# Kaledin classes and formality criteria

Coline Emprin  Coline Emprin, Département de mathématiques et applications,
École normale supérieure, 45 rue d’Ulm, 75230 Paris, France.
cccccccccccccccccccccccccccccccccccccccccccccccc Laboratoire de Géométrie,
Analyse et Applications, Université Sorbonne Paris Nord, UMR 7539, 93430,
Villetaneuse, France<EMAIL_ADDRESS>
(Date: April 26, 2024)

###### Abstract.

We develop a general obstruction theory to the formality of algebraic
structures over any commutative ground ring. It relies on the construction of
Kaledin obstruction classes that faithfully detect the formality of
differential graded algebras over operads or properads, possibly colored in
groupoids. The present treatment generalizes the previous obstruction classes
in two directions: outside characteristic zero and including a wider range of
algebraic structures. This enables us to establish novel formality criteria,
including formality descent with torsion coefficients, formality in families,
intrinsic formality, and criteria in terms of chain-level lifts of homology
automorphism.

2024 _Mathematics Subject Classification._ 18D50, 18G55, 17B60 16W25, 13D10 .
The author was supported by the project ANR-20-CE40-0016 HighAGT and École
Normale Supérieure.

###### Contents

  1. 1 Kaledin classes in differential graded Lie algebras
  2. 2 Operadic Kaledin classes over a commutative ground ring
  3. 3 Kaledin classes in the properadic setting
  4. 4 Formality criteria

## Introduction

The notion of formality originated in the field of rational homotopy theory.
In this context, a topological space $X$ is _formal_ if one can recover all
its rational homotopy type from its cohomology ring, i.e. if there exists a
zig-zag relating the singular cochains to its cohomology

$C_{\mathrm{sing}}^{*}(X;\mathbb{Q})\overset{\sim}{\longleftarrow}\cdot\overset{\sim}{\longrightarrow}\cdots\overset{\sim}{\longleftarrow}\cdot\overset{\sim}{\longrightarrow}H_{\mathrm{sing}}^{*}(X;\mathbb{Q})\
,$

through quasi-isomorphisms, i.e. morphisms of associative algebras inducing
isomorphisms in cohomology. When dealing with a formal space, the singular
cochains can be replaced by the cohomology, which is usually a more calculable
object. For instance, the formality of spheres allowed Sullivan in [Sul77] to
recover the calculation of rational homotopy groups of spheres established by
Serre in [Ser53]. A seminal formality result was given by Deligne, Griffiths,
Morgan, and Sullivan who proved in [DGMS75], that any compact Kähler manifold
is formal. The formality of topological spaces appears as a particular case of
a much more general notion: the one of algebraic structures. Let $A$ be a
chain complex over a commutative ring $R$. Let us consider a certain
differential graded (dg) algebra structure $\varphi$ over $A$ (e.g. a dg
associative algebra, a dg Hopf algebra, a dg Poisson algebra,etc.). It induces
a structure of the same type $\varphi_{*}$ at the homology level. The same
question then arises : does this induced structure keeps all the homotopical
information that was at the level of the original algebra? In other words,
does there exist a zig-zag of quasi-isomorphisms preserving the algebraic
structures and relating the algebra to the induced structure in homology? If
so, the algebra is said formal.

#### Massey products

In [Mas58], Massey associated to any dg associative algebra, higher order
homology operations, known today as the _Massey products_. These products turn
out to be obstructions to formality: the non-vanishing of only one of them
prevents a dg associative algebra from being formal, see [DGMS75, Theorem
4.1]. Unfortunately, the converse is false: the vanishing of all the Massey
products does not imply formality, see e.g. Halperin–Stasheff [HS79]. To prove
the formality of compact Kähler manifolds, the authors of [DGMS75] mainly use
Hodge theory. Nevertheless, their intuition came from Massey products and
Weil’s conjectures insights. Let $(A,\varphi)$ be a dg associative algebra and
let $\alpha$ be a unit of infinite order in $R$. The grading automorphism
$\sigma_{\alpha}$ is the homology automorphism defined as the multiplication
by $\alpha^{k}$ in homological degree $k$. Sullivan proved in [Sul77] that if
$(A,\varphi)$ satisfies a purity property, i.e. if $\sigma_{\alpha}$ admits a
chain level lift, the algebra is formal. The heuristic is the following one:
if there exists such a lift, the Massey products have to be compatible with
it. Thus, they are going to intertwine multiplication by different powers of
$\alpha$. If the unit has infinite order, all Massey products have to vanish.
This is nonetheless not sufficient to ensure formality, and the proof of
[Sul77] do not make use of Massey products. Sullivan’s result was latter
widely generalized to other types algebraic structures and to any coefficient
ring in [GSNPR05], [CH20], [DCH21] and [CH22].

Kaledin classes. In order to construct a refinement of the Massey products
which would characterize formality, Kaledin introduced in [Kal07] new
obstruction classes, nowadays called after his name. He associates to any dg
associative algebra its _Kaledin class_ in Hochschild cohomology, which
vanishes if and only if the algebra is formal. He was motivated by
establishing formality criteria for families of dg associative algebras, for
which the methods of [DGMS75] do not apply. The work of Kaledin was extended
by Lunts in [Lun07] for homotopy associative algebras over a
$\mathbb{Q}$-algebra and by Melani and Rubió in [MR19] for algebras over a
binary Koszul operad in characteristic zero.

One can ask for the formality of a wide range of other algebraic structures:
operads themselves, structures involving operations with several inputs but
also several outputs such as dg Frobenius bialgebras, dg involutive Lie
bialgebras etc. Such structures are encoded by generalizations of operads:
colored operads and properads. The present article generalizes the Kaledin
classes construction to study formality

  * $\centerdot$

of any algebra encoded by a groupoid colored operad or properad;

  * $\centerdot$

over any commutative ground ring $R$.

We use the resulting Kaledin classes to establish new formality criteria:
formality descent, an intrinsic formality criterium, formality in families,
and formality criteria in terms of chain level lifts of certain homology
automorphisms. On the one hand, this enables us to recover and incorporate
previous results into a single theory. On the other hand, this allows us to
address new formality problems, such as the formality of algebras over
properads and formality results with coefficients in any commutative ring.

Formality as a deformation problem. The Kaledin classes generalization relies
on an operadic approach which reduces the formality to a deformation problem.
In general, the induced structure forgets a part of the homotopical
information: most structures are not formal. Under some hypotheses, for
instance if $R$ is a field, there is another way to transfer an algebraic
structure to the homology without loss of homotopical information. This is the
homotopy transfer theorem, see e.g. [LV12, Section 10.3], [Ber14], [HLV20,
Section 4], [GRiL23, Section 5.5] and references therein. This leads to a
transferred structure on the homology

$\varphi_{t}\coloneqq\varphi_{*}+\varphi^{(2)}+\varphi^{(3)}+\cdots$

made up of the induced structure $\varphi_{*}$ and higher order products. The
algebra is then formal if these higher order products are trivial in a
suitable sense. Thus, formality boils down to the following deformation
problem:

_Is the deformation $\varphi_{t}$ of the induced structure $\varphi_{*}$
trivial?_

The philosophy of deformation theory which goes back to Deligne, Gerstenhaber,
Goldman, Grothendieck, Millson, Schlessinger, Stasheff, and many others, tells
that any deformation problem is encoded by a dg Lie algebra, see [Pri10] and
[Lur11]. This is the case for the formality of algebraic structures. Thanks to
the operadic calculs, one can treat algebraic structures of a certain type
$\mathcal{C}$ on the homology $H(A)$ as Maurer–Cartan elements of a
convolution dg Lie algebra $\mathfrak{g}_{\mathcal{C},H(A)}$ see [LV12,
Section 10] and [MV09a, Proposition 11]. This dg Lie algebra is for instance
the Hochschild complex in the case of associative structures or the Harrison
complex in the case of commutative structures. As a dg Lie algebra, the set of
degree zero elements of the convolution dg Lie algebra forms a group under the
Backer–Campbell–Hausdorff formula, called the gauge group. Its acts on the set
of Maurer–Cartan elements through the gauge action. Two elements in the same
orbit are said gauge equivalent. This leads to the following definition. A
differential graded algebra structure $(A,\varphi)$ admitting a transferred
structure is

  * $\centerdot$

_gauge formal_ if the structures $\varphi_{t}$ and $\varphi_{*}$ are gauge
equivalent;

  * $\centerdot$

_gauge $n$-formal_ for $n\geqslant 0$, if $\varphi_{t}$ is gauge equivalent to
a structure of the form

$\psi=\varphi_{*}+\psi^{(n+2)}+\cdots$

Over a characteristic zero field, an algebra is formal if an only if it is
gauge formal. In the general case, these two notions are related but none of
them imply the other, see Proposition 2.21. Gauge formality generalizes to any
weight-graded dg Lie algebra: a Maurer–Cartan element in a weight-graded dg
Lie algebras is gauge formal if it is gauge equivalent to its weight one
component.

Outline of the paper. In Section 1, we study the gauge formality of any
Maurer–Cartan element $\varphi$ in a weight-graded dg Lie algebra over a
$\mathbb{Q}$-algebra. To so do, we construct a certain homology class, its
_Kaledin class_ $K_{\varphi}$, which vanishes if and only if $\varphi$ is
gauge formal, see Theorem 1.31. Over a $\mathbb{Q}$-algebra, the gauge
formality of algebraic structures can be addressed using Theorem 1.31 in
$\mathfrak{g}_{\mathcal{C},H(A)}$. This particular dg Lie algebra comes
equipped with additional structures: a convolution product and a Lie-
admissible product whose skew-symmetrization defines the Lie bracket. By
reformulating the problem in this setting, it turns out that the
characteristic zero can be dispensed with. In Section 2, after having
generalized operadic calculus results to any coefficient ring, we establish
the following theorem.

Theorem 2.44. Let $R$ be a commutative ring. Let $\mathbb{V}$ be a groupoid
and let $\mathcal{C}$ be a $\mathbb{V}$-colored reduced weight-graded dg
cooperad over $R$. Let $(A,\varphi)$ be a $\Omega\mathcal{C}$-algebra
structure admitting a transferred structure.

  1. (1)

Let $n\geqslant 1$ be an integer such that $n!$ is a unit in $R$. The algebra
$(A,\varphi)$ is gauge $n$-formal if and only if the reduction of Kaledin
class modulo $\hbar^{n}$ is zero.

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra, the algebra $(A,\varphi)$ is gauge formal if
and only if its Kaledin class vanishes.

In Section 3, we establish a properadic version of Theorem 2.44 by defining
Kaledin classes for algebras over properads. Theorem 2.44 and its properadic
version are used in Section 4 to establish new formality criteria and results.
Section 4.1 establishes formality descent results which apply in particular
for colored operads and properads or with torsion coefficients. This allows us
to prove new formality results with torsion coefficients, see e.g. Theorem 4.2
dealing with $\mathbb{Z}_{(p)}$ coefficients. Section 4.2 studies the problem
motivating the introduction of Kaledin classes in the first place: formality
in families. Section 4.3 presents an intrinsic formality criterion, allowing
us to recover standard intrinsic formality results, such as the one at the
heart of Tamarkin’s proof of Kontsevich formality theorem, see [Hin03].

Section 4.4 is dedicated to formality criteria in terms of chain-level lifts
of certain homology automorphisms. Using Kaledin classes, we generalize the
grading automorphism criteria to any algebra encoded by a groupoid colored
operad or a properad. We also address the following question: is the grading
automorphism $\sigma_{\alpha}$, the only homology automorphism satisfying this
property? In Theorem 4.10, we settle conditions on an homology automorphism so
that the existence a chain level lift implies formality. This condition is
satisfied by the Frobenius action in the $\ell$-adic cohomology of any smooth
projective variety thanks to the Weil conjectures and Riemann hypothesis for
finite fields. This leads to the following result.

Theorem 4.17. Let $\mathbb{V}$ be a groupoid and let $\mathcal{P}$ be a
$\mathbb{V}$-colored operad in sets. Let $p$ be a prime number. Let $K$ be a
finite extension of $\mathbb{Q}_{p}$ and let $K\hookrightarrow\mathbb{C}$ be
an embedding. Let $X$ be a $\mathcal{P}$-algebra in the category of smooth and
proper schemes over $K$ of good reduction, i.e. for which there exists a
smooth and proper model $\mathcal{X}$ over the ring of integers
$\mathcal{O}_{K}$. The dg $\mathcal{P}$-algebra of singular chains
$C_{*}(X_{\mathrm{an}},\mathbb{Q})$ is formal.

As an application to this theorem, one recovers the formality of the cyclic
operad of moduli spaces of stable algebraic curves established in [GSNPR05].
This also opens the doors to establishing new formality results for algebraic
structures in smooth and proper schemes, which will be the subject of a
forthcoming article.

Notations and conventions.

  * $\centerdot$

We work over a commutative ground ring $R$. All tensor products are taken over
$R$ and every morphism is $R$-linear unless otherwise specified.

  * $\centerdot$

We adopt a homological point of view and we work in the symmetric monoidal
category of chain complexes over $R$ with the Koszul sign rule.

  * $\centerdot$

If $A$ is a chain complex and $x\in A$ is a homogeneous element, we denote by
$|x|$ its homological degree.

  * $\centerdot$

The abbreviation “dg” stands for the words “differential graded”.

  * $\centerdot$

We use the notations of [LV12] for operads and the ones of [HLV20] for
properads.

Acknowledgments. I would like to thank my advisors, Geoffroy Horel and Bruno
Vallette for their constant support, the many fruitful discussions and their
invaluable guidance. I am grateful to Victor Roca i Lucio for useful
discussions of results of [GRiL23] and to Olivier Benoist and Arnaud Vanhaecke
for enlightening conversations around [SGA].

##  1\. Kaledin classes in differential graded Lie algebras

In this section, we construct Kaledin classes in any weight-graded dg Lie
algebra. This allows us to detect whether a given Maurer–Cartan element is
gauge equivalent to its first component. In all this section, the ground ring
$R$ is a $\mathbb{Q}$-algebra.

###  1.1. Deformation theory with dg Lie algebras

We start by recalling the deformation theory controlled by a dg Lie algebra,
see e.g. [DSV23, Chapter 1] for more details.

######  Definition 1.1 (Differential graded Lie algebra).

A _differential graded (dg) Lie algebra_ $\left(\mathfrak{g},[-,-],d,\right)$
is a graded $R$-module $\mathfrak{g}$ equipped with

  * $\centerdot$

a Lie bracket $[-,-]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$ , i.e.
homogeneous map of degree $0$ , satisfying the Jacoby identity and the
antisymmetry properties;

  * $\centerdot$

a differential $d:\mathfrak{g}\to\mathfrak{g}$ , i.e. a degree $-1$ derivation
that squares to zero.

A dg Lie algebra $\mathfrak{g}$ said _complete_ if it admits a complete
descending filtration $\mathcal{F}$ , i.e. a decreasing chain of sub-complexes

$\mathfrak{g}=\mathcal{F}^{1}\mathfrak{g}\supset\mathcal{F}^{2}\mathfrak{g}\supset\mathcal{F}^{3}\mathfrak{g}\supset\cdots$

compatible with the bracket
$\left[\mathcal{F}^{n}\mathfrak{g},\mathcal{F}^{m}\mathfrak{g}\right]\subset\mathcal{F}^{n+m}\mathfrak{g}$
and such that the canonical projections
$\pi_{n}:\mathfrak{g}\twoheadrightarrow\mathfrak{g}/\mathcal{F}^{n}\mathfrak{g}$
induce an isomorphism

$\pi:\mathfrak{g}\to\lim_{n\in\mathbb{N}}\mathfrak{g}/\mathcal{F}^{n}\mathfrak{g}\
.$

In the sequel, we will use the same notation for the underlying chain complex
$\mathfrak{g}$ and the full data of a complete dg Lie algebra.

######  Example 1.2.

Let $\mathfrak{g}$ be a dg Lie algebra. The set of its derivations forms a dg
Lie algebra denoted $\mathrm{Der}(\mathfrak{g})$. Its grading comes from the
degree of derivations, the bracket is defined by

$\\{\delta,\delta^{\prime}\\}\coloneqq\delta\circ\delta^{\prime}-(-1)^{|\delta||\delta^{\prime}|}\delta^{\prime}\circ\delta\
,$

and the differential is given by $\delta\mapsto\\{d,\delta\\}$. The dg Lie
subalgebra of _inner derivations_
$\mathrm{ad}(\mathfrak{g})\subset\mathrm{Der}(\mathfrak{g})$ is the image of
the dg Lie algebras morphism

$\mathrm{ad}:\mathfrak{g}\to\mathrm{Der}(\mathfrak{g}),\quad
x\mapsto\mathrm{ad}_{x}(-)\coloneqq[x,-]\ .$

######  Definition 1.3 (Maurer–Cartan element).

A _Maurer–Cartan element_ of a dg Lie algebra $\mathfrak{g}$ is a degree $-1$
element $\varphi\in\mathfrak{g}_{-1}$ satisfying the Maurer–Cartan equation:

$d\left(\varphi\right)+\tfrac{1}{2}\left[\varphi,\varphi\right]=0\ .$

The set of Maurer–Cartan elements in $\mathfrak{g}$ is denoted by
$\mathrm{MC}(\mathfrak{g})$ .

######  Proposition 1.4.

Let $\varphi\in\mathrm{MC}(\mathfrak{g})$ be a Maurer–Cartan elements in a dg
Lie algebra $\mathfrak{g}$. The map $d^{\psi}\coloneqq d+\mathrm{ad}_{\psi}$
is a differential and defines a _twisted dg Lie algebra_

$\mathfrak{g}^{\psi}\coloneqq\left(\mathfrak{g},[-,-],d^{\psi}\right)\ .$

###### Proof.

See [DSV23, Proposition 1.43]. ∎

######  Definition 1.5 (Baker–Campbell–Hausdorff formula).

The _Baker–Campbell–Hausdorff_ formula is the element in the associative
algebra of formal power series on two variables $x$ and $y$ given by
$\mathrm{BCH}(x,y)\coloneqq\mathrm{log}(e^{x}e^{y})\ .$

######  Proposition 1.6.

The _gauge group_ of a complete dg Lie algebra $\mathfrak{g}$ is the group
obtained from the set of degree $0$ elements via the Baker–Campbell–Hausdorff
formula :

$\left(\mathfrak{g}_{0},\mathrm{BCH},0\right)\ .$

It acts on the set of Maurer–Cartan elements through the _gauge action_
defined by

$\lambda\cdot\varphi\coloneqq
e^{\mathrm{ad}_{\lambda}}(\varphi)-\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(d\lambda)\
,$

for every Maurer–Cartan element $\varphi\in\mathrm{MC}(\mathfrak{g})$ and
every gauge $\lambda\in\mathfrak{g}_{0}$.

###### Proof.

See [DSV23, Theorem 1.53]. ∎

######  Remark 1.7.

The gauge action’s formula comes from the flow of certain vector fields on the
set of Maurer–Cartan elements $\mathrm{MC}(\mathfrak{g})$ which is a variety
(an intersection of quadrics) in the finite dimensional case, see [DSV23,
Theorem 1.42]. For every $\lambda\in\mathfrak{g}_{0}$, one can consider the
vector field $\Upsilon_{\lambda}$ characterized for all
$\varphi\in\mathrm{MC}(\mathfrak{g})$ by

(1) $\Upsilon_{\lambda}(\varphi)\coloneqq d\lambda+[\varphi,\lambda]\ .$

The integration of the flow associated to $\Upsilon_{-\lambda}$, with the
initial condition $\gamma_{\lambda}(0)=\varphi$, leads to

$\gamma_{\lambda}(t)=e^{t\mathrm{ad}_{\lambda}}(\varphi)-\frac{e^{t\mathrm{ad}_{\lambda}}-\mathrm{id}}{t\mathrm{ad}_{\lambda}}(td\lambda)\in\mathrm{MC}(\mathfrak{g})\
.$

Thus, we have $\lambda\cdot\varphi=\psi$ if and only if the previous
integration gives $\psi$ at time $t=1$ .

######  Definition 1.8 (Gauge equivalence).

Two Maurer–Cartan elements $\varphi,\psi\in\mathrm{MC(\mathfrak{g})}$ in a
complete dg Lie algebra $\mathfrak{g}$ are _gauge equivalent_ if there exists
an element $\lambda\in\mathfrak{g}_{0}$ such that

$\lambda\cdot\varphi=\psi\ .$

We say that $\lambda$ is a _gauge_ between $\varphi$ and $\psi$ . The _moduli
space of Maurer–Cartan elements_ $\mathcal{MC}(\mathfrak{g})$ is the coset of
Maurer–Cartan elements modulo the gauge action.

