text:
\begin{document}

\maketitle

\begin{abstract}
In paper \cite{EnhancedLie}, we constructed a symmetric monoidal
category $\tLie$ whose objects are shifted (and filtered) $L_{\infty}$-algebras.
Here, we fix a cooperad $\cC$ and show that algebras over the operad
$\Cobar(\cC)$ naturally form a category enriched over $\tLie$.
Following \cite{EnhancedLie}, we ``integrate'' this $\tLie$-enriched category
to a simplicial category $\HoAlg^{\sD}_{\cC}$ whose mapping spaces
are Kan complexes. The simplicial category $\HoAlg^{\sD}_{\cC}$ gives us
a particularly nice model of an $(\infty,1)$-category of $\Cobar(\cC)$-algebras.
We show that the homotopy category of $\HoAlg^{\sD}_{\cC}$ is
the localization of the category of $\Cobar(\cC)$-algebras and
$\infty$-morphisms with respect to $\infty$ quasi-isomorphisms.
Finally, we show that the Homotopy Transfer Theorem is
a simple consequence of the Goldman-Millson theorem.
\end{abstract}

\section{Introduction}
This work is motivated by Dmitry Tamarkin's answer \cite{dgcat} to Vladimir Drinfeld's
question ``What do dg categories form?''  \cite{Drinfeld-dgcat}, and by papers \cite{Berglund},
\cite{DotsPoncin}, \cite{Ezra-infty}, \cite{Hinich:1997}, and \cite{Lazarev}.
Here, we give an answer to the question ``What do homotopy algebras form?''

Homotopy algebras and their generalizations appear in constructions of string topology, in
rational homotopy theory, symplectic topology, deformation quantization, and quantum field
theory.  For a gentle introduction to this topic, we refer the reader to paper
\cite{Bruno-plus-equals}. For a more detailed exposition, a standard reference is
book \cite{LV} by Jean-Louis Loday and Bruno Vallette.

Despite an important role of homotopy algebras, it is not clear what higher
categorical structure stands behind the homotopy theory of homotopy algebras.
A possible way to answer to this question is to use closed model categories
and this approach is undertaken in paper \cite{Bruno-HT} by Bruno Vallette.
Here, we suggest a different approach which is based on the use
of the convolution $L_{\infty}$-algebra and the Getzler-Hinich construction
\cite{Ezra-infty}, \cite{Hinich:1997}.

By a homotopy algebra structure on a cochain complex of
$\bbk$-vector\footnote{In this paper, we assume that
  $\textrm{char}(\bbk) = 0$.}  spaces $A$ we mean a
$\Cobar(\cC)$-algebra structure on $A$, where $\cC$ is a differential
graded (dg) cooperad satisfying some technical conditions. In this
paper, we fix such a cooperad $\cC$ and show that
$\Cobar(\cC)$-algebras form a $\tLie$-enriched category
$\HoAlg_{\cC}$, where $\tLie$ is the enhancement of the symmetric
monoidal category of shifted $L_{\infty}$-algebras introduced
in\footnote{See also Section \ref{sec:tLie} of this paper.} \cite{EnhancedLie}.
Then, using a generalization of the Getzler-Hinich construction
\cite{Ezra-infty}, \cite{Hinich:1997}, we show that the
$\tLie$-category of $\Cobar(\cC)$-algebras can be ``integrated'' to
the category $\HoAlg^{\sD}_{\cC}$ enriched over $\infty$-groupoids
(a.k.a. Kan complexes).

Our approach is conceptually similar to a standard construction of the
simplicial category of chain complexes i.e., ``trivial'' homotopy
algebras. There, instead of constructing the simplicial mapping
spaces directly, one exploits the simple fact that chain complexes
are naturally enriched over chain complexes i.e., abelian
$L_{\infty}$-algebras. The simplicial category is then constructed
by first truncating the mapping complexes and applying the
Dold-Kan functor. It has been noted, for example by E.\ Getzler in \cite{Ezra-infty}, that the
integration of a $L_{\infty}$-algebra is, up to homotopy, a non-abelian
analogue of the Dold-Kan functor. Hence,
to construct the simplicial category $\HoAlg^{\sD}_{\cC}$ of non-trivial homotopy algebras,
we proceed via analogy by first considering a category of homotopy algebras enriched
over $L_{\infty}$-algebras i.e., ``non-abelian complexes''.

Furthermore, we prove that the homotopy category of
$\HoAlg^{\sD}_{\cC}$ {\it is} the localization of the category of
$\Cobar(\cC)$-algebras with respect to $\infty$ quasi-isomorphisms, and
this is indeed the correct homotopy category of homotopy algebras (see
for example \cite[Thm.\ 11.4.12]{LV}).
Thus the simplicial category $\HoAlg^{\sD}_{\cC}$ is the sought higher
categorical structure which stands behind the homotopy category of
homotopy algebras.

For simplicity of the exposition, we present all constructions in the ``1-colored'' setting.
However, it is not hard to see the all the statements can be easily extended to the multi-colored
setting.

Our paper is organized as follows. Section \ref{sec:prereq-HA} is
devoted to prerequisites on homotopy algebras. In this section,
we describe the convolution $L_{\infty}$-algebra $\Conv(V,A)$ associated to
a $\cC$-coalgebra $V$ and a $\Cobar(\cC)$-algebra $A$, and recall \cite{DotsPoncin}
that, for every pair of $\Cobar(\cC)$-algebras $A,B$, Maurer-Cartan (MC)
elements of $\Conv(\cC(A), B)$ are in bijection with $\infty$-morphisms
from $A$ to $B$.  Section \ref{sec:HoAlg-cC} starts with a brief reminder of
the symmetric monoidal category $\tLie$ introduced in \cite{EnhancedLie}.
Then we construct the
$\tLie$-enriched category $\HoAlg_{\cC}$ and the corresponding simplicial
category $\HoAlg^{\sD}_{\cC}$.  In Section \ref{sec:HT}, we show that
$\pi_0 \big( \HoAlg^{\sD}_{\cC} \big)$ is the homotopy category of
$\Cobar(\cC)$-algebras, i.e. every $\infty$ quasi-isomorphism of
$\Cobar(\cC)$-algebras gives an invertible morphism in $\pi_0 \big( \HoAlg^{\sD}_{\cC} \big)$
and the category $\pi_0 \big( \HoAlg^{\sD}_{\cC} \big)$ has the desired universal
property.  Also in Section \ref{sec:HT}, we characterize $\infty$
  quasi-isomorphisms as precisely those morphisms which
  induce homotopy equivalences between mapping spaces via (pre)composition.
Finally, in Section \ref{sec:HTT}, we give a very concise proof
of the Homotopy Transfer Theorem for homotopy algebras. This proof is
based on a construction from \cite{DefHomotInvar} and a version of the
Goldman-Millson theorem from \cite{GMtheorem}.

\paragraph{On a more general definition of a homotopy algebra.}
One could also define a homotopy algebra as an algebra over
an operad $\cO$ which is freely generated by a collection $\cC$
when viewed as an operad in the category of graded vector spaces,
where the collection $\cC$ and the differential on $\cO$ are subject
to some technical conditions. It is not hard to generalize all the constructions of
our paper to this more general setting. However, for simplicity of
the exposition, we decided to present the whole story for the
case when $\cO$ is the cobar construction applied to a fixed
cooperad.

\paragraph{Is the $\tLie$-enriched category of $L_{\infty}$-algebras
related to the rational homotopy theory?} It is not surprising that the
answer to this question is ``yes''. However, a precise formulation of
the answer requires some technical conditions on $L_{\infty}$-algebras
and some amendments to the definition of the mapping space.
So separate note \cite{RHT} is devoted to this question.

\subsubsection*{Related work on this subject}
While this preprint was in preparation, paper \cite{KhPQ} appeared on arXiv.org.
In this paper, the authors constructed a quasi-category of Leibniz $\infty$-algebras also
using the Getzler-Hinich construction \cite{Ezra-infty}, \cite{Hinich:1997}.

Our paper agrees in spirit with treatise \cite{h-alg} by Jacob Lurie.
In this treatise, J. Lurie develops a systematic approach to algebraic
structures on objects in a fixed symmetric monoidal $\infty$-category.
This approach allows him to define $E_{\infty}$-ring as a commutative algebra object
in the $\infty$-category of spectra. Even though, the framework of the
presentation of \cite{h-alg} is very general, we could not find a particular
statement in the current version of draft \cite{h-alg} from which all statements
proved in our paper will follow. One of the reasons for this is that, in \cite{h-alg},
J. Lurie usually assumes the all the (co)chain complexes under consideration
satisfy some kind of boundedness condition whereas in our paper we
do not impose this assumption.

In paper \cite{Bruno-HT}, Bruno Vallette used the simplicial localization
methods of Dwyer and Kan \cite{DK} to upgrade the category of
$\Cobar(\cC)$-algebras with $\infty$-morphisms (for a cooperad $\cC$
satisfying some conditions) to a simplicial category (see Theorem 3.5 in \cite{Bruno-HT}).
The homotopy category of this simplicial category is isomorphic to that of
$\HoAlg^{\sD}_{\cC}$ and we believe that there is a more precise link between these
two simplicial categories.

~\\

\noindent
\textbf{Acknowledgements:} We would like to thank Thomas Willwacher
for useful discussions, and Jim Stasheff for helpful comments.
V.A.D. and C.L.R. acknowledge NSF grant DMS-1161867
for a partial support. C.L.R. also acknowledges support from the German Research Foundation
(Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the
University of G\"ottingen. Finally, we would like to thank an anonymous referee for reading carefully
our manuscript and for many helpful suggestions.

\subsubsection*{Notation and conventions}
A big part of our conventions is borrowed from \cite{EnhancedLie}.
For example, the ground field $\bbk$ has characteristic zero and
$\Ch_{\bbk}$ denotes the category of unbounded cochain complexes
of $\bbk$-vector spaces.  Any $\bbZ$-graded vector space $V$ is tacitly considered
as the cochain complex with the zero differential. Following
\cite{EnhancedLie}  we frequently use the ubiquitous
abbreviation ``dg'' (differential graded) to refer to algebraic objects
in $\Ch_{\bbk}$\,. The notation $\bs V$ (resp. by $\bsi V$) is reserved
for the suspension (resp. the
desuspension) of a cochain complex $V$\,, i.e.
$\big(\bs V\big)^{\bul} = V^{\bul-1}$ and $\big(\bs^{-1} V\big)^{\bul} = V^{\bul+1}$\,.
Finally, for a pair $V$, $W$ of $\bbZ$-graded vector spaces we denote by
$$
\Hom (V,W)
$$
the corresponding inner-hom object in the category of $\bbZ$-graded vector spaces.

Following \cite{EnhancedLie} we tacitly assume the Koszul sign rule. In particular,
$$
(-1)^{\ve(\si; v_1, \dots, v_m)}
$$
will always denote the sign factor corresponding to the permutation $\si \in S_m$ of
homogeneous vectors $v_1, v_2, \dots, v_m$. Namely,
\begin{equation}
\label{ve-si-vvv}
(-1)^{\ve(\si; v_1, \dots, v_m)} := \prod_{(i < j)} (-1)^{|v_i | |v_j|}\,,
\end{equation}
where the product is taken over all inversions $(i < j)$ of $\si \in S_m$.

For a finite group $G$ acting on a cochain complex
(or a graded vector space) $V$, we denote by
$$
V^G,  \qquad \textrm{ and } \qquad  V_G,
$$
respectively, the subcomplex of $G$-invariants in $V$ and
the quotient complex of $G$-coinvariants. Using the advantage of
the zero characteristic, we often identify $V_G$ with $V^G$
via this isomorphism
\begin{equation}
\label{coinvar-invar}
v ~\mapsto~ \sum_{g \in G} g (v) \maps V_G \to V^G\,.
\end{equation}

For a graded vector space (or a cochain complex) $V$
the notation $S(V)$ (resp. $\und{S}(V)$) is reserved for the
underlying vector space of the
symmetric algebra (resp. the truncated symmetric algebra) of $V$:
$$
S(V) = \bbk \oplus V \oplus S^2(V) \oplus S^3(V) \oplus \dots\,,
$$
$$
\und{S}(V) =  V \oplus S^2(V) \oplus S^3(V) \oplus \dots\,,
$$
where
$$
S^n(V) = \big( V^{\otimes_{\bbk}\, n} \big)_{S_n}\,.
$$

We denote by $\As$, $\Com$, $\Lie$ the operads governing
associative, commutative (and associative), and Lie algebras,
respectively. We set $\As(0) = \Com(0) =\bfzero$\,. In other words, algebras
over $\As$ and $\Com$ are non-unital. We denote by $\As_{\infty}$,
$\Com_{\infty}$, and $\Lie_{\infty}$ the dg operads which
govern homotopy versions of the corresponding algebras.
Furthermore, we denote by $\coAs$, $\coCom$, and $\coLie$,
the cooperads which are obtained from $\As$, $\Com$, and $\Lie$ respectively,
by taking the linear dual.

For an augmented operad $\cO$,
we denote by $\cO_{\c}$ the kernel of the augmentation.
Dually, for a coaugmented cooperad $\cC$
we denote by $\cC_{\c}$ the cokernel of the coaugmentation.
Recall that for every augmented operad $\cO$ (resp.  coaugmented cooperad $\cC$)
the collection  $\cO_{\c}$ (resp. $\cC_{\c}$) is naturally a pseudo-operad
(resp. pseudo-cooperad) in the sense of \cite[Section 3.2]{notes} (resp.
\cite[Section 3.4]{notes}).

For an operad (resp. a cooperad) $P$ and a cochain complex $V$ we denote by
$P(V)$ the free $P$-algebra (resp. the cofree\footnote{In this paper we only consider
nilpotent coalgebras.} $P$-coalgebra) generated by $V$:
\begin{equation}
\label{P-Schur-V}
P(V) : = \bigoplus_{n \ge 0} \Big( P(n) \otimes V^{\otimes \, n} \Big)_{S_n}\,.
\end{equation}
For example,
$$
\Com(V) = \und{S}(V) \qquad \textrm{and} \qquad \coCom(V) = \und{S}(V)\,.
$$

For a cooperad $\cC$, it is sometimes more convenient to
work with the cofree $\cC$-coalgebra defined as the direct sum
of the space of invariants (instead of coinvariants):
\begin{equation}
\label{cC-V-invar}
\bigoplus_{n \ge 0} \Big( \cC(n) \otimes V^{\otimes \, n} \Big)^{S_n}\,.
\end{equation}
For example, it is more natural to define a $\cC$-coalgebra
structure on a graded vector space (or a cochain complex) $V$ as
a collection of comultiplication maps
\begin{equation}
\label{Delta-n}
\D_n : V \to \Big( \cC(n) \otimes V^{\otimes \, n} \Big)^{S_n}
\end{equation}
satisfying some natural coassociativity axioms which are
obtained by dualizing the corresponding axioms for algebras
over an operad.

We denote this direct sum by
\begin{equation}
\label{cC-V-inv}
\cC(V)^{\inv} : = \bigoplus_{n \ge 0} \Big( \cC(n) \otimes V^{\otimes \, n} \Big)^{S_n}
\end{equation}
and keep in mind that $\cC(V)^{\inv}$ is isomorphic to $\cC(V)$ via map \eqref{coinvar-invar}.

Following \cite{EnhancedLie}, $\mS$ and $\mS^{-1}$ denote the underlying
collections of the operads
$$
\End_{\bsi \bbk}\,, \qquad \textrm{and} \qquad \End_{\bs \bbk}\,,
$$
respectively. Furthermore, for a dg (co)operad $P$, we denote
by $\mS P$ (resp. $\mS^{-1} P$) the dg (co)operad which is obtained from
$P$ by tensoring with $\mS$ (resp. $\mS^{-1}$):
$$
\mS P : = \mS \otimes P\,, \qquad
\mS^{-1} P : = \mS^{-1} \otimes P\,.
$$
For example, $\sLie$-algebras are algebra over the dg operad
\begin{equation}
\label{sLie-oper}
\sLie : =\Cobar(\coCom)
\end{equation}
and $L_{\infty}$-algebras are algebras over the dg operad
\begin{equation}
\label{Lie-inf-oper}
\Lie_{\infty} : =\Cobar(\mS^{-1} \coCom)\,.
\end{equation}

Just as in \cite{EnhancedLie},
we often call $\sLie$-algebras {\it shifted $L_{\infty}$-algebras}.
Although a $\sLie$-algebra structure on a cochain complex $V$ is the same thing
as an $L_{\infty}$ structure on $\bs V$\,, working with $\sLie$-algebras has
important technical advantages. This is why we prefer to deal with shifted
$L_{\infty}$ structures on $V$ versus original $L_{\infty}$ structures on $\bs V$.

Following \cite{EnhancedLie}, we denote the tensor product of $\infty$-morphisms
of $\sLie$-algebras by $\otimes$ even though the tensor product of the
$\sLie$-algebras is denoted by $\oplus$.

We often use the plain arrow $\to$ for $\infty$-morphisms of homotopy algebras.
Of course, it should be kept in mind that in general such morphisms are maps of
the corresponding coalgebras but not the underlying cochain complexes.

The abbreviation ``MC'' is reserved for the term ``Maurer-Cartan''.

