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# Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces

Marvin Verstraete

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Marvin Verstraete. Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces. 2025. hal-05151553

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# Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces

Marvin VERSTRAETE

July 4, 2025

## Abstract

The purpose of this memoir is to study pre-Lie algebras up to homotopy with divided powers, and to use this algebraic structure for the study of mapping spaces in the category of operads. We define a new notion of algebra called $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra which characterizes the notion of $\Gamma(\mathcal{P}\mathcal{L}_{\infty}, -)$-algebra. We also define a notion of a Maurer-Cartan element in complete $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebras which generalizes the classical definition in Lie algebras. We prove that for every complete brace algebra $A$, and for every $n \ge 0$, the tensor product $A \otimes \Sigma N^*(\Delta^n)$ is endowed with the structure of a complete $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra, and define the simplicial Maurer-Cartan set $\mathcal{M}\mathcal{C}_{\bullet}(A)$ associated to $A$ as the Maurer-Cartan set of $A \otimes \Sigma N^*(\Delta^{\bullet})$. We compute the homotopy groups of this simplicial set, and prove that the functor $\mathcal{M}\mathcal{C}_{\bullet}(-)$ satisfies a homotopy invariance result, which extends the Goldman-Millson theorem in dimension 0. As an application, we give a description of mapping spaces in the category of non-symmetric operads in terms of this simplicial Maurer-Cartan set. We establish a generalization of the latter result for symmetric operads.

## Contents

<table><tr><td><strong>Introduction</strong></td><td style="text-align: right;"><strong>2</strong></td></tr><tr><td>State-of-the-art in characteristic 0</td><td style="text-align: right;">3</td></tr><tr><td>Objective and Results</td><td style="text-align: right;">5</td></tr><tr><td>Organization of the memoir</td><td style="text-align: right;">8</td></tr><tr><td>Acknowledgements</td><td style="text-align: right;">10</td></tr><tr><td><strong>1 Conventions and notations</strong></td><td style="text-align: right;"><strong>10</strong></td></tr><tr><td>1.1 The category dgMod<sub>K</sub></td><td style="text-align: right;">10</td></tr><tr><td>1.2 The notion of an operad and a cooperad</td><td style="text-align: right;">13</td></tr><tr><td>1.3 On trees and the operad Brace</td><td style="text-align: right;">18</td></tr><tr><td>1.4 On the Barratt-Eccles and the surjection operads</td><td style="text-align: right;">20</td></tr><tr><td>1.5 Appendix: basic results on permutations</td><td style="text-align: right;">23</td></tr><tr><td><strong>2 On PreLie<sub>∞</sub>-algebras with divided powers</strong></td><td style="text-align: right;"><strong>24</strong></td></tr><tr><td>2.1 Recollections on pre-Lie algebras up to homotopy</td><td style="text-align: right;">25</td></tr><tr><td>2.2 The category $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$</td><td style="text-align: right;">27</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>2.3</td>
      <td>Symmetric weighted braces and Maurer-Cartan elements in ΓΛP<span style="text-decoration: overline">L</span><sub>∞</sub></td>
      <td>33</td>
    </tr>
    <tr>
      <td>2.4</td>
      <td>Pre-Lie algebras up to homotopy with divided powers and ΓΛP<span style="text-decoration: overline">L</span><sub>∞</sub></td>
      <td>39</td>
    </tr>
    <tr>
      <td><b>3</b></td>
      <td><b>A morphism from PreLie<sub>∞</sub> to Brace ⊗ <span style="text-decoration: overline">E</span><sub>H</sub></b></td>
      <td><b>44</b></td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>A morphism from B<sup>c</sup>(Λ<sup>-1</sup>Brace<sup>√</sup>) to <span style="text-decoration: overline">E</span></td>
      <td>44</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>On the twisted coderivation of Brace<sup>√</sup>(ΣN<sup>*</sup>(Δ<sup>n</sup>))</td>
      <td>47</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>A morphism from PreLie<sub>∞</sub> to Brace ⊗ B<sup>c</sup>(Λ<sup>-1</sup>Brace<sup>√</sup>)</td>
      <td>59</td>
    </tr>
    <tr>
      <td><b>4</b></td>
      <td><b>The simplicial Maurer-Cartan set of a complete brace algebra</b></td>
      <td><b>62</b></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>The simplicial set MC<sub>•</sub>(A)</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Connected components and homotopy groups of MC<sub>•</sub>(A)</td>
      <td>65</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Remarks: interpretation of the low dimensional twisting cod derivations</td>
      <td>85</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>A Goldman-Millson theorem.</td>
      <td>93</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Comparison with the deformation theory of shifted Lie<sub>∞</sub>-algebras</td>
      <td>98</td>
    </tr>
    <tr>
      <td><b>5</b></td>
      <td><b>A mapping space in the category of non-symmetric operads</b></td>
      <td><b>102</b></td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>The free operad functor and the model structure on Op</td>
      <td>102</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>A cosimplicial frame for B<sup>c</sup>(C)</td>
      <td>106</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Computation of Map<sub>Op</sub>(B<sup>c</sup>(C), P)</td>
      <td>108</td>
    </tr>
    <tr>
      <td><b>6</b></td>
      <td><b>A mapping space in the category of symmetric connected operads</b></td>
      <td><b>112</b></td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>The free symmetric operad functor and the model structure on ΣOp<sup>0</sup></td>
      <td>112</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>A cosimplicial frame for B<sup>c</sup>(C ⊗ <b>Sur</b><sub>K</sub>)</td>
      <td>116</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Computation of Map<sub>ΣOp<sup>0</sup></sub>(B<sup>c</sup>(C), P)</td>
      <td>123</td>
    </tr>
    <tr>
      <td><b>References</b></td>
      <td></td>
      <td><b>128</b></td>
    </tr>
  </tbody>
</table>

Introduction

The usual category of topological spaces comes equipped with a functor $\mathrm{Map}_{\mathcal{T}_{op}}(-, -) : \mathcal{T}_{op}^{op} \times \mathcal{T}_{op} \to \mathrm{sSet}$ which endows $\mathcal{T}_{op}$ with the structure of a simplicial category (see for instance [Fre17b, §2.1.1]). This functor can be used in order to handle higher homotopies in the category $\mathcal{T}_{op}$. For every topological spaces $X, Y$, the connected components of $\mathrm{Map}_{\mathcal{T}_{op}}(X, Y)$ are in bijection with homotopy classes of morphisms $X \to Y$, while the homotopy groups encode higher homotopy relations. This approach allows us to use tools from algebraic topology in order to study homotopy morphisms from $X$ to $Y$. The functor $\mathrm{Map}_{\mathcal{T}_{op}}(-, -)$ is defined as follows. For every $X \in \mathcal{T}_{op}$, we define two functors $X \otimes - : \mathrm{sSet} \to \mathcal{T}_{op}$ and $X^{-} : \mathrm{sSet}^{op} \to \mathcal{T}_{op}$ by

$$
X \otimes K := X \times |K| ; \quad X^K := \mathrm{Mor}_{\mathcal{T}_{op}}(|K|, X),
$$

for every $X \in \mathcal{T}_{op}$ and $K \in \mathrm{sSet}$, where $|K| \in \mathcal{T}_{op}$ denotes the geometric realization of the
simplicial set $K$. For every $X, Y \in \mathcal{T}_{op}$ and $K \in \mathrm{sSet}$, we have the isomorphism

$$
\mathrm{Mor}_{\mathcal{T}_{op}}(X \otimes K, Y) \simeq \mathrm{Mor}_{\mathcal{T}_{op}}(X, Y^K).
$$
We then define $Map_{\mathcal{T}_{op}}(X, Y)$ as the simplicial set $Mor_{\mathcal{T}_{op}}(X \otimes \Delta^{\bullet}, Y)$, where, for every $n \ge 0$, we denote by $\Delta^n$ the fundamental $n$-simplex.

In a general model category $C$, we have an analogue of the functors $X \otimes -$ and $X^-$. Such functors are defined by giving the image of $\Delta^n$ for every $n \ge 0$. The cosimplicial set $X \otimes \Delta^{\bullet}$ is called a *cosimplicial frame* associated to $X$, while $X^{\Delta^{\bullet}}$ is called a *simplicial frame* associated to $X$ (see for instance [Fre17b, §3.2.2, §3.2.7]). However, we only have a zig-zag of weak-equivalences of simplicial sets between $Mor_C(X \otimes \Delta^{\bullet}, Y)$ and $Mor_C(X, Y^{\Delta^{\bullet}})$ instead of an isomorphism, provided that $X$ is cofibrant and $Y$ is fibrant. This still allows us to construct a simplicial set $Map_C(X, Y)$, which is unique up to a zig-zag of weak-equivalences. As in $\mathcal{T}_{op}$, the connected components of $Map_C(X, Y)$ are in bijection with homotopy classes of morphisms $X \to Y$.

In this memoir, we provide an approach in order to study the homotopy of such mapping spaces in the category of non-symmetric operads and in the category of symmetric operads, where in both cases we consider operads defined in the category of differential graded $\mathbb{K}$-modules (dg $\mathbb{K}$-modules for short).

The category of operads in dg $\mathbb{K}$-modules inherits a model structures (see [Hin97] and [Hin03]). Therefore, we can use the above theory to construct mapping spaces in the category of operads. We aim to give an explicit description of these mapping spaces, extending known results in characteristic zero to the case where the ground ring $\mathbb{K}$ is a field of positive characteristic. More precisely, we describe these mapping spaces as simplicial Maurer-Cartan spaces associated to some pre-Lie algebra up to homotopy with divided powers, a notion that we also define and study in this memoir.

We review the known results in characteristic 0 before explaining our results in details in positive characteristic.

## State-of-the-art in characteristic 0

A comprehensive study of the homotopy type of a mapping spaces in the category of symmetric operads has already been done in the case $char(\mathbb{K}) = 0$. Let $\mathcal{C}$ be a coaugmented connected cooperad and $\mathcal{P}$ be an augmented connected operad. The computation of the homotopy groups can be deduced from a description of a mapping space $Map_{\Sigma_{\mathcal{O}p^0}}(B^c(\mathcal{C}), \mathcal{P})$ given in [Yal16], in the context of properads. For this purpose, we use an explicit simplicial frame $\mathcal{P}^{\Delta^{\bullet}}$ associated to the operad $\mathcal{P}$, given by:

$$ \mathcal{P}^{\Delta^{\bullet}} := \mathcal{P} \otimes \Omega^*(\Delta^{\bullet}), $$

where, for every $n \ge 0$, we denote by $\Omega^*(\Delta^n)$ the Sullivan algebra of de Rham polynomial forms on $\Delta^n$ (see for instance [BG76, §2.1]). The $n$-simplices of $Map_{\Sigma_{\mathcal{O}p^0}}(B^c(\mathcal{C}), \mathcal{P})$ then correspond to elements in $Hom_{\Sigma_{\mathcal{O}eq_k}}(\bar{\mathcal{C}}, \bar{\mathcal{P}}) \hat{\otimes} \Omega^*(\Delta^n)$ which satisfy some equations, where $\hat{\otimes}$ is the complete tensor product associated to the complete filtered dg modules $Hom_{\Sigma_{\mathcal{O}eq_k}}(\bar{\mathcal{C}}, \bar{\mathcal{P}})$ and $\Omega^*(\Delta^n)$. These equations can be written by using the Lie algebra structure on $Hom_{\Sigma_{\mathcal{O}eq_k}}(\bar{\mathcal{C}}, \bar{\mathcal{P}})$ induced by its pre-Lie algebra structure (see for instance [LV12, §6.4.4] for a definition of the pre-Lie product). We recall the definitions in the paragraphs to follow.
Recall that if $L$ is a complete Lie algebra, then a Maurer-Cartan element is an element $\tau \in L_{-1}$ such that

$$d(\tau) + \frac{1}{2}[\tau, \tau] = 0.$$

We denote by $\mathcal{MC}(L)$ the set of Maurer-Cartan elements in $L$. Note that every $\tau \in \mathcal{MC}(L)$ induces a differential $d_{\tau}$ defined by

$$d_{\tau}(x) = d(x) + [x, \tau].$$

We let $L^{\tau}$ be the dg $\mathbb{K}$-module $L$ endowed with the differential $d_{\tau}$. Using that $\Omega^{*}(\Delta^{n})$ is endowed with the structure of a commutative algebra for every $n \ge 0$, the dg $\mathbb{K}$-module $L \hat{\otimes} \Omega^{*}(\Delta^{n})$ is endowed with the structure of a Lie algebra. We define the simplicial Maurer-Cartan set associated to $L$ as

$$\mathcal{MC}_{\bullet}(L) = \mathcal{MC}(L \hat{\otimes} \Omega^{*}(\Delta^{\bullet})).$$

From [Yal16, Theorem 3.12], for every coaugmented cooperad $\mathcal{C}$, and for every augmented operad $\mathcal{P}$, we obtain the following description:

$$\mathrm{Map}_{\Sigma\mathcal{O}_p^0}(B^c(\mathcal{C}), \mathcal{P}) = \mathcal{MC}_{\bullet}(\mathrm{Hom}_{\Sigma\mathrm{Seq}_{\mathbb{K}}}(\bar{C}, \bar{P})).$$

The computation of the homotopy groups of the simplicial set $\mathrm{Map}_{\Sigma\mathcal{O}_p}(B^c(\mathcal{C}), \mathcal{P})$ can be deduced from the general computations of the homotopy groups of $\mathcal{MC}_{\bullet}(L)$ associated to a given complete Lie algebra $L$. These computations have been made in [Ber15, Theorem 1.1]. Explicitly, if $L$ is a complete Lie algebra, then, for every $\tau \in \mathcal{MC}(L)$ and $k \ge 0$, we have the isomorphism

$$\pi_{k+1}(\mathcal{MC}_{\bullet}(L), \tau) \simeq H_k(L^{\tau}),$$

where $H_0(L^{\tau})$ is endowed with the group structure $BCH$ given by the Baker-Campbell-Hausdorff formula.

The computation of the connected components of $\mathrm{Map}_{\Sigma\mathcal{O}_p}(B^c(\mathcal{C}), \mathcal{P})$ can be achieved by using the pre-Lie deformation theory developed in [DSV16]. Recall that a pre-Lie algebra is a dg $\mathbb{K}$-module $L$ endowed with a linear morphism $\star : L \otimes L \to L$ such that

$$(x \star y) \star z - x \star (y \star z) = (-1)^{|y||z|}((x \star z) \star y - x \star (z \star y)).$$

In particular, any pre-Lie algebra $L$ is endowed with the structure of a Lie algebra with the bracket $[x, y] = x \star y - (-1)^{|x||y|}y \star x$. In [DSV16], the author generalized the Lie deformation theory to the pre-Lie context. Explicitly, a Maurer-Cartan element $\tau$ in a pre-Lie algebra $L$ is an element $\tau \in L_{-1}$ such that

$$d(\tau) + \tau \star \tau = 0.$$

The gauge group $(L_0, BCH, 0)$ can also be written in terms of pre-Lie operations. We consider the subset $1 + L_0 \subset \mathbb{K} \oplus L$. Under some convergence hypothesis, we define the circular product $\odot : L \times (1 + L_0) \to L$ by

$$x \odot (1 + y) = \sum_{n \ge 0} \frac{1}{n!} x \underbrace{\{y, \dots, y\}}_{n},$$
for every $x \in L$ and $y \in L_0$, where we denote by $-\{-, \dots, -\}$ the symmetric brace operations associated to $L$ (see [OG08] or [LM05]). We can restrict this product to an operation on $1 + L_0$ defined by

$$ (1+x) \odot (1+y) = 1+y + \sum_{n \ge 0} \frac{1}{n!} x \{ \underbrace{y, \dots, y}_{n} \} $$

for every $x, y \in L_0$. Then the triple $(1 + L_0, \odot, 1)$ is a group isomorphic to the gauge group (see [DSV16, Theorem 2]). The group $(1 + L_0, \odot, 1)$ also acts on $\mathcal{MC}(L)$ via

$$ (1 + \mu) \cdot \tau = (\tau + \mu \star \tau - d(\tau)) \odot (1 + \mu)^{\otimes -1}. $$

We define the Deligne groupoid Deligne($L$) as the category with $\mathcal{MC}(L)$ as set of objects, and $(1 + L_0, \odot, 1)$ as hom-sets.

Using a cylinder object associated to $B^c(C)$ (see [Fre17b, Theorem 3.2.14]) and [DSV16, Corollary 2], we obtain a bijection

$$ \pi_0 \mathrm{Map}_{\Sigma O_{p^0}}(B^c(C), \mathcal{P}) \simeq \pi_0 \mathrm{Deligne}(\mathrm{Hom}_{\Sigma \mathrm{Seq}_K}(\bar{C}, \bar{P})), $$

where the right hand-side denotes the set of isomorphism classes of Deligne($\mathrm{Hom}_{\Sigma \mathrm{Seq}_K}(\bar{C}, \bar{P})$).

## Objectives and Results

If $\mathrm{char}(K) > 0$, then the simplicial set $\mathcal{P} \otimes \Omega^*(\Delta^\bullet)$ given in [Yal16] is no longer a simplicial frame associated to $\mathcal{P}$, as the cohomology of $\Omega^*(\Delta^n)$ is not 0 for every $n \ge 0$.

The first description of $\pi_0 \mathrm{Map}_{\Sigma O_{p^0}}(B^c(C), \mathcal{P})$ has been generalized to the positive characteristic context in [Ver23] by using a $\Gamma(\mathrm{PreLie}, -)$-algebra structure on $\mathrm{Hom}_{\Sigma \mathrm{Seq}_K}(\bar{C}, \bar{P})$. Recall briefly that a $\Gamma(\mathrm{PreLie}, -)$-algebra is a dg module endowed with operations $-\{-, \dots, -\}_{r_1, \dots, r_n}$, defined for every integers $r_1, \dots, r_n \ge 0$, and which mimic the operations

$$ x\{y_1, \dots, y_n\}_{r_1, \dots, r_n} = \frac{1}{\prod_i r_i!} x\{\underbrace{y_1, \dots, y_1}_{r_1}, \dots, \underbrace{y_n, \dots, y_n}_{r_n}\}. $$

This notion has been studied in the non-graded context in [Ces18], and generalized to the graded context in [Ver23]. Following the formulas of [DSV16], the pre-Lie deformation theory can be generalized to a deformation theory controlled by $\Gamma(\mathrm{PreLie}, -)$-algebras, which is valid over a ring with positive characteristic. For every $\Gamma(\mathrm{PreLie}, -)$-algebra $L$, we thus have a notion of Deligne groupoid Deligne($L$) (see [Ver23, Proposition-Definition 2.30]). Using a $\Gamma(\mathrm{PreLie}, -)$-algebra structure on $\mathrm{Hom}_{\Sigma \mathrm{Seq}_K}(\bar{C}, \bar{P})$ (see [Ver23, Corollary 2.18]), we retrieve, by [Ver23, Theorem 3.6], a bijection

$$ \pi_0 \mathrm{Map}_{\Sigma O_{p^0}}(B^c(C), \mathcal{P}) \simeq \pi_0 \mathrm{Deligne}(\mathrm{Hom}_{\Sigma \mathrm{Seq}_K}(\bar{C}, \bar{P})). $$

In this memoir, we construct an explicit cosimplicial frame associated to $B^c(C)$, in the case where $C$ is a non-symmetric cooperad. Explicitly, for ever $n \ge 0$, we construct a twisting derivation $\partial^n$ on the operad $\mathcal{F}(\bar{C} \otimes \Sigma^{-1} N_*(\Delta^n))$ such that

$$ B^c(C) \otimes \Delta^\bullet := (\mathcal{F}(\bar{C} \otimes \Sigma^{-1} N_*(\Delta^\bullet)), \partial^\bullet) $$
is a cosimplicial frame associated to $B^c(C)$ where $N_*(\Delta^n)$ is the normalized chain complex of the simplicial set $\Delta^n$. The $n$-simplices of a mapping space from $B^c(C)$ to $\mathcal{P}$ can then be identified with elements of $\mathrm{Hom}_{\mathrm{Seq}_k}(\bar{C}, \bar{P}) \otimes \Sigma N^*(\Delta^n)$ which satisfy some equations. Our purpose is to interpret these equations as Maurer-Cartan equations. Our main ideas are the following. We deal with $\Gamma(\mathrm{PreLie}_\infty, -)$-algebra structures, where $\mathrm{PreLie}_\infty$ denotes the operad that governs pre-Lie algebras up to homotopy. The key point is that if $A$ is a brace algebra and if $N$ is an algebra over the Barratt-Eccles operad $\mathcal{E}$, then $A \otimes N$ is a $\Gamma(\mathrm{PreLie}_\infty, -)$-algebra. Using this result with $A = \mathrm{Hom}_{\mathrm{Seq}_k}(\bar{C}, \bar{P})$ (which is a brace algebra by [LV12, Proposition 6.4.2] and [GV95, Proposition 1]) and $N = N^*(\Delta^n)$ (see [BF04]) precisely give the desired equations.

The $\mathrm{PreLie}_\infty$-algebras, also called pre-Lie algebras up to homotopy, have been studied in [CL01]. The author characterized the data of a $\mathrm{PreLie}_\infty$-algebra structure on $L$ as the data of brace operations which satisfy some identities. We denote these brace operations by $-\{\}$ in this memoir, and we assume that these operations defined on the suspension $\Sigma L$. As for the study of the monad $\Gamma(\mathrm{PreLie}, -)$ in [Ces18], we prove that giving a $\Gamma(\mathrm{PreLie}_\infty, -)$-algebra structure on $L$ is equivalent to giving weighted brace operations $-\{\}$ on the suspension $\Sigma L$ which are similar to the operations

$$x\{y_1, \dots, y_n\}_{r_1, \dots, r_n} = \frac{1}{\prod_i r_i!} x\{\underbrace{y_1, \dots, y_1}_{r_1}, \dots, \underbrace{y_n, \dots, y_n}_{r_n}\}.$$

We give another characterization of such objects that will emphasize a notion of $\infty$-morphism. For any graded $\mathbb{K}$-module $V$, we set

$$\Gamma \mathrm{Perm}^c(V) = \bigoplus_{n \ge 0} V \otimes (V^{\otimes n})^{\Sigma_n}.$$

We prove that $\Gamma \mathrm{Perm}^c(V)$ is endowed with a coproduct $\Delta_{\Gamma \mathrm{Perm}}$ which, in some sense, is compatible with the coproduct defined in [CL01, §2.3] on $\mathrm{Perm}^c(V)$. We then define the category $\Gamma \Lambda \mathcal{P} \mathcal{L}_\infty$ formed by pairs $(V, Q)$ where $V$ is a graded $\mathbb{K}$-module and $Q$ a codерivation on $\Gamma \mathrm{Perm}^c(V)$ of degree $-1$ such that $Q^2 = 0$. A morphism in $\Gamma \Lambda \mathcal{P} \mathcal{L}_\infty$, also called an $\infty$-morphism, is a morphism of coalgebras which preserve the codiverations. We prove that $L$ is a $\Gamma(\mathrm{PreLie}_\infty, -)$-algebra if and only if $\Sigma L \in \Gamma \Lambda \mathcal{P} \mathcal{L}_\infty$. We also define a notion of (complete) filtered $\Gamma \Lambda \mathcal{P} \mathcal{L}_\infty$-algebra. The category of complete $\Gamma \Lambda \mathcal{P} \mathcal{L}_\infty$-algebras is denoted by $\Gamma \Lambda \widehat{\mathcal{P} \mathcal{L}}_\infty$.

Given an object $V \in \Gamma \Lambda \widehat{\mathcal{P} \mathcal{L}}_\infty$, a Maurer-Cartan element is a degree 0 element $x \in V$ such that

$$d(x) + \sum_{n \ge 1} x\{x\}_n = 0.$$

We denote by $\mathcal{MC}(V)$ the set formed by these objects. We prove that any $\infty$-morphism $\phi: V \rightsquigarrow W$ induces a map

$$\mathcal{MC}(\phi) : \mathcal{MC}(V) \to \mathcal{MC}(W)$$

so that $\mathcal{MC} : \Gamma \Lambda \widehat{\mathcal{P} \mathcal{L}}_\infty \to \mathrm{Set}$ is a functor.

The motivation for using $\Gamma(\mathrm{PreLie}_\infty, -)$-algebras is given by the following theorem.
**Theorem A.** Let Brace be the operad which governs brace algebras (see [Cha02, Proposition 2]).
There exists an operad morphism $\mathcal{PreLie}_\infty \to \mathcal{Brace} \otimes \mathcal{E}$ which fits in a commutative square

<figure><img src="image_3.png" /></figure>

As brace algebras are endowed with the structure of a $\Gamma(\mathcal{PreLie}, -)$-algebra (see [Ver23, Theorem 2.15]), Theorem A implies that every $\mathcal{Brace} \otimes \mathcal{E}$-algebra $L$ is a $\Gamma(\mathcal{PreLie}_\infty, -)$-algebra, via the composite

$$ \Gamma(\mathcal{PreLie}_\infty, L) \to \Gamma(\mathcal{Brace} \otimes \mathcal{E}, L) \stackrel{\sim}{\leftarrow} S(\mathcal{Brace} \otimes \mathcal{E}, L) \to L. $$

Using that the normalized cochain complex $N^*(X)$ of a simplicial set $X$ admits the structure of an algebra over the Barratt-Eccles operad (see [BF04]) and Theorem A, we define the simplicial Maurer-Cartan set associated to a complete brace algebra $A$ as

$$ \mathcal{MC}_\bullet(\mathcal{A}) = \mathcal{MC}(\mathcal{A} \otimes \Sigma N^*(\Delta^\bullet)). $$

In particular, the 0-vertices are identified with Maurer-Cartan elements in $A$, when using its underlying $\Gamma(\mathcal{PreLie}, -)$-algebra structure (see [Ver23, Theorem 2.15]). We explicitly compute the connected components and the homotopy groups of $\mathcal{MC}_\bullet(\mathcal{A})$.

**Theorem B.** For every complete brace algebra $A$, the simplicial set $\mathcal{MC}_\bullet(\mathcal{A})$ is a Kan complex.
Moreover, we have the following computations for every $\tau \in \mathcal{MC}(\mathcal{A})$.

* $\pi_0(\mathcal{MC}_\bullet(\mathcal{A})) \simeq \pi_0(\text{Deligne}(\mathcal{A}))$, where Deligne($\mathcal{A}$) denotes the Deligne groupoid associated to the $\Gamma(\mathcal{PreLie}, -)$-algebra $A$ (see [Ver23, Proposition-Definition 2.30]);

* $\pi_1(\mathcal{MC}_\bullet(\mathcal{A}), \tau) \simeq \{\mu \in A_0 \mid d(\mu) = \tau + \mu\langle\tau\rangle - \tau \odot (1+\mu)\} / \sim_\tau$, where $\sim_\tau$ is the equivalence relation such that $\mu \sim_\tau \mu'$ if and only if there exists $\psi \in A_1$ such that

$$ \mu - \mu' = d(\psi) + \psi\langle\tau\rangle + \sum_{p,q \ge 0} \tau \underbrace{\langle\mu, \dots, \mu, \psi, \mu', \dots, \mu'\rangle}_{p}^{q}. $$

* $\pi_2(\mathcal{MC}_\bullet(\mathcal{A}), \tau) \simeq (H_1(A^\tau), *_{\tau}, 0)$, where $*_{\tau}$ is the group structure on $H_1(A^\tau)$ such that

$$ [\mu] *_{\tau} [\mu'] = [\mu + \mu' + \tau \langle\mu, \mu'\rangle]. $$

* $\pi_{n+1}(\mathcal{MC}_\bullet(\mathcal{A}), \tau) \simeq H_n(A^\tau)$ for every $n \ge 3$.

We have the following homotopy invariance result, which extends the Goldman-Millson theorem in dimension 0.

**Theorem C.** Let $\Theta: A \to B$ be a morphism of complete brace algebras such that $\Theta$ is a weak equivalence in $\mathrm{dgMod}_k$. Then $\mathcal{MC}_\bullet(\Theta): \mathcal{MC}_\bullet(\mathcal{A}) \to \mathcal{MC}_\bullet(\mathcal{B})$ is a weak equivalence.
We use this new deformation theory for the study of the homotopy of mapping spaces in the category of non symmetric operads. For every non-symmetric coaugmented cooperad $\mathcal{C}$ such that $\mathcal{C}(0) = 0$, and for every $n \ge 0$, we construct a twisting derivation $\partial^n$ on the operad $\mathcal{F}(\bar{\mathcal{C}} \otimes \Sigma^{-1}N_*(\Delta^n))$ such that

$$B^c(\mathcal{C}) \otimes \Delta^\bullet := (\mathcal{F}(\bar{\mathcal{C}} \otimes \Sigma^{-1}N_*(\Delta^\bullet)), \partial^\bullet)$$

is a cosimplicial frame associated to $B^c(\mathcal{C})$. This leads to the following theorem.

**Theorem D.** Let $\mathcal{C}$ be a coaugmented cooperad and $\mathcal{P}$ be an augmented operad such that $\mathcal{C}(0) = \mathcal{P}(0) = 0$ and $\mathcal{C}(1) = \mathcal{P}(1) = \mathbb{K}$. Then we have an isomorphism of simplicial sets

$$\mathrm{Map}_{\mathcal{O}_p}(B^c(\mathcal{C}), \mathcal{P}) \simeq \mathcal{MC}_{\bullet}(\mathrm{Hom}_{\mathrm{Seq}_{\mathbb{K}}}(\bar{\mathcal{C}}, \bar{\mathcal{P}})).$$

The computation of the connected components and the homotopy groups of $\mathrm{Map}_{\mathcal{O}_p}(B^c(\mathcal{C}), \mathcal{P})$ can then be achieved by using Theorem B.

