# A Lie theoretic approach to the twisting procedure and Maurer-Cartan
simplicial sets over arbitrary rings

Niek de Kleijn  Delft Institute of Applied Mathematics, Technical University
Delft, Mekelweg 4, Delft, The Netherlands<EMAIL_ADDRESS>Felix
Wierstra  Korteweg-de Vries Institute for Mathematics, University of
Amsterdam, Science Park 105-107, Amsterdam, The Netherlands,
<EMAIL_ADDRESS>
###### Abstract

The Deligne-Getzler-Hinich–$\infty$-groupoid or Maurer-Cartan simplicial set
of an $L_{\infty}$-algebra plays an important role in deformation theory and
many other areas of mathematics. Unfortunately, this construction only works
over a field of characteristic $0$. The goal of this paper is to show that the
notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be
defined for a much larger number of operads than just the $L_{\infty}$-operad.
More precisely, we show that the Koszul dual of every unital Hopf cooperad (a
cooperad in the category of unital associative algebras) with an arity $0$
operation admits a twisting procedure, a natural notion of Maurer-Cartan
equation and under some mild additional assumptions can also be integrated to
a Maurer-Cartan simplicial set. In particular, we show that the Koszul dual of
the Barratt-Eccles operad and its $\mathcal{E}_{n}$-suboperads admit Maurer-
Cartan simplicial sets. In this paper, we will work over arbitrary rings.

##  1 Introduction

Lie algebras up to homotopy, better known as $L_{\infty}$-algebras, play an
important role in many areas of mathematics like deformation quantization and
deformation theory, mathematical physics, symplectic geometry, rational
homotopy theory and many others. It is a well known philosophy that, over a
field of characteristic $0$, all deformation problems are controlled by the
Maurer-Cartan elements in an $L_{\infty}$-algebra. This philosophy goes back
to Deligne, Drinfeld, Feigin, Hinich, Kontsevich-Soibelman, Manetti, and many
others, and was made precise by Lurie in [21] and Pridham in [26]. The
information about such a deformation problem can be conveniently organized in
a simplicial set, which is known as the Deligne-Getzler-
Hinich–$\infty$-groupoid, Maurer-Cartan simplicial set or nerve. This
simplicial set was first defined by Getzler in the case of
$L_{\infty}$-algebra in [13].

When working over fields of general characteristic, the situation is a lot
more complicated. Deformation problems or formal moduli problems are no longer
controlled by $L_{\infty}$-algebras but by $E_{n}$-algebras. More explicitly,
Lurie showed that for $n<\infty$, there is an equivalence between the
$\infty$-category of of formal $E_{n}$-moduli problems and the
$\infty$-category of augmented $E_{n}$-algebras. This was later extended to
$n=\infty$ by Brantner and Mathew [5] and Brantner, Campos and Nuiten [4].

Although all these approaches are great from a theoretical perspective, there
has not been an analog of the Deligne-Hinich-Getzler–$\infty$-groupoid. With
this we mean a simplicial set which encodes the homotopical information of the
deformation problem, in which the simplices (Maurer-Cartan elements) are
defined by explicit equations and operations like the twist are defined by
explicit formulas. In [6], we made a first step towards this by showing that
$A_{\infty}$-algebras (also known as $E_{1}$-algebras) admit a Maurer-Cartan
simplicial set. In this paper, we show that a much larger number of operads
have a natural notion of Maurer-Cartan equation, twisting procedure and
Maurer-Cartan simplicial set.

To construct our Maurer-Cartan simplicial set over a general ring $R$, we use
Kontsevich’s perspective of formal pointed manifolds from [18] and generalize
it to what could be seen as "formal pointed Lie groups". Over a field of
characteristic $0$, Kontsevich defined an $L_{\infty}$-algebra on a graded
vector space $V$ as a degree $-1$ coderivation $Q:C(V)\rightarrow C(V)$ which
squares to zero, where $C(V)$ denotes the cofree conilpotent cocommutative
coalgebra cogenerated by $V$. The coderivation $Q$ is seen as a "vector field"
on $\mathcal{C}(V)$. In this paper, we generalize this idea by taking more
general cooperads $\mathcal{C}$ instead of the cocommutative cooperad. If we
furthermore assume that $\mathcal{C}$ is a unital Hopf cooperad (a cooperad in
the category of unital associative algebras), which admits an arity zero
operation encoding the counit, then we define an $\Omega\mathcal{C}$-algebra
on a graded $R$-module $V$ as a square-zero coderivation
$Q:\mathcal{C}(V)\rightarrow\mathcal{C}(V)$ on the cofree conilpotent
$\mathcal{C}$-coalgebra cogenerated by $V$. Here, $\Omega\mathcal{C}$ denotes
the operadic cobar construction on $\mathcal{C}$ and the correspondence
between $\Omega\mathcal{C}$-coalgebras and coderivations follows from the
Rosetta Stone Theorem (see [20], Theorem 10.1.13). To ensure that certain
infinite sums converge, we need to introduce filtrations and restrict
ourselves to complete or pro-nilpotent $\Omega\mathcal{C}$-algebras. To ensure
that the theory works over arbitrary rings, we further need to assume that in
arity $r$, $\mathcal{C}(r)$ is free as an $R[\mathbf{S}_{r}]$-module, where
$\mathbf{S}_{r}$ denotes the symmetric group on $r$ elements.

The class of operads that satisfy these assumptions is quite large. For
example, it contains the normalized cochains on any operad $\mathcal{P}$ in
simplicial sets, such that $\mathcal{P}(0)=*$, $\mathcal{P}$ has a free
symmetric group action in each arity and finitely many non-degenerate
simplices in each arity. This is because the cochains on a topological operad
have the cochain level cup product making it a unital Hopf cooperad. Some
particular examples of such operads are the commutative operad, the
associative operad, the Barratt-Eccles operad and the
$\mathcal{E}_{n}$-suboperads of the Barratt-Eccles operad.

By a theorem of Moerdijk [24], we can equip $\mathcal{C}(V)$ with an
associative product

$\star:\mathcal{C}(V)\otimes\mathcal{C}(V)\rightarrow\mathcal{C}(V)$

which is a generalization of the shuffle product on the cofree conilpotent
cocommutative coalgebra and is therefore called the generalized shuffle
product. This product turns $\mathcal{C}(V)$ into a
$\mathcal{C}$-$\mathcal{ASS}$-bialgebra, which can be interpreted as the
analog of a Lie group, where $\star$ corresponds to the group structure and
the differential $Q:\mathcal{C}(V)\rightarrow\mathcal{C}(V)$ corresponds to
the manifold structure. The chain complex $V$ corresponds in this case to the
tangent space or "Lie algebra".

This "Lie group" has many similarities to the theory of Lie groups and we
generalize certain concepts from classical Lie theory to this new setting. We
start by defining the analog of the exponential map, which is a map

$\exp:V_{0}\rightarrow\mathcal{C}(V)$

from the "Lie algebra" $V$ to the "Lie group" $\mathcal{C}(V)$ which shares
many properties with the ordinary exponential map. This exponential map allows
us to define a twisting procedure which twists the differential
$Q:\mathcal{C}(V)\rightarrow\mathcal{C}(V)$ by a degree $0$ element $v\in
V_{0}$, in a way that is analogous to the adjoint representation of a
classical Lie group. More precisely, if $x\in\mathcal{C}(V)$ then the twisted
differential $Q^{v}:\mathcal{C}(V)\rightarrow\mathcal{C}(V)$ is defined as

$Q^{v}:=\exp(-v)\star Q(\exp(v)\star x).$

The Maurer-Cartan equation then naturally follows as a flatness condition for
this twist. This further gives an alternative to the twisting procedure from
[8], which works for differential graded operads and over arbitrary rings
instead of fields of characteristic $0$.

If the cooperad $\mathcal{C}$ comes equipped with a map
$\mathcal{E}_{\infty}\rightarrow\mathcal{C}$, where $\mathcal{E}_{\infty}$
denotes the cochains on the Barratt-Eccles operad $\mathcal{E}_{\infty}$, we
can integrate every $\Omega\mathcal{C}$-algebra to a simplicial set which we
call the Maurer-Cartan simplicial set. The main examples which satisfy this
requirement are the Barratt-Eccles cooperad and its $E_{n}$-subcooperads. We
finish by showing that this simplicial set is Kan complex and therefore an
$\infty$-groupoid.

### Acknowledgements

The authors would like to thank Daniel Robert-Nicoud, Martin Markl, Igor
Khavkine, Dion Leijnse, Noah Olander, Sergey Shadrin and José Moreno-Fernández
for useful conversations and comments. The second author is supported by Dutch
Research Organisation (NWO) grant number VI.Veni.202.046.

##  2 Preliminaries and conventions

In this section, we recall the necessary preliminaries and establish our
notation and conventions.

###  2.1 Conventions and notation

In this paper, we work in the category of chain complexes over a general ring
$R$ with unit. We use a homological grading, so the differentials are of
degree $-1$. Unless stated otherwise, all tensor products are taken over $R$
and we implicitly assume the Koszul sign rule, i.e. if $V$ and $W$ are two
chain complexes then the switch map $V\otimes W\rightarrow W\otimes V$ induces
an additional sign. The chain complex of $R$-linear maps between two chain
complexes $V$ and $W$ is denoted by $\hom_{R}(V,W)$, often we drop the $R$ and
write just $\hom(V,W)$.

Let $G$ be a finite group and $V$ a chain complex with a linear $G$-action,
then we denote the coinvariants with respect to the $G$ action by $V_{G}$. The
invariants with respect to the $G$-action are denoted by $V^{G}$. The
symmetric group in $r$ elements is denoted by $\mathbf{S}_{r}$, by convention
we say that both $\mathbf{S}_{1}$ and $\mathbf{S}_{0}$ are the trivial group.

###  2.2 Conventions on operads and cooperads

We assume that the reader is familiar with the basic theory of algebraic
operads and cooperads and otherwise refer the reader to [9], [10] and [20].
Although the book [20] is written in characteristic $0$, most of the results
hold over arbitrary rings with only small modifications which are explained
later in this paper.

Unless stated otherwise, all operads are in chain complexes over a ring $R$
(except for Section 6 where we also consider operads in simplicial sets).
Since we work over a general ring $R$, and not a field of characteristic $0$,
there are a few special conventions we need in this paper that are not super
standard and we recall those here. For more details, we mainly refer to [10],
which is one of the few references that works in this generality.

The unit operad and the unit cooperad are both denoted by $I$ and defined as
$I(r)=0$ if $r\neq 1$ and $I(1)=R$, with the appropriate (co)operad structure.
From the context it should be clear whether we mean the operad or cooperad. An
operad $\mathcal{P}$ is called augmented if there is an additional map
$\epsilon:\mathcal{P}\rightarrow I$ which is a splitting for the operadic
unit. A cooperad is coaugmented if there is a splitting
$\eta:I\rightarrow\mathcal{C}$ for the cooperadic counit.