######  Lemma 1.9.

For every gauge $\lambda\in\mathfrak{g}_{0}$ in a complete dg Lie algebra
$\mathfrak{g}$, the morphism

$e^{\mathrm{ad}_{\lambda}}:\mathfrak{g}^{\varphi}\rightarrow\
\mathfrak{g}^{\lambda\cdot\varphi},\quad x\mapsto
e^{\mathrm{ad}_{\lambda}}(x)\ ,$

is an isomorphism of dg Lie algebras whose inverse is given by
$e^{\mathrm{ad}_{-\lambda}}$.

###### Proof.

The morphism $e^{\mathrm{ad}_{\lambda}}$ preserves the Lie bracket and is a
morphism of Lie algebras as the exponential of a derivation. Furthermore, for
all $x\in\mathfrak{g}$ , one has

$\begin{split}d^{\lambda\cdot\varphi}\left(e^{\mathrm{ad}_{\lambda}}(x)\right)&=\sum_{n\geqslant
0}\frac{1}{n!}d\left(\mathrm{ad}_{\lambda}^{n}(x)\right)-\sum_{k,l\geqslant
0}\frac{1}{l!(k+1)!}\left[\mathrm{ad}_{\lambda}^{k}(d\lambda),\mathrm{ad}_{\lambda}^{l}(x)\right]+e^{\mathrm{ad}_{\lambda}}(\left[\varphi,x\right]).\end{split}$

Since $d$ is a derivation, one can prove by induction on $n\geqslant 1$ , that

(2)
$d\left(\mathrm{ad}_{\lambda}^{n}(x)\right)=\mathrm{ad}^{n}_{\lambda}(dx)+\sum_{k+l=n-1}\frac{n!}{l!(k+1)!}\left[\mathrm{ad}_{\lambda}^{k}(d\lambda),\mathrm{ad}_{\lambda}^{l}(x)\right]$

for all $x\in\mathfrak{g}$ . This leads to

$d^{\lambda\cdot\varphi}\left(e^{\mathrm{ad}_{\lambda}}(x)\right)=\sum_{k\geqslant
0}\frac{1}{k!}\mathrm{ad}_{\lambda}^{k}(d(x))+e^{\mathrm{ad}_{\lambda}}\left(\left[\varphi,x\right]\right)=e^{\mathrm{ad}_{\lambda}}\left(d^{\varphi}(x)\right)\
,$

and $e^{\mathrm{ad}_{\lambda}}$ is an isomorphism of dg Lie algebras whose
inverse is given by $e^{\mathrm{ad}_{-\lambda}}$ . ∎

Let $R[\\![\hbar]\\!]$ be the ring of formal power series in the variable
$\hbar$ and let $\mathfrak{g}$ be any dg Lie algebra. We consider the complete
dg Lie algebra of formal power series

$\mathfrak{g}[\\![\hbar]\\!]:=\mathfrak{g}\ \widehat{\otimes}\
R[\\![\hbar]\\!]\ .$

We write any element $x\in\mathfrak{g}[\\![\hbar]\\!]$ as $x=\sum_{i\geqslant
0}x_{i}\hbar^{i}\ .$

######  Definition 1.10 (Formal deformations).

Let $\varphi_{0}$ a Maurer–Cartan element in $\mathfrak{g}$ . A _formal
deformation of $\varphi_{0}$_ is a Maurer–Cartan element

$\Phi=\varphi_{0}+\varphi_{1}\hbar+\varphi_{2}\hbar^{2}+\cdots\in\mathrm{MC}(\mathfrak{g}[\\![\hbar]\\!])$

which reduces to $\varphi_{0}$ modulo the maximal ideal $(\hbar)$ . We denote
by $\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$ the set of all formal
deformations of $\varphi_{0}$ . Two formal deformations $\Phi$ and $\Psi$ are
_gauge equivalent_ , if there exists a gauge
$\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ such that

$\Psi=\lambda\cdot\Phi\coloneqq
e^{\mathrm{ad}_{\lambda}}(\Phi)-\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(d\lambda)\
.$

######  Example 1.11.

Every $\varphi_{0}\in\mathrm{MC}(\mathfrak{g})$ admits a trivial formal
deformation $\varphi_{0}+0\in\mathrm{MC}(\mathfrak{g}[\\![\hbar]\\!])$ .

###  1.2. Kaledin classes for formal deformations

Let $\varphi_{0}$ be a Maurer–Cartan element in a dg Lie algebra
$\mathfrak{g}$ and let $\Phi$ be a formal deformation of $\varphi_{0}$. Is
$\Phi$ gauge equivalent to the trivial deformation? To answer this question,
we generalize the construction of [Lun07, Section 7], which addresses the
particular case $\varphi_{0}=0$ .

######  Definition 1.12 (Differentiation operator).

For any graded $R$-module $V$ , the _differentiation operator_
$\partial_{\hbar}:V[\\![\hbar]\\!]\longrightarrow V[\\![\hbar]\\!]$ is defined
by

$\sum_{i\geqslant 0}x_{i}\hbar^{i}\longmapsto\sum_{i\geqslant
1}ix_{i}\hbar^{i-1}\ .$

######  Remark 1.13.

Let $\mathfrak{g}$ be a dg Lie algebra,

  * $\centerdot$

the operator
$\partial_{\hbar}:\mathfrak{g}[\\![\hbar]\\!]\to\mathfrak{g}[\\![\hbar]\\!]$
is a derivation of degree zero,

  * $\centerdot$

the underlying differential $d$ of $\mathfrak{g}$ , viewed in
$\mathrm{Der}(\mathfrak{g})[\\![\hbar]\\!]$ , satisfies
$\partial_{\hbar}(d)=0$ .

######  Proposition 1.14.

Any formal deformation $\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$
satisfies

$\partial_{\hbar}\Phi\in
Z_{-1}\left(\mathfrak{g}[\\![\hbar]\\!]^{\Phi}\right)\ ,$

i.e. the image of $\Phi$ under the operator $\partial_{\hbar}$ is a cycle in
the twisted dg Lie algebra $\mathfrak{g}[\\![\hbar]\\!]^{\Phi}$ .

###### Proof.

Let us decompose $\Phi$ as $\varphi_{0}+\tilde{\Phi}$ where
$\tilde{\Phi}\in\mathrm{MC}(\mathfrak{g}[\\![\hbar]\\!]^{\varphi_{0}})$. The
result now follows from the following computation:

$\begin{split}d^{\Phi}\left(\partial_{\hbar}\Phi\right)&=d\left(\partial_{\hbar}\tilde{\Phi}\right)+\left[\varphi_{0}+\tilde{\Phi},\partial_{\hbar}\tilde{\Phi}\right]\\\
&=\partial_{\hbar}\left(d\tilde{\Phi}\right)+\left[\varphi_{0},\partial_{\hbar}\tilde{\Phi}\right]+\left[\tilde{\Phi},\partial_{\hbar}\tilde{\Phi}\right]\\\
&=\partial_{\hbar}\left(d\tilde{\Phi}\right)+\partial_{\hbar}\left[\varphi_{0},\tilde{\Phi}\right]+\tfrac{1}{2}\partial_{\hbar}\left[\tilde{\Phi},\tilde{\Phi}\right]\\\
&=\partial_{\hbar}\left(d\tilde{\Phi}+\left[\varphi_{0},\tilde{\Phi}\right]+\tfrac{1}{2}\left[\tilde{\Phi},\tilde{\Phi}\right]\right)\\\
&=0\ .\qed\end{split}$

######  Definition 1.15 (Kaledin class).

The _Kaledin class_ of a formal deformation
$\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$ is the homology class

$K_{\Phi}\coloneqq\left[\partial_{\hbar}\Phi\right]\in
H_{-1}\left(\mathfrak{g}[\\![\hbar]\\!]^{\Phi}\right)\ .$

######  Example 1.16.

The Kaledin class of the trivial deformation is trivial, i.e.
$K_{\varphi_{0}}=0$ .

######  Theorem 1.17.

Let $\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$ be a formal
deformation. Its Kaledin class $K_{\Phi}$ is zero if and only if $\Phi$ is
gauge equivalent to the trivial deformation $\varphi_{0}$ , i.e. there exists
a gauge $\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ such that

$\lambda\cdot\Phi=\varphi_{0}\in\mathfrak{g}[\\![\hbar]\\!]\ .$

######  Remark 1.18.

The heuristic behind this obstruction class is the following one. The Kaledin
class $K_{\Phi}$ is zero if and only if $\partial_{\hbar}\Phi$ is a boundary,
i.e. there exits $\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ such that

$\partial_{\hbar}\Phi=d^{\Phi}(\lambda)=d\lambda+[\Phi,\lambda]\ .$

One recognizes the vector field (1) characterizing the gauge action.

The proof of Theorem 1.17 requires the following lemma on the invariance of
the Kaledin class under the gauge group action.

######  Lemma 1.19.

Every $\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ induces an isomorphism of dg
Lie algebras,

$e^{\mathrm{ad}_{\lambda}}:\mathfrak{g}[\\![\hbar]\\!]^{\Phi}\longrightarrow\
\mathfrak{g}[\\![\hbar]\\!]^{\lambda\cdot\Phi},\quad x\mapsto
e^{\mathrm{ad}_{\lambda}}(x)\ .$

Furthermore, the classes $K_{\lambda\cdot\Phi}$ and
$e^{\mathrm{ad}_{\lambda}}\left(K_{\Phi}\right)$ are equal in

$H_{-1}\left(\mathfrak{g}[\\![\hbar]\\!]^{\lambda\cdot\Phi}\right)\ ,$

where we still denote by $e^{\mathrm{ad}_{\lambda}}$ the induced isomorphism
in homology.

###### Proof.

By the proof of Lemma 1.9, the exponential

$e^{\mathrm{ad}_{\lambda}}:\mathfrak{g}[\\![\hbar]\\!]^{\Phi}\rightarrow\
\mathfrak{g}[\\![\hbar]\\!]^{\lambda\cdot\Phi}$

is an isomorphism of dg Lie algebras by the proof of which holds _mutatis
mutandis_ by $\hbar$-linearity. Let us prove that the cycles
$e^{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\Phi)$ and
$\partial_{\hbar}(\lambda\cdot\Phi)$ are homologous. We have

$\partial_{\hbar}(\lambda\cdot\Phi)=\partial_{\hbar}\left(e^{\mathrm{ad}_{\lambda}}(\Phi)-\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(d\lambda)\right)\
.$

The operator $\partial_{\hbar}$ is a derivation and it satisfies an equation
similar to Equation (2). In other words, for all $n\geqslant 1$ and all
$x\in\mathfrak{g}[\\![\hbar]\\!]$ , one can prove that

$\partial_{\hbar}\left(\mathrm{ad}_{\lambda}^{n}(x)\right)=\mathrm{ad}^{n}_{\lambda}\left(\partial_{\hbar}x\right)+\sum_{k+l=n-1}\frac{n!}{l!(k+1)!}\left[\mathrm{ad}_{\lambda}^{k}(\partial_{\hbar}\lambda),\mathrm{ad}_{\lambda}^{l}(x)\right].$

Using twice this identity, one gets

$\begin{split}\partial_{\hbar}\left(e^{\mathrm{ad}_{\lambda}}(\Phi)\right)-e^{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\Phi)&=-\left[e^{\mathrm{ad}_{\lambda}}(\Phi),\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\lambda)\right]\
,\ \mathrm{and}\end{split}$
$\begin{split}\partial_{\hbar}\left(\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(d\lambda)\right)&=\sum_{n\geqslant
0}\frac{1}{(n+1)!}\mathrm{ad}^{n}_{\lambda}(\partial_{\hbar}(d\lambda))+\sum_{\begin{subarray}{c}n\geqslant
0\\\
k+l=n\end{subarray}}\frac{1}{n(l!)(k+1)!}\left[\mathrm{ad}_{\lambda}^{k}(\partial_{\hbar}\lambda),\mathrm{ad}_{\lambda}^{l}(d\lambda)\right]\end{split}\
.$

In the latter equation, the element
$\mathrm{ad}^{n}_{\lambda}(\partial_{\hbar}(d\lambda))=\mathrm{ad}^{n}_{\lambda}(d(\partial_{\hbar}\lambda))$
can be decomposed using the equation (2). This leads to

$\begin{split}\partial_{\hbar}\left(\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(d\lambda)\right)&=d\left(\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\lambda)\right)-\left[\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(d\lambda),\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\lambda)\right]\
.\end{split}$

In conclusion, the following equality holds

$\begin{split}\partial_{\hbar}(\lambda\cdot\Phi)-e^{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\Phi)&=d^{\lambda\cdot\Phi}\left(-\frac{e^{\mathrm{ad}_{\lambda}}-\mathrm{id}}{\mathrm{ad}_{\lambda}}(\partial_{\hbar}\lambda)\right)\
,\end{split}$

and the classes $K_{\lambda\cdot\Phi}$ and
$e^{\mathrm{ad}_{\lambda}}\left(K_{\Phi}\right)$ are equal in homology. ∎

All the above definitions and properties can be adapted over the truncated
ring $R[\\![\hbar]\\!]/(\hbar^{n})$ , for all $n\geqslant 1$ . This leads to
the following definition of the $n^{\text{th}}$-truncated Kaledin class.

######  Definition 1.20 ($n^{\text{th}}$-truncated Kaledin class).

For all $n\geqslant 1$ , the _$n^{\text{th}}$ -truncated Kaledin class_ of the
formal deformation $\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$ is
the homology class

$K_{\Phi}^{n}\coloneqq\left[\varphi_{1}+2\varphi_{2}\hbar+\cdots+n\varphi_{n}\hbar^{n-1}\right]\in
H_{-1}\left(\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\right)\
,$

where $\overline{\Phi}^{n}$ denotes the projection in
$\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\ .$

######  Remark 1.21.

Let $\Phi$ be a formal deformation. For all $n\geqslant 1$ , the projection

$\mathfrak{g}[\\![\hbar]\\!]^{\Phi}\twoheadrightarrow\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}$

is a morphism of dg Lie algebras. In particular, if the Kaledin class
$K_{\Phi}$ is zero, then all the truncated Kaledin classes $K^{n}_{\Phi}$ are
zero. Similarly, if the $n^{\text{th}}$-truncated Kaledin class $K^{n}_{\Phi}$
is zero then, for all $m\leqslant n$ , the class $K^{m}_{\Phi}$ is zero.

######  Lemma 1.22.

Every $\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ induces an isomorphism of dg
Lie algebras,

$e^{\mathrm{ad}_{\lambda}}:\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\longrightarrow\
\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\lambda\cdot\Phi}^{n}}\
.$

The classes $K_{\lambda\cdot\Phi}^{n}$ and
$e^{\mathrm{ad}_{\lambda}}\left(K_{\Phi}^{n}\right)$ are equal in

$H_{-1}\left(\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\lambda\cdot\Phi}^{n}}\right)\
,$

where we still denote by $e^{\mathrm{ad}_{\lambda}}$ the induced isomorphism
on homology.

###### Proof.

The proof is similar to the one of Lemma 1.19. ∎

######  Theorem 1.23.

Let $\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$ be a formal
deformation. For all $n\geqslant 1$, the following propositions are
equivalent.

  1. (1)

The $n^{\text{th}}$-truncated Kaledin class $K_{\Phi}^{n}$ is zero.

  2. (2)

There exists $\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ such that
$\lambda\cdot\Phi\equiv\varphi_{0}\pmod{\hbar^{n+1}}\ .$

###### Proof.

If Point (2) holds, Lemma 1.22 implies that

$K_{\Phi}^{n}=e^{\mathrm{ad}_{-\lambda}}\left(K_{\lambda\cdot\Phi}^{n}\right)=e^{\mathrm{ad}_{-\lambda}}\left(K_{\varphi_{0}}^{n}\right)=0\
.$

Let us prove the converse result by induction on $n$. If the first Kaledin
class

$K^{1}_{\Phi}=\left[\varphi_{1}\right]\in
H_{-1}\left(\mathfrak{g}^{\varphi_{0}}\right)$

is zero, there exists $\phi_{0}\in\mathfrak{g}_{0}$ such that
$d^{\varphi_{0}}(\phi_{0})=\varphi_{1}\ .$ The element
$\lambda\coloneqq\phi_{0}h$ satisfies

$\lambda\cdot\Phi\equiv\varphi_{0}+\varphi_{1}h-d^{\varphi_{0}}(\phi_{0})h\equiv\varphi_{0}\pmod{\hbar^{2}}\
.$

Suppose that $n>1$ and that the result holds for $n-1$. If the class
$K_{\Phi}^{n}$ is zero, so does $K_{\Phi}^{n-1}$. By the induction hypothesis,
there exists $\nu\in\mathfrak{g}_{0}\otimes(\hbar)$ such that

$\nu\cdot\Phi\equiv\varphi_{0}\pmod{\hbar^{n}}\ .$

Considering $\Psi\coloneqq\nu\cdot\Phi\ ,$ Lemma 1.22 implies that
$e^{\mathrm{ad}_{\nu}}(K_{\Phi}^{n})=K_{\Psi}^{n}=0$ . By construction

$\Psi\equiv\varphi_{0}+\psi_{n}\hbar^{n}\pmod{\hbar^{n+1}}\quad\mbox{and}\quad
K_{\Psi}^{n}=\left[n\psi_{n}\hbar^{n-1}\right]=0\ .$

Thus, there exists
$\phi\coloneqq\phi_{0}+\phi_{1}\hbar+\cdots+\phi_{n-1}\hbar^{n-1}\in\mathfrak{g}[\\![\hbar]\\!]$
of degree zero such that

$d^{\varphi_{0}}(\phi)\equiv n\psi_{n}\hbar^{n-1}\pmod{\hbar^{n}}\ .$

Looking at the coefficient of $\hbar^{n-1}$ on both sides, the following
equality holds

$n\psi_{n}=d(\phi_{n-1})+[\varphi_{0},\phi_{n-1}]\ .$

The element
$\upsilon\coloneqq\frac{1}{n}\phi_{n-1}\hbar^{n}\in\mathfrak{g}_{0}\otimes(\hbar)$
satisfies

$\upsilon\cdot\Psi\equiv\varphi_{0}\pmod{\hbar^{n+1}}\ .$

Finally, the gauge $\lambda\coloneqq\mathrm{BCH}(\upsilon,\nu)$ is such that
$\lambda\cdot\Phi=\upsilon\cdot(\nu\cdot\Phi)\equiv\varphi_{0}\pmod{\hbar^{n+1}}\
.$ ∎

###### Proof of Theorem 1.17.

If $\Phi$ is gauge equivalent to $\varphi_{0}$, Lemma 1.19 implies that

$K_{\Phi}=K_{\varphi_{0}}=0\ .$

Conversely, suppose that $K_{\Phi}=0$ . If $\Phi=\varphi_{0}$ , the result is
immediate. Otherwise, let $n_{1}\geqslant 1$ be the smallest integer such that
$\varphi_{n_{1}}$ is not equal to zero. Since $K_{\Phi}$ is zero, so does the
class $K_{\Phi}^{n_{1}}$. By Theorem 1.23, there exists
$\lambda^{1}\in\mathfrak{g}_{0}\otimes(\hbar)$ such that

$\lambda^{1}\cdot\Phi\equiv\varphi_{0}\pmod{\hbar^{n_{1}+1}}\ .$

The proof of Theorem 1.23 actually shows that one can find a gauge of the form
$\lambda^{1}=x_{n_{1}}\hbar^{n_{1}}$ . Let
$\Phi^{2}\coloneqq\lambda^{1}\cdot\Phi$ and let $n_{2}$ be the smallest
integer ${n_{2}}>{n_{1}}$ such that $(\Phi^{2})_{n_{2}}$ is not zero. By Lemma
1.22, we have

$K_{\Phi^{2}}^{n_{2}}=e^{\mathrm{ad}_{\lambda^{1}}}(K^{n_{2}}_{\Phi})=0\ .$

Again, there exists a gauge of the form $\lambda^{2}=x_{n_{2}}\hbar^{n_{2}}$
such that

$\lambda^{2}\cdot\Phi^{2}\equiv\varphi_{0}\pmod{\hbar^{n_{2}+1}}\ .$

By repeating the procedure, we obtain a increasing sequence

$n_{1}<n_{2}<n_{3}<\cdots$

of integers and a sequence of gauges of the form
$\lambda^{i}=x_{n_{i}}\hbar^{n_{i}}$ , for all $i$ , such that

$\lambda^{i}\cdot\Phi^{i}\equiv\varphi_{0}\pmod{\hbar^{n_{i}+1}}$

where $\Phi^{i}\coloneqq\lambda^{{i-1}}\cdot\Phi^{i-1}$ for all $n\geqslant
2$. It follows from the construction that

$\lambda=\mathrm{BCH}(\cdots\mathrm{BCH}(\lambda^{3},\mathrm{BCH}(\lambda^{2},\lambda^{1}))\cdots)\
,$

is well defined, since each gauge $\lambda^{i}$ only has components in
$\hbar^{n_{i}}$ , and is a desired gauge. ∎

######  Proposition 1.24.

Let $\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$ be a formal
deformation. The Kaledin class $K_{\Phi}$ is zero if and only if, for all
$n\geqslant 1$, the truncation $K^{n}_{\Phi}$ is zero.