In the symmetric context, the derivation $\partial^n$ constructed above on $\mathcal{F}(\bar{\mathcal{C}} \otimes \Sigma^{-1}N_*(\Delta^n))$ does not preserve the action of the symmetric group for every $n \ge 2$. We instead consider a $\Sigma_*$-cofibrant replacement of $B^c(\mathcal{C})$ given by the map $B^c(\mathcal{C}) \xrightarrow{\sim} B^c(\mathcal{C} \otimes \mathbf{Sur}_{\mathbb{K}})$, where $\mathbf{Sur}_{\mathbb{K}}$ is the surjection cooperad defined in [BCN23, Theorem A.1]. Using that the action of $\Sigma_n$ on $\bar{\mathcal{C}}(n) \otimes \overline{\mathbf{Sur}_{\mathbb{K}}}(n)$ is free for every $n \ge 1$, we construct a twisting derivation $\partial^n$ on the operad $\mathcal{F}(\bar{\mathcal{C}} \otimes \mathbf{Sur}_{\mathbb{K}} \otimes \Sigma^{-1}N_*(\Delta^n))$ such that

$$B^c(\mathcal{C} \otimes \mathbf{Sur}_{\mathbb{K}}) \otimes \Delta^\bullet := (\mathcal{F}(\bar{\mathcal{C}} \otimes \overline{\mathbf{Sur}_{\mathbb{K}}}) \otimes \Sigma^{-1}N_*(\Delta^\bullet)), \partial^\bullet)$$

is a cosimplicial frame associated to $B^c(\mathcal{C} \otimes \mathbf{Sur}_{\mathbb{K}})$. We deduce the following theorem.

**Theorem E.** Let $\mathcal{C}$ be a symmetric coaugmented cooperad and $\mathcal{P}$ be a symmetric augmented operad such that $\mathcal{C}(0) = \mathcal{P}(0) = 0$ and $\mathcal{C}(1) = \mathcal{P}(1) = \mathbb{K}$. Then $\Sigma\mathrm{Hom}_{\Sigma\mathrm{Seq}_{\mathbb{K}}}(\bar{\mathcal{C}} \otimes \mathbf{Sur}_{\mathbb{K}} \otimes N_*(\Delta^\bullet), \bar{\mathcal{P}})$ is endowed with the structure of a $\Gamma\Lambda\Pi\mathcal{L}_\infty$-algebra such that we have an isomorphism of simplicial sets

$$\mathrm{Map}_{\Sigma\mathcal{O}p^0}^h(B^c(\mathcal{C}), \mathcal{P}) \simeq \mathcal{MC}(\Sigma\mathrm{Hom}_{\Sigma\mathrm{Seq}_{\mathbb{K}}}(\bar{\mathcal{C}} \otimes \overline{\mathbf{Sur}_{\mathbb{K}}}, \otimes N_*(\Delta^\bullet), \mathcal{P})),$$

$$\text{where } \mathrm{Map}_{\Sigma\mathcal{O}p^0}^h(B^c(\mathcal{C}), \mathcal{P}) := \mathrm{Map}_{\Sigma\mathcal{O}p^0}(B^c(\mathcal{C} \otimes \mathbf{Sur}_{\mathbb{K}}), \mathcal{P})$$

## Organization of the memoir

In the first part of this memoir, we recall notions that will be useful for explaining our results. In §1.1, we explain our conventions on the context of differential graded $\mathbb{K}$-modules (dg $\mathbb{K}$-modules) in which we carry out our constructions. We examine in particular the definition of dg $\mathbb{K}$-modules which are complete with respect to a filtration and which we use in the definition of Maurer-Cartan elements. In §1.2, we review our conventions on operads, and recall the precise definition of algebras with divided powers over an operad. In §1.3, we recall the definition of the operad that governs brace algebras and its expression in terms of $\mathbb{K}$-modules of planar rooted trees. In §1.4, we recall the definition of the Barratt-Eccles and the definition of the action of this operad on the cochain algebra of simplicial sets through an intermediate operad given by an operad of surjections.
In the second part, we study the structure of a $\Gamma(\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty}, -)$-algebra. In §2.1, we recall the construction of the operad $\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty}$ and a characterization of $\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty}$-algebras in terms of twisting coderivation on cofree Perm-coalgebras. In §2.2, we explain the definition of the category $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$. In §2.3, we explain the definition of the weighted brace operations $-\{-, \dots, -\}_{r_1, \dots, r_n}$ and of the notion of a Maurer-Cartan element in a complete $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_{\infty}$-algebra. In §2.4, we explain the equivalence between $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebras and $\Gamma(\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty}, -)$-algebras (up to a shift).

The goal of part 3 is to define the morphism $\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty} \to \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}\otimes_{\mathcal{H}}\mathcal{E}$ and to prove Theorem A. We actually obtain this morphism as a composite $\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty} \xrightarrow{(1)} \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}\otimes_{\mathcal{H}} B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}^{\vee}) \xrightarrow{(2)} \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}\otimes_{\mathcal{H}}\mathcal{E}$ where (2) is induced by a morphism $B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}^{\vee}) \to \mathcal{E}$. In §3.1, we explain the construction of the latter morphism $B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}^{\vee}) \to \mathcal{E}$. Then, from the general bar duality theory of algebras over operads, every $\mathcal{E}$-algebra $A$ comes with a twisting morphism on $\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}^{\vee}(\Sigma A)$. In §3.2, we make this twisting morphism explicit in the case $A = N^*(\Delta^n)$ for every $n \ge 0$. We will use this description to control the $\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty}$-algebra structure on $A \otimes N^*(\Delta^n)$ for every $n \ge 0$, when we study the simplicial Maurer-Cartan set associated to brace algebras. In §3.3, we explain the definition of (1) to complete our construction of the morphism $\mathcal{P}\mathcal{r}\mathcal{e}\mathcal{L}\mathcal{e}_{\infty} \to \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{c}\mathcal{e}\otimes_{\mathcal{H}}\mathcal{E}$ and the proof of Theorem A.

In the fourth part, we define and study our notion of a simplicial Maurer-Cartan set associated to a complete brace algebra. In §4.1, we define this simplicial set and prove that it is a Kan complex. In §4.2, we prove Theorem B, which gives a computation of the connected component and the homotopy groups of the simplicial Maurer-Cartan set associated to a complete brace algebra. In §4.3, we give an interpretation of the first differentials computed in §3.2 by computing the first simplices of the simplicial Maurer-Cartan set associated to a chosed complete brace algebra defined in the context of associated algebras up to homotopy. In §4.4, we prove Theorem C, which is an extension of the classical Goldman-Millson theorem for Lie algebras. In §4.5, we prove that, in characteristic 0, our simplicial Maurer-Cartan set is related to the simplicial Maurer-Cartan set defined for Lie algebras via a zig-zag of weak-equivalences.

In the fifth part, we show that we can describe a mapping space from the cobar construction of a coaugmented non-symmetric cooperad to an augmented non-symmetric operad as a simplicial Maurer-Cartan set associated to a complete brace algebra. In §5.1, we recall the definition of the free operad generated by a sequence in terms of planar rooted trees with inputs, and recall the model structure on the category of operads. In §5.2, we construct a cosimplicial frame associated to the cobar construction of a coaugmented cooperad. In §5.3, we prove Theorem D, which shows that we can describe a mapping space from the cobar construction of a coaugmented non-symmetric cooperad $\mathcal{C}$ to an augmented non-symmetric operad $\mathcal{P}$ as the simplicial Maurer-Cartan set associated to the complete brace algebra $\text{Hom}_{\text{Seq}_k}(\bar{\mathcal{C}}, \bar{\mathcal{P}})$.

In the last part of this memoir, we show that we can describe a mapping space from the cobar construction of a coaugmented symmetric cooperad to an augmented symmetric operad as a degree-wise Maurer-Cartan set of some $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_{\infty}$-algebras. In §6.1, we recall the definition of the free operad generated by a symmetric sequence in terms of planar rooted trees with inputs, and recall the model structure on the category of symmetric connected operads. In §6.2, we construct a cosimplicial frame associated to the cobar construction of a symmetric coaugmented cooperad. In §6.3, we
prove Theorem E, we show that we can describe a mapping space from the cobar construction of a coaugmented cooperad $\mathcal{C}$ to an augmented operad $\mathcal{P}$ as a degree-wise Maurer-Cartan set of some $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebras.

## Acknowledgements

I acknowledge support from the Labex CEMPI (ANR-11-LABX-0007-01).

I also acknowledge the strong support of Benoit Fresse for the reading and discussions around this memoir.

# 1 Conventions and notations

The goal of this section is to give recollections that will be needed in this memoir, and to set on our notations and conventions.

In §1.1, we recall basic definitions on $\mathrm{dg}\ \mathbb{K}$-modules, and give the notation used in this memoir. We also give our definitions and notation on the notion of a (complete) filtered $\mathrm{dg}\ \mathbb{K}$-module, with underlying category $\mathrm{dg}\ \widehat{\mathrm{Mod}}_{\mathbb{K}}$.

In §1.2, we recall the notion of an operad and a cooperad. We also recall the operation of the Hadamard tensor product, which is widely used in this memoir. From the definition, we recall the notion of (co)operadic suspension, and study the (co)algebras over suspensions. We finally recall the notion of a $\Gamma(\mathcal{P}, -)$-algebra associated to an operad $\mathcal{P}$ such that $\mathcal{P}(0) = 0$.

In §1.3, we recall the definition of the operad *Brace* which governs brace algebras in terms of planar rooted trees. We also set on our notations and conventions on planar rooted trees in this subsection.

In §1.4, we recall the definition of the Barratt-Eccles and surjection operads, following the conventions of [BF04]. We also recall an important example of an algebra over such operads, given by the normalized cochain complex of simplicial sets.

In §1.5, we recall notions and notations on permutations and symmetric groups. More precisely, we recall the notion of shuffle permutations which gives a set of representatives of $\Sigma_m/\Sigma_{r_1} \times \cdots \times \Sigma_{r_n}$ where $r_1 + \cdots + r_n = m$.

## 1.1 The category $\mathrm{dgMod}_{\mathbb{K}}$

Let $\mathbb{K}$ be a field. In this memoir, we work in the category $\mathrm{dgMod}_{\mathbb{K}}$ that we aim to define in this subsection.

A graded $\mathbb{K}$-module is a $\mathbb{K}$-module $V$ equipped with a decomposition

$$V \simeq \bigoplus_{n \in \mathbb{Z}} V_n.$$
Given such a decomposition, an element $x \in V$ is homogeneous if $x \in V_n$ for some $n \in \mathbb{Z}$. The integer $n$ is called the degree of $x$, and denoted by $|x|$. A morphism of graded $\mathbb{K}$-modules of degree $d$ is a morphism $f: V \to W$ of $\mathbb{K}$-modules such that $f(V_n) \subset W_{n+d}$. We denote by $\operatorname{Hom}(V, W)_d$ the $\mathbb{K}$-module formed by such morphisms. We set

$$ \operatorname{Hom}(V, W) := \bigoplus_{d \in \mathbb{Z}} \operatorname{Hom}(V, W)_d. $$

We denote by $\operatorname{gVect}_{\mathbb{K}}$ the category formed by graded $\mathbb{K}$-modules with as set of morphisms from $V$ to $W$ the $\mathbb{K}$-module $\operatorname{Hom}(V, W)_0$. The dual graded $\mathbb{K}$-module of $V$, denoted by $V^{\vee}$, is defined by $V^{\vee} = \operatorname{Hom}(V, \mathbb{K})$ where $\mathbb{K}$ is the graded $\mathbb{K}$-module with only one degree 0 component given by $\mathbb{K}$. Explicitly, if we denote by $E^{\vee}$ the dual $\mathbb{K}$-module of a $\mathbb{K}$-module $E$, then we have

$$ V^{\vee} \simeq \bigoplus_{n \in \mathbb{Z}} V_{-n}^{\vee}. $$

If $V$ is finite dimensional, then, given a basis $x_1, \dots, x_n$ of $V$, we endow $V^{\vee}$ with the basis $x_1^{\vee}, \dots, x_n^{\vee}$ where, for every $1 \le i \le n$, the linear form $x_i^{\vee} \in \operatorname{Hom}(V, \mathbb{K})$ is defined by

$$ x_i^{\vee}(x_j) = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{else} \end{cases}. $$

A differential on $V$ is a degree $-1$ morphism $d_V: V \to V$ such that $d_V \circ d_V = 0$. We usually omit the index $V$ if there is no ambiguity on the ambient $\mathbb{K}$-module. The pair $(V, d_V)$ is called a differential graded $\mathbb{K}$-module (or dg $\mathbb{K}$-module). A morphism of dg $\mathbb{K}$-modules is a morphism of graded $\mathbb{K}$-modules which commutes with the differentials. If $V$ and $W$ are dg $\mathbb{K}$-modules, then the graded $\mathbb{K}$-module $\operatorname{Hom}(V, W)$ comes equipped with a differential $d = d_{\operatorname{Hom}(V,W)}$ defined by

$$ d(f) = d_W \circ f - (-1)^{|f|} f \circ d_V. $$

We denote by $\operatorname{dgMod}_{\mathbb{K}}$ the category formed by dg $\mathbb{K}$-modules with as hom-sets the previous dg $\mathbb{K}$-module. This category is endowed with the structure of a symmetric monoidal category: the tensor product $V \otimes W$ of two elements $V, W \in \operatorname{dgMod}_{\mathbb{K}}$ is the usual tensor product of $\mathbb{K}$-modules, with as degree $n$ component

$$ (V \otimes W)_n = \bigoplus_{p+q=n} V_p \otimes W_q. $$

The differential on $V \otimes W$ is defined by

$$ d_{V \otimes W}(v \otimes w) = d_V(v) \otimes w + (-1)^{|v|} v \otimes d_W(w). $$

The symmetry operator $\tau: V \otimes W \to W \otimes V$ is defined by

$$ \tau(v \otimes w) = (-1)^{|v||w|} w \otimes v. $$

The tensor product $f \otimes g$ of two morphisms of dg $\mathbb{K}$-modules $f: V \to V'$ and $g: W \to W'$ is defined by

$$ (f \otimes g)(v \otimes w) = (-1)^{|g||v|} f(v) \otimes g(w). $$
Note that, as in the non-graded setting, we have an isomorphism of dg $\mathbb{K}$-modules

$$
\operatorname{Hom}(U \otimes V, W) \xrightarrow{\simeq} \operatorname{Hom}(U, \operatorname{Hom}(V, W))
$$

for every dg $\mathbb{K}$-modules $U, V$ and $W$, defined by sending a morphism $f : U \otimes V \to W$ to the
morphism which sends $u \in U$ to the morphism $v \in V \mapsto f(u \otimes v) \in W$.

If $V$ is finite dimensional, we also have an isomorphism of dg $\mathbb{K}$-modules

$$
\operatorname{Hom}(V, W) \xrightarrow{\simeq} W \otimes V^{\vee}.
$$

This morphism is defined by sending $f \in \operatorname{Hom}(V, W)$ to $\sum_{i=1}^{n} f(e_i) \otimes e_i^{\vee}$, where $e_1, \dots, e_n$ is a
chosen basis of $V$. Using the two above isomorphisms gives

$$
(V \otimes W)^{\vee} \simeq W^{\vee} \otimes V^{\vee},
$$

provided that $V$ is finite dimensional.

If we set $V = A^{\otimes n}, V' = C^{\otimes k}$ and $W = B^{\otimes n}, W' = D^{\otimes k}$ for some dg $\mathbb{K}$-modules $A, B, C, D$ and $n, k \ge 0$, then $f \otimes g$ is a morphism from $A^{\otimes n} \otimes B^{\otimes n}$ to $C^{\otimes k} \otimes D^{\otimes k}$. For our needs, it will sometimes be more convenient to see $f \otimes g$ as a morphism from $(A \otimes B)^{\otimes n}$ to $(C \otimes D)^{\otimes k}$.

**Definition 1.1.** Let $f : A^{\otimes n} \to C^{\otimes k}$ and $g : B^{\otimes n} \to D^{\otimes k}$. We denote by $f \tilde{\otimes} g : (A \otimes B)^{\otimes n} \to (C \otimes D)^{\otimes k}$ the morphism defined by the following commutative diagram:

$$
\begin{tikzcd}
A^{\otimes n} \otimes B^{\otimes n} \arrow[r, "f \otimes g"] & C^{\otimes k} \otimes D^{\otimes k} \arrow[d, "\simeq"] \\
(A \otimes B)^{\otimes n} \arrow[r, "f \otimes g"] & (C \otimes D)^{\otimes k}
\end{tikzcd}
$$

where we consider the isomorphisms given by the symmetry operator.

We now recall the definition of the suspension of dg $\mathbb{K}$-modules.

**Definition 1.2.** Let $k \in \mathbb{Z}$ and $V \in \operatorname{dgMod}_{\mathbb{K}}$. We denote by $\Sigma^k$ the dg $\mathbb{K}$-module generated by one degree $k$ element $\Sigma^k \in \Sigma^k$ with 0 as differential. We define the $k$-suspension of $V$ as

$$
\Sigma^k V = \Sigma^k \otimes V.
$$

For every $v \in V$, we set $\Sigma^k v := \Sigma^k \otimes v$. We also set $\Sigma^1 = \Sigma$.

For every $n, k \ge 0$, we have an isomorphism of $\mathbb{K}$-modules $(\Sigma^k V)_n \simeq V_{n-k}$. Besides, giving a
degree $k$ morphism $V \to W$ is equivalent to giving a degree 0 morphism $V \to \Sigma^k W$, and also
equivalent to giving a degree 0 morphism $\Sigma^{-k} V \to W$.

Note that, for every $k \in \mathbb{Z}$, the $k$-suspension defines an endofunctor in the category $\operatorname{dgMod}_{\mathbb{K}}$: for
every $f \in \operatorname{Hom}(V, W)$, we define $\Sigma^k f \in \operatorname{Hom}(\Sigma^k V, \Sigma^k W)$ by

$$
(\Sigma^k f)(\Sigma^k v) = (-1)^{k|f|} \Sigma^k f(v)
$$

for every $v \in V$.

We now make explicit the notion of filtration that we consider in this memoir.
**Definition 1.3.** Let $V \in \mathrm{dgMod}_K$. A filtration on $V$ is a sequence $(F_nV)_{n \ge 1}$ of sub dg $K$-modules of $V$ such that

$$ \dots \subset F_n V \subset F_{n-1} V \subset \dots \subset F_1 V = V. $$

A dg $K$-module endowed with a filtration is called a filtered dg $K$-module. A filtered dg $K$-module $V$ is said to be complete if we have an isomorphism

$$ V \simeq \lim_{\leftarrow} V/F_nV. $$

For every filtered dg $K$-module $V$, the completion of $V$ with respect to its filtration is the filtered dg $K$-module defined by

$$ \hat{V} = \lim_{\leftarrow} V/F_nV, $$

with as filtration

$$ F_m \hat{V} \vee = \lim_{\leftarrow} F_m V / (F_n V \cap F_m V). $$

We immediately see that $\hat{V}$ is complete.

**Remark 1.4.** If $V$ is complete with respect to the filtration $(F_nV)_{n \ge 1}$, then $\bigcap_{n \ge 1} F_nV = 0$. This implies that if $x \in V$ is such that $x \in F_kV \implies x \in F_{k+1}V$ for every $k \ge 1$, then $x = 0$.

Let $V, W \in \mathrm{dgMod}_K$ be two complete filtered dg $K$-modules. We say that a morphism $f: V \to W$ preserves the filtrations if it satisfies, for all $n \ge 1$,

$$ f(F_nV) \subset F_nW. $$

The complete filtered dg $K$-modules together with the filtration preserving morphisms define a category denoted by $\widehat{\mathrm{dgMod}}_K$. If $V$ and $W$ are filtered, then their tensor product $V \otimes W$ is also filtered with

$$ F_n(V \otimes W) = \sum_{p+q=n} F_p V \otimes F_q W. $$

However, this filtered dg $K$-module is not complete in general, even if $V$ and $W$ are so. We therefore define the complete tensor product by

$$ V \hat{\otimes} W = \lim_{\leftarrow} (V \otimes W)/F_n(V \otimes W). $$

We can check that the category $\widehat{\mathrm{dgMod}}_K$ endowed with $\hat{\otimes}$ is a symmetric monoidal category.

## 1.2 The notion of an operad and a cooperad

We briefly recall the notion of an operad and its dual notion, the notion of a cooperad. We will mostly follow [Fre17b] and [LV12].

Let $\mathrm{Seq}_K$ be the category whose objects are sequences in $\mathrm{dgMod}_K$. For every $M, N \in \mathrm{Seq}_K$, we denote by $\mathrm{Hom}(M, N)$ the sequence defined for every $n \ge 0$ by

$$ \mathrm{Hom}(M, N)(n) = \mathrm{Hom}(M(n), N(n)). $$
**Definition 1.5.** A symmetric sequence is a sequence $M \in \mathrm{Seq}_{\mathbb{K}}$ such that, for every $n \ge 0$, the dg $\mathbb{K}$-module $M(n)$ comes equipped with an action of $\Sigma_n$ on it. A morphism of symmetric sequences is a morphism of sequences which preserves the actions of the symmetric groups.

We denote by $\Sigma\mathrm{Seq}_{\mathbb{K}}$ the subcategory of symmetric sequences. Note that if $M, N \in \Sigma\mathrm{Seq}_{\mathbb{K}}$, then $\mathrm{Hom}(M, N) \in \Sigma\mathrm{Seq}_{\mathbb{K}}$ with the action defined by

$$ (\sigma \cdot f)(x) = \sigma \cdot f(\sigma^{-1} \cdot x) $$

for every $n \ge 0$, $f \in \mathrm{Hom}(M(n), N(n))$, $x \in M(n)$ and $\sigma \in \Sigma_n$.

**Definition 1.6.**

* An operad is a symmetric sequence $\mathcal{P} \in \Sigma\mathrm{Seq}_{\mathbb{K}}$ endowed with composition products

$$ \gamma : \mathcal{P}(n) \otimes \mathcal{P}(r_1) \otimes \cdots \otimes \mathcal{P}(r_n) \to \mathcal{P}\left(\sum_i r_i\right), $$

which satisfy associativity, unitality and symmetry axioms. The underlying category is denoted by $\Sigma\mathcal{Op}$.

* Dually, a cooperad is a symmetric sequence $\mathcal{C} \in \Sigma\mathrm{Seq}_{\mathbb{K}}$ endowed with composition coproducts

$$ \Delta : \mathcal{C}\left(\sum_i r_i\right) \to \mathcal{C}(n) \otimes \mathcal{C}(r_1) \otimes \cdots \otimes \mathcal{C}(r_n), $$

which satisfy coassociativity, counitality and symmetry axioms. The underlying category is denoted by $\Sigma\mathcal{Op}^c$.

By forgetting the action of the symmetric groups, and the symmetry axioms, we have the notion of a non-symmetric (co)operad. We denote by $\mathcal{Op}$ and $\mathcal{Op}^c$ the underlying categories.

**Remark 1.7.** If $\mathcal{P}$ is an operad such that $\mathcal{P}(n)$ is finite dimensional for every $n \ge 0$ and $\mathcal{P}(0) = 0$, then the symmetric sequence

$$ \mathcal{P}^{\vee}(n) := \mathcal{P}(n)^{\vee} $$

is endowed with the structure of a cooperad given by the dualization of the operadic structure of $\mathcal{P}$.

If $\mathcal{P}$ is an operad, then we define, for every $n, m \ge 0$ and $1 \le i \le n$, the $i$-th partial composition morphism by

$$ \circ_i : \mathcal{P}(p) \otimes \mathcal{P}(q) \xrightarrow{\simeq} \mathcal{P}(p) \otimes \mathbb{K} \otimes \cdots \otimes \mathcal{P}(q) \otimes \cdots \otimes \mathbb{K} \xrightarrow{\gamma} \mathcal{P}(p+q-1) $$

where we plug operadic units in places $j \ne i$. Dually, if $\mathcal{C}$ is a cooperad, we define the $i$-th partial decomposition morphism by

$$ \Delta_i : \mathcal{C}(p+q-1) \xrightarrow{\Delta} \mathcal{C}(p) \otimes \mathbb{K} \otimes \cdots \otimes \mathcal{C}(q) \otimes \cdots \otimes \mathbb{K} \xrightarrow{\simeq} \mathcal{C}(p) \otimes \mathcal{C}(q) $$

where we plug cooperadic counits in places $j \ne i$.
**Example 1.8.** For every $n \ge 0$, we set $\text{Com}(n) = \mathbb{K}$ endowed with the trivial $\Sigma_n$-action. The isomorphism $\mathbb{K} \otimes \mathbb{K} \simeq \mathbb{K}$ endows $\text{Com}$ with the structure of an operad called the commutative operad.

**Example 1.9.** Let $V \in \text{dgMod}_{\mathbb{K}}$. We define the symmetric sequences $\text{End}_V$ and $\text{CoEnd}_V$ by

$$ \begin{aligned} \text{End}_V(n) &= \text{Hom}(V^{\otimes n}, V); \\ \text{CoEnd}_V(n) &= \text{Hom}(V, V^{\otimes n}). \end{aligned} $$

These symmetric sequences are endowed with the structure of a symmetric operad defined as follows. Let $f \in \text{End}_V(p), g \in \text{End}_V(q)$ and $1 \le i \le p$. We set

$$ f \circ_i g = f \circ (\text{id}_V \otimes \cdots \otimes g \otimes \cdots \otimes \text{id}_V). $$

Let $\phi \in \text{CoEnd}_V(p), \psi \in \text{CoEnd}_V(q)$ and $1 \le i \le p$. We set,

$$ \phi \circ_i \psi = (-1)^{|\phi| |\psi|} (\text{id}_V \otimes \cdots \otimes \psi \otimes \cdots \otimes \text{id}_V) \circ \phi. $$

These operads are called respectively called the endomorphism and coendomorphism operads generated by $V$.

**Remark 1.10.** Let $n \ge 0$ and $\mathcal{P}$ be an operad. The elements of $\mathcal{P}(n)$ are seen as operations with abstract variables labeled by $1, \dots, n$. In this memoir, we often label these variables by elements of a finite set $X$ with $n$ elements. This can be formalized as follows. Let $X$ be a set with $n$ elements, and $\Sigma(n, X)$ be the set of bijections from $[1, n]$ to $X$. We set

$$ \mathcal{P}(X) = (\mathcal{P}(n) \otimes \mathbb{K}[\Sigma(n, X)])_{\Sigma_n}, $$

where we make coincide the action of $\Sigma_n$ on $\mathcal{P}(n)$ with its action by right translation on $\Sigma(n, X)$. The group of permutations on $X$ acts on $\mathcal{P}(X)$ by left translation on $\Sigma(n, X)$. Note that, for every finite sets $X, Y$ with $n$ elements, every bijection $u : X \to Y$ induces a morphism $u_* : \mathcal{P}(X) \to \mathcal{P}(Y)$, so that $\mathcal{P}$ defines a functor from the category of finite sets to the category of dg $\mathbb{K}$-modules.

For our needs, we apply the above construction to totally ordered finite sets. In this setting, we can shape operadic compositions on finite sets in the following way. Let $X = x_1 < \cdots < x_n$ and $Y = y_1 < \cdots < y_n$ be two disjoint totally ordered sets. We denote by $x : [1, n] \to X$ and $y : [1, m] \to Y$ the unique order preserving maps. Let $1 \le i \le n$. We set

$$ X \sqcup_i Y = x_1 < \cdots < x_{i-1} < y_1 < \cdots < y_n < x_{i+1} < \cdots < x_n. $$

Let $z : [1, n+m-1] \to X \sqcup_i Y$ be the order preserving map. We define

$$ \circ_i : \mathcal{P}(X) \otimes \mathcal{P}(Y) \to \mathcal{P}(X \sqcup_i Y) $$

by the following commutative diagram:

$$ \begin{aligned} \mathcal{P}(n) \otimes \mathcal{P}(m) &\xrightarrow{\circ_i} \mathcal{P}(n+m-1) \\ x_* \otimes y_* \big|_{\mathcal{P}(X) \otimes \mathcal{P}(Y)} &\cong z_* \big|_{\mathcal{P}(X \sqcup_i Y)} \end{aligned} $$
**Definition 1.11.** Let $\mathcal{P}$ be an operad. A $\mathcal{P}$-algebra $A$ (respectively $\mathcal{P}$-coalgebra $C$) is the data of a $dg$ $\mathbb{K}$-module $A$ (resp. $C$) together with an operad morphism $\mathcal{P} \to \text{End}_A$ (respectively $\mathcal{P} \to \text{CoEnd}_C$).

If $A$ is a $\mathcal{P}$-algebra with associated morphism $\phi : \mathcal{P} \to \text{End}_A$, for every $p \in \mathcal{P}(n)$, we set $p^A := \phi(p) \in \text{Hom}(A^{\otimes n}, A)$.

Analogously, if $C$ is a $\mathcal{P}$-coalgebra with associated morphism $\phi : \mathcal{P} \to \text{CoEnd}_A$, we set $p^C := \phi(p) \in \text{Hom}(C, C^{\otimes n})$.

**Remark 1.12.** The above definition is such that the dual $dg$ $\mathbb{K}$-module $C^\vee$ of every $\mathcal{P}$-coalgebra $C$ is endowed with the structure of a $\mathcal{P}$-algebra. We define an operad morphism $\phi^\vee : \mathcal{P} \to \text{End}_{C^\vee}$ as follows. Let $\phi : \mathcal{P} \to \text{CoEnd}_C$ be the coalgebra structure given by $\mathcal{P}$ on $C$. For every $n \ge 0$, $p \in \mathcal{P}(n)$ and $u_1, \dots, u_n \in C^\vee$, we set

$$ \phi^{\vee}(p)(u_1 \otimes \cdots \otimes u_n) = \phi(p)^{\vee}(u_1 \otimes \cdots \otimes u_n). $$

In the following, we also use the notion of a complete $\mathcal{P}$-algebra. A filtered $\mathcal{P}$-algebra is a filtered $dg$ $\mathbb{K}$-module $A$ endowed with the structure of a $\mathcal{P}$-algebra such that, for every $n \ge 0$ and $p \in \mathcal{P}(n)$, the morphism $p^A : A^{\otimes n} \to A$ preserves the filtrations. A complete $\mathcal{P}$-algebra is a filtered $\mathcal{P}$-algebra which is complete with respect to its filtration.

We define a monoidal structure on $\Sigma\mathcal{O}p$ and $\Sigma\mathcal{O}p^c$ given by the *Hadamard tensor product*.

**Definition 1.13.** Let $\mathcal{P}, \mathcal{Q}$ be two operads and $\mathcal{C}, \mathcal{D}$ be two cooperads.

* The Hadamard tensor product of $\mathcal{P}$ and $\mathcal{Q}$ is the operad $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{Q}$ defined by

$$ (\mathcal{P} \otimes_{\mathbb{H}} \mathcal{Q})(n) = \mathcal{P}(n) \otimes \mathcal{Q}(n) $$

and equipped with the tensor-wise operadic composition product and the diagonal action of $\Sigma_n$.