An operad $\mathcal{P}$ (resp. cooperad $\mathcal{C}$) is called unitary if
$\mathcal{P}(0)=R$ (resp. $\mathcal{C}(0)=R$), the unitary operation will be
denoted by $\mathbbm{1}$. We denote unitary operads by $u\mathcal{P}$ (resp.
unitary cooperads by $u\mathcal{C}$) to indicate that they are unitary. Every
unitary operad (resp. cooperad) has a non-unitary suboperad (resp. non-unitary
quotient cooperad) which is denoted by $\mathcal{P}$ (resp. $\mathcal{C}$) and
is defined as $\mathcal{P}(0):=0$ and $\mathcal{P}(r):=u\mathcal{P}(r)$ for
$r\geq 1$ (resp. $\mathcal{C}(0):=0$ and $\mathcal{C}(r):=u\mathcal{C}(r)$ for
$r\geq 1$, the coproduct is called the reduced coproduct). A unitary operad
$\mathcal{P}$ (resp. unitary cooperad $\mathcal{C}$) is called reduced if
$\mathcal{P}(1)=R$ is the operadic unit (resp. $\mathcal{C}(1)=R$ is the
cooperadic counit). A non-unitary operad $\mathcal{P}$ (resp. cooperad
$\mathcal{C}$) is called reduced if $\mathcal{P}(0)=\mathcal{P}(1)=0$ (resp.
$\mathcal{C}(0)=\mathcal{C}(1)=0$). Note that every reduced operad (resp.
cooperad) is canonically augmented (resp. coaugmented). The non-unitary part
of every reduced unitary operad has a natural weight grading which coincides
with the arity grading.

We further assume for simplicity that all operads and cooperads we consider
are of finite $R$-type, i.e. they are finitely generated $R$-modules in each
degree in each arity. All operads and cooperads are assumed to be projective
as $R$-modules in each arity. Finally, we also assume that the non-unitary
part of all cooperads is conilpotent. With these assumptions we can freely
dualize between operads and cooperads. Note that the dual of a unitary operad
is usually not conilpotent.

###  2.3 (Co)Algebras over (co)operads

In this section, we recall the details about (co)algebras over (co)operads and
their up to homotopy versions.

####  2.3.1 Invariants vs coinvariants

Since we are working over general rings instead of a field of characteristic
$0$, there are a few subtleties in the definition of algebras over symmetric
operads. The main issue is that when working over an arbitrary ring, there are
two ways to consider the symmetric group actions, we can either take
invariants or coinvariants with respect to the symmetric group actions. As is
explained in [9], over a field characteristic $0$ these choices are
isomorphic, but over an arbitrary ring these choices can be very different. In
this section, we recall the necessary definitions and fix our conventions
about algebras and coalgebras over operads and cooperads, for more details see
[9].

Given two symmetric sequences $M$ and $N$, the tensor product of $M$ and $N$
is defined as

$\left(M\otimes
N\right)(r)=\bigoplus_{i+j=r}\operatorname{Ind}^{\mathbf{S}_{r}}_{\mathbf{S}_{i}\times\mathbf{S}_{j}}M(i)\otimes
N(j),$

where $\operatorname{Ind}$ denotes the induced representation. Besides the
tensor product, there are also two version of the composition product. The
first version, which is just called the composition product, is defined as

$M\circ N=\bigoplus_{r\geq 0}\left(M(r)\otimes N^{\otimes
r}\right)_{\mathbf{S}_{r}}.$

The composition product with divided symmetries is defined as

$M\tilde{\circ}N=\bigoplus_{r\geq 0}\left(M(r)\otimes N^{\otimes
r}\right)^{\mathbf{S}_{r}},$

so instead of using coinvariants we are using invariants with respect to the
symmetric groups $\mathbf{S}_{r}$.

Using these two versions of the composition product we can define algebras
(resp. coalgebras) over operads (resp. cooperads), again we have two different
versions here, one using invariants and one using coinvariants.

Let $V$ be a chain complex and $\mathcal{P}$ an operad. Then we call $V$ a
$\mathcal{P}$-algebra if there exists a map

$\gamma_{V}:\mathcal{P}\circ V\rightarrow V$

satisfying the usual axioms. The chain complex $V$ is called a
$\mathcal{P}$-algebra with divided symmetries if there exists a map

$\tilde{\gamma_{V}}:\mathcal{P}\tilde{\circ}V\rightarrow V,$

again satisfying the usual axioms.

Let $\mathcal{C}$ be a (non-unitary) reduced operad, the cofree conilpotent
$\mathcal{C}$-coalgebra on a chain complex $V$, denoted $\mathcal{C}(V)$, is
defined as

$\mathcal{C}(V):=\bigoplus_{r\geq 1}\left(\mathcal{C}(r)\otimes V^{\otimes
r}\right)_{\mathbf{S}_{r}}.$

Note that the direct sum starts at $1$ because we assumed that $\mathcal{C}$
is reduced. The coproduct is induced by the cooperad structure of
$\mathcal{C}$ (see [20] for more details).

Let $M$ and $N$ be two symmetric sequences, assume that $M(r)$ is free as an
$R[\mathbf{S}_{r}]$-module in each arity $r\geq 0$, then we there is an
isomorphism between the composition product and the composition product with
divided powers. This isomorphism is induced by the norm map. If $X$ is an
chain complex with an $\mathbf{S}_{r}$ action then there is a natural map,
called the norm map, from the coinvariants to the invariants given by

$Tr:X_{\mathbf{S}_{r}}\rightarrow X^{\mathbf{S}_{r}}$
$Tr(x)=\sum_{\sigma\in\mathbf{S}_{r}}\sigma x,$

with $x\in X_{\mathbf{S}_{r}}$. The norm map is in general not an isomorphism,
but if $X$ is free as an $R[\mathbf{S}_{r}]$-module then the norm map is an
isomorphism. When $\mathbb{Q}\subseteq R$ there is a variation of the norm map
given by $Tr(x)=\sum_{\sigma\in\mathbf{S}_{r}}\frac{\sigma x}{r!}$, which is
also an isomorphism. Since we assumed that $M$ is free as an
$R[\mathbf{S}_{r}]$-module, the norm map induces an isomorphism

$Tr_{M,N}:M\circ N\rightarrow M\tilde{\circ}N.$ (1)

So in case that $\mathcal{P}$ (resp. $\mathcal{C})$ is an operad (resp.
cooperad) which free as an $R[\mathbf{S}_{r}]$-module in each arity, there is
no difference between algebras (resp. coalgebras) and algebras with divided
symmetries (resp. coalgebras with divided symmetries).

####  2.3.2 Homotopy algebras and the bar and cobar construction

In this section, we discuss algebras up to homotopy and set up the situation
in which things can be twisted.

Let $\mathcal{P}$ be a reduced operad (without unitary operation), if
$\mathbb{Q}\nsubseteq R$, we further assume that the action of
$\mathbf{S}_{r}$ is free in each arity $r$. Let $\mathcal{C}$ be a reduced
cooperad (also non-unitary) together with a Koszul twisting morphism
$\tau:\mathcal{C}\rightarrow\mathcal{P}$. Then we define a
$\mathcal{P}_{\infty}$-algebra as an algebra over $\Omega\mathcal{C}$, the
operadic cobar construction of $\mathcal{C}$.

######  Remark 2.1.

This definition varies a little bit from the definition of a
$\mathcal{P}_{\infty}$-algebra from [20], in their definition the operad
$\mathcal{P}$ would be assumed to be quadratic and Koszul and $\mathcal{C}$
would be the quadratic dual. For preciseness, the cooperad $\mathcal{C}$ could
have been included in the notation of a $\mathcal{P}_{\infty}$-algebra but in
this paper it will always be clear which cooperad $\mathcal{C}$ we refer to.

Alternatively, we can also define a $\mathcal{P}_{\infty}$-algebra on agraded
$R$-module $V$ as a square-zero coderivation on the cofree conilpotent
$\mathcal{C}$-coalgebra with divided symmetries $\mathcal{C}(V)$. This follows
from the Rosetta Stone Theorem (Theorem 10.3.1 in [20]), because we assumed
that all the symmetric group actions are free, this extends to arbitrary
rings.

######  Proposition 2.1.

Let $V$ be a chain complex and $\tau:\mathcal{C}\rightarrow\mathcal{P}$ be a
Koszul twisting morphism between a reduced cooperad $\mathcal{C}$ and a
reduced operad $\mathcal{P}$, if $\mathbb{Q}\nsubseteq R$ further assume that
$\mathcal{C}$ and $\mathcal{P}$ are free as $R[\mathbf{S}_{r}]$-modules in
each arity. Then the notion of a $\mathcal{P}_{\infty}$-algebra, i.e. a
morphism of operads
$\gamma:\mathcal{P}_{\infty}=\Omega\mathcal{C}\rightarrow\operatorname{End}(V)$
is equivalent to a square-zero coderivation
$Q:\mathcal{C}(V)\rightarrow\mathcal{C}(V)$ of degree $-1$, where
$\mathcal{C}(V)$ is the cofree conilpotent $\mathcal{C}$-coalgebra.

Since $\mathcal{C}(V)$ is cofree, it further turns out that the coderivation
$Q$ is determined by its image on cogenerators.

######  Lemma 2.1.

Under the assumptions of Proposition 2.1, the coderivation
$Q:\mathcal{C}(V)\rightarrow\mathcal{C}(V)$ is determined by
$\tilde{Q}:\mathcal{C}(V)\rightarrow V$, the projection onto the cogenerators
of $\mathcal{C}(V)$.

The proof is completely analogous to the proof of Proposition 6.3.8 of [20]
and is therefore omitted. The converse of the lemma is not true, not every map
$f:\mathcal{C}(V)\rightarrow V$ determines a square-zero coderivation. Every
degree $-1$ map $f:\mathcal{C}(V)\rightarrow V$ does determine a coderivation
but this might not square to zero.

The map $\tilde{Q}:\mathcal{C}(V)\rightarrow V$ can be decomposed to define
the set of multiplications on $V$. We denote the arity $r$-component by
$\tilde{Q}_{r}:\mathcal{C}(r)\otimes V^{\otimes r}\rightarrow V$. Every
element $\delta\in\mathcal{C}(r)$ defines a multiplication
$Q_{\delta}:V^{\otimes r}\rightarrow V$ by first taking the inclusion
$\mathcal{C}(r)\otimes V^{\otimes r}\hookrightarrow\bigoplus_{r\geq
1}\left(\mathcal{C}(r)\otimes V^{\otimes r}\right)_{\mathbf{S}_{r}}$ and then
the projection onto $V$.

####  2.3.3 Filtrations and completions

In the rest of this paper, we need to consider certain infinite sums, we
therefore need to include certain filtrations. We mainly use the conventions
from [6] (except that in this paper we use a homological grading instead of
the cohomological one in [6]).

Let $V$ be a chain complex then we will from now on assume that $V$ is
equipped with a descending filtration

$V=F^{1}V\supset F^{2}V\supset F^{3}V\supset...,$

whichis respected by the differential and satisfies $\cap_{k}F^{k}V=\\{0\\}$.
We further assume that the filtration is complete, i.e. $V=\varprojlim
F^{i}V$.