\paragraph{Conventions about trees.} By a {\it tree} we mean
a connected graph without cycles with a marked vertex
called {\it the root}.  In this paper, we assume that the root of
every tree has valency $1$ (such trees are sometimes called  {\it planted}).
The edge adjacent to the root is called the {\it root edge}.
Non-root vertices of valency $1$ are
called {\it leaves}.  A vertex is called {\it internal} if it is
neither a root nor a leaf. We always orient trees in the
direction towards the root. Thus every internal vertex
has at least $1$ incoming edge and exactly $1$ outgoing edge.
An edge adjacent to a leaf is called {\it external}.
A tree $\bt$ is called {\it planar} if, for every internal vertex $v$ of $\bt$, the set
of edges terminating at $v$ carries a total order.

Let us recall \cite[Section 2]{notes} that for every  planar tree $\bt$
the set $V(\bt)$ of all its vertices is equipped with a natural total order
such that the root is the smallest vertex of the tree.

For a non-negative integer $n$, an $n$-labeled planar tree $\bt$ is
a planar tree equipped with
an injective map
\begin{equation}
\label{labeling}
\ml : \{1,2, \dots, n\} \to L(\bt)
\end{equation}
from the set $\{1,2, \dots, n\}$ to the set $L(\bt)$ of leaves of $\bt$\,.
Although the set $L(\bt)$ has a natural total order we do not require
that map \eqref{labeling} is monotonous.

The set $L(\bt)$ of leaves of an $n$-labeled planar tree $\bt$
splits into the disjoint union of the image $\ml(\{1,2, \dots, n\})$
and its complement. We call leaves in the image of $\ml$
{\it labeled}.

A vertex $x$ of an $n$-labeled planar tree $\bt$ is called
{\it nodal} if it is neither the root, nor a labeled leaf.
We denote by $V_{\nod}(\bt)$ the set of all nodal vertices of
$\bt$. Keeping in mind the canonical total order on
the set of all vertices of $\bt$ we can say things like
``the first nodal vertex'', ``the second nodal vertex'', and
``the $i$-th nodal vertex''.

It is convenient to talk about (co)operads and pseudo-(co)operads
using the groupoid  $\Tree(n)$ of $n$-labeled planar trees.
Objects of $\Tree(n)$ are $n$-labeled planar trees and
morphisms are \und{non-planar} isomorphisms of the corresponding
(non-planar) trees compatible with labeling.

Following \cite[Section 3.2, 3.4]{notes}, for an $n$-labelled planar tree $\bt$ and
pseudo-operad $P$ (resp. pseudo-cooperad $Q$) the notation
$\mu_{\bt}$ (resp. the notation $\D_{\bt}$) is reserved for the multiplication
map
\begin{equation}
\label{mu-bt}
\mu_{\bt} :  P (r_1) \otimes P(r_2) \otimes \dots \otimes P(r_k) \to P(n)
\end{equation}
and the comultiplication map
\begin{equation}
\label{Delta-bt}
\D_{\bt} : Q(n) \to Q(r_1) \otimes Q(r_2) \otimes \dots \otimes Q(r_k)\,.
\end{equation}
respectively. Here, $k$ is the number of nodal vertices of the planar tree $\bt$
and $r_i$ is the number of edges (of $\bt$) which terminate at the
$i$-th nodal vertex of $\bt$\,.

For example, if $\bt_{n,k,i}$ is the labeled planar tree shown
on figure\footnote{On figures, small white circles are used for
nodal vertices and small black circles are used for
all the remaining vertices.} \ref{fig:bt-n-k-i} then the map
\begin{equation}
\label{mu-bt-nki}
\mu_{\bt_{n,k,i}} :  P (n) \otimes P(k) \to P(n+k-1)
\end{equation}
is precisely the $i$-th elementary insertion
\begin{equation}
\label{circ-i}
\circ_i :   P (n) \otimes P(k) \to P(n+k-1)\,.
\end{equation}
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5]
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=3, inner sep=1]

\node[b] (v1) at (0, 2) {};
\draw (0,2.6) node[anchor=center] {{\small $1$}};
\node[b] (v2) at (1.5, 2) {};
\draw (1.5,2.6) node[anchor=center] {{\small $2$}};
\draw (2.8,2) node[anchor=center] {{\small $\dots$}};
\node[b] (v1i) at (4, 2) {};
\draw (4,2.6) node[anchor=center] {{\small $i-1$}};
\node[w] (vv) at (6, 2) {};

\node[b] (vi) at (4.5, 4) {};
\draw (4.5,4.6) node[anchor=center] {{\small $i$}};
\draw (5.8,4) node[anchor=center] {{\small $\dots$}};
\node[b] (v1ik) at (7, 4) {};
\draw (7.6,4.6) node[anchor=center] {{\small $i+k-1$}};

\node[b] (vik) at (8, 2) {};
\draw (8,2.6) node[anchor=center] {{\small $i+k$}};
\draw (10,2) node[anchor=center] {{\small $\dots$}};
\node[b] (v1nk) at (12, 2) {};
\draw (12.6,2.6) node[anchor=center] {{\small $n+k-1$}};

\node[w] (v) at (6, 0.5) {};
\node[b] (r) at (6, -0.5) {};

\draw (vv) edge (vi);
\draw (vv) edge (v1ik);
\draw (v) edge (v1);
\draw (v) edge (v2);
\draw (v) edge (v1i);
\draw (v) edge (vv);
\draw (v) edge (vik);
\draw (v) edge (v1nk);
\draw (r) edge (v);
\end{tikzpicture}
\caption{The $(n+k-1)$-labeled planar tree $\bt_{n,k,i}$}
\label{fig:bt-n-k-i}
\end{figure}

Recall that a (co)operad $P$ is {\it reduced} if it
satisfies this technical condition
\begin{equation}
\label{reduced}
P(0)= \bfzero\,.
\end{equation}

When we deal with reduced (co)operads, we may discard all labeled trees which have
at least one nodal vertex with no incoming edges. In other words, one
 may safely assume that nodal vertices of a labeled tree
are precisely its internal vertices, i.e. map \eqref{labeling} is a bijection.

\section{Prerequisites on homotopy algebras}
\label{sec:prereq-HA}

Let $\cC$ be a dg coaugmented cooperad satisfying the following technical condition:
\begin{cond}
\label{cond:cC-circ-filtered}
The pseudo-cooperad $\cC_{\c}$ carries an ascending filtration
\begin{equation}
\label{cC-circ-filtr}
\bfzero =  \cF^0 \cC_{\c} \subset \cF^1 \cC_{\c} \subset  \cF^2 \cC_{\c} \subset  \cF^3 \cC_{\c} \subset \dots
\end{equation}
which is compatible with the  pseudo-cooperad  structure in the following sense:
\begin{equation}
\label{D-bt-filtr}
\D_{\bt} \big( \cF^m \cC_{\c} (n) \big) \subset
\bigoplus_{m_1 + \dots + m_k  = m}
\cF^{m_1} \cC_{\c} (r_1) \otimes  \cF^{m_2} \cC_{\c} (r_2) \otimes  \dots
\otimes  \cF^{m_k} \cC_{\c} (r_k)\,,
\end{equation}
$$
 \bt \in \Tree(n)\,,
$$
where $k$ is the number of nodal vertices of the planar tree $\bt$
and $r_i$ is the number of edges (of $\bt$) which terminate at the
$i$-th nodal vertex of $\bt$\,. We also assume that $\cC_{\c}$ is cocomplete
with respect to filtration \eqref{cC-circ-filtr}, i.e.
\begin{equation}
\label{cocomplete}
\cC_{\c} = \bigcup_{m} \cF^m \cC_{\c}\,.
\end{equation}
\end{cond}

We will use the following pedestrian definition of
homotopy algebras of a given type:
\begin{defi}
\label{dfn:infinity}
Homotopy algebras of type $\cC$ are
algebras in $\Ch_{\bbk}$ over the operad
$\Cobar(\cC)$\,.
\end{defi}
Thus, $A_{\infty}$-, $L_{\infty}$-, and
$\Com_{\infty}$-algebras are examples of
homotopy algebras. Indeed,  $A_{\infty}$-algebras
are algebras over the operad $\Cobar(\mS^{-1} \coAs)$,
$L_{\infty}$-algebras are algebras over \\ $\Cobar(\mS^{-1}\coCom)$,
and $\Com_{\infty}$-algebras are algebras over $\Cobar(\mS^{-1} \coLie)$\,.

Let $A$ be a cochain complex and let
\begin{equation}
\label{coDer-cC-A}
\coDer\big( \cC(A) \big)
\end{equation}
be the dg Lie algebra of coderivations of the $\cC$-coalgebra $\cC(A)$\,.

It is known \cite[Corollary 5.3]{notes}, \cite[Proposition 2.15]{GJ},
that $\Cobar(\cC)$-algebra structures on $A$ are in bijection
with MC elements of the dg Lie subalgebra
\begin{equation}
\label{coDer-pr-cC-A}
\coDer'\big( \cC(A) \big)  \subset \coDer\big( \cC(A) \big)
\end{equation}
of coderivations $Q$ satisfying the condition
\begin{equation}
\label{coder-condition}
Q \Big|_{A} = 0\,.
\end{equation}

Thus every homotopy algebra $A$ of type $\cC$ gives
us a (dg) $\cC$-coalgebra
\begin{equation}
\label{cC}
\Big( \cC(A), \pa + Q  \Big)
\end{equation}
where $\pa$ is the differential on $\cC(A)$ induced
by the ones on $A$ and $\cC$\,.

This observation is used to define a notion of
$\infty$-morphism of homotopy algebras of type $\cC$\,.
In other words,
\begin{defi}
\label{dfn:morphism}
Let $A$ and $B$ be homotopy algebras of type $\cC$
and let $Q_A$ (resp. $Q_B$) be the MC element of
$\coDer(\cC(A))$ (resp. $\coDer(\cC(B))$) corresponding
to the homotopy algebra structure on $A$ (resp. on $B$).
Then, an $\infty$-morphism from $A$ to $B$ is
a homomorphism of dg $\cC$-coalgebras
\begin{equation}
\label{U-morph}
U ~:~ \Big( \cC(A), \pa + Q_A  \Big) ~\to~ \Big( \cC(B), \pa + Q_B  \Big)\,.
\end{equation}
\end{defi}
Recall that any such homomorphism $U$ is uniquely determined
by its composition
\begin{equation}
\label{U-pr}
U' : = p_B \circ U
\end{equation}
with the canonical projection $p_B : \cC(B) \to B$.
In this paper, we often use this convention: {\it $U'$ denotes composition
\eqref{U-pr} corresponding to a homomorphism of
$\cC$-coalgebras $U$ \eqref{U-morph}.}

Given an $\infty$-morphism $U$ from $A_1$ to $A_2$ and
an $\infty$-morphism $\ti{U}$ from $A_2$ to $A_3$, their composition
is defined, in the obvious way, as the composition of the corresponding
homomorphisms of dg $\cC$-coalgebras. We denote by
$$
\Cat_{\cC}
$$
the category whose objects are homotopy algebras of type $\cC$ (a.k.a. $\Cobar(\cC)$-algebras)
and whose morphisms are  $\infty$-morphisms with the above obvious composition.

\begin{remark}
\label{rem:why-homot-alg}
Due to \cite[Proposition 38]{MVnado11}, Condition \ref{cond:cC-circ-filtered}
implies that $\Cobar(\cC)$ is a cofibrant object in the closed model category
of dg operads. The same condition also guarantees
that homotopy algebras of type $\cC$ enjoy the obvious version of
the Homotopy Transfer Theorem (see \cite[Theorem 10.3.2]{LV}).
A very concise proof of this important theorem in
given in Section \ref{sec:HTT} of this paper.
\end{remark}

\begin{remark}
\label{rem:Markl}
Another way to define the notion of $\infty$-morphism
of homotopy algebras is to resolve a $2$-colored operad which
governs pairs of algebras with a morphism between them.
This different approach to the ``higher category'' of
homotopy algebras is initiated in works \cite{Doubek}, \cite{Markl-h-alg}
of M. Doubek and M. Markl.
\end{remark}

\subsection{The convolution $\sLie$-algebra}

Let $V$ be a $\cC$-coalgebra and  $A$ be a homotopy
algebra of type $\cC$ (i.e. an algebra over $\Cobar(\cC)$)\,.

On the graded vector space
\begin{equation}
\label{Conv-V-A}
\Hom(V, A)
\end{equation}
we define the following multi-brackets:
\begin{equation}
\label{1-bracket}
\{f\}(v) = \pa_{A}\, f(v) -(-1)^{|f|} f(\pa_{V}\, v)  +  p_A  \circ Q_A\big( 1 \otimes f(\D_1(v)) \big)
\end{equation}
\begin{equation}
\label{m-bracket}
\{f_1, \dots, f_m\} (v) =
p_A \circ Q_A \big( 1 \otimes f_{1}\otimes \dots \otimes f_{m}
(\D_m (v))  \big)\,, \qquad m \ge 2\,,
\end{equation}
where $\D_m$ is the $m$-th component of the comultiplication
$$
\D_m : V \to \Big( \cC(m) \otimes V^{\otimes\, m} \Big)^{S_m}
$$
and $p_A$ is the canonical projection
$$
p_A : \cC(A) \to A\,.
$$

Note that, since $Q_A$ has degree $1$, each multi-bracket in \eqref{m-bracket}
also carries degree $1$\,.

We claim that
\begin{prop}
\label{prop:Conv-V-A}
For every $\cC$-coalgebra $V$ and a $\Cobar(\cC)$-algebra $A$,
multi-brackets \eqref{1-bracket}, \eqref{m-bracket} equip the graded vector
space  $\Hom(V, A)$ with a structure of a $\sLie$-algebra.
\end{prop}
The proof of this proposition is given in Appendix \ref{app:proof-Conv}.

\begin{defi}
\label{dfn:Conv}
Let $V$ be a $\cC$-coalgebra and  $A$ be a homotopy
algebra of type $\cC$\,. Then $\sLie$-algebra
\eqref{Conv-V-A} is called the convolution algebra of the pair
$(V, A)$\,. We use the notation:
$$
\Conv(V,A) : = \Hom(V, A)\,.
$$
\end{defi}

\subsubsection{Convolution $\sLie$-algebra and $\infty$-morphisms}

For a pair $A$, $B$ of  homotopy algebras of type $\cC$, we
consider the convolution $\sLie$-algebra
\begin{equation}
\label{Conv-cCA-B}
L = \Hom(\cC(A), B)\,,
\end{equation}
where the $\cC$-coalgebra $\cC(A)$ is considered with
the differential $\pa + Q_{A}$\,, and $\pa$ comes from the
differential on $A$ and the differential on $\cC$.

We observe that the $\sLie$-algebra carries the following
descending filtration
\begin{equation}
\label{Conv-filter}
\begin{split}
\cF^{\ari}_0 L & \supset \cF^{\ari}_{1} L \supset \cF^{\ari}_{2}L \supset \cdots  \\[0.3cm]
\cF^{\ari}_{n}L & = \{ f  \in \Hom(\cC(A), B) ~\vert ~
f \big|_{\cC(m) \otimes_{S_m} A^{\otimes \, m}} =0
~~ \forall ~ m < n  \}.
\end{split}
\end{equation}

It is also easy to check that:
\begin{prop}
\label{prop:filtered}
The convolution $\sLie$-algebra structure given by \eqref{1-bracket} and \eqref{m-bracket}
is compatible with filtration \eqref{Conv-filter} i.e.\
\[
\Bigl \{\cF^{\ari}_{i_{1}}L,\cF^{\ari}_{i_{2}}L,\ldots,\cF^{\ari}_{i_{k}}L \Bigr \} \subseteq
\cF^{\ari}_{i_{1} + i_{2} + \cdots + i_{k}} L \quad \forall~~ k >1,
\]
Moreover, the $\sLie$-algebra $L = \Hom(\cC(A), B)$ is complete
with respect to this filtration, i.e.
$$
L =  \varprojlim_{k} L / \cF^{\ari}_{k}L\,.
$$
\qed
\end{prop}

The notion of the convolution $\sLie$-algebra is partially justified by
the following lemma:
\begin{lem}
\label{lem:infty-morph}
Let $A$ and $B$ be homotopy algebras of type $\cC$. If the cooperad
$\cC$ satisfies condition
\begin{equation}
\label{cC-reduced}
\cC(0) = \bfzero
\end{equation}
then
\begin{equation}
\label{Conv-AB-pronilp}
\Hom(\cC(A), B) = \cF^{\ari}_1 \Hom(\cC(A), B)\,.
\end{equation}
In particular, the $\sLie$-algebra $\Hom(\cC(A), B)$ is pro-nilpotent.
Furthermore, the assignment
$$
U ~\mapsto~ U' : = p_{B} \circ U
$$
is a bijection between the set of $\infty$-morphisms from $A$ to $B$ and
the set of MC elements of the $\sLie$-algebra $\Hom(\cC(A), B)$\,.
\end{lem}
\begin{proof}
The first statement of the lemma follows directly from
the definition of filtration \eqref{Conv-filter} on the
$\sLie$-algebra $\Hom(\cC(A), B)$. Since  $\Hom(\cC(A), B)$ is complete
with respect to this filtration, we conclude that the $\sLie$-algebra $\Hom(\cC(A), B)$
is pronilpotent.