* The Hadamard tensor product of $\mathcal{C}$ and $\mathcal{D}$ is the cooperad $\mathcal{C} \otimes_{\mathbb{H}} \mathcal{D}$ defined by

$$ (\mathcal{C} \otimes_{\mathbb{H}} \mathcal{D})(n) = \mathcal{C}(n) \otimes \mathcal{D}(n) $$

and equipped with the tensor-wise cooperadic composition coproduct and the diagonal action of $\Sigma_n$.

We now define the notion of suspension in the category of operads and cooperads.

**Definition 1.14.** Let $\mathcal{P}$ be an operad and $\mathcal{C}$ be a cooperad. We set $\Lambda^k = \text{End}_{\Sigma^k}$ and $\Lambda := \Lambda^1$.

* The $k$-operadic suspension of $\mathcal{P}$ is the operad $\Lambda^k \mathcal{P}$ defined by

$$ \Lambda^k \mathcal{P} = \Lambda^k \otimes_{\mathbb{H}} \mathcal{P}. $$

* The $k$-cooperadic suspension of $\mathcal{C}$ is the cooperad $\Lambda^k \mathcal{C}$ defined by

$$ \Lambda^k \mathcal{C} = (\Lambda^k)^{\vee} \otimes_{\mathbb{H}} \mathcal{C}. $$
Accordingly, $\Lambda^k \mathcal{P}(n) \simeq \Sigma^{k(1-n)} \mathcal{P}(n)$ and $\Lambda^k \mathcal{C}(n) \simeq \Sigma^{k(1-n)} \mathcal{C}(n)$.

We have an isomorphism of operads $\Lambda^{-k} \Lambda^k \mathcal{P} \to \mathcal{P}$ given by $(\Sigma^{-k(1-n)}(\Sigma^{k(1-n)}p)) \mapsto -(-1)^{\frac{n(n+1)}{2}k} p$ for every $p \in \mathcal{P}(n)$.

Note that, for every $k \in \mathbb{Z}$, the dg $\mathbb{K}$-module $\Sigma^k$ is a $\Lambda^k$-algebra. We thus have the following.

**Proposition 1.15.** Let $\mathcal{P}$ be an operad and $\mathcal{C}$ be a cooperad. Let $V$ be a dg $\mathbb{K}$-module.

* Giving a structure of $\mathcal{P}$-algebra on $V$ is equivalent to giving a structure of $\Lambda^k \mathcal{P}$-algebra on $\Sigma^k V$.

* Giving a structure of $\mathcal{C}$-coalgebra on $V$ is equivalent to giving a structure of $\Lambda^k \mathcal{C}$-coalgebra on $\Sigma^k V$.

Any operad $\mathcal{P}$ gives a monad $\mathcal{S}(\mathcal{P}, -) : \mathrm{dgMod}_{\mathbb{K}} \to \mathrm{dgMod}_{\mathbb{K}}$ called the *Schur functor* and defined by

$$ \mathcal{S}(\mathcal{P}, V) = \bigoplus_{n \ge 0} \mathcal{P}(n) \otimes_{\Sigma_n} V^{\otimes n}, $$

where we consider the action of $\Sigma_n$ on $\mathcal{P}(n)$, and the action of $\Sigma_n$ on $V^{\otimes n}$ by permutation. The monadic structure is given by the composite

$$ \mathcal{S}(\mathcal{P}, \mathcal{S}(\mathcal{P}, V)) \xrightarrow{\simeq} \mathcal{S}(\mathcal{P} \circ \mathcal{P}, V) \xrightarrow{\mathcal{S}(\gamma, V)} \mathcal{S}(\mathcal{P}, V), $$

where we denote by $\circ$ the composition of symmetric sequences. Note that the algebras over the monad $\mathcal{S}(\mathcal{P}, -)$ are precisely the $\mathcal{P}$-algebras.

If $\mathcal{P}(0) = 0$, we also have a monad $\Gamma(\mathcal{P}, -) : \mathrm{dgMod}_{\mathbb{K}} \to \mathrm{dgMod}_{\mathbb{K}}$ defined by

$$ \Gamma(\mathcal{P}, V) = \bigoplus_{n \ge 1} \mathcal{P}(n) \otimes_{\Sigma_n} V^{\otimes n}. $$

We refer to [Fre00, §1.1.18] for the description of this monadic structure. We only note that we have a morphism of monads

$$ \mathrm{Tr} : \mathcal{S}(\mathcal{P}, V) \to \Gamma(\mathcal{P}, V) $$

given by the trace map. This is an isomorphism as soon as $\mathrm{char}(\mathbb{K}) = 0$. It is however no longer an isomorphism in general when $\mathrm{char}(\mathbb{K}) \ne 0$.

**Definition 1.16.** Let $\mathcal{P}$ be an operad such that $\mathcal{P}(0) = 0$. A $\mathcal{P}$-algebra with divided powers is a $\Gamma(\mathcal{P}, -)$-algebra.

Note that every $\mathcal{P}$-algebra with divided powers is in particular a $\mathcal{P}$-algebra through the trace map.

**Proposition 1.17.** Let $\mathcal{P}$ be an operad such that $\mathcal{P}(0) = 0$ and $V$ be a dg $\mathbb{K}$-module. Let $k \in \mathbb{Z}$. Then $V$ is a $\Gamma(\mathcal{P}, -)$-algebra if and only if $\Sigma^k V$ is a $\Gamma(\Lambda^k \mathcal{P}, -)$-algebra.
Proof. Let $V$ be a $\Gamma(\mathcal{P}, -)$-algebra. We endow $\Sigma^k V$ with the structure of a $\Gamma(\Lambda^k \mathcal{P}, -)$-algebra via the composite

$$ \Gamma(\Lambda^k \mathcal{P}, \Sigma^k V) \to \Sigma^k \Gamma(\mathcal{P}, V) \to \Sigma^k V, $$

where the first morphism comes from the fact that $\text{End}_{\Sigma^k}(n) \otimes (\Sigma^k)^{\otimes n}$ is isomorphic to $\Sigma^k$ endowed with the trivial $\Sigma_n$ action. The fact that this endows $\Sigma^k V$ with a $\Gamma(\Lambda^k \mathcal{P}, -)$-algebra structure is an immediate verification. $\square$

## 1.3 On trees and the operad Brace

In this section, we recall the notion of a tree and define the operad Brace. The notion of a brace algebra was introduced in [GV95, Definition 1], while an explicit construction of their governing operad Brace is given in [Cha02, §2.1-2.2].

**Definition 1.18.** We call (planar) $n$-tree any simply connected graph endowed with a special vertex called the root and a labeling of its set of vertices from 1 to $n$. We put the root at the bottom by convention:

<figure><img src="image_4.png" /></figure>

We denote by $\mathcal{PRT}(n)$ the set of planar rooted trees with $n$ vertices. For every $T \in \mathcal{PRT}(n)$, we set $|T| = n$ and $r(T)$ denotes the root of the tree $T$.

In some situation, it is more convenient to label an $n$-tree by a finite set with $n$ elements endowed with a total ordered relation. If $X$ is such a set, we denote by $\mathcal{PRT}(X)$ the set of $n$-trees labeled with elements of $X$. Note that since there is a unique order preserving bijection $[1, n] \to X$, there is a canonical bijection $\mathcal{PRT}(n) \to \mathcal{PRT}(X)$. For instance, the tree $T$ shown in the above definition can be seen in $\mathcal{PRT}(a_1 < \dots < a_7)$ as

<figure><img src="image_5.png" /></figure>

**Proposition 1.19** ([Cha02, Proposition 2]). Let Brace be the symmetric sequence defined by Brace = $\mathbb{K}[\mathcal{PRT}(n)]$. Then Brace is endowed with the structure of an operad. Its algebras are given by $\text{dg}\,\mathbb{K}$-modules $A$ endowed with morphisms $-\langle - , \dots , - \rangle : A^{\otimes n+1} \to A$ for any $n \ge 0$ such that $x\langle\rangle = x$ and

$$ x\langle y_1, \dots, y_n \rangle\langle z_1, \dots, z_p \rangle = \sum \pm x\langle Z_1, y_1\langle Z_2 \rangle, \dots, Z_{2n-1}, y_n\langle Z_{2n} \rangle, Z_{2n+1} \rangle $$

for every $x, y_1, \dots, y_n, z_1, \dots, y_p \in A$, where the sum runs over all consecutive subsets such that $Z_1 \sqcup \dots \sqcup Z_{2n+1} = (z_1, \dots, z_p)$.

Note that every tree $T$ with $|T| \ge 2$ can be uniquely written as $T = \gamma(F_n(\textcircled{1}, T_1, \dots, T_n))$ where we denote by

$$ F_n = \textcircled{2} \cdots \textcircled{n+1} $$
the corolla with $n$ leaves.

In the next sections, in order to have formulas which preserve the symmetric groups actions on *Brace*, we pick an explicit set of representatives of *Brace(n)* as a free $\Sigma_n$-set. We achieve this by setting a total order relation on the set of vertices $V_T$ which we call the *canonical order*. For every $a \in \mathbb{N}^*$, we set $V_{\circled{a}} = a$, and define by induction,

$$V_{\gamma(F_n((\circ), T_1, \dots, T_n))} = a < V_{T_1} < \dots < V_{T_n}$$

for every tree $T_1, \dots, T_n$. For instance, if we set

<figure><img src="image_6.png" /></figure>

then $V_T = 5 < 6 < 3 < 7 < 1 < 4 < 2$.

**Definition 1.20.** A tree $T \in \mathcal{PT}(a_1 < \dots < a_n)$ is canonical (or in the canonical order) if

$$V_T = a_1 < \dots < a_n.$$

We let $\sigma_T \in \Sigma_{|T|}$ to be the unique permutation such that $\sigma_T^{-1} \cdot T$ is in the canonical order.

For instance, if we consider the above tree, then $\sigma_T = (5637142)$ and

<figure><img src="image_7.png" /></figure>

is in the canonical order in $\mathcal{PT}(1 < \dots < 7)$.

**Definition 1.21.** Let $X$ be a totally finite ordered set and $T \in \mathcal{PT}(X)$.

* A subtree $S \subset T$ of $T$ is an induced simply connected subgraph of $T$ whose set of vertices is seen as a subset $Y$ of $X$ endowed with the induced order relation. Note that $V_S \subset V_T$ as ordered sets.

* If $S \subset T$, we define the tree $T/S \in \mathcal{PT}(X \setminus Y \cup \{S\})$ obtained from $T$ by contracting the tree $S$ on the root of $S$, denoted by $S$ in the labeling of $T/S$. The totally ordered set $X \setminus Y \cup \{S\}$ is obtained by changing $r(S)$ into $S$, and removing all the non-root vertices of $S$ in $X$.

A subtree $S \subset T$ is non-trivial if neither $|S| \neq 1$ nor $|T/S| \neq 1$.

**Remark 1.22.** Let $X$ be a totally finite ordered set. Let $T \in \mathcal{PT}(X)$ and $S \subset T$. If $T$ is canonical, then so are $S$ and $T/S$.
Example: If

<figure><img src="image_8.png" /></figure>

then

<figure><img src="image_9.png" /></figure>

is a subtree of T such that

<figure><img src="image_10.png" /></figure>

## 1.4 On the Barratt-Eccles and the surjection operads

We devote this subsection to recollections on the Barratt-Eccles operad and the surjection operad. We will mostly follow conventions of [BF04].

**Definition 1.23.** We let $\mathcal{E}(r)_d$ to be the $\mathbb{K}$-module spanned by $(d+1)$-tuples

$$ (w_0, \dots, w_d) \in (\Sigma_r)^{d+1} $$

with the identification $(w_0, \dots, w_d) \equiv 0$ if $w_i = w_{i+1}$ for some $i$. We denote by $\mathcal{E}(r)$ the $\mathrm{dg}\,\mathbb{K}$-module with $\mathcal{E}(r)_d$ as degree $d$ component. The differential on $\mathcal{E}(r)$ is defined by

$$ d(w_0, \dots, w_d) = \sum_{i=0}^{d} (-1)^i (w_0, \dots, \hat{w}_i, \dots, w_d). $$

We also have an action of $\Sigma_r$ on $\mathcal{E}(r)$ given by the diagonal action and the left translation of $\Sigma_r$ on itself.

**Proposition 1.24.** The symmetric sequence $\mathcal{E}$ is an operad called the Barratt-Eccles operad.

We refer to [BF04, §1.1] for an explicit description of the composition product. We have an operad morphism $\mathcal{E} \to \mathrm{Com}$ obtained by sending each degree 0 element to 1, and sending each non-degree 0 element to 0. This morphism is a weak equivalence arity-wise.

**Remark 1.25.** The operad $\mathcal{E}$ has the structure of a Hopf operad. Namely, we have an operad morphism $\Delta_{\mathcal{E}}: \mathcal{E} \to \mathcal{E} \otimes \mathcal{E}$ defined by

$$ \Delta_{\mathcal{E}}(w_0, \dots, w_d) = \sum_{k=0}^{d} (w_0, \dots, w_k) \otimes (w_k, \dots, w_d). $$
We now aim to define the surjection operad $\chi$.

**Definition 1.26.** Let $r, d \ge 0$. A surjective map $u : [[1, r + d]] \to [[1, r]]$ is degenerate if $u(i) = u(i + 1)$ for some $i \in [[1, r + d - 1]]$. We let $\chi(r)_d$ to be the $\mathbb{K}$-module spanned by non-degenerate surjective maps from $[[1, r + d]]$ to $[[1, r]]$.

In practice, we represent a surjection $u : [[1, r + d]] \to [[1, r]]$ by a sequence of values:

$$ (u(1) \cdots u(r+d)). $$

**Definition 1.27.** Let $u \in \chi(r)_d$. An integer $k \in [[1, r + d]]$ is called a caesura if $u(k)$ does not represent the last occurrence of its value in $u$.

We sometimes represent a surjection by its table arrangement, which is defined as follows. Let $u \in \chi(r)_d$. We cut $u$ at the caesuras, in the sense that we set

$$ u = (u_0(1) \cdots u_0(r_0)) \cdots (u_d(1) \cdots u_d(r_d)), $$

where $\sum_i r_i = r + d$, and where $u_0(r_0), \dots, u_{d-1}(r_{d-1})$ represent caesuras of $u$. We then write $u$ as

$$ u = \begin{vmatrix} u_0(1) & \cdots & u_0(r_0) \\ \vdots & & \vdots \\ u_d(1) & \cdots & u_d(r_d) \end{vmatrix}. $$

We have an obvious action of $\Sigma_r$ on $\chi(r)_d$ given by the pre-composition.

**Proposition 1.28** (see [BF04, §1.2]). The symmetric sequence $\chi$ is endowed with the structure of a symmetric operad and is called the surjection operad.

In fact, the surjection operad $\chi$ is a quotient of the Barratt-Eccles operad $\mathcal{E}$. The quotient map is called the table reduction morphism.

**Proposition 1.29.** There exists an operad morphism $TR : \mathcal{E} \to \chi$ called the table reduction morphism which is surjective arity-wise.

We refer to [BF04] for more details on the morphism $TR$. We only recall its definition. Let $w = (w_0, \dots, w_d) \in \mathcal{E}(r)_d$. We set

$$ TR(w) = \sum_{r_0+\cdots+r_d=r+d} \begin{vmatrix} w'_0(1) & \cdots & w'_0(r_0-1) & w'_0(r_0) \\ \vdots & & \vdots & \vdots \\ w'_d(1) & \cdots & w'_d(r_d-1) & w'_d(r_d) \end{vmatrix} $$

where each row $w'_i(1) \cdots w'_i(r_i)$ represents the first $r_i$ integers occurring in the permutation $w_i$ such that the values $w'_i(1) \cdots w'_i(r_i - 1)$ do not occur in

$$ \begin{vmatrix} w'_0(1) & \cdots & w'_0(r_0-1) \\ \vdots & & \vdots \\ w'_{i-1}(1) & \cdots & w'_{i-1}(r_i-1) \end{vmatrix}. $$

An important example of $\chi$-algebra is given by the normalized cochain complex associated to a simplicial set.
**Definition 1.30.** Let $X$ be a simplicial set and, for every $k \ge 0$, let $C_k(X)$ be the $\mathbb{K}$-module spanned by $X_k$. We define a differential on $C_*(X)$ by setting, for every $x \in X_k$,

$$d(x) = \sum_{i=0}^{k} (-1)^i d_i(x),$$

where we denote by $d_0, \dots, d_k : X_k \to X_{k-1}$ the faces maps. We then set

$$N_k(X) = C_k(X) / \left( \sum_{i=0}^{k} s_i C_{k-1}(X) \right),$$

where we denote by $s_0, \dots, s_{k-2} : X_{k-1} \to X_k$ the degeneracy maps. The dg $\mathbb{K}$-module $N_*(X)$ is called the normalized chain complex of $X$. Its dual dg $\mathbb{K}$-module, denoted by $N^*(X)$, is called the normalized cochain complex of $X$.

Note that $N_*$ and $N^*$ are functors from $sSet$ to $dgMod_{\mathbb{K}}$.

**Theorem 1.31** ([BF04, §2]). Let $X \in sSet$. Then $N_*(X)$ is a $\chi$-coalgebra, given by the interval cut operations, which is natural in $X$. As a consequence, the dg $\mathbb{K}$-module $N^*(X)$ is endowed with the structure of a $\chi$-algebra.

We refer to [BF04, §2.2.1, §2.2.4] for an explicit description of the interval cut operations. In particular, for every simplicial set $X$, the dg $\mathbb{K}$-module $N^*(X)$ is endowed with the structure of a $\mathcal{E}$-algebra through the table reduction morphism $TR : \mathcal{E} \to \chi$. In this memoir, we mostly consider the case $X = \Delta^n$ for some $n \ge 0$. The elements of $N_d(\Delta^n)$ are linear combination of non-decreasing sequences $a_0 < \dots < a_d$ of integers in $[1, n]$, which we denote by $a_0 \cdots a_d$. The normalized chain complex of $\Delta^n$ has the following fundamental property.

**Proposition 1.32.** Let $n \ge 0$ and $0 \le k \le n$. Then there exists a deformation retract

$$h_n^k \left( N_*(\Delta^n) \xrightarrow{p_n} \underset{i_n^k}{\longleftarrow} N_*(\Delta^0) \right),$$

where $i_n^k : N_*(\Delta^0) \to N_*(\Delta^n)$ is the morphism which sends $\emptyset$ to $k$, and $p_n : N_*(\Delta^n) \to N_*(\Delta^0)$ is the morphism which sends every vertex to $\emptyset$.

The claim is that we have the identities

$$\begin{aligned} p_n i_n^k &= id_{N_*(\Delta^0)}; \\ id_{N_*(\Delta^n)} - i_n^k p_n &= dh_n^k + h_n^k d. \end{aligned}$$

We set $\varphi_n^k = i_n^k p_n$. The homotopy $h_n^k$ can be explicitly defined as follows. Let $a_0 \cdots a_r \in N_r(\Delta^n)$ be a non-zero element. If this sequence contains $k$, then we set $h_n^k(a_0 \cdots a_r) = 0$. Otherwise we set

$$h_n^k(a_0 \cdots a_r) = (-1)^i a_0 \cdots k \cdots a_r,$$
where $i$ is the unique possible position to insert $k$ in $a_0 \cdots a_r$ so that we have a non decreasing sequence of integers.

By taking linear duals, we have a similar deformation retract on $N^*(\Delta^n)$. We will keep the same notation $h_n^k : N^*(\Delta^n) \to N^{*-1}(\Delta^n)$ and $\varphi_n^k : N^*(\Delta^n) \to N^*(\Delta^n)$ for the linear duals of $h_n^k : N_*(\Delta^n) \to N_{*+1}(\Delta^n)$ and $\varphi_n^k : N_*(\Delta^n) \to N_*(\Delta^n)$.

The dg $\mathbb{K}$-module $I = N^*(\Delta^1)$ can be used to model intervals. We indeed have a decomposition of the diagonal map $\Delta : \mathbb{K} \to \mathbb{K}^2$ as

$$ \mathbb{K} = N^*(\Delta^0) \xrightarrow[\Delta]{\sim} N^*(\Delta^1) \xrightarrow[(d_0, d_1)]{\sim} N^*(\Delta^0) \times N^*(\Delta^0) = \mathbb{K}^2 $$

where $s_0 = (p_1)^\vee$ and $d_0 = (i_1^0)^\vee$, $d_1 = (i_1^1)^\vee$. We can lift such a diagram in the category of $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebras for any operad $\mathcal{P}$ to get a construction of a path-object. Recall that a path object for a $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebra $R$ is a $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebra $R^I$ such that the diagonal map $\Delta : R \to R \times R$ can be described as a composite

$$ R \xrightarrow[\Delta]{\sim} R^I \xrightarrow[(d_0, d_1)]{\sim} R \times R . $$

**Proposition 1.33** (see [BF04, §3.1.4, §3.1.9]). Let $\mathcal{P}$ be an operad, and $R$ be a $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebra. Then

$$ R^I = R \otimes N^*(\Delta^1) $$

is a path object in the category of $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebras. The $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebra structure on $R^I$ is given by the composite

$$ \mathcal{P} \otimes_{\mathbb{H}} \mathcal{E} \xrightarrow{id \otimes \Delta_{\xi}} \mathcal{P} \otimes_{\mathbb{H}} \mathcal{E} \otimes_{\mathbb{H}} \mathcal{E} \longrightarrow \mathrm{End}_{\mathcal{R} \otimes \mathcal{N}^*(\Delta^1)}, $$

where we use the $\mathcal{P} \otimes_{\mathbb{H}} \mathcal{E}$-algebra structure on $R$, and the $\mathcal{E}$-algebra structure on $N^*(\Delta^1)$.

## 1.5 Appendix: basic results on permutations

In this appendix, we recall basic definitions and notations on permutations and the symmetric groups. Our conventions will follow those given in [Fre17a, §1.1.7]. Let $n \ge 0$. We denote by $\Sigma_n$ the symmetric group on the elements $1, \dots, n$. For every $m, n \ge 0$, we denote by $\llbracket m, n \rrbracket$ the set of integers $k$ such that $m \le k \le n$. We denote by $\mathrm{id}$ the relevant identity permutation, and we write any permutation $\sigma \in \Sigma_n$ as its sequence of values $(\sigma(1) \cdots \sigma(n))$.

For every $p, q \in \mathbb{N}$ and $\sigma \in \Sigma_p, \tau \in \Sigma_q$, we let $\sigma \oplus \tau \in \Sigma_{p+q}$ to be the permutation defined, for every $1 \le i \le p+q$, by

$$ (\sigma \oplus \tau)(i) = \begin{cases} \sigma(i) & \text{if } 1 \le i \le p \\ \tau(p+i) & \text{if } p+1 \le i \le p+q \end{cases}. $$
The operation $\oplus$ is associative in $\coprod_{n \ge 0} \Sigma_n$, so that we can generalize the definition of $\oplus$ to a direct sum of $k \ge 1$ permutations $\sigma_1 \oplus \cdots \oplus \sigma_k$.

Let $r_1, \dots, r_n \ge 0$ and $\sigma \in \Sigma_n$. We set $\mathbf{r}_i = r_1 + \cdots + r_{i-1} + 1 < \cdots < r_1 + \cdots + r_{i-1} + r_i$. We define the block permutation induced by $\sigma$ of type $(r_1, \dots, r_r)$ by

$$ \sigma_*(r_1, \dots, r_n) = \mathbf{r}_{\sigma(1)} \cdots \mathbf{r}_{\sigma(n)}. $$

**Lemma 1.34** ([Fre17a, Proposition 1.1.8]). Let $\sigma \in \Sigma_n$ and $\tau_1 \in \Sigma_{r_1}, \dots, \tau_n \in \Sigma_{r_n}$. Then

$$ (\tau_1 \oplus \cdots \oplus \tau_n) \cdot \sigma_*(r_1, \dots, r_n) = \sigma_*(r_1, \dots, r_n) \cdot (\tau_{\sigma(1)} \oplus \cdots \oplus \tau_{\sigma(n)}). $$

Let $\sigma \in \Sigma_n$ and $\tau_1 \in \Sigma_{r_1}, \dots, \tau_n \in \Sigma_{r_n}$. We define the permutation $\sigma(\tau_1, \dots, \tau_n) \in \Sigma_{r_1+\dots+r_n}$ by

$$ \sigma(\tau_1, \dots, \tau_n) = (\tau_1 \oplus \cdots \oplus \tau_n) \cdot \sigma_*(r_1, \dots, r_n). $$

In operads theory, one needs a set of representatives of the quotient $\Sigma_m/\Sigma_{r_1} \times \cdots \times \Sigma_{r_n}$ for every $r_1, \dots, r_n \ge 0$ such that $r_1 + \cdots + r_n = m$. This leads us to the notion of *shuffle permutation*. A $(r_1, \dots, r_n)$-shuffle permutation is a permutation in $\Sigma_{r_1+\dots+r_n}$ which preserves the order on each block $\mathbf{r}_1, \dots, \mathbf{r}_n$. We denote by $\mathrm{Sh}(r_1, \dots, r_n)$ the set of such permutations. A shuffle permutation $\omega \in \mathrm{Sh}(r_1, \dots, r_n)$ is *pointed* if it satisfies $\omega(1) < \omega(r_1+1) < \cdots < \omega(r_1+\cdots+r_{n-1}+1)$. We denote by $\mathrm{Sh}_*(r_1, \dots, r_n)$ the set of such permutations.

The following results allow us to write any permutations in terms of a product of a shuffle permutation with a composite of a direct sum and a block permutation.

**Proposition 1.35.** Let $n \ge 0$ and $r_1, \dots, r_n \ge 0$.

* Every $\sigma \in \Sigma_{r_1+\dots+r_n}$ admits a unique decomposition of the form

$$ \sigma = \omega \cdot (\tau_1 \oplus \cdots \oplus \tau_n) $$

where $\tau_i \in \Sigma_{r_i}$ and $\omega \in \mathrm{Sh}(r_1, \dots, r_n)$.

* Every $\sigma \in \Sigma_{r_1+\dots+r_n}$ admits a unique decomposition of the form

$$ \sigma = \omega \cdot \sigma(\tau_1, \dots, \tau_n) $$

where $\tau_i \in \Sigma_{r_i}$, $\sigma \in \Sigma_n$ and $\omega \in \mathrm{Sh}_*(r_1, \dots, r_n)$.

## 2 On $\mathcal{P}reLie_\infty$-algebras with divided powers

In this section, we study the structure of $\Gamma(\mathcal{P}reLie_\infty, -)$-algebras. The operad $\mathcal{P}reLie_\infty$ and its algebras have been explicitly described in [CL01], using the computation of the Koszul dual operad of $\mathcal{P}reLie$ given in [Cha01].

In §2.1, we recall this explicit construction of $\mathcal{P}reLie_\infty$. We also focus on the characterization of the structure of a $\mathcal{P}reLie_\infty$-algebra as an algebraic structure on the suspension which we call a
$\Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebra.

In §2.2, we define the notion of a $\Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebra which will be the analogue, in the divided power framework, of a $\Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebra, and we define a notion of $\infty$-morphism of $\Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebras.

In §2.3, we define the symmetric weighted braces associated to a $\Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebra and the notion of a Maurer-Cartan element in the complete framework. We also prove that giving the structure of a $\Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebra is equivalent to giving symmetric brace operations.

In §2.4, we prove that giving a structure of a $\Gamma(\mathcal{P} \mathcal{L}_{\infty}, -)$-algebra is equivalent to giving a structure of a $\Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$-algebra on the suspension.

## 2.1 Recollections on pre-Lie algebras up to homotopy

We begin this section by some recollections on the operad $\text{Perm}$, which was introduced by Chapoton in [Cha01]. Let $\text{Perm}(n) = \mathbb{K}^n$. We denote by $(e_i^n)_{1 \le i \le n}$ the canonical basis of $\text{Perm}(n)$. The group $\Sigma_n$ acts on $\text{Perm}(n)$ by

$$ \sigma \cdot e_i^n = e_{\sigma^{-1}(i)}^n. $$

**Proposition 2.1.** *The symmetric sequence $\text{Perm}$ is an operad with as compositions*

$$ e_i^n (e_{j_1}^{n_1}, \dots, e_{j_k}^{n_k}) = e_{n_1 + \dots + n_{i-1} + j_i}^{n_1 + \dots + n_k}. $$

**Theorem 2.2** (see [Cha01]). *The operad $\mathcal{P} \mathcal{L}$ is Koszul and its Koszul dual operad is $\mathcal{P} \mathcal{L}'$ = $\text{Perm}^c$.*

This theorem implies that the operad $\mathcal{P} \mathcal{L}_{\infty} = B^c(\Lambda^{-1}\text{Perm}^c)$ gives a model for $\mathcal{P} \mathcal{L}$-algebras up to homotopy. Such algebras have been described by Chapoton and Livernet in [CL01]. We recall this description in the following paragraphs. Let $V \in \text{gVect}_{\mathbb{K}}$. We set

$$ \mathcal{S}(V) := \bigoplus_{n \ge 0} (V^{\otimes n})_{\Sigma_n}, $$

where we consider the usual action of $\Sigma_n$ on $V^{\otimes n}$ by permutation. Note that $\mathcal{S}(V) \simeq \mathbb{K} \oplus \bar{\mathcal{S}}(V)$ with

$$ \bar{\mathcal{S}}(V) := \bigoplus_{n \ge 1} (V^{\otimes n})_{\Sigma_n}. $$

**Definition 2.3** (see [CL01, §2.3]). *The free $\text{Perm}$-coalgebra generated by $V$ is the graded $\mathbb{K}$-module $\text{Perm}^c(V) = V \otimes \mathcal{S}(V)$ endowed with the following comultiplication:*

$$ \Delta_{\text{Perm}}(v_0 \otimes 1) = 0; $$

$$ \Delta_{\text{Perm}}(v_0 \otimes v_1 \cdots v_n) = \sum_{\substack{0 \le k \le n-1 \\ \sigma \in \text{Sh}(k, 1, n-k-1)}} \pm (v_0 \otimes v_{\sigma(1)} \cdots v_{\sigma(k)}) \otimes (v_{\sigma(k+1)} \otimes v_{\sigma(k+2)} \cdots v_{\sigma(n)}) $$

for every $v_0, \dots, v_n \in V$, where the sign in the sum is produced by permutations of the factors.