The tensor product of two chain complexes is defined by first taking the
tensor product of the filtrations and then completing with respect to this
induced filtration (for more details see [12] Section 7.3). We further need
that if $W$ is a finitely generated chain complex and $V$ a filtered chain
complex then $\hom(W,V)$ is also filtered with filtration

$F^{n}\hom(W,V):=\hom(W,F^{n}V).$

If the filtration on $V$ is complete then so is the filtration on $\hom(W,V)$.
Similary, we can equip the tensor product of a filtered chain complex $V$ and
an unfiltered finitely generated chain complex $W$ with a filtration given by

$F^{n}(W\otimes V):=W\otimes F^{n}V.$

Again if the filtration on $V$ is complete then so is the filtration on
$W\otimes V$.

Using this completed tensor product, the $r$-fold tensor product of a
dg-$R$-module $V$ with itself also becomes a complete $R$-module. For a
cooperad $\mathcal{C}$, we can extend this induced filtration to
$\mathcal{C}(V)$.

An $\Omega\mathcal{C}$-algebra is called complete or pro-nilpotent if the
square-zero coderivation $\tilde{Q}:u\mathcal{C}(V)\rightarrow V$ respects the
filtrations, i.e. if

$\tilde{Q}_{r}\left(\mathcal{C}(r)\otimes F^{i_{1}}V\otimes...\otimes
F^{i_{r}}V\right)\subseteq F^{i_{1}+...+i_{r}}V.$

######  Convention 2.1.

From now on we will always assume that the objects we are working with have a
filtration and are complete with respect to this filtration.

####  2.3.4 Curved $\mathcal{P}_{\infty}$-algebras and $\infty$-morphisms

If we assume that $u\mathcal{C}$ is reduced unitary operad and that all our
chain complexes are equipped with complete filtrations, then we can generalize
$\mathcal{P}_{\infty}$-algebras to curved $\mathcal{P}_{\infty}$-algebras.
Curved algebras where first introduced by Positselski in [25], where he showed
the remarkable fact that there is a Koszul duality between unital algebras and
curved coalgebras.

######  Definition 2.1.

Let $u\mathcal{C}$ be a unitary reduced cooperad with non-unitary part
$\mathcal{C}$ and $V$ a filtered graded $R$-module. Since $u\mathcal{C}$ is
reduced, its non-unitay part $\mathcal{C}$ automatically has a weight grading.
We denote by $\widehat{u\mathcal{C}(V)}$ the completion of $\bigoplus_{r\geq
0}\left(u\mathcal{C}(r)\otimes V^{\otimes r}\right)_{\mathbf{S}_{r}}$ with
respect to this weight grading, where we put $\mathbbm{1}\in u\mathcal{C}(0)$
in weight $0$. We call this the completed conilpotent cofree
$\mathcal{C}$-coalgebra.

By using the cooperadic decomposition map $u\mathcal{C}(V)$ becomes a
$u\mathcal{C}$-coalgebra.

######  Definition 2.2.

Let $u\mathcal{C}$ be a unitary cooperad and $\mathcal{C}$ its non-unitary
part. A curved $\Omega\mathcal{C}$-algebra on a graded $R$-module $V$ is
defined as a square-zero coderivation $Q:u\mathcal{C}(V)\rightarrow
u\mathcal{C}(V)$. Similar to Lemma 2.1, every square-zero coderivation is
determined by its image on the cogenerators
$\tilde{Q}:u\mathcal{C}(V)\rightarrow V$.

The curvature $\mathcal{R}$ of a curved $\Omega\mathcal{C}$-algebra
$(V,\tilde{Q}:u\mathcal{C}(V)\rightarrow V)$ is defined as
$\mathcal{R}:=\tilde{Q}(\mathbbm{1})\in V$, where $\mathbbm{1}$ is the unitary
operation of $u\mathcal{C}$. We call a curved $\Omega\mathcal{C}$-algebra flat
if $\mathcal{R}=\tilde{Q}(\mathbbm{1})=0$.

######  Remark 2.2.

For simplicity, we have chosen to only use the definition of curved coalgebras
as coderivations on the cofree conilpotent unitary $u\mathcal{C}$-coalgebra.
There is also a theory of curved coalgebras using curved operads, it seems
plausible that most of the theory would also work in that description but that
is beyond the scope of this paper. For more details, see for example [1].

For the definition of the exponential map we further need the notion of
tangent vector of an element

######  Definition 2.3.

Let $\tilde{Q}:u\mathcal{C}(V)\rightarrow V$ be a curved complete
$\Omega\mathcal{C}$-algebra, then the cooperadic counit map
$\epsilon:u\mathcal{C}\rightarrow I$ induces a map
$T:u\mathcal{C}(V)\rightarrow I(V)\cong V$, where $I(V)$ is the cofree
coalgebra generated by the cooperad $I$. We call this the tangent space of
$u\mathcal{C}(V)$. If $v\in u\mathcal{C}(V)$ then we call $T(v)$ the tangent
vector of $v$.

######  Lemma 2.2.

Let $V$ be a flat curved $\Omega\mathcal{C}$-algebra, then $V$ is an algebra
over the operad $\Omega\mathcal{C}$.

###### Proof.

Since $V$ is a flat curved $\Omega\mathcal{C}$-coalgebra it is defined by a
coderivation $\tilde{Q}:u\mathcal{C}(V)\rightarrow V$ with
$\tilde{Q}(\mathbbm{1})=0$. We can therefore restrict the coderivation
$\tilde{Q}$ to a coderivation $\tilde{Q}^{\prime}:\mathcal{C}(V)\rightarrow
V$, this coderivation also squares to zero because $\tilde{Q}$ squared to zero
and $R=0$. It then follows from the Rosetta Stone (Proposition 2.1) that this
is an $\Omega\mathcal{C}$-algebra. ∎

######  Definition 2.4.

A curved $\Omega\mathcal{C}$-algebra is called nilpotent if there exists an
$N\in\mathbb{N}$ such that the arity $r$-part of the coderivation
$\tilde{Q}:u\mathcal{C}(V)\rightarrow V$ is zero for all elements of arity
$r>N$.

Curved $\Omega\mathcal{C}$-algebras have two types of morphisms, strict
morphisms and $\infty$-morphisms. Let $A$ and $B$ be two curved
$\Omega\mathcal{C}$-algebras together with coderivations
$\tilde{Q}^{A}:u\mathcal{C}(A)\rightarrow A$ and
$\tilde{Q}^{B}:u\mathcal{C}(B)\rightarrow B$, then a strict morphism is a
linear map $f:A\rightarrow B$ that strictly commutes with all structure maps
and filtrations, i.e. if $a_{1},...,a_{r}\in A$ and $\delta\in
u\mathcal{C}(r)$ then $f\tilde{Q}^{A}_{r}\left(\delta\otimes
a_{1}\otimes...\otimes a_{r}\right)=Q^{B}_{r}\left(\delta\otimes
f(a_{1})\otimes...\otimes f(a_{r})\right)$. The more interesting notion of a
morphism is called an $\infty$-morphism and is defined as follows. An
$\infty$-morphism $\Phi:A\rightsquigarrow B$ is a $u\mathcal{C}$-coalgebra map

$\Phi:u\mathcal{C}(A)\rightarrow u\mathcal{C}(B),$

such that $\Phi\circ Q^{A}=Q^{b}\circ\Phi$. Since $u\mathcal{C}(B)$ is cofree,
this is equivalent to a sequence of maps

$\phi_{r}:u\mathcal{C}(r)\otimes A^{r}\rightarrow B,$

satisfying a certain equation coming from the cooperadic decompositon map. We
call the map $\phi_{r}$ the arity $r$ component of $\Phi$. Note that an
$\infty$-morphism for which $\phi_{r}=0$ for all $r\neq 1$ is the same as a
strict morphism.

######  Remark 2.3.

There is a second notion of an $\infty$-morphism, which is given by replacing
$u\mathcal{C}(A)$ and $u\mathcal{C}(B)$ by their completions
$\widehat{u\mathcal{C}(A)}$ and $\widehat{u\mathcal{C}(B)}$. These notions can
be very different, for example in the non completed case the only grouplike
element in $u\mathcal{C}(A)$ (resp. $u\mathcal{C}(B)$) is $\mathbbm{1}_{A}$
(resp. $\mathbbm{1}_{B}$) so $\Phi(\mathbbm{1}_{A})=\mathbbm{1}_{B}$. In the
completed case, this is no longer the case and the homotopy theory of these
algebras behaves very differently (see for example [16] in the case of
modules). Since the non-completed notion of $\infty$-morphism coincides with
the classical notion of $\infty$-morphism we have chosen to only focus on this
notion.

##  3 The (generalized) shuffle product

In this section, we recall a generalization of the shuffle product, but before
we can do that we first recall the notion of a Hopf (co)operad. For more
details see for example [24] or [19].

###  3.1 Hopf (co)operads and the tensor product of (co)algebras

Let $\mathcal{P}$ be an operad and $A$ and $B$ two $\mathcal{P}$-algebras, in
general the tensor product $A\otimes B$ does not have the structure of a
$\mathcal{P}$-algebra but only the structure of a
$\mathcal{P}\otimes\mathcal{P}$-algebra, where $\mathcal{P}\otimes\mathcal{P}$
denotes the aritywise tensor product of the operad $\mathcal{P}$ with itself.
There is a special class of operads called Hopf operads for which the tensor
product has a natural $\mathcal{P}$-algebra structure.

A Hopf operad $\mathcal{P}$ is an operad in the category of coassociative
coalgebras, i.e. in each arity $r$ we have a coproduct
$\Delta_{r}:\mathcal{P}(r)\rightarrow\mathcal{P}(r)\otimes\mathcal{P}(r)$ such
that the operadic composition maps commute with the coproducts. A Hopf operad
is called counital if the coproducts $\Delta^{r}$ are all counital.

Suppose that $A$ and $B$ are algebras over a Hopf operad $\mathcal{P}$ then we
can equip the tensor product $A\otimes B$ with a $\mathcal{P}$-algebra
structure where the multiplication map

$\mathcal{P}(r)\otimes\left(A\otimes B\right)^{\otimes r}\rightarrow A\otimes
B$

is given by the following map

$\mathcal{P}(r)\otimes\left(A\otimes B\right)^{\otimes
r}\xrightarrow{\Delta_{r}\otimes\operatorname{id}_{\left(A\otimes
B\right)^{\otimes r}}}\mathcal{P}(r)\otimes\mathcal{P}(r)\left(A\otimes
B\right)^{\otimes r}\xrightarrow{\tau}\mathcal{P}(r)\otimes A^{\otimes
r}\otimes\mathcal{P}(r)\otimes B^{\otimes
r}\xrightarrow{\gamma_{A}\otimes\gamma_{B}}A\otimes B,$

where $\tau$ is the map shuffling the tensor factors and $\gamma_{A}$ and
$\gamma_{B}$ are the structure maps of $A$ and $B$.