This conclusion allows us to write the MC equation
in the $\sLie$-algebra $\Hom(\cC(A), B)$ for any degree zero element.

According to Definition \ref{dfn:morphism}, an  $\infty$-morphism from
$A$ to $B$ is a homomorphism of dg $\cC$-coalgebras
$$
U ~:~ \Big( \cC(A), \pa + Q_A  \Big) ~\to~ \Big( \cC(B), \pa + Q_B  \Big)\,.
$$
Since the $\cC$-coalgebra $\cC(B)$ is cofree (over graded vector spaces),
the homomorphism $U$ is uniquely determined by its composition
\begin{equation}
\label{pB-U}
U' : = p_{B} \circ U :   \cC(A) \to B\,.
\end{equation}
Furthermore, the compatibility of $U$ with the differentials $\pa  + Q_A$
and $\pa + Q_B$ is equivalent to
the equation
\begin{multline}
\label{U-and-diffs}
\pa \circ U' (X; a_1, \dots, a_m) \\
 + Q_B \circ (1 \otimes U') \circ \D_1 (X; a_1, \dots, a_m) -  U' \circ (\pa + Q_A) (X; a_1, \dots, a_m)    \\
+ \sum_{k=2}^{\infty} \frac{1}{k!} Q_B \circ (1 \otimes (U')^{\otimes k})
\circ \D_k (X; a_1, \dots, a_m) = 0\,,
\end{multline}
where $(X; a_1, \dots, a_m)$ represents a vector in $\cC(A)$ and the factor
$1/k!$ in the last sum comes from the identification of $\cC(B)^{\inv}$ with
$\cC(B)$ via the inverse of isomorphism \eqref{coinvar-invar}.

Using the definition of multi-brackets \eqref{1-bracket}, \eqref{m-bracket} on
$\Hom(\cC(A), B)$, we see that \eqref{U-and-diffs} is precisely the MC equation
for $U'$ in the $\sLie$-algebra $\Hom(\cC(A), B)$\,.
Thus
$$
U ~~ \leftrightarrow ~~ U' : =  p_{B} \circ U
$$
is a desired bijection between the set of $\infty$-morphisms from $A$ to $B$
and the set of MC elements of the $\sLie$-algebra $\Hom(\cC(A), B)$\,.
\end{proof}
\begin{remark}
\label{rem:Dots-Poncin}
A version of Lemma \ref{lem:infty-morph} is proved in \cite{DotsPoncin}.
See Proposition 3 in \cite[Section 1.3]{DotsPoncin}.
\end{remark}
\begin{example}
\label{exam:A-infinity}
Let us recall that $\coAs$ is the cooperad which governs coassociative coalgebras
without counit and the dg operad $\Cobar(\mS^{-1} \coAs)$ governs (flat) $A_{\infty}$-algebras.
It is easy to see that, for every $A_{\infty}$-algebra $A$,
\begin{equation}
\label{Conv-cC-A-A}
\Hom(\mS^{-1} \coAs(A),  A )
\end{equation}
is the completed version of the truncated Hochschild cochain complex of $A$
\begin{equation}
\label{Hoch-A}
\prod_{n \ge 1} \bs^{n-1} \Hom(A^{\otimes \, n}, A)
\end{equation}
and the shifted $L_{\infty}$-algebra on \eqref{Hoch-A} is obtain by
symmetrizing the cup product and its higher analogues. It is worthy of mentioning that
the Hochschild differential on \eqref{Hoch-A} is obtained by twisting the differential
on \eqref{Conv-cC-A-A} by the MC element corresponding to the identity map $\id: A \to A$.
\end{example}

\section{The $\tLie$-enriched category $\HoAlg_{\cC}$ and the simplicial category  $\HoAlg^{\sD}_{\cC}$}
\label{sec:HoAlg-cC}

\subsection{A brief reminder of the symmetric monoidal category $\tLie$}
\label{sec:tLie}

Let us recall \cite{EnhancedLie} that a $\sLie$-algebra $(L, \pa, \{\cdot,\cdot\},
\{\cdot,\cdot,\cdot\},\ldots)$ is \emph{filtered} if the underlying
complex $(L, \pa)$ is equipped with a complete descending filtration,
\begin{equation}
\label{filtr-L}
L = \cF_{1}L \supset \cF_{2}L \supset  \cF_{3}L  \cdots
\end{equation}
\begin{equation}
\label{L-complete}
L =\varprojlim_{k} L/\cF_{k}L\,,
\end{equation}
which is compatible with the brackets, i.e.
\[
\Bigl \{\cF_{i_{1}}L,\cF_{i_{2}}L,\ldots,\cF_{i_{m}}L \Bigr \} \subseteq
\cF_{i_{1} + i_{2} + \cdots + i_{m}} L \quad \forall ~~ m >1.
\]

For example, if $A$ and $B$ are $\Cobar(\cC)$-algebras and the cooperad
$\cC$ satisfies the condition $\cC(0) = \bfzero$, then the filtration ``by arity''
\eqref{Conv-filter} on $\Hom(\cC(A), B)$ satisfies the above conditions.
In other words,  $\Hom(\cC(A), B)$ is a filtered $\sLie$-algebra.

The filtration \eqref{filtr-L} induces a natural descending
filtration and hence a topology on $\und{S}(L)$. Just as in \cite{EnhancedLie},
we tacitly assume that $\infty$-morphisms of filtered $\sLie$-algebras
are continuous with respect to this topology.

Let us also recall \cite{EnhancedLie} that an \emph{enhanced morphism}
\[
L_1 \xto{(\al, F)} L_2
\]
between filtered $\sLie$-algebras is a pair consisting of a MC element
$\alpha \in L_2$ and a (continuous) $\infty$-morphism
$F \maps L_1 \to L^{\alpha}_2$, where $L^{\alpha}_2$ is obtained from
$L_2$ via twisting\footnote{See \cite[Section 2]{EnhancedLie} for details on twisting
by MC elements.} by the MC element $\al$.
The composition of two enhanced morphisms $L_1 \xto{(\al_2, F)} L_2$
and $L_2 \xto{(\al_3, G)} L_3$ is the pair
\begin{equation}
\label{composition}
(\al_3 + G_*(\al_2), G^{\al_2} \circ F)\,,
\end{equation}
where the $\infty$-morphism $G^{\al_2}$ is obtained from
$G$ via twisting by the MC element $\al_2 \in L_2$.

Following \cite{EnhancedLie}, we denote by $\tLie$ the
category of filtered $\sLie$-algebras with the above enhanced
morphisms.

Given two filtered $\sLie$ algebras $(L, \{\cdot,\cdot\},
\{\cdot,\cdot, \cdot\},\ldots)$ and $(\wt{L}, \{\cdot,\cdot \}\wt{~},
\{\cdot,\cdot, \cdot\}\wt{~},\ldots)$, one obtains a
filtered $\sLie$ structure on the direct sum $L \oplus \wt{L}$
by setting
\[
\{x_1 + x'_1, x_2 + x'_2,\ldots, x_k + x'_k\} :=
\{x_1, x_2,\ldots, x_k\} + \{x'_1, x'_2,\ldots, x'_k\}\wt{~}\,,
\]
and
$$
\cF_k  (L \oplus \wt{L}) : = (\cF_k L) \oplus  (\cF_k \wt{L})\,.
$$

If $\al$ and $\wt{\al}$ are MC elements of $L$ and $\wt{L}$, respectively,
then $\al + \wt{\al} \in L \oplus \wt{L}$ is clearly a MC element of
the $\sLie$-algebra $L\oplus \wt{L}$. Furthermore, the operation of
twisting (by a MC element) is compatible with $\oplus$, i.e. the $\sLie$-algebra
$L^{\al} \oplus \wt{L}^{\wt{\al}}$ is canonically isomorphic to the $\sLie$-algebra
$(L\oplus \wt{L})^{\al + \wt{\al}}$\,.

Using these observations, we show, in \cite[Section 3.1]{EnhancedLie}, that
the assignment
$$
(L, \wt{L}) ~\mapsto~ L \oplus \wt{L}
$$
can be upgraded to a structure of a symmetric monoidal category on $\tLie$
with $\bfzero$ being the unit object.

\subsection{The $\tLie$-enriched category of $\Cobar(\cC)$-algebras}
Given two $\Cobar(\cC)$-algebras $A$ and $B$\,, we denote by
\begin{equation}
\label{Map-A-B}
\map(A, B) : =  \Hom(\cC(A), B)
\end{equation}
the convolution $\sLie$-algebra defined in Proposition \ref{prop:Conv-V-A}.

In this section, we construct a $\tLie$-enriched category \cite{EnhancedLie}, \cite{Kelly} $\HoAlg_{\cC}$
whose objects are homotopy algebras $A,B, \dots$ of type $\cC$ and whose
mapping spaces are $\sLie$-algebras \eqref{Map-A-B}.

We start with defining a degree $0$ linear map
\begin{equation}
\label{comp_map}
\begin{split}
\cU' : \und{S}\big( \Hom(\cC(A_2), A_3) \oplus  \Hom(\cC(A_1), A_2)  \big)
\to  \Hom(\cC(A_1), A_3)
\end{split}
\end{equation}
by using the identification
\begin{align*}
\und{S}\bigl( \Hom(\cC(A_2), A_3) \oplus \Hom(\cC(A_1), A_2)  \bigr)
 & = \und{S}\bigl(\Hom(\cC(A_2), A_3) \bigr) \, \oplus \,
 \und{S}\bigl(\Hom(\cC(A_1), A_2) \bigr )
\\  & \quad \oplus \,
\Big( \und{S}\bigl( \Hom(\cC(A_2), A_3) \bigr) \tensor \und{S}\bigl( \Hom(\cC(A_1), A_2) \bigr) \Big)
\end{align*}
and the formulas:
\begin{equation}
\label{cU-pr-nonzero}
\cU' \big( g \otimes (f_1 \dots f_n) \big) (X) =
g \big( 1 \otimes f_{1} \otimes \dots \otimes f_{n} \, (\D_n(X)) \big)\,,
\end{equation}
for all $X \in \cC(A_1)$, $g \in \Hom(\cC(A_2), A_3)$,
$f_1, \dots, f_n \in \Hom(\cC(A_1), A_2)$,
and
\begin{equation}
\label{cU-pr-zero}
\begin{array}{c}
\cU' \big|_{ \und{S}(\Hom(\cC(A_2), A_3)) } ~ = ~
\cU' \big|_{ \und{S}(\Hom(\cC(A_1), A_2)) }  ~ = ~ 0\,, \\[0.5cm]
\cU' \big|_{ \und{S}^{m \neq 1} ( \Hom(\cC(A_2), A_3) ) \otimes
\und{S} (\Hom(\cC(A_1), A_2) ) }   ~ = ~ 0\,.
\end{array}
\end{equation}

We claim that
\begin{prop}
\label{prop:comp}
The vector $\cU'$ defined by the above formulas
is a MC element of the $\sLie$-algebra
\begin{equation}
\label{Conv-for-composition}
\Hom \Big(\und{S} \big( \Hom(\cC(A_2), A_3) \oplus \Hom(\cC(A_1), A_2)
\big) \,,\,
 \Hom(\cC(A_1), A_3) \Big)\,.
\end{equation}
\end{prop}
\begin{proof}
Let us denote by $L_{ij}$ the $\sLie$-algebra $\Hom(\cC(A_i), A_j)$
and by $d_{ij}$ the differential on $L_{ij}$. We also denote by
$Q_{ij}$ the degree $1$ coderivation on the $\coCom$-coalgebra
\begin{equation}
\label{und-S-Lij}
\und{S} \big( L_{ij} \big)
\end{equation}
corresponding to the  $\sLie$-algebra $L_{ij}$\,.
By abuse of notation, $d_{ij}$ also denotes the differential on
\eqref{und-S-Lij} coming from the one on $L_{ij}$.

In terms of this notation, our goal is to prove that $\cU'$ satisfies
the equation\footnote{We sometimes use the subscript $m$ in $\{~,~,\dots, ~\}_m$
to denote the number of entries of the corresponding multi-bracket.}
\begin{equation}
\label{MC-U-pr}
d_{13} \circ \cU' - \cU' \circ \big( (d_{23} + Q_{23}) \otimes 1 + 1 \otimes (d_{12} +Q_{12})  \big) +
\sum_{m=2}^{\infty} \frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m = 0\,.
\end{equation}

We will present the most bulky part of the proof of \eqref{MC-U-pr}.
Namely, we will show in detail that the sum
\begin{equation}
\label{sum-UUU-brack}
\sum_{m=2}^{\infty} \frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m
\end{equation}
cancels with the term $-\cU' \circ (Q_{23} \otimes 1)$ in \eqref{MC-U-pr}.
The remaining cancellations are much more straightforward and
we leave them to the reader.

In the computations given below, we do not specify explicitly sign factors
coming from the Koszul sign rule. We address this
issue by a short comment at the end of the proof.

Let $g_1, \dots, g_k \in L_{23}$ and $f_1, \dots, f_n \in L_{12}$.
Due to \eqref{cU-pr-zero},
$$
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_k, f_1, \dots, f_n) = 0
$$
if $k \neq m$ or $n < k$\,.

Unfolding the term
\begin{equation}
\label{UUU-brack}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n)
\end{equation}
with $n \ge m$, we get
\begin{multline}
\label{UUU-brack-unfold}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) = \\
\sum_{\substack{k_1 + \dots + k_m = n \\[0.1cm] k_j \ge 1}}~
\sum_{\tau \in S_m}
\sum_{\si \in \Sh_{k_1, k_2, \dots, k_m}} \frac{\pm 1}{m!}
\big\{
\cU'(g_{\tau(1)}, f_{\si(1)}, \dots, f_{\si(k_1)}),
\cU'(g_{\tau(2)}, f_{\si(k_1+1)}, \dots, f_{\si(k_1+k_2)}),
\dots  \\[0.1cm]
\dots \cU'(g_{\tau(m)}, f_{\si(n-k_m+1)}, \dots, f_{\si(n)})  \big\}^{L_{13}}_m\,,
\end{multline}
where $\{~, ~, \dots, ~\}^{L_{13}}_m$ denotes the corresponding multi-bracket
on $L_{13}$ and the sign factors in the right hand side are determined by the Koszul rule.

Since  $\{~, ~, \dots, ~\}^{L_{13}}_m$ is (graded) symmetric in its argument, we can simplify
term \eqref{UUU-brack} further:
\begin{multline}
\label{UUU-brack-better}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) = \\
\sum_{\substack{k_1 + \dots + k_m = n \\[0.1cm] k_j \ge 1}}~
\sum_{\si \in \Sh_{k_1, k_2, \dots, k_m}} \pm
\big\{
\cU'(g_{1}, f_{\si(1)}, \dots, f_{\si(k_1)}),
\cU'(g_{2}, f_{\si(k_1+1)}, \dots, f_{\si(k_1+k_2)}),
\dots  \\[0.1cm]
\dots \cU'(g_{m}, f_{\si(n-k_m+1)}, \dots, f_{\si(n)})  \big\}^{L_{13}}_m\,.
\end{multline}

Let $X \in \cC(A_1)$ and
\begin{equation}
\label{Delta-X}
\D_m (X) = \sum_{\al} (\ga_{\al}; X_{\al,1}, X_{\al, 2}, \dots, X_{\al, m})  \in \big( \cC(m) \otimes \cC(A_1)^{\otimes\, m} \big)^{S_m}\,.
\end{equation}

Then, applying equation \eqref{UUU-brack-better} and using the definition of the
multi-bracket on $L_{13} = \Hom( \cC(A_1), A_3)$, we get
\begin{multline}
\label{UUU-brack-applied}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) (X) = \\[0.1cm]
\sum_{\al}
\sum_{\substack{k_1 + \dots + k_m = n \\[0.1cm] k_j \ge 1}}~
\sum_{\si \in \Sh_{k_1, k_2, \dots, k_m}}
\pm
Q_{A_3}
\big( \ga_{\al} ;
\cU'(g_{1}, f_{\si(1)}, \dots, f_{\si(k_1)}) (X_{\al,1}),   \\[0.1cm]
\cU'(g_{2}, f_{\si(k_1+1)}, \dots, f_{\si(k_1 + k_2)}) (X_{\al, 2}),
\dots,
\cU'(g_{m}, f_{\si(n-k_m+1)}, \dots, f_{\si(n)}) (X_{\al, m}) \big)\,,
\end{multline}
where $Q_{A_3}$ denotes the coderivation of $\cC(A_3)$ corresponding
to the $\Cobar(\cC)$-algebra structure on $A_3$.