###### Proof.

If Kaledin class $K_{\Phi}$ is zero then all the truncations $K^{n}_{\Phi}$
are zero by Remark 1.21. Conversely, suppose that for all $n\geqslant 1$ , the
truncation $K^{n}_{\Phi}$ is zero. If $\Phi=\varphi_{0}$ , the result is
immediate. Otherwise, let $n_{1}$ be the smallest integer such that
$\varphi_{n_{1}}\neq 0$ . As in the proof of Theorem 1.23, we can construct a
sequence of integers $n_{i}$ and a sequence of gauges
$\lambda^{i}=x_{n_{i}}\hbar^{n_{i}}$ for all $i$ , such that

$\lambda^{i}\cdot\Phi^{i}\equiv\varphi_{0}\pmod{\hbar^{n_{i}+1}}$

where $\Phi^{i}\coloneqq\lambda^{{i-1}}\cdot\Phi^{i-1}$ for $i\geqslant 2$ and
$\Phi^{1}\coloneqq\Phi$. The infinite composition

$\lambda=\mathrm{BCH}(\cdots\mathrm{BCH}(\lambda^{3},\mathrm{BCH}(\lambda^{2},\lambda^{1}))\cdots)\
,$

is then such that $\lambda\cdot\Phi=\varphi_{0}\in\mathfrak{g}[\\![\hbar]\\!]$
and $K_{\Phi}=0$ by Theorem 1.17. ∎

######  Theorem 1.25 (Descent).

Let $S$ be a faithfully flat commutative $R$-algebra. Let
$\varphi_{0}\in\mathrm{MC}(\mathfrak{g})$ be a Maurer–Cartan element. A formal
deformation $\Phi$ of $\varphi_{0}$ is gauge equivalent to $\varphi_{0}$ if
and only if $\Phi\otimes 1$ is gauge equivalent to the trivial deformation
$\varphi_{0}\otimes 1$ in $\mathfrak{g}[\\![\hbar]\\!]\otimes S$.

###### Proof.

If there exists $\lambda\in\mathfrak{g}_{0}\otimes(\hbar)$ satisfying
$\lambda\cdot\Phi=\varphi_{0}$ , then

$(\lambda\otimes 1)\cdot(\Phi\otimes 1)=\varphi_{0}\otimes 1\ .$

Conversely, suppose that $\Phi\otimes 1$ is gauge equivalent to the trivial
deformation $\varphi_{0}\otimes 1$ . For all $n\geqslant 1$ , Theorem 1.23
implies that the Kaledin classes $K^{n}_{\Phi\otimes 1}$ are zero. The
isomorphism

$(\mathfrak{g}\otimes
S)[\\![\hbar]\\!]/(\hbar^{n})\cong\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\otimes_{R[\\![\hbar]\\!]/(\hbar^{n})}S[\\![\hbar]\\!]/(\hbar^{n})$

leads to an isomorphism

$H_{-1}\left(\left((\mathfrak{g}\otimes
S)[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}\otimes
1}\right)\cong
H_{-1}\left(\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\right)\otimes_{R[\\![\hbar]\\!]}S[\\![\hbar]\\!]/(\hbar^{n})\
,$

since $R[\\![\hbar]\\!]/(\hbar^{n})\to S[\\![\hbar]\\!]/(\hbar^{n})$ is flat.
By faithful flatness, the class $K^{n}_{\Phi\otimes 1}$ is zero on the left
hand-side if and only if $K_{\Phi}^{n}\otimes 1$ is zero on the right hand-
side. Thus, the Kaledin class

$K_{\Phi}^{n}\in
H_{-1}\left(\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\right)$

is zero for all $n$ . Proposition 1.24 and Theorem 1.23 allow us to conclude.
∎

The following result is well-known, see e.g. [DSV23, Theorem 1.66], we give it
a new proof using the Kaledin classes.

######  Theorem 1.26 (Rigidity).

Let $\mathfrak{g}$ be a dg Lie algebra and let
$\varphi_{0}\in\mathrm{MC}(\mathfrak{g})$ . If
$H_{-1}(\mathfrak{g}^{\varphi_{0}})=0$ , then $\varphi_{0}$ is rigid, i.e.
every formal deformation $\Phi\in\mathrm{Def}_{\varphi_{0}}(R[\\![\hbar]\\!])$
is gauge equivalent to $\varphi_{0}$ .

###### Proof.

We prove by induction on $n$ that all the truncated classes $K_{\Phi}^{n}$ are
zero. The class

$K^{1}_{\Phi}\in H_{-1}\left(\mathfrak{g}^{\varphi_{0}}\right)$

is necessarily zero. Suppose that $K^{n-1}_{\Phi}=0$ for $n>1$. There exists
$\nu\in\mathfrak{g}_{0}\otimes(\hbar)$ such that

$\nu\cdot\Phi\equiv\varphi_{0}\pmod{\hbar^{n}}\ ,$

by Theorem 1.23. The element $\Psi\coloneqq\nu\cdot\Phi$ satisfies
$d^{\overline{\Psi}^{n}}=d^{\varphi_{0}}$ . By assumption, the group

$H_{-1}\left(\left(\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Psi}^{n}}\right)\cong
H_{-1}\left(\mathfrak{g}^{\varphi_{0}}\right)[\\![\hbar]\\!]/(\hbar^{n})$

is trivial. Thus the class $K^{n}_{\Psi}$ is zero and by Lemma 1.22

$K^{n}_{\Phi}=e^{\mathrm{ad}_{-\nu}}(K^{n}_{\Psi})=0\ \leavevmode\nobreak\ .$

Using Proposition 1.24 and Theorem 1.17, this implies that $\Phi$ is
equivalent to $\varphi_{0}$ . ∎

###  1.3. The case of weight-graded dg Lie algebras

This section detects whether a given Maurer–Cartan element is gauge equivalent
to its first component in a weight-graded dg Lie algebra. In this setting, the
Kaledin classes of Section 1.2 and [Lun07, Section 7] do not directly apply,
see Remark 1.32. Hence, we refine the previous constructions to deal with any
$\delta$-weight-graded dg Lie algebra, for $\delta\geqslant 1$ an integer.

######  Definition 1.27 ($\delta$-weight-graded dg Lie algebra).

A complete dg Lie algebra $\left(\mathfrak{g},[-,-],d\right)$ is a _$\delta$
-weight-graded dg Lie algebra_ if it has an additional weight grading such
that

$\mathfrak{g}\cong\prod_{k\geqslant 1}\mathfrak{g}^{(k)},\quad
d\left(\mathfrak{g}^{(k)}\right)\subset\mathfrak{g}^{(k+\delta)},\quad\left[\mathfrak{g}^{(k)},\mathfrak{g}^{(l)}\right]\subset\mathfrak{g}^{(k+l)}\
.$

Every element $\varphi$ in $\mathfrak{g}$ decomposes as
$\varphi=\varphi^{(1)}+\varphi^{(2)}+\varphi^{(3)}+\cdots$ , where
$\varphi^{(k)}\in\mathfrak{g}^{(k)}$ , for all $k\geqslant 1$ . It is complete
for the descending filtration defined by

$\mathcal{F}^{n}\mathfrak{g}\coloneqq\prod_{k\geqslant n}\mathfrak{g}^{(k)}\
.$

######  Definition 1.28 (Prismatic decomposition).

Let $\mathfrak{g}$ be a $\delta$-weight-graded dg Lie algebra. The _prismatic
decomposition_ is the morphism of graded vector spaces

$\mathfrak{D}:\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\to\mathfrak{g}[\\![\hbar]\\!]\
,$

defined for all $\varphi\in\mathfrak{g}_{-1}$ and all
$\lambda\in\mathfrak{g}_{0}$ , by

$\begin{split}&\mathfrak{D}(\varphi)\coloneqq\varphi^{(1)}+\cdots+\varphi^{(\delta)}+\varphi^{(\delta+1)}\hbar+\varphi^{(\delta+2)}\hbar^{2}+\cdots\\\
&\mathfrak{D}(\lambda)\coloneqq\lambda^{(1)}\hbar+\lambda^{(2)}\hbar^{2}+\lambda^{(3)}\hbar^{3}+\cdots\
.\end{split}$

######  Lemma 1.29.

Let $\mathfrak{g}$ be a $\delta$-weight-graded dg Lie algebra, let
$\varphi\in\mathrm{MC}\left(\mathcal{F}^{\delta}\mathfrak{g}\right)$ and let
$\lambda\in\mathfrak{g}_{0}$.

  1. (1)

The prismatic decomposition $\mathfrak{D}(\varphi)$ is a formal deformation of
$\varphi^{(\delta)}$, i.e.

$\mathfrak{D}(\varphi)=\varphi^{(\delta)}+\varphi^{(\delta+1)}\hbar+\varphi^{(\delta+2)}\hbar^{2}+\cdots\in\mathrm{Def}_{\varphi^{(\delta)}}(R[\\![\hbar]\\!])\
.$

  2. (2)

The prismatic decomposition of $\lambda\cdot\varphi$ is given by the action of
$\mathfrak{D}(\lambda)$ on $\mathfrak{D}(\varphi)$ in
$\mathfrak{g}[\\![\hbar]\\!]$:

$\mathfrak{D}(\lambda\cdot\varphi)=\mathfrak{D}(\lambda)\cdot\mathfrak{D}(\varphi)\
.$

###### Proof.

Since $\varphi\in\mathrm{MC}(\mathfrak{g})$ is a Maurer–Cartan element, the
component $\varphi^{(\delta)}$ is also a Maurer–Cartan element. Point (1)
follows from the fact that the coefficient of $\hbar^{n}$ in the equation

$d\left(\mathfrak{D}(\varphi)\right)+\tfrac{1}{2}\left[\mathfrak{D}(\varphi),\mathfrak{D}(\varphi)\right]$

is equal to the component of weight $n+2\delta$ of
$d(\varphi)+\tfrac{1}{2}\left[\varphi,\varphi\right]$. For Point (2), one can
prove that the coefficient of $\hbar^{n}$ in

$\mathfrak{D}(\lambda)\cdot\mathfrak{D}(\varphi)$

coincides with the component of weight $n+\delta$ of $\lambda\cdot\varphi$. ∎

######  Definition 1.30 (Kaledin classes).

Let $\mathfrak{g}$ be a $\delta$-weight-graded dg Lie algebra and let

$\varphi\in\mathrm{MC}\left(\mathcal{F}^{\delta}\mathfrak{g}\right)$

be a Maurer–Cartan element.

  * $\centerdot$

The _Kaledin class_ $K_{\varphi}$ is the Kaledin class
$K_{\mathfrak{D}(\varphi)}$ of its prismatic decomposition.

  * $\centerdot$

Its _$n^{\text{th}}$ -truncated Kaledin class_ $K_{\varphi}^{n}$ is the
$n^{\text{th}}$-truncated Kaledin class $K_{\mathfrak{D}(\varphi)}^{n}$ of its
prismatic decomposition.

######  Theorem 1.31.

Let $R$ be a $\mathbb{Q}$-algebra and let $\delta\geqslant 1$ be an integer.
Let $\varphi\in\mathrm{MC}\left(\mathcal{F}^{\delta}\mathfrak{g}\right)$ be a
Maurer–Cartan element in a $\delta$-weight-graded dg Lie algebra
$\mathfrak{g}$ over $R$ .

  1. (1)

The Kaledin class $K_{\varphi}$ is zero if and only if there exists
$\lambda\in\mathfrak{g}_{0}$ such that
$\lambda\cdot\varphi=\varphi^{(\delta)}$ .

  2. (2)

Let $\psi\in\mathrm{MC}(\mathfrak{g})$ be a Maurer–Cartan element concentrated
in weight $\delta$. If

$H_{-1}\left(\mathfrak{g}^{\psi}\right)=0\ ,$

then $\psi$ is _rigid_ , i.e. every Maurer–Cartan element
$\varphi\in\mathrm{MC}\left(\mathcal{F}^{\delta}\mathfrak{g}\right)$ such that
$\varphi^{(\delta)}=\psi$ , is gauge equivalent to $\psi$ .

  3. (3)

Let $S$ be a faithfully flat commutative $R$-algebra. The element $\varphi$ is
gauge equivalent to $\varphi^{(\delta)}$ in $\mathfrak{g}$ if and only if
$\varphi\otimes 1$ is gauge equivalent to $\varphi^{(\delta)}\otimes 1$ in
$\mathfrak{g}\otimes S$ .

######  Remark 1.32.

If the Kaledin class $K_{\varphi}$ is zero, then there exists
$\lambda\in\mathfrak{g}_{0}\otimes(\hbar)\ ,$ such that
$\lambda\cdot\mathcal{D}(\varphi)=\varphi^{(\delta)}$ by Theorem 1.17. This is
not sufficient _a priori_ to get a gauge $\nu\in\mathfrak{g}_{0}$ such that
$\nu\cdot\varphi=\varphi^{(\delta)}$. For all $k\geqslant 1$ , each
$\lambda_{i}$ may have a non-trivial component of weight $k$ and nothing
asserts that the sum

$\lambda_{1}^{(k)}+\lambda_{2}^{(k)}+\lambda_{3}^{(k)}+\cdots$

is well-defined. Therefore, we need to refine the proof of the Theorem 1.17.

######  Proposition 1.33.

Let $\varphi\in\mathrm{MC}\left(\mathcal{F}^{\delta}\mathfrak{g}\right)$ be a
Maurer–Cartan element in a $\delta$-weight-graded dg Lie algebra
$\mathfrak{g}$ over $R$ . For all $n\geqslant 1$ , the following propositions
are equivalent.

  1. (1)

The $n^{\text{th}}$-truncated Kaledin class $K_{\varphi}^{n}$ is zero.

  2. (2)

There exists $\lambda\in\mathfrak{g}_{0}$ such that
$(\lambda\cdot\varphi)^{(k)}=0$ , for all $\ \delta+1\leqslant
k\leqslant\delta+n$ .

###### Proof.

If Point (2) holds, Lemma 1.29 implies that

$\mathfrak{D}(\lambda)\cdot\mathfrak{D}(\varphi)\equiv\varphi^{(\delta)}\pmod{\hbar^{n+1}}\
,$

and $K_{\varphi}^{n}=0$ by Theorem 1.23. Let us prove the converse result by
induction on $n\geqslant 1$ . If

$K^{1}_{\varphi}=\left[\varphi^{(\delta+1)}\right]\in
H_{-1}\left(\mathfrak{g}^{\varphi^{(\delta)}}\right)$

is zero, there exists $\phi\in\mathfrak{g}_{0}$ such that

$d^{\varphi^{(\delta)}}(\phi)=\varphi^{(\delta+1)}\ .$

Then, the element $\lambda\coloneqq\phi^{(1)}\in\mathfrak{g}_{0}$ is such that
$(\lambda\cdot\varphi)^{(\delta+1)}=0$ . Suppose that $n>1$ and that the
result holds for $n-1$ . If the class $K_{\varphi}^{n}$ is zero, so does the
class $K_{\varphi}^{n-1}$. By the induction hypothesis, there exists
$\nu\in\mathfrak{g}_{0}$ such that

$(\nu\cdot\varphi)^{(k)}=0\leavevmode\nobreak\ ,\quad\mbox{for all }\
\delta+1\leqslant k\leqslant\delta+n-1\leavevmode\nobreak\ .$

Let us denote $\psi\coloneqq\nu\cdot\varphi$. We have

$\mathfrak{D}\left(\psi\right)\equiv\varphi^{(\delta)}+\psi^{(\delta+n)}\hbar^{n}\pmod{\hbar^{n+1}}\quad\mbox{and}\quad
K_{\psi}^{n}=\left[n\psi^{(\delta+n)}\hbar^{n-1}\right]\ .$

Since
$\mathfrak{D}\left(\psi\right)=\mathfrak{D}\left(\nu\right)\cdot\mathfrak{D}\left(\varphi\right)$
by Lemma 1.29, we have

$K_{\psi}^{n}=e^{\mathrm{ad}_{\mathfrak{D}\left(\nu\right)}}(K_{\varphi}^{n})=0\
\leavevmode\nobreak\ ,$

by Lemma 1.22. Thus, there exists
$\phi\coloneqq\phi_{0}+\phi_{1}h+\dots+\phi_{n-1}\hbar^{n-1}\in\mathfrak{g}[\\![\hbar]\\!]/(\hbar^{n})$
such that

$d^{\varphi^{(\delta)}}(\phi)\equiv
n\psi^{(\delta+n)}\hbar^{n-1}\pmod{\hbar^{n}}\ .$

Looking at the coefficient of $\hbar^{n-1}$ and the component of weight
$\delta+n$ on both sides, we have

$d^{\varphi^{(\delta)}}\left(\phi_{n-1}^{(n)}\right)=n\psi^{(\delta+n)}\ .$

The element $\upsilon\coloneqq\frac{1}{n}\phi_{n-1}^{(n)}$ in
$\mathfrak{g}_{0}^{(n)}$ thus satisfies

$(\upsilon\cdot\psi)^{(k)}=0\leavevmode\nobreak\ ,\quad\mbox{for all }\
\delta+1\leqslant k\leqslant\delta+n-1\leavevmode\nobreak\ .$

Finally, the element
$\lambda\coloneqq\mathrm{BCH}(\upsilon,\nu)\in\mathfrak{g}_{0}$ satisfies
Point (2). ∎

###### Proof of Theorem 1.31.

Let us prove Point (1). Suppose that there exists $\lambda\in\mathfrak{g}_{0}$
such that $\lambda\cdot\varphi=\varphi^{(\delta)}$ . By Lemma 1.29, this
implies that

$\mathfrak{D}(\lambda)\cdot\mathfrak{D}(\varphi)=\varphi^{(\delta)}\ ,$

and $K_{\Phi}=0$ by Theorem 1.17. Conversely, we proceed as in the proof of
Theorem 1.17. If $K_{\Phi}=0$ , we can use Proposition 1.33 to construct a
sequence of integers $(n_{i})$ greater that $1$ and a sequence of gauges
$(\lambda^{i})$ such that such that

$(\lambda^{i}\cdot\psi^{i})^{(k)}=0\leavevmode\nobreak\ ,\quad\mbox{for all }\
\delta+1\leqslant k\leqslant\delta+n_{i}\leavevmode\nobreak\ ,$

where $\psi^{i+1}\coloneqq\lambda^{{i}}\cdot\psi^{i}$ and
$\psi^{1}\coloneqq\varphi$ . The proof of Proposition 1.33 actually shows that
one can find gauges such that $\lambda^{i}\in\mathfrak{g}^{(n_{i})}_{0}$ , for
all $i$ . It follows from the construction that the gauge

$\lambda\coloneqq\mathrm{BCH}(\cdots\mathrm{BCH}(\lambda^{3},\mathrm{BCH}(\lambda^{2},\lambda^{1}))\cdots)\
,$

satisfies $\lambda\cdot\varphi=\varphi^{(\delta)}$ . The proofs of points (2)
and (3) make use of the Kaledin classes and Point (1) in a way similar to the
proofs of Theorem 1.26 and Corollary 1.25 respectively. ∎

Theorem 1.31 for $\delta=1$ precisely corresponds to the formality setting
addressed in Section 2.