Similarly, we can define Hopf cooperads, these are cooperads in the category
of associative algebras. More precisely, a Hopf cooperad $\mathcal{C}$ is a
cooperad together with maps
$\mu_{r}:\mathcal{C}(r)\otimes\mathcal{C}(r)\rightarrow\mathcal{C}(r)$, such
that each $\mu_{r}$ forms an associative product on $\mathcal{C}(r)$ and the
maps $\mu_{r}$ commute with the cooperadic decomposition. We remark that
technically this should be called a co-Hopf cooperad (as it is called in
[14]), but by a small abuse of terminology we just call it a Hopf cooperad. A
Hopf cooperad is called unital if all the products $\mu_{r}$ are unital, the
units are denote by $\eta_{r}\in\mathcal{C}(r)$. Since the cooperadic
decomposition maps respect the units, every unitary unital Hopf cooperad
$u\mathcal{C}$ comes equipped with a map $\eta:u\mathcal{COCOM}\rightarrow
u\mathcal{C}$, where $u\mathcal{COCOM}$ is the unitary cocommutative cooperad.
The map is given by sending the arity $r$ operation of $u\mathcal{COCOM}$ to
$\eta_{r}$.

Similar to algebras, it turns out that the tensor product of two coalgebras
with divided symmetries $C$ and $D$ over a Hopf cooperad $\mathcal{C}$ is
naturally a $\mathcal{C}$-coalgebra with divided symmetries. The coproduct

$\delta_{r}^{C\otimes D}:C\otimes
D\rightarrow\left(\mathcal{C}(r)\otimes\left(C\otimes D\right)^{\otimes
r}\right)_{\mathbf{S}_{r}}$

is defined in a dual way as in the algebra case and is explicitly given by the
following composition

$C\otimes
D\xrightarrow{\delta^{C}_{r}\otimes\delta^{D}_{r}}\left(\mathcal{C}(r)\otimes
C^{\otimes r}\otimes\mathcal{C}(r)\otimes D^{\otimes
r}\right)_{\mathbf{S}_{r}}\xrightarrow{\tau}\left(\mathcal{C}(r)\otimes\mathcal{C}(r)\otimes\left(C\otimes
D\right)^{\otimes r}\right)_{\mathbf{S}_{r}}$
$\xrightarrow{\mu_{r}\otimes\operatorname{id}_{\left(C\otimes
D\right)^{\otimes r}}}\left(\mathcal{C}(r)\otimes\left(C\otimes
D\right)^{\otimes r}\right)_{\mathbf{S}_{r}},$

where $\delta^{C}_{r}$ and $\delta^{D}_{r}$ are the the coproduct maps of $C$
and $D$ and $\tau$ is the map that permutes the tensor factors.

###  3.2 The generalized shuffle product

It is a well known fact that the counital cofree conilpotent cocommutative
coalgebra on a chain complex $V$ is not just a cocommutative coalgebra but
also carries an additional product called the shuffle product. With this
shuffle product, the cofree conilpotent cocommutative coalgebra becomes a Hopf
algebra. In this section, we recall a generalization of the shuffle product
which was originally due to Moerdijk (see Example 2.3 of [24]). Moerdijk
showed that the free algebra over a Hopf operad carries a coassociative
coproduct. For the constructions in this paper, we need the dual of Moerdijk’s
construction which we describe here. For more details see also Section 3.2.1
of [19].

Let $u\mathcal{C}$ be a reduced unitary Hopf cooperad and $V$ a chain complex.
Since $u\mathcal{C}$ is reduced it is canonically augmented. Then
$u\mathcal{C}(V)$, the cofree conilpotent coassociative coalgebra on $V$, has
an associative product

$\star:u\mathcal{C}(V)\otimes u\mathcal{C}(V)\rightarrow u\mathcal{C}(V)$

which is called the generalized shuffle product and is defined as follows.
Because $u\mathcal{C}$ is a Hopf cooperad, the tensor product
$u\mathcal{C}(V)\otimes u\mathcal{C}(V)$ becomes a $u\mathcal{C}$-coalgebra as
well. Since $u\mathcal{C}(V)$ is cofree, we only need to specify what the
image is on the cogenerators of $u\mathcal{C}(V)$. The coaugmentation
$\eta:I\rightarrow u\mathcal{C}$ of $u\mathcal{C}$ induces a map $V\cong
I(V)\rightarrow u\mathcal{C}(V)$, by abuse of notation we will denote the
image of this map by $V$. The restriction to cogenerators
$\tilde{\star}:u\mathcal{C}(V)\otimes u\mathcal{C}(V)\rightarrow V$ of the
generalized shuffle product is defined by

$\mathbbm{1}\star v=v\star\mathbbm{1}=v,$

for $v\in V$ and zero otherwise. By cofreeness this extends to a product
$\star:u\mathcal{C}(V)\otimes u\mathcal{C}(V)\rightarrow u\mathcal{C}(V)$. We
call this product the generalized shuffle product.

The generalized shuffle product has the following compatibility with the
coproducts of $u\mathcal{C}(V)$.

######  Lemma 3.1.

Let $u\mathcal{C}$ be a unitary Hopf cooperad and $V$ a graded $R$-module. The
generalized shuffle $\star:u\mathcal{C}(V)\otimes u\mathcal{C}(V)\rightarrow
u\mathcal{C}(V)$ product is a morphism of $u\mathcal{C}$-coalgebras, in
particular the following diagram commutes

${u\mathcal{C}(V)\otimes
u\mathcal{C}(V)}$${u\mathcal{C}(u\mathcal{C}(V)\otimes
u\mathcal{C}(V))}$${u\mathcal{C}(V)}$${u\mathcal{C}(u\mathcal{C}(V)),}$$\scriptstyle{\Delta_{u\mathcal{C}(V)\otimes
u\mathcal{C}(V)}}$$\scriptstyle{\star}$$\scriptstyle{u\mathcal{C}(\star)}$$\scriptstyle{\Delta_{u\mathcal{C}(V)}}$

where $\Delta_{u\mathcal{C}(V)\otimes u\mathcal{C}(V)}$ and
$\Delta_{u\mathcal{C}(V)}$ are the coalgebra maps.

The proof of the lemma is omitted since it is a straightforward consequence of
the definition of the generalized shuffle product and the compatibility of the
Hopf product with the cooperad structure. By the same arguments as in Appendix
A of [6], we can extend the generalized shuffle product to the completed
$u\mathcal{C}$-coalgebras.

######  Lemma 3.2.

Let $u\mathcal{C}$ be a reduced unitary Hopf cooperad and $V$ a graded
$R$-module. The generalized shuffle product extends to the completed cofree
$u\mathcal{C}$-coalgebras

$\star:\widehat{u\mathcal{C}(V)}\otimes\widehat{u\mathcal{C}(V)}\rightarrow\widehat{u\mathcal{C}(V)}.$

Both $u\mathcal{C}(V)$ and $\widehat{u\mathcal{C}(V)}$ therefore become
$u\mathcal{C}$-$\mathcal{ASS}$-bialgebras.

##  4 The twisting procedure for algebras over Hopf operads

In the previous section, we saw that $\widehat{u\mathcal{C}(V)}$, the
completed cofree conilpotent $u\mathcal{C}$-coalgebra on a graded $R$-module
$V$ over a Hopf cooperad $u\mathcal{C}$, becomes a
$u\mathcal{C}$-$\mathcal{ASS}$-bialgebra with the generalized shuffle product.
In this section, we explain how this bialgebra structure can be interpreted as
the analog of a Lie group. In this interpretation, the "group structure" is
given by the generalized shuffle product and the manifold structure is given
by the $u\mathcal{C}(V)$-coalgebra structure and its differential
$Q:u\mathcal{C}(V)\rightarrow u\mathcal{C}(V)$. We assume that the reader is
familiar with the basics of classical Lie theory, otherwise we refer the
reader to for example [17] for an introduction to classical Lie theory.

In this analogy, the tangent space of the "Lie group" $u\mathcal{C}(V)$ is
given by $V$. It turns out that many concepts from the classical Lie theory of
manifolds also extend to our setting. For example, the exponential map and the
adjoint representation both have analogs in this setting and together they
give rise to the twisting procedure.

Most of the ideas in this section can be seen as a generalization of
Dolgushev’s approach to the twisting procedure of $L_{\infty}$-algebras from
[7]. This approach to the twist was later extended to $A_{\infty}$-algebras by
the authors.

###  4.1 The exponential map

In this section, we define the exponential map and show that it has properties
similar to the exponential map in the theory of classical Lie groups. Recall
that for a Lie group $G$ with Lie algebra $\mathfrak{g}$, the exponential map
is a function $\exp:\mathfrak{g}\rightarrow G$ that maps $\mathfrak{g}$ to $G$
and captures its local structure. For an element $v\in\mathfrak{g}$, it is
explicitly defined by first considering the unique one parameter subgroup
$\gamma_{v}:\mathbb{R}\rightarrow G$ whose tangent vector at the identity is
equal to $v$. The exponential map of $v$ is then defined as
$\exp(v)=\gamma_{v}(1)$, where $1\in\mathbb{R}$ is the multiplicative unit of
$\mathbb{R}$.

We want to generalize this construction to
$u\mathcal{C}$-$\mathcal{ASS}$-bialgebras. For this we first need to find the
analog of the real line, which is given by $R[R]$, the group ring of $R$ as an
additive group with coefficients in $R$. This needs to be turned into a
$u\mathcal{C}$-$\mathcal{ASS}$-bialgebra where the $u\mathcal{C}$-bialgebra
structure has divided symmetries. The group ring $R[R]$ has a natural
cocommutative coproduct given by defining the elements of the additive group
$R$ as grouplike, this needs to be extended to a $u\mathcal{C}$-coalgebra
structure. To do this we use that every unitary unital Hopf cooperad
$u\mathcal{C}$ comes with a natural map $\eta:u\mathcal{COCOM}\rightarrow
u\mathcal{C}$.

####  4.1.1 The analog of the real line

As mentioned earlier, the analog of the real line $\mathbb{R}$ in our setting
is played by the group ring $R[R]$. In the setting of classical Lie groups,
the real line $\mathbb{R}$ has a group structure given by addition and a
manifold structure given by the usual manifold structure on $\mathbb{R}$. In
our setting the group ring $R[R]$ has a "group" structure (strictly speaking
monoid structure) but not yet a $u\mathcal{C}$-coalgebra structure, which
would be the analog of the manifold structure on $\mathbb{R}$.

To avoid confusion between $R$ as a coefficient ring and $R$ as an additive
group, we will from now on denote $R$ as an additive group by $G$. The
elements of $G$ are denoted by $g_{\lambda}$ with $\lambda\in G$, since $G$
and $R$ are the same as additive groups we will occasionally also use $R$ as
an indexing set for the elements $g_{\lambda}$. The additive unit of $R$ is
denoted by $e$ and the multiplicative unit by $g_{\mathbbm{1}}$. With this
notation the group ring has a basis given by the elements $g_{\lambda}$ and
the ring structure of the group ring is defined on these basis elements by
$g_{\lambda}\cdot g_{\mu}=g_{\lambda+\mu}$, with $\lambda,\mu\in G$. The group
ring $R[G]$ further becomes a cocommutative coalgebra with the classical
coproduct

$\Delta:R[G]\rightarrow R[G]\otimes R[G]$

defined on the basis $\\{g_{\lambda}\\}$ by

$\Delta(g_{\lambda})=g_{\lambda}\otimes g_{\lambda}$

for $g_{\lambda}\in G$.