Using \eqref{cU-pr-nonzero}, we deduce that
\begin{multline}
\label{UUU-brack1}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) (X) = \\[0.1cm]
\sum_{\substack{k_1 + \dots + k_m = n \\[0.1cm] k_j \ge 1}}~
\sum_{\si \in \Sh_{k_1, k_2, \dots, k_m}}
\pm Q_{A_3} \circ
(1 \otimes g_{1}\otimes f_{\si(1)} \otimes \dots \otimes f_{\si(k_1)}
\otimes \dots \otimes g_{m} \otimes
f_{\si(n-k_m+1)} \otimes \dots \otimes f_{\si(n)})  \\[0.1cm]
\circ (1 \otimes \D_{k_1} \otimes \D_{k_2} \otimes \dots \otimes \D_{k_m} ) \circ \D_m (X)\,.
\end{multline}

By the axioms of the $\cC$-coalgebra structure on $\cC(A_1)$, we have
\begin{equation}
\label{Delta-Delta-X}
(1 \otimes \D_{k_1} \otimes \D_{k_2} \otimes \dots \otimes \D_{k_m} ) \circ \D_m (X) =
\mb \circ (\D_{\bt^{\pf}_{k_1, \dots, k_m}} \otimes 1^{\otimes\, n} )  \circ  \D_n(X)\,,
\end{equation}
where $n = k_1 + k_2 + \dots + k_m$,
$\D_{\bt^{\pf}_{k_1, \dots, k_m}}$ is the cooperadic comultiplication
$$
\D_{\bt^{\pf}_{k_1, \dots, k_m}} : \cC(n) \to \cC(m) \otimes \cC(k_1) \otimes
\cC(k_2) \otimes \dots \otimes \cC(k_m)
$$
corresponding to the planar tree $\bt^{\pf}_{k_1, \dots, k_m}$ shown on
figure \ref{fig:fork-kkk}, and $\mb$ is the braiding isomorphism which
``changes the positions'' of tensor factors appropriately.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5]
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=3, inner sep=1]

\node[b] (l1) at (0, 8) {};
\draw (0,8.6) node[anchor=center] {{\small $1$}};
\draw (1.5,8) node[anchor=center] {{\small $\dots$}};
\node[b] (lk1) at (3, 8) {};
\draw (3,8.6) node[anchor=center] {{\small $k_1$}};
\node[w] (n2) at (1.5, 6) {};

\node[b] (lk11) at (6, 8) {};
\draw (6,8.6) node[anchor=center] {{\small $k_1+1$}};
\draw (7.5,8) node[anchor=center] {{\small $\dots$}};
\node[b] (lk1k2) at (9, 8) {};
\draw (9.2,8.6) node[anchor=center] {{\small $k_1+k_2$}};
\node[w] (n3) at (7.5, 6) {};

\draw (11.3,7) node[anchor=center] {{\small $\dots$}};

\node[b] (lnkm1) at (14, 8) {};
\draw (13.6,8.6) node[anchor=center] {{\small $n-k_m+1$}};
\draw (15.5,8) node[anchor=center] {{\small $\dots$}};
\node[b] (ln) at (17, 8) {};
\draw (17,8.6) node[anchor=center] {{\small $n$}};
\node[w] (nm1) at (15.5, 6) {};

\node[w] (n1) at (8.5, 3) {};
\node[b] (r) at (8.5, 2) {};

\draw (n2) edge (l1);
\draw (n2) edge (lk1);
\draw (n3) edge (lk11);
\draw (n3) edge (lk1k2);
\draw (nm1) edge (lnkm1);
\draw (nm1) edge (ln);
\draw (n1) edge (n2);
\draw (n1) edge (n3);
\draw (n1) edge (nm1);
\draw (r) edge (n1);
\end{tikzpicture}
\caption{The labeled planar tree $\bt^{\pf}_{k_1, \dots, k_m}$}
\label{fig:fork-kkk}
\end{figure}

Unfolding $\D_n (X)$
$$
\D_n(X) =  \sum_{\beta} (\ti{\ga}_{\beta}; \ti{X}_{\beta, 1}, \dots, \ti{X}_{\beta, n})
$$
and using the fact that $\D_n$ lands in the space of $S_n$-invariants, we rewrite
the expression $(\D_{\bt^{\pf}_{k_1, \dots, k_m}} \otimes 1^{\otimes\, n} )  \circ  \D_n(X)$
as follows:
\begin{multline}
\label{compute-Delta-Delta}
(\D_{\bt^{\pf}_{k_1, \dots, k_m}} \otimes 1^{\otimes\, n} )  \circ  \D_n(X) =  \\[0.1cm]
\sum_{\beta} \pm
(\D_{\bt^{\pf}_{k_1, \dots, k_m} } \otimes 1^{\otimes\, n} )
( \si^{-1} (\ti{\ga}_{\beta}); \ti{X}_{\beta, \si(1)}, \dots, \ti{X}_{\beta, \si(n)})
=   \\[0.1cm]
\sum_{\beta} \pm
(\D_{\si(\bt^{\pf}_{k_1, \dots, k_m}) } \otimes 1^{\otimes\, n} )
( \ti{\ga}_{\beta}; \ti{X}_{\beta, \si(1)}, \dots, \ti{X}_{\beta, \si(n)})
\end{multline}
for any $\si \in S_n$\,.

Combining this observation with \eqref{Delta-Delta-X}, we conclude that
\begin{multline}
\label{UUU-brack11}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) (X)  \\[0.1cm]
= \sum_{\beta}
\sum_{\substack{k_1 + \dots + k_m = n \\[0.1cm] k_j \ge 1}}~
\sum_{\si \in \Sh_{k_1, k_2, \dots, k_m}}
\pm Q_{A_3} \circ (1 \otimes g_1 \otimes \dots \otimes g_m)  \\[0.1cm] \circ
(\D_{\si(\bt^{\pf}_{k_1, \dots, k_m}) } \otimes 1^{\otimes\, n} )
( \ti{\ga}_{\beta}; f_{\si(1)}(\ti{X}_{\beta, \si(1)}), \dots, f_{\si(n)}(\ti{X}_{\beta, \si(n)})) \,.
\end{multline}

On the other hand,
\begin{multline}
\label{line-Delta-Delta}
\sum_{\beta}
\sum_{\substack{k_1 + \dots + k_m = n \\[0.1cm] k_j \ge 1}}~
\sum_{\si \in \Sh_{k_1, k_2, \dots, k_m}}
\pm (\D_{\si(\bt^{\pf}_{k_1, \dots, k_m}) } \otimes 1^{\otimes\, n} )
( \ti{\ga}_{\beta}; f_{\si(1)}(\ti{X}_{\beta, \si(1)}), \dots, f_{\si(n)}(\ti{X}_{\beta, \si(n)}))  \\[0.1cm]
= \D_m \Big(  \sum_{\beta} \pm (\ti{\ga}_{\beta}; f_{1}(\ti{X}_{\beta, 1}), \dots, f_n(\ti{X}_{\beta, n}))  \Big)\,.
\end{multline}

Therefore, by definition of the $\sLie$-structure on $L_{23} = \Hom(\cC(A_2), A_3)$, we have
\begin{multline}
\label{UUU-brack-OK}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) (X) = \\[0.1cm]
 \cU'  (\{g_1, \dots, g_m \}^{L_{23}}_m, f_1, \dots, f_n) \, (X)\,.
\end{multline}

Let us also observe that, due to  \eqref{cU-pr-zero},
$$
\big( \cU' \circ (Q_{23} \otimes 1) (g_1, \dots, g_m, f_1, \dots, f_n) \big) (X) =
\cU'  (\{g_1, \dots, g_m \}^{L_{23}}_m, f_1, \dots, f_n) (X)\,.
$$

Thus,
\begin{multline}
\label{UUU-brack-OK1}
\frac{1}{m!} \{\cU', \cU', \dots, \cU' \}_m (g_1, \dots, g_m, f_1, \dots, f_n) (X) = \\[0.1cm]
\big( \cU' \circ  (Q_{23} \otimes 1)  (g_1, \dots, g_m, f_1, \dots, f_n) \big) (X)
\end{multline}
which implies the desired cancellation of sum
\eqref{sum-UUU-brack} with the term $- \cU' \circ (Q_{23} \otimes 1)$
in \eqref{MC-U-pr}.

Let us now address the issue of sign factors.
The sign factor in front of the term
\begin{equation}
\label{term-UUU-brack}
Q_{A_3} \circ (1 \otimes g_1 \otimes \dots \otimes g_m)  \\[0.1cm] \circ
(\D_{\si(\bt^{\pf}_{k_1, \dots, k_m}) } \otimes 1^{\otimes\, n} )
( \ti{\ga}_{\beta}; f_{\si(1)}(\ti{X}_{\beta, \si(1)}), \dots, f_{\si(n)}(\ti{X}_{\beta, \si(n)}))
\end{equation}
in \eqref{UUU-brack11} comes from rearranging the homogeneous vectors
\begin{equation}
\label{stand-order}
f_1, f_2, \dots, f_n,  \ti{\ga}_{\beta}, \ti{X}_{\beta, 1}, \ti{X}_{\beta, 2},
\dots \ti{X}_{\beta, n}
\end{equation}
from their standard order in \eqref{stand-order} to the order in which they appear
in \eqref{term-UUU-brack}. It is easy to see that we get the same sign factors
in front of the corresponding terms, when we unfold the right hand
side of \eqref{UUU-brack-OK}.

Proposition \ref{prop:comp} is proved.
\end{proof}

Combining Lemma \ref{lem:infty-morph} with Proposition \ref{prop:comp}
we deduce that
\begin{cor}
\label{cor:cU-intro}
The map $\cU'$ defined by equations \eqref{cU-pr-nonzero}
and \eqref{cU-pr-zero} lifts to an $\infty$-morphism
\begin{equation}
\label{cU-comp}
\cU \maps \map(A_2, A_3) \oplus \map(A_1, A_2) \to \map(A_1, A_3)\,.
\end{equation}
\qed
\end{cor}
\begin{remark}
\label{rem:U-pr-again}
We would like to remark that the map
$$
\cU'  \maps \und{S} \big( \Hom(\cC(A_2), A_3) \oplus  \Hom(\cC(A_1), A_2)  \big)
\to  \Hom(\cC(A_1), A_3),
$$
can be equivalently defined by the single formula:
\begin{equation}
\label{U-pr-again}
\cU' \bigl( (g_1 \oplus f_1),(g_2  \oplus f_2),\ldots,(g_n  \oplus f_n) \bigr) =
\sum_{i=1}^{n} \pm g_{i} \circ  \bigl(1 \tensor f_1 \tensor \cdots \tensor
\widehat{f_{i}} \tensor \cdots \tensor f_{n} \bigr) \circ \D_{n-1}
\end{equation}
where $(g_i \oplus f_i) \in \Hom(\cC(A_2), A_3)
\oplus  \Hom(\cC(A_1), A_2)$ and $\pm$ is the usual Koszul sign factor.
\end{remark}

The following theorem shows that the composition in
$\HoAlg_{\cC}$ given by $\infty$-morphism \eqref{cU-comp}
is associative.
\begin{thm}
\label{thm:comp-assoc}
Let $A_1, \dots, A_4$ be $\Cobar(\cC)$-algebras and let
$$
\cU_{i_1 i_2 i_3} \maps \map(A_{i_2} , A_{i_3}) \oplus \map(A_{i_1}, A_{i_2}) \to \map(A_{i_1}, A_{i_3})
$$
be the $\infty$-morphism given in Corollary \ref{cor:cU-intro}. Then the
following diagram
\begin{equation}
\label{pent}
\begin{tikzpicture}[ext/.style={rectangle, minimum size=4, inner sep=1}]
\node[ext] (123-4) at (0,1.5) {$\map(A_3, A_4) \oplus \big( \map(A_2,A_3) \oplus \map(A_1,A_2) \big)$};
\node[ext] (1-234) at (0,0) {$\big( \map(A_3, A_4) \oplus  \map(A_2,A_3) \big) \oplus \map(A_1,A_2)$};
\node[ext] (134) at (6, 3) {$\map(A_3, A_4) \oplus \map(A_1,A_3)$};
\node[ext] (124) at (6, -1.5) {$\map(A_2, A_4) \oplus \map(A_1,A_2)$};
\node[ext] (14) at (10, 0.75) {$\map(A_1, A_4) $};
\path[->,font=\scriptsize]
(0,1.1) edge  node[auto] {$\cong$} (0, 0.4)
(1,2)  edge  node[auto] {$~~\id \otimes \cU_{123}$} (3,2.8)
(1,-0.5) edge  node[below] {$ \cU_{234} \otimes \id~~$} (3,-1.3)
(8,2.5) edge  node[auto] {$ \cU_{134}$} (9.5,1.2)
(8,-1) edge  node[below] {$~~~~\cU_{124}$} (9.5,0.3) ;
\end{tikzpicture}
\end{equation}
commutes.
\end{thm}
\begin{proof}
Let  $h \in \map(A_3,A_4)$,   $g_{1}, \ldots, g_{m} \in \map(A_2,A_3)$,
and $f_{1}, \ldots, f_{n} \in \map(A_1,A_2)$\,. Composing the
lower arrows in \eqref{pent} with the canonical projection
\begin{equation}
\label{p-14}
p_{14} :  \und{S}(\map(A_1,A_4)) \to  \map(A_1, A_4),
\end{equation}
we get
\begin{equation}
\label{enrich_eq1}
\begin{split}
&p_{14}\circ  \cU_{124} \circ  (\cU_{234} \tensor \id )(h, g_1, \dots, g_m,
f_1, \dots, f_n ) =\\
& \quad  h
\Bigl( (1 \tensor g_1 \tensor \cdots \tensor g_{m})\circ  \Delta_{m} \circ
 (1 \tensor f_{1} \tensor \cdots \tensor f_{n}) \circ \Delta_{n} \Bigr)\,.
\end{split}
\end{equation}

Similarly, composing the upper arrows in \eqref{pent}
with canonical projection \eqref{p-14}, we get
\begin{equation}
\label{enrich_eq2}
\begin{split}
& p_{14} \circ \cU_{134} \circ (\id \tensor \, \cU_{123})(h, g_1, \dots, g_m, f_1, \dots, f_n) = \\
&\sum_{\substack{k_1+\cdots + k_{m} =n \\ \sigma \in
    \Sh(k_1,\ldots,k_m)}} \pm
h \Bigl ( 1 \tensor g_1(1 \tensor f_{\sigma(1)} \tensor
f_{\sigma(2)} \tensor   \cdots \tensor f_{\sigma(k_1)})
 \tensor  g_2(1 \tensor f_{\sigma(k_1 +1)} \tensor \cdots \tensor f_{\sigma(k_1+
  k_2)}) \tensor
 \cdots  \\
&\quad \quad \tensor g_m(1 \tensor f_{\sigma(n-k_m +1)} \tensor \cdots
 \tensor f_{\sigma(n)})  \circ (1 \tensor \Delta_{k_1} \tensor
 \Delta_{k_2} \tensor  \cdots \tensor \Delta_{k_m} ) \Bigr) \Delta_m,
\end{split}
\end{equation}
where the sign factors are determined by the Koszul rule.

Using equation \eqref{Delta-Delta-X}, computation \eqref{compute-Delta-Delta}, and
equation \eqref{line-Delta-Delta} from the proof of Proposition \ref{prop:comp},
we conclude that the left hand side of
\eqref{enrich_eq1} coincides with the left hand side of \eqref{enrich_eq2}.
Since any $\infty$-morphism to $\map(A_1, A_4)$ is uniquely determined by
its composition with projection \eqref{p-14}, we deduce that
$$
 \cU_{124} \circ  (\cU_{234} \tensor \id ) =  \cU_{134} \circ (\id \tensor\, \cU_{123})\,.
$$

Thus diagram \eqref{pent} is indeed commutative.
\end{proof}

\subsection{The proof of the unit axiom}

Let us recall that $\bfzero$ is the unit object in the category $\tLie$ and
observe that for every $\Cobar(\cC)$-algebra $A$ we have a canonical
enhanced morphism
\begin{equation}
\label{unit-morphism}
\bfzero ~\xto{(\id_A, \, 0)}~ \map(A,A)\,,
\end{equation}
where $\id_A$ is the MC element of $\map(A,A) = \Hom(\cC(A), A)$
corresponding to the identity $\infty$-morphism
$$
\id_{\cC(A)} : \cC(A) \to  \cC(A)
$$
and $0$ is the unique $\sLie$-morphism from $\bfzero $ to $\map(A,A)$.

We claim that
\begin{prop}
\label{prop:unit}
For every pair $A, B$ of homotopy algebras of type $\cC$,
the diagrams
\begin{equation}
\label{unit-right}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em]
{\map(A,B) \oplus \map(A,A) & \map(A,B)   ~ \\
 \map (A,B) \oplus \bfzero & ~ \\ };
\path[->, font=\scriptsize]
(m-1-1) edge node[above] {$\cU$} (m-1-2)
(m-2-1) edge  node[auto] {$ \id_{\map(A,B)} \otimes (\id_A, \, 0)$} (m-1-1)
edge (m-1-2);
\end{tikzpicture}
\end{equation}
\begin{equation}
\label{unit-left}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=3em, column sep=3em]
{\map(B,B) \oplus \map(A,B) & \map(A,B)   ~ \\
  \bfzero \oplus \map(A,B) & ~ \\ };
\path[->, font=\scriptsize]
(m-1-1) edge node[above] {$\cU$} (m-1-2)
(m-2-1) edge  node[auto] {$(\id_B, \, 0) \otimes  \id_{\map(A,B)}$} (m-1-1)
edge (m-1-2);
\end{tikzpicture}
\end{equation}
commute.
\end{prop}
\begin{proof}
Let us denote by
$$
\cK ~:~  \map (A,B) \oplus \bfzero ~\to~  \map (A,B)
$$
the composition of the vertical arrow and the horizontal arrow in  \eqref{unit-right}
and let $\cK'$ be the corresponding element in
$$
\Hom \big( \und{S}( \map (A,B) \oplus \bfzero),\, \map(A,B) \big)\,.
$$

Unfolding the definition of  composition of enhanced morphisms in $\tLie$ (see Proposition 3.4
in \cite{EnhancedLie}), we get that
\begin{equation}
\label{cK-pr}
\cK'(g_1, \dots, g_m) = \sum_{n \ge 0} \frac{1}{n!}
 \cU'\big( (g_1, \dots, g_m) \otimes \id^{n}_A \big)\,,
\end{equation}
where $g_1, \dots, g_m \in \map (A,B)$.