######  Remark 1.34 (Weight zero setting).

One can prove another version of Theorem 1.31 with non-trivial weight zero
component, a differential preserving the weight-grading, and the following
definition of the prismatic decomposition of $\varphi\in\mathfrak{g}_{-1}$,

$\mathfrak{D}\left(\varphi\right)\coloneqq\varphi^{(0)}+\varphi^{(1)}\hbar+\varphi^{(2)}\hbar^{2}+\cdots\in\mathrm{Def}_{\varphi^{(0)}}(R[\\![\hbar]\\!])\
.$

More precisely, let $\mathfrak{g}$ be a differential graded Lie algebra with a
compatible extra weight grading such that

$\mathfrak{g}\cong\prod_{k=0}^{\infty}\mathfrak{g}^{(k)},\quad
d\left(\mathfrak{g}^{(k)}\right)\subset\mathfrak{g}^{(k)},\quad\left[\mathfrak{g}^{(k)},\mathfrak{g}^{(l)}\right]\subset\mathfrak{g}^{(k+l)}\
.$

Then for every Maurer–Cartan element $\varphi\in\mathrm{MC}(\mathfrak{g})$ ,
one can prove that the Kaledin class $K_{\mathfrak{D}\left(\varphi\right)}$ is
zero if and only if there exists $\lambda\in\mathcal{F}^{1}\mathfrak{g}_{0}$
such that $\lambda\cdot\varphi=\varphi^{(0)}$ .

######  Remark 1.35 (The complete setting).

The Kaledin classes cannot be defined without the weight-grading assumption.
Nonetheless, in the spirit of Halperin and Stasheff’s approach [HS79], one
forgo treating everything in a single class. In [Emp24, Section 1], sequences
of obstructions are constructed in order to detect gauge equivalences between
a Maurer–Cartan element and its first component and more generally if two
Maurer–Cartan elements are gauge equivalent in a complete dg Lie algebra.

##  2\. Operadic Kaledin classes over a commutative ground ring

In this section, we apply the Kaledin classes developed in Section 1 to study
formality of algebras encoded by colored operads. In this particular context,
we adapt the construction to deal with any commutative ground ring $R$. This
generalizes the article [MR19] of Melani and Rubió to any commutative ring and
to algebras over any reduced colored operad.

Let us fix $R$ a commutative ground ring and let $A$ be a differential graded
$R$-module. Let $\mathcal{C}$ be a reduced weight-graded differential graded
cooperad over $R$, i.e. $\mathcal{C}(0)=0$ and

$\mathcal{C}=\mathrm{I}\oplus\mathcal{C}^{(1)}\oplus\mathcal{C}^{(2)}\oplus\cdots\oplus\mathcal{C}^{(n)}\oplus\cdots\quad\mbox{and}\quad
d_{\mathcal{C}}\left(\mathcal{C}^{(k)}\right)\subset\mathcal{C}^{(k-1)}\ .$

Let us denote the coaugmentation coideal
$\overline{\mathcal{C}}\coloneqq\bigoplus_{k\geqslant 1}\mathcal{C}^{(k)}$ so
that $\mathcal{C}\cong\mathrm{I}\oplus\overline{\mathcal{C}}$ .

######  Examples 2.1.

Among possible choices for such cooperad $\mathcal{C}$, we have:

  1. (1)

The bar construction $\mathcal{C}\coloneqq\mathrm{B}\mathcal{P}$ of a reduced
operad $\mathcal{P}$, see [LV12, Section 6.5] .

  2. (2)

The Koszul dual cooperad
$\mathcal{C}:=\mathcal{P}^{\text{\raisebox{0.0pt}{\textexclamdown}}}$ of a
homogeneous Koszul operad $\mathcal{P}$, see [LV12, Section 7.4], or an
inhomogeneous quadratic Koszul operad $\mathcal{P}$, see [LV12, Section 7.6].

  3. (3)

The quasi-planar cooperad
$\mathcal{C}=\mathrm{B}\left(\mathcal{E}\otimes\mathcal{P}\right)$ where
$\mathcal{E}$ denotes the Barratt-Eccles operad and $\mathcal{P}$ is any
reduced operad, see [GRiL23].

###  2.1. Deformation complex of algebras over an operad

This section recalls the definition of a $\Omega\mathcal{C}$-algebra structure
on $A$ , as a Maurer–Cartan element of its associated convolution dg pre-Lie
algebra $\mathfrak{g}_{A}$. We refer the reader to [DSV16, Section 5] for more
details.

######  Definition 2.2 (dg pre-Lie algebra).

A _dg pre-Lie algebra_ is a chain complex $(\mathfrak{g},d)$ equipped with a
linear map of degree zero
$\star:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$ , the pre-Lie product,
satisfying

$\forall x,y,z\in\mathfrak{a},\,(x\star y)\star z-x\star(y\star
z)=(-1)^{|y||z|}((x\star z)\star y-x\star(z\star y))\ .$

Its set of _Maurer–Cartan elements_ is defined as

$\mathrm{MC(\mathfrak{g})}\coloneqq\\{x\in\mathfrak{g}_{-1}\mid dx+x\star
x=0\\}\ .$

######  Remark 2.3.

In a dg pre-Lie algebra, the skew-symmetrized bracket

$[x,y]:=x\star y-(-1)^{|x||y|}y\star x\ ,$

induces a dg Lie algebra structure. If $2$ is invertible in $R$ , we have
$\varphi\star\varphi=\frac{1}{2}[\varphi,\varphi]$ and the Maurer–Cartan
equation coincides with usual one.

######  Definition 2.4 (Convolution dg pre-Lie algebra).

The _convolution dg pre-Lie algebra_ associated to $\mathcal{C}$ and $A$ is
the dg pre-Lie algebra

$\mathfrak{g}_{A}:=\left(\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right),\star,d\right)\
,$

where the underlying space is

$\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right):=\prod_{p\geqslant
0}\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}}(p),\mathrm{End}_{A}(p)\right)\
,$

equipped with the pre-Lie product

$\varphi\star\psi\coloneqq\overline{\mathcal{C}}\xrightarrow{\Delta_{(1)}}\overline{\mathcal{C}}\circ_{(1)}\overline{\mathcal{C}}\xrightarrow{\varphi\circ_{(1)}\psi}\mathrm{End}_{A}\circ_{(1)}\mathrm{End}_{A}\xrightarrow{\gamma_{(1)}}\mathrm{End}_{A}\
,$

and the differential

$d(\varphi)=d_{\mathrm{End}_{A}}\circ\varphi-(-1)^{|\varphi|}\varphi\circ
d_{\overline{\mathcal{C}}}\ .$

It embeds inside the dg pre-Lie algebra made up of all the maps from
$\mathcal{C}\cong\mathrm{I}\oplus\overline{\mathcal{C}}$, i.e.

$\mathfrak{g}_{A}\hookrightarrow\mathfrak{a}_{A}\coloneqq\left(\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}_{A}\right),\star,d,1\right)$

where the left unit is given by $1:\mathrm{I}\mapsto\mathrm{id}_{A}$ .

######  Proposition 2.5.

A _$\Omega\mathcal{C}$ -algebra structure_ $\varphi$ on $A$ is a Maurer–Cartan
element in $\mathfrak{g}_{A}$ , i.e.

$\varphi\in\mathrm{MC}(\mathfrak{g}_{A})=\\{\varphi\in\mathfrak{g}_{A}\mid|\varphi|=-1,\;d(\varphi)+\varphi\star\varphi=0\\}\
.$

The _deformation complex_ of an $\Omega\mathcal{C}$-algebra
$\varphi\in\mathrm{MC}(\mathfrak{g}_{A})$ is the twisted dg pre-Lie algebra

$\mathfrak{g}_{A}^{\varphi}:=\left(\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right),\star\
,d^{\varphi}\coloneqq d+\mathrm{ad}_{\varphi}\right)\ .$

###### Proof.

See [LV12, Proposition 10.1.1]. ∎

######  Examples 2.6.

The homology groups of the complex $\mathfrak{g}_{A}^{\varphi}$ is often
called the _tangent homology._

  1. (1)

In the case of the cooperad
$\mathcal{C}=A^{\text{\raisebox{0.0pt}{\textexclamdown}}}$ and an associative
algebra structure $\varphi$, the tangent homology is isomorphic to the
Hochschild cohomology with coefficients into itself (up to a suspension) with
Gerstenhaber bracket, see [LV12, Section 9.1.6].

  2. (2)

In the case of the cooperad $Com^{\text{\raisebox{0.0pt}{\textexclamdown}}}$
and a commutative algebra structure $\varphi$, one recovers the Harrison
cohomology of commutative algebras, see [LV12, Section 13.1.7].

  3. (3)

For the operad $Lie^{\text{\raisebox{0.0pt}{\textexclamdown}}}$ and a Lie
algebra structure $\varphi$, one gets the Chevalley–Eilenberg homology of Lie
algebras, see [LV12, Section 12.3].

######  Proposition 2.7.

The algebra $\mathfrak{a}_{A}$ is an associative algebra with the product
defined by

$\varphi\circledcirc\psi\coloneqq\mathcal{C}\xrightarrow{\Delta}\mathcal{C}\circ\mathcal{C}\xrightarrow{\varphi\
\circ\
\psi}\mathrm{End}_{A}\circ\mathrm{End}_{A}\xrightarrow{\gamma}\mathrm{End}_{A}\
.$

###### Proof.

See [LV12, Section 10.2.3]. ∎

######  Definition 2.8 ($\infty$-morphism).

Let $\varphi$ and $\psi$ be two $\Omega\mathcal{C}$-algebra structures on $A$
. An _$\infty$ -morphism_ $\varphi\rightsquigarrow\psi$ is a map
$f\in\mathfrak{a}_{A}$ of degree $0$ satisfying

$f\star\varphi-\psi\circledcirc f=d(f)\ .$

The composite of two $\infty$-morphisms $f:\varphi\rightsquigarrow\psi$ and
$g:\psi\rightsquigarrow\phi$ is given by $f\circledcirc g$.

######  Remark 2.9.

Let $\varphi\in\mathrm{MC}(\mathfrak{g}_{A})$ and
$\psi\in\mathrm{MC}(\mathfrak{g}_{B})$ be two $\Omega\mathcal{C}$-algebra
structures. In this context, an $\infty$-morphism
$\varphi\rightsquigarrow\psi$ can be defined as a morphism

$f\in\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}^{A}_{B}\right)$

of degree $0$ satisfying a similar identity, see [LV12, Theorem 10.2.3] for
more details.

Since the cooperad $\mathcal{C}$ is weight-graded, the pre-Lie algebra
$\mathfrak{a}_{A}$ admits a weight-grading,

$\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}_{A}\right)\cong\prod_{n\geqslant
0}\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C}^{(n)},\mathrm{End}_{A}\right)\ .$

Every element
$f\in\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}_{A}\right)$
decomposes as

$f=f^{(0)}+f^{(1)}+f^{(2)}+\cdots$

where $f^{(i)}$ is the restriction of $f$ to $\mathcal{C}^{(i)}$. The first
component

$f^{(0)}:\mathrm{I}\to\mathrm{End}_{A}$

of $\infty$-morphism is equivalent to a chain map in $\mathrm{End}_{A}$ that
we still denote by $f^{(0)}$. The convolution dg pre-Lie algebra
$\mathfrak{g}_{A}$ has a similar underlying weight grading and every element
$\varphi$ decomposes as

$\varphi=\varphi^{(1)}+\varphi^{(2)}+\varphi^{(3)}+\cdots\ .$

######  Definition 2.10.

An $\infty$-morphism $f:\varphi\rightsquigarrow\psi$ between two
$\Omega\mathcal{C}$-algebra structures on $A$ is

  * $\centerdot$

an _$\infty$ -quasi-isomorphism_ if $f^{(0)}\in\mathrm{End}_{A}$ is a quasi-
isomorphism;

  * $\centerdot$

an _$\infty$ -isomorphism_ if $f^{(0)}\in\mathrm{End}_{A}$ is an isomorphism;

  * $\centerdot$

an _$\infty$ -isotopy_ if $f^{(0)}=\mathrm{id}_{A}$ .

######  Lemma 2.11.

The $\infty$-isomorphisms are the isomorphisms in the category of
$\Omega\mathcal{C}$-algebra structures on $A$. We will denote by $f^{-1}$ the
unique inverse of an $\infty$-isomorphism $f$ with respect to the product
$\circledcirc$.

###### Proof.

See [LV12, Theorem 10.4.1]. ∎

######  Theorem 2.12 ([DSV16, Theorem 3]).

Let $R$ be a $\mathbb{Q}$-algebra and let $A$ be a chain complex over $R$. The
pre-Lie exponential map induces an isomorphism

$\mathrm{exp}:((\mathfrak{g}_{A})_{0},\mathrm{BCH},0)\overset{\cong}{\longrightarrow}(\infty-\mathrm{iso},\circledcirc,1)$

between the gauge group and the group of $\infty$-isotopies.

###  2.2. Gauge formality

This section introduces the gauge formality notion studied in this article. It
coincides with the standard formality definition in most cases, see
Proposition 2.21. Let us consider the operad

(3)
$\mathcal{P}\coloneqq\mathcal{T}\left(s^{-1}\mathcal{C}^{(1)}\right)/\left(d_{\Omega\mathcal{C}}\left(s^{-1}\mathcal{C}^{(2)}\right)\right)\
,$

which comes with a twisting morphism $\mathcal{C}\to\mathcal{P}$.

######  Proposition 2.13.

Any $\Omega\mathcal{C}$-algebra structure $(A,\varphi)$ induces a canonical
$\mathcal{P}$-algebra structure

$(H(A),\varphi_{*})\ .$

###### Proof.

The homology of $(A,\varphi)$ carries a natural $\Omega\mathcal{C}$-algebra
structure

$\psi:\mathcal{C}\longrightarrow\mathrm{End}_{H(A)}\ ,$

see [LV12, Proposition 6.3.5]. Since the homology has no differential, the
component

$\varphi_{*}\coloneqq\psi^{(1)}$

is a $\Omega\mathcal{C}$-algebra structure which appears as a
$\mathcal{P}$-algebra over the operad $\mathcal{P}$ defined by (3). ∎

######  Examples 2.14.

Examples 2.1 all comes equipped with a Koszul morphism
$\kappa:\mathcal{C}\to\mathcal{P}$.

  1. (1)

The bar construction $\mathcal{C}\coloneqq\mathrm{B}\mathcal{P}$ of a reduced
operad $\mathcal{P}$ in $\mathrm{grMod}_{R}$ has an associated universal
twisting morphism $\pi:\mathrm{B}\mathcal{P}\to\mathcal{P}$ , see [LV12,
Section 6.5] .

  2. (2)

The Koszul dual cooperad
$\mathcal{C}:=\mathcal{P}^{\text{\raisebox{0.0pt}{\textexclamdown}}}$ has the
canonical twisting morphism
$\kappa:\mathcal{P}^{\text{\raisebox{0.0pt}{\textexclamdown}}}\to\mathcal{P}$
.

  3. (3)

The quasi-planar cooperad
$\mathcal{C}=\mathrm{B}\left(\mathcal{E}\otimes\mathcal{P}\right)$ where
$\mathcal{P}$ is any reduced operad has an associated canonical twisting
morphism $\mathcal{C}\to\mathcal{P}$, see [GRiL23].

######  Definition 2.15 (Formality and gauge formality).

A $\Omega\mathcal{C}$-algebra structure $(A,\varphi)$ is

  1. $\centerdot$

_formal_ if there exists a zig-zag of quasi-isomorphisms of
$\Omega\mathcal{C}$-algebras

$(A,\varphi)\;\overset{\sim}{\longleftarrow}\;\cdot\;\overset{\sim}{\longrightarrow}\;\cdots\;\overset{\sim}{\longleftarrow}\;\cdot\;\overset{\sim}{\longrightarrow}\;(H(A),\varphi_{*})\
;$

  2. $\centerdot$

_gauge formal_ if there exists an $\infty$-quasi-isomorphism of
$\Omega\mathcal{C}$-algebras

${(A,\varphi)}$${(H(A),\varphi_{*})\ .}$$\scriptstyle{\sim}$

The formality and the gauge formality coincide in most examples, see
Proposition 2.21.

######  Definition 2.16 (Transferred structure).

A $\Omega\mathcal{C}$-algebra $(A,\varphi)$ admits a _transferred structure_
if there exist a $\Omega\mathcal{C}$-algebra structure $(H(A),\varphi_{t})$,
extending the $\mathcal{P}$-algebra structure $\varphi_{*}$

$\varphi_{t}\coloneqq\varphi_{*}+\varphi_{t}^{(2)}+\varphi_{t}^{(3)}+\cdots\in\mathrm{MC}\left(\mathfrak{g}_{H(A)}\right)\
,$

and two $\infty$-quasi-isomorphisms

$(A,\varphi)$$(H(A),\varphi_{t})\ .$$p_{\infty}$$i_{\infty}$

######  Proposition 2.17.

Let $(A,\varphi)$ be $\Omega\mathcal{C}$-algebra structure. If there exists a
transferred structure $(H(A),\varphi_{t})$, the algebra $(A,\varphi)$ is gauge
formal if and only if there exists an $\infty$-isotopy

${f:(H(A),\varphi_{t})}$${(H(A),\varphi_{*})\ .}$$\scriptstyle{=}$

###### Proof.

The transferred structure comes equipped with two $\infty$-quasi-isomorphisms

$(A,\varphi)$$(H(A),\varphi_{t})\ .$$p_{\infty}$$i_{\infty}$

The algebra $(A,\varphi)$ is gauge formal if and only if there exists an
$\infty$-quasi-isomorphism

${f:(H(A),\varphi_{t})}$${(H(A),\varphi_{*})\ .}$$\scriptstyle{\sim}$

It is an $\infty$-isomorphism since the homology $H(A)$ has no differential.
By definition of an $\infty$-morphism, the equation
$f\star\varphi_{t}-\varphi_{*}\circledcirc f=d(f)$ holds and gives in weight
1,

$f^{(0)}\star\left(\varphi_{t}\right)^{(1)}-\varphi_{*}\circledcirc
f^{(0)}=-f^{(0)}\circ d_{\mathcal{C}}\ .$

Since $\varphi_{*}=\left(\varphi_{t}\right)^{(1)}$ , the component $f^{(0)}$
is an $\infty$-automorphism of $(H(A),\varphi_{*})$ and the element

$\left(f^{(0)}\right)^{-1}\circledcirc f$

is an $\infty$-isotopy between $(H(A),\varphi_{t})$ and $(H(A),\varphi_{*})$.
∎

######  Definition 2.18.

A chain complex $(A,d)$ is related to $H(A)$ via a _contraction_ if there
exists

$(A,d)$$(H(A),0)$$p$$i$$h$

such that $ip-\mathrm{id}_{A}=d_{A}h+hd_{A}\ ,$ $pi=\mathrm{id}_{H(A)}\ ,$
$h^{2}=0\ ,$ $ph=0\ ,$ $hi=0\ .$

######  Example 2.19.

If $R$ is a field, any chain complex $A$ is related to its homology $H(A)$ via
a contraction. This is also the case when $A$ and $H(A)$ are degreewise
projective and $R$ is an hereditary ring.

######  Assumptions 1.

Let $R$ be a commutative ground ring and let $A$ be a chain complex related to
$H(A)$ via a contraction. Let $\mathcal{C}$ be a reduced weight-graded dg
cooperad over $R$.

  * $\centerdot$

If $\mathcal{C}$ is a symmetric cooperad, we assume that $R$ is a
$\mathbb{Q}$-algebra.

  * $\centerdot$

If $\mathcal{C}$ is a symmetric quasi-planar cooperad, we assume that $R$ is a
field.

######  Theorem 2.20 (Homotopy transfer theorem).

Under Assumptions 1, for any $\Omega\mathcal{C}$-algebra structure, there
exists a transferred structure $(H(A),\varphi_{t})$ such that the embedding
$i:H(A)\to A$ and the projection $p:A\to H(A)$ extend to $\infty$-quasi-
isomorphisms

$(A,\varphi)$$(H(A),\varphi_{t})\ .$$p_{\infty}$$i_{\infty}$

The transferred structure is independent of the choice of contraction in the
following sense: any two such transferred structures are related by an
$\infty$-isotopy.

###### Proof.

We refer the reader to [Ber14], [LV12, Section 10.3] and references therein.
In the case of a symmetric quasi-planar cooperad, we refer the reader to
[GRiL23, Section 5.5], ∎

######  Proposition 2.21.

Let $\mathcal{C}$ be a reduced weight-graded dg cooperad over $R$.

  1. (1)

If $R$ is a characteristic zero field, an $\Omega\mathcal{C}$-algebra
structure is formal if and only if it is gauge formal.

  2. (2)

Suppose that $\mathcal{C}$ is non-symmetric.

    1. i.

If $R$ is a field, then a formal $\Omega\mathcal{C}$-algebra structure on $A$
is also gauge formal.

    2. ii.

