_{4}+m_{3}^{\text{ht}}\circ m_{2}-m_{2}\circ
m_{3}^{\text{ht}}=0\,,$ (201)

In order for Equation (201) to hold, one should verify that the terms
containing $h_{B}$ (i.e., $\partial_{B}m_{4}$) and those containing $h_{A}$
(the last two) vanish separately. This would imply that $\partial_{B}m_{4}=0$,
which is not satisfied. We conclude that

$\displaystyle\mathrm{d}^{\mathsf{H}}\mu\neq 0\ ,$ (202)

so that there is no deformation connecting the homotopy transfer
$A_{\infty}$-algebra and the Homotopy Module-Induced one. ∎

######  Example 2 (Associative algebras).

The failure of the infinitesimal deformation prescription happens even at the
level of associative algebras. In particular, let
$\left(B,m_{2}^{\text{ht}}\right)$ and
$\left(B,\left.m_{2}\right|_{k_{1}=1,k_{2}=0}\right)$ be associative algebras,
where the product of the latter algebra is defined by fixing the coefficients
of Equation (78) to $k_{1}=1,k_{2}=0.$ We assume the same conditions for the
maps $Z$ and $Y$ as in Definition (2). In order to check whether the product
$\left.m_{2}\right|_{k_{1}=1,k_{2}=0}$ can be obtained via a deformation of
$m_{2}^{\text{ht}}$, we need to check if

$\displaystyle\mu\coloneqq\left.m_{2}\right|_{k_{1}=1,k_{2}=0}-m_{2}^{\text{ht}}$
(203)

is closed under the action of the Hochschild differential given by
$\mathrm{d}^{\mathsf{H}}=\left[m_{2}^{\text{ht}},\,\cdot\,\right],$ as
discussed in the proof of Proposition 18. Note that if the closure condition
of $\mu$ with respect to $\mathrm{d}^{\mathsf{H}}$ held, then there would
exist a one-parameter family of maps $m(t)\in\mathsf{Hom}\left(B^{\otimes
2},B\right)[t]$ at most linear in $t$ such that $m(0)=m_{2}^{\text{ht}}$ and
$m(1)=\left.m_{2}\right|_{k_{1}=1,k_{2}=0}$. Explicitly, we have

$\displaystyle\mathrm{d}^{\mathsf{H}}\mu$
$\displaystyle=-m_{2}^{\text{ht}}\circ(\left.m_{2}\right|_{k_{1}=1,k_{2}=0}-m_{2}^{\text{ht}})-(\left.m_{2}\right|_{k_{1}=1,k_{2}=0}-m_{2}^{\text{ht}})\circ
m_{2}^{\text{ht}}$ (204)
$\displaystyle=-m_{2}^{\text{ht}}\circ\left.m_{2}\right|_{k_{1}=1,k_{2}=0}-\left.m_{2}\right|_{k_{1}=1,k_{2}=0}\circ
m_{2}^{\text{ht}}\ ,$ (205)

which gives

$\displaystyle\mathrm{d}^{\mathsf{H}}\mu\left(b\otimes b^{\prime}\otimes
b^{\prime\prime}\right)=$ $\displaystyle-Y\left(Z\left(Z(b)\rhd
b^{\prime}\right)\wedge Z(b^{\prime\prime})\right)+Y\left(Z(b)\wedge
Z\left(Z\left(b^{\prime}\right)\rhd b^{\prime\prime}\right)\right)$ (206)
$\displaystyle-Z\circ Y\left(Z(b)\wedge Z(b^{\prime})\right)\rhd
b^{\prime\prime}+Y\left(Z(b)\wedge Z(b^{\prime})\wedge
Z(b^{\prime\prime})\right)$ (207) $\displaystyle=$ $\displaystyle-Z\circ
Y\left(Z(b)\wedge Z(b^{\prime})\right)\rhd b^{\prime\prime}+Y\left(Z(b)\wedge
Z(b^{\prime})\wedge Z(b^{\prime\prime})\right)\neq 0\ .$ (208)

This shows that $\mu$ is not a Hochschild cocycle, thus there is no
infinitesimal deformation connecting the two algebraic structures.

## References

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