Hence, using \eqref{cU-pr-nonzero} and \eqref{cU-pr-zero} we deduce that
$$
\cK'(g_1, \dots, g_m) = 0 \qquad \forall ~~ m \ge 2
$$
and, for every $X \in \cC(A)$,
\begin{equation}
\label{cK-pr-unfold}
\cK'(g_1) (X) =  \sum_{n \ge 0} \frac{1}{n!} \,
g_1 \big( (1\otimes  \id^{n}_A)   \D_n(X) \big) =  g_1(X)\,,
\end{equation}
where the last equality follows from the identification
of $\cC(\cC(A) )^{\inv}$ and $\cC(\cC(A))$ via the inverse
of isomorphism \eqref{coinvar-invar}.

Thus diagram \eqref{unit-right} indeed commutes.

The proof of the commutativity of \eqref{unit-left} is easier.
So we leave it to the reader.
\end{proof}

\subsection{The simplicial category $\HoAlg_{\cC}^{\sD}$ of homotopy algebras}

Let $\Omega_{n}=\Omega^{\bullet}(\Delta^{n})$ denote the polynomial de
Rham complex on the $n$-simplex with coefficients in $\bbk$, and
$\{\Omega_{n} \}_{n \geq 0}$ the associated simplicial
dg commutative $\bbk$-algebra. Let us recall \cite[Proposition 4.1]{EnhancedLie} that
for every filtered $\sLie$-algebra $L$ the simplicial set $\mMC_{\bul}(L)$
with\footnote{Recall that $\MC(L)$ denotes the set of MC elements
of a filtered $\sLie$-algebras $L$ \cite{EnhancedLie}.}
$$
\mMC_n(L) : = \MC(L \hotimes \Om_n )
$$
is a Kan complex (a.k.a. an $\infty$-groupoid). We call the simplicial set
$\mMC_{\bul}(L)$ the {\it Deligne-Getzler-Hinich (DGH) $\infty$-groupoid}.

Let us also recall (see Theorem 4.2 in \cite{EnhancedLie}) that applying
the functor $\mMC_{\bul}$ to mapping spaces of any $\tLie$-enriched category,
we get a simplicial category whose mapping spaces are Kan complexes.
Thus, applying \cite[Theorem 4.2]{EnhancedLie} to the $\tLie$-enriched category
$\HoAlg_{\cC}$ and using Lemma \ref{lem:infty-morph} we deduce the
following theorem:
\begin{thm}
\label{thm:HoAlg-Delta}
Let $\cC$ be a coaugmented dg cooperad satisfying
Conditions \eqref{cond:cC-circ-filtered} and \eqref{cC-reduced}.
Then the assignment
$$
(A, B) ~\in~ \Objects(\Cat_{\cC})  \times  \Objects(\Cat_{\cC}) ~~ \mapsto ~~
\mMC_{\bul}(\map(A,B))
$$
gives us a category enriched over $\infty$-groupoids (a.k.a. Kan complexes).
Moreover, for every pair of $\Cobar(\cC)$-algebras $A, B$, the set $\mMC_{0}(\map(A,B))$
is in bijection with the set of $\infty$-morphisms from $A$ to $B$\,. \qed
\end{thm}

\section{$\pi_0 \big(\HoAlg^{\sD}_{\cC} \big)$ is a correct homotopy category of homotopy algebras}
\label{sec:HT}

Let $A$ and $B$ be homotopy algebras of type $\cC$ and
$F$ be an $\infty$-morphism from $A$ to $B$:
$$
F ~:~ \Big( \cC(A), \pa + Q_A  \Big) ~\to~ \Big( \cC(B), \pa + Q_B  \Big)\,.
$$

Composing $F$ with a canonical projection $p_B :  \cC(B) \to B$,
and restricting this composition to $A \subset \cC(A)$, we get
a map of cochain complexes:
\begin{equation}
\label{F-lin-term}
p_B  \circ F \Big|_{A} : A \to B\,.
\end{equation}
We refer to \eqref{F-lin-term} as {\it the linear term} of the $\infty$-morphism $F$.
Recall that an $\infty$-morphism $F$ is called an $\infty$ {\it quasi-isomorphism} if
its linear term \eqref{F-lin-term} is a quasi-isomorphism of cochain complexes.

Let us recall that $\Cat_{\cC}$ is the category of $\Cobar(\cC)$-algebras with
morphisms being $\infty$-morphisms, and observe that we have the obvious functor
\begin{equation}
\label{mF}
\mF : \Cat_{\cC} \to \pi_0 \big(\HoAlg^{\sD}_{\cC} \big)
\end{equation}
which acts by identity on objects and assigns to every $\infty$-morphism
the isomorphism class of the corresponding MC element.

The goal of this section is to prove that the category $\pi_0 \big(\HoAlg^{\sD}_{\cC} \big)$ is
the homotopy category for $\Cat_{\cC}$. Namely,
\begin{thm}
\label{thm:pi-0-HoAlg}
The functor $\mF$ sends $\infty$ quasi-isomorphisms to isomorphisms
and it is a universal functor with this property. I.e., if  $\mG : \Cat_{\cC} \to \mD$ is a functor
which sends $\infty$ quasi-isomorphisms to isomorphisms in $\mD$ then
there exists a unique functor
$$
\mG' : \pi_0\big( \HoAlg^{\sD}_{\cC} \big) \to \mD
$$
such that $\mG = \mG' \circ \mF$\,.
\end{thm}

\subsection{``Inverting'' $\infty$ quasi-isomorphisms}
Let $A_1, A_2, A_3$ be $\Cobar(\cC)$-algebras, $F$ be an
$\infty$-morphism from $A_1$ to $A_2$, and $F'$ be the MC element of $\map(A_1, A_2)$
corresponding to $F$. Let us denote by $F' \oplus 0$, $0 \oplus F'$ the corresponding
MC elements of the $\sLie$-algebras
$$
\map(A_1, A_2) \oplus \map(A_3 , A_1)
$$
and
$$
\map(A_2, A_3) \oplus \map(A_1, A_2)\,,
$$
respectively.

Twisting the $\sLie$-morphisms
$$
\cU : \map(A_1, A_2) \oplus \map(A_3 , A_1)  \to \map(A_3, A_2)
$$
and
$$
\cU : \map(A_2, A_3) \oplus \map(A_1, A_2)  \to \map(A_1, A_3)
$$
by the MC elements  $F' \oplus 0$, $0 \oplus F'$, respectively, and
composing the resulting  $\sLie$-morphisms with the canonical maps
$$
f \mapsto 0 \oplus f ~: ~
\map(A_3, A_1)  \to   \map(A_1, A_2) \oplus \map(A_3 , A_1)
$$
$$
f \mapsto f \oplus 0 ~: ~
 \map(A_2, A_3)  \to  \map(A_2, A_3) \oplus \map(A_1, A_2)
$$
we get two $\sLie$-morphisms
\begin{equation}
\label{comp-A-312}
\cU_{A_3 A_1 A_2} :  \map(A_3, A_1) \to  \map(A_3, A_2)
\end{equation}
and
\begin{equation}
\label{comp-A-123}
\cU_{A_1 A_2  A_3} :  \map(A_2, A_3) \to \map(A_1, A_3)\,.
\end{equation}

The following proposition says that the induced maps of MC elements
\begin{equation}
\label{comp-F-312}
(\cU_{A_3 A_1 A_2})_* : \MC( \map(A_3, A_1) ) \to \MC(  \map(A_3, A_2)  )
\end{equation}
and
\begin{equation}
\label{comp-F-123}
(\cU_{A_1 A_2 A_3})_* : \MC( \map(A_2, A_3) ) \to \MC( \map(A_1, A_3) )
\end{equation}
correspond to the composition (resp. the pre-composition) of
an $\infty$-morphism from $A_3$ to $A_1$ (resp. from $A_2$ to $A_3$)
with $F$:
\begin{prop}
\label{prop:cU-star-comp}
If $G$ is an $\infty$-morphism from $A_3$ to $A_1$ and $G'$ is the
corresponding MC element of $\map(A_3, A_1)$ then the MC element
$(\cU_{A_3 A_1 A_2})_* (G')$ of $\map(A_3, A_2)$ corresponds to
the composition $F \circ G$. Similarly, if $G$ is an $\infty$-morphism from $A_2$ to
$A_3$ and $G'$ is the
corresponding MC element of $\map(A_2, A_3)$ then the MC element
$(\cU_{A_1 A_2 A_3})_* (G')$ of $ \map(A_1, A_3) $ corresponds to
the composition $G \circ F$\,.
\end{prop}
\begin{proof} The proof of these statements is straightforward. \\
\end{proof}

We also claim that
\begin{prop}
\label{prop:cU-F-q-iso}
$\sLie$-morphisms \eqref{comp-A-312} and \eqref{comp-A-123}
are compatible with the filtrations $\cF^{\ari}_{\bul}$ from \eqref{Conv-filter}.
Furthermore,  if $F$ is an $\infty$ quasi-isomorphism,
then \eqref{comp-A-312} and \eqref{comp-A-123} give us
$\sLie$ quasi-isomorphisms
\begin{equation}
\label{q-iso-cF-m}
\cF^{\ari}_{m}  \map(A_3, A_1) \to \cF^{\ari}_{m} \map(A_3, A_2)
\quad \textrm{and} \quad
\cF^{\ari}_{m} \map(A_2, A_3) \to \cF^{\ari}_{m} \map(A_1, A_3)
\end{equation}
respectively, for every $m \ge 1$\,.
\end{prop}
\begin{proof}
Due to Remark \ref{rem:U-pr-again}, the composition
$$
\cU'_{A_3 A_1 A_2} : = p_{\map(A_3, A_2)} \circ  \cU_{A_3 A_1 A_2} :  \und{S}\big(\map(A_3, A_1)\big) \to \map(A_3, A_2)
$$
is given by the formula
\begin{equation}
\label{cU-A-312-pr}
\cU'_{A_3 A_1 A_2} (g_1, \dots, g_n) (X) =
F' \big( 1 \otimes g_{1} \otimes \dots \otimes g_{n} \, (\D_n(X)) \big)\,,
\end{equation}
where $X \in \cC(A_3)$ and $g_1, \dots, g_n \in \map(A_3, A_1)$\,.
Similarly, the composition
$$
\cU'_{A_1 A_2 A_3} : = p_{\map(A_1, A_3)} \circ \cU_{A_1 A_2 A_3} :
\und{S}\big( \map(A_2, A_3) \big) \to \map(A_1, A_3)
$$
is given by the formula
 \begin{equation}
\label{cU-A-123-pr}
\cU'_{A_1 A_2 A_3} (h_1, \dots, h_n) (Y) =
\begin{cases}
\displaystyle \sum_{m \ge 1}
\frac{1}{m!} h_1 \big( 1 \otimes F' \otimes \dots \otimes F' \, (\D_m(Y)) \big)   \qquad {\rm if} ~~  n=1 \,, \\[0.5cm]
 0 \qquad {\rm otherwise}\,,
\end{cases}
\end{equation}
where $Y \in \cC(A_1)$ and $h_1, \dots, h_n \in \map(A_2, A_3)$\,.

The compatibility of map $\cU'_{A_3 A_1 A_2}$ and $\cU'_{A_1 A_2 A_3}$ with the filtrations
$\cF^{\ari}_{\bul}$ can be checked directly by unfolding the right hand side
of \eqref{cU-A-312-pr} and the right hand side of \eqref{cU-A-123-pr}, respectively.

We also see that the linear terms
\begin{equation}
\label{cU-1-A-312}
\cU_{1, A_3 A_1 A_2} :  \map(A_3, A_1) \to  \map(A_3, A_2)
\end{equation}
and
\begin{equation}
\label{cU-1-A-123}
\cU_{1, A_1 A_2 A_3} :   \map(A_2, A_3) \to \map(A_1, A_3)
\end{equation}
of the $\sLie$-morphisms $\cU_{A_3 A_1 A_2}$ and $\cU_{A_1 A_2 A_3}$ are
given by the formulas:
$$
\cU_{1, A_3 A_1 A_2} (g) (X) = F' \big( (1 \otimes g) \circ \D_1(X) \big)
$$
and
$$
\cU_{1, A_1 A_2 A_3}(h) (Y) = \sum_{k \ge 1}
\frac{1}{k!} h \big( 1 \otimes F' \otimes \dots \otimes F' \, (\D_k(Y)) \big)\,,
$$
respectively, where $g \in \map(A_3, A_1)$, $h \in \map(A_2, A_3)$  $X \in \cC(A_3)$ and $Y \in \cC(A_1)$.

Let $\vf :  A_1 \to A_2$ be the linear term
\begin{equation}
\label{vf}
\vf : = p_{A_2} \circ F \big|_{A_1}
\end{equation}
of the $\infty$-morphism $F$ and assume that
$\vf$ is a quasi-isomorphism of cochain complexes.

To prove that $\cU_{1, A_1 A_2 A_3}$ induces an isomorphism
$$
H^{\bul} \big( \cF^{\ari}_{m} \map(A_2, A_3) \big) \to H^{\bul} \big( \cF^{\ari}_{m} \map(A_1, A_3) \big)
$$
for every $m$, we observe that, for $m \ge 2$
$$
\cF^{\ari}_{m} \map(W, A_3)  =   \cF^{\ari}_{m} \Hom(\cC_{\c}(W), A_3)
$$
and  $\map(W, A_3) = \cF^{\ari}_{1} \map(W, A_3)$ splits into the direct sum of
cochain complexes
\begin{equation}
\label{Map-W-A3-splits}
\map(W, A_3) =  \Hom(W, A_3) ~\oplus~ \Hom(\cC_{\c}(W), A_3)\,,
\end{equation}
where $W$ is either $A_2$ or $A_1$, and $\cC_{\c}$ is the cokernel of
the coaugmentation of $\cC$.

We also observe that the chain map $\cU_{1, A_1 A_2 A_3}$ is
compatible with decomposition \eqref{Map-W-A3-splits} and
the corresponding chain map
$$
\Hom(A_2, A_3) \to  \Hom(A_1, A_3)
$$
is a quasi-isomorphism because the functor $\Hom(-,-)$ preserves
quasi-isomorphisms in $\Ch_{\bbk}$\,.

So we should now prove that the chain map
\begin{equation}
\label{cU1-123-cC-circ}
\cU_{1, A_1 A_2 A_3} \Big|_{\cF^{\ari}_{m} \Hom(\cC_{\c}(A_2), A_3) }   ~:~ \cF^{\ari}_{m} \Hom(\cC_{\c}(A_2), A_3)
\to   \cF^{\ari}_{m} \Hom(\cC_{\c}(A_1), A_3)
\end{equation}
induces an isomorphism on cohomology for every $m \ge 1$\,.

For this purpose, we equip the cochain complex $\cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3)$
with the following descending filtration
$$
\cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3) = \cF^{\cC}_0 \cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3)
\supset  \cF^{\cC}_1 \cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3)  \supset
$$
$$
\supset  \cF^{\cC}_2 \cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3) \supset \dots
$$
\begin{equation}
\label{cF-cC-Map}
\cF^{\cC}_q \, \cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3) ~ : = ~
\end{equation}
$$
\big\{ g \in  \cF^{\ari}_{m}   \Hom(\cC_{\c}(W), A_3) ~~ \big|~~ g (X)  = 0 ~~\forall~ X \in \cF^{q}\cC_{\c}(W) \big\}\,,
$$
where $\cF^{\bul} \cC_{\c}$ is the ascending filtration on the pseudo-cooperad
$\cC_{\c}$ from \eqref{cC-circ-filtr} and $W$ is, as above, either $A_1$ or $A_2$.

Due to inclusion \eqref{D-bt-filtr}, the differential on $\map(W,A_3)$ is
compatible with the filtration. Moreover, condition \eqref{cocomplete}
implies that  $\cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3)$ is complete with respect to
this filtration, i.e.
$$
\cF^{\ari}_{m}   \Hom(\cC_{\c}(W), A_3) = \lim_q ~ \cF^{\ari}_{m}   \Hom(\cC_{\c}(W), A_3)  ~
\big/ ~\cF^{\cC}_q \cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3)\,.
$$

Let us denote by $E_{m,q}(\cC, W)$ the following cochain complex
\begin{equation}
\label{E-m-q}
E_{m,q}(\cC_{\c}, W) : =
\bigoplus_{n \ge m} \Big(  \big( \cF^q \cC_{\c}(n) / \cF^{q-1} \cC_{\c}(n) \big) \,  \otimes \, W^{\otimes\, n}  \Big)_{S_n} \,.
\end{equation}

It is clear that,  the associated graded complex
$$
\Gr_{\cF^{\cC}} \, \cF^{\ari}_m  \Hom(\cC_{\c}(W), A_3)
$$
is isomorphic to
\begin{equation}
\label{assoc-Gr-m-ge2}
\bigoplus_{q \ge 1} \Hom\big(E_{m,q}(\cC_{\c}, W) ,   A_3  \big)\,.
\end{equation}
Furthermore, the differential $\pa_{\Gr}$ on \eqref{assoc-Gr-m-ge2} comes from
those on $W$, $\cC_{\c}$ and $A_3$\,.