If $\mathcal{C}$ and $\Omega\mathcal{C}$ are aritywise and degreewise flat,
then a gauge formal $\Omega\mathcal{C}$-algebra is formal.

  3. (3)

[GRiL23, Proposition 43] Suppose that $\mathcal{C}$ is a symmetric quasi-
planar cooperad. If $R$ is a field, then an $\Omega\mathcal{C}$-algebra is
formal if and only if it is gauge formal.

###### Proof.

Let $(A,\varphi)$ be a formal $\Omega\mathcal{C}$-algebra structure. If $R$ is
a field and the hypotheses of Theorem 2.20 are satisfied, we claim that it is
also gauge formal. It suffices to prove that given a quasi-isomorphism of
$\Omega\mathcal{C}$-algebras

$(A,\varphi)\longleftarrow(B,\psi)\ ,$

there exists an $\infty$-quasi-isomorphism in the other direction. By Theorem
2.20, there exists $\infty$-quasi-isomorphisms

${f:(H(B),\psi_{t})}$${(B,\psi)}$${(A,\varphi)}$${(H(A),\varphi_{t})\
.}$$\scriptstyle{i_{\infty}}$$\scriptstyle{\sim}$$\scriptstyle{p_{\infty}}$

The $\infty$-isomorphism $f$ admits an inverse $f^{-1}$ by Lemma 2.11. The
composite

${(A,\phi)}$${(H(A),\varphi_{t})}$${(H(B),\psi_{t})}$${(B,\psi)\
.}$$\scriptstyle{p_{\infty}\
}$$\scriptstyle{f^{-1}}$$\scriptstyle{i_{\infty}}$

produces the required $\infty$-quasi-isomorphism. This proves (2).i. and the
direct implications of (1) and (3). Conversely, let $(A,\varphi)$ be a gauge
formal $\Omega\mathcal{C}$-algebra structure and let

${f:(A,\varphi)}$${(H(A),\varphi_{*})\ .}$$\scriptstyle{\sim}$

be an $\infty$-quasi-isomorphism. By [LV12, Proposition 11.3.1], the universal
Koszul morphism $\iota:\mathcal{C}\to\Omega\mathcal{C}$ induces an adjunction

$\Omega_{\iota}:\\{\mbox{dg }\mathcal{C}\mbox{-coalgebras}\\}$$\\{\mbox{dg
}\Omega\mathcal{C}\mbox{-algebras}\\}:\mathrm{B}_{\iota}\ .$

$\perp$

If $R$ is a characteristic zero field, this leads to a zig-zag of
$\Omega\mathcal{C}$-algebras quasi-isomorphisms

(4)
$(A,\varphi)\underset{\sim}{\xleftarrow{\epsilon_{\iota}(A,\varphi)}}\Omega_{\iota}\mathrm{B}_{\iota}(A,\varphi)\underset{\sim}{\xrightarrow{\Omega_{\iota}\mathrm{B}_{\iota}(f)}}\Omega_{\iota}\mathrm{B}_{\iota}(H(A),\varphi_{*})\underset{\sim}{\xrightarrow{\epsilon_{\iota}(H(A),\varphi_{*})}}(H(A),\varphi_{*})\
,$

where the first and the third quasi-isomorphisms are given by the counit
$\epsilon_{\iota}$ of the adjunction, see [LV12, Theorem 11.3.3] and
$\Omega_{\iota}\mathrm{B}_{\iota}(f)$ is a quasi-isomorphism by [LV12, Theorem
11.4.7]. This proves the converse implication of (1).

The ground ring $R$ is assumed to be a characteristic zero field in [LV12,
Section 11] so that the operadic Künneth formula [LV12, Proposition 6.2.3]
holds true. If $\mathcal{C}$ and $\Omega\mathcal{C}$ are non-symmetric and
aritywise and degreewise flat, there is a version of this proposition over any
ring $R$, as mentioned in the proof of [DCH21, Proposition 1.18]. Thus, under
the hypotheses of (2).ii., the zig-zag (4) also holds and $(A,\varphi)$ is
formal. ∎

######  Definition 2.22 (Gauge $n$-formality).

Let $(A,\varphi)$ be $\Omega\mathcal{C}$-algebra such that there exists a
transferred structure $(H(A),\varphi_{t})$. It is called _gauge $n$-formal_ if
there exist an $\Omega\mathcal{C}$-algebra $(H(A),\psi)$ such that
$\psi^{(k)}=0$ , for all $2\leqslant k\leqslant n+1$ and an $\infty$-isotopy

${(H(A),\varphi_{t})}$${(H(A),\psi)\ .}$$\scriptstyle{=}$

######  Remark 2.23.

The notion of gauge $n$-formality depends on the choice of weight-grading on
$\mathcal{C}$. If the weight-grading is the homological degree of
$\mathcal{C}$, it coincides with [DCH21, Definition 1.22]. If the weight-
grading is given by the arity, one recovers the approach taken for instance in
[Lun07], [Sal17], [MR19], etc.

###  2.3. Pre-Lie calculus in operadic convolution dg algebras

As explained in Section 2.1, the pre-Lie algebra $\mathfrak{a}_{A}$ is
equipped with a pre-Lie product $\star$ and an associative product
$\circledcirc$. These two operations satisfy compatibility relations, some of
them are proved over a characteristic zero field in [DSV16, Proposition 5]. In
this section, we establish these relations over any commutative ground ring
$R$ .

######  Definition 2.24.

For all $f,g,h\in\mathfrak{a}_{A}$ , we define
$f\circledcirc\left(g;h\right)\in\mathfrak{a}_{A}$ as the following
composition

$f\circledcirc\left(g;h\right)\coloneqq\mathcal{C}\xrightarrow{\Delta}\mathcal{C}\circ\mathcal{C}\xrightarrow{\mathrm{id}_{\mathcal{C}}\circ^{\prime}\mathrm{id}_{\mathcal{C}}}\mathcal{C}\circ\left(\mathcal{C};\mathcal{C}\right)\xrightarrow{f\circ(g;h)}\mathrm{End}_{A}\circ\left(\mathrm{End}_{A};\mathrm{End}_{A}\right)\xrightarrow{\pi}\mathrm{End}_{A}\
,$

where the map $\circ^{\prime}$ is the infinitesimal composition of morphisms
and where the map $\pi$ is

$\mathrm{End}_{A}\circ\left(\mathrm{End}_{A};\mathrm{End}_{A}\right)\rightarrowtail\mathrm{End}_{A}\circ\left(\mathrm{End}_{A}\oplus\mathrm{End}_{A}\right)\twoheadrightarrow\mathrm{End}_{A}\circ\mathrm{End}_{A}\xrightarrow{\gamma}\mathrm{End}_{A}\
,$

see [LV12, Section 6.1.1] for more details.

######  Lemma 2.25.

For all $f,g,h,k\in\mathfrak{a}_{A}$ , the following identities hold

  1. (a)

$\left(f\circledcirc g\right)\star h=f\circledcirc\left(g;g\star h\right)\ ,$

  2. (b)

$\left(f\circledcirc\left(g;h\right)\right)\circledcirc
k=f\circledcirc\left(g\circledcirc k;h\circledcirc k\right)\ ,$

  3. (c)

$f\star h=f\circledcirc\left(1;h\right)\ .$

If furthermore, the element $g$ is invertible with respect to $\circledcirc$ ,
then

  1. (d)

$f\circledcirc\left(g;h\right)=\left(f\star\left(h\circledcirc
g^{-1}\right)\right)\circledcirc g\ ,$

  2. (e)

$\left(\left(f\circledcirc g\right)\star h\right)\circledcirc
g^{-1}=f\star\left(\left(g\star h\right)\circledcirc g^{-1}\right)\ .$

###### Proof.

By coassociativity of the decomposition map $\Delta$ , the following diagram
is commutative

${\mathcal{C}}$${\mathcal{C}\circ\mathcal{C}}$${\mathcal{C}\circ\left(\mathcal{C};\mathcal{C}\right)}$${(\mathcal{C}\circ\mathcal{C})\circ(\mathrm{I};\mathcal{C})}$${\mathcal{C}\circ\left(\mathcal{C};\mathcal{C}\circ\left(\mathrm{I};\mathcal{C}\right)\right)}$${(\mathrm{End}_{A}\circ\mathrm{End}_{A})\circ(\mathrm{I};\mathrm{End}_{A})}$${\mathrm{End}_{A}\circ\left(\mathrm{End}_{A};\mathrm{End}_{A}\circ\left(\mathrm{I};\mathrm{End}_{A}\right)\right)}$${\mathrm{End}_{A}\circ(\mathrm{I};\mathrm{End}_{A})}$${\mathrm{End}_{A}\circ\left(\mathrm{End}_{A};\mathrm{End}_{A}\right)}$${\mathrm{End}_{A}\circ\mathrm{End}_{A}}$${\mathrm{End}_{A}}$$\scriptstyle{\Delta}$$\scriptstyle{\mathrm{id}_{\mathcal{C}}\circ^{\prime}\mathrm{id}_{\mathcal{C}}}$$\scriptstyle{\mathrm{id}_{\mathcal{C}}\circ\left(\mathrm{id}_{\mathcal{C}},\Delta_{(1)}\right)}$$\scriptstyle{\Delta\circ\left(\eta,\mathrm{id}_{\mathcal{C}}\right)}$$\scriptstyle{\cong}$$\scriptstyle{(f\circ
g)\circ(\mathrm{id}_{\mathrm{I}};h)}$$\scriptstyle{f\circ\left(g;g\circ(\mathrm{id};h)\right)}$$\scriptstyle{\cong}$$\scriptstyle{\gamma\circ(\mathrm{id}_{\mathrm{I}};\mathrm{id}_{\mathrm{End}_{A}})}$$\scriptstyle{\mathrm{id}_{\mathrm{End}_{A}}\circ\left(\mathrm{id}_{\mathrm{End}_{A}};\gamma_{(1)}\right)}$$\scriptstyle{\gamma}$

The left-hand side is equal to $\left(f\circledcirc g\right)\star h$ and the
right-hand side is equal to $f\circledcirc\left(g;g\star h\right)$, which
gives Formula (a). The proof of Formula (b) is similar to the one of (a) and
relies on the isomorphism

$\left(\mathcal{C}\circ\left(\mathcal{C};\mathcal{C}\right)\right)\circ\mathcal{C}\cong\mathcal{C}\circ\left(\mathcal{C}\circ\left(\mathcal{C}\circ\mathcal{C};\mathcal{C}\circ\mathcal{C}\right)\right)\
.$

Formula (c) follows from (a) with $g=1$ . Considering $k\coloneqq g^{-1}$ in
Formula (b) and using Formula (c), we obtain Formula (d) :

$f\circledcirc\left(g;h\right)\overset{(b)}{=}\left(f\circledcirc\left(1;h\circledcirc
g^{-1}\right)\right)\circledcirc
g\overset{(c)}{=}\left(f\star\left(h\circledcirc
g^{-1}\right)\right)\circledcirc g\ .$

Formulas (a) and (d) give Formula (e) :

$\left(\left(f\circledcirc g\right)\star h\right)\circledcirc
g^{-1}\overset{(a)}{=}\left(f\circledcirc\left(g;g\star
h\right)\right)\circledcirc g^{-1}\overset{(d)}{=}f\star\left(\left(g\star
h\right)\circledcirc g^{-1}\right)\ .\qed$

######  Lemma 2.26.

For all $f,g\in\mathfrak{a}_{A}$ , the following identities hold

  1. (i)

$d\left(f\star g\right)=d(f)\star g+(-1)^{|f|}f\star d(g)\ ,$

  2. (j)

If $g$ is of degree zero, then $d\left(f\circledcirc
g\right)=d\left(f\right)\circledcirc
g+(-1)^{|f|}f\circledcirc\left(g;d(g)\right)\ ,$

  3. (k)

$d(1)=0\ .$

If furthermore, the element $g$ is invertible with respect to $\circledcirc$ ,
then

  1. (l)

$d\left(g^{-1}\right)=-g^{-1}\star\left(d(g)\circledcirc g^{-1}\right)\ .$

###### Proof.

For points (i) and (j), we refer the reader to [LV12, Section 6.4] . Formula
$(k)$ follows from Formula (i) applied to $f=g=1$. By Formula (j) applied to
$\left(g^{-1}\circledcirc g\right)\circ d_{\mathcal{C}}=1\circ
d_{\mathcal{C}}=0$ , the following equality holds

$g^{-1}\circ d_{\mathcal{C}}=-\left(g^{-1}\circledcirc\left(g;g\circ
d_{\mathcal{C}}\right)\right)\circledcirc g^{-1}\ .$

Formula (d) of Lemma 2.25 implies formula (l). ∎

######  Proposition 2.27.

Let $\varphi$ and $\psi$ be two $\Omega\mathcal{C}$-algebra structures on $A$.
Every $\infty$-isomorphism $f:\varphi\rightsquigarrow\psi$ induces an
isomorphism of dg Lie algebras

$\mathrm{Ad}_{f}:\mathfrak{g}_{A}^{\varphi}\to\mathfrak{g}_{A}^{\psi},\quad
x\mapsto(f\star x)\circledcirc f^{-1}\ ,$

whose inverse is given by $\mathrm{Ad}_{f^{-1}}$ .

###### Proof.

The bracket is defined by the skew-symmetrization of the pre-Lie product, i.e.
for all $x,y\in\mathfrak{g}_{H}$ , we have

$\mathrm{Ad}_{f}\left([x,y]\right)=\left(f\star\left(x\star
y\right)\right)\circledcirc f^{-1}-(-1)^{|x||y|}\left(f\star\left(y\star
x\right)\right)\circledcirc f^{-1}\ .$

The definition of a pre-Lie product gives

$f\star\left(x\star y\right)=\left(f\star x\right)\star
y-(-1)^{|x||y|}\left(\left(f\star y\right)\star
x\right)+(-1)^{|x||y|}\left(f\star\left(y\star x\right)\right)\ .$

This shows that

$\mathrm{Ad}_{f}\left([x,y]\right)=\left(\left(\mathrm{Ad}_{f}(x)\circledcirc
f\right)\star y\right)\circledcirc
f^{-1}-(-1)^{|x||y|}\left(\left(\mathrm{Ad}_{f}(y)\circledcirc f\right)\star
x\right)\circledcirc f^{-1}\ .$

Formula (e) of Lemma 2.25 implies that

$\mathrm{Ad}_{f}\left([x,y]\right)=[\mathrm{Ad}_{f}(x),\mathrm{Ad}_{f}(y)]\ .$

Formula (i) of Lemma 2.26 implies that

$\mathrm{Ad}_{f}\left(dx\right)\overset{(i)}{=}d\left(f\star
x\right)\circledcirc f^{-1}-\left(d\left(f\right)\star x\right)\circledcirc
f^{-1}\ .$

By Formula (j) of Lemma 2.25, we have

$d\left(f\star x\right)\circledcirc
f^{-1}\overset{(j)}{=}d\left(\mathrm{Ad}_{f}\left(x\right)\right)-(-1)^{|x|}\left(f\star
x\right)\circledcirc\left(f^{-1};d\left(f^{-1}\right)\right)\ .$

By Formula (l) and (a), we get

$\left(f\star
x\right)\circledcirc\left(f^{-1};d\left(f^{-1}\right)\right)=-\mathrm{Ad}_{f}\left(x\right)\star\left(d\left(f\right)\circledcirc
f^{-1}\right)$

Formula (e) of Lemma 2.25 implies

$\left(d(f)\star x\right)\circledcirc f^{-1}=\left(d\left(f\right)\circledcirc
f^{-1}\right)\star\mathrm{Ad}_{f}\left(x\right)\ .$

This leads to

$\mathrm{Ad}_{f}\left(d(x)\right)=d\left(\mathrm{Ad}_{f}\left(x\right)\right)-\left[d\left(f\right)\circledcirc
f^{-1},\mathrm{Ad}_{f}\left(x\right)\right]\ .$

Thus, for all $x\in\mathfrak{g}_{H}$ , we have

$\begin{split}\mathrm{Ad}_{f}\left(d^{\varphi}(x)\right)&=\mathrm{Ad}_{f}\left(d(x)\right)+\mathrm{Ad}_{f}\left(\left[\varphi,x\right]\right)\\\
&=d\left(\mathrm{Ad}_{f}\left(x\right)\right)-\left[d\left(f\right)\circledcirc
f^{-1},\mathrm{Ad}_{f}\left(x\right)\right]+\left[\mathrm{Ad}_{f}\left(\varphi\right),\mathrm{Ad}_{f}\left(x\right)\right]\
.\end{split}$

By definition of an $\infty$-morphism, we have $f\star\varphi-\psi\circledcirc
f=d(f)$ and thus

$\psi=\mathrm{Ad}_{f}\left(\varphi\right)-d(f)\circledcirc f^{-1}\ .$

We get
$\mathrm{Ad}_{f}\left(d^{\varphi}(x)\right)=d^{\psi}\left(\mathrm{Ad}_{f}(x)\right)\
.$ Finally, Formula (e) of Lemma 2.25 implies that

$\mathrm{Ad}_{f^{-1}}\left(\mathrm{Ad}_{f}\left(x\right)\right)=\left(f^{-1}\star\left((f\star
x)\circledcirc f^{-1}\right)\right)\circledcirc
f\overset{(e)}{=}\left(\left(\left(f^{-1}\circledcirc f\right)\star
x\right)\circledcirc f^{-1}\right)\circledcirc f=x\ .$

Thus $\mathrm{Ad}_{f}$ is an isomorphism of dg Lie algebras whose inverse is
given by $\mathrm{Ad}_{f^{-1}}$ . ∎

######  Theorem 2.28.

  1. (1)

The set of degree zero elements $f\in\mathfrak{a}_{A}$ such that
$f^{(0)}\in\mathrm{End}_{A}$ is an isomorphism forms a group with respect to
the product $\circledcirc$.

  2. (2)

It acts on the set of Maurer–Cartan elements $\mathrm{MC}(\mathfrak{g}_{A})$
under

$f\cdot\varphi=\mathrm{Ad}_{f}\left(\varphi\right)-d(f)\circledcirc f^{-1}\ .$

###### Proof.

Point (1) follows from Lemma 2.11. We have $1\cdot\varphi=\varphi$. Let $f$
and $g$ in $\mathfrak{a}_{A}$ whose weight-zero components are isomorphisms.
Then, we have

$f\cdot(g\cdot\varphi)=\mathrm{Ad}_{f}\circ\mathrm{Ad}_{g}\left(\varphi\right)-\mathrm{Ad}_{f}\left(d(g)\circledcirc
g^{-1}\right)-d(f)\circledcirc f^{-1}\ .$

Formula (e) of Lemma 2.25 implies that

$\mathrm{Ad}_{f}\circ\mathrm{Ad}_{g}\left(\varphi\right)=\mathrm{Ad}_{f\circledcirc
g}\left(\varphi\right)\ .$

Formula (j) of Lemma 2.26 gives

$d\left(f\circledcirc g\right)\circledcirc g^{-1}\circledcirc
f^{-1}=f\circledcirc\left(g;d(g)\right)\circledcirc g^{-1}\circledcirc
f^{-1}+d\left(f\right)\circledcirc f^{-1}\ .$

By Formula (b) and (c) of Lemma 2.25, we have

$f\circledcirc\left(g;d(g)\right)\circledcirc g^{-1}\circledcirc
f^{-1}\overset{(b)}{=}f\circledcirc\left(1;d(g)\circledcirc
g^{-1}\right)\circledcirc
f^{-1}\overset{(c)}{=}\mathrm{Ad}_{f}\left(d(g)\circledcirc g^{-1}\right)\ .$

This proves that $f\cdot(g\cdot\varphi)=(f\circledcirc g)\cdot\varphi\ .$ ∎

######  Lemma 2.29.

Let $H$ be a graded $R$-module and let $\varphi$ and $\psi$ be two
$\Omega\mathcal{C}$-algebra structures. Let $n\geqslant 1$ be an integer and
suppose that there exists an $\infty$-isomorphism
$f:\varphi\rightsquigarrow\psi$ such that

  * $\centerdot$

$\varphi^{(k)}=0$ , for $2\leqslant k\leqslant n$ and

  * $\centerdot$

$f^{(k)}=0$ , for $1\leqslant k\leqslant n-1$ .

The weight components of $\psi$ are then given by

  1. $\centerdot$

$\psi^{(1)}=\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(1)}\right)+\left(f^{(0)}\circ
d_{\mathcal{C}}\right)\circledcirc\left(f^{(0)}\right)^{-1}$ ,

  2. $\centerdot$

$\psi^{(k)}=0$ , for $2\leqslant k\leqslant n$ , and

  3. $\centerdot$

$\psi^{(n+1)}=\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(n+1)}\right)-d^{\psi^{(1)}}\left(f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right)$
.