The coproduct $\Delta_{\text{Perm}}$ satisfies the following identities (see [CL01, §2.3]):

$$ (\text{id} \otimes \Delta_{\text{Perm}}) \Delta_{\text{Perm}} = (\Delta_{\text{Perm}} \otimes \text{id}) \Delta_{\text{Perm}}; $$

$$ (\text{id} \otimes \Delta_{\text{Perm}}) \Delta_{\text{Perm}} = (\text{id} \otimes \tau)(\text{id} \otimes \Delta_{\text{Perm}}) \Delta_{\text{Perm}}. $$
**Remark 2.4.** Let $\Delta_{S(V)}: S(V) \to S(V) \otimes S(V)$ be the coproduct defined by $\Delta_{S(V)}(1) = 1 \otimes 1$ and

$$ \Delta_{S(V)}(v_1 \cdots v_n) = \sum_{k=0}^{n} \sum_{\sigma \in Sh(k,n-k)} \pm (v_{\sigma(1)} \cdots v_{\sigma(k)}) \otimes (v_{\sigma(k+1)} \cdots v_{\sigma(n)}) $$

for every $v_1, \dots, v_n \in V$. Then $\Delta_{\text{Perm}}$ is given by the composite

$$ V \otimes S(V)^{id \otimes \Delta_{S(V)}} \xrightarrow{\text{id} \otimes \Delta_{S(V)}} V \otimes S(V) \otimes S(V) \longrightarrow V \otimes S(V) \otimes \bar{S}(V)^{id \otimes \text{id} \otimes i_V} \xrightarrow{\text{id} \otimes \text{id} \otimes i_V} (V \otimes S(V)) \otimes (V \otimes S(V)) $$

where $i_V: \bar{S}(V) \to V \otimes S(V)$ is defined by

$$ v_1 \cdots v_n \mapsto \sum_{k=1}^{n} \pm v_k \otimes v_1 \cdots \hat{v_k} \cdots v_n $$

for every $v_1, \dots, v_n \in V$.

Let $\pi_V: \text{Perm}^c(V) \to V$ be the projection on the first factor.

**Proposition 2.5** ([CL01, §2.4]). The map

$$ \begin{array}{ccc} \text{Coder}(\text{Perm}^c(V)) & \to & \text{Hom}(\text{Perm}^c(V), V) \\ d & \mapsto & \pi_V \circ d \end{array} $$

is a bijection.

*Proof.* We only recall the construction of the inverse bijection $\Psi$. Let $l \in \text{Hom}(\text{Perm}^c(V), V)$. We define $\Psi(l) \in \text{Coder}(\text{Perm}^c(V))$ as the sum of the composite

$$ \Psi_1(l) : V \otimes S(V) \xrightarrow{\text{id} \otimes \Delta_{S(V)}} V \otimes S(V) \otimes S(V) \xrightarrow{l \otimes i} V \otimes S(V) $$

and of the composite

$$ \Psi_2(l) : V \otimes S(V) \xrightarrow{\Delta_{\text{Perm}}} (V \otimes S(V)) \otimes (V \otimes S(V)) \xrightarrow{\text{id} \otimes l} V \otimes S(V) \otimes V \longrightarrow V \otimes S(V) $$

where the last morphism is given by the projection from $S(V) \otimes V$ to $S(V)$. One can check that we retrieve the definition given in the proof of [CL01, §2.4]. $\square$

**Proposition 2.6** ([CL01, §2.5]). Let $L \in \text{gVect}_K$. Giving a structure of pre-Lie algebra up to homotopy on $L$ is equivalent to giving a degree $-1$ morphism $l \in \text{Hom}(\text{Perm}^c(\Sigma L), \Sigma L)$ such that, for every $x, y_1, \dots, y_n \in \Sigma L$, we have

$$ \begin{aligned} & \sum_{i=0}^{n} \sum_{\sigma \in \text{Sh}(i, n-i)} \pm l(l(x \otimes y_{\sigma(1)} \cdots y_{\sigma(i)}) \otimes y_{\sigma(i+1)} \cdots y_{\sigma(n)}) \\ & + \sum_{i=0}^{n} \sum_{\sigma \in \text{Sh}(1, i, n-i-1)} \pm l(x \otimes l(y_{\sigma(1)} \otimes y_{\sigma(2)} \cdots y_{\sigma(i+1)}) \cdot y_{\sigma(i+2)} \cdots y_{\sigma(n)}) = 0 \end{aligned} $$

where the signs $\pm$ are produced by the permutations of the elements $y_1, \dots, y_n$.
In particular, if $L$ is a $\mathcal{P}re\mathcal{L}ie$-algebra up to homotopy, then $\Sigma L$ is endowed with a differential $d$ given by the restriction $l: \Sigma L \to \Sigma L$.

In the following, we adopt the following notation:

$$x\{y_1, \dots, y_n\} := l(x \otimes y_1 \cdots y_n).$$

We call such operations the symmetric braces associated to the $\mathcal{P}re\mathcal{L}ie_{\infty}$-algebra $L$.

**Remark 2.7.** We have an operad morphism $\mathcal{P}re\mathcal{L}ie_{\infty} \to \mathcal{P}re\mathcal{L}ie$ which sends $e_1^2$ to the pre-Lie product, and the other $e_1^n$'s to 0. Thus, every $\mathcal{P}re\mathcal{L}ie$-algebra has a canonical $\mathcal{P}re\mathcal{L}ie_{\infty}$-algebra structure. Beware that the symmetric braces in pre-Lie algebras have nothing to do with the symmetric braces in $\mathcal{P}re\mathcal{L}ie_{\infty}$-algebras.

Equivalently, Proposition 2.6 asserts that giving a structure of a $\mathcal{P}re\mathcal{L}ie_{\infty}$-algebra on $L$ is equivalent to giving a coderivation $Q \in \mathcal{C}oder(\mathrm{Perm}^c(\Sigma L))$ such that $Q^2 = 0$.

**Definition 2.8.** We define the category $\Lambda\mathcal{P}L_{\infty}$ with as set of objects the pairs $(V, Q)$, where $V \in \mathrm{gVect}_{\mathbb{K}}$ and $Q \in \mathcal{C}oder(\mathrm{Perm}^c(V))$ is a degree $-1$ element such that $Q^2 = 0$. A morphism $\phi: (V, Q) \to (V', Q')$ in $\Lambda\mathcal{P}L_{\infty}$ is a morphism of coalgebras $\phi: \mathrm{Perm}^c(V) \to \mathrm{Perm}^c(V')$ which preserves the cod derivations $Q$ and $Q'$.

Usually, a morphism in $\Lambda\mathcal{P}L_{\infty}$ from $(V, Q)$ to $(V', Q')$ is denoted by $\phi: V \rightsquigarrow V'$ and is called an $\infty$-morphism.

**Theorem 2.9.** A $dg\mathbb{K}$-module $L$ is a $\mathcal{P}re\mathcal{L}ie_{\infty}$-algebra if and only if $\Sigma L \in \Lambda\mathcal{P}L_{\infty}$. Moreover, any morphism of $\mathcal{P}re\mathcal{L}ie_{\infty}$-algebras $\phi: L \to L'$ gives rise to a morphism $\Sigma\phi: \Sigma L \to \Sigma L'$ in $\Lambda\mathcal{P}L_{\infty}$ which preserves the symmetric braces.

## 2.2 The category $\Gamma\Lambda\mathcal{P}L_{\infty}$

In this subsection, we aim to define an analogue of the category $\Lambda\mathcal{P}L_{\infty}$, denoted by $\Gamma\Lambda\mathcal{P}L_{\infty}$, which will characterize the $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebras as in Theorem 2.9.

Let $V$ be a graded $\mathbb{K}$-module. We define $\Gamma(V)$ by

$$\Gamma(V) := \bigoplus_{n \ge 0} (V^{\otimes n})^{\Sigma_n}.$$

We have $\Gamma(V) \simeq \mathbb{K} \oplus \bar{\Gamma}(V)$ with

$$\bar{\Gamma}(V) := \bigoplus_{n \ge 1} (V^{\otimes n})^{\Sigma_n}.$$

Note that we have a morphism $\mathrm{Tr}: S(V) \to \Gamma(V)$ called the trace map and defined by $\mathrm{Tr}(1) = 1$ and

$$\mathrm{Tr}(v_1 \cdots v_n) = \sum_{\sigma \in \Sigma_n} \pm v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(n)}$$

for every $v_1, \dots, v_n \in V$ and $n \ge 1$.
**Definition 2.10.** For every $V \in g\mathrm{Vect}_{\mathbb{K}}$, we set

$$ \Gamma\mathrm{Perm}^c(V) := V \otimes \Gamma(V). $$

Our goal is to construct a coproduct $\Delta_{\Gamma\mathrm{Perm}} : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(V) \otimes \Gamma\mathrm{Perm}^c(V)$ which is compatible, in some sense, with the coproduct $\Delta_{\mathrm{Perm}} : \mathrm{Perm}^c(V) \to \mathrm{Perm}^c(V) \otimes \mathrm{Perm}^c(V)$. For this purpose, we first consider the tensor algebra generated by $V$:

$$ T(V) := \bigoplus_{n \ge 0} V^{\otimes n}. $$

We have a coproduct $\Delta_{T(V)} : T(V) \to T(V) \otimes T(V)$ defined by $\Delta_{T(V)}(1) := 1 \otimes 1$ and

$$ \Delta_{T(V)}(v_1 \otimes \cdots \otimes v_n) := \sum_{k=0}^{n} (v_1 \otimes \cdots \otimes v_k) \otimes (v_{k+1} \otimes \cdots \otimes v_n). $$

for every $v_1, \dots, v_n \in V$. Consider

$$ \bar{T}(V) := \bigoplus_{n \ge 1} V^{\otimes n}. $$

We also have a coproduct on $\bar{T}(V)$, defined, for every $v_1, \dots, v_n \in V$, by $\Delta_{\bar{T}(V)}(v_1) := 0$ and,

$$ \Delta_{\bar{T}(V)}(v_1 \otimes \cdots \otimes v_n) := \sum_{k=1}^{n-1} (v_1 \otimes \cdots \otimes v_k) \otimes (v_{k+1} \otimes \cdots \otimes v_n). $$

We embed $V \otimes \Gamma(V) \subset \bar{T}(V)$. Note that

$$ \Delta_{\bar{T}(V)}(V \otimes \Gamma(V)) \subset V \otimes \Gamma(V) \otimes \bar{\Gamma}(V). $$

By applying the embedding $(V^{\otimes n})^{\Sigma_n} \subset V \otimes (V^{\otimes n-1})^{\Sigma_{n-1}}$ for each $n \ge 2$, we have the inclusion $\bar{\Gamma}(V) \subset V \otimes \Gamma(V)$. We thus have obtained a coproduct

$$ \Delta_{\Gamma\mathrm{Perm}} : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(V) \otimes \Gamma\mathrm{Perm}^c(V). $$

We can also identify $\Delta_{\Delta_{\Gamma\mathrm{Perm}}}$ with the composite

$$ \begin{tikzcd} \Delta_{\Gamma\mathrm{Perm}} : V \otimes \Gamma(V) \arrow[r, "id \otimes \Delta_{T(V)}"] & V \otimes \Gamma(V) \otimes \Gamma(V) \arrow[d] \\ & V \otimes \Gamma(V) \otimes \bar{\Gamma}(V) \arrow[r, "(V \otimes \Gamma(V)) \otimes (V \otimes \Gamma(V))"'] & \end{tikzcd} $$

**Lemma 2.11.** The morphism $\Delta_{\Gamma\mathrm{Perm}} : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(V) \otimes \Gamma\mathrm{Perm}^c(V)$ satisfies the identities:

$$ (id \otimes \Delta_{\Gamma\mathrm{Perm}})\Delta_{\Gamma\mathrm{Perm}} = (\Delta \otimes id)\Delta_{\Gamma\mathrm{Perm}}; $$

$$ (id \otimes \Delta_{\Gamma\mathrm{Perm}})\Delta_{\Gamma\mathrm{Perm}} = (id \otimes \tau)(id \otimes \Delta_{\Gamma\mathrm{Perm}})\Delta_{\Gamma\mathrm{Perm}}. $$

Moreover, we have the following commutative diagram:
<figure><img src="image_11.png" /></figure>

Proof. The proof of this lemma comes from straightforward computations. □

**Remark 2.12.** The relation $(id \otimes \Delta_{\Gamma\text{Perm}})\Delta_{\Gamma\text{Perm}} = (id \otimes \tau)(id \otimes \Delta_{\Gamma\text{Perm}})\Delta_{\Gamma\text{Perm}}$ implies that, for every $k \ge 1$,

$$(\Delta_{\Gamma\text{Perm}})^k(\Gamma\text{Perm}^c(V)) \subset \Gamma\text{Perm}^c(V) \otimes (\Gamma\text{Perm}^c(V)^{\otimes k})^{\Sigma_k}.$$

As a consequence, since $(\Delta_{\overline{T}(V)})^k$ reduces to the identity on $V^{\otimes k+1}$, and by definition of $\Delta_{\Gamma\text{Perm}}$, we have the following commutative diagram:

<figure><img src="image_12.png" /></figure>

where, for every $k \ge 0$, we denote by $\pi_{V^{\otimes k}} : \overline{T}(V) \to V^{\otimes k}$ the projection onto $V^{\otimes k}$.

**Definition 2.13.** An endomorphism $d$ of $\Gamma\text{Perm}^c(V)$ is called a coderivation if it satisfies

$$\Delta_{\Gamma\text{Perm}}d = (d \otimes id + id \otimes d)\Delta_{\Gamma\text{Perm}}.$$

We let $\text{Coder}(\Gamma\text{Perm}^c(V))$ to be the $K$-module spanned by coderivations.

Our goal is to prove that any coderivation is characterized by its composite with $\pi_V$. We rely on the following definition.

**Definition 2.14.** Let $w, v_1, \dots, v_r \in V$. We define $\text{Sh} : T(V) \otimes V \to T(V)$ by

$$\text{Sh}(v_1 \otimes \cdots \otimes v_n; w) = \sum_{i=0}^{n} \pm v_1 \otimes \cdots \otimes v_i \otimes w \otimes v_{i+1} \otimes \cdots \otimes v_n$$

where the sign is given by the permutation $v_1 \otimes \cdots \otimes v_n \otimes w \mapsto \pm v_1 \otimes \cdots \otimes v_i \otimes w \otimes v_{i+1} \otimes \cdots \otimes v_n$ for every $0 \le i \le n$. We also define analogously $\text{Sh} : V \otimes T(V) \to T(V)$.

We immediately see that $\text{Sh}(\Gamma(V) \otimes V) \subset \Gamma(V)$.

**Proposition 2.15.** The map

$$\begin{array}{ccc} \text{Coder}(\Gamma\text{Perm}^c(V)) & \to & \operatorname{Hom}(\Gamma\text{Perm}^c(V), V) \\ d & \mapsto & \pi_V \circ d \end{array}$$
is a bijection. We denote by $\tilde{\Psi}$ its inverse. If we consider the inverse $\Psi$ given in the proof of Proposition 2.5, then $\tilde{\Psi}$ is compatible with $\Psi$ in the following sense. Let $\tilde{l} \in \operatorname{Hom}(\Gamma\operatorname{Perm}^c(V), V)$. We define $l \in \operatorname{Hom}(\operatorname{Perm}^c(V), V)$ by the composite

$$l : \operatorname{Perm}^c(V) \xrightarrow{\mathrm{id} \otimes \mathrm{Tr}} \Gamma\operatorname{Perm}^c(V) \xrightarrow{\tilde{l}} V.$$

Then the following diagram is commutative:

$$\begin{array}{ccc} \Gamma\operatorname{Perm}^c(V) & \xrightarrow{\tilde{\Psi}(\tilde{l})} & \Gamma\operatorname{Perm}^c(V) \\ \xrightarrow{\mathrm{id} \otimes \mathrm{Tr}} & & \xrightarrow{\mathrm{id} \otimes \mathrm{Tr}} \\ \operatorname{Perm}^c(V) & \xrightarrow{\Psi(l)} & \operatorname{Perm}^c(V) \end{array}$$

*Proof.* Let $\tilde{l} : \Gamma\operatorname{Perm}^c(V) \to V$ be a morphism. We define an endomorphism $\tilde{\Psi}(\tilde{l})$ of $\Gamma\operatorname{Perm}^c(V)$ by the sum of the composite

$$\tilde{\Psi}_1(\tilde{l}) : V \otimes \Gamma(V) \xrightarrow{\mathrm{id} \otimes \Delta_T(V)} V \otimes \Gamma(V) \otimes \Gamma(V) \xrightarrow{\tilde{l} \otimes \mathrm{id}} V \otimes \Gamma(V)$$

and of the composite

$$\tilde{\Psi}_2(\tilde{l}) : V \otimes \Gamma(V) \xrightarrow{\Delta_{\Gamma\operatorname{Perm}}} (V \otimes \Gamma(V)) \otimes (V \otimes \Gamma(V)) \xrightarrow{\mathrm{id} \otimes \mathrm{id} \otimes \tilde{l}} V \otimes \Gamma(V) \otimes V \xrightarrow{\mathrm{id} \otimes \mathrm{Sh}(-;-)} V \otimes \Gamma(V).$$

Let $x \in V$ and $Y \in \Gamma(V)$. We write the coproduct $\Delta_{\Gamma\operatorname{Perm}}(x \otimes Y)$ by using the Sweedler notation without the sum symbol, as

$$\Delta_{\Gamma\operatorname{Perm}}(x \otimes Y) = (x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes Y_{(2)}),$$

where $x_{(2)} \in V$ and $Y_{(1)}, Y_{(2)} \in \Gamma(V)$. Note that we have

$$(\mathrm{id} \otimes \Delta_{T(V)})(x \otimes Y) = (x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes Y_{(2)}) + (x \otimes Y) \otimes 1.$$

Then, by definition of $\tilde{\Psi}_1(\tilde{l})$ and $\tilde{\Psi}_2(\tilde{l})$, we have

$$\tilde{\Psi}_1(\tilde{l})(x \otimes Y) = \tilde{l}(x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes Y_{(2)}) + \tilde{l}(x \otimes Y) \otimes 1$$

and

$$\tilde{\Psi}_2(\tilde{l})(x \otimes Y) = \pm x \otimes \mathrm{Sh}(Y_{(1)}; \tilde{l}(x_{(2)} \otimes Y_{(2)})),$$

so that

$$\tilde{\Psi}(\tilde{l})(x \otimes Y) = \tilde{l}(x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes Y_{(2)}) + \tilde{l}(x \otimes Y) \otimes 1 \pm x \otimes \mathrm{Sh}(Y_{(1)}; \tilde{l}(x_{(2)} \otimes Y_{(2)})).$$

We set

$$\Delta_{\Gamma\operatorname{Perm}}(x \otimes Y_{(1)}) = (x \otimes Y_{(11)}) \otimes (x_{(1)} \otimes Y_{(12)});$$

$$\Delta_{\Gamma\operatorname{Perm}}(x_{(2)} \otimes Y_{(2)}) = (x_{(2)} \otimes Y_{(21)}) \otimes (x_{(22)} \otimes Y_{(22)}).$$

We then have
$$ (\tilde{\Psi}(\tilde{l}) \otimes id) \Delta_{\Gamma\text{Perm}}(x \otimes Y) = \tilde{l}(x \otimes Y_{(11)}) \otimes (x_{(1)} \otimes Y_{(12)}) \otimes (x_{(2)} \otimes Y_{(2)}) \\ + \tilde{l}(x \otimes Y_{(1)}) \otimes 1 \otimes (x_{(2)} \otimes Y_{(2)}) \\ \pm (x \otimes \text{Sh}(Y_{(11)}; \tilde{l}(x_{(1)} \otimes Y_{(12)}))) \otimes (x_{(2)} \otimes Y_{(2)}); $$

$$ (id \otimes \tilde{\Psi}(\tilde{l})) \Delta_{\Gamma\text{Perm}}(x \otimes Y) = \pm(x \otimes Y_{(1)}) \otimes (\tilde{l}(x_{(2)} \otimes Y_{(21)}) \otimes (x_{(22)} \otimes Y_{(22)})) \\ \pm(x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes \text{Sh}(Y_{(21)}; \tilde{l}(x_{(22)} \otimes Y_{(22)}))) \\ \pm(x \otimes Y_{(1)}) \otimes \tilde{l}(x_{(2)} \otimes Y_{(2)}) \otimes 1. $$

We now compute $\Delta_{\Gamma\text{Perm}}\tilde{\Psi}(\tilde{l})(x \otimes Y)$. The term $\Delta_{\Gamma\text{Perm}}\tilde{\Psi}_1(\tilde{l})(x \otimes Y)$ gives

$$ \tilde{l}(x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes Y_{(21)}) \otimes (x_{(22)} \otimes Y_{(22)}) + \tilde{l}(x \otimes Y_{(1)}) \otimes 1 \otimes (x_{(2)} \otimes Y_{(2)}). $$

By using the first identity of Lemma 2.11 which gives

$$ (x \otimes Y_{(11)}) \otimes (x_{(1)} \otimes Y_{(12)}) \otimes (x_{(2)} \otimes Y_{(2)}) = (x \otimes Y_{(1)}) \otimes (x_{(2)} \otimes Y_{(21)}) \otimes (x_{(22)} \otimes Y_{(22)}), $$

we have that $\Delta_{\Gamma\text{Perm}}\tilde{\Psi}_1(\tilde{l})(x \otimes Y)$ is

$$ \tilde{l}(x \otimes Y_{(11)}) \otimes (x_{(1)} \otimes Y_{(12)}) \otimes (x_{(2)} \otimes Y_{(2)}) + \tilde{l}(x \otimes Y_{(1)}) \otimes 1 \otimes (x_{(2)} \otimes Y_{(2)}), $$

which is exactly the first two lines occurring in $(\tilde{\Psi}(\tilde{l}) \otimes id) \Delta_{\Gamma\text{Perm}}(x \otimes Y)$. The term $\Delta_{\Gamma\text{Perm}}\tilde{\Psi}_2(\tilde{l})(x \otimes Y)$ gives

$$ \pm(x \otimes \text{Sh}(Y_{(11)}; \tilde{l}(x_{(2)} \otimes Y_{(2)}))) \otimes (x_{(1)} \otimes Y_{(12)}) $$

$$ \pm(x \otimes Y_{(11)}) \otimes (\tilde{l}(x_{(2)} \otimes Y_{(2)}) \otimes Y_{(12)}) $$

$$ \pm(x \otimes Y_{(11)}) \otimes (x_{(1)} \otimes \text{Sh}(Y_{(12)}; \tilde{l}(x_{(2)} \otimes Y_{(2)}))) $$

$$ \pm(x \otimes Y_{(1)}) \otimes (\tilde{l}(x_{(2)} \otimes Y_{(2)}) \otimes 1). $$

From the second formula given in Lemma 2.11, which gives

$$ (x \otimes Y_{(11)}) \otimes (x_{(1)} \otimes Y_{(12)}) \otimes (x_{(2)} \otimes Y_{(2)}) = \pm(x \otimes Y_{(11)}) \otimes (x_{(2)} \otimes Y_{(2)}) \otimes (x_{(1)} \otimes Y_{(12)}), $$

we obtain that $\Delta_{\Gamma\text{Perm}}\tilde{\Psi}_2(\tilde{l})(x \otimes Y)$ is given by

$$ \pm(x \otimes \text{Sh}(Y_{(11)}; \tilde{l}(x_{(1)} \otimes Y_{(12)}))) \otimes (x_{(2)} \otimes Y_{(2)}) $$

$$ \pm(x \otimes Y_{(11)}) \otimes (\tilde{l}(x_{(1)} \otimes Y_{(12)}) \otimes Y_{(2)}) $$

$$ \pm(x \otimes Y_{(11)}) \otimes (x_{(2)} \otimes \text{Sh}(Y_{(2)}; \tilde{l}(x_{(1)} \otimes Y_{(12)}))) $$

$$ \pm(x \otimes Y_{(1)}) \otimes (\tilde{l}(x_{(2)} \otimes Y_{(2)}) \otimes 1). $$

The first line is the remaining term in $(\tilde{\Psi}(\tilde{l}) \otimes id) \Delta_{\Gamma\text{Perm}}(x \otimes Y)$, while the remaining lines give $(id \otimes \tilde{\Psi}(\tilde{l})) \Delta_{\Gamma\text{Perm}}(x \otimes Y)$ when using again the first formula of Lemma 2.11. We thus have proved that $\tilde{\Psi}(\tilde{l}) \in \text{Coder}(\Gamma\text{Perm}^c(V))$, and $\pi_V \circ \tilde{\Psi}(\tilde{l}) = \tilde{l}$. We now prove that $\tilde{\Psi}(\tilde{l})$ is the only codерivation $Q$ such that $\pi_V \circ Q = l$. We use Remark 2.12, which gives

$$ \pi_{V \otimes k+1} Q = \pi_V^{\otimes k+1} \Delta_{\Gamma\text{Perm}}^k Q = \sum_{i=1}^{k+1} (\pi_V^{\otimes i-1} \otimes \tilde{l} \otimes \pi_V^{\otimes k-i+1}) \Delta_{\Gamma\text{Perm}}^k $$
which proves that $Q$ is fully determined by $\tilde{l}$. We thus have $Q = \tilde{\Psi}(\tilde{l})$, and then that $\Psi$ is the desired bijection. We now prove the commutativity of the diagram. We note that, by definition of $\Psi(l)$, $\tilde{\Psi}(\tilde{l})$ and $l$, the following diagram is commutative:

<figure><img src="image_13.png" /></figure>

By Lemma 2.11, and by the formula

$$ \mathrm{Tr}(v_1 \cdots v_n \cdot w) = \mathrm{Sh}(\mathrm{Tr}(v_1 \cdots v_n); w), $$

we also obtain the following commutative diagram:

<figure><img src="image_14.png" /></figure>

which proves the theorem. $\square$

**Definition 2.16.** We define the category $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$ with as objects the pairs $(V, Q)$ where $V$ is a graded $\mathbb{K}$-module and $Q$ a cod derivation of degree $-1$ on $\Gamma\mathrm{Perm}^c(V)$ such that $Q^2 = 0$; a morphism $\phi : (V, Q) \to (V', Q')$ is a morphism of coalgebras $\phi : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(V_2)$ which commutes with the cod derivations $Q$ and $Q'$.

We usually denote a morphism $\phi : (V, Q) \to (V', Q')$ by $\phi : V \rightsquigarrow V'$ when there is no ambiguity on $Q$ and $Q'$, and call it an $\infty$-morphism.

If $\phi : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(V')$ is a morphism of graded $\mathbb{K}$-modules, then we set, for all $k, n \ge 0$,

$$ \phi_k : V \otimes (V^{\otimes k})^{\Sigma_k} \longrightarrow \Gamma\mathrm{Perm}^c(V) \xrightarrow{\phi} \Gamma\mathrm{Perm}^c(V'); $$

$$ \phi^n : \Gamma\mathrm{Perm}^c(V) \xrightarrow{\phi} \Gamma\mathrm{Perm}^c(V) \xrightarrow{\pi_{V^{\otimes n+1}}} V \otimes (V^{\otimes n})^{\Sigma_n}. $$

Using these notations, a degree $-1$ cod derivation $Q$ on $\Gamma\mathrm{Perm}^c(V)$ is such that $Q^2 = 0$ if and only if for all $n \ge 0$,

$$ \sum_{k=0}^{n} Q_k^0 Q_n^k = 0. $$
In particular, $Q_0^0$ is a differential on $V$. From now on, we endow $V \in \Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$ with the structure of a dg $\mathbb{K}$-module with differential $d = Q_0^0$. We also have that a morphism of graded $\mathbb{K}$-modules $\phi : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(V_2)$ is a morphism of coalgebras if and only if

$$ \Delta_{\Gamma\mathrm{Perm}}\phi^n = \sum_{p+q=n-1} (\phi^p \otimes \phi^q) \Delta_{\Gamma\mathrm{Perm}}. $$

**Proposition 2.17.** Every $\infty$-morphism $\phi$ is fully determined by the composite $\phi^0 = \pi_V \circ \phi$.

*Proof.* Let $\phi$ be an $\infty$-morphism. We have that

$$ \phi^k = \pi_V^{\otimes k+1} \Delta_{\Gamma\mathrm{Perm}}^k \phi^k = (\phi^0)^{\otimes k} \Delta_{\Gamma\mathrm{Perm}}^k, $$

which gives, for every $v \in \Gamma\mathrm{Perm}^c(V)$,

$$ \phi^k(v) = \phi^0(v_{(1)}) \otimes \cdots \otimes \phi^0(v_{(k)}) $$

where we use the Sweedler notation in the coalgebra $\Gamma\mathrm{Perm}^c(V)$. We then see that $\phi$ is fully determined by $\phi^0$. $\square$

**Remark 2.18.** This proposition implies that giving an $\infty$-morphism $\phi : V \rightsquigarrow W$ is equivalent to giving a morphism $\phi^0 : \Gamma\mathrm{Perm}^c(V) \to W$ such that the morphism $\phi : \Gamma\mathrm{Perm}^c(V) \to \Gamma\mathrm{Perm}^c(W)$ constructed in Proposition 2.17 satisfies

$$ \sum_{k=0}^{n} (Q')_k^0 \phi_n^k = \sum_{k=0}^{n} \phi_k^0 Q_n^k $$

for every $n \ge 0$. In particular, $\phi_0^0 : V \to W$ is a morphism of dg $\mathbb{K}$-modules.

**Definition 2.19.** An $\infty$-morphism $\phi : V \rightsquigarrow W$ is strict if $\phi_k^0 = 0$ for all $k \ge 1$.

Equivalently, a strict morphism $\phi : V \to W$ is the data of a morphism of dg $\mathbb{K}$-modules $\phi : V \to W$ such that

$$ (Q')_n^0 \phi^{\otimes n+1} = \phi Q_n^0 $$

for every $n \ge 0$.

## 2.3 Symmetric weighted braces and Maurer-Cartan elements in $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$

In this subsection, we define weighted brace operations for $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$-algebras, and prove that giving a structure of a $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$-algebra is equivalent to giving such operations. These operations will be analogue to the operations given in [Ver23, Theorem 2.6] for $\Gamma(\mathcal{P}\mathrm{re}\mathcal{L}\mathrm{ie}, -)$-algebras. We also define the notion of Maurer-Cartan element in complete $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$-algebras.