The coproduct of $R[G]$ naturally lands in the invariants and not in the
coinvariants, so this coalgebra does not have divided symmetries. Since we
want to look at morphisms from $R[G]$ to $u\mathcal{C}(V)$ we need to give
$R[G]$ a $u\mathcal{C}$-coalgebra structure with divided symmetries.

First, we turn $R[G]$ into a $u\mathcal{C}$-coalgebra (without divided
symmetries) by using the morphism $\eta:u\mathcal{COCOM}\rightarrow
u\mathcal{C}$, so we have a map

$R[G]\rightarrow\widehat{\bigoplus_{r\geq 0}\left(u\mathcal{C}(r)\otimes
R[G]^{\otimes r}\right)^{\mathbf{S}_{r}}}.$ (2)

Since we assumed that $u\mathcal{C}$ is free as an $R[\mathbf{S}_{r}]$-module
in each arity, the invariants and coinvariants are isomorphic. We can
therefore apply this isomorphism to Equation 2 to get a map

$R[G]\rightarrow\widehat{\bigoplus_{r\geq 0}\left(u\mathcal{C}(r)\otimes
R[G]^{\otimes r}\right)_{\mathbf{S}_{r}}},$

which turns the group algebra into a $u\mathcal{C}$-coalgebra with divided
symmetries.

######  Remark 4.1.

When $\mathbb{Q}\subseteq R$, the norm map is always invertible and in this
case our constructions work for all unitary reduced unital Hopf cooperads.

####  4.1.2 The exponential map

The exponential map is defined as follows. We want to define a map which
generalizes the exponential map of classical Lie algebras. In particular, we
want to associate to each degree $0$ element $v\in V_{0}$ the unique "one
parameter subgroup" of $\widehat{u\mathcal{C}(V)}$ with tangent vector $v$.
With this we mean a $u\mathcal{C}$-$\mathcal{ASS}$-bialgebra map
$\gamma_{v}:R[G]\to\widehat{u\mathcal{C}(V)}$ from $R[G]$, the analog of the
real line, to $\widehat{u\mathcal{C}(V)}$ with $\gamma(e)=\mathbbm{1}$ and
$T(\gamma_{v}(g_{\mathbbm{1}}))=v$, i.e. the tangent vector at the
multiplicative unit $g_{\mathbbm{1}}\in G$ is given by $v$ (see Definition
2.3). This does not necessarily make the map $\gamma_{v}$ unique, but it
becomes unique once we further require that $T(\gamma_{v}(\lambda
g_{\mathbbm{1}}))=\lambda T(\gamma_{v}(g_{\mathbbm{1}}))$, for $\lambda\in R$.
So in particular if the "speed" of the exponential becomes $\lambda$ times
bigger, the tangent vector needs to become $\lambda$ times bigger as well.

######  Proposition 4.1.

Let $v\in V_{0}$, there is a unique map
$\gamma_{v}:R[G]\rightarrow\widehat{u\mathcal{C}(V)}$ of
$u\mathcal{C}$-$\mathcal{ASS}$-bialgebras determined by the following
properties:

  1. 1.

The tangent vector of $\gamma_{v}:R[G]\rightarrow\widehat{u\mathcal{C}(V)}$ at
the unit is given by $T(\gamma_{v}(e))=v$ and $\lambda\in R$ we have that
$T(\gamma_{v}(\lambda\cdot e))=\lambda\cdot v$.

  2. 2.

The map $\gamma_{v}:R[G]\rightarrow\widehat{u\mathcal{C}(V)}$ is a morphism of
bialgebras, i.e. we have
$\gamma_{v}(\alpha\cdot\beta)=\gamma_{v}(\alpha)\star\gamma_{v}(\beta)$, with
$\alpha,\beta\in R[G]$.

###### Proof.

Since $\widehat{u\mathcal{C}(V)}$ is completed cofree every map of
$u\mathcal{C}$-coalgebras is determined by its image on the cogenerators. As a
morphism of $u\mathcal{C}$-coalgebras, the map $\gamma_{v}$ is therefore
uniquely defined by the $R$-linear map

$\widetilde{\gamma_{v}}:R[G]\rightarrow V$

which is defined on basis elements $g_{\lambda}\in R[G]$ (with $\lambda\in R$)
by

$\widetilde{\gamma_{v}}(g_{\lambda})=\lambda\cdot v.$

This determines $\gamma_{v}$ uniquely as a $u\mathcal{C}$-coalgebra map, what
is left to check is whether this map is also a bialgebra map. So we need to
check that it commutes with the associative products of $R[G]$ and
$\widehat{u\mathcal{C}(V)}$.

To show that the exponential map is a morphism of bialgebras, we need to check
that

$\gamma_{v}(g_{\alpha}\cdot
g_{\beta})=\gamma_{v}(g_{\alpha})\star\gamma_{v}(g_{\beta}).$

This is equivalent to showing that the following diagram of
$u\mathcal{C}$-coalgebra maps commutes:

${R[G]\otimes R[G]}$${u\mathcal{C}(V)\otimes
u\mathcal{C}(V)}$${R[G]}$${u\mathcal{C}(R[G]),}$$\scriptstyle{\mu_{R[G]}}$$\scriptstyle{\gamma_{v}\otimes\gamma_{v}}$$\scriptstyle{\star}$$\scriptstyle{\gamma_{v}}$

where $\star$ is the generalized shuffle product and $\mu_{R[G]}$ is the
multiplication of the group ring $R[G]$. Since $u\mathcal{C}(R[G])$ is cofree
in the completed sense, every map is determined by its image on the
cogenerators. We therefore only need to check that the maps
$\gamma_{v}\circ\mu_{R_{[G]}}$ and
$\star\circ\left(\gamma_{v}\otimes\gamma_{v}\right)$ have the same tangent
vector, which is equal to its image on the cogenerators. In other words, we
have to show that for $\alpha\otimes\beta\in R[G]\otimes R[G]$

$T\left(\gamma_{v}\circ\mu_{R_{[G]}}\right)=T\left(\star\circ\left(\gamma_{v}\otimes\gamma_{v}\right)\right).$
(3)

If we do this explicit computation, we see that the left hand side of Equation
3 is equal to

$T(\gamma_{v}\circ\mu_{R_{[G]}}(\alpha\otimes\beta))=\alpha+\beta.$

The right hand side of Equation 3 is computed as follows. First of all, the
elements $\gamma_{v}(\alpha)$ (resp. $\gamma_{v}(\beta)$) are of the form
$1+\alpha+\mbox{higher order terms}$ (resp. $1+\beta+\mbox{higher order
terms}$), where the higher order terms are given by coproducts of arity $2$
and greater. The generalized shuffle product is then given by
$\gamma_{v}(\alpha)\star\gamma_{v}(\beta)=1+\alpha+\beta+\mbox{higher order
terms}$, so after projecting on cogenerators it is given by $\alpha+\beta$.
The maps therefore have the same projection onto the cogenerators and are
therefore equal. The map $\gamma_{v}$ is therefore a map of bialgebras. ∎

From these properties, it follows that $\gamma_{v}$ is determined by the image
of $g_{\mathbbm{1}}$.

######  Lemma 4.1.

From the properties of Properties 4.1, it also follows that the
$u\mathcal{C}$-$\mathcal{ASS}$-bialgebra map
$\gamma_{v}:R[G]\rightarrow\widehat{u\mathcal{C}(V)}$ is equivalent to a
$u\mathcal{C}$-coalgebra map
$\gamma_{v}^{\prime}:R\rightarrow\widehat{u\mathcal{C}(V)}$.

###### Proof.

It is clear that a $u\mathcal{C}$-$\mathcal{ASS}$-bialgebra map
$\gamma_{v}:R[G]\rightarrow\widehat{u\mathcal{C}(V)}$ is determines a
$u\mathcal{C}$-coalgebra map
$\gamma_{v}^{\prime}:R\rightarrow\widehat{u\mathcal{C}(V)}$. This map is given
by $\gamma_{v}^{\prime}(\lambda):=\gamma_{v}(\lambda g_{\mathbbm{1}})$, with
$\lambda\in R$.

To go in the other direction, we assume that we have a map
$\gamma_{v}^{\prime}:R\rightarrow\widehat{u\mathcal{C}(V)}$ and extend it to a
map $\gamma_{v}:R[G]\rightarrow\widehat{u\mathcal{C}(V)}$. Since
$\widehat{u\mathcal{C}(V)}$ is cofree, we only need to define its projection
on cogenerators, the image of the generators $g_{\lambda}\mathbb{R}[G]$ by the
following formula

$T(\gamma_{v}(g_{\lambda})):=T(\gamma_{v}^{\prime}(\lambda)).$

It follows from the properties of Propositions 4.1 that this can indeed be
extended to a bialgebra map. ∎

Now that we have the definition of the analog of a "one-parameter subgroup" we
can define the exponential map in exactly the same way as it is done in
classical Lie theory.

######  Definition 4.1.

The exponential map is defined by

$\exp:V_{0}\rightarrow\widehat{u\mathcal{C}(V)}$
$exp(v):=\gamma_{v}(g_{\mathbbm{1}}).$

Similar to the classical exponential map, our exponential map has the
following properties.

######  Proposition 4.2.

Let $v\in V_{0}$, $\kappa,\lambda\in R$ and $x\in u\mathcal{C}(V)$. The
exponential map has the following properties:

  1. 1.

$\exp(\kappa v)\star\exp(\lambda v)=exp((\kappa+\lambda)v)$.

  2. 2.

$\exp(-v)\star\exp(v)=\mathbbm{1}=\exp(v)\star\exp(-v)$ or alternatively
$\exp(-v)$ is a multiplicative inverse to $\exp(v)$.

  3. 3.

The map
$\exp(v)\star-:\widehat{u\mathcal{C}(V)}\rightarrow\widehat{u\mathcal{C}(V)}$
is a morphism of $u\mathcal{C}$-coalgebras.

###### Proof.

Part $1$ and part $2$ follow immediately from Proposition 4.1 and their proof
is omitted. We prove Part $3$ as follows. We need to show that the map
$\exp(v)\star-:\widehat{u\mathcal{C}(V)}\rightarrow\widehat{u\mathcal{C}(V)}$
commutes with the coproduct, so if $x\in\widehat{u\mathcal{C}(V)}$ then we
need to show that

$\Delta_{\widehat{u\mathcal{C}(V)}}(\exp(v)\star
x)=\Delta(\star)(\Delta_{\widehat{u\mathcal{C}(v)}\otimes\widehat{u\mathcal{C}(V)}}(\exp(v)\otimes
x)),$ (4)

with $x\in\widehat{u\mathcal{C}(V)}$ and $\Delta(\star)$ is the coproduct of
the generalized shuffle product $\star$. To show that Equation 4 holds, we
need two ingredients. The first one is Lemma 3.1 and the second one is that
$\exp(v)$ satisfies a grouplike property. Since the element $g_{\mathbbm{1}}$
is grouplike in $R[G]$, its coproduct is given by
$\Delta(g_{\mathbbm{1}})=\sum_{r\geq 0}\eta_{r}\otimes
g_{\mathbbm{1}}^{\otimes r}$ ,where $\eta_{r}$ is the image of
$u\mathcal{COCOM}(r)$ in $u\mathcal{C}(r)$. Since $\gamma_{v}$ is a morphism
of $u\mathcal{C}$-bialgebras, the coproduct of
$\Delta(\exp(v))=\Delta(\gamma_{v}(g_{\mathbbm{1}}))=\sum_{r\geq
0}\eta_{r}\otimes\exp(v)^{\otimes r}$. If we now use that $\eta_{r}$ the unit
is for the product $\mu_{r}:u\mathcal{C}(r)\otimes u\mathcal{C}(r)\rightarrow
u\mathcal{C}(r)$ and Lemma 3.1, we see that Equation 4 holds. ∎

The following corollary follows almost immediately from Proposition 4.2. Note
that in second part of the following proposition we do not use the completion
of the coalgebra $u\mathcal{C}(V)$.