In particular, if $f$ is an $\infty$-isotopy, then $\psi$ decomposes as

$\psi=\varphi^{(1)}+\varphi^{(n+1)}-d^{\varphi^{(1)}}\left(f^{(n)}\right)+\psi^{(n+2)}+\cdots\
.$

###### Proof.

By definition of an $\infty$-morphism, we have

(5) $\psi\circledcirc f=f\star\varphi+f\circ d_{\mathcal{C}}\ .$

The assumptions on $\varphi$ and on $f$ imply that

$f\star\varphi\equiv
f^{(0)}\star\varphi^{(1)}+f^{(0)}\star\varphi^{(n+1)}+f^{(n)}\star\varphi^{(1)}\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\
,$ $\psi\circledcirc f\equiv\psi\circledcirc
f^{(0)}+\psi^{(1)}\circledcirc\left(f^{(0)};f^{(n)}\right)\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\
,$ $\mbox{and}\quad\quad f\circ d_{\mathcal{C}}\equiv f^{(0)}\circ
d_{\mathcal{C}}+f^{(n)}\circ
d_{\mathcal{C}}\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\ .$

Equation (5) implies that

$\begin{split}\psi\ \equiv\
&\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(1)}\right)+\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(n+1)}\right)+\left(f^{(n)}\star\varphi^{(1)}\right)\circledcirc\left(f^{(0)}\right)^{-1}+\left(f^{(0)}\circ
d_{\mathcal{C}}\right)\circledcirc\left(f^{(0)}\right)^{-1}\\\
&+\left(f^{(n)}\circ
d_{\mathcal{C}}\right)\circledcirc\left(f^{(0)}\right)^{-1}-\left(\psi^{(1)}\circledcirc\left(f^{(0)};f^{(n)}\right)\right)\circledcirc\left(f^{(0)}\right)^{-1}\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\
.\end{split}$

This gives the desired formulas for the components $\psi^{(k)}$ for
$1\leqslant k\leqslant n$. Furthermore, by Formula (e) of Lemma 2.25, we have

$\left(f^{(n)}\star\varphi^{(1)}\right)\circledcirc\left(f^{(0)}\right)^{-1}\overset{(e)}{=}\left(f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right)\star\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(1)}\right)$

and it follows from Formulas (b) and (c) of Lemma 2.25, that

$\left(\psi^{(1)}\circledcirc\left(f^{(0)};f^{(n)}\right)\right)\circledcirc\left(f^{(0)}\right)^{-1}=\psi^{(1)}\star\left(f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right)\
.$

Formula (j) of Lemma 2.26 implies that

$\left(f^{(n)}\circ
d_{\mathcal{C}}\right)\circledcirc\left(f^{(0)}\right)^{-1}\overset{(j)}{=}\left(f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right)\circ
d_{\mathcal{C}}-f^{(n)}\circledcirc\left(\left(f^{(0)}\right)^{-1};\left(f^{(0)}\right)^{-1}\circ
d_{\mathcal{C}}\right)\ .$

By Formula (l) of Lemma 2.26 and Formula (a) of Lemma 2.25, we have

$-f^{(n)}\circledcirc\left(\left(f^{(0)}\right)^{-1};\left(f^{(0)}\right)^{-1}\circ
d_{\mathcal{C}}\right)\overset{(a)}{=}\left(f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right)\star\left(\left(f^{(0)}\circ
d_{\mathcal{C}}\right)\circledcirc\left(f^{(0)}\right)^{-1}\right)\ .$

We obtain that

$\psi^{(n+1)}=\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(n+1)}\right)-\left[\psi^{(1)},f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right]+\left(f^{(n)}\circledcirc\left(f^{(0)}\right)^{-1}\right)\circ
d_{\mathcal{C}}\ .\qed$

###  2.4. Operadic Kaledin classes

Let $(H,\varphi)$ be an $\Omega\mathcal{C}$-algebra structure on a graded
$R$-module $H$. In the spirit of gauge formality and Proposition 2.17, this
section addresses the following problem: does there exist an $\infty$-isotopy
between $\varphi$ and $\varphi^{(1)}$ ? If $R$ is a $\mathbb{Q}$-algebra, this
is equivalent to the existence of a gauge $\lambda\in\mathfrak{g}_{H}$ such
that

$\lambda\cdot\varphi=\varphi^{(1)}\ ,$

through the isomorphism given by Theorem 2.12. This last problem is fully
characterized by the Kaledin classes, applying Theorem 1.31 to the $1$-weight-
graded dg Lie algebra $\mathfrak{g}_{H}$ . Nonetheless, we aim at studying
this question for any base rings $R$. If $R$ that is not a
$\mathbb{Q}$-algebra, the gauge group do not even exist, neither the
isomorphism of Theorem 2.12. However, we can still ask for the existence of an
$\infty$-isotopy $\varphi\rightsquigarrow\varphi^{(1)}$. The goal of this
section is to adapt the methods of Section 1.3 to deal with any commutative
ground ring.

######  Remark 2.30.

For the sake of clarity, we deal with the case $\delta=1$. But if
$d_{\mathcal{C}}=0$ then $\mathfrak{g}_{H}$ is also $\delta$-weight-graded dg
Lie algebra for all $\delta\geqslant 1$. All what follows adapts directly to
detect if an $\Omega\mathcal{C}$-algebra structure
$\varphi\in\mathcal{F}^{\delta}(\mathfrak{g}_{H})$ is related to
$\varphi^{\delta}$ by an $\infty$-isotopy.

######  Definition 2.31 (Operadic Kaledin classes).

Let $\varphi\in\mathrm{MC}(\mathfrak{g}_{H})$ be an
$\Omega\mathcal{C}$-algebra structure on a graded $R$-module $H$ and let
$\mathfrak{D}(\varphi)$ be its prismatic decomposition. The _Kaledin class_
$K_{\varphi}$ is the Kaledin class of $\mathfrak{D}(\varphi)$, i.e.

$K_{\varphi}=\left[\partial_{\hbar}\mathfrak{D}(\varphi)\right]\in
H_{-1}\left(\mathfrak{g}_{H}[\\![\hbar]\\!]^{\Phi}\right)\ .$

Its _$n^{\text{th}}$ -truncated Kaledin class_ $K_{\varphi}^{n}$ is the
$n^{\text{th}}$-truncated Kaledin class of $\mathfrak{D}(\varphi)$, i.e.

$K^{n}_{\varphi}=\left[\varphi^{(2)}+2\varphi^{(3)}\hbar+\dots+n\varphi^{(n+1)}\hbar^{n-1}\right]\in
H_{-1}\left(\left(\mathfrak{g}_{H}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\right)\
.$

######  Proposition 2.32.

Let $R$ be a commutative ground ring and let $\mathcal{C}$ be a reduced
weight-graded dg cooperad over $R$. Let
$\varphi\in\mathrm{MC}(\mathfrak{g}_{H})$ be an $\Omega\mathcal{C}$-algebra
structure on a graded $R$-module $H$ .

  1. (1)

Let $n\geqslant 1$ be an integer such that $n!$ is a unit in $R$. The
$n^{\text{th}}$-truncated Kaledin class $K^{n}_{\varphi}$ is zero if and only
if there exists an $\infty$-isotopy $f:\varphi\rightsquigarrow\psi$ where
$\psi\in\mathrm{MC}(\mathfrak{g}_{H})$ is such that $\psi^{(k)}=0$ , for
$2\leqslant k\leqslant n+1$.

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra, the Kaledin class $K_{\varphi}$ vanishes if
and only if there exists an $\infty$-isotopy
$f:\varphi\rightsquigarrow\varphi^{(1)}$ .

######  Remark 2.33.

If $R$ is a $\mathbb{Q}$-algebra, Proposition 2.32 is a direct corollary of
Proposition 1.33 and Theorem 1.31 through the isomorphism of Theorem 2.12. In
order to deal with any commutative ground ring, we are going to adapt the
proofs, mainly by using $\infty$-isotopies of the form $1+\lambda$ , instead
of exponential maps $e^{\lambda}$ .

######  Definition 2.34.

The _prismatic decomposition_ of $f\in\mathfrak{a}_{H}$ in degree zero is
defined by

$\mathfrak{D}\left(f\right)\coloneqq
f^{(0)}+f^{(1)}\hbar+f^{(2)}\hbar^{2}+\cdots\in\mathfrak{a}_{H}[\\![\hbar]\\!]\
.$

######  Remark 2.35.

The pre-Lie product $\star$ and the circle product $\circledcirc$ extend by
$\hbar$-linearity to $\mathfrak{a}_{H}[\\![\hbar]\\!]$. For all
$f,g\in\mathfrak{a}_{H}$ of degree zero, we have

$\mathfrak{D}\left(f\star
g\right)=\mathfrak{D}\left(f\right)\star\mathfrak{D}\left(g\right)\quad\mbox{and}\quad\mathfrak{D}\left(f\circledcirc
g\right)=\mathfrak{D}\left(f\right)\circledcirc\mathfrak{D}\left(g\right)\ .$

Furthermore, if $f:\varphi\rightsquigarrow\psi$ is an $\infty$-isomorphism
then $\mathfrak{D}\left(f\right)$ satisfies

$\mathfrak{D}\left(\psi\right)=\left(\mathfrak{D}\left(f\right)\star\mathfrak{D}\left(\varphi\right)\right)\circledcirc\mathfrak{D}\left(f\right)^{-1}+\left(\mathfrak{D}\left(f\right)\circ
d_{\mathcal{C}}\right)\circledcirc\mathfrak{D}\left(f\right)^{-1}\ .$

######  Lemma 2.36.

For all $F,G\in\mathfrak{a}_{H}[\\![\hbar]\\!]$ , the following identities
hold:

  1. (1)

$\partial_{\hbar}(F\star G)=\partial_{\hbar}F\star G+F\star\partial_{\hbar}G\
,$

  2. (2)

$\partial_{\hbar}(F\circledcirc G)=\partial_{\hbar}F\circledcirc
G+F\circledcirc(G;\partial_{\hbar}G)\ ,$

If furthermore, the element $F$ is invertible with respect to $\circledcirc$ ,
then

  1. (3)

$\partial_{\hbar}(F^{-1})=-F^{-1}\star\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\ .$

###### Proof.

Points (1) and (2) follow from the $\hbar$-linearity of the pre-Lie product.
Furthermore, by Point (2), applied to $\partial_{\hbar}(F^{-1}\circledcirc
F)=\partial_{\hbar}\left(1\right)=0\ ,$ the following equality holds

$\partial_{\hbar}\left(F^{-1}\right)=-\left(F^{-1}\circledcirc(F;\partial_{\hbar}F)\right)\circledcirc
F^{-1}\ .$

Formula (d) of Lemma 2.25 then implies

$\partial_{\hbar}\left(F^{-1}\right)\overset{(d)}{=}-F^{-1}\star\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\ .\qed$

######  Lemma 2.37.

Let $n\geqslant 1$ and let $f:\varphi\rightsquigarrow\psi$ be an
$\infty$-isomorphism. Let us denote $\Phi\coloneqq\mathfrak{D}(\varphi)$ ,
$\Psi\coloneqq\mathfrak{D}(\psi)$ and $F\coloneqq\mathfrak{D}(f)$. There are
isomorphisms of dg Lie algebras

$\mathrm{Ad}_{F}:\mathfrak{g}_{H}[\\![\hbar]\\!]^{\Phi}\to\mathfrak{g}_{H}[\\![\hbar]\\!]^{\Psi}\quad\mbox{and}\quad\mathrm{Ad}_{F}^{n}:\left(\mathfrak{g}_{H}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\to\left(\mathfrak{g}_{H}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Psi}^{n}}$

induced by the mapping $x\mapsto(F\star x)\circledcirc F^{-1}$.

###### Proof.

The proof of Proposition 2.27 holds _mutatis mutandis_ by $\hbar$-linearity. ∎

######  Lemma 2.38 (Invariance of Kaledin classes under
$\infty$-isomorphisms).

Let $n\geqslant 1$ and let $f:\varphi\rightsquigarrow\psi$ be an
$\infty$-isomorphism. Let us denote $\Phi\coloneqq\mathfrak{D}(\varphi)$ ,
$\Psi\coloneqq\mathfrak{D}(\psi)$ and $F\coloneqq\mathfrak{D}(f)$.

  1. (1)

The classes $K_{\Psi}$ and $\mathrm{Ad}_{F}\left(K_{\Phi}\right)$ are equal in

$H_{-1}\left(\mathfrak{g}_{H}[\\![\hbar]\\!]^{\Psi}\right)\ ,$

where we still denote by $\mathrm{Ad}_{F}$ the induced isomorphism on
homology.

  2. (2)

The classes $K_{\Psi}^{n}$ and $\mathrm{Ad}_{F}^{n}\left(K_{\Phi}^{n}\right)$
are equal in

$H_{-1}\left(\left(\mathfrak{g}_{H}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Psi}^{n}}\right)\
,$

where we still denote by $\mathrm{Ad}_{F}^{n}$ the induced isomorphism on
homology.

###### Proof.

Let us prove point $(1)$ . By definition of an $\infty$-isomorphism, we have

$\psi=\left(f\star\varphi\right)\circledcirc f^{-1}+\left(f\circ
d_{\mathcal{C}}\right)\circledcirc f^{-1}$

and thus $\Psi=\mathrm{Ad}_{F}(\Phi)+\left(F\circ
d_{\mathcal{C}}\right)\circledcirc F^{-1}$ . Let us prove that the cycles
$\mathrm{Ad}_{F}(\partial_{\hbar}\Phi)$ and $\partial_{\hbar}\Psi$ are
homologous. The relations of Lemma 2.36 imply that

$\partial_{\hbar}\left(\mathrm{Ad}_{F}(\Phi)\right)=\left(\partial_{\hbar}F\star\Phi\right)\circledcirc
F^{-1}+\left(F\star\partial_{\hbar}\Phi\right)\circledcirc
F^{-1}+(F\star\Phi)\circledcirc\left(F^{-1};\partial_{\hbar}(F^{-1})\right)\
.$

By Formula (e) of Lemma 2.25, the following equality holds

$\left(\partial_{\hbar}F\star\Phi\right)\circledcirc
F^{-1}=\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\star\mathrm{Ad}_{F}(\Phi)\ .$

Using the formula of $\partial_{\hbar}(F^{-1})$ given by Lemma 2.36 and
Formula (a) of Lemma 2.25, we have

$\begin{split}(F\star\Phi)\circledcirc\left(F^{-1};\partial_{\hbar}(F^{-1})\right)&=-(F\star\Phi)\circledcirc\left(F^{-1};F^{-1}\star\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\right)\\\
&\overset{(a)}{=}-\mathrm{Ad}_{F}(\Phi)\star\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\ .\end{split}$

This shows that
$\partial_{\hbar}\left(\mathrm{Ad}_{F}(\Phi)\right)=\mathrm{Ad}_{F}(\partial_{\hbar}\Phi)-\left[\mathrm{Ad}_{F}(\Phi),\partial_{\hbar}F\circledcirc
F^{-1}\right]\ .$ Formula (2) of Lemma 2.36 implies that

$\partial_{\hbar}\left(\left(F\circ d_{\mathcal{C}}\right)\circledcirc
F^{-1}\right)=\left(\partial_{\hbar}F\circ d_{\mathcal{C}}\right)\circledcirc
F^{-1}+\left(F\circ
d_{\mathcal{C}}\right)\circledcirc\left(F^{-1};\partial_{\hbar}(F^{-1})\right)\
.$

Using the expression of $\partial_{\hbar}(F^{-1})$ and Formula (a) of Lemma
2.25, we have

$\left(F\circ
d_{\mathcal{C}}\right)\circledcirc\left(F^{-1};\partial_{\hbar}(F^{-1})\right)\overset{(a)}{=}-\left(\left(F\circ
d_{\mathcal{C}}\right)\circledcirc
F^{-1}\right)\star\left(\partial_{\hbar}F\circledcirc F^{-1}\right)\ .$

Formula (j) of Lemma 2.26 implies that

$\left(\partial_{\hbar}F\circ d_{\mathcal{C}}\right)\circledcirc
F^{-1}=\left(\partial_{\hbar}F\circledcirc F^{-1}\right)\circ
d_{\mathcal{C}}-\partial_{\hbar}F\circledcirc\left(F^{-1};F^{-1}\circ
d_{\mathcal{C}}\right)\ .$

Using the expression of $F^{-1}\circ d_{\mathcal{C}}$ given by Lemma 2.26 and
Formula (a) of Lemma 2.25, we have

$\partial_{\hbar}F\circledcirc\left(F^{-1};F^{-1}\circ
d_{\mathcal{C}}\right)\overset{(a)}{=}-\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\star\left(\left(F\circ d_{\mathcal{C}}\right)\circledcirc
F^{-1}\right)\ .$

This leads to

$\partial_{\hbar}\left(\left(F\circ d_{\mathcal{C}}\right)\circledcirc
F^{-1}\right)=\left(\partial_{\hbar}F\circledcirc F^{-1}\right)\circ
d_{\mathcal{C}}-\left[\left(F\circ d_{\mathcal{C}}\right)\circledcirc
F^{-1},\partial_{\hbar}F\circledcirc F^{-1}\right]\leavevmode\nobreak\
,\quad\mbox{and}$ (6)
$\mathrm{Ad}_{F}(\partial_{\hbar}\Phi)-\partial_{\hbar}\Psi=d^{\Psi}\left(\partial_{\hbar}F\circledcirc
F^{-1}\right)\ .$

The proof of Point (2) is similar. ∎

###### Proof of Proposition 2.32.

Let us prove Point (1). Let $\psi\in\mathrm{MC}(\mathfrak{g}_{H})$ be such
that $\psi^{(k)}=0$ , for $2\leqslant k\leqslant n+1$ , and let
$\varphi\rightsquigarrow\psi$ be an $\infty$-isotopy. Point (2) of Lemma 2.38
implies that

$K_{\Phi}^{n}=\left(\mathrm{Ad}_{F}^{n}\right)^{-1}\left(K_{\Psi}^{n}\right)=\left(\mathrm{Ad}_{F}^{n}\right)^{-1}\left(K_{\varphi^{(1)}}^{n}\right)=0\
.$

Let us prove the converse result by induction on $n\geqslant 1$. Suppose that

$K^{1}_{\Phi}=0\in H_{-1}\left(\mathfrak{g}^{\varphi^{(1)}}_{H}\right)\ .$

There exists $\upsilon\in\mathfrak{g}_{H}$ of degree zero such that
$d^{\varphi^{(1)}}(\upsilon)=\varphi^{(2)}\ .$ Let us set $f\coloneqq
1+\upsilon^{(1)}$ and

$\psi\coloneqq(f\star\varphi)\circledcirc f^{-1}+\left(f\circ
d_{\mathcal{C}}\right)\circledcirc f^{-1}\ .$

By construction, $\psi\in\mathrm{MC}(\mathfrak{g}_{H})$ and $f$ determines an
$\infty$-isotopy $\varphi\rightsquigarrow\psi\ .$ Lemma 2.29 gives

$\psi^{(2)}=\varphi^{(2)}-d^{\varphi^{(1)}}\left(\upsilon^{(1)}\right)=0\ .$

Suppose that $n>1$ and that the result holds for $n-1$ . If the class
$K_{\Phi}^{n}$ is zero, so does the class $K_{\Phi}^{n-1}$. By the induction
hypothesis, there exist $\psi\in\mathrm{MC}(\mathfrak{g}_{H})$ such that
$\psi^{(k)}=0$ for $2\leqslant k\leqslant n$ and an $\infty$-isotopy
$g:\varphi\rightsquigarrow\psi$ . Point (2) of Lemma 2.38 implies that

$\mathrm{Ad}_{\mathfrak{D}(g)}^{n}(K^{n}_{\Phi})=K^{n}_{\mathfrak{D}(\psi)}=[n\psi^{(n+1)}\hbar^{n-1}]=0\
.$

There exists
$\upsilon\coloneqq\upsilon_{0}+\upsilon_{1}h+\dots+\upsilon_{n-1}\hbar^{n-1}\in\mathfrak{g}_{H}[\\![\hbar]\\!]$
of degree zero such that

$d^{\varphi^{(1)}}(\upsilon)\equiv n\psi^{(n+1)}\hbar^{n-1}\pmod{\hbar^{n}}\
.$

Looking at the coefficient of $\hbar^{n-1}$ and in weight $n+1$ on both sides,
we have