The map between the associated graded complexes
\begin{equation}
\label{assoc-Gr-map}
\cU^{\Gr}_{1, A_1 A_2 A_3} ~ : ~
\Hom\big(E_{m,q}(\cC_{\c}, A_2) ,   A_3  \big) \to  \Hom\big(E_{m,q}(\cC_{\c}, A_1) ,   A_3  \big)
\end{equation}
induced by \eqref{cU1-123-cC-circ} is given by the formula
\begin{equation}
\label{assoc-Gr-map-dfn}
\cU^{\Gr}_{1, A_1 A_2 A_3} (g)(\ga; a_1, \dots, a_n)  = g (\ga; \vf(a_1), \dots, \vf(a_n))\,,
\end{equation}
where $\vf$ is the linear term of the $\infty$-morphism $F$ and
$\ga \in  \cF^q \cC_{\c}(n) / \cF^{q-1} \cC_{\c}(n)$\,.

Using the K\"unneth theorem, the fact that
$\vf$ induces an isomorphism $H^{\bul}(A_1) \to H^{\bul}(A_2)$,
and $\textrm{char}(\bbk) = 0$ we conclude that the chain map
$$
E_{m,q}(\cC_{\c}, A_1)  \to E_{m,q}(\cC_{\c}, A_2)
$$
induced by $\vf$ is a quasi-isomorphism.

Therefore, since the functor $\Hom(-,-)$ preserves quasi-isomorphisms
in $\Ch_{\bbk}$, we deduce chain map \eqref{assoc-Gr-map} between the
associated graded complexes is a quasi-isomorphism.

Since  $\cF^{\ari}_{m}  \Hom(\cC_{\c}(W), A_3)$ is complete with respect to filtration
\eqref{cF-cC-Map},  and filtration \eqref{cF-cC-Map} is bounded from the
left, applying Lemma D.1 from \cite{DeligneTw} to the cone of chain map
\eqref{cU1-123-cC-circ}, we conclude that  \eqref{cU1-123-cC-circ}, and hence
\eqref{cU-1-A-123}, is indeed a quasi-isomorphism
of cochain complexes.

A similar argument shows that map \eqref{cU-1-A-312} is a quasi-isomorphism
of cochain complexes, provided so is $\vf$ \eqref{vf}.

Proposition \ref{prop:cU-F-q-iso} is proved.
\end{proof}

Let $A$ and $B$ be $\Cobar(\cC)$-algebras.
We will now prove that every $\infty$ quasi-morphism $F$ from $A$ to $B$
is ``invertible'' in the following sense:
\begin{cor}
\label{cor:invertible}
Let $F$ be an $\infty$ quasi-isomorphism from $A$ to $B$.
Then there exists an $\infty$-morphism $G$ from $B$ to $A$
such that the MC element $(G \circ F)'$ (resp.  $(F \circ G)'$)
of the $\sLie$-algebra $\map(A,A)$ (resp. $\map(B,B)$)
corresponding to the composition $G \circ F$ (resp. $F \circ G$)
is isomorphic in $\mMC_{\bul}(\map(A,A))$ (resp. in $\mMC_{\bul}(\map(B,B))$)
to the MC element $\id'_A$ (resp. $\id'_B$) corresponding to the
identity morphism $\id_A$ (resp. $\id_B$). If $\wt{G}$ is another
$\infty$-morphism from $B$ to $A$ satisfying the above properties then
the MC element $\wt{G}' \in \map(B,A)$ corresponding to $\wt{G}$ is isomorphic
to $G'$ in $\mMC_{\bul}(\map(B,A))$\,.
\end{cor}
\begin{proof} Let us start with the question of existence of $G$.

Due to Proposition \ref{prop:cU-star-comp}, it suffices to
prove that there exists a MC element $G'$ of $\map(B,A)$
such that the $0$-cell
\begin{equation}
\label{G-pr-to-A}
(\cU_{A B A})_* (G') \in \mMC_0(\map(A,A))
\end{equation}
is connected to $\id'_A$ and the $0$-cell
\begin{equation}
\label{G-pr-to-B}
(\cU_{B A B})_* (G') \in \mMC_0(\map(B,B))
\end{equation}
is connected to $\id'_B$.

Proposition \ref{prop:cU-F-q-iso} implies that the $\sLie$-morphism
$$
\cU_{A B A} : \map(B,A) \to \map(A,A)
$$
is a quasi-isomorphism and, moreover, it satisfies
the necessary conditions of Theorem 2.2 from \cite{GMtheorem}.
This theorem, in turn, implies that $\cU_{A B A}$  induces a bijection of sets
\begin{equation}
\label{cU-ABA-pi0}
\pi_0\Big( \mMC_{\bul}(\map(B,A)) \Big)
 \xto{\cong}
\pi_0\Big( \mMC_{\bul}(\map(A,A)) \Big) \,.
\end{equation}

Therefore, there exists a MC element $G' \in \map(B,A)$ such
that the $0$-cell
\begin{equation}
\label{GF-pr}
(G \circ F)'= (\cU_{A B A})_* (G')
\end{equation}
is connected to $\id'_A$\,.

To prove that the $0$-cell $(F\circ G)' = (\cU_{B A B})_* (G')$ is connected to $\id'_B$,
we consider the $\sLie$-morphisms
\begin{equation}
\label{ABB}
\cU_{A  B  B} :  \map(B, B)  \to \map(A, B)\,,
\end{equation}
and
\begin{equation}
\label{AAB}
\cU_{AAB} :  \map(A, A)  \to \map(A, B)
\end{equation}
constructed, as above, using the $\infty$-morphism $F$.

Proposition \ref{prop:cU-star-comp} implies that, if $K'$ is a MC element
of $\map(B, B)$ (resp.  $\map(A, A)$) corresponding to an $\infty$-morphism
$K$ from $B$ to $B$ (resp. from $A$ to $A$) then the MC element
$(\cU_{A B B})_*(K')$ (resp. $(\cU_{A A B})_*(K')$)
corresponds to the composition $K \circ F$ (resp. $F \circ K$).

Let us now consider the composition
$F \circ  G$ and denote by $(F\circ G)'$ the $0$-cell of $\mMC_{\bul}(\map(B,B))$ corresponding
to the $\infty$-morphism $F \circ G$.

Due to the above observation, the MC element
$(F\circ G \circ F)' \in \map(A, B)$ corresponding to the $\infty$-morphism
$F\circ G \circ F$ satisfies the equations
\begin{equation}
\label{MC-FGF-FG}
(F\circ G \circ F)'  = (\cU_{A B B})_*\, (F\circ G)'
\end{equation}
and
\begin{equation}
\label{MC-FGF-GF}
(F\circ G \circ F)'  = (\cU_{A A B})_* \, (G\circ F)'\,.
\end{equation}

Therefore, since the $0$-cell $(G \circ F)'$ is connected to the
$0$-cell $\id'_A$, the $0$-cell $(F\circ G \circ F)'$ is connected
to $F'$ in $\mMC_{\bul} (\map(A,B))$\,.

On other hand, $F' = (\cU_{A B B})_* (\id'_B)$, and hence
the $0$-cells
$$
(\cU_{A B B })_* (\id'_B) \qquad \textrm{and} \qquad  (\cU_{A B B})_*\, (F\circ G)'
$$
are connected in  $\mMC_{\bul} (\map(A,B))$\,.

Since $F$ is an $\infty$ quasi-isomorphism,  Proposition \ref{prop:cU-F-q-iso} and
\cite[Theorem 2.2]{GMtheorem} imply that $(\cU_{A B B })_*$ induces
a bijection
$$
\pi_0 \Big( \mMC_{\bul} (\map(B,B)) \Big) \xto{\cong}  \pi_0 \Big( \mMC_{\bul} (\map(A,B)) \Big)\,.
$$

Thus the $0$-cells $(F\circ G)'$ and $\id'_B$ are also connected in $\mMC_{\bul} (\map(B,B))$
and the existence of a desired $\infty$-morphism $G$ is proved.

Let $\wt{G}$ be another $\infty$-morphism from $B$ to $A$ such
that $0$-cells \eqref{GF-pr} and $\id'_A$ are connected in
$\mMC_{\bul}( \map(A,A) )$.

Therefore, since $\cU_{ABA}$ induces bijection \eqref{cU-ABA-pi0}
and the $0$-cells $(\cU_{ABA})_* (G')$ and $\id'_A$ are connected,
we conclude that the $0$-cells $\wt{G}'$ and $G'$ are also connected.
\end{proof}

Thus we proved the first part of Theorem \ref{thm:pi-0-HoAlg}.

We would like to conclude this subsection with the observation that
mapping spaces of the simplicial category $\HoAlg^{\sD}_{\cC}$ enjoy
the following remarkable property
\begin{cor}
\label{cor:q-iso-gives}
Let $A_1, A_2, A_3$ be $\Cobar(\cC)$-algebras and $F$ be
an $\infty$ quasi-isomorphism from $A_1$ to $A_2$. Then the composition
(resp. pre-composition) with $F$ induces the weak equivalences of simplicial sets:
$$
\mMC_{\bul}(\map(A_3, A_1)) \to \mMC_{\bul}(\map(A_3, A_2))\,,
$$
$$
\mMC_{\bul}(\map(A_2, A_3)) \to \mMC_{\bul}(\map(A_1, A_3))\,.
$$
\end{cor}
\begin{proof}
The desired statement is a direct consequence of Proposition \ref{prop:cU-F-q-iso}
and \cite[Theorem 2.2]{GMtheorem}.
\end{proof}

\subsection{The functor $\mF$ from Theorem \ref{thm:pi-0-HoAlg} has the desired universal property}

The proof of the universal property of the functor $\mF$ is based on
the following proposition:
\begin{prop}
\label{prop:homot-implies-OK}
Let $\mG : \Cat_{\cC} \to \mD$ be a functor which sends $\infty$ quasi-isomorphisms
to isomorphisms in $\mD$. Let $A$, $B$ be $\Cobar(\cC)$-algebras and $F, G$ be
$\infty$-morphisms from $A$ to $B$. If the corresponding
$0$-cells $F'$ and $G'$ of $\mMC_{\bul}(\map(A,B))$ are connected
then
$$
\mG(F) = \mG(G)\,.
$$
\end{prop}
\begin{proof}
By the condition of the proposition, there exists a $1$-cell
\begin{equation}
\label{K-pr}
K'  \in \Hom(\cC(A), B) \hotimes \bbk[t, \, dt]
\end{equation}
such that
\begin{equation}
\label{proj-0}
K' \Big|_{t=d t = 0} = F'
\end{equation}
and
\begin{equation}
\label{proj-1}
K' \Big|_{t=1, ~ d t = 0} = G'\,.
\end{equation}

Since
$$
\cC(A) = \bigoplus_n \big( \cC(n) \otimes A^{\otimes \, n} \big)_{S_n}
$$
and $\Hom \big(\cC(A), B \big)$ is considered with the
topology coming from filtration \eqref{Conv-filter},
we have the natural strict $\sLie$-morphism
\begin{equation}
\label{isom-hotimes}
\Hom \big(\cC(A), B \big)  \hotimes \bbk[t, \, dt]  ~\to~
\Hom \big( \cC(A), B \otimes  \bbk[t, \, dt] \big) \,.
\end{equation}

Therefore the $1$-cell $K'$ gives us an $\infty$-morphism
$K$ from $A$ to $B \otimes   \bbk[t, \, dt] $ which fits into
the following commutative diagram
\begin{equation}
\label{diag-K}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=2em, column sep=4em]
{   ~& ~  & B \\
A & B \otimes   \bbk[t, \, dt] & ~ \\
~ & ~ & B\,,\\ };
\path[->, font=\scriptsize]
(m-2-1) edge node[above] {$K$} (m-2-2)
edge[bend left=15] node[above] {$F$} (m-1-3)  edge[bend right=15] node[below] {$G$} (m-3-3)
(m-2-2) edge node[above] {$p_0$} (m-1-3)  edge node[below] {$p_1$} (m-3-3);
\end{tikzpicture}
\end{equation}
where $ B \otimes   \bbk[t, \, dt]$ is considered with the differential
$\pa_{B} + dt \,\pa_t $ and the natural $\Cobar(\cC)$-structure coming
from the one on $B$. Moreover,  $p_0$ and $p_1$ are the obvious (strict)
morphisms of $\Cobar(\cC)$-algebras
\begin{equation}
\label{p-0}
p_0 (v) : = v \big|_{t=dt=0}  ~:~  B \otimes   \bbk[t, \, dt] \to B
\end{equation}
and
\begin{equation}
\label{p-1}
p_1 (v) : = v \big|_{t=1,~ dt=0}  ~:~  B \otimes   \bbk[t, \, dt] \to B\,.
\end{equation}

Let us observe that the maps $p_0$ and $p_1$ fit into the
commutative diagram
\begin{equation}
\label{diag-p0p1}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=2em, column sep=4em]
{   ~& ~  & B \\
B & B \otimes   \bbk[t, \, dt] & ~ \\
~ & ~ & B\,,\\ };
\path[->, font=\scriptsize]
(m-2-1) edge node[above] {$~~i$} (m-2-2)
edge[bend left=15] node[above] {$\id_B$} (m-1-3)  edge[bend right=15] node[below] {$\id_B$} (m-3-3)
(m-2-2) edge node[above] {$p_0$} (m-1-3)  edge node[below] {$p_1$} (m-3-3);
\end{tikzpicture}
\end{equation}
where $i: B \to  B \otimes   \bbk[t, \, dt]$ is the natural embedding given by
$$
i (v) : = v \otimes 1\,.
$$

Applying the functor $\mG$ to \eqref{diag-p0p1}, we get
$$
\mG(p_0) \circ \mG(i) = \mG(p_1) \circ \mG(i) = \id_{\mG(B)}\,.
$$
Hence, since $i$ is obviously a quasi-isomorphism, we deduce that
$$
\mG(p_0) = \mG(p_1)\,.
$$

Finally, applying  the functor $\mG$ to \eqref{diag-K}, we get
$$
\mG(p_0) \circ \mG(K) = \mG(F)\,, \qquad
\mG(p_1) \circ \mG(K) = \mG(G)
$$
which implies that $\mG(F) = \mG(G)$\,.
\end{proof}

Proposition \ref{prop:homot-implies-OK} motivates the following
definition\footnote{For several other justifications of this definition, we refer the
reader to paper \cite{DotsPoncin} by V. Dotsenko and N. Poncin.}
\begin{defi}
\label{dfn:homotopy}
Let $A,B$ be $\Cobar(\cC)$-algebras. We say that $\infty$-morphisms $F,G$
from $A$ to $B$ are \emph{homotopic} if the corresponding
$0$-cells $F'$ and $G'$ are connected in $\mMC_{\bul}(\map(A,B))$.
\end{defi}

We can now prove the universal property of functor \eqref{mF}.

Indeed, let $\mG$ be a functor from $\Cat_{\cC}$ to some category $\mD$
which sends $\infty$ quasi-isomorphisms to isomorphisms.

For objects of $A,B, \dots$ of $\pi_0 \big( \HoAlg^{\sD}_{\cC} \big)$ we set
$$
\mG'(A) : = \mG(A)\,.
$$

Next, given two $\Cobar(\cC)$-algebras $A,B$ and
an isomorphism class
$$
[F'] \in \pi_0 \big( \mMC_{\bul}(\map(A,B)) \big)
$$
of a MC element $F'$ in $\map(A,B)$ we set
\begin{equation}
\label{mG-pr-morph}
\mG([F']) : = \mG(F)\,,
\end{equation}
where $F$ is the $\infty$-morphism from $A$ to $B$ corresponding
to the MC element $F'$.

Due to Proposition \ref{prop:homot-implies-OK}, the right hand side of
\eqref{mG-pr-morph} does not depend on the choice of the MC element
$F'$ in its isomorphism class $[F']$.

It is clear that, this way, we get a functor
$$
\mG' : \pi_0 \big( \HoAlg^{\sD}_{\cC} \big) \to \mD
$$
satisfying $\mG' \circ \mF = \mG$\,.

It is also clear that such a functor $\mG'$ is unique and the
proof of Theorem \ref{thm:pi-0-HoAlg} is complete.
We can use the above results to obtain the following converse to
Cor.\ \ref{cor:q-iso-gives}. Together they give a
recognition principal for $\infty$ quasi-isomorphisms, in analogy
with the characterization of weak equivalences via function complexes in a simplicial model category.

\begin{cor}
\label{cor:q-iso-gives-converse}
For every $\infty$-morphism $F \maps A \to B$ of $\Cobar(\cC)$
algebras, the following statements are equivalent:
\begin{enumerate}

\item  $F$ is an $\infty$-quasi-isomorphism.

\item The $\sLie$-morphisms
\[
\cU_{AAB} \maps \map(A,A) \to \map(A,B)  \quad \textrm{ and } \quad
\cU_{BAB} \maps \map(B,A) \to \map(B,B)
\]
induce homotopy equivalences of simplicial sets
\[
\mMC_{\bul}(\map(A, A)) \xto{\sim} \mMC_{\bul}(\map(A, B))
~ \textrm{ and } ~
\mMC_{\bul}(\map(B, A)) \xto{\sim} \mMC_{\bul}(\map(B, B))
\]
respectively.