We first need an explicit basis of $\Gamma\mathrm{Perm}^c(V)$. We use the same arguments as in [Ver23, §2.1.1]. Let $\mathcal{B}$ be a basis of $V$ composed of homogeneous elements. For every $n \ge 0$, this gives a basis on $V^{\otimes n}$ which we denote by $\mathcal{B}^{\otimes n}$. We consider the action of $\Sigma_n$ on $\mathcal{B}^{\otimes n}$ by permutation of the factors
without the Koszul sign rule. For every $t \in \mathcal{B}^{\otimes n}$, we denote by $X_t$ the orbit of $t$ under this action.
We then have the unequivariant identity

$$V^{\otimes n} = \bigoplus_{t \in \mathcal{B}^{\otimes n} / \Sigma_n} \mathbb{K}[X_t].$$

For every $t \in \mathcal{B}^{\otimes n}$, we set $\mathbb{K}[X_t]^{\pm} = \mathbb{K}[X_t]$ with underlying action

$$\sigma \cdot x = \varepsilon(\sigma, x)x$$

for every $\sigma \in \Sigma_n$ and $x \in X_t$, where we denote by $\varepsilon(\sigma, x) \in \mathbb{K}$ the Koszul sign which appears after
the action of $\sigma$ on $x$. We then have the identification of $\Sigma_n$-representations:

$$V^{\otimes n} = \bigoplus_{t \in \mathcal{B}^{\otimes n} / \Sigma_n} \mathbb{K}[X_t]^{\pm}.$$

Let $(\mathcal{B}^{\otimes n})^s$ be the subset of $\mathcal{B}^{\otimes n}$ given by elements $t \in \mathcal{B}^{\otimes n}$ such that there exists $\sigma \in \text{Stab}_{\Sigma_n}(t)$ with $\varepsilon(\sigma, t) \neq 1$. We set $(\mathcal{B}^{\otimes n})^r = \mathcal{B}^{\otimes n} \setminus (\mathcal{B}^{\otimes n})^s$. Note that, if $\text{char}(\mathbb{K}) = 2$, then $(\mathcal{B}^{\otimes n})^r = \mathcal{B}^{\otimes n}$, else, the subset $(\mathcal{B}^{\otimes n})^r$ is given by tensors of the form $x_1^{\otimes r_1} \otimes \cdots \otimes x_n^{\otimes r_n}$ with $x_1, \dots, x_n \in \mathcal{B}$ pairwise distinct and $r_1, \dots, r_n \ge 0$ such that if $x_i$ has an odd degree for some $i$, then $r_i = 1$. We let $\mathcal{S}^r(V)$ to be given by the projections of $(\mathcal{B}^{\otimes n})^r$ on $\mathcal{S}(V)$.

**Proposition 2.20.** The map $\mathcal{O}: \mathcal{S}^r(V) \to \Gamma(V)$ defined by

$$\mathcal{O}(x_1 \cdots x_n) = \sum_{\sigma \in \Sigma_n / \text{Stab}_{\Sigma_n}(x_1 \cdots x_n)} \pm x_{\sigma^{-1}(1)} \otimes \cdots \otimes x_{\sigma^{-1}(n)}$$

is an isomorphism.

*Proof.* It is the same arguments as in [Ver23, Proposition 2.5]. □

In the following, in order to handle both the cases $\text{char}(\mathbb{K}) = 2$ and $\text{char}(\mathbb{K}) \neq 2$, when taking
elements with associated weights, we will tacitly suppose that if $\text{char}(\mathbb{K}) \neq 2$, then all odd degree
elements will have an associated weight equal to 1.

**Lemma 2.21.** Let $x \in V, y_1, \dots, y_n \in \mathcal{B}$ and $r_1, \dots, r_n \ge 0$. Then

$$\Delta(x \otimes \mathcal{O}(y_1^{\otimes r_1} \cdots y_n^{\otimes r_n})) = \sum_{k=1}^{n} \sum_{\substack{p_i+q_i=r_i, i \neq k \\ p_k+q_k=r_k-1}} \pm (x \otimes \mathcal{O}(y_1^{\otimes p_1} \cdots y_n^{\otimes p_n})) \otimes (y_k \otimes \mathcal{O}(y_1^{\otimes q_1} \cdots y_n^{\otimes q_n})).$$

where the sign is yielded by the shuffle

$$x \otimes y_1^{\otimes r_1} \otimes \cdots \otimes y_n^{\otimes r_n} \mapsto \pm x \otimes y_1^{\otimes p_1} \otimes \cdots \otimes y_n^{\otimes p_n} \otimes y_k \otimes y_1^{\otimes q_1} \otimes \cdots \otimes y_n^{\otimes q_n}.$$

*Proof.* Straightforward computations. □

**Theorem 2.22.** Let $V \in \Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$. Then $V$ comes equipped with operations, called weighted braces,
which have the following form.
- If $\mathrm{char}(\mathbb{K}) = 2$, then weighted braces are maps

$$- \{ \llbracket - , \dots , - \rrbracket_{r_1 , \dots , r_n } : V^{ \times n + 1 } \to V ,$$

defined for any collections of integers $r_1, \dots, r_n \ge 0$, which preserve the grading in the sense that

$$V_k \{ \llbracket V_{k_1} , \dots , V_{k_n } \rrbracket_{r_1 , \dots , r_n } \subset V_{k + k_1 r_1 + \dots + k_n r_n } .$$

- If $\mathrm{char}(\mathbb{K}) \ne 2$, by setting $V^{\mathrm{ev}} = \bigoplus_{n \in \mathbb{Z}} V_{2n}$ and $V^{\mathrm{odd}} = \bigoplus_{n \in \mathbb{Z}} V_{2n+1}$, weighted brace are maps

$$- \{ \underbrace{- , \dots , -}_{p} , \underbrace{- , \dots , -}_{q} \}_{r_1 , \dots , r_p , 1 , \dots , 1} : V \times (V^{\mathrm{ev}})^{\times p} \times (V^{\mathrm{odd}})^{\times q} \to V,$$

defined for any collection of integers $p, q, r_1, \dots, r_n \ge 0$ which preserve the grading.

In addition, in both cases, the weighted brace operations satisfy the following formulas:

$$(i) \ x \{ \llbracket y_{\sigma(1)} , \dots , y_{\sigma(n)} \rrbracket_{r_{\sigma(1)} , \dots , r_{\sigma(n)}} = \pm x \{ \llbracket y_1 , \dots , y_n \rrbracket_{r_1 , \dots , r_n} ,$$

$$(ii) \ x \{ \llbracket y_1 , \dots , y_{i-1} , y_i , y_{i+1} , \dots , y_n \rrbracket_{r_1 , \dots , r_{i-1} , 0 , r_{i+1} , \dots , r_n}$$

$$= x \{ \llbracket y_1 , \dots , y_{i-1} , y_{i+1} , \dots , y_n \rrbracket_{r_1 , \dots , r_{i-1} , r_{i+1} , \dots , r_n} ,$$

$$(iii) \ x \{ \llbracket y_1 , \dots , \lambda y_i , \dots , y_n \rrbracket_{r_1 , \dots , r_i , \dots , r_n} = \lambda^{r_i} x \{ \llbracket y_1 , \dots , y_i , \dots , y_n \rrbracket_{r_1 , \dots , r_i , \dots , r_n} ,$$

$$(iv) \ x \{ \llbracket y_1 , \dots , y_i , y_i , \dots , y_n \rrbracket_{r_1 , \dots , r_i , r_{i+1} , \dots , r_n}$$

$$= \left( \frac{r_i + r_{i+1}}{r_i} \right) x \{ \llbracket y_1 , \dots , y_i , \dots , y_n \rrbracket_{r_1 , \dots , r_{i-1} , r_i + r_{i+1} , r_{i+2} , \dots , r_n} ,$$

$$(v) \ x \{ \llbracket y_1 , \dots , y_i + \tilde{y}_i , \dots , y_n \rrbracket_{r_1 , \dots , r_i , \dots , r_n} = \sum_{s=0}^{r_i} x \{ \llbracket y_1 , \dots , y_i , \tilde{y}_i , \dots , y_n \rrbracket_{r_1 , \dots , s , r_i - s , \dots , r_n} ,$$

$$(vi) \ \sum_{p_i + q_i = r_i} \pm x \{ \llbracket y_1 , \dots , y_n \rrbracket_{p_1 , \dots , p_n} \{ \llbracket y_1 , \dots , y_n \rrbracket_{q_1 , \dots , q_n}$$

$$+ \sum_{k=1}^{n} \sum_{\substack{p_i + q_i = r_i, i \neq k \\ p_k + q_k = r_k - 1}} \pm x \{ \llbracket y_k \{ \llbracket y_1 , \dots , y_n \rrbracket_{p_1 , \dots , p_n} , y_1 , \dots , y_n \rrbracket_{1, q_1 , \dots , q_n} \} = 0.$$

In the converse direction, if a graded $\mathbb{K}$-module $V$ admits such operations, then $V \in \Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$.

In particular, the operation $d(x) := x \{ \| \| $ is a differential. We usually endow $V \in \Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$ with the structure of a dg $\mathbb{K}$-module with differential $d$.

*Proof.* Let $V \in \Gamma \Lambda \mathcal{P} \mathcal{L}_{\infty}$. The strategy is the same as in [Ver23, Theorem 2.6] or [Ces18, Proposition 5.10]. Let $x, y_1, \dots, y_n \in V$ be homogeneous elements, and $e_1, \dots, e_n$ be formal elements with
the same degrees as $y_1, \dots, y_n$. We let $E$ to be graded $\mathbb{K}$-module spanned by $Y_1, \dots, Y_n$. Let $\psi : \Gamma(E) \to \Gamma(V)$ be the morphism which sends the $Y_i$'s to the $y_i$'s. We immediately see that $\psi$ is a morphism of coalgebras. We set

$$x \llbracket y_1, \dots, y_n \rrbracket_{r_1, \dots, r_n} := Q_{\sum_i r_i}^0 (x \otimes \psi \mathcal{O}(Y_1^{\otimes r_1} \cdots Y_n^{\otimes r_n})).$$

Formulas (i) – (v) are consequences of straightforward computations. We prove formula (vi). Since $\psi$ is a morphism of coalgebras, Lemma 2.21 gives

$$Q(x \otimes \psi \mathcal{O}(Y_1^{r_1} \cdots Y_n^{r_n})) = \sum_{p_i+q_i=r_i} \pm Q^0(x \otimes \psi \mathcal{O}(Y_1^{p_1} \cdots Y_n^{p_n})) \otimes \psi(\mathcal{O}(Y_1^{q_1} \cdots Y_n^{q_n})) \\ + \sum_{k=1}^n \sum_{\substack{p_i+q_i=r_i, i \neq k \\ p_k+q_k=r_k-1}} \pm x \otimes \mathrm{Sh}(Q^0(y_k \otimes \psi \mathcal{O}(Y_1^{p_1} \cdots Y_n^{p_n})); \psi \mathcal{O}(Y_1^{q_1} \cdots Y_n^{q_n})).$$

For fixed $p_i$'s and $q_i$'s, we have

$$Q^0(Q^0(x \otimes \psi \mathcal{O}(Y_1^{p_1} \cdots Y_n^{p_n})) \otimes \psi \mathcal{O}(Y_1^{q_1} \cdots Y_n^{q_n})) = x \llbracket y_1, \dots, y_n \rrbracket_{p_1, \dots, p_n} \llbracket y_1, \dots, y_n \rrbracket_{q_1, \dots, q_n},$$

by definition of weighted brace operations. Concerning the second line, for fixed $p_i$'s, $q_i$'s and $k$, let $Z$ be a formal element with the same degree as $Q^0(y_k \otimes \psi \mathcal{O}(Y_1^{p_1}, \dots, Y_n^{p_n}))$. We extend $\psi$ to $\psi : \Gamma(E \oplus \mathbb{K}Z) \to \Gamma(V)$ by sending $Z$ to $Q^0(y_k \otimes \psi \mathcal{O}(Y_1^{p_1}, \dots, Y_n^{p_n}))$. We then have

$$\mathrm{Sh}(Q^0(y_k \otimes \psi \mathcal{O}(Y_1^{p_1} \cdots Y_n^{p_n})); \psi \mathcal{O}(Y_1^{q_1} \cdots Y_n^{q_n})) = \psi \mathcal{O}(Z \cdot Y_1^{q_1} \cdots Y_n^{q_n}).$$

Taking the image under $Q^0$ thus gives

$$Q^0(x \otimes \mathrm{Sh}(Q^0(y_k \otimes \psi \mathcal{O}(Y_1^{p_1} \cdots Y_n^{p_n})); \psi \mathcal{O}(Y_1^{q_1} \cdots Y_n^{q_n}))) \\ = x \llbracket y_k \llbracket y_1, \dots, y_n \rrbracket_{p_1, \dots, p_n}, y_1, \dots, y_n \rrbracket_{1, q_1, \dots, q_n}.$$

Since $Q^0Q = 0$, formula (vi) follows. We now prove the converse direction. Suppose that $V$ is a $\mathbb{K}$-module equipped with operations $\llbracket - , \dots, - \rrbracket_{r_1, \dots, r_n}$ for all $r_1, \dots, r_n \ge 0$ which satisfy the formulas given in the theorem. We pick a basis $\mathcal{B}$ of $V$ composed of homogeneous elements. Let $x, y_1, \dots, y_n \in \mathcal{B}$. For all $r_1, \dots, r_n \ge 0$, we set

$$Q^0(x \otimes \mathcal{O}(y_1^{r_1} \cdots y_n^{r_n})) = x \llbracket y_1, \dots, y_n \rrbracket_{r_1, \dots, r_n}$$

where we consider the orbit map $\mathcal{O}$ associated to the basis $\mathcal{B}$. By formulas (iii) – (v) and the same computations as in [Ces18, Lemma 5.15], this definition does not depend on the choice of $\mathcal{B}$. Let $Q = \tilde{\Psi}(Q^0)$ be the cod derivation associated to $Q^0 \in \mathrm{Hom}(\Gamma \mathrm{Perm}^c(V), V)$ given by Proposition 2.15. We need to prove that $Q^2 = 0$, which is equivalent to prove that $Q^0Q = 0$. By Lemma 2.21, we have that

$$Q(x \otimes \mathcal{O}(y_1^{r_1} \cdots y_n^{r_n})) = \sum_{p_i+q_i=r_i} \pm Q^0(x \otimes \mathcal{O}(y_1^{p_1} \cdots y_n^{p_n})) \otimes \mathcal{O}(y_1^{q_1} \cdots y_n^{q_n}) \\ + \sum_{k=1}^n \sum_{\substack{p_i+q_i=r_i, i \neq k \\ p_k+q_k=r_k-1}} \pm x \otimes \mathrm{Sh}(Q^0(y_k \otimes \mathcal{O}(y_1^{p_1} \cdots y_n^{p_n})); \mathcal{O}(y_1^{q_1} \cdots y_n^{q_n})).$$

Applying $Q^0$ to this identity gives $Q^0Q = 0$. $\square$
**Remark 2.23.** A strict morphism $\phi: V \to W$ preserves the braces in the sense that

$$ \phi(x(\llbracket y_1, \dots, y_n \rrbracket_{r_1, \dots, r_n}) = \pm \phi(x)(\llbracket \phi(y_1), \dots, \phi(y_n) \rrbracket_{r_1, \dots, r_n}), $$

where $\pm$ is produced by the commutation of $\phi$ with $x$ and the $y_i^{\otimes r_i}$'s.

We aim to define the notion of a Maurer-Cartan element. To achieve this, we define the notion of a complete $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra.

**Definition 2.24.** A filtered $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra is a $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra $V$ endowed with a filtration $(F_nV)_{n \ge 1}$ such that

$$ F_m V (\llbracket F_{p_1} V, \dots, F_{p_n} V \rrbracket_{r_1, \dots, r_n}) \subset F_{m+p_1 r_1 + \dots + p_n r_n} V, $$

for all $m, p_1, \dots, p_n \ge 1$ and $r_1, \dots, r_n \ge 0$. An $\infty$-morphism $\phi: V \to V'$ between two filtered $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebras is an $\infty$-morphism such that

$$ \phi_n^0(F_k(V^{\otimes n+1}) \cap \Gamma\mathrm{Perm}^c(V)) \subset F_k(V'). $$

for every $k \ge 1$, where we consider the filtration associated to a tensor product (see the end of §1.1). A filtered $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra is complete if the map $V \to \lim_{n \ge 1} V/F_nV$ is an isomorphism.

We denote by $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_{\infty}$ the category formed by complete filtered $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebras with as morphisms the $\infty$-morphisms which preserve the filtrations.

**Remark 2.25.** If $V$ is a filtered $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra, then its completion $\widehat{V}$ admits the structure of a complete filtered $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra.

**Definition 2.26.** Let $V \in \Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_{\infty}$. A Maurer-Cartan element is an element $x \in V_0$ such that

$$ d(x) + \sum_{n \ge 1} x(\llbracket x \rrbracket)_n = 0. $$

We denote by $\mathcal{MC}(V)$ the set composed of Maurer-Cartan elements.

**Proposition 2.27.** Let $V, V' \in \Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_{\infty}$ and $\phi: V \to V'$. Then $\phi$ induces a map

$$ \mathcal{MC}(\phi): \begin{array}{ccc} \mathcal{MC}(V) & \to & \mathcal{MC}(V') \\ x & \mapsto & \sum_{n \ge 0} \phi_n^0(x \otimes x^{\otimes n}) \end{array} $$

such that $\mathcal{MC}(-): \Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_{\infty} \to \text{Set}$ is a functor. Moreover, if $\phi_0^0$ is an isomorphism, then $\mathcal{MC}(\phi)$ is a bijection.

**Proof.** Let $x \in \mathcal{MC}(V)$. We first prove that $y = \sum_{n \ge 0} \phi_n^0(x^{\otimes n+1}) \in \mathcal{MC}(V')$. We have

$$ \sum_{m \ge 0} (Q')_m^0 (y^{\otimes m+1}) = \sum_{m \ge 0} \sum_{k \ge m} (Q')_m^0 \left( \sum_{p_0 + \dots + p_m = k - m} \phi_{p_0}^0 (x^{\otimes p_0+1}) \otimes \dots \otimes \phi_{p_m}^0 (x^{\otimes p_m+1}) \right). $$

By using the proof of Proposition 2.17, we have
$$
\begin{align*}
\sum_{m \ge 0} (Q')_m^0 (y^{\otimes m+1}) &= \sum_{k \ge 0} \sum_{m=0}^k (Q')_m^0 \phi_k^m (x^{\otimes k+1}) \\
&= \sum_{k \ge 0} \sum_{m=0}^k \phi_m^0 Q_k^m (x^{\otimes k+1}) \\
&= \phi^0 \left( \sum_{k \ge 0} Q_k (x^{\otimes k+1}) \right).
\end{align*}
$$

By using that $Q = \tilde{\Psi}(Q^0)$ (see the proof of Proposition 2.15), we obtain

$$
\sum_{k \ge 0} Q_k (x^{\otimes k+1}) = \sum_{q \ge 0} \left( \sum_{p \ge 0} Q_p^0 (x^{\otimes p+1}) \right) \otimes x^{\otimes q} \pm \sum_{p \ge 0} x \otimes \mathrm{Sh} \left( x^{\otimes p}; \sum_{q \ge 0} Q_q^0 (x^{\otimes q+1}) \right) = 0
$$

since $x \in \mathcal{MC}(V)$. The map $\mathcal{MC}(\phi)$ is thus well defined. Suppose now that $\phi_0^0$ is an isomorphism,
and let $y \in \mathcal{MC}(V')$. We search $x \in \mathcal{MC}(V)$ such that

$$
\sum_{n \ge 0} \phi_n^0 (x^{\otimes n+1}) = y,
$$

which is equivalent to

$$
x = (\phi_0^0)^{-1} \left( y - \sum_{n \ge 1} \phi_n^0 (x^{\otimes n+1}) \right).
$$

We set $x_0 = (\phi_0^0)^{-1}(y)$. We define a Cauchy sequence $(x_k)_k$ by induction by

$$
x_{k+1} = (\phi_0^0)^{-1} \left( y - \sum_{n \ge 1} \phi_n^0 (x_k^{\otimes n+1}) \right).
$$

We denote by $x$ its limit. We show that $x \in \mathcal{MC}(V)$. For every $W \in \Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$, we set

$$
\mathcal{R}(w) = d(w) + \sum_{n \ge 1} w \|w\|_n
$$

for every $w \in W_0$. We apply $\mathcal{R}$ on the identity $\sum_{n \ge 0} \phi_n^0 (x^{\otimes n+1}) = y$, and use that $y \in \mathcal{MC}(V')$:

$$
\sum_{p \ge 0} (Q')_p^0 \left( \left( \sum_{n \ge 0} \phi_n^0 (x^{\otimes n+1}) \right)^{\otimes p+1} \right) = 0.
$$

This can be written as

$$
\sum_{n \ge 0} \sum_{p=0}^{n} (Q')_p^0 \phi_n^p (x^{\otimes n+1}) = 0.
$$
Using that $\phi$ is a morphism in $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$, we obtain that

$$ \sum_{n \ge 0} \sum_{p=0}^{n} \phi_p^0 Q_n^p (x^{\otimes n+1}) = 0 $$

which gives

$$ \mathcal{R}(x) = -(\phi_0^0)^{-1} \left( \sum_{n \ge 1} \sum_{p=1}^{n} \phi_p^0 Q_n^p (x^{\otimes n+1}) \right). $$

We use the computation of $Q$ as $Q = \tilde{\Psi}(Q^0)$:

$$ \mathcal{R}(x) = -(\phi_0^0)^{-1} \left( \sum_{n \ge 1} \sum_{p=1}^{n} \phi_p^0 (Q_{n-p}^0 (x^{\otimes n-p+1}) \otimes x^{\otimes p} + x \otimes \mathrm{Sh}(x^{\otimes p-1}; Q_{n-p}^0 (x^{\otimes n-p+1}))) \right). $$

We finally obtain

$$ \mathcal{R}(x) = -(\phi_0^0)^{-1} \left( \sum_{p \ge 1} \phi_p^0 (\mathcal{R}(x) \otimes x^{\otimes p} + x \otimes \mathrm{Sh}(x^{\otimes p-1}; \mathcal{R}(x))) \right). $$

From this identity, and because $\phi$ preserves the filtrations on $V$ and $V'$, we have that if $\mathcal{R}(x) \in F_k V$ for some $k \ge 1$, then $\mathcal{R}(x) \in F_{k+1} V$. Since $\mathcal{R}(x) \in F_1 V$, it follows that $\mathcal{R}(x) \in \bigcap_{k \ge 1} F_k V = 0$ so that $x \in \mathcal{MC}(V)$, and $\mathcal{MC}(\phi)(x) = y$ by construction. The map $\mathcal{MC}(\phi)$ is then surjective. We now prove that it is injective. Suppose that there exists $x_1, x_2 \in \mathcal{MC}(V)$ such that $\mathcal{MC}(\phi)(x_1) = \mathcal{MC}(\phi)(x_2)$. Then

$$ x_1 - x_2 = (\phi_0^0)^{-1} \left( \sum_{n \ge 1} (x_2^{\otimes n+1} - x_1^{\otimes n+1}) \right). $$

Suppose that $x_1 - x_2 \in F_k V$ for some $k \ge 1$. Then there exists $\alpha_k \in F_k V$ such that $x_1 = x_2 + \alpha_k$. By definition of the filtration on tensor products, and because $x_2 \in F_1 V$, for every $n \ge 0$, we have $x_1^{\otimes n+1} = x_2^{\otimes n+1} + \alpha'_k$ where $\alpha'_k \in F_{k+1} V$ so that $x_2^{\otimes n+1} - x_1^{\otimes n+1} \in F_{k+1} V$. Since that $\phi$ preserves the filtrations, this implies $x_1 - x_2 \in F_{k+1} V$. We thus have $x_1 = x_2$, so that $\mathcal{MC}(\phi)$ is injective. $\square$

## 2.4 Pre-Lie algebras up to homotopy with divided powers and $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$

In this subsection, we show that giving a structure of a $\Gamma(\mathrm{PreLie}_{\infty}, -)$-algebra is equivalent to giving the structure of a $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra up to a shift.

Let $L$ be a dg $\mathbb{K}$-module. We make explicit a choice of a basis for $\Gamma(\mathrm{PreLie}_{\infty}, L)$ so that we can apply [Ver23, Lemma 2.3]. Let $\mathcal{B}$ be a basis of $L$. As a basis for $B^c(\Lambda^{-1}\mathrm{Perm}^c)(n)$, we consider tree monomials in $\mathcal{F}(\Sigma^{-1}\Lambda^{-1}\mathrm{Perm}^c)(n)$ with as vertices elements of the form $\Sigma^{j-2}e_i^j$ where $j \ge 2$ and $1 \le i \le j$ (see [DK10, §3.1] for a definition of these trees, or also Definition 6.4). We denote by $\mathcal{T}\mathcal{M}(n)$ the set of tree monomials with $n$ inputs. This gives a basis of $\mathrm{PreLie}_{\infty}(n) \otimes L^{\otimes n}$ which we denote by $\mathcal{T}\mathcal{M}(n) \otimes \mathcal{B}^{\otimes n}$. We consider the action of $\Sigma_n$ on $\mathcal{T}\mathcal{M}(n)$ given by the action of $\Sigma_n$ on $B^c(\Lambda^{-1}\mathrm{Perm}^c)$ where we omit the Koszul sign rule obtained after using the equivariance axioms
for trees in $B^c(\Lambda^{-1}\text{Perm}^c)$ in order to obtain a tree monomial. We also consider the action of $\Sigma_n$ on $B^{\otimes n}$ by permutations. We deduce an action of $\Sigma_n$ on $\mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n}$ defined as the diagonal action which uses the two previous actions of $\Sigma_n$ on $\mathcal{T}\mathcal{M}(n)$ and $B^{\otimes n}$. Given such an action, we can write

$$ \mathcal{P}re\mathcal{L}ie_{\infty}(n) \otimes L^{\otimes n} = \bigoplus_{t \in (\mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n})/\Sigma_n} \mathbb{K}[X_t] $$

where we denote by $X_t$ the orbit of the element $t \in \mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n}$ under the above action. Now, for every $t \in \mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n}$, $\sigma \in \Sigma_n$ and $x \in X_t$, we denote by $\varepsilon(\sigma, x) \in \mathbb{K}$ the Koszul sign which appears after the action of $\sigma$ on $x$, using the usual actions of $\Sigma_n$ on $\mathcal{T}\mathcal{M}(n)$ and $B^{\otimes n}$. We define the $\Sigma_n$-representation $\mathbb{K}[X_t]^{\pm}$ as $\mathbb{K}[X_t]$ endowed with the $\Sigma_n$-action given by

$$ \sigma \cdot x^{\pm} = \varepsilon(\sigma, x)(\sigma \cdot x)^{\pm}. $$

We obtain the following identification of $\Sigma_n$-representations:

$$ \mathcal{P}re\mathcal{L}ie_{\infty}(n) \otimes L^{\otimes n} = \bigoplus_{t \in (\mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n})/\Sigma_n} \mathbb{K}[X_t]^{\pm}. $$

**Lemma 2.28.** For every $n \ge 0$, let $(\mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n})^r$ be the subset of $(\mathcal{T}\mathcal{M}(n) \otimes B^{\otimes n})^{\otimes r}$ formed by elements $x$ such that, if $\sigma \cdot x = x$ for some $\sigma \in \Sigma_n$, then $\varepsilon(\sigma, x) = 1$. Let $S^r(\mathcal{P}re\mathcal{L}ie_{\infty}, L)$ be the subspace of $S(\mathcal{P}re\mathcal{L}ie_{\infty}, L)$ given by these elements. Then we have an isomorphism

$$ \mathcal{O} : S^r(\mathcal{P}re\mathcal{L}ie_{\infty}, L) \to \Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, L). $$

*Proof.* This comes from the previous analysis and [Ver23, Lemma 2.3]. See also the proof of [Ver23, Proposition 2.5]. □

**Lemma 2.29.** Let $L$ be a dg $\mathbb{K}$-module. Denote by $\mu : S(B^c(\text{Perm}^c), S(B^c(\text{Perm}^c), L)) \to S(B^c(\text{Perm}^c), L)$ and $\tilde{\mu} : \Gamma(B^c(\text{Perm}^c), \Gamma(B^c(\text{Perm}^c), L)) \to \Gamma(B^c(\text{Perm}^c), L)$ the monadic compositions. Let $x \in L$ and $B_1, \dots, B_n \in S^r(B^c(\text{Perm}^c), L)$ be basis elements. Then

$$ \tilde{\mu}(\mathcal{O}\Sigma^{-1}e_1^{n+1}(x, \mathcal{O}B_1, \dots, \mathcal{O}B_n)) = \mathcal{O}(\mu(\Sigma^{-1}e_1^{n+1}(x, B_1, \dots, B_n))). $$

*Proof.* The proof is identical to the proofs given in [Ces18, Theorem 1.5.1, Lemma 1.5.2]. □

**Theorem 2.30.** A dg $\mathbb{K}$-module $(L, d)$ is a $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebra if and only if $\Sigma L \in \Gamma\mathcal{A}\mathcal{P}\mathcal{L}_{\infty}$ with $Q_0^0 = \Sigma d$. Moreover, every morphism of $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebras $\phi : L \to L'$ gives rise to a strict morphism $\Sigma\phi : \Sigma L \to \Sigma L'$ in $\Gamma\mathcal{A}\mathcal{P}\mathcal{L}_{\infty}$.