$d^{\varphi^{(1)}}\left(\upsilon_{n-1}^{(n)}\right)=n\psi^{(n+1)}\ .$

Let us consider $\lambda\coloneqq\frac{1}{n}\upsilon_{n-1}^{(n)}$ and
$f\coloneqq 1+\lambda$ . We set

$\psi^{\prime}\coloneqq(f\star\psi)\circledcirc f^{-1}+\left(f\circ
d_{\mathcal{C}}\right)\circledcirc f^{-1}\in\mathrm{MC}(\mathfrak{g}_{H})\ .$

By construction, $f\circledcirc g$ determines an $\infty$-isotopy
$\varphi\rightsquigarrow\psi^{\prime}\ .$ Lemma 2.29 implies that

$\left(\psi^{\prime}\right)^{(k)}=0\leavevmode\nobreak\
,\quad\mbox{for}\;2\leqslant k\leqslant n\leavevmode\nobreak\
,\quad\mbox{and}\quad(\psi^{\prime})^{(n+1)}=\psi^{(n+1)}-d^{\varphi^{(1)}}\left(\lambda\right)=0\
,$

and we get the desired result. Let us prove Point (2). If there exists an
$\infty$-isotopy $f:\varphi\rightsquigarrow\varphi^{(1)}$ , Point (1) of Lemma
2.38 implies that

$K_{\Phi}=\mathrm{Ad}_{F}^{-1}\left(K_{\varphi^{(1)}}\right)=0\ .$

Conversely, suppose that $K_{\Phi}=0$ . If $\Phi=\varphi^{(1)}$ , the result
is immediate. Otherwise, let $n_{1}>1$ be the smallest integer such that
$\varphi^{(n_{1})}$ is not zero. Since $K_{\Phi}=0$ , the class
$K_{\Phi}^{n_{1}}$ is also zero. By the proof of Point (1), there exist
$\psi\in\mathrm{MC}(\mathfrak{g}_{H})$ such that $\psi^{(k)}=0$ for
$2\leqslant k\leqslant n_{1}+1$ and an $\infty$-isotopy between $\varphi$ and
$\psi$ of the form

$f_{n_{1}}\coloneqq 1+\lambda_{1}\ ,\quad\mbox{where
}\lambda_{1}\in\mathfrak{g}_{H}^{(n_{1})}\ .$

We can repeat the procedure with $\psi_{n_{1}}\coloneqq\psi$ . If $R$ is a
$\mathbb{Q}$-algebra, we obtain a sequence

$n_{1}<n_{2}<n_{3}<\cdots$

of integers and a sequence of $\Omega\mathcal{C}$-algebra structures
$\psi_{n_{i}}$ such that $\psi_{n_{i}}$ is zero in weight $k$ for $2\leqslant
k\leqslant n_{i}+1$ associated to a sequence of $\infty$-isotopies

${\varphi}$${\psi_{n_{1}}}$${\psi_{n_{2}}}$${\cdots\
.}$$\scriptstyle{f_{n_{1}}}$$\scriptstyle{f_{n_{2}}}$$\scriptstyle{f_{n_{3}}}$

where $f_{n_{i}}$ is of the form $1+\lambda_{i}$ with $\lambda_{i}$
concentrated in weight $n_{i}$. The composite

$f\coloneqq\cdots f_{n_{3}}\circledcirc f_{n_{2}}\circledcirc f_{n_{1}}$

is well-defined by construction and induces an $\infty$-isotopy
$f:\varphi\rightsquigarrow\varphi^{(1)}$ . ∎

######  Proposition 2.39.

Let $H$ be a graded $R$-module and let $(H,\varphi)$ be an
$\Omega\mathcal{C}$-algebra structure. If $R$ is a $\mathbb{Q}$-algebra, the
Kaledin class $K_{\varphi}$ is zero if and only if, for all $n\geqslant 1$,
the $n^{\text{th}}$-truncated Kaledin class $K^{n}_{\varphi}$ is zero.

###### Proof.

The proof makes use of the Kaledin classes and Proposition 2.32 in a way
similar to the proof of Proposition 1.24. ∎

###  2.5. Kaledin classes in the colored setting

All the previous results apply _mutatis mutandis_ to operads colored in
groupoids. The notion of a colored operad is a generalization of the notion of
operads in which each input or output comes with a color chosen in a given
set. A composition is possible whenever the colors of the corresponding input
and output involved match. While an operad acts on a single chain complex,
colored operads encode operations acting on various chain complexes. In order
to encode symmetries, one can also consider groupoid colored operads where
colors are chosen in a given groupoid $\mathbb{V}$. The associated Koszul
duality theory was developed by Ward in [War21]. We refer the reader to
[RiL22, Chapter 2, Section 6.4] for more details.

######  Example 2.40 (Endomorphism operad).

Let $A$ be a $\mathbb{V}$-module, i.e. a functor
$A:\mathbb{V}\to\mathrm{dgMod_{R}}$. The associated endomorphism
$\mathbb{V}$-colored operad is given by

$\mathrm{End}_{A}\left(v_{0};v_{1},\dots,v_{r}\right)\coloneqq\mathrm{Hom}\left(A(v_{0});A(v_{1})\otimes\cdots\otimes
A(v_{r})\right)$

with the $\mathbb{S}_{r}$ action given by permuting the inputs.

We fix $\mathbb{V}$ a groupoid and $\mathcal{C}$ a reduced
$\mathbb{V}$-colored weight-graded dg cooperad over $R$. For instance, let
$\mathcal{C}=\mathcal{P}^{\text{\raisebox{0.0pt}{\textexclamdown}}}$ be the
dual $\mathbb{V}$-colored cooperad of a Koszul $\mathbb{V}$-colored operad
$\mathcal{P}$, see [War21, Definition 2.45].

######  Example 2.41.

We denote by $\mathbb{S}$ the groupoid defined by
$\mathbb{S}\coloneqq\bigsqcup_{n\geqslant 0}\mathrm{B}\mathbb{S}_{n}$ where
$\mathrm{B}\mathbb{S}_{n}$ is the category with one object $\\{*\\}$ and
$\mathrm{Aut}(*)=\mathbb{S}_{n}$ . As proved by Ward in [War21], there exists
an $\mathbb{S}$-colored Koszul operad encoding symmetric operads.

The previous sections generalize directly to this setting by working in the
following colored version of the convolution dg pre-Lie algebra.

######  Definition 2.42 (Colored convolution dg pre-Lie algebra).

The _convolution dg pre-Lie algebra_ associated to $\mathcal{C}$ and to a
$\mathbb{V}$-module $A$ is the dg pre-Lie algebra

$\mathfrak{g}_{A}:=\left(\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right),\star,d\right)\
,$

where the underlying space is

$\mathrm{Hom}_{\mathcal{S}_{\mathbb{V}}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right):=\prod_{\mathcal{S}_{\mathbb{V}}}\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}}\left(v_{0};v_{1},\dots,v_{r}\right),\mathrm{End}_{A}\left(v_{0};v_{1},\dots,v_{r}\right)\right)\
,$

where
$\mathcal{S}_{\mathbb{V}}\coloneqq\\{\left(v_{1},\dots,v_{r};v_{0}\right)\in\mathrm{ob}\left(\mathbb{V}\right)^{r+1}\\}$.
It is equipped with the pre-Lie product

$\varphi\star\psi\coloneqq\overline{\mathcal{C}}\xrightarrow{\Delta_{(1)}}\overline{\mathcal{C}}\circ_{(1)}\overline{\mathcal{C}}\xrightarrow{\varphi\circ_{(1)}\psi}\mathrm{End}_{A}\circ_{(1)}\mathrm{End}_{A}\xrightarrow{\gamma_{(1)}}\mathrm{End}_{A}\
,$

and the differential

$d(\varphi)=d_{\mathrm{End}_{A}}\circ\varphi-(-1)^{|\varphi|}\varphi\circ
d_{\overline{\mathcal{C}}}\ .$

######  Assumptions 2.

Let $R$ be a commutative ground ring and let $A$ be a chain complex related to
$H(A)$ via a contraction. Let $\mathbb{V}$ be a groupoid and let $\mathcal{C}$
be a reduced weight-graded $\mathbb{V}$-colored dg cooperad where $R$ is
either

  * $\centerdot$

any commutative ground ring if $\mathbb{V}$ is a set and $\mathcal{C}$ is a
non-symmetric;

  * $\centerdot$

a characteristic zero field if $\mathbb{V}$ is a groupoid.

######  Theorem 2.43 (Homotopy transfer theorem).

Under Assumptions 2, for any $\Omega\mathcal{C}$-algebra structure, there
exists a transferred structure $(H(A),\varphi_{t})$. Ihis is independent of
the choice of contraction, i.e. any two such transferred structures are
related by an $\infty$-isotopy.

###### Proof.

The groupoid colored version of the homotopy transfer theorem is given in
[War21] for a characteristic zero field. In the case of set colored non-
symmetric operads, one can deal with any coefficient ring see e.g. [DV15,
Theorem 5]. ∎

######  Theorem 2.44.

Let $R$ be a commutative ring. Let $\mathbb{V}$ be a groupoid and let
$\mathcal{C}$ be a $\mathbb{V}$-colored reduced weight-graded dg cooperad over
$R$. Let $(A,\varphi)$ be a $\Omega\mathcal{C}$-algebra structure such that
there exists a transferred structure $(H(A),\varphi_{t})$.

  1. (1)

Let $n\geqslant 1$ be an integer such that $n!$ is a unit in $R$. The algebra
$(A,\varphi)$ is gauge $n$-formal if and only if the truncated class
$K^{n}_{\varphi_{t}}$ is zero.

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra, the algebra $(A,\varphi)$ is gauge formal if
and only if the class $K_{\varphi_{t}}$ is zero.

###### Proof.

This follows from the straightforward colored counterpart of Proposition 2.32,
by using Definition 2.22 and Proposition 2.17. ∎

######  Remark 2.45 (Naturality in the algebra).

Let us consider $\infty$-quasi-isomorphic $\Omega\mathcal{C}$-algebras
$(A,\varphi)$ and $(B,\psi)$, admitting transferred structures. This leads to
an $\infty$-isomorphism

${f:(H(A),\varphi_{t})}$${(A,\varphi)}$${(B,\psi)}$${(H(B),\psi_{t})\ .}$

By setting $\psi_{t}^{\prime}=\left(f^{(0)}\right)^{-1}\cdot\psi_{t}$, we have
an $\infty$-isotopy

${g:(H(A),\varphi_{t})}$${(H(B),\psi_{t})}$${(H(A),\psi_{t}^{\prime})\
,}$$\scriptstyle{f}$$\scriptstyle{\left(f^{(0)}\right)^{-1}}$

see Theorem 2.28. By Lemma 2.38, their respective Kaledin classes are related
by

$K_{\psi_{t}^{\prime}}=\mathrm{Ad}_{\mathfrak{D}(g)}(K_{\varphi_{t}})\quad\mbox{and}\quad
K_{\psi_{t}^{\prime}}^{n}=\mathrm{Ad}_{\mathfrak{D}(g)}^{n}(K_{\varphi_{t}}^{n})\
.$

##  3\. Kaledin classes in the properadic setting

The formality question also arises for algebraic structures involving
operations with several inputs but also several outputs such as dg Frobenius
bialgebras, dg involutive Lie bialgebras etc. Such structures are encoded by a
generalization of operads introduced in [Val07] called properads. The purpose
of this section is to recall the associated deformation theory and apply the
methods of Section 1 in order to study the formality of algebras over a
properad.

In all this section, $R$ denotes a characteristic zero field. Let $A$ be a
chain complex and let $\mathcal{C}$ be a reduced weight-graded dg coproperad
over $R$, i.e.

$\mathcal{C}=\mathcal{I}\oplus\mathcal{C}^{(1)}\oplus\mathcal{C}^{(2)}\oplus\cdots\oplus\mathcal{C}^{(n)}\oplus\cdots\quad\
,\quad
d_{\mathcal{C}}\left(\mathcal{C}^{(k)}\right)\subset\mathcal{C}^{(k-1)}$

and $\mathcal{C}(0,0)=0$. Let us denote the coaugmentation coideal
$\overline{\mathcal{C}}\coloneqq\bigoplus_{k\geqslant 1}\mathcal{C}^{(k)}$ so
that $\mathcal{C}\cong\mathrm{I}\oplus\overline{\mathcal{C}}$ .

######  Example 3.1.

Among possible choices for such a coproperad $\mathcal{C}$, we have:

  1. (1)

The bar construction $\mathcal{C}\coloneqq\mathrm{B}\mathcal{P}$ of a reduced
properad $\mathcal{P}$ in $\mathrm{grMod}_{R}$, see [Val07] .

  2. (2)

The Koszul dual coproperad
$\mathcal{C}:=\mathcal{P}^{\text{\raisebox{0.0pt}{\textexclamdown}}}$, if
$\mathcal{P}$ is a homogeneous Koszul properad, see [Val07], or an
inhomogeneous quadratic Koszul properad, see [GCTV12, Appendix A].

###  3.1. Recollections on properadic homological algebra

The Koszul duality theory of properads, the associated deformation theory and
the homotopy theory of algebras over a properad were respectively developed in
[Val07], [MV09a, MV09b] and [HLV20]. In this section, we recall the definition
of $\Omega\mathcal{C}$-algebra structures on $A$, as Maurer–Cartan elements of
the associated convolution algebra $\mathfrak{g}_{A}$. We refer the reader to
[HLV20] for more details.

######  Definition 3.2 (dg Lie-admissible algebra).

A _dg Lie-admissible algebra_ is a chain complex $\mathfrak{g}$ equipped with
a linear map of degree zero
$\star:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$ such that $d$ is a
derivation and such that the skew-symmetrized bracket induces a dg Lie algebra
structure.

######  Proposition 3.3 ([MV09a, Proposition 11]).

The _convolution dg Lie-admissible algebra_ associated to the coproperad
$\mathcal{C}$ and the chain complex $A$ is the Lie-admissible algebra

$\mathfrak{g}_{A}\coloneqq\left(\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right),\partial,\star\right)\
,$

where the underlying space is

$\mathrm{Hom}_{\mathbb{S}}\left(\overline{\mathcal{C}},\mathrm{End}_{A}\right)\coloneqq\prod_{m,n\geqslant
0}\mathrm{Hom}_{\mathbb{S}_{m}^{op}\times\mathbb{S}_{n}}\left(\overline{\mathcal{C}}(m,n),\mathrm{End}_{A}(m,n)\right)\
,$

equipped with the Lie-admissible product

$\varphi\star\psi\coloneqq\overline{\mathcal{C}}\xrightarrow{\Delta_{(1,1)}}\overline{\mathcal{C}}\underset{(1,1)}{\boxtimes}\overline{\mathcal{C}}\xrightarrow{\varphi\underset{(1,1)}{\boxtimes}\psi}\mathrm{End}_{A}\underset{(1,1)}{\boxtimes}\mathrm{End}_{A}\xrightarrow{\gamma_{(1,1)}}\mathrm{End}_{A}\
,$

and the differential

$d(\varphi)\coloneqq
d_{\mathrm{End}_{A}}\circ\varphi-(-1)^{|\varphi|}\varphi\circ
d_{\overline{\mathcal{C}}}\ .$

######  Remark 3.4.

It embeds into the dg Lie-admissible algebra made up of maps from
$\mathcal{C}$ , i.e.

$\mathfrak{g}_{A}\hookrightarrow\mathfrak{a}_{A}\coloneqq\left(\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}_{A}\right),\star,d\right)\
.$

######  Proposition 3.5 ([HLV20, Proposition 3.19]).

The space $\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}_{A}\right)$
is equipped with the following associative and unital product

$\varphi\circledcirc\psi\coloneqq\mathcal{C}\xrightarrow{\Delta}\mathcal{C}\boxtimes\mathcal{C}\xrightarrow{\varphi\boxtimes\psi}\mathrm{End}_{A}\boxtimes\mathrm{End}_{A}\xrightarrow{\gamma}\mathrm{End}_{A}\
,$

where the unit is given by $1:\mathcal{I}\mapsto\mathrm{id}_{A}$ and $\Delta$
and $\gamma$ stand respectively for the decomposition map of the coproperad
$\mathcal{C}$ and the composition map of the endomorphism properad
$\mathrm{End}_{A}$ .

######  Definition 3.6.

A _$\Omega\mathcal{C}$ -algebra structure_ on $A$ is a Maurer–Cartan element
$\varphi\in\mathrm{MC}(\mathfrak{g}_{A})$. An _$\infty$ -morphism_
$\varphi\rightsquigarrow\psi$ between two $\Omega\mathcal{C}$-algebra
structures $\varphi$ and $\psi$ on $A$ is a map $f\in\mathfrak{a}_{A}$ of
degree $0$ satisfying

$f\rhd\varphi-\psi\lhd f=d(f)\ \ ,$

where the left and right actions are respectively defined by

$\begin{array}[]{ccccccccc}\psi\lhd
f&\coloneqq&\overline{\mathcal{C}}&\xrightarrow{\Delta_{(1,*)}}&\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\mathcal{C}&\xrightarrow{\psi\underset{(1,*)}{\boxtimes}f}&\mathrm{End}_{A}\underset{(1,*)}{\boxtimes}\mathrm{End}_{A}&\longrightarrow&\mathrm{End}_{A}\
,\\\
&&\mathcal{I}&\overset{\cong}{\longrightarrow}&\mathcal{I}\boxtimes\mathcal{I}&\xrightarrow{\varphi\;\boxtimes\;f}&\mathrm{End}_{A}\boxtimes\mathrm{End}_{A}&\longrightarrow&\mathrm{End}_{A}\
,\\\ &&&&&&&&\\\
f\rhd\varphi&\coloneqq&\overline{\mathcal{C}}&\xrightarrow{\Delta_{(*,1)}}&\mathcal{C}\underset{(*,1)}{\boxtimes}\overline{\mathcal{C}}&\xrightarrow{f\underset{(*,1)}{\boxtimes}\varphi}&\mathrm{End}_{A}\underset{(*,1)}{\boxtimes}\mathrm{End}_{A}&\longrightarrow&\mathrm{End}_{A}\
,\\\
&&\mathcal{I}&\overset{\cong}{\longrightarrow}&\mathcal{I}\boxtimes\mathcal{I}&\xrightarrow{f\;\boxtimes\;\varphi}&\mathrm{End}_{A}\boxtimes\mathrm{End}_{A}&\longrightarrow&\mathrm{End}_{A}\
.\end{array}$

######  Remark 3.7.

Let $\varphi\in\mathrm{MC}(\mathfrak{g}_{A})$ and
$\psi\in\mathrm{MC}(\mathfrak{g}_{B})$ be two $\Omega\mathcal{C}$-algebra
structures. In this context, an $\infty$-morphism
$\varphi\rightsquigarrow\psi$ can be defined as a morphism

$f\in\mathrm{Hom}_{\mathbb{S}}\left(\mathcal{C},\mathrm{End}^{A}_{B}\right)$

satisfying a similar identity, see [HLV20, Proposition 3.16] for more details.

Since the coproperad $\mathcal{C}$ is weight-graded, the Lie-admissible
algebra $\mathfrak{a}_{A}$ admits a weight-grading. Every element
$f\in\mathfrak{a}_{A}$ decomposes as

$f=f^{(0)}+f^{(1)}+f^{(2)}+\cdots$

where $f^{(i)}$ is the restriction of $f$ to $\mathcal{C}^{(i)}$ . The first
component

$f^{(0)}:\mathcal{I}\to\mathrm{Hom}(A,A)$

is equivalent to the data of an endomorphism $f^{(0)}(1)\in\mathrm{End}_{A}$
that we simply denote by $f^{(0)}$ . The $f$ is an $\infty$-morphism, this
element $f^{(0)}$ is a chain map. The convolution dg Lie-admissible algebra
$\mathfrak{g}_{A}$ has the same underlying weight grading and every
$\varphi\in\mathfrak{g}_{A}$ decomposes as
$\varphi=\varphi^{(1)}+\varphi^{(2)}+\cdots\ .$

######  Definition 3.8.

An $\infty$-morphism $f:\varphi\rightsquigarrow\psi$ between two
$\Omega\mathcal{C}$-algebra structures on $A$ is

  * $\centerdot$

an _$\infty$ -quasi-isomorphism_ if $f^{(0)}:A\to A$ is a quasi-isomorphism;

  * $\centerdot$

an _$\infty$ -isomorphism_ if $f^{(0)}:A\to A$ is an isomorphism;

  * $\centerdot$

an _$\infty$ -isotopy_ if $f^{(0)}=\mathrm{id}_{A}$ .