\item  The $\sLie$-morphisms
\[
\cU_{ABA} \maps \map(B,A) \to \map(A,A)
\quad \textrm{ and } \quad
\cU_{ABB} \maps \map(B, B) \to \map(A, B)
\]
induce homotopy equivalences of simplicial sets
\[
\mMC_{\bul}(\map(B, A)) \xto{\sim} \mMC_{\bul}(\map(A, A))
~ \textrm{ and } ~
\mMC_{\bul}(\map(B, B)) \xto{\sim} \mMC_{\bul}(\map(A, B))
\]
respectively.
\end{enumerate}
\end{cor}

\begin{proof}
The implications $1 \Rightarrow 2$ and $1 \Rightarrow 3$ are particular
cases of  Corollary \ref{cor:q-iso-gives}. Let us prove the implication $2 \Rightarrow 1$.

Since $F$ induces a homotopy equivalence of simplicial sets
\[
\mMC_{\bul}(\map(B, A)) \xto{\sim} \mMC_{\bul}(\map(B, B)),
\]
we have an isomorphism
\[
\pi_0 \Bigl(\mMC_{\bul}(\map(B, A))\Bigr ) \cong \pi_0
\Bigl(\mMC_{\bul}(\map(B, B)) \Bigr),
\]
which implies that there exists an $\infty$-morphism $G \maps B \to A$
such that $F \circ G$ and $\id_B$ are homotopic, in the sense of Def.\ \ref{dfn:homotopy}.
As in the proof of Prop.\ \ref{prop:homot-implies-OK}, this gives a
commutative diagram of $\Cobar(\cC)$ algebras and $\infty$-morphisms:
\begin{equation*}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=2em, column sep=4em]
{   ~& ~  & B \\
B & B \otimes   \bbk[t, \, dt] & ~ \\
~ & ~ & B\,\\ };
\path[->, font=\scriptsize]
(m-2-1) edge node[above] {$H$} (m-2-2)
edge[bend left=15] node[above] {$F\circ G$} (m-1-3)  edge[bend right=15] node[below] {$\id_B$} (m-3-3)
(m-2-2) edge node[above] {$p_0$} (m-1-3)  edge node[below] {$p_1$} (m-3-3);
\end{tikzpicture}
\end{equation*}
By taking the linear terms of the $\infty$-morphisms, this diagram, in turn, gives a diagram of cochain complexes
\begin{equation*}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes, row sep=2em, column sep=4em]
{   ~& ~  & B \\
B & B \otimes   \bbk[t, \, dt] & ~ \\
~ & ~ & B\,,\\ };
\path[->, font=\scriptsize]
(m-2-1) edge node[above] {$h$} (m-2-2)
edge[bend left=15] node[above] {$f\circ g$} (m-1-3)  edge[bend right=15] node[below] {$\id_B$} (m-3-3)
(m-2-2) edge node[above] {$p_0$} (m-1-3)  edge node[below] {$p_1$} (m-3-3);
\end{tikzpicture}
\end{equation*}
where $h$ and $f\circ g$ are the linear terms $p_{B\tensor \bbk[t,dt]}
\circ H \vert_{B}$, and $p_{B} \circ F\circ G \vert_{B}$, respectively.
Let $I \maps B\tensor \bbk[t,dt] \to B$ denote  ``integration over the fiber'', i.e.
the degree $-1$ linear map
\[
I\Bigl( b \tensor q(t) + \tilde{b} \tensor \tilde{q}(t)dt \Bigr) = (-1)^{\vert
  \tilde{b} \vert} ~ \tilde{b}\int^{1}_{0} \tilde{q}(t)dt.
\]
Then one can show that the map $s_B \maps B \to B$ defined as
\[
s_B : = I \circ h
\]
is a chain homotopy
\[
\id_B - f \circ g  = \partial_B s_B + s_B \partial_B.
\]
Hence, the map $f$ is surjective on cohomology.

Now consider the $\sLie$-morphism
\[
\cU_{ABB} \maps \map(B,B) \to \map(A,B)
\]
which sends a MC element $K'$ to $(K \circ F)'$. This gives a map
between sets
\[
\pi_0 \Bigl(\mMC_{\bul}(\map(B, B))\Bigr ) \to \pi_0\Bigl(\mMC_{\bul}(\map(A, B)) \Bigr).
\]
Since the MC elements $(F \circ G)'$ and $(\id_B)'$ are connected by a $1$-cell,
we conclude that $F'$ is connected to $(F \circ G \circ F)'$.
By the first part of statement 2, the $\sLie$-morphism
\[
\cU_{AAB} \maps \map(A,A) \to \map(A,B)
\]
induces a bijection
\[
\pi_0 \Bigl(\mMC_{\bul}(\map(A, A))\Bigr ) \cong \pi_0
\Bigl(\mMC_{\bul}(\map(A, B)) \Bigr).
\]
sending $[(\id_A)']$ to $[F']$,
and $[(G \circ F)']$ to $[(F \circ G \circ F)']=[F']$. Hence, we
see that $\id_A$ is homotopic to $G \circ F$. We produce a chain
homotopy $s_A \maps A \to A$  using the same construction as before,
which shows that $f$ is also injective on cohomology. Hence, $F \maps A \to
B$ is an $\infty$-quasi-isomorphism.

The proof of the implication $3 \Rightarrow 1$ is very similar to that of $2 \Rightarrow 1$.
So we leave it to the reader.
\end{proof}

\section{The Homotopy Transfer Theorem is a simple consequence of the Goldman-Millson theorem}
\label{sec:HTT}

In this section we give an elegant proof of the Homotopy Transfer Theorem
for $\Cobar(\cC)$-algebras. This proof is based on a construction from \cite{DefHomotInvar}
and a version of the Goldman-Millson theorem from  \cite{GMtheorem}.

Let us consider a dg cooperad $\cC$ for which $\cC_{\c}$ carries
ascending filtration \eqref{cC-circ-filtr} satisfying condition \eqref{cocomplete}
and let $A$, $B$ be cochain complexes.

In  \cite[Section 3.1]{DefHomotInvar}, we equipped the cochain complex
\begin{equation}
\label{Cyl-cC-A-B}
\Cyl(\cC, A, B) : = \bsi \Hom(\cC_{\c}(A), A) ~\oplus~ \Hom(\cC(A), B)~ \oplus~
 \bsi \Hom(\cC_{\c}(B), B)
\end{equation}
with a natural $\sLie$-algebra structure\footnote{In \cite{DefHomotInvar}, we actually
introduce an $L_{\infty}$-structure on the suspension of \eqref{Cyl-cC-A-B}. But the latter is,
of course, equivalent to introducing a $\sLie$-algebra structure on  \eqref{Cyl-cC-A-B}.}

According to \cite[Section 3.1]{DefHomotInvar}, MC elements of $\Cyl(\cC, A, B)$ are
triples:
\begin{itemize}

\item a $\Cobar(\cC)$-algebra structure on $A$,

\item a $\Cobar(\cC)$-algebra structure on $B$, and

\item an $\infty$-morphism from $A$ to $B$\,.

\end{itemize}

In particular, any chain map $\vf : A \to B$ gives a MC element $Q_{\vf}$
corresponding to the zero  $\Cobar(\cC)$-algebra structures on $A$, $B$,
and the strict $\infty$-morphism from $A$ to $B$.

Let us twist the $\sLie$-algebra structure on  \eqref{Cyl-cC-A-B},
and denote the new $\sLie$-algebra by
\begin{equation}
\label{Cyl-cC-AB-twisted}
\Cyl(\cC, A, B)^{Q_{\vf}}\,.
\end{equation}

It is not hard to see that the graded subspace
\begin{equation}
\label{Cyl-cC-0-AB}
\Cyl_{\c}(\cC, A, B)^{Q_{\vf}} : = \bsi \Hom(\cC_{\c}(A), A) ~\oplus~ \Hom(\cC_{\c}(A), B)~ \oplus~
 \bsi \Hom(\cC_{\c}(B), B)
\end{equation}
is a $\sLie$-subalgebra of \eqref{Cyl-cC-AB-twisted} and
filtration  \eqref{cC-circ-filtr} on $\cC_{\c}$ allows us to equip
\eqref{Cyl-cC-0-AB} with a natural complete descending filtration
$\cF_{\bul} \Cyl_{\c}(\cC, A, B)^{Q_{\vf}}$ (see Remark 2 in \cite[Section 3.2]{DefHomotInvar})
such that
\begin{equation}
\label{Cyl-cC-0-AB-OK}
\Cyl_{\c}(\cC, A, B)^{Q_{\vf}} = \cF_{1} \Cyl_{\c}(\cC, A, B)^{Q_{\vf}}\,.
\end{equation}
In other words, \eqref{Cyl-cC-0-AB} is a filtered $\sLie$-algebra.

Furthermore, according to \cite[Section 3.2]{DefHomotInvar},
MC elements of \eqref{Cyl-cC-0-AB} are in bijection with triples:
\begin{itemize}

\item a $\Cobar(\cC)$-algebra structure on $A$,

\item a $\Cobar(\cC)$-algebra structure on $B$, and

\item an $\infty$-morphism $F$ from $A$ to $B$ whose linear term is $\vf$.

\end{itemize}

The Homotopy Transfer Theorem can be now formulated as follows:
\begin{thm}
\label{thm:HTT}
Let $B$ be a $\Cobar(\cC)$-algebra, $A$ be a cochain complex
and $\vf : A \to B$ be a quasi-isomorphism of cochain complexes.
Then there exists a  $\Cobar(\cC)$-algebra structure $Q_A$ on $A$,
a $\Cobar(\cC)$-algebra structure $Q_B$ on $B$ (which is homotopy
equivalent to the original one) and an $\infty$-morphism $F$ from
$(A, Q_A)$ to $(B,Q_B)$ whose linear term is $\vf$. If
$(\ti{Q}_A, \ti{Q}_B, \ti{F})$ is another triple
satisfying the above properties then the MC elements
corresponding to $(Q_A, Q_B, F)$ and  $(\ti{Q}_A, \ti{Q}_B, \ti{F})$
are isomorphic in
$$
\mMC_{\bul} \big(\, \Cyl_{\c}(\cC, A, B)^{Q_{\vf}}\, \big)\,.
$$
\end{thm}
\begin{proof}
Let us identify the graded vector space of
the convolution Lie algebra
$$
\Conv(\cC_{\c}, \End_B)
$$
with $\Hom(\cC_{\c}(B), B)$ and consider $\bsi \Hom(\cC_{\c}(B), B)$ with
the corresponding $\sLie$-algebra structure.

Due to \cite[Proposition 3.2]{DefHomotInvar}, the canonical projection
\begin{equation}
\label{pi-B}
\pi_B ~:~   \Cyl_{\c}(\cC, A, B)^{Q_{\vf}} \to \bsi \Hom(\cC_{\c}(B), B)
\end{equation}
is a strict quasi-isomorphism of $\sLie$-algebras which is
obviously compatible with the descending filtrations coming
from  \eqref{cC-circ-filtr}.

Using the same arguments, as in the proof of  \cite[Proposition 3.2]{DefHomotInvar},
it is easy to see that
$$
\pi_B ~:~   \cF_m \Cyl_{\c}(\cC, A, B)^{Q_{\vf}} \to \bsi \cF_m \Hom(\cC_{\c}(B), B)
$$
is a quasi-isomorphism of cochain complexes for every $m \ge 1$.

Therefore, applying Theorem 1.1 from \cite{GMtheorem} to \eqref{pi-B},
we conclude that $\pi_B$ induces a weak equivalence of simplicial sets
$$
\mMC_{\bul} \big(\, \Cyl_{\c}(\cC, A, B)^{Q_{\vf}}\, \big) \to
\mMC_{\bul} \big( \bsi \Hom(\cC_{\c}(B), B) \big)
$$
and hence a bijection
$$
\pi_0\Big( \mMC_{\bul} \big(\, \Cyl_{\c}(\cC, A, B)^{Q_{\vf}}\, \big) \Big) \to
\pi_0\Big( \mMC_{\bul} \big( \bsi \Hom(\cC_{\c}(B), B) \big) \Big)\,.
$$

Thus Theorem \ref{thm:HTT} is a simple consequence of the fact
that MC elements of the $\sLie$-algebra
\begin{equation}
\label{bsi-Hom-cCB-B}
\bsi \Hom(\cC_{\c}(B), B)
\end{equation}
are in bijection with
$\Cobar(\cC)$-algebra structures on $B$. Moreover, homotopy equivalent
$\Cobar(\cC)$-algebra structures on $B$ correspond precisely to isomorphic
MC elements of \eqref{bsi-Hom-cCB-B}.
\end{proof}

\begin{remark}
\label{rem:HTT}
In the usual version of the Homotopy Transfer Theorem (see  \cite[Theorem 10.3.2]{LV})
one constructs a $\Cobar(\cC)$-algebra structure on $A$ and an $\infty$ quasi-isomorphism
$F$ from $A$ to $B$ with the original $\Cobar(\cC)$-algebra structure, while in the
above theorem, $F$ lands in $(B, Q_B)$ where $Q_B$ is only homotopy equivalent to
the original one. On the other hand, given an isomorphism connecting two MC elements
$\ti{Q}_B$ and $Q_B$ in
$$
\mMC_{\bul} \big( \bsi \Hom(\cC_{\c}(B), B) \big)\,,
$$
it is easy to construct an $\infty$-morphism $G$ from $(B, Q_B)$ to
$(B, \ti{Q}_B)$ whose linear term is $\id_B$\,. So the ``usual'' version of
the Homotopy Transfer Theorem follows from Theorem \ref{thm:HTT}.
\end{remark}

\appendix

\section{Proof of Proposition \ref{prop:Conv-V-A}}
\label{app:proof-Conv}

Although very similar claims to Proposition \ref{prop:Conv-V-A}
appeared in the literature (see, for example, \cite{Berglund},
\cite{FMYau}, \cite{KhPQ}, \cite{MVnado1}) we still decided to give its proof
for convenience of the reader.

Let us denote by $Q'_A$ the composition
$$
p_A \circ Q_A : \cC(A) \to A\,.
$$

To prove that multi-bracket \eqref{m-bracket} is symmetric
in its arguments, we let
\begin{equation}
\label{Delta-m}
\D_m(v) = \sum_{\al} (\ga_{\al}; v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m)
\end{equation}
and recall that $\D_m(v)$ lands in $S_m$-invariants of $ \cC(m) \otimes V^{\otimes\, m} $.
In other words, for every $\si \in S_m$ we have
\begin{equation}
\label{S-m-invar}
\sum_{\al} \big( \si^{-1}(\ga_{\al}); v^{\al}_{\si(1)}, v^{\al}_{\si(2)}, \dots, v^{\al}_{\si(m)} \big) =
 \sum_{\al} (\ga_{\al}; v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m)\,.
\end{equation}

Therefore, for every $\si \in S_m$, we have
\begin{multline*}
\{f_{\si(1)}, \dots, f_{\si(m)}\} (v) = \\
\sum_{\al} Q'_A \circ (1 \otimes f_{\si(1)}\otimes \dots \otimes f_{\si(m)} )
 \big( \si^{-1}(\ga_{\al}); v^{\al}_{\si(1)}, v^{\al}_{\si(2)}, \dots, v^{\al}_{\si(m)} \big)=\\
\sum_{\al} \pm\, Q'_A \big( \si^{-1}(\ga_{\al});   f_{\si(1)}(v^{\al}_{\si(1)}),
f_{\si(2)}(v^{\al}_{\si(2)}), \dots, f_{\si(m)}(v^{\al}_{\si(m)}) \big)\,.
\end{multline*}

Thus, since $Q_A$ is compatible with the action of the symmetric group,
\begin{multline*}
\{f_{\si(1)}, \dots, f_{\si(m)}\} (v) =\\
\sum_{\al} \pm\, Q'_A \big( \si^{-1}(\ga_{\al});   f_{\si(1)}(v^{\al}_{\si(1)}),
f_{\si(2)}(v^{\al}_{\si(2)}), \dots, f_{\si(m)}(v^{\al}_{\si(m)}) \big) =\\
\sum_{\al} \pm\, Q'_A \big( \ga_{\al};   f_{1}(v^{\al}_{1}),
f_{2}(v^{\al}_{2}), \dots, f_{m}(v^{\al}_{m}) \big) =\\
 \sum_{\al} \pm Q'_A \circ (1 \otimes f_{1}\otimes \dots \otimes f_{m} )
 (\ga_{\al}; v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m) =
\pm \{f_{1}, \dots, f_{m}\} (v)\,.
\end{multline*}
So multi-bracket \eqref{m-bracket} is indeed symmetric\footnote{In the above calculations,
the sign factors can be easily deduced from the Koszul rule of signs.}
in its arguments.

Let us now prove that the operation
\begin{equation}
\label{diff-Conv}
f \mapsto \{f\}
\end{equation}
is a differential on  $\Hom(V, A)$, i.e. $\{\{f\}\} =0$ for every
$f \in  \Hom(V, A)$.