######  Corollary 4.1.

The element $\exp(v)$ is invertible in $\widehat{u\mathcal{C}(V)}$ with the
generalized shuffle product, its inverse is given by $\exp(-v)$. The morphism

$\exp(v)\star-:u\mathcal{C}(V)\rightarrow u\mathcal{C}(V)$

is therefore an isomorphism with inverse $\exp(-v)\star-$.

###### Proof.

The fact that $\exp(v)\star-$ is invertible follows immediately from
Proposition 4.2. The fact that the map $\exp(v)$ preserves the subspace
$u\mathcal{C}(V)\subset\widehat{u\mathcal{C}(V)}$ follows from the same
arguments as in Appendix A of [6]. ∎

###  4.2 The twisting procedure and the Maurer-Cartan equation

Using the exponential from the previous section, we can define the twisting
procedure for $\Omega\mathcal{C}$-algebras. This is similar to the adjoint
representation of a Lie group on its Lie algebra.

######  Definition 4.2.

Let $(u\mathcal{C}(V),Q)$ be a curved $\Omega\mathcal{C}$-algebra and $v\in
V_{0}$. Then we define $Q^{v}:u\mathcal{C}(V)\rightarrow u\mathcal{C}(V)$, the
differential of $(u\mathcal{C}(V),Q)$ twisted by $v$ as

$Q^{v}:u\mathcal{C}(V)\rightarrow u\mathcal{C}(V)$ $Q^{v}(x):=\exp(-v)\star
Q\left(\exp(v)\star x\right),$

with $x\in u\mathcal{C}(V)$.

######  Theorem 4.1.

The twisted differential $Q^{v}$ is a coderivation and squares to zero. The
twist of a curved $\Omega\mathcal{C}$-algebra is therefore again a curved
$\Omega\mathcal{C}$-algebra.

###### Proof.

We need to show that the twisted differential $\mathbb{Q}^{v}$ is again a
coderivation and squares to zero. First note that by Proposition 4.2,
multiplication by $\exp(v)$ is a $u\mathcal{C}$-coalgebra isomorphism with
inverse $\exp(-v)$. Since the conjugation of a coderivation by an isomorphism
is again a coderivation, the map $Q^{v}$ is again a coderivation.

The map $Q^{v}$ squares to zero since

$\displaystyle(Q^{v})^{2}(x)$ $\displaystyle=\exp(-v)\star
Q(\exp(v)\star\exp(-v)\star Q(\exp(v)\star x))$ $\displaystyle=\exp(-v)\star
Q(\mathbbm{1}\star Q(\exp(v)\star x))$ $\displaystyle=\exp(-v)\star
Q^{2}(\exp(v)\star x)$ $\displaystyle=0$

where we used that multiplication by $\mathbbm{1}$ is the identity. So the
twisted differential squares again to zero which proves the theorem. ∎

Using this notion of the twist, we can define Maurer-Cartan elements as those
elements that produce a flat $\Omega\mathcal{C}$-algebra after twisting.

######  Definition 4.3.

Let $(u\mathcal{C}(V),Q)$ be a curved $\Omega\mathcal{C}$-algebra and let
$v\in V_{0}$, then the Maurer-Cartan equation is defined as

$\tilde{Q}(exp(v))=0.$

An element is called a Maurer-Cartan element if it satisfies the Maurer-Cartan
equation, the set of Maurer-Cartan elements is denoted by
$\operatorname{MC}(V)\subseteq V$.

The Maurer-Cartan equation can be made more explicit by using the explicit
morphism $\eta:u\mathcal{COCOM}\rightarrow u\mathcal{C}$. If
$\eta_{r}^{\prime}$ denotes the basis element for $u\mathcal{COCOM}(r)$ and
$\eta_{r}=\varphi(\eta_{r}^{\prime})$ then the Maurer-Cartan equation can be
rewritten as

$\sum_{r\geq 0}\tilde{Q}_{r}\left(\eta_{r}\otimes v^{\otimes r}\right)=0.$

Using the same arguments as in Section 4 of [6], we get the following
propostion which shows that the Maurer-Cartan equation is indeed a flatness
equation.

######  Proposition 4.3.

Let $(u\mathcal{C}(V),Q)$ be a curved $\Omega\mathcal{C}$-algebra, then
$(u\mathcal{C}(V),Q^{v})$, the twist of $Q$ by an element $v\in V_{0}$, is
flat if and only if $v$ is a Maurer-Cartan element.

##  5 The Maurer-Cartan simplicial set

In the theory of $L_{\infty}$\- and $A_{\infty}$-algebras, the set of Maurer-
Cartan elements can be extended to a simplicial set which encodes all the
relevant gauges. These Maurer-Cartan simplicial sets have many applications in
deformation theory, rational homotopy theory and related fields. In this
section, we show that these Maurer-Cartan simplicial sets can be constructed
in much greater generality. In particular, we show that one can construct a
Maurer-Cartan simplicial set for every unitary reduced unital Hopf cooperad
$u\mathcal{C}$ with a map $\mathcal{E}_{\infty}\rightarrow\mathcal{C}$, where
$\mathcal{E}_{\infty}$ denotes the cochains on the Barratt-Eccles operad and
$\mathcal{C}$ is the non-unitary part of $\mathcal{C}$.

######  Convention 5.1.

From now on we only work with flat $\Omega\mathcal{C}$-algebras.

###  5.1 The Maurer-Cartan simplicial set

We construct the Maurer-Cartan simplicial set as follows. Suppose that
$u\mathcal{C}$ is a unitary reduced unital Hopf cooperad, then we have seen in
Section 4 that there exists a morphism $u\mathcal{COCOM}\rightarrow
u\mathcal{C}$ and that we have an exponential map and a Maurer-Cartan
equation. We further saw that if $V$ is a curved $\Omega\mathcal{C}$-algebra,
then the exponential map is defined via the "one parameter subgroup"
$\gamma_{v}:R[G]\rightarrow u\mathcal{C}(V)$, which in turn was determined by
a morphism of $u\mathcal{C}$-coalgebras $R\rightarrow u\mathcal{C}(V)$.

Since $R$ can be identified with $N_{*}(\Delta^{0};R)$, the chains on
$\Delta^{0}$ with coefficients in $R$, the set of Maurer-Cartan elements is
equal to the set of morphisms of $N_{*}(\Delta^{0};R)$, considered as a
$u\mathcal{C}$-coalgebra using the morphism $u\mathcal{COCOM}\rightarrow
u\mathcal{C}$. to $u\mathcal{C}(V)$.

To define the higher simplices of the Maurer-Cartan simplicial set, we use the
higher standard simplices $\Delta^{n}$. It is well known that the collection
of standard simplices forms a cosimplicial object in the category of
simplicial sets (see [15] for example). So if we apply $N_{*}(-,R)$, the
normalized chains functor with coefficients in $R$, we get a cosimplicial
object in the category of chain complexes. To shorten the notation a little,
we will from now on omit the coefficients in the notation for the normalized
chains and always implicitly assume they are the ring $R$. To define the
Maurer-Cartan simplicial set, we need to turn this into a cosimplicial object
in the category of $\mathcal{C}$-coalgebras. The Maurer-Cartan simplicial set
is then morally defined as

$\operatorname{MC}_{n}(V):=\hom_{\mathcal{C}-\mbox{coalg}}(N_{*}(\Delta^{n}),\widehat{u\mathcal{C}(V)}).$
(5)

To equip $N_{*}(\Delta^{n})$ with a $\mathcal{C}$-coalgebra structure, we
first use the fact that it is an $\mathcal{E}_{\infty}$-coalgebra. There are
several choices possible for an $\mathcal{E}_{\infty}$-coalgebra structure on
the normalized chains. There are for example the surjection operad and the
Barratt-Eccles operad (see [22] and [2]). In the rest of this paper, we use
the Barratt-Eccles operad from [2] and instead of working with coalgebras over
operads, we work with the dual of the Barratt-Eccles operad, which we denote
by $\mathcal{E}_{\infty}$. Since the Barratt-Eccles operad is of finite type,
coalgebras over an operad are equivalent to coalgebras over the dual cooperad.
So to turn $N_{*}(\Delta^{n})$ into a $\mathcal{C}$-coalgebra, we need to
assume that we have a morphism of cooperads
$\varphi:\mathcal{E}_{\infty}\rightarrow\mathcal{C}$. Since the Barratt-Eccles
cooperad has a free symmetric group action, we can identify invariants and
coinvariants so there is no difference between coalgebras with divided
symmetries and ordinary coalgebras.

Since we assumed that all $\Omega\mathcal{C}$-algebras are flat, we see that a
flat $\Omega\mathcal{C}$-algebra structure on a chain complex $V$ is
equivalent to a square-zero coderivation on $\mathcal{C}(V)$ (so the non
counital version). In this case, the set of $n$-simplices of the Maurer-Cartan
simplicial set from Equation 5 is given by the set of maps of
$\mathcal{C}$-coalgebras

$\operatorname{MC}_{n}(V):=\hom_{\mathcal{C}-\mbox{coalg}}(N_{*}(\Delta^{n}),\widehat{\mathcal{C}(V)}).$
(6)

This is because we can identify $\mathcal{C}(V)$ with the relative bar
construction on $V$ (see [20], Chapter 11). Recall that since
$(\mathcal{C}(V),Q)$ is cofree, every map to it is determined by its image on
cogenerators. The converse is however not true since not every map commutes
with the differential $Q$. We can however form the convolution algebra
$\hom_{R}\left(N_{*}(\Delta^{n}),V\right)$, which becomes an algebra over the
convolution operad $\hom(\mathcal{C},\Omega\mathcal{C})$ (see [3]). As is
explained in Chapter 11 of [20], there is a Maurer-Cartan equation in this
convolution algebra whose solutions correspond to the maps that commute with
the differential $Q$. To define this Maurer-Cartan equation, we first define
the $\star_{\tau}$-operator.