*Proof.* Let $L$ be a $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebra. Then $\Sigma L$ is a $\Gamma(\mathcal{A}\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebra by Proposition 1.17. Since $\mathcal{A}\mathcal{P}re\mathcal{L}ie_{\infty} \simeq B^c(\text{Perm}^c)$, we have a morphism

$$ l : \Gamma(\Sigma^{-1}\overline{\text{Perm}}^c, \Sigma L) \to \Sigma L. $$

We then set, for homogeneous elements $x, y_1, \dots, y_n \in \Sigma L$, and $r_1, \dots, r_n \ge 0$,

$$ x[\![y_1, \dots, y_n]\!]_{r_1, \dots, r_n} := l(\mathcal{O}(\Sigma^{-1}e_1^{r_1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n})), $$
where $r = r_1 + \cdots + r_n$ and where the considered orbit map is using a basis which includes $x, y_1, \ldots, y_n$. We check all formulas given in Theorem 2.22. Formulas (i) – (v) come from straight-forward computations. We prove formula (vi). We compute

$$d(x[\!\!\llap{\vphantom{x}}y_1, \ldots, y_n\!\!\llap{\vphantom{x}}]_{r_1, \ldots, r_n}) = ld(\mathcal{O}(\Sigma^{-1}e_1^{r_1+1} \otimes x \otimes y_1^{\otimes r_1} \otimes \cdots \otimes y_n^{\otimes r_n})).$$

We have

$$\mathcal{O}(\Sigma^{-1}e_1^{r_1+1} \otimes x \otimes y_1^{\otimes r_1} \otimes \cdots \otimes y_n^{\otimes r_n}) = \sum_{\sigma \in \mathrm{Sh}(1, r_1, \ldots, r_n)} \sigma \cdot (\Sigma^{-1}e_1^{r_1+1} \otimes x \otimes y_1^{\otimes r_1} \otimes \cdots \otimes y_n^{\otimes r_n}).$$

Let $\partial$ be the differential of $B^c(\mathrm{Perm}^c)$. Then

$$d(x[\!\!\llap{\vphantom{x}}y_1, \ldots, y_n\!\!\llap{\vphantom{x}}]_{r_1, \ldots, r_n}) = l \left( \sum_{\sigma \in \mathrm{Sh}(1, r_1, \ldots, r_n)} \sigma \cdot (\partial(\Sigma^{-1}e_1^{r_1+1}) \otimes x \otimes y_1^{\otimes r_1} \otimes \cdots \otimes y_n^{\otimes r_n}) \right) \\ + d(x)[\!\!\llap{\vphantom{x}}y_1, \ldots, y_k, d(y_k), \ldots, y_n\!\!\llap{\vphantom{x}}]_{r_1, \ldots, r_k-1, 1, \ldots, r_n} + \sum_{k=1}^{n} \pm x[\!\!\llap{\vphantom{x}}y_1, \ldots, y_k, d(y_k), \ldots, y_n\!\!\llap{\vphantom{x}}]_{r_1, \ldots, r_k-1, 1, \ldots, r_n}.$$

We compute the first sum. Recall from the operadic composition in $\mathrm{Perm}$ (see Proposition 2.1) and from the definition of the differential in the cobar construction of a coaugmented cooperad that we have

$$\partial(\Sigma^{-1}e_1^{r_1+1}) = \sum_{\substack{p+q=r+2 \\ p,q \ge 2}} \left( \sum_{\omega \in \mathrm{Sh}_*(q, 1, \ldots, 1)} \omega \cdot T_{p,q}^{1,1} + \sum_{k=2}^{p} \sum_{i=1}^{q} \sum_{\omega \in \mathrm{Sh}_*(1, \ldots, q, \ldots, 1)} \omega \cdot T_{p,q}^{k,i} \right),$$

where we have set

$$T_{p,q}^{k,i} = 1 \xrightarrow{k \to k+1} \sum_{i=1}^{k-1} \Sigma^{-1} e_i^q \xrightarrow{k+1 \to p+q-1} \Sigma^{-1} e_1^p.$$

Let $z_1, \ldots, z_{r+1} = x, \underbrace{y_1, \ldots, y_1}_{r_1}, \ldots, \underbrace{y_n, \ldots, y_n}_{r_n}$. For given $p, q \ge 2$ such that $p+q=r+2$, we need to compute the sums,

$$S_{p,q}^1 := \sum_{\sigma \in \mathrm{Sh}(1, r_1, \ldots, r_n)} \sum_{\omega \in \mathrm{Sh}_*(q, 1, \ldots, 1)} \pm \sigma \omega \cdot (T_{p,q}^{1,1} \otimes z_{\omega(1)} \otimes \cdots \otimes z_{\omega(r+1)});$$

$$S_{p,q}^2 := \sum_{k=2}^{p} \sum_{i=1}^{q} \sum_{\sigma \in \mathrm{Sh}(1, r_1, \ldots, r_n)} \sum_{\omega \in \mathrm{Sh}_*(1, \ldots, q, \ldots, 1)} \pm \sigma \omega \cdot (T_{p,q}^{k,i} \otimes z_{\omega(1)} \otimes \cdots \otimes z_{\omega(r+1)}).$$

We first compute $S_{p,q}^1$. We claim that

$$S_{p,q}^1 = \sum_{\substack{p_i+q_i=r_i \\ p_1+\cdots+p_n=p-1 \\ q_1+\cdots+q_n=q-1}} \pm \mathcal{O}(T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes q_1} \otimes \cdots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \cdots \otimes y_n^{\otimes p_n}).$$
Let $\sigma \in \mathrm{Sh}(r_1, \dots, r_n)$ and $\omega \in \mathrm{Sh}_*(q, 1, \dots, 1)$. In particular we have $\omega \in \mathrm{Sh}(1, q-1, p-1)$ with $\omega(1) = 1$. Then there exist $p_1, \dots, p_n, q_1, \dots, q_n$ with $p_i + q_i = r_i, p_1 + \dots + p_n = p-1$ and $q_1 + \dots + q_n = q-1$ such that

$$z_{\omega(1)} \otimes \dots \otimes z_{\omega(r+1)} = \pm x \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}.$$

Thus, every term in the left hand-side sum is part of the sum in the right hand-side. We now consider an element which occurs in the expansion of $O(T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n})$ for some $p_i$'s, $q_i$'s as above. Let $\beta \in \Sigma_{r+1}$ be a blocks permutation which sends $x \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}$ to $\pm x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}$. Then

$$O(T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}) = \pm O(\beta \cdot T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}).$$

Let $\tau \in \Sigma_{r+1}$. We write $\tau = \sigma\eta$ where $\sigma \in \mathrm{Sh}(1, r_1, \dots, r_n)$ and $\eta \in \Sigma_1 \times \Sigma_{r_1} \times \dots \times \Sigma_{r_n}$. Then

$$\tau \cdot (\mu \cdot T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}) = \sigma \cdot (\eta\beta \cdot T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}).$$

We write $\eta\beta = \omega\nu$ where $\omega \in \mathrm{Sh}(1, q-1, p-1)$ and $\nu \in \Sigma_1 \times \Sigma_{q-1} \times \Sigma_{p-1}$. Since $\eta\beta(1) = 1$, we have $\omega(1) = 1$ so that $\omega \in \mathrm{Sh}_*(q, 1, \dots, 1)$. We finally have

$$\tau \cdot (\mu \cdot T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}) = \sigma \cdot (\omega \cdot T_{p,q}^{1,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n})$$

so that every term in the right hand-side is part of $S_{p,q}^1$. We thus have proved the first identity.

We now compute $S_{p,q}^2$. We claim that

$$S_{p,q}^2 = \sum_{j=1}^{n} \sum_{\substack{p_i+q_i=r_i, i \neq j \\ p_j+q_j=r_j-1 \\ p_1+\dots+p_n=p-1 \\ q_1+\dots+q_n=q-1}} \pm O(T_{p,q}^{2,1} \otimes x \otimes y_j \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}).$$

Since there exists $\nu \in \Sigma_{r+1}$ such that $T_{p,q}^{k,i} = \nu \cdot T_{p,q}^{2,1}$, we can apply the same arguments as before to show that every term which occurs in $S_{p,q}^2$ is part of the right-hand side sum. Now consider some term $T_{p,q}^{2,1} \otimes x \otimes y_j \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}$. Let $\beta \in \Sigma_{r+1}$ be a blocks permutation which sends $x \otimes y_j \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}$ to $\pm x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}$. Then

$$O(T_{p,q}^{2,1} \otimes x \otimes y_j \otimes y_1^{\otimes q_1} \otimes \dots \otimes y_n^{\otimes q_n} \otimes y_1^{\otimes p_1} \otimes \dots \otimes y_n^{\otimes p_n}) = \pm O(\beta \cdot T_{p,q}^{2,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}).$$

Now let $\tau \in \Sigma_{r+1}$. We write $\tau = \sigma\eta$ where $\sigma \in \mathrm{Sh}(1, r_1, \dots, r_n)$ and $\eta \in \Sigma_1 \times \Sigma_{r_1} \times \dots \times \Sigma_{r_n}$. Then

$$\tau \cdot (\beta \cdot T_{p,q}^{2,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}) = \sigma \cdot (\eta\beta \cdot T_{p,q}^{2,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}).$$

We finally write $\eta\beta = \omega \cdot \nu_*(1, \gamma, 1, \dots, 1)$ where $\nu \in \Sigma_p, \gamma \in \Sigma_q, \omega \in \mathrm{Sh}_*(1, \dots, q, \dots, 1)$ and $k = \nu(2)$. Since $\eta\beta(1) = 1$, we have $\nu(1) = 1$. We thus obtain

$$\tau \cdot (\beta \cdot T_{p,q}^{2,1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n}) = \sigma \cdot (\omega \cdot T_{p,q}^{k,\gamma(1)} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n})$$

so that every term in the right-hand side is part of $S_{p,q}^2$.
From Lemma 2.29, we deduce that formula (vi) of Theorem 2.22 is satisfied so that $\Sigma L \in \Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$. Suppose now that $L$ is such that $\Sigma L \in \Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$. We prove that $L$ is a $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebra, or equivalently, that $\Sigma L$ is a $\Gamma(B^c(\text{Perm}^c), -)$-algebra (see Proposition 1.17). We first define

$$l: \bigoplus_{n \ge 0} (\Sigma^{-1} \overline{\text{Perm}^c}(n) \otimes (\Sigma L)^{\otimes n})^{\Sigma_n} \to \Sigma L$$

by setting, for every basis elements $x, y_1, \dots, y_n \in \Sigma L$,

$$l(\mathcal{O}(\Sigma^{-1}e_1^{\sum_i r_i+1} \otimes x \otimes y_1^{\otimes r_1} \otimes \dots \otimes y_n^{\otimes r_n})) := x[\{y_1, \dots, y_n\}]_{r_1, \dots, r_n}.$$

We then extend $l: \Gamma(B^c(\text{Perm}^c), \Sigma L) \to \Sigma L$ by Lemma 2.29. By the same identities as before, we can show that $l$ preserves the differentials, giving a structure of a $\Gamma(\Lambda\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebra on $\Sigma L$. $\square$

**Remark 2.31.** This theorem implies that the category of the $\Gamma(\Lambda\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebras is a full sub-category of $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$. However, a morphism in $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$ does not necessarily preserves the monadic structure of $\Gamma(\Lambda\mathcal{P}re\mathcal{L}ie_{\infty}, -)$.

**Corollary 2.32.** For every complete $\Gamma(\mathcal{P}re\mathcal{L}ie, -)$-algebra $L$, the $\text{dg}\mathbb{K}$-module $\Sigma L$ is endowed with the structure of a $\overline{\Gamma\Lambda\mathcal{P}\mathcal{L}}_{\infty}$-algebra such that $\mathcal{M}\mathcal{C}(L)$ is in bijective correspondance with $\mathcal{M}\mathcal{C}(\Sigma L)$.

*Proof.* The operad morphism $\mathcal{P}re\mathcal{L}ie_{\infty} \to \mathcal{P}re\mathcal{L}ie$ given in Remark 2.7 gives rise to a monad morphism $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -) \to \Gamma(\mathcal{P}re\mathcal{L}ie, -)$. Then, every $\Gamma(\mathcal{P}re\mathcal{L}ie, -)$-algebra $L$ is endowed with the structure of a $\Gamma(\mathcal{P}re\mathcal{L}ie_{\infty}, -)$-algebra. By Theorem 2.30, the $\text{dg}\mathbb{K}$-module $\Sigma L$ is a $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra. If we denote by $-\{-, \dots, -\}_{r_1, \dots, r_n}$ the weighted brace operations given by the $\Gamma(\mathcal{P}re\mathcal{L}ie, -)$-algebra structure on $L$, then, by definition of the weighted brace operations given in the proof of Theorem 2.30, we have

$$\Sigma x[\{\Sigma y_1, \dots, \Sigma y_n\}]_{r_1, \dots, r_n} = \begin{cases} -\Sigma d(x) & \text{if } r_1 + \dots + r_n = 0 \\ (-1)^{|x|} \Sigma x\{y_1\}_1 & \text{if } r_1 = 1, r_2 = \dots = r_n = 0 \\ 0 & \text{if } r_1 + \dots + r_n \ge 2 \end{cases}$$

for every $x, y_1, \dots, y_n \in L$. Since the operations $-\{-, \dots, -\}_{r_1, \dots, r_n}$ preserve the filtration on $L$, we have that the operations $-[\{-, \dots, -\}]_{r_1, \dots, r_n}$ also preserve the filtration on $\Sigma L$ so that $\Sigma L \in \overline{\Gamma\Lambda\mathcal{P}\mathcal{L}}_{\infty}$. Moreover, the previous computation of the braces shows that $x \in \mathcal{M}\mathcal{C}(L)$ if and only if $\Sigma x \in \mathcal{M}\mathcal{C}(\Sigma L)$, which proves the corollary. $\square$

**Corollary 2.33.** Suppose that $\text{char}(\mathbb{K}) = 0$. Then every $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra $V$ is endowed with the structure of a $\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra such that

$$x[\{y_1, \dots, y_n\}]_{r_1, \dots, r_n} = \frac{1}{\prod_i r_i!} x[\underbrace{y_1, \dots, y_1}_{r_1}, \dots, \underbrace{y_n, \dots, y_n}_{r_n}]$$

for every $x \in V$ and $y_1, \dots, y_n \in V$ with associated weights $r_1, \dots, r_n \ge 0$, where we consider the symmetric braces $-[\{-, \dots, -\}]$ defined in Proposition 2.6.
Proof. Let $V$ be a $\Gamma\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra. By Theorem 2.30, we have that $\Sigma^{-1}V$ is a $\Gamma(\mathcal{P}\mathcal{L}\mathcal{e}_{\infty}, -)$-algebra. Since we have a morphism of monads given by the trace map

$$Tr : \mathcal{S}(\mathcal{P}\mathcal{L}\mathcal{e}_{\infty}, -) \to \Gamma(\mathcal{P}\mathcal{L}\mathcal{e}_{\infty}, -),$$

$\Sigma^{-1}V$ is endowed with a $\mathcal{P}\mathcal{L}\mathcal{e}_{\infty}$-algebra structure, so that $V$ is a $\Lambda\mathcal{P}\mathcal{L}_{\infty}$-algebra. By using the definition of weighted brace operations, we obtain the desired relation. $\square$

# 3 A morphism from $\mathcal{P}\mathcal{L}\mathcal{e}_{\infty}$ to $\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}\otimes\mathcal{E}$

In this section, we construct an operad morphism from $\mathcal{P}\mathcal{L}\mathcal{e}_{\infty}$ to $\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}\otimes\mathcal{E}$. We will define this morphism as a composite of the form $\mathcal{P}\mathcal{L}\mathcal{e}_{\infty} \to \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}\otimes B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}^{\vee}) \to \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}\otimes\mathcal{E}$.

In §3.1, we construct an operad morphism $B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}^{\vee}) \to \mathcal{E}$, which will be given as a lift of some diagram.

In §3.2, we give a computation of the twisted coderivation on $\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}^{\vee}(\Sigma^{-1}N_*(\Delta^n))$ induced by the morphism constructed in §3.1 and the $\mathcal{E}$-coalgebra structure on $N_*(\Delta^n)$ by induction on $n \ge 0$.

In §3.3, we construct the morphism $\mathcal{P}\mathcal{L}\mathcal{e}_{\infty} \to \mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}\otimes B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}^{\vee})$ and deduce Theorem A.

## 3.1 A morphism from $B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}^{\vee})$ to $\mathcal{E}$

Let $\mathcal{Com}$ be the commutative operad. Recall that if we consider the model structure on the category of symmetric operads $\mathcal{P}$ such that $\mathcal{P}(0) = 0$ (see [Hin03, §3.3]), then we have an acyclic fibration $\mathcal{E} \xrightarrow{\sim} \mathcal{Com}$. We thus have there exists a lift of the following diagram:

$$ \begin{tikzcd} & \mathcal{E} \arrow[r, "\exists"] \arrow[d, "\sim"'] & \\ B^c(\Lambda^{-1}\mathcal{B}\mathcal{r}\mathcal{a}\mathcal{e}^{\vee}) \arrow[r, "\sim"'] & \mathcal{Com} \arrow[d, "\sim"'] & \\ & & \end{tikzcd} $$

The goal of this subsection is to give an explicit choice of such a lift. Equivalently, we are searching for elements $\mu_T \in \mathcal{E}(|T|)|_{T|-2}$ for every $T \in \mathcal{P}\mathcal{R}\mathcal{T}$ such that

$$d(\Lambda\mu_T) + \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S = 0.$$

For every $\sigma \in \Sigma_r$, let $h_{\mathcal{E}}^{\sigma}: \mathcal{E}(r)_d \to \mathcal{E}(r)_{d+1}$ be the morphism such that, for every $w_0, \dots, w_d \in \Sigma_r$,

$$h_{\mathcal{E}}^{\sigma}(w_0, \dots, w_r) = (\sigma, w_0, \dots, w_r).$$

We can check that this morphism is a homotopy between the identity map of $\mathcal{E}(r)_d$ and the morphism $\varphi_{\mathcal{E}}^{\sigma}: \mathcal{E}(r)_d \to \mathcal{E}(r)_d$ defined by

$$\varphi_{\mathcal{E}}^{\sigma}(w_0, \dots, w_d) = \begin{cases} \sigma & \text{if } d=0 \\ 0 & \text{else} \end{cases}.$$
Accordingly,

$$dh_{\mathcal{E}}^{\sigma} + h_{\mathcal{E}}^{\sigma} d = id_{\mathcal{E}} - \varphi_{\mathcal{E}}^{\sigma}.$$

By functoriality, we have that $\Lambda h_{\mathcal{E}}^{\sigma}$ is a homotopy between the identity map and $\Lambda \varphi_{\mathcal{E}}^{\sigma}$.

**Construction 3.1.** We set $\mu_{\bigcirc} = 0$, $\mu_{\bigcirc} = (12)$ and $\mu_{\bigcirc} = (21)$. For every $T \in \mathcal{PT}$, we define $\mu_T$ by induction on $|T|$ by

$$\Lambda \mu_T = -\Lambda h_{\mathcal{E}}^{\sigma_T} \left( \sum_{S \subset T} \Lambda \mu_{T/S} \circ_S \Lambda \mu_S \right),$$

where we take $\mu_{T/S} \in \mathcal{E}(V_{T/S})$ and $\mu_S \in \mathcal{E}(V_S)$ (see Remark 1.10).

**Example 3.2.** Let us make explicit the $\mu_T$'s for $T$ a tree with 3 vertices.

* If $T = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}$};
\end{tikzpicture}$, then we have two non trivial trees $S_1 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}$};
\end{tikzpicture}$ and $S_2 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 2 \end{smallmatrix}$};
\end{tikzpicture}$, which are such that
$$T/S_1 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 3 \\ S_1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 2 \\ S_2 \end{smallmatrix}$};
\end{tikzpicture} \text{ and } T/S_2 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}$};
\end{tikzpicture}.$$ We thus have

$$\sum_{S \subset T} \Lambda \mu_{T/S} \circ_S \Lambda \mu_S = \Sigma^{-1}(S_1 3) \circ_{S_1} \Sigma^{-1}(12) + \Sigma^{-1}(S_2 2) \circ_{S_2} \Sigma^{-1}(13) \\ = \Sigma^{-1}(123) - \Sigma^{-1}(132).$$

We then deduce $\Lambda \mu_T = \Sigma^{-1}(123, 132)$.

* If $T = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
\end{tikzpicture}$, then we have two non trivial trees $S_1 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 2 \end{smallmatrix}$};
\end{tikzpicture}$ and $S_2 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 2 \end{smallmatrix}$};
\end{tikzpicture}$, which are such that
$$T/S_1 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 3 \\ S_1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 2 \\ S_2 \end{smallmatrix}$};
\end{tikzpicture} \text{ and } T/S_2 = \begin{tikzpicture}[baseline=(current bounding box.center)]
  
  \node[circle, fill, inner sep=1.5pt] (1) at (0,0) {};
  \node[circle, fill, inner sep=1.5pt] (2) at (1,0) {};
  \node[circle, fill, inner sep=1.5pt] (3) at (2,0) {};
  \draw (1) -- (2) node[left] {$\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}$};
  \draw (2) -- (3) node[right] {$\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}$};
\end{tikzpicture}.$$ We thus have

$$\sum_{S \subset T} \Lambda \mu_{T/S} \circ_S \Lambda \mu_S = \Sigma^{-1}(S_1 3) \circ_{S_1} \Sigma^{-1}(12) + \Sigma^{-1}(S_2 2) \circ_{S_2} \Sigma^{-1}(23) \\ = \Sigma^{-1}(123) - \Sigma^{-1}(123) \\ = 0.$$

We then deduce $\Lambda \mu_T = 0$.

**Theorem 3.3.** For every tree $T \in \mathcal{PT}$, we have

$$d(\Lambda \mu_T) + \sum_{S \subset T} \Lambda \mu_{T/S} \circ_S \Lambda \mu_S = 0$$

where we consider the elements $\mu_T$'s defined in Construction 3.1. Then we have an explicit lift

<figure><img src="image_15.png" /></figure>

given by the morphism
$$ \begin{array}{ccc} B^c(\Lambda^{-1}Brace^\vee) & \to & \mathcal{E} \\ \Sigma^{-1}\Lambda^{-1}T^\vee & \mapsto & \mu_T. \end{array} $$

*Proof.* The theorem obviously holds if $|T| = 1, 2$, and also if $|T| = 3$ by Example 3.2. We now suppose that $n := |T| \ge 4$. We use the identity $d\Lambda h_{\mathcal{E}}^{\sigma_T} + \Lambda h_{\mathcal{E}}^{\sigma_T} d = id_{\Lambda\mathcal{E}} - \Lambda\varphi_{\mathcal{E}}^{\sigma_T}$:

$$ d(\Lambda\mu_T) = \Lambda h_{\mathcal{E}}^{\sigma_T} d \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) + \Lambda\varphi_{\mathcal{E}}^{\sigma_T} \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) - \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S. $$

By an immediate induction, for every non trivial tree $T$, we have that $\Lambda\mu_T \in \Lambda\mathcal{E}(n)_{-1}$, or equivalently $\mu_T \in \mathcal{E}(n)_{n-2}$. This then gives $|\mu_{T/S} \circ_S \mu_S| = n - 3 > 0$. Thus, by definition of $\varphi_{\mathcal{E}}^{\sigma_T}$, we have that

$$ \Lambda\varphi_{\mathcal{E}}^{\sigma_T} \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) = 0. $$

We now prove that

$$ d \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) = 0. $$

We compute:

$$ d \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) = \sum_{S \subset T} d(\Lambda\mu_{T/S}) \circ_S \Lambda\mu_S - \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S d(\Lambda\mu_S), $$

because the differential of the Barratt-Eccles operad is compatible with its operad structure. Now, because $\mu_S = 0$ if $S$ has only one vertex, we can consider subtrees of $T$ with at most $n-1$ vertices. We can then use the induction hypothesis.

$$ d \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) = - \sum_{S \subset T} \sum_{U \subset T/S} \left( \Lambda\mu_{(T/S)/U} \circ_U \Lambda\mu_U \right) \circ_S \Lambda\mu_S + \sum_{S \subset T} \sum_{U \subset S} \Lambda\mu_{T/S} \circ_S \left( \Lambda\mu_{S/U} \circ_U \Lambda\mu_U \right) $$

We have two types of subtrees $U$ of $T/S$: either $U$ does not contain the vertex $S$, so that $U$ can be canonically seen as a subtree of $T$ such that $V_U \cap V_S = \emptyset$, or $U$ contains the vertex $S$, so that $U$ can be seen as a subtree of $T$ such that $S \subset U$. We thus have

$$ d \left( \sum_{S \subset T} \Lambda\mu_{T/S} \circ_S \Lambda\mu_S \right) = - \sum_{S \subset T} \sum_{S \subset U \subset T} \left( \Lambda\mu_{T/U} \circ_{U/S} \Lambda\mu_{U/S} \right) \circ_S \Lambda\mu_S \\ - \sum_{\substack{S, U \subset T \\ V_S \cap V_U = \emptyset}} \left( \Lambda\mu_{(T/S)/U} \circ_U \Lambda\mu_U \right) \circ_S \Lambda\mu_S \\ + \sum_{S \subset T} \sum_{U \subset S} \Lambda\mu_{T/S} \circ_S \left( \Lambda\mu_{S/U} \circ_U \Lambda\mu_U \right). $$
In the second line, by exchanging the roles of $S$ and $U$, we have a sum of terms of the form

$$ (\Lambda\mu_{(T/S)/U} \circ U \Lambda\mu_U) \circ_S \Lambda\mu_S + (\Lambda\mu_{(T/U)/S} \circ_S \Lambda\mu_S) \circ_U \Lambda\mu_U $$

which is 0, because of the operadic axioms and because $|\Lambda\mu_U| = |\Lambda\mu_S| = -1$. We thus obtain

$$ \begin{aligned} d \left( \sum_{S \subset T} \Lambda \mu_{T/S} \circ_S \Lambda \mu_S \right) &= - \sum_{U \subset T} \sum_{S \subset U} (\Lambda \mu_{(T/U)} \circ_{U/S} \Lambda \mu_{(U/S)}) \circ_S \Lambda \mu_S \\ &\quad + \sum_{S \subset T} \sum_{U \subset S} \Lambda \mu_{T/S} \circ_S (\Lambda \mu_{(S/U)} \circ_U \Lambda \mu_U) \\ &= - \sum_{S \subset T} \sum_{U \subset S} (\Lambda \mu_{T/S} \circ_{S/U} \Lambda \mu_{(S/U)}) \circ_U \Lambda \mu_U \\ &\quad + \sum_{S \subset T} \sum_{U \subset S} \Lambda \mu_{T/S} \circ_S (\Lambda \mu_{(S/U)} \circ_U \Lambda \mu_U) \\ &= 0, \end{aligned} $$

using again the associativity of the operadic composition.

□

## 3.2 On the twisted coderivation of $\mathcal{B}race^{\vee}(\Sigma N^{*}(\Delta^{n}))$

Every $\mathcal{E}$-algebra $E$ inherits a $B^c(\Lambda^{-1}\mathcal{B}race^{\vee})$-algebra structure induced by the morphism $B^c(\Lambda^{-1}\mathcal{B}race^{\vee}) \to \mathcal{E}$ given by Theorem 3.3. This algebraic structure is equivalent to giving a twisted coderivation on the free $\mathcal{B}race^{\vee}$-coalgebra $\mathcal{B}race^{\vee}(\Sigma E)$ generated by $\Sigma E$. Recall that

$$ \mathcal{B}race^{\vee}(\Sigma E) = \bigoplus_{k \ge 1} \bigoplus_{T \in \mathcal{P}\mathcal{R}\mathcal{T}(k)} (T^{\vee} \otimes (\Sigma E)^{\otimes k})^{\Sigma_k}, $$

where we equalize the action of $\Sigma_k$ on $T^{\vee}$ with the action of $\Sigma_k$ by permutation on $(\Sigma E)^{\otimes k}$. In the following, we identify the $k = 1$ component with $\Sigma E$.

If $E$ is finite dimensional, giving such a coderivation is equivalent to giving a twisting morphism $\partial^E$ on the free complete brace algebra

$$ \widehat{\mathcal{B}race}(\Sigma^{-1}E^{\vee}) = \prod_{k \ge 1} \bigoplus_{T \in \mathcal{P}\mathcal{R}\mathcal{T}(k)} (T^{\vee} \otimes (\Sigma^{-1}E^{\vee})^{\otimes k})^{\Sigma_k}. $$

This completion is obtained from the free brace algebra $\mathcal{B}race(\Sigma^{-1}E^{\vee})$ endowed with the filtration

$$ F_p \mathcal{B}race(\Sigma^{-1}E^{\vee}) = \bigoplus_{k \ge p+1} \bigoplus_{T \in \mathcal{P}\mathcal{R}\mathcal{T}(k)} (T^{\vee} \otimes (\Sigma^{-1}E^{\vee})^{\otimes k})^{\Sigma_k}. $$

One can check that the brace algebra structure of $\mathcal{B}race(\Sigma^{-1}E^{\vee})$ preserves this filtration so that the completion $\widehat{\mathcal{B}race}(\Sigma^{-1}E^{\vee})$ is endowed with a brace algebra structure. The derivation $\partial^E$ is thus given by a generating function

$$ \partial^E = \sum_{\substack{T \in \mathcal{P}\mathcal{R}\mathcal{T} \\ T \text{ canonical}}} T \otimes \Lambda \mu_T^{E^{\vee}} $$
where each $w_k''(1), \dots, w_k''(r_k')$ are obtained from $w_k(1), \dots, w_k(r)$ by taking the first $r_k'$ terms which are not among

$$ 
\begin{array}{ccc}
w_0''(1) & \cdots & w_0''(r_0' - 1) \\
\vdots & & \vdots \\
w_{k-1}''(1) & \cdots & w_{k-1}''(r_{k-1}' - 1)
\end{array}
$$

then

$$ \pi_k(TR(w)) = \sum_{\substack{r_0+\cdots+r_d=r+d-k+1 \\ 1 \le r_i \le r}} \begin{vmatrix} w_0'(1) & \cdots & w_0'(r_0) \\ \vdots & & \vdots \\ w_d'(1) & \cdots & w_d'(r_d) \end{vmatrix} $$

where, as above, each $w_k'(1), \dots, w_k'(r_k)$ are obtained from $w_k(1), \dots, w_k(r)$ by taking the first $r_k$ terms which are not among

$$ 
\begin{array}{ccc}
1 & \cdots & k-1 \\
w_0'(1) & \cdots & w_0'(r_0-1) \\
\vdots & & \vdots \\
w_{k-1}'(1) & \cdots & w_{k-1}'(r_{k-1}-1)
\end{array}
$$

We then deduce

$$ \sum_{k=1}^{r} 1 \cdots k \cdot \pi_k(TR(w)) = \sum_{k=1}^{r} \sum_{\substack{r_0+\cdots+r_d=r+d-k+1 \\ 1 \le r_i \le r}} \begin{vmatrix} 1 & \cdots & k \\ w_0'(1) & \cdots & w_0'(r_0) \\ \vdots & & \vdots \\ w_d'(1) & \cdots & w_d'(r_d) \end{vmatrix} $$

which proves the lemma.

□

**Lemma 3.5.** Let $X$ be a totally ordered finite set and $T \in \mathcal{PT}(X)$ with $|T| \ge 3$. We let $b_T$ to be the number of vertices in the first branch of $T$ (without the root).