Indeed, using the identities $\pa^2_A =0$, $\pa^2_{V} = 0$
and the compatibility of $\D_1$ with the differentials
on $V$ and $\cC(1) \otimes V$, we get
\begin{multline}
\label{1-brack-square}
\{\{f\}\}(v) = \pa_{A}\, \{f\}(v) -(-1)^{|f|+1} \{f\}(\pa_{V}\, v)  +  Q'_A\big( 1 \otimes \{f\} (\D_1(v)) \big) =\\
\pa_A \circ Q'_A \circ (1 \otimes f) \circ \D_1(v) +
(-1)^{|f|} Q'_A \circ (1 \otimes f) \big( (\pa_{\cC} + \pa_V) \D_1(v) \big) \\
+ Q'_A \circ (1\otimes (\pa_A \circ f)) \big(\D_1 (v) \big)
-(-1)^{|f|} Q'_A \circ (1\otimes (f\circ \pa_V)) \big(\D_1 (v) \big) \\
+ Q'_A \circ (1 \otimes Q'_A) \circ (1\otimes 1 \otimes f) \big( (1\otimes \D_1)\circ \D_1 (v) \big)
\end{multline}
for every $v \in V$\,.

Using the axioms of a coalgebra over a cooperad, we rewrite the
expression $(1\otimes \D_1)\circ \D_1 (v) $ in \eqref{1-brack-square} as
follows
\begin{equation}
\label{D1D1}
(1\otimes \D_1)\circ \D_1 (v) = (\D^{\cC}  \otimes 1) \circ \D_1 (v)\,,
\end{equation}
where $\D^{\cC}$ is the cooperadic comultiplication
$$
\D^{\cC} : \cC(1) \to \cC(1) \otimes \cC(1)\,.
$$

Therefore the expression $\{\{f\}\}(v)$ can be rewritten as
\begin{equation}
\label{1-brack-sq-easy}
\{\{f\}\}(v) =  \big(\pa_{A} \circ Q'_A + Q'_A \circ (\pa_{\cC} + \pa_A) + Q'_A \circ Q_A
\big) \circ (1 \otimes f) \circ \D_1 (v) \,.
\end{equation}

Thus the identity $ \{\{f\}\} =0 $ is a consequence of the
MC equation for $Q'_A$.

Our goal now is to prove the relation
\begin{equation}
\label{goal-brack}
\sum_{p=1}^{m}
\sum_{\si \in \Sh_{p, m-p} }
(-1)^{\ve(\si; f_1, \dots, f_m)}
\{ \{f_{\si(1)}, \dots, f_{\si(p)} \}, f_{\si(p+1)}, \dots, f_{\si(m)} \} (v) =0
\end{equation}
for every $m \ge 2$ and $v \in V$\,, where the sign factor
$(-1)^{\ve(\si; f_1, \dots, f_m)}$ is defined in \eqref{ve-si-vvv}.

In our calculations below, we often put $\pm$ instead of
the precise sign factor. These sign factors can be easily deduced
from the Koszul rule of signs.

Unfolding \eqref{goal-brack} we get
\begin{multline}
\label{goal-unfolded}
\sum_{p=1}^{m}
\sum_{\si \in \Sh_{p, m-p} }
(-1)^{\ve(\si; f_1, \dots, f_m)}
\{ \{f_{\si(1)}, \dots, f_{\si(p)} \}, f_{\si(p+1)}, \dots, f_{\si(m)} \} (v) = \\
\pa_{A} \big( \{f_1, \dots, f_m\} (v)\big) + (-1)^{|f_1| + \dots + |f_m|}
 \{f_1, \dots, f_m\} \big(\pa_{V}(v) \big) \\
+ \sum_{i=1}^{m} (-1)^{|f_1| + \dots + |f_{i-1}|} \{f_1, \dots, (\pa_{A} \circ f_i - (-1)^{|f_i|} f_i \circ \pa_{V} ), \dots,  f_m\} (v)\\
+ \sum_{ \substack{1 \le  p \le  m\\[0.1cm]
\si \in \Sh_{p, m-p} } } (-1)^{\ve(\si; f_1, \dots, f_m)}
 Q'_{A}\big(1; Q'_{A}\big( (1; f_{\si(1)}, \dots, f_{\si(p)}) \D_{p}(-)\big) , f_{\si(p+1)}, \dots, f_{\si(m)} (\D_{m-p+1}(v))\big)\,.
\end{multline}

Expanding the expression $\pa_{A} \big( \{f_1, \dots, f_m\} (v)\big)$, we get
\begin{eqnarray*}
\pa_{A} \big( \{f_1, \dots, f_m\} (v)\big) &=& \pa_{A}Q'_A \big( 1; f_{1},\dots, f_{m}
(\D_m (v))  \big)\\
&=& \sum_{\al}\pa_{A}Q'_A \big( (1; f_{1}, \dots, f_{m}  )
( \ga_{\al}; v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m))\\
&=& \sum_{\al}\pm\pa_{A}Q'_A \big( \ga_{\al}; f_{1}(v^{\al}_1), \dots, f_{m}(v^{\al}_m)).
\end{eqnarray*}

Using
$$
\D_m(\pa_V(v)) =
\sum_{\al} (\pa_{\cC}(\ga_{\al}); v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m) +
\sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}}\pm(\ga_{\al}; v^{\al}_1, \dots, \pa_V(v^{\al}_{i}),\dots, v^{\al}_m)\,,
$$
we expand $\{f_1, \dots, f_m\} \big(\pa_{V}(v) \big)$ obtaining
\begin{eqnarray*}
\{f_1, \dots, f_m\} \big(\pa_{V}(v) \big) &=& Q'_A \big( 1; f_{1}, \dots, f_{m}
(\D_m (\pa_{V}(v)))  \big)\\
&=&\sum_{\al} Q'_A \big( (1; f_{1}, \dots, f_{m} )
(\pa_{\cC}(\ga_{\al}); v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m)  \big)\\
&+&  \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}}\pm Q'_A \big( (1; f_{1}, \dots, f_{m} )(\ga_{\al}; v^{\al}_1, \dots, \pa_V(v^{\al}_{i}),\dots, v^{\al}_m)\big)\\
&=&\sum_{\al} \pm Q'_A \big( \pa_{\cC}(\ga_{\al}); f_{1}(v^{\al}_1), \dots, f_{m}(v^{\al}_m) \big) \\
&+& \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}} \pm Q'_A \big(\ga_{\al}; f_{1}(v^{\al}_1), \dots, f_i(\pa_V(v^{\al}_{i})),\dots, f_{m}(v^{\al}_m)\big).
\end{eqnarray*}

Expanding the sum
$$
\sum_{i=1}^{m} (-1)^{|f_1| + \dots + |f_{i-1}|} \{f_1, \dots, (\pa_{A} \circ f_i - (-1)^{|f_i|} f_i \circ \pa_{V} ), \dots,  f_m\} (v)
$$
we obtain
\begin{multline}
\label{brack-f-df-fd-f}
\sum_{i=1}^{m} (-1)^{|f_1| + \dots + |f_{i-1}|} \{f_1, \dots, (\pa_{A} \circ f_i - (-1)^{|f_i|} f_i \circ \pa_{V} ), \dots,  f_m\} (v) = \\
 \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}} \pm Q'_A \big( 1; f_{1},\dots,
  (\pa_{A} \circ f_i - (-1)^{|f_i|} f_i \circ \pa_{V} ), \dots, f_{m} \big)(\ga_{\al}; v^{\al}_1, v^{\al}_{2}, \dots, v^{\al}_m) \\
 = \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}}\pm Q'_A \big( \ga_{\al}; f_{1}(v^{\al}_1),\dots,
   (\pa_{A} \circ f_i - (-1)^{|f_i|} f_i \circ \pa_{V} ) (v^{\al}_i), \dots, f_{m}(v^{\al}_m) \big)\\
 = \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}}\pm Q'_A \big( \ga_{\al}; f_{1}(v^{\al}_1),\dots, \pa_{A} (f_i(v^{\al}_i)),\dots,f_{m}(v^{\al}_m) \big)\\
 - \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}}\pm Q'_A \big( \ga_{\al};f_{1}(v^{\al}_1),\dots, f_i(\pa_{V}v^{\al}_i),\dots, f_{m}(v^{\al}_m) \big).
\end{multline}

The last sum in the R.H.S. of \eqref{goal-unfolded} is expanded as follows:
\begin{multline}
\label{QQ-fff}
\sum_{ \substack{1 \le  p \le  m\\[0.1cm]
\si \in \Sh_{p, m-p} } } (-1)^{\ve(\si; f_1, \dots, f_m)}
 Q'_{A}\big(1; Q'_{A}\big( (1; f_{\si(1)}, \dots, f_{\si(p)}) \D_{p}(-)\big) , f_{\si(p+1)}, \dots, f_{\si(m)} (\D_{m-p+1}(v))\big) \\
= \sum_{ \substack{1 \le  p \le  m\\[0.1cm]
\si \in \Sh_{p, m-p} } }\pm
Q'_A ( 1 \otimes Q'_A  \otimes 1^{\otimes (m-p)}) \big( 1; (1; f_{\si(1)}, \dots , f_{\si(p)})
, f_{\si(p+1)},\dots , f_{\si(m)} \big) \\
\big( (1 \otimes \D_p \otimes 1^{\otimes (m-p)} ) \circ  \D_{m-p+1} (v)  \big)\,.
\end{multline}

Using the axioms of $\cC$-coalgebra structure on $V$, we can rewrite the term
$(1 \otimes \D_p \otimes 1^{\otimes (m-p)} )\circ \D_{m-p+1} (v) $
as follows:
\begin{equation}
\label{Del-Del-v}
(1 \otimes \D_p \otimes 1^{\otimes (m-p)} ) \circ \D_{m-p+1} (v) =
(\D^{\cC}_{\bt_{m, p}} \otimes 1^{\otimes \, m})\circ \D_m (v)\,,
\end{equation}
where $\bt_{m,p}$ is the labeled planar tree depicted in Figure
\ref{fig:bt-m-p-i}.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5]
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=3, inner sep=1]
\node[w] (l1) at (0, 3) {};
\draw (6,3) node[anchor=center] {{\small $\dots$}};
\node[b] (limp) at (3, 3) {};
\draw (3,3.6) node[anchor=center] {{\small $p+1$}};

\node[b] (limp1) at (-2, 5) {};
\draw (-2.6,5.6) node[anchor=center] {{\small $1$}};
\draw (0,5) node[anchor=center] {{\small $\dots$}};
\node[b] (li) at (2, 5) {};
\draw (2.4,5.6) node[anchor=center] {{\small $p$}};

\node[b] (lm) at (9, 3) {};
\draw (9,3.6) node[anchor=center] {{\small $m$}};

\node[w] (v1) at (6, 0) {};
\node[b] (r) at (6, -1) {};

\draw (r) edge (v1);
\draw (v1) edge (l1);
\draw (v1) edge (limp);
\draw (v1) edge (lm);
\draw (l1) edge (limp1);
\draw (l1) edge (li);
\end{tikzpicture}
\caption{\label{fig:bt-m-p-i} The labeled planar tree $\bt_{m,p}$}
\end{figure}
and let $\ga^{\si}_{\al, \beta, 1}$ and $\ga^{\si}_{\al, \beta, 2}$ be the tensor factors in
\begin{equation}
\label{Del-tau-bt-mpi}
\D^{\cC}_{\si(\bt_{m,p})}(\ga_{\al}) = \sum_{\beta} \ga^{\si}_{\al, \beta, 1} \otimes \ga^{\si}_{\al, \beta, 2}\,,
\end{equation}
where $\si(\bt_{m,p})$ is the tree corresponding to the $(p,m-p)$-shuffle $\si$.

Using the axioms of the cooperadic comultiplication $\D^{\cC}$, we get
$$
\textrm{Sum \eqref{QQ-fff}} =
$$
\begin{equation}
\label{the-sum111}
\sum^{\al, \beta}_{ \substack{1 \le  p \le  m \\[0.1cm]
\si \in \Sh_{p, m-p} } }
\pm Q'_A( 1 \otimes Q'_A \otimes 1^{\otimes (m-p)})
\end{equation}
$$
\Big(\, \ga^{\si}_{\al, \beta, 1}\,; ( \,\ga^{\si}_{\al, \beta, 2}; f_{\si(1)}(v^{\al}_{\si(1)}),\dots, f_{\si(p)}(v^{\al}_{\si(p)})),
f_{\si(p+1)} (v^{\al}_{\si(p+1)}), \dots,
f_{\si(m)} (v^{\al}_{\si(m)})  \Big)\,,
$$
which can be rewritten in terms of the bracket on
the convolution Lie algebra $\Conv(\cC_{\c}, \End_{A})$ (Prop. 4.1, \cite{notes}).
Namely,
$$
\textrm{Sum \eqref{QQ-fff} } =
$$
\begin{equation}
\label{the-sum-OK}
\frac{1}{2} \sum_{\al} [Q'_A , Q'_A] (\ga_{\al}; f_1(v^{\al}_1), \dots, f_m(v^{\al}_m))\,,
\end{equation}
where we tacitly identify the composition $Q'_A = p_A \circ Q_A$ with
the corresponding element in
\begin{equation}
\label{Conv-cC-End-A}
\Conv(\cC_{\c}, \End_{A}) = \prod_{n \ge 1}
\Hom_{S_n} \big(\cC_{\c}(n),   \End_{A}(n) \big)\,.
\end{equation}

Collecting the expanded terms of the R.H.S. of \eqref{goal-unfolded} we obtain
\begin{equation*}
\sum_{\al}\pm\pa_{A}Q'_A \big( \ga_{\al}; f_{1}(v^{\al}_1), \dots, f_{m}(v^{\al}_m))
+ \sum_{\al} \pm Q'_A \big( \pa_{\cC}(\ga_{\al}); f_{1}(v^{\al}_1), \dots, f_{m}(v^{\al}_m) ) \\
\end{equation*}
\begin{equation}
\label{pa_V-1}
+ \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}} \pm Q'_A \big(\ga_{\al}; f_{1}(v^{\al}_1), \dots, f_i(\pa_V(v^{\al}_{i})),
\dots,  f_{m}(v^{\al}_m)\big)\\
\end{equation}
\begin{equation}
\label{pa_V-2}
 - \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}}\pm Q'_A \big( \ga_{\al}; f_{1}(v^{\al}_1), \dots, f_i(\pa_{V}v^{\al}_i), \dots, f_{m}(v^{\al}_m) \big)\\
 \end{equation}
\begin{equation*}
 \sum_{ \substack{\al \\[0.1cm] 1\leq i\leq m}} \pm Q'_A \big( \ga_{\al}; f_{1}(v^{\al}_1), \dots, \pa_{A} (f_i(v^{\al}_i)), \dots, f_{m}(v^{\al}_m) \big)
 + \frac{1}{2} \sum_{\al} [Q'_A , Q'_A] (\ga_{\al}; f_1(v^{\al}_1), \dots, f_m(v^{\al}_m))\,
\end{equation*}

Canceling terms \eqref{pa_V-1} and \eqref{pa_V-2}, we see that the right hand side
of \eqref{goal-unfolded} can be rewritten as
$$
\textrm{The R.H.S. of  \eqref{goal-unfolded} }  =
$$
\begin{equation}
\Big( \pa_A  \circ Q'_A  + Q'_A \circ (\pa_{\cC} + \pa_A)   + \frac{1}{2} [Q'_A, Q'_A] \Big)  (1;f_1,\ldots,f_m)(\D_m(v))\,,
\label{RHS-goal-unfolded}
\end{equation}
where, as above, we identify $Q'_A$ with the corresponding element in
\eqref{Conv-cC-End-A}.

Thus desired equations \eqref{goal-brack} are satisfied due to the fact that
$Q'_A$ is a MC element of dg Lie algebra \eqref{Conv-cC-End-A}.

We conclude this proof by a comment about the sign factors.

All sign factors in the above computations are subject to the usual Koszul rule.
For example, to show explicitly that terms \eqref{pa_V-1} and \eqref{pa_V-2} cancel
each other, we need to verify that the corresponding contributions from the term
\begin{equation}
\label{term1}
(-1)^{|f_1| + \dots + |f_m|}  \{f_1,\dots,f_m\}(\pa_{V}(v))
\end{equation}
matches with the contributions from
\begin{equation}
\label{term2}
(-1)^{|f_1| + \dots + |f_i|} \{f_1,\dots,f_i \circ  \pa_{V},\dots,f_m\}(v)\,.
\end{equation}
in equation \eqref{goal-unfolded}.

It is easy to see that the contribution
\begin{equation}
\label{contribution}
Q'_{A}(\ga_{\al};f_1(v_{1}^{\al}),\dots,f_i \circ \pa_V(v_{i}^{\al}),\dots f_m(v_{m}^{\al}))
\end{equation}
from \eqref{term1} and from \eqref{term2} has the same sign factor
$$
(-1)^{\ve+ \ve'},
$$
where
$$
\ve = |\ga_{\al}| (1 + |f_1| + \dots + |f_m|) + |v_1^{\al}| + \dots + |v_{i-1}^{\al}| +
|f_{1}| + \dots + |f_i|\,,
$$
and
$$
\ve' = |v_1^{\al}|(|f_2|+\dots +|f_{m}|) + \dots + |v^{\al}_{m-1}| |f_{m}|\,.
$$

Proposition \ref{prop:Conv-V-A} is proved.  \qed

config: arxiv