Let $\mathcal{C}$ be a cooperad and $\mathcal{P}$ an operad both with free
symmetric group actions. Let $(C,\Delta_{C})$ be a $\mathcal{C}$-coalgebra and
$(V,\gamma_{V})$ a $\mathcal{P}$-algebra, since the symmetric group actions
are free there is no difference between divided symmetries and no divided
symmetries. Further suppose that we have an operadic twisting morphism
$\tau:\mathcal{C}\rightarrow\mathcal{P}$. In our case $\mathcal{P}$ is given
by $\Omega\mathcal{C}$ and $\tau$ is given by the canonical twisting morphism
$\iota:\mathcal{C}\rightarrow\Omega\mathcal{C}$. The $\star_{\tau}$-operator
is the (non-linear) map of degree $-1$

$\star_{\tau}:\hom_{R}(C,V)\rightarrow\hom_{R}(C,V)$

defined as the composite

$\star_{\tau}(\psi):=C\xrightarrow{\Delta_{C}}\mathcal{C}\circ
C\xrightarrow{\tau\circ\psi}\mathcal{P}\circ V\xrightarrow{\gamma_{V}}V.$

We further define, for $r\geq 2$, the operations

$\mu_{r}:\hom_{R}(C,V)^{\otimes r}\rightarrow\hom_{R}(C,V)$

by

$\mu_{r}(\psi_{1},...,\psi_{r}):=\gamma_{V}\circ(\psi_{1}\otimes...\otimes\psi_{r})\circ\Delta^{r}_{C},$

with $\psi_{1},...,\psi_{r}\in\hom_{R}(C,V)$ and
$\Delta^{r}_{C}:C\rightarrow\left(\mathcal{C}(r)\otimes C^{\otimes
r}\right)_{\mathbf{S}_{r}}$ is the arity $r$ part of the coproduct of $C$. The
operator $\star_{\tau}$ can be written as $\star_{\tau}(\psi)=\sum_{r\geq
2}\mu_{r}(\psi,...,\psi)$.

A Maurer-Cartan element is a degree $0$ morphism $\psi\in\hom_{R}(C,V)$ which
satisfies the Maurer-Cartan equation, which is given by

$\partial(\psi)+\star_{\tau}(\psi)=0,$ (7)

where $\partial$ is the differential of $\hom_{R}(C,V)$. The set of solutions
to the Maurer-Cartan equation is denoted by
$\operatorname{MC}\left(\hom_{R}(C,V)\right)$. From Chapter 11 of [20] we get
the following proposition.

######  Proposition 5.1 ([20], Proposition 11.3.1).

Under the previous assumptions we have the following bijection

$\operatorname{MC}\left(\hom_{R}(C,V)\right)\cong\hom_{\mathcal{C}-\mbox{coalg}}\left(C,(\mathcal{C}(V),Q)\right).$

In our specific situation, the twisting morphism is given by the canonical
twisting morphism $\iota:\mathcal{C}\rightarrow\Omega\mathcal{C}$ (see [20],
Section 6.5). Because of Proposition 5.1, we see that have an equivalence
between the set of $n$-simplices from Equation 6 and the Maurer-Cartan
elements in $\hom_{R}(N_{*}(\Delta^{n}),V)$. Further notice that when $n=0$,
the chains on $\Delta^{0}$ are isomorphic to $R$. So there is an isomorphism
$\hom_{R}(N_{*}(\Delta^{0}),V)\cong V$. It is straightforward to see that
under this isomorphism the Maurer-Cartan equation in
$\hom_{R}(N_{*}(\Delta^{0}),V)$ given by $\partial(\psi)+\star_{\iota}(\psi)$
is equivalent to the Maurer-Cartan equation from Definition 4.3. An equivalent
formulation of the Maurer-Cartan simplicial set is then given by

$\operatorname{MC}_{n}(V):=\operatorname{MC}\left(\hom_{R}\left(N_{*}(\Delta^{n},V\right)\right).$
(8)

The face and degeneracy maps are the maps induced by the face and degeneracy
maps of $\\{\Delta^{n}\\}_{n\geq 0}$.

######  Lemma 5.1.

Under the earlier assumptions from this section, the induced face maps
$d_{i}:\operatorname{MC}_{n}(V)\rightarrow\operatorname{MC}_{n-1}(V)$ and
$s_{j}:\operatorname{MC}(V)\rightarrow\operatorname{MC}_{n+1}(V)$ preserve
Maurer-Cartan elements.

This follows from the fact that $\operatorname{MC}$ is a bifunctor in the
coalgebra and the algebra. Since the cosimplicial maps are coalgebra maps they
preserve the Maurer-Cartan equation. The Maurer-Cartan simplicial set
therefore becomes indeed a well defined simplicial set.

######  Remark 5.1.

In characteristic $0$, the convolution algebra becomes canonically an
$L_{\infty}$-algebra with the $L_{\infty}$-structure defined in [27]. This is
unfortunately not the case when working over rings that do not contain
$\mathbb{Q}$ as a subring.

######  Remark 5.2.

In our definition of the Maurer-Cartan simplicial set we have chosen to use
the chains on the standard simplex. But this construction could of course also
be done for other models of the simplex. It is a natural question to ask
whether other models would give homotopy equivalent Maurer-Cartan simplicial
sets. It seems highly likely that the methods of Milham and Rogers (see [23])
would also apply to this more general setting, but this is beyond the scope of
this paper.

###  5.2 The Maurer-Cartan simplicial set is a Kan complex

In this section, we prove that the Maurer-Cartan simplicial set is a Kan
complex.

######  Theorem 5.1.

Let $V$ be a complete $\Omega\mathcal{C}$-algebra then the Maurer-Cartan
simplicial set $\operatorname{MC}_{\bullet}(V)$ is a Kan complex.

To prove the theorem, we follow Getzler’s proof of the fact that the Maurer-
Cartan simplicial set or Deligne-Getzler-Hinich groupoid associated to an
$L_{\infty}$-algebra (see [13], Section 4). The main idea behind Getzler’s
proof is that the chains on $\Delta^{n}$ come with a retraction onto each of
the vertices of $\Delta^{n}$. Given a horn $\Lambda^{n}_{k}$ in $V$, we can
use this contraction to inductively build a horn filler. In [6], we
generalized Getzler’s proof to the case of $A_{\infty}$-algebras by using the
cochains on the simplices. In the case of a general
$\Omega\mathcal{C}$-algebra, we need to replace the cochains by the chains and
replace the tensor product by the mapping space. The biggest difference is
that in the $A_{\infty}$\- and $L_{\infty}$-case the tensor product with the
cochains (resp. polynomial de Rham forms) is again an $A_{\infty}$ (resp.
$L_{\infty}$-algebra). In our case, the tensor product is replaced by the
convolution algebra which is not naturally an $\Omega\mathcal{C}$-algebra. We
therefore need to work with the more general Maurer-Cartan equation from
Equation 7, which has as a consequence that our formulas are slightly
different from Getzler’s proof in [13].

Before we prove Theorem 5.1, we first need a contraction on the chains of the
simplex $\Delta^{n}$. Since all the arguments are completely analogues to the
arguments in [6], we have left the proofs to the reader. For what follows we
use the following notation. The $k$-dimensional subsimplex of $\Delta^{n}$
with vertices $i_{0},...,i_{k}$ is denoted by $e_{i_{0},...,i_{k}}$, with
$0\leq i_{0}<...<i_{k}\leq n$. A set of generators for $N_{k}(\Delta^{n})$ as
a chain complex is then given by $\\{e_{I}\\}$ where $I\subseteq\\{0,...,n\\}$
runs over all subsets of order $k+1$. The differential is then given by

$d(e_{i_{0},...,i_{k}})=\sum_{j=0}^{k}(-1)^{j}e_{i_{0},...,\widehat{i_{j}},...,i_{k}},$

where $\widehat{i_{j}}$ means that we omit the $i_{j}$th index.

The counit of $N_{*}(\Delta^{n})$ is the map induced by the map of simplicial
sets $\Delta^{n}\rightarrow\Delta^{0}$. Explicitly the counit
$\epsilon:N_{*}(\Delta^{n})\rightarrow R$ is defined on generators by
$\epsilon(e_{k})=\mathbbm{1}$ and zero otherwise. The inclusion of the $k$th
vertex is denoted by $p^{k}_{n}:\Delta^{0}\rightarrow\Delta^{n}$ and is given
by $p^{k}_{n}(e_{0})=e_{k}$.

The composition $p_{n}^{k}\circ\epsilon:N_{*}(\Delta^{n})\rightarrow
N_{*}(\Delta^{n})$ is homotopic to the identity
$\operatorname{id}:N_{*}(\Delta^{n})\rightarrow N_{*}(\Delta^{n})$ via the
chain homotopy $h^{k}_{n}:N_{*}(\Delta^{n})\rightarrow N_{*+1}(\Delta^{n})$
given by

$h^{k}_{n}(e_{I}):=(-1)^{s}e_{I\cup k},$

with $e_{I}\in N_{*}(\Delta^{n})$ and where $I\cup k$ is defined as $0$ if $k$
was already an element of $I$. The sign $s$ is given by the number of elements
in $I$ smaller than $k$.

The map $h^{k}_{n}$ is a chain homotopy between $p_{n}^{k}$ and the identity,
i.e. it satisfies the following equation

$dh^{k}_{n}+h^{k}_{n}d=\operatorname{id}_{N_{*}(\Delta^{n})}-p_{n}^{k}.$ (9)

The map $p^{k}_{n}$ induces a map

$\tilde{P_{n}^{k}}:\hom_{R}(N_{*}(\Delta^{n}),V)_{d}\rightarrow\hom_{R}(N_{*}(\Delta^{0}),V)_{d}\cong
V$

given by

$\tilde{P_{n}^{k}}(\varphi):=\varphi\circ p^{k}_{n}$

with $\varphi\in\hom_{R}(N_{*}(\Delta^{n}),V)_{d}$. Similarly, the counit
$\epsilon:N_{*}(\Delta^{n})\rightarrow R$ induces a map

$E:V\cong\hom_{R}(N_{*}(\Delta^{0}),V)_{d}\rightarrow\hom_{R}(N_{*}(\Delta^{n}),V)_{d}$

given by

$E(\phi):=\phi\circ\epsilon$

with $\phi\in\hom_{R}(N_{*}(\Delta^{0}),V)_{d}$. The map

$P^{k}_{n}:\hom_{R}(N_{*}(\Delta^{n}),V)_{d}\rightarrow\hom_{R}(N_{*}(\Delta^{n}),V)_{d}$

is defined by

$P_{n}^{k}:=E\circ\tilde{P^{k}_{n}}.$

The homotopy $h^{k}_{n}$ induces a similar homotopy on the level of
convolution algebras

$H^{k}_{n}:\hom_{R}(N_{*}(\Delta^{n}),V)_{d}\rightarrow\hom_{R}(N_{*}(\Delta^{n}),V)_{d+1}.$

given by

$H^{k}_{n}(\varphi):=\varphi\circ h^{k}_{n}$

with $\varphi\in\hom_{R}(N_{*}(\Delta^{n}),V)_{d}$. The maps $\partial$, $E$,
$P_{n}^{k}$ and $H^{k}_{n}$ satisfy the following identity

$\partial
H^{k}_{n}+H^{k}_{n}\partial=\operatorname{id}_{\hom_{R}(N_{*}(\Delta^{n}),V)}-P_{n}^{k}.$
(10)

We further define the map

$R_{n}^{k}:\hom_{R}(N_{*}(\Delta^{n}),V)_{d}\rightarrow\hom_{R}(N_{*}(\Delta^{n}),V)_{d}$

as

$R^{k}_{n}:=\partial\circ H^{k}_{n}.$

Using these maps we can now prove Theorem 5.1.