* If $b_T \ge 2$, then $TR(\mu_T) = 0$.

* If $b_T = 1$, then, if we denote by $r$ the root of $T$ and $s$ the second element of $V_T$, then there exists $u_T \in \chi(X \setminus \{r\})$ such that $TR(\mu_T) = rs \cdot u_T$.

*Proof.* It is sufficient to prove the lemma for a canonical tree $T \in \chi(1 < \cdots < |T|)$. We prove the lemma by induction on $|T|$. If $|T| = 3$, then Example 3.2 implies that the assertion of the lemma is true. We now suppose that $|T| \ge 4$. Suppose first that $b_T \ge 2$. By Lemma 3.4,

$$ TR(\mu_T) = - \sum_{k=2}^{|T|} \sum_{S \subset T} \pm 1 \cdots k \cdot \pi_k(TR(\mu_{T/S}) \circ_S TR(\mu_S)). $$

Note that the sum begins at $k=2$, since that, by an immediate induction, the elements $TR(\mu_T)$'s begins at 1. Let $S \subset T$. Our goal is to prove that, for every $2 \le k \le |T|,$

$$ 1 \cdots k \cdot \pi_k(TR(\mu_{T/S}) \circ_S TR(\mu_S)) = 0 $$

We distinguish several cases. If either $b_S \ge 2$ or $b_{T/S} \ge 2$, then the identity holds by induction hypothesis. Suppose now that $b_S = b_{T/S} = 1$. Since $b_T \ge 2$, we have these different cases.
* If $r(S)$ is one of the vertex of the first branch of $T$, then, because $b_{T/S} = 1$, the tree $S$ is the full first branch of $T$. In this situation, the first permutation occurring in $\mu_{T/S} \circ_S \mu_S$ is the identity permutation, so that taking the image of this element under $h_{\mathcal{E}}^{id}$ gives 0.

* If $r(S) = 1$, since $b_S = b_{T/S} = 1$, we have $b_T = 2$, and the second vertex of $S$ is 2. Then, by induction hypothesis,

$$TR(\mu_S) = 12 \cdot u_S; \\ TR(\mu_{T/S}) = S3 \cdot u_{T/S},$$

where $u_S \in \chi(V_S \setminus \{1\})$ and $u_{T/S} \in \chi(V_{T/S} \setminus \{S\})$. We thus have

$$TR(\mu_{T/S}) \circ_S TR(\mu_S) = 12 \cdot u_S \cdot 3 \cdot u_{T/S}.$$

Since 2 occurs in the surjection $u_S$, taking the image of such an element under $\pi_k$ will give 0 for every $k \ge 3$. If $k = 2$, then the corresponding term is

$$12 \cdot \pi_2(12 \cdot u_S \cdot 3 \cdot u_{T/S}) = 122 \cdot u_S \cdot 3 \cdot u_{T/S} = 0.$$

* If $r(S)$ is neither 1 nor an element of the first branch of $T$, then we cannot have $b_{T/S} = 1$, since we have supposed $b_T \ge 2$.

This concludes the case $b_T \ge 2$. We now suppose that $b_T = 1$. We use again the identity up to signs

$$TR(\mu_T) = - \sum_{k=2}^{|T|} \sum_{S \subset T} \pm 1 \cdots k \cdot \pi_k(TR(\mu_{T/S}) \circ_S TR(\mu_S)),$$

given by Lemma 3.4. Our goal is to prove that the terms of this sum with $k \ge 3$ are 0. Let $S \subset T$ be such that $b_S = b_{T/S} = 1$. We have three cases.

* If $|S|, |T/S| \ne 2$, then, since $b_T = 1$, we cannot have $r(S) = 2$ so that 2 is the second occurrence of either the surjection $TR(\mu_S)$ or the surjection $TR(\mu_{T/S})$. By induction hypothesis, in any case, taking the image of $\pi_k$ for every $k \ge 3$ of the corresponding element gives 0.

* Suppose now that $|S| = 2$ and $|T/S| \ne 2$. If $S$ does not contain 2, then the above argument gives a resulting element equal to 0 in the sum, for $k \ge 3$. Else, we have $r(S) = 1$ so that $TR(\mu_S) = 12$. We thus have, by induction hypothesis, that $TR(\mu_{T/S}) \circ_S TR(\mu_S)$ is of the form $123 \cdot u_{T/S}$ where $u_{T/S} = 0$ or $u_{T/S} \in \chi(V_{T/S} \setminus \{S\})$. The image of this element under $\pi_k$ is 0 for every $k \ge 4$. If $k = 3$, then we have $123 \cdot \pi_3(123 \cdot u_{T/S}) = 1233 \cdot \pi_3(u_{T/S}) = 0$.

* Suppose now that $|T/S| = 2$ and $|S| \ne 2$. Because $b_T = 1$, we have $r(S) = 1$. We then obtain that $TR(\mu_{T/S}) = S\alpha$ for some $1 \le \alpha \le |T|$. Therefore, by induction hypothesis, the composite $TR(\mu_{T/S}) \circ_S TR(\mu_S)$ is given by $1\Sigma \cdot u_S \cdot \alpha$, where $\Sigma$ is the second vertex of $S$. If $\alpha \ne 2$, then $\Sigma = 2$ and $u_S$ contains an occurrence of 2. Taking the image of such element under $\pi_k$ for every $k \ge 3$ gives 0. If $\alpha = 2$, then $\Sigma = 3$. The image under $\pi_k$ of the resulting composite gives 0 for every $k \ge 4$. If $k = 3$, then the resulting term in the sum is $123 \cdot \pi_3(13 \cdot u_S \cdot 2) = 1233 \cdot \pi_3(u_S) = 0$.
The lemma is proved.

This lemma allows us to compute $\partial^0$.

**Lemma 3.6.** We have the following identity:

$$ \partial^0(\Sigma^{-1}\underline{0}) = \begin{array}{c|c} \Sigma^{-1}\underline{0} & \\ & \Sigma^{-1}\underline{0} \end{array}. $$

*Proof.* For every canonical $n$-tree $T$ with $n \ge 3$, we have that $\Lambda\mu_T(\Sigma^{-1}\underline{0}) = 0$. Indeed, by Lemma 3.5, in this case, the number 2 occurs at least two times in the surjection $TR(\mu_T)$, so that its coevaluation on $\underline{0}$ gives 0 by definition of the interval cuts operations. There only remains the case $n = 2$. The associated canonical tree is $T = \begin{array}{c} \underline{2} \\ \underline{1} \end{array}$, which gives $TR(\mu_T) = (12)$ by definition. By definition of the interval cuts operations, the coevaluation of the surjection (12) on $\underline{0}$ is $\underline{0} \otimes \underline{0}$, which gives the result. □

For every $n, k \ge 0$, let $\partial_{(k)}^n$ be the composite of $\partial^n$ with the projection on trees with $k + 1$ vertices. Our goal is to compute $\partial_{(k)}^n$ by induction on $n, k \ge 0$. Recall, from after Proposition 1.32, the two morphisms $\varphi_n^0 : N_*(\Delta^n) \to N_*(\Delta^n)$ and $h_n^0 : N_*(\Delta^n) \to N_{*+1}(\Delta^n)$, which satisfy

$$ dh_n^0 + h_n^0 d = id_{N_*(\Delta^n)} - \varphi_n^0. $$

We keep the notations $\varphi_n^0$ and $h_n^0$ for the two induced morphisms on $\Sigma^{-1}N_*(\Delta^n)$, which also satisfy the same homotopy relation. We extend such morphisms on the tensor algebra of $\Sigma^{-1}N_*(\Delta^n)$ by

$$ (\varphi_n^0)(x_1 \otimes \cdots \otimes x_p) = \varphi_n^0(x_1) \otimes \cdots \otimes \varphi_n^0(x_p); $$

$$ (h_n^0)(x_1 \otimes \cdots \otimes x_p) = \sum_{i=1}^{p} \pm \varphi_n^0(x_1) \otimes \cdots \otimes \varphi_n^0(x_{i-1}) \otimes h_n^0(x_i) \otimes x_{i+1} \otimes \cdots \otimes x_p, $$

for every $x_1, \dots, x_p \in \Sigma^{-1}N_*(\Delta^n)$. We extend $\varphi_n^0$ and $h_n^0$ on $Brace(\Sigma^{-1}N_*(\Delta^n))$ by setting, for every canonical tree $T$ and $X \in (\Sigma^{-1}N_*(\Delta^n))^{\otimes|T|}$,

$$ \Phi_n^0(T \otimes X) = T \otimes \varphi_n^0(X); $$

$$ H_n^0(T \otimes X) = T \otimes h_n^0(X). $$

These definitions are extended to any tree $T$ by symmetry. We then obtain the homotopy relation:

$$ \partial_{(0)}^n H_n^0 + H_n^0 \partial_{(0)}^n = id - \Phi_n^0. $$
**Theorem 3.7.** Let $n, k \ge 1$. Then

$$ \partial_{(k)}^{n}(\Sigma^{-1}\underline{0\cdots n}) = -H_{n}^{0}\left(\sum_{\substack{p+q=k \\ p \neq 0}} \partial_{(p)}^{n} \partial_{(q)}^{n}(\Sigma^{-1}\underline{0\cdots n})\right). $$

In particular, we have the induction relation:

$$ \partial_{(k)}^{n}(\Sigma^{-1}\underline{0\cdots n}) = H_{n}^{0}d^{0}\partial_{(k)}^{n-1}(\Sigma^{-1}\underline{0\cdots(n-1)}) - H_{n}^{0}\left(\sum_{\substack{p+q=k \\ p, q \neq 0}} \partial_{(p)}^{n}\partial_{(q)}^{n}(\Sigma^{-1}\underline{0\cdots n})\right). $$

Starting from Lemma 3.6, this theorem allows us to compute the elements $\partial_{(k)}^{n}(\Sigma^{-1}\underline{0\cdots n})$ by induction on $n, k \ge 0$.

*Proof.* We have

$$ \partial_{(0)}^{n}\partial_{(k)}^{n}(\Sigma^{-1}\underline{0\cdots n}) = -\sum_{\substack{p+q=k \\ p \neq 0}} \partial_{(p)}^{n}\partial_{(q)}^{n}(\Sigma^{-1}\underline{0\cdots n}). $$

Applying $H_n^0$ on this equality gives

$$ \partial_{(k)}^{n}(\Sigma^{-1}\underline{0\cdots n}) = -H_{n}^{0}\left(\sum_{\substack{p+q=k \\ p \neq 0}} \partial_{(p)}^{n} \partial_{(q)}^{n}(\Sigma^{-1}\underline{0\cdots n})\right) + \Phi_{n}^{0}\left(\sum_{\substack{p+q=k \\ p \neq 0}} \partial_{(p)}^{n} \partial_{(q)}^{n}(\Sigma^{-1}\underline{0\cdots n})\right) - \partial_{(0)}^{n}H_{n}^{0}\partial_{(k)}^{n}(\Sigma^{-1}\underline{0\cdots n}) $$

The second term on the right hand-side vanishes, since the differentials $\partial^n$ are compatible with the simplicial structure of $\mathcal{B}race(\Sigma^{-1}N_*(\Delta^\bullet))$ defined tensor-wise and $\Phi_n^0(\Sigma^{-1}\underline{0\cdots n}) = 0$ since $n \ge 1$. We now deal with the last term. Let $T$ be a canonical tree with $|T| = k+1$ and $b_T = 1$. By Lemma 3.5, there exists $u_T \in \chi([2, k+1])$ such that

$$ TR(\mu_T) = 12 \cdot u_T. $$

Then, up to a sign, the elements occurring in each terms of $TR(\mu_T)(\underline{0\cdots n})$ are on the form

$$ \underline{0\cdots k} \otimes \underline{x} \otimes X, $$
where $\underline{x} \in N_*(\Delta^n)$ has a length which is at least 2, and $X$ is a tensor product of elements in $N_*(\Delta^n)$. The image of such elements by $h_n^0$ is 0. We thus deduce that $H_n^0 \partial_{(k)}(\Sigma^{-1}\underline{0\cdots n}) = 0$, which proves the first formula.

We now prove the induction relation. For every $n, k \ge 1$, we have

$$ \partial_{(k)}^n (\Sigma^{-1} \underline{0 \cdots n}) = -H_n^0 \partial_{(k)}^n \partial_{(0)}^n (\Sigma^{-1} \underline{0 \cdots n}) - H_n^0 \left( \sum_{\substack{p+q=k \\ p,q \ne 0}} \partial_{(p)}^n \partial_{(q)}^n (\Sigma^{-1} \underline{0 \cdots n}) \right). $$

We only need to compute the first term. We have

$$ \partial_{(k)}^n \partial_{(0)}^n (\Sigma^{-1} \underline{0 \cdots n}) = - \sum_{i=0}^n (-1)^i d^i \partial_{(k)}^{n-1} (\Sigma^{-1} \underline{0 \cdots (n-1)}). $$

For every $1 \le i \le n$ we have $H_n^0 d^i = d^i H_{n-1}^0$. From the first formula that we have proved, we deduce that the element $\partial_{(k)}^{n-1}(\Sigma^{-1}\underline{0\cdots(n-1)})$ is in the image of $H_{n-1}^0$. Since $H_{n-1}^0 H_{n-1}^0 = 0$, we obtain at the end

$$ H_n^0 \partial_{(k)}^n \partial_{(0)}^n (\Sigma^{-1} \underline{0 \cdots n}) = H_n^0 d^0 \partial_{(k)}^{n-1} (\Sigma^{-1} \underline{0 \cdots (n-1)}), $$

which proves the theorem. $\square$

**Corollary 3.8.** Let $n \ge 0$. Then

$$ \partial_{(1)}^n (\Sigma^{-1} \underline{0 \cdots n}) = \sum_{k=0}^{n} (-1)^k \frac{\Sigma^{-1} \underline{k \cdots n}}{\Sigma^{-1} \underline{0 \cdots k}}. $$

*Proof.* We prove the corollary by induction on $n \ge 0$. Lemma 3.6 proves the case $n = 0$. We now suppose that $n \ge 1$. Theorem 3.7 gives

$$ \partial_{(1)}^n (\Sigma^{-1} \underline{0 \cdots n}) = H_n^0 d^0 \partial_{(1)}^{n-1} (\Sigma^{-1} \underline{0 \cdots (n-1)}). $$

By induction hypothesis on $n$, we have

$$ d^0 \partial_{(1)}^{n-1} (\Sigma^{-1} \underline{0 \cdots (n-1)}) = - \sum_{k=1}^{n} (-1)^k \frac{\Sigma^{-1} \underline{k \cdots n}}{\Sigma^{-1} \underline{1 \cdots k}}. $$

This gives

$$ \partial_{(1)}^n (\Sigma^{-1} \underline{0 \cdots n}) = \sum_{k=1}^{n} (-1)^k \frac{\Sigma^{-1} \underline{k \cdots n}}{\Sigma^{-1} \underline{0 \cdots k}} + \frac{\Sigma^{-1} \underline{0 \cdots n}}{\Sigma^{-1} \underline{0}} $$

which gives the result. $\square$
<figure><img src="image_16.png" /></figure>

Proof. We first compute $\partial^1$. Lemma 3.6 and Corollary 3.8 give

$$ \partial^1(\underline{0}) = \begin{array}{l} \underline{0} \\ | \\ \underline{0} \end{array} ; \partial^1(\underline{1}) = \begin{array}{l} \underline{1} \\ | \\ \underline{1} \end{array} ; \partial^1(\underline{01}) = \begin{array}{l} \underline{01} \\ | \\ \underline{0} \end{array} - \begin{array}{l} \underline{1} \\ | \\ \underline{01} \end{array}. $$

We compute $\partial^1_{(k)}(\underline{01})$ by induction on $k \ge 1$. We give the details for the case $k = 2$. By Theorem 3.7, we have

$$ \partial^1_{(2)}(\underline{01}) = -H^0_1 \partial^1_{(2)} \partial^1_{(0)}(\underline{01}) - H^0_1 \partial^1_{(1)} \partial^1_{(1)}(\underline{01}). $$

Because the differentials $\partial^n$'s preserve the face maps, we have

$$ \begin{aligned} \partial^1_{(2)} \partial^1_{(0)}(\underline{01}) &= d^0 \partial^0_{(2)}(\underline{0}) - d^1 \partial^0_{(2)}(\underline{0}) \\ &= 0, \end{aligned} $$
since $\partial_{(2)}^0 = 0$. Now, by the Leibniz rule in $\mathcal{B}race(\Sigma^{-1}N_*(\Delta^1))$,

$$ \partial_{(1)}^1 \partial_{(1)}^1(\underline{01}) = \begin{array}{c|c|c|c} \underline{01} & \partial_{(2)}^1(\underline{01}) & \underline{1} & \partial_{(2)}^1(\underline{1}) \\ \hline \partial_{(1)}^1(\underline{0}) & \underline{0} & \underline{0} & \partial_{(2)}^1(\underline{01}) \end{array} . $$

These terms give

$$ \begin{array}{rcl} \begin{array}{c} \underline{01} \\ \hline \partial_{(1)}^1(\underline{0}) \end{array} & = & \begin{array}{c} \underline{01} \\ \hline \underline{0} + \underline{0} \end{array} \begin{array}{c} \underline{01} \\ \hline \underline{0} \end{array} + \begin{array}{c} \underline{01} \\ \hline \underline{0} \end{array} ; \\[1.5em] \begin{array}{c} \partial_{(2)}^1(\underline{01}) \\ \hline - \underline{0} \end{array} & = & \begin{array}{c} \underline{01} \\ \hline \underline{0} + \underline{01} \end{array} ; \\[1.5em] \begin{array}{c} \underline{1} \\ \hline - \partial_{(2)}^1(\underline{01}) \end{array} & = & \begin{array}{c} \underline{01} \\ \hline - \underline{0} + \underline{01} \end{array} \begin{array}{c} \underline{1} \\ \hline \begin{array}{c} \underline{1} \\ \hline \begin{array}{c} \underline{1} \\ \hline \begin{array}{c} \underline{1} \\ \hline \begin{array}{c} \underline{1} \\ \hline \underline{01} \end{array} \end{array} \end{array} \end{array} - \begin{array}{c} \underline{01} \\ \hline \underline{0} \end{array} ; \\[1.5em] \begin{array}{c} \partial_{(2)}^1(\underline{1}) \\ \hline - \underline{01} \end{array} & = & \begin{array}{c} \underline{1} \\ \hline - \underline{01} \end{array} . \end{array} $$

The boxed tree is the only tree which gives a non-zero element when applying $H_1^0$. This gives

$$ \partial_{(2)}^1(\underline{01}) = \begin{array}{c} \underline{01} \\ \hline \underline{0} \end{array} . $$

We now suppose that $k \ge 3$. By definition, and since $\partial_{(k-1)}^1 \partial_{(0)}^1(\underline{01}) = 0$ because $\partial_{(k-1)}^0 = 0$, we have

$$ \partial_{(k)}^1(\underline{01}) = -H_1^0 \partial_{(k-1)}^1 \partial_{(1)}^1(\underline{01}) - H_1^0 \partial_{(1)}^1 \partial_{(k-1)}^1(\underline{01}) - \sum_{\substack{p+q=k \\ p,q \ne 0,1}} H_1^0 \partial_{(p)}^1 \partial_{(q)}^1(\underline{01}). $$
By induction hypothesis, for every $p, q \neq 0, 1$ such that $p+q=k$, the term $\partial_{(p)}^1 \partial_{(q)}^1(\underline{01})$ will only give trees with as vertices $\underline{0}$ or $\underline{01}$, so that $H_1^0 \partial_{(p)}^1 \partial_{(q)}^1(\underline{01}) = 0$. We now look at the remaining terms. Since $k-1 \geq 2$, we have by induction hypothesis and by the Leibniz rule

$$ \partial_{(1)}^1 \partial_{(k-1)}^1(\underline{01}) = \sum_{p+q+r=k-1} \underbrace{\underline{01}}_{\textcircled{0}} \underbrace{\big| \big| \big| \big|}_{\textcircled{0}} \underbrace{\underline{01}}_{\textcircled{0}} + \sum_{p+q=k-1} \underbrace{\underline{01}}_{\textcircled{0}} \underbrace{\big| \big| \big| \big|}_{\textcircled{0}} \underbrace{\underline{01}}_{\textcircled{0}} - \sum_{p+q=k-1} \underbrace{\underline{01}}_{\textcircled{0}} \underbrace{\big| \big| \big| \big|}_{\textcircled{0}} \underbrace{\underline{01}}_{\textcircled{0}}, $$

which gives $H_1^0 \partial_{(1)}^1 \partial_{(k-1)}^1(\underline{01}) = 0$. We finally have

$$ \partial_{(k-1)}^1 \partial_{(1)}^1(\underline{01}) = - \underbrace{\underline{01}}_{\textcircled{0}} \underbrace{\big| \big| \big| \big|}_{\textcircled{0}} - \sum_{p+q=k-2} \underbrace{\underline{01}}_{\textcircled{0}} \underbrace{\big| \big| \big| \big|}_{\textcircled{0}} \underbrace{\underline{01}}_{\textcircled{0}} - \sum_{p+q=k-1} \underbrace{\underline{01}}_{\textcircled{0}} \underbrace{\big| \big| \big| \big|}_{\textcircled{0}} \underbrace{\underline{01}}_{\textcircled{0}}. $$

All the terms occurring in the right hand-side give elements in the kernel of $H_1^0$, except for the last sum with $p=0$ and $q=k-1$, which gives

$$ \partial_{(k)}^1(\underline{01}) = \begin{array}{c} \underline{01} \\ \textcircled{0} \end{array}. $$

The computation of $\partial^1$ is proven. We now compute $\partial^2$. By Lemma 3.6, we have

$$ \partial^2(\underline{0}) = \begin{array}{c|c|c} \underline{0} & \underline{1} & \underline{2} \\ \hline \underline{0} & \underline{1} & \underline{2} \end{array}; \partial^2(\underline{1}) = \begin{array}{c|c|c} \underline{0} & \underline{1} & \underline{2} \\ \hline \underline{0} & \underline{1} & \underline{2} \end{array}; \partial^2(\underline{2}) = \begin{array}{c|c|c} \underline{0} & \underline{1} & \underline{2} \\ \hline \underline{0} & \underline{1} & \underline{2} \end{array}. $$

By Corollary 3.8, we have

$$ \partial_{(1)}^2(\underline{01}) = \begin{array}{c|c|c} \underline{01} & \underline{1} & \underline{2} \\ \hline \underline{0} & \underline{01} & \underline{02} \end{array}; \partial_{(1)}^2(\underline{02}) = \begin{array}{c|c|c} \underline{02} & \underline{2} & \underline{2} \\ \hline \underline{0} & \underline{02} & \underline{02} \end{array}; \partial_{(1)}^2(\underline{12}) = \begin{array}{c|c|c} \underline{12} & \underline{2} & \underline{2} \\ \hline \underline{1} & \underline{12} & \underline{12} \end{array}, $$
and

$$ \partial_{(1)}^2(\underline{012}) = \begin{array}{@{}c@{\,}c@{\,}c@{\,}c@{}} \begin{array}{@{}c@{}} \underline{012} \\ \underline{0} \end{array} & \begin{array}{@{}c@{}} \underline{12} \\ \underline{01} \end{array} & \begin{array}{@{}c@{}} \underline{2} \\ \underline{012} \end{array} & \begin{array}{@{}c@{}} \underline{0} \\ \underline{012} \end{array} \end{array}. $$

As before, we can compute by hand $\partial_{(2)}^2(\underline{012})$ by using Theorem 3.7. We obtain

$$ \partial_{(2)}^1(\underline{012}) = - \begin{array}{@{}c@{\,}c@{\,}c@{\,}c@{}} \begin{array}{@{}c@{}} \underline{12} \\ \underline{01} \end{array} & \begin{array}{@{}c@{}} \underline{12} \\ \underline{012} \end{array} & \begin{array}{@{}c@{}} (\underline{01} + \underline{12}) \\ \underline{0} \end{array} & \begin{array}{@{}c@{}} \underline{02} \\ \underline{012} \end{array} \end{array}. $$

We now compute $\partial_{(k)}^2(\underline{012})$ for every $k \ge 3$. We use that

$$ \partial_{(k)}^2(\underline{012}) = -H_2^0 \partial_{(k)}^2 \partial_{(0)}^2(\underline{012}) - H_2^0 \partial_{(k-1)}^2 \partial_{(1)}^2(\underline{012}) - \sum_{\substack{p+q=k \\ q \ne 0,1 \\ p \ne 0}} H_2^0 \partial_{(p)}^2 \partial_{(q)}^2(\underline{012}). $$

We have

$$ -H_2^0 \partial_{(k)}^2 \partial_{(0)}^2(\underline{012}) = - \begin{array}{@{}c@{}} \underline{12} \\ \underline{01} \end{array} + \begin{array}{@{}c@{}} \underline{012} \\ \underline{0} \end{array} \begin{array}{@{}c@{}} \underline{12} \\ \underline{0} \end{array}. $$

by induction hypothesis. We now compute $H_2^0 \partial_{(k-1)}^2 \partial_{(1)}^2(\underline{012})$. We have

$$ -H_2^0 \partial_{(k-1)}^2 \begin{pmatrix} \underline{012} \\ \underline{0} \end{pmatrix} = 0, $$

$$ H_2^0 \partial_{(k-1)}^2 \begin{pmatrix} \underline{12} \\ \underline{01} \end{pmatrix} = \begin{array}{@{}c@{}} \underline{012} \\ \underline{0} \end{array} \begin{array}{@{}c@{}} \underline{01} \\ \underline{-1} \end{array}, $$

and

$$ -H_2^0 \partial_{(k-1)}^2 \begin{pmatrix} \underline{2} \\ \underline{012} \end{pmatrix} = \sum \begin{array}{@{}c@{\,}c@{\,}c@{\,}c@{}} \begin{array}{@{}c@{}} \underline{02} \\ \underline{012} \end{array} & \begin{array}{@{}c@{}} \underline{02} \\ \underline{012} \end{array} & \begin{array}{@{}c@{}} (\underline{01} + \underline{12} + \sum \begin{array}{@{}c@{}} \underline{12} \\ \underline{01} \end{array}) \\ \dots \end{array} & \begin{array}{@{}c@{}} (\underline{01} + \underline{12} + \sum \begin{array}{@{}c@{}} \underline{12} \\ \underline{01} \end{array}) \\ \dots \end{array} \end{array}. $$
where $\mathcal{B}(n)$ denotes a basis of $\mathcal{P}(n)$, and where $\mathcal{L}ie_{\infty} = \mathcal{B}^c(\Lambda^{-1}\mathcal{Com}^{\vee})$ is the operad which governs
Lie algebras up to homotopy. Recall also that we have a morphism of operads $\mathcal{L}ie_{\infty} \to \mathcal{PreL}ie_{\infty}$
given by the morphism of symmetric sequences

$$
\begin{array}{ccc}
\mathrm{Perm} & \to & \mathrm{Com} \\
e_1^n & \mapsto & 1.
\end{array}
$$

**Lemma 3.10.** *There exists an explicit lift of the diagram*

$$
\begin{tikzcd}
& & \mathcal{L}ie_{\infty} \arrow[r] \arrow[d] & \mathcal{B}race \otimes_{\mathbb{H}} \mathcal{B}^c(\Lambda^{-1}\mathcal{B}race^{\vee}) \arrow[d] \\
& & \mathcal{P}re\mathcal{L}ie_{\infty} &
\end{tikzcd}
$$

*Proof.* Giving a morphism $\mathcal{P}re\mathcal{L}ie_{\infty} \to \mathcal{B}race \otimes_{\mathbb{H}} \mathcal{B}^c(\Lambda^{-1}\mathcal{B}race^{\vee})$ is equivalent to giving a Maurer-Cartan element $f$ in the pre-Lie algebra $\mathrm{Hom}_{\Sigma\mathrm{Seq}_k}(\overline{\mathrm{Perm}}^{\vee}, \mathcal{B}race \otimes_{\mathbb{H}} \mathcal{B}^c(\mathcal{B}race^{\vee}))$ (see for instance [LV12, Theorem 6.5.10]). By symmetry, it is sufficient to give the image of $e_1^n$ for every $n \ge 1$. We set

$$
f(e_1^n) = - \sum_{\substack{T \in \mathcal{PRT}(n) \\ r(T)=1}} T \otimes \Sigma^{-1} T^{\vee}.
$$

We check that $d(f)(e_1^n) + (f \star f)(e_1^n) = 0$, where $\star$ denotes the pre-Lie product of $\mathrm{Hom}_{\Sigma\mathrm{Seq}_k}(\overline{\mathrm{Perm}}^{\vee}, \mathcal{B}race \otimes_{\mathbb{H}} \mathcal{B}^c(\mathcal{B}race^{\vee}))$. We first have

$$
d(f)(e_1^n) = d(f(e_1^n)) = - \sum_{T \in \mathcal{PRT}(n) \atop r(T)=1} \sum_{S \subseteq T} T \otimes (\Sigma^{-1}(T/S)^{\vee} \circ_S \Sigma^{-1} S^{\vee}).
$$

We now compute $(f \star f)(e_1^n)$. Recall that

$$
\Delta_1(e_1^n) = \sum_{\substack{p+q=n+1 \\ p,q \ge 2}} e_1^p \otimes e_1^q;
$$

$$
\forall k \neq 1, \Delta_k(e_1^n) = \sum_{\substack{p+q=n+1 \\ p,q \ge 2}} \sum_{i=1}^{q} e_1^p \otimes e_i^q.
$$

The $\Delta_1$ part gives

$$
\sum_{\substack{p+q=n+1 \\ p,q \ge 2}} \sum_{\omega \in \mathrm{Sh}_*(q,1,\dots,1)} \sum_{\substack{U \in \mathcal{PRT}(p) \\ r(U)=1}} \sum_{\substack{V \in \mathcal{PRT}(q) \\ r(V)=1}} \omega \cdot (U \circ_1 V) \otimes \omega \cdot (\Sigma^{-1} U^{\vee} \circ_1 \Sigma^{-1} V^{\vee}),
$$

and the $\Delta_k$'s part for $k \neq 1$ gives

$$
\sum_{p+q=n+1} \sum_{k=2 \atop p,q \ge 2} \sum_{\omega \in \mathrm{Sh}_*(1,\dots,q,\dots,1)} \sum_{\substack{U \in \mathcal{PRT}(p) \\ r(U)=1}} \sum_{\substack{V \in \mathcal{PRT}(q) \\ r(V)=1}} \omega \cdot (U \circ_k V) \otimes \omega \cdot (\Sigma^{-1} U^{\vee} \circ_k \Sigma^{-1} V^{\vee}).
$$
Let $T \in \mathcal{PT}(n)$ be such that $r(T) = 1$. Consider a term $\omega \cdot (U \circ_k V)$ occurring in one of the two previous sums, with $k \ge 1$, $\omega \in Sh_*(1, \dots, q, \dots, 1)$, $U \in \mathcal{PT}(p)$, $V \in \mathcal{PT}(q)$, such that $T$ occurs in the expansion of $\omega \cdot (U \circ_k V)$. We see $U$ and $V$ as $U \in \mathcal{PT}(1 < \dots < k-1 < V < k+q < \dots < p)$ and $V \in \mathcal{PT}(k < \dots < k+q-1)$. Because $\omega \in Sh_*(1, \dots, q, \dots, 1)$, the composite $\omega \cdot (U \circ_k V)$ is equal to the composite $(\omega \cdot U) \circ_V (\omega \cdot V)$ where $\omega \cdot U$ is $U$ seen in $\mathcal{PT}(1 < \dots < k-1 < V < \omega(k+q) < \dots < \omega(p))$ and $\omega \cdot V$ is $V$ seen in $\mathcal{PT}(k = \omega(k) < \dots < \omega(k+q-1))$. Thus, by definition of the operadic composition in $\mathcal{Brace}$, the tree $\omega \cdot V$ can be seen as a subtree $S \subset T$ such that $T/S = \omega \cdot U$.