###### Proof of Theorem 5.1.

As mentioned earlier, we use a variation of Getzler’s proof in which we start
with a certain element in $\hom_{R}(N_{*}(\Delta^{n}),V)$ and extend this to a
Maurer-Cartan element.

Suppose that we have a $k$-horn in $\operatorname{MC}_{\bullet}(V)$, i.e. a
map $\varphi:N_{*}(\Lambda^{n}_{k})\rightarrow V$, then we need to construct a
horn filler, i.e. a map $\psi:N_{*}(\Delta^{n})\rightarrow V$ that fills
$\varphi$. We define $\psi$ inductively and start with $\psi_{1}$ which we
define as $\psi_{1}=\xi+\rho$, where

$\xi(e_{k}):=\varphi(e_{k})$

and zero otherwise and

$\rho(e_{I}):=\begin{cases}\varphi(e_{I})&\mbox{if }e_{I}\neq
e_{k},e_{0...\hat{k}...n}\mbox{ or }e_{01...n},\\\ \sum_{i\neq
k}\varphi(e_{0...\hat{i}...n})&\mbox{if }e_{I}=e_{0...\hat{k}...n},\\\
0&\mbox{if }e_{I}=e_{k}\mbox{ or }e_{01...n}.\end{cases}$

Since $\varphi$ is a Maurer-Cartan element, the element $\xi$ is also a
Maurer-Cartan element. It is further straightforward to see that
$\partial\rho=0$, so it is a cycle.

In most cases, the element $\psi_{1}$ is not a Maurer-Cartan element. Since
$\xi$ is a Maurer-Cartan element and $\rho$ a cycle, it does however satisfy
the Maurer-Cartan equation in

$F_{1}\hom(N_{*}(\Delta^{n}),V)/F_{2}\hom(N_{*}(\Delta^{n}),V),$

i.e. it satisfies the Maurer-Cartan equation modulo elements of filtration
degree $2$. We proceed by adding a "correction" term $\gamma_{1}$ that defines
an element $\psi_{2}$ which satisfies the Maurer-Cartan equation up to
elements of filtration degree $3$. First, we define

$\gamma_{1}:=H^{k}_{n}(\partial\psi_{1}+\star_{\iota}(\psi_{1})).$

The element $\gamma_{1}$ is of filtration degree greater or equal $2$ because
both $\star_{\iota}(\psi_{1})$ and $\partial\psi_{1}$ are of filtration degree
$\geq 2$. We define $\psi_{2}$ as

$\psi_{2}:=\psi_{1}-\gamma_{1}.$

Next we show that the element $\psi_{2}$ satisfies the Maurer-Cartan equation
up to filtration degree $3$. We have

$\partial(\psi_{2})+\star_{\iota}(\psi_{2})=\partial\psi_{1}-\partial\gamma_{1}+\star_{\iota}(\psi_{1}-\gamma_{1}).$
(11)

Since $\gamma_{1}$ is of filtration degree $\geq 2$, the element
$\star_{\iota}(\psi_{1}-\gamma_{1})$ can be rewritten as

$\star_{\iota}(\psi_{1}-\gamma_{1})=\star_{\iota}(\psi_{1})+\mbox{terms of
filtration degree $\geq 3$}.$

So modulo elements of filtration degree $\geq 3$, Equation 11 reduces to

$\partial\psi_{1}-\partial\gamma_{1}+\star_{\iota}(\psi_{1}).$ (12)

If we apply Equation 10 to $\partial\gamma_{1}$, we get

$\displaystyle\partial\gamma_{1}$ $\displaystyle=\partial
H^{k}_{n}(\partial\psi_{1}+\star_{\iota}\psi_{1})$ (13)
$\displaystyle=\partial\psi_{1}+\star_{\iota}(\psi_{1})-P_{n}^{k}(\partial(\psi_{1}))-P_{n}^{k}(\star_{\iota}(\psi_{1}))-H^{k}_{n}(\partial^{2}\psi_{1})-H_{n}^{k}(\partial(\star_{\iota}(\psi_{1}))).$
(14)

So if we combine this with Equation 12, we get

$-P_{n}^{k}(\partial(\psi_{1}))-P_{n}^{k}(\star_{\iota}(\psi_{1}))-H^{k}_{n}(\partial^{2}\psi_{1})-H_{n}^{k}(\partial(\star_{\iota}(\psi_{1}))).$
(15)

Because $\xi$ satisfies the Maurer-Cartan equation, the terms
$-P_{n}^{k}(\partial(\psi_{1}))-P_{n}^{k}(\star_{\iota}(\psi_{1}))$ are zero.
The term $-H^{k}_{n}(\partial^{2}\psi_{1})$ is also zero since it involves
$\partial^{2}$. So we are left with the term
$-H_{n}^{k}(\partial(\star_{\iota}(\psi_{1})))$ and we need to show that this
is of filtration degree $\geq 3$. If we use the Leibniz rule for algebras over
operads we get

$-H_{n}^{k}(\partial(\star_{\iota}(\psi_{1})))=\sum_{r\geq
2}\partial(\mu_{r})(\psi_{1}^{\otimes r})+\sum_{r\geq 2,0\leq l\leq
r}\mu_{r}(\psi_{1}^{\otimes
l}\otimes\partial(\psi_{1})\otimes\psi_{1}^{\otimes r-l-1}).$

Since $\partial\psi_{1}$ is of filtration degree $\geq 2$, all the terms of
the form $\mu_{r}(\psi_{1}^{\otimes
l}\otimes\partial(\psi_{1})\otimes\psi_{1}^{\otimes r-l-1})$ are of filtration
degree $\geq 3$. Further, for $r\geq 3$, the terms
$\partial(\mu_{r})(\psi_{1}^{\otimes r})$ are all of filtration degree $\geq
3$. The term $\partial(\mu_{2})(\psi_{1}\otimes\psi_{1})=0$ because $\mu_{2}$
is the arity two component of an operadic twisting morphism. The element
$\psi_{2}$ is therefore Maurer-Cartan up to terms of filtration degree $3$.

We continue inductively by defining the next "correction" terms as

$\gamma_{i}:=H_{n}^{k}\left(\partial\psi_{i}+\star_{\iota}(\psi_{i})\right)$

and

$\psi_{i+1}=\psi_{i}-\gamma_{i}.$

By exactly the same arguments it follows that $\psi_{i}$ is Maurer-Cartan up
to elements of filtration degree $\geq i+1$. The element $\psi$ is then
defined as

$\psi:=\lim\psi_{i}$

by completeness this limit actually converges and is a Maurer-Cartan element.
It can further be shown that it satisfies the properties of a horn filler for
$\varphi$ and therefore proves that $\operatorname{MC}_{\bullet}(V)$ is a Kan
complex.

∎

##  6 Comparison to other approaches and examples

Recently, in [8] another approach to the twisting procedure was described by
using the gauge group. It is currently unclear how to their twisting procedure
exactly compares to ours. In both cases, the unitary operation plays an
essential role but it is not clear how the unital Hopf (co)operad condition
compares to their unital extendability condition. However, the approach in
this paper has multiple advantages compared to [8]. First of all, our
constructions work over arbitrary rings and not just fields of characteristic
$0$ or rings that contain $\mathbb{Q}$. Second, the constructions of this
paper also apply to differential graded operads and not just operads defined
by quadratic data.

We finish this paper by showing that the Koszul dual of every unitary operad
$\mathcal{P}$ in simplicial sets admits a twisting procedure. By unitary in
simplicial sets we mean $\mathcal{P}(0)=*$. We further show that one can
construct a Maurer-Cartan simlicial set for the Barratt-Eccles operad and its
$\mathcal{E}_{n}$-suboperads.

######  Theorem 6.1.

Let $\mathcal{P}$ be a unitary operad in simplicial sets with finitely many
non-degenerate simplices in each arity, then $N^{*}(\mathcal{P})$ satisfies
the conditions of Theorem 4.1 and therefore admits a twisting procedure.

###### Proof.

We need to show that $N^{*}(\mathcal{P})$ is a unital Hopf cooperad and that
there is a map of cooperads $u\mathcal{COCOM}\rightarrow N^{*}(\mathcal{P})$.
Since $\mathcal{P}$ is an operad of finite type, the cochains on $\mathcal{P}$
are naturally a cooperad. The Hopf structure comes from the chain level cup
product, it follows from the properties of the cochain functor that the cup
product is unital and is compatible with the cooperad structure. Since
$u\mathcal{COM}$ is the terminal operad in simplicial sets, every operad
$\mathcal{P}$ comes equipped with a unique map $\mathcal{P}\rightarrow
u\mathcal{COM}$. The induced map on cochains is the map
$u\mathcal{COCOM}\rightarrow N^{*}(\mathcal{P})$ we need. ∎

The main example of a class of cooperads that admit a Maurer-Cartan simplicial
set are the $\mathcal{E}_{n}$-subcooperads of the dual Barratt-Eccles operad.
The Barratt-Eccles operad $\mathcal{BE}_{\infty}$ is an operad in simplicial
sets which naturally acts on the chains and cochains of a simplicial set (see
[2] for more details). It comes with a sequence of suboperads

$\mathcal{BE}_{1}\hookrightarrow\mathcal{BE}_{2}\hookrightarrow\mathcal{BE}_{3}\hookrightarrow...\mathcal{BE}_{\infty},$
(16)

where each $\mathcal{BE}_{n}$ models the chains on the little $n$-disks
operad. We denote the normalized cochains on $\mathcal{BE}_{n}$ by
$\mathcal{E}_{n}:=N^{*}(\mathcal{BE}_{n})$. Since the cochains are
contravariant, the sequence of maps from Equation 16 induces a sequence of
maps of cooperads

$\mathcal{E}_{\infty}\rightarrow...\mathcal{E}_{3}\rightarrow\mathcal{E}_{2}\rightarrow\mathcal{E}_{1}.$

So every $\mathcal{E}_{n}$-cooperad has a map
$\mathcal{E}_{\infty}\rightarrow\mathcal{E}_{n}$ and therefore satisfies the
conditions from Section 5. The Koszul dual of $\mathcal{E}_{n}$ is given by
$\Omega\mathcal{E}_{n}$ and Fresse showed in [11] that $\Omega\mathcal{E}_{n}$
is weakly equivalent to $\Lambda^{-n}\mathcal{E}_{n}^{\vee}$, where
$\Lambda^{-n}$ denotes the operadic desuspension of $\mathcal{E}_{n}$ (note
that we exchanged $\mathcal{E}_{n}$ and $\mathcal{E}_{n}^{\vee}$ from [11]).
By the results of Lurie from [21], it turns out that these Maurer-Cartan
simplicial sets control $\mathcal{E}_{n}$ deformation problems.

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