In the converse direction, let $S \subset T$. Let $k = \min(V_S)$ and $q = |S|$. Let $\omega_S : \llbracket k, k+q-1 \rrbracket \to V_S$ and $\omega_{T/S} : \llbracket 1, n \rrbracket \setminus \llbracket k+1, k+q-1 \rrbracket \to V_{T/S}$ be the unique order preserving maps between the two considered finite sets. Then, by definition, $\omega = \omega_{T/S} \circ_k \omega_S \in Sh_*(1, \dots, q, \dots, 1)$. We finally set $U = \omega^{-1} \cdot (T/S)$ and $V = \omega^{-1} \cdot S$. Because $T/S \circ_S S$ obviously contains the tree $T$, we have that $T$ occurs in the composite $\omega \cdot (U \circ_k V)$.

We thus have proved that

$$ (f \star f)(e_1^n) = \sum_{\substack{T \in \mathcal{PT}(n) \\ r(T)=1}} T \otimes \left( \sum_{S \subset T} \Sigma^{-1}(T/S)^\vee \circ_S \Sigma^{-1} S^\vee \right). $$

The identity $d(f)(e_1^n) + (f \star f)(e_1^n) = 0$ follows. $\square$

We now prove Theorem A.

**Theorem 3.11.** There exists an operad morphism $\mathcal{PreLie}_\infty \to \mathcal{Brace} \otimes_H \mathcal{E}$ which fits in a commutative square

<figure><img src="image_17.png" /></figure>

*Proof.* The morphism $\mathcal{PreLie}_\infty \to \mathcal{Brace} \otimes_H \mathcal{E}$ is given by the composite of the morphism $\mathcal{PreLie}_\infty \to \mathcal{Brace} \otimes_H B^c(\Lambda^{-1}\mathcal{Brace}^\vee)$ given by Lemma 3.10 with the morphism $\mathcal{Brace} \otimes_H B^c(\Lambda^{-1}\mathcal{Brace}^\vee) \to \mathcal{Brace} \otimes_H \mathcal{E}$ given by applying the morphism $B^c(\Lambda^{-1}\mathcal{Brace}^\vee) \to \mathcal{E}$ defined in Theorem 3.3 on the second tensor. The commutative diagram is an immediate check. $\square$

**Corollary 3.12.** Every $\mathcal{Brace} \otimes_H \mathcal{E}$-algebra $L$ admits the structure of a $\Gamma(\mathcal{PreLie}_\infty, -)$-algebra.

*Proof.* Using that the action of $\Sigma_n$ on $(\mathcal{Brace} \otimes_H \mathcal{E})(n)$ is free and the previous theorem, we define, for every $\mathcal{Brace} \otimes_H \mathcal{E}$-algebras $L$, the composite

$$ \Gamma(\mathcal{PreLie}_\infty, L) \to \Gamma(\mathcal{Brace} \otimes_H \mathcal{E}, L) \xleftarrow{\simeq} S(\mathcal{Brace} \otimes_H \mathcal{E}, L) \to L. $$
This gives a $\Gamma(\mathcal{P}re\mathcal{L}ie_\infty, -)$-algebra structure on $L$. $\square$

In particular, if $L = A \otimes \Sigma E$ where $A$ is a brace algebra and $E$ a $\mathcal{E}$-algebra, then $A \otimes \Sigma E$ is a $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$-algebra. We can compute the weighted braces of $A \otimes \Sigma E$ as follows. Let $l: S(Brace, A) \to A$ be the brace algebra structure on $A$, and let $\partial^E$ be the twisting morphism on $Brace^\vee(\Sigma E)$ induced by the $B^c(\Lambda^{-1}Brace^\vee)$-algebra structure of $E$ (see Theorem 3.3). Then, for every $a, b_1, \dots, b_n \in A, x, y_1, \dots, y_n \in \Sigma E$ and $r_1, \dots, r_n \ge 0$, we have

$$ a \otimes x \{b_1 \otimes y_1, \dots, b_n \otimes y_n\}_{r_1, \dots, r_n} = \sum_{\substack{\sigma \in Sh(r_1, \dots, r_n) \\ T \in \mathcal{P}\mathcal{R}\mathcal{T}(r+1) \\ \text{canonical}}} \pm l(T \otimes a \otimes c_{\sigma(1)} \otimes \dots \otimes c_{\sigma(r)}) \otimes \partial^E(T^{\vee} \otimes x \otimes z_{\sigma(1)} \otimes \dots \otimes z_{\sigma(r)}), $$

where we have set $r = r_1 + \dots + r_n$, $c_1, \dots, c_r = \underbrace{b_1, \dots, b_1, \dots, b_n, \dots, b_n}_{r_1}, \underbrace{b_n, \dots, b_n, \dots, b_n}_{r_n}$ and $z_1, \dots, z_r = \underbrace{x_1, \dots, x_1, \dots, x_n, \dots, x_n}_{r_1}, \underbrace{x_n, \dots, x_n, \dots, x_n}_{r_n}$. The sign is given by the permutation of the $c_i$'s with $x$ and the $z_i$'s, and the permutation of $\partial^E$ with $a$ and the $c_i$'s.

# 4 The simplicial Maurer-Cartan set of a complete brace algebra

The goal of this section is to define the notion of a simplicial Maurer-Cartan set $\mathcal{MC}_\bullet(A)$ associated to a brace algebra $A$, and to study the homotopy type of this simplicial set. Explicitly, the $n$-component $\mathcal{MC}_n(A)$ will be defined as the Maurer-Cartan set of $A \otimes \Sigma N^*(\Delta^n)$ for the $\Gamma\Lambda\mathcal{P}\mathcal{L}_\infty$-algebra structure given by Corollary 3.12.

In §4.1, we define the simplicial set $\mathcal{MC}_\bullet(A)$ and prove the first part of Theorem B which asserts that it is a Kan complex.

In §4.2, we prove the remaining part of Theorem B, which gives a computation of the connected components and the homotopy groups of $\mathcal{MC}_\bullet(A)$. More precisely, we first compute the connected components, whose computation is similar to [Ver23, Theorem 3.6], before computing the $\pi_1, \pi_2$ and then the $\pi_n$ for $n \ge 3$.

In §4.4, we prove Theorem C, which is a higher version of the Goldman-Millson theorem (see [GM88, §2.4]). Our proof basically follows the proof found in [MR23, §6], which will be adapted to our context.

In §4.5, we compare our simplicial notion of Maurer-Cartan set defined for complete brace algebras to the notion of simplicial Maurer-Cartan set associated to a complete Lie algebra, and prove that in fact, these two simplicial sets are weakly equivalent.
## 4.1 The simplicial set $MC_•(A)$

Let $A$ be a complete brace algebra. By Corollary 3.12, and using that $N^*(\Delta^n)$ is a $\mathcal{E}$-algebra, we obtain that $A \otimes \Sigma N^*(\Delta^n) = A \otimes \Sigma N^*(\Delta^n)$ is a $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_\infty$-algebra with the filtration

$$F_k(A \otimes \Sigma N^*(\Delta^n)) = F_k A \otimes \Sigma N^*(\Delta^n).$$

where we denote by $(F_k A)_{k \ge 1}$ the filtration on $A$.

**Definition 4.1.** Let $A$ be a complete brace algebra. Its simplicial Maurer-Cartan set is the simplicial set $MC_•(A)$ such that

$$MC_•(A) = MC(A \otimes \Sigma N^*(\Delta^•)).$$

**Proposition 4.2.** The previous definition defines a functor $MC_•$ from the category of complete brace algebras to the category of sets.

*Proof.* This follows directly from the fact that every brace algebra morphism $f: A \to B$ which preserves the filtrations gives rise to a morphism of $\mathcal{B}race \otimes \Lambda\mathcal{E}$-algebras $f \otimes \mathrm{id}: A \otimes \Sigma N^*(\Delta^n) \to B \otimes \Sigma N^*(\Delta^n)$ which preserves the filtrations, and then to a strict morphism of $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_\infty$-algebras. $\square$

We aim to prove that $MC_•(A)$ is a Kan complex. We will basically follow the proof of the analogous theorem in [KW21]. Recall from Proposition 1.32 that we have morphisms $\varphi_n^i: N^*(\Delta^n) \to N^*(\Delta^n)$ and $h_n^i: N^*(\Delta^n) \to N^{i-1}(\Delta^n)$ which satisfy

$$dh_n^i + h_n^i d = \mathrm{id} - \varphi_n^i.$$

These relations can be carried to $A \otimes \Sigma N^*(\Delta^n)$ by setting $H_n^i = \mathrm{id} \otimes \Sigma h_n^i$ and $\Phi_n^i = \mathrm{id} \otimes \Sigma \varphi_n^i$. Accordingly, we have

$$dH_n^i + H_n^i d = \mathrm{id} - \Phi_n^i.$$

Note that if $x \in MC(A \otimes \Sigma N^*(\Delta^n))$, then the identity

$$d(x) + \sum_{k \ge 1} x[x]_k = 0$$

gives, after composing by $H_n^i$,

$$x = \Phi_n^i(x) + dH_n^i(x) - \sum_{k \ge 1} H_n^i(x[x]_k).$$

We give the following lemma, which is the analogue of [KW21, Lemma 6.5]. For our needs, we need an analogue valid for every complete $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_\infty$-algebra structure on $A \otimes \Sigma N^*(\Delta^n)$.

**Lemma 4.3.** Let $n \ge 0$ and $0 \le i \le n$. Let $A \in \mathrm{dg}\widehat{\mathrm{Mod}}_k$. Consider any complete $\Gamma\Lambda\widehat{\mathcal{P}\mathcal{L}}_\infty$-algebra structure on $A \otimes \Sigma N^*(\Delta^n)$ such that $\Phi_n^i: A \otimes \Sigma N^*(\Delta^n) \to A \otimes \Sigma N^*(\Delta^n)$ is a strict morphism. Then the map
$$ \mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^n)) \xrightarrow{x} (\mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^n)) \cap \mathrm{Im}(\Phi_n^i)) \times \mathrm{Im}(dH_n^i) \\ (\Phi_n^i(x), dH_n^i(x)) $$

is a bijection.

*Proof.* We first note that the above map is well defined since $\Phi_n^i$ is a strict morphism by hypothesis. Now, let $e \in \mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^n)) \cap \mathrm{Im}(\Phi_n^i)$ and $r \in \mathrm{Im}(dH_n^i)$. We set:

$$ \left\{ \begin{array}{l} \alpha_0 = e+r \\ \forall k \ge 0, \alpha_{k+1} = e+r - \sum_{l \ge 1} H_n^i(\alpha_k \llbracket \alpha_k \rrbracket_l) . \end{array} \right. $$

This defines a Cauchy sequence $(\alpha_k)_k$. Let $\alpha$ be its limit. We then have

$$ \alpha = e + r - \sum_{l \ge 1} H_n^i(\alpha \llbracket \alpha \rrbracket_l). $$

From this identity, we deduce $\Phi_n^i \alpha = e$ and $dH_n^i \alpha = r$. We just need to check that $\alpha \in \mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^n))$. Using the relation

$$ dH_n^i + H_n^i d = id - \Phi_n^i, $$

we obtain

$$ d(\alpha) = d(e) + \sum_{l \ge 1} H_n^i(d(\alpha \llbracket \alpha \rrbracket_l)) - \sum_{l \ge 1} \alpha \llbracket \alpha \rrbracket_l + \Phi_n^i \left( \sum_{l \ge 1} \alpha \llbracket \alpha \rrbracket_l \right), $$

and then, because $\Phi_n^i$ is a strict morphism and that $e \in \mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^n))$,

$$ d(\alpha) + \sum_{l \ge 1} \alpha \llbracket \alpha \rrbracket_l = \sum_{l \ge 1} H_n^i(d(\alpha \llbracket \alpha \rrbracket_l)). $$

Let $\mathcal{R}(\alpha) = d(\alpha) + \sum_{l \ge 1} \alpha \llbracket \alpha \rrbracket_l$. We use the identity

$$ \sum_{p+q=l} \alpha \llbracket \alpha \rrbracket_p \llbracket \alpha \rrbracket_q + \sum_{p+q=l-1} \alpha \llbracket \alpha \rrbracket_p \llbracket \alpha \rrbracket_{1,q} = 0 $$

for every $l \ge 1$. We thus have

$$ d(\alpha \llbracket \alpha \rrbracket_l) = - \sum_{\substack{p+q=l \\ q \ne 0}} \alpha \llbracket \alpha \rrbracket_p \llbracket \alpha \rrbracket_q - \sum_{p+q=l-1} \alpha \llbracket \alpha \rrbracket_p \llbracket \alpha \rrbracket_{1,q} $$

so that

$$ \sum_{l \ge 1} d(\alpha \llbracket \alpha \rrbracket_l) = - \sum_{q \ge 1} \mathcal{R}(\alpha) \llbracket \alpha \rrbracket_q - \sum_{q \ge 0} \alpha \llbracket \mathcal{R}(\alpha), \alpha \rrbracket_q. $$

This leads finally to the identity

$$ \mathcal{R}(\alpha) = - \sum_{q \ge 1} H_n^i(\mathcal{R}(\alpha) \llbracket \alpha \rrbracket_q) - \sum_{q \ge 0} H_n^i(\alpha \llbracket \mathcal{R}(\alpha), \alpha \rrbracket_{1,q}). $$

It follows from this identity that if $\mathcal{R}(\alpha) \in F_k(A) \otimes \Sigma N^*(\Delta^n)$ for some $k \ge 1$, then $\mathcal{R}(\alpha) \in F_{k+1}(A) \otimes \Sigma N^*(\Delta^n)$. We thus have $\mathcal{R}(\alpha) = 0$ so that $\alpha \in \mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^n))$. We then have a bijection which has as inverse this previous construction. $\square$
**Definition 4.4.** A simplicial $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebra is a simplicial object in the category $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$. A simplicial $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebra $A$ is strict if the face and degeneracy maps of $A$ are strict morphisms of $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebras. A morphism of simplicial $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebras $\phi: A \to B$ is strict if, for every $n \ge 0$, the map $\phi_n: A_n \to B_n$ is a strict morphism of $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebras.

**Theorem 4.5.** Let $A, B \in \operatorname{dgMod}_{\mathbb{K}}$ be such that $A \otimes \Sigma N^*(\Delta^\bullet)$ and $B \otimes \Sigma N^*(\Delta^\bullet)$ are endowed with the structure of simplicial $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebras. Let $f: A \to B$ be a surjective morphism in $\operatorname{dgMod}_{\mathbb{K}}$ such that $f \otimes \mathrm{id}: A \otimes \Sigma N^*(\Delta^\bullet) \to B \otimes \Sigma N^*(\Delta^\bullet)$ is a strict morphism. Then $\mathcal{M}\mathcal{C}(f \otimes \mathrm{id}): \mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^\bullet)) \to \mathcal{M}\mathcal{C}(B \otimes \Sigma N^*(\Delta^\bullet))$ is a Kan fibration.

*Proof.* Since the $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebra structures are compatible with the simplicial structures on $A \otimes \Sigma N^*(\Delta^\bullet)$ and $B \otimes \Sigma N^*(\Delta^\bullet)$, we can follow the same proof as in [KW21, Proposition 6.6] to obtain the result. $\square$

Applying this result to $B = 0$ thus gives the following corollary.

**Corollary 4.6.** For every $A \in \operatorname{dgMod}_{\mathbb{K}}$ such that $A \otimes \Sigma N^*(\Delta^\bullet)$ is a strict simplicial $\Gamma\Lambda\widehat{P\mathcal{L}}_\infty$-algebra, the simplicial set $\mathcal{M}\mathcal{C}(A \otimes \Sigma N^*(\Delta^\bullet))$ is a Kan complex.
In particular, for every complete brace algebra $A$, the simplicial set $\mathcal{M}\mathcal{C}_\bullet(A)$ is a Kan complex.

## 4.2 Connected components and homotopy groups of $\mathcal{M}\mathcal{C}_\bullet(A)$

We are now able to compute the connected components and the homotopy groups of $\mathcal{M}\mathcal{C}_\bullet(A)$ for a given complete brace algebra $A$. For this purpose, recall from [Ver23, Theorem 2.15] that any brace algebra $A$ is endowed with the structure of a $\Gamma(\mathcal{P}\mathcal{R}\mathcal{L}, -)$-algebra via the composite

$$ \Gamma(\mathcal{P}\mathcal{R}\mathcal{L}, A) \to \Gamma(\mathcal{B}\mathcal{R}\mathcal{L}, A) \xleftarrow{\sim} \mathcal{S}(\mathcal{B}\mathcal{R}\mathcal{L}, A) \to A. $$

In this setting, we recall from [Ver23, Definition 2.19] the operation $\odot$ defined by

$$ x \odot (1 + y) := \sum_{n \ge 0} x \underbrace{\langle y, \dots, y \rangle}_{n} $$

for every $x \in A$ and $y \in A_0$. By [Ver23, Theorem 2.24], we have that this operation induces a group structure on the set $G = 1 + A_0$ with the product

$$ (1 + x) \odot (1 + y) = 1 + x + y + \sum_{n \ge 1} x \underbrace{\langle y, \dots, y \rangle}_{n}. $$

This group is called the *gauge group* associated to the brace algebra $A$. In the following, we use the operation $\bar{\odot}$ defined by

$$ x \bar{\odot} y := x + y + \sum_{n \ge 1} x \underbrace{\langle y, \dots, y \rangle}_{n}, $$

for every $x \in A$ and $y \in A_0$. Note that the group $(1+A_0, \odot, 1)$ is isomorphic to the group $(A_0, \bar{\odot}, 0)$. Using this identification and [Ver23, Theorem 2.29], we have an action of $(A_0, \bar{\odot}, 0)$ on $\mathcal{M}\mathcal{C}(A)$ by

$$ x \cdot \tau = (\tau + x \langle \tau \rangle - d(x)) \bar{\odot} x^{\bar{\odot}-1} $$
for every $x \in A_0$ and $\tau \in \mathcal{MC}(A)$.

By Corollary 3.9, we have an obvious identification $\mathcal{MC}_0(A) = \mathcal{MC}(A)$, using the Maurer-Cartan set of a $\Gamma(\mathcal{PreLie}, -)$-algebra (see [Ver23, Definition 2.17]). This identification is given by sending $\tau \in \mathcal{MC}(A)$ to $-\tau \otimes \Sigma\underline{0}^\vee \in \mathcal{MC}_0(A)$.

In this subsection, in order to write easier formulas, for every $\underline{x} \in N_*(\Delta^n)$, we drop the desuspension $\Sigma^{-1}$ on the element $\Sigma^{-1}\underline{x} \in \Sigma^{-1}N_*(\Delta^n)$. Analogously, we drop the suspension $\Sigma$ on elements of $\Sigma N^*(\Delta^n)$.

### 4.2.1 Connected components

We first compute the $\pi_0$. We begin by the following lemma.

**Lemma 4.7.** Let $\tau_0, \tau_1 \in \mathcal{MC}(A)$. Then every element $\alpha \in \mathcal{MC}_1(A)$ such that $d_0\alpha = \tau_0$ and $d_1\alpha = \tau_1$ are written

$$ \alpha = -\tau_1 \otimes \underline{0}^{\vee} - \tau_0 \otimes \underline{1}^{\vee} - h \otimes \underline{01}^{\vee} $$

where $h \in A_0$ is such that

$$ d(h) = \tau_0 + h\langle\tau_0\rangle - \tau_1 \odot (1+h). $$

*Proof.* Let $\alpha \in \mathcal{MC}_1(A)$ be such that $d_0\alpha = \tau_0$ and $d_1\alpha = \tau_1$. We write

$$ \alpha = -\tau_1 \otimes \underline{0}^{\vee} - \tau_0 \otimes \underline{1}^{\vee} - h \otimes \underline{01}^{\vee} $$

for some $h \in A_0$. We make explicit the Maurer-Cartan condition on $\alpha$. We have

$$ d(\alpha) = -d(\tau_1) \otimes \underline{0}^{\vee} - d(\tau_0) \otimes \underline{1}^{\vee} + (-d(h) + \tau_0 - \tau_1) \otimes \underline{01}^{\vee}. $$

Let $p \ge 1$. By formula (v) of Theorem 2.22, we have that

$$ \alpha[\alpha]_p = \sum_{p_1+p_2+p_3=p} -\tau_1 \otimes \underline{0}^{\vee}[\!\langle -\tau_1 \otimes \underline{0}^{\vee}, -\tau_0 \otimes \underline{1}^{\vee}, -h \otimes \underline{01}^{\vee} \rangle]_{p_1,p_2,p_3} \\ - \sum_{p_1+p_2+p_3=p} \tau_0 \otimes \underline{1}^{\vee}[\!\langle -\tau_1 \otimes \underline{0}^{\vee}, -\tau_0 \otimes \underline{1}^{\vee}, -h \otimes \underline{01}^{\vee} \rangle]_{p_1,p_2,p_3} \\ - \sum_{p_1+p_2+p_3=p} h \otimes \underline{01}^{\vee}[\!\langle -\tau_1 \otimes \underline{0}^{\vee}, -\tau_0 \otimes \underline{1}^{\vee}, -h \otimes \underline{01}^{\vee} \rangle]_{p_1,p_2,p_3}. $$

By the computation of $\partial^1$ in Corollary 3.9, the first sum gives non-zero elements only for the case $p_1 = 1$ and $p_2 = p_3 = 0$, and the case $p_1 = p_2 = 0$. This then gives

$$ -\tau_1 \langle \tau_1 \rangle \otimes \underline{0}^{\vee} - \sum_{n \ge 1} \tau_1 \langle h, \dots, h \rangle \otimes \underline{01}^{\vee}. $$

The second sum gives non-zero elements only for the case $p_2 = 1$ and $p_1 = p_3 = 0$. We obtain the term

$$ -\tau_0 \langle \tau_0 \rangle \otimes \underline{1}^{\vee}. $$
Finally, the third sum gives non-zero elements only for the case $p_2 = 1$ and $p_1 = p_3 = 0$. We have the term

$$h\langle\tau_0\rangle \otimes \underline{01}^{\vee}.$$

At the end, we have

$$\sum_{p \ge 0} \alpha[\alpha]_p = \left( -d(h) + \tau_0 + h\langle\tau_0\rangle - \sum_{n \ge 0} \tau_1 \underbrace{\langle h, \dots, h \rangle}_n \right) \otimes \underline{01}^{\vee}.$$

Then, the Maurer-Cartan condition on $\alpha$ is equivalent to the equation

$$d(h) = \tau_0 + h\langle\tau_0\rangle - \tau_1 \odot (1+h)$$

which proves the lemma. □

Recall that the Deligne groupoid associated to $A$ is the category formed by Maurer-Cartan elements with as morphisms the elements of the gauge group (see [Ver23, Proposition-Definition 2.22]).

**Theorem 4.8.** Let $A$ be a complete brace algebra. We have a bijection

$$\pi_0(\mathcal{MC}_{\bullet}(A)) \simeq \pi_0\mathrm{Deligne}(A),$$

where we denote by $\pi_0\mathrm{Deligne}(A)$ the set of objects in $\mathrm{Deligne}(A)$ up to isomorphisms.

*Proof.* Recall that

$$\pi_0(\mathcal{MC}_{\bullet}(A)) = \mathcal{MC}_0(A) / \sim,$$

where $\sim$ denotes the homotopy relation in sSet. Consider the projection $f$ of $\mathcal{MC}_0(A)$ on $\mathcal{MC}(A)/G$, where $G$ denotes the gauge group of the $\Gamma(\mathcal{PreLie}, -)$-algebra $A$. Let $\tau_0, \tau_1 \in \mathcal{MC}(A)$. By Lemma 4.7, the elements $-\tau_1 \otimes \underline{0}^{\vee}$ and $-\tau_0 \otimes \underline{1}^{\vee}$ are homotopic in $\mathcal{MC}_0(A)$ if and only if there exists $h \in A_0$ such that $h \cdot \tau_0 = \tau_1$, which proves that $f$ induces a bijection $\bar{f}: \pi_0(\mathcal{MC}_{\bullet}(A)) \to \pi_0\mathrm{Deligne}(A)$. □

### 4.2.2 The group $\pi_1(\mathcal{MC}_{\bullet}(A), \tau)$

We now compute $\pi_1(\mathcal{MC}_{\bullet}(A), \tau)$ for a given $\tau \in \mathcal{MC}(A)$. Let $\mathrm{Aut}_{\mathrm{Deligne}(A)}(\tau) = \{h \in A_0 \mid d(h) = \tau + h\langle\tau\rangle - \tau \odot (1+h)\}$. We have the following lemma.

**Lemma 4.9.** Let $A$ be a complete brace algebra and $\tau \in \mathcal{MC}(A)$. For every $h, h' \in \mathrm{Aut}_{\mathrm{Deligne}(A)}(\tau)$, we write $h \sim_{\tau} h'$ if there exists $\psi \in A_1$ such that

$$h - h' = d(\psi) + \psi\langle\tau\rangle + \sum_{p,q \ge 0} \tau \underbrace{\langle h, \dots, h \rangle}_p \psi \underbrace{\langle h', \dots, h' \rangle}_q.$$

Then $\sim_{\tau}$ is an equivalence relation on the set $\mathrm{Aut}_{\mathrm{Deligne}(A)}(\tau)$. Moreover, the circular product $\odot$ is compatible with $\sim_{\tau}$, so that the triple $(\mathrm{Aut}_{\mathrm{Deligne}(A)}(\tau) / \sim_{\tau}, \bar{\circ}, 0)$ is a group.
Proof. The relation $\sim_{\tau}$ is reflexive (just take $\psi = 0$ so that $h \sim_{\tau} h$ for all $h \in \mathrm{Aut}_{\mathrm{Deligne}(A)}(\tau)$). We prove that this relation is transitive. Let $h, h', h'' \in A_0$ be such that $h \sim_{\tau} h'$ and $h' \sim_{\tau} h''$. Then there exist $\psi, \psi' \in A_1$ such that

$$ (1) \qquad h - h' = d(\psi) + \psi\langle\tau\rangle + \sum_{p,q \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle; $$

$$ (2) \qquad h' - h'' = d(\psi') + \psi'\langle\tau\rangle + \sum_{p,q \ge 0} \tau\langle\underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{q}\rangle. $$

We set $\psi'' := \psi + \psi' + \sum_{p,q,r \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle$, and prove that

$$ h - h'' = d(\psi'') + \psi''\langle\tau\rangle + \sum_{p,q \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi'', \underbrace{h'', \dots, h'''}_{q}\rangle. $$

Let us analyze the right hand-side. We analyze the terms given by $d(\psi'')$ and compare it with the others given either by $\psi''\langle\tau\rangle$ or by the terms of the form $\tau\langle h, \dots, h, \psi'', h'', \dots, h''\rangle$. We first have

$$ d(\psi) + d(\psi') = h - h'' - \psi\langle\tau\rangle - \psi'\langle\tau\rangle - \sum_{p,q \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q}\rangle - \sum_{q,r \ge 0} \tau\langle\underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle. $$

We now differentiate the sum which occurs in the definition of $\psi''$. By the Leibniz rule in the brace algebra $A$, and by applying the differential on $\tau \in \mathcal{MC}(A)$, we, in particular, obtain the sum

$$ - \sum_{p,q,r \ge 0} \tau\langle\tau\rangle\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle. $$

This can be computed by using the brace algebra structure of $A$:

$$ \begin{aligned} - \sum_{p,q,r \ge 0} \tau\langle\tau\rangle\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle = & \\ & - \sum_{p_1, p_2, q, r \ge 0} \tau\langle\underbrace{h, \dots, h}_{p_1}, \tau\odot(1+h), \underbrace{h, \dots, h}_{p_2}, \psi, \underbrace{h', \dots, h'}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle \\ & - \sum_{p, q, r, s, t \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \tau\langle\underbrace{h, \dots, h}_{s}, \psi, \underbrace{h', \dots, h'}_{t}, \psi', \underbrace{h'', \dots, h'''}_{q}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle \\ & + \sum_{p, q_1, q_2, r \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q_1}, \tau\odot(1+h'), \underbrace{h', \dots, h'}_{q_2}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle \\ & + \sum_{p, q, r, s, t \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{q}, \tau\langle\underbrace{h', \dots, h'}_{s}, \psi', \underbrace{h'', \dots, h'''}_{t}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle \\ & - \sum_{p, q, r_1, r_2 \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \psi, \underbrace{h', \dots, h'}_{r}, \psi', \underbrace{h'', \dots, h'''}_{r_1}, \tau\odot(1+h''), \underbrace{h'', \dots, h'''}_{r_2}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle \\ & - \sum_{p, q, r, s, t \ge 0} \tau\langle\underbrace{h, \dots, h}_{p}, \tau\langle\underbrace{h, \dots, h}_{q}, \psi, \underbrace{h', \dots, h'}_{s}, \psi', \underbrace{h'', \dots, h'''}_{t}, \psi', \underbrace{h'', \dots, h'''}_{r}\rangle, \end{aligned} $$