The set of all the $\infty$-isotopies is denoted by $\infty-\mathrm{iso}$ .

######  Theorem 3.9 ([HLV20, Theorem 3.21]).

The $\infty$-isomorphisms are the isomorphisms in the category of
$\Omega\mathcal{C}$-algebra structures on $A$ , i.e. every
$\infty$-isomorphism $f:\varphi\rightsquigarrow\psi$ admits a unique inverse,
denoted $f^{-1}$, with respect to the product $\circledcirc$.

######  Theorem 3.10 ([CV24, Theorem 2.16]).

The gauge group associated to the convolution dg Lie admissible algebra
$\mathfrak{g}_{A}$ and the group of $\infty$-isotopies are isomorphic though
the _graph exponential/logarithm maps_ ,

$\mathrm{exp}:((\mathfrak{g}_{A})_{0},\mathrm{BCH},0)\cong(\infty-\mathrm{iso},\circledcirc,1):\mathrm{log}\
.$

###  3.2. Gauge formality of algebras and properadic Kaledin classes

In this section, we define gauge formality for algebras over a properad and we
introduce Kaledin classes characterizing it. Let us consider the properad

$\mathcal{P}\coloneqq\mathcal{T}\left(s^{-1}\mathcal{C}^{(1)}\right)/\left(d_{\Omega\mathcal{C}}\left(s^{-1}\mathcal{C}^{(2)}\right)\right)\
,$

which comes with a twisting morphism $\mathcal{C}\to\mathcal{P}$.

######  Proposition 3.11.

Any $\Omega\mathcal{C}$-algebra structure $(A,\varphi)$ induces a canonical
$\mathcal{P}$-algebra structure

$(H(A),\varphi_{*})\ .$

###### Proof.

Similar to the one of Proposition 2.13. ∎

######  Definition 3.12.

A $\Omega\mathcal{C}$-algebra structure $(A,\varphi)$ is

  1. $\centerdot$

_formal_ if there exists a zig-zag of quasi-isomorphisms of
$\Omega\mathcal{C}$-algebras

$(A,\varphi)\;\overset{\sim}{\longleftarrow}\;\cdot\;\overset{\sim}{\longrightarrow}\;\cdots\;\overset{\sim}{\longleftarrow}\;\cdot\;\overset{\sim}{\longrightarrow}\;(H(A),\varphi_{*})\
;$

  2. $\centerdot$

_gauge formal_ if there exists an $\infty$-quasi-isomorphism

${(A,\varphi)}$${(H(A),\varphi_{*})\ .}$$\scriptstyle{\sim}$

######  Theorem 3.13 ([HLV24, Theorem 1.11]).

An $\Omega\mathcal{C}$-algebra structure $(A,\varphi)$ is formal if and only
if it is gauge formal.

###### Proof.

As for operads, the existence of a zig-zag of quasi-isomorphisms implies an
$\infty$-quasi-isomorphism relies on the invertibility of $\infty$-quasi-
isomorphisms, see [HLV20, Corollary 4.19]. Since the rectification method do
not admit direct generalization to the context of $\Omega\mathcal{C}$-algebras
over a properad, the converse is nonetheless more subtle. ∎

######  Theorem 3.14 ([HLV20, Theorem 4.14]).

Let $(A,\varphi)$ be an $\Omega\mathcal{C}$-algebra structure. There exists a
transferred structure on the homology, i.e. a $\Omega\mathcal{C}$-algebra
structure $(H(A),\varphi_{t})$ extending the induced structure $\varphi_{*}$
such that the embedding $i:H(A)\to A$ and the projection $p:A\to H(A)$ extend
to $\infty$-quasi-isomorphisms. The transferred structure is independent of
the choice of contraction in the following sense: any two such transferred
structures are related by an $\infty$-isotopy.

######  Proposition 3.15.

Let $(A,\varphi)$ be $\Omega\mathcal{C}$-algebra structure and let
$(H(A),\varphi_{t})$ be a transferred structure. The algebra $(A,\varphi)$ is
formal if and only if there exists an $\infty$-isotopy

${(H(A),\varphi_{t})}$${(H(A),\varphi_{*})\ .}$$\scriptstyle{=}$

###### Proof.

By Theorem 3.13, the algebra is formal if and only if it is gauge formal. The
proof is then similar to the one of Proposition 2.17. ∎

######  Definition 3.16.

Let $(A,\varphi)$ be $\Omega\mathcal{C}$-algebra such that there exists a
transferred structure $(H(A),\varphi_{t})$. It is called _gauge $n$-formal_ if
there exist an $\Omega\mathcal{C}$-algebra $(H(A),\psi)$ such that
$\psi^{(k)}=0$ , for all $2\leqslant k\leqslant n+1$ and an $\infty$-isotopy

${(H(A),\varphi_{t})}$${(H(A),\psi)\ .}$$\scriptstyle{=}$

######  Definition 3.17 (Properadic Kaledin classes).

Let $\varphi\in\mathrm{MC}(\mathfrak{g}_{H})$ be an
$\Omega\mathcal{C}$-algebra structure on a graded $R$-module $H$ and let
$\mathfrak{D}(\varphi)$ be its prismatic decomposition. The _Kaledin class_
$K_{\varphi}$ is the Kaledin class of $\mathfrak{D}(\varphi)$, i.e.

$K_{\varphi}=\left[\partial_{\hbar}\mathfrak{D}(\varphi)\right]\in
H_{-1}\left(\mathfrak{g}_{H}[\\![\hbar]\\!]^{\Phi}\right)\ .$

Its _$n^{\text{th}}$ -truncated Kaledin class_ $K_{\varphi}^{n}$ is the
$n^{\text{th}}$-truncated Kaledin class of $\mathfrak{D}(\varphi)$, i.e.

$K^{n}_{\varphi}=\left[\varphi^{(2)}+2\varphi^{(3)}\hbar+\dots+n\varphi^{(n+1)}\hbar^{n-1}\right]\in
H_{-1}\left(\left(\mathfrak{g}_{H}[\\![\hbar]\\!]/(\hbar^{n})\right)^{\overline{\Phi}^{n}}\right)\
.$

######  Theorem 3.18.

Let $R$ be a characteristic zero field. Let $\mathcal{C}$ be a reduced weight-
graded differential graded coproperad over $R$. Let $(A,\varphi)$ be a
$\Omega\mathcal{C}$-algebra structure and let $(H(A),\varphi_{t})$ be a
transferred structure.

  1. (1)

The algebra $(A,\varphi)$ is gauge $n$-formal if and only if the truncated
class $K^{n}_{\varphi_{t}}$ is zero.

  2. (2)

The algebra $(A,\varphi)$ is formal if and only if the Kaledin class
$K_{\varphi_{t}}$ is zero.

###### Proof.

Point (1) is an immediate corollary of Proposition 1.33, by Proposition 3.15
and the isomorphism of Theorem 3.10. Similarly, Point (2) follows from Point
(1) of Theorem 1.31. ∎

######  Remark 3.19.

If $R$ is not a characteristic zero field, the properadic Kaledin classes also
adapts using the same arguments as in Section 2. One can also prefer to work
with the obstruction sequences developed in [Emp24, Section 2], without a
bound on the characteristic.

###  3.3. Properadic calculus

This section generalizes some lemmas established in Section 2.3 to the
properadic case. These results will be used in Section 4.4 to prove the
vanishing of properadic Kaledin classes.

######  Lemma 3.20.

Let $g\in\mathfrak{a}_{H}$ be an element concentrated in weight $0$ and
invertible with respect to $\circledcirc$. Let $f$ and $h$ in
$\mathfrak{a}_{A}$ and let $(f;h)$ be the standard notation of [HLV20] for $f$
applied everywhere except at one place where $h$ is applied. The following
identities holds

  1. (a)

$\left(f\lhd g\right)\star h=f\lhd\left(g;g\star h\right)$;

  2. (b)

$f\star h=f\lhd(1;h)$;

  3. (c)

$\left(g;f\lhd g\right)\rhd h=f\lhd\left(g;g\rhd h\right)$;

If furthermore, the element $f^{(0)}$ is invertible with respect to
$\circledcirc$ , then

  1. (d)

$\left(f\lhd\left(g;h\right)\right)\lhd g^{-1}=f\star\left(h\lhd
g^{-1}\right)$;

  2. (e)

$\left(f\lhd g\right)\lhd g^{-1}=f$;

  3. (f)

$\left(\left(g;f\right)\rhd h\right)\lhd g^{-1}=\left(f\lhd
g^{-1}\right)\star\left(\left(g\rhd h\right)\lhd g^{-1}\right)$ .

###### Proof.

The proof is similar to the one of Lemma 2.25 and relies on the fact that $g$
is zero outside of $\mathcal{C}^{(0)}=\mathcal{I}$. Formula (a) follows from
the isomorphism

$\left(\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\mathcal{I}\right)\underset{(1,1)}{\boxtimes}\overline{\mathcal{C}}\cong\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\left(\mathcal{I};\mathcal{I}\underset{(1,1)}{\boxtimes}\overline{\mathcal{C}}\right)\
.$

Formula (b) follows from (a) with $g=1$. Formula (c) boils down to the
isomorphism

$\left(\mathcal{I};\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\mathcal{I}\right)\underset{(*,1)}{\boxtimes}\overline{\mathcal{C}}\cong\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\left(\mathcal{I};\mathcal{I}\underset{(*,1)}{\boxtimes}\overline{\mathcal{C}}\right)\
.$

We have that $\left(f\lhd\left(g;h\right)\right)\lhd g^{-1}=f\lhd\left(g\lhd
g^{-1};h\lhd g^{-1}\right)$ by the isomorphism

$\left(\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\left(\mathcal{I};\overline{\mathcal{C}}\right)\right)\underset{(1,*)}{\boxtimes}\mathcal{I}\cong\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\left(\mathcal{I}\underset{(1,*)}{\boxtimes}\mathcal{I};\overline{\mathcal{C}}\underset{(1,*)}{\boxtimes}\mathcal{I}\right)\
.$

Using the fact that $g\lhd g^{-1}=g\circledcirc g^{-1}=1$, Formula (d) then
follows from (b). Formula (e) is a particular case of (d) with $h=g$. Using
Formulas (e) and (c), we have

$\left(g;f\right)\rhd h=\left(g;\left(f\lhd g^{-1}\right)\lhd g\right)\rhd
h=\left(f\lhd g^{-1}\right)\lhd\left(g;g\rhd h\right)\ .$

Formula (f) then follows from Formula (d) since

$\left(\left(g;f\right)\rhd h\right)\lhd g^{-1}=\left(\left(f\lhd
g^{-1}\right)\lhd\left(g;g\rhd h\right)\right)\lhd
g^{-1}\overset{(d)}{=}\left(f\lhd g^{-1}\right)\star\left(\left(g\rhd
h\right)\lhd g^{-1}\right)\ .\qed$

######  Lemma 3.21.

For all $f$ and $g$ in $\mathfrak{a}_{A}$ of degree $0$, the following
identities hold

  1. (i)

$(f\rhd g)\circ d_{\mathcal{C}}=f\rhd(g\circ d_{\mathcal{C}})+(f;f\circ
d_{\mathcal{C}})\rhd g$;

  2. (j)

$\left(f\lhd g\right)\circ d_{\mathcal{C}}=\left(f\circ
d_{\mathcal{C}}\right)\lhd g+f\lhd\left(g;g\circ d_{\mathcal{C}}\right)\ .$

If furthermore, the element $f^{(0)}$ is invertible with respect to
$\circledcirc$ , then for all $x\in\mathfrak{a}_{H}$

  1. (k)

$\left(f^{(0)}\right)^{-1}\circ
d_{\mathcal{C}}=-\left(f^{(0)}\right)^{-1}\star\left(\left(f^{(0)}\circ
d_{\mathcal{C}}\right)\lhd\left(f^{(0)}\right)^{-1}\right)\ .$

###### Proof.

Points (i) and (j) follows from the fact that $\mathcal{C}$ is a dg
coproperad. By Formula (j) and Formula (e) of Lemma 3.20, we have

$\left(f^{(0)}\right)^{-1}\circ
d_{\mathcal{C}}=-\left(\left(f^{(0)}\right)^{-1}\lhd\left(f^{(0)};f^{(0)}\circ
d_{\mathcal{C}}\right)\right)\lhd\left(f^{(0)}\right)^{-1}\ .$

Point (k) then follows from Point (d) of Lemma 3.20. ∎

######  Lemma 3.22.

Let $n\geqslant 1$ be an integer and let $f:\varphi\rightsquigarrow\psi$ be an
$\infty$-isomorphism such that

  * $\centerdot$

$\varphi^{(k)}=0$ , for $2\leqslant k\leqslant n$ and

  * $\centerdot$

$f^{(k)}=0$ , for $1\leqslant k\leqslant n-1$ .

Then, the weight components of $\psi$ are given by

  * $\centerdot$

$\psi^{(1)}=\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(1)}\right)+\left(f^{(0)}\circ
d_{\mathcal{C}}\right)\lhd\left(f^{(0)}\right)^{-1}$ ,

  * $\centerdot$

$\psi^{(k)}=0$ , for $2\leqslant k\leqslant n$ , and

  * $\centerdot$

$\psi^{(n+1)}=\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(n+1)}\right)-d^{\psi^{(1)}}\left(f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right)$
,

where $\mathrm{Ad}_{f^{(0)}}(x)\coloneqq\left(f^{(0)}\rhd
x\right)\lhd\left(f^{(0)}\right)^{-1}$. In particular, if $f$ is an
$\infty$-isotopy, we have

$\psi=\varphi^{(1)}+\varphi^{(n+1)}-d^{\varphi^{(1)}}\left(f^{(n)}\right)+\psi^{(n+2)}+\cdots\
.$

###### Proof.

By definition of an $\infty$-morphism, we have

(7) $\psi\lhd f=f\rhd\varphi+f\circ d_{\mathcal{C}}\ .$

The assumptions on $\varphi$ and on $f$ imply that

$\begin{split}f\rhd\varphi+f\circ d_{\mathcal{C}}\ \equiv\
&f^{(0)}\rhd\varphi^{(1)}+f^{(0)}\rhd\varphi^{(n+1)}+\left(f^{(0)};f^{(n)}\right)\rhd\varphi^{(1)}\\\
&+f^{(0)}\circ d_{\mathcal{C}}+f^{(n)}\circ
d_{\mathcal{C}}\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\ .\end{split}$

Similarly, we have

$\psi\lhd f\equiv\psi\lhd
f^{(0)}+\psi^{(1)}\lhd\left(f^{(0)};f^{(n)}\right)\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\
.$

Equation (7) implies that

$\begin{split}\psi\ \equiv\
&\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(1)}\right)+\left(\left(f^{(0)};f^{(n)}\right)\rhd\varphi^{(1)}\right)\lhd\left(f^{(0)}\right)^{-1}+\left(f^{(0)}\circ
d_{\mathcal{C}}+f^{(n)}\circ
d_{\mathcal{C}}\right)\lhd\left(f^{(0)}\right)^{-1}\\\
&+\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(n+1)}\right)-\left(\psi^{(1)}\lhd\left(f^{(0)};f^{(n)}\right)\right)\lhd\left(f^{(0)}\right)^{-1}\pmod{\mathcal{F}^{n+2}\mathfrak{g}_{H}}\
.\end{split}$

This gives the desired formulas for the components $\psi^{(k)}$ for
$1\leqslant k\leqslant n$. Furthermore, by Point (d) of Lemma 3.20, we have

$\left(\psi^{(1)}\lhd\left(f^{(0)};f^{(n)}\right)\right)\lhd\left(f^{(0)}\right)^{-1}=\psi^{(1)}\star\left(f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right)\
.$

By Point (f) of Lemma 3.20, we have

$\left(\left(f^{(0)};f^{(n)}\right)\rhd\varphi^{(1)}\right)\lhd\left(f^{(0)}\right)^{-1}=\left(f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right)\star\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(1)}\right)\
.$

Point (j) of Lemma 3.21 leads to

$\left(f^{(n)}\circ
d_{\mathcal{C}}\right)\lhd\left(f^{(0)}\right)^{-1}=\left(f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right)\circ
d_{\mathcal{C}}-f^{(n)}\lhd\left(\left(f^{(0)}\right)^{-1};\left(f^{(0)}\right)^{-1}\circ
d_{\mathcal{C}}\right)$

while Point (k) of Lemma 3.21 and Point (a) of Lemma 3.20 imply that

$-f^{(n)}\lhd\left(\left(f^{(0)}\right)^{-1};\left(f^{(0)}\right)^{-1}\circ
d_{\mathcal{C}}\right)=\left(f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right)\star\left(\left(f^{(0)}\circ
d_{\mathcal{C}}\right)\lhd\left(f^{(0)}\right)^{-1}\right)\ .$

We obtain that

$\psi^{(n+1)}=\mathrm{Ad}_{f^{(0)}}\left(\varphi^{(n+1)}\right)-\left[\psi^{(1)},f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right]+\left(f^{(n)}\lhd\left(f^{(0)}\right)^{-1}\right)\circ
d_{\mathcal{C}}\ .$

∎

##  4\. Formality criteria

This section exploits the obstruction theory set out above in order to produce
explicit formality criteria. We establish formality descent results in Section
4.1 and study formality of families in Section 4.2. In Section 4.3, we
establish a criterion of intrinsic formality. Section 4.4 concludes with
criteria in terms of chain level lifts of certain homology automorphisms. This
generalizes the previously known results to any coefficient ring and to
algebras over a colored operad or a properad.

######  Assumptions 3.

Let $A$ be a chain complex over a commutative ring $R$. Let $n\geqslant 1$ be
an integer such that $n!$ is a unit in $R$. Let $(A,\varphi)$ be a
$\Omega\mathcal{C}$-algebra structure where $\mathcal{C}$ is either

  * $\centerdot$

a reduced weight-graded dg cooperad over $R$;

  * $\centerdot$

a $\mathbb{V}$-colored reduced weight-graded dg cooperad over $R$, where
$\mathbb{V}$ is a given groupoid;

  * $\centerdot$

a reduced weight-graded dg coproperad over a characteristic zero field.

Suppose that there exists a transferred structure $(H(A),\varphi_{t})$, e.g.
Assumptions 1 are satisfied in the case of a cooperad or Assumptions 2 are
satisfied in the colored case.

We denote by $\mathfrak{g}\coloneqq\mathfrak{g}_{H(A)}$ the convolution dg Lie
algebra associated to $\mathcal{C}$ and $H(A)$ and by $\tilde{\mathfrak{g}}$
is the subcomplex of $\mathfrak{g}$ whose underlying space is given by

$\sum_{k\geqslant 1}\mathfrak{g}^{(k)}\ .$

The constraint on the integer $n$ in positive characteristic and the
requirement of $R$ being a characteristic zero field for properads come from
the use of Kaledin classes to establish the following criteria. In the
forthcoming paper [Emp24], we refine the Kaledin classes construction in order
to remove these assumptions.

###  4.1. Formality descent

Let $R\to S$ be any commutative ring morphism. If the
$\Omega\mathcal{C}$-algebra $(A,\varphi)$ is formal, then the algebra
$(A\otimes_{R}S,\varphi\otimes 1)$ stays formal. This section gives sufficient
conditions under which the converse holds true.

######  Theorem 4.1 (Formality descent).

Under Assumptions 3, let $S$ be a faithfully flat commutative $R$-algebra.
Suppose that $H(A)$ is an $R$-module of finite presentation. Let us denote

$A_{S}\coloneqq A\otimes_{R}S\ .$

  1. (1)

The $\Omega\mathcal{C}$-algebra $(A,\varphi)$ is gauge $n$-formal if and only
if $(A_{S},\varphi\otimes 1)$ is gauge $n$-formal.

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra, the $\Omega\mathcal{C}$-algebra
$(A,\varphi)$ is gauge formal if and only if $(A_{S},\varphi\otimes 1)$ is
gauge formal.

###### Proof.

In the properadic setting, this is a direct corollary of Point (3) of Theorem
1.31, using in Theorem 3.18. Let us prove the result in the operadic case. For
both points, the direct implications are immediate. Conversely, suppose that
$(A_{S},\varphi\otimes 1)$ is gauge $n$-formal. By Theorem 2.44, the
$k^{\text{th}}$-truncated Kaledin class of $\varphi_{t}\otimes 1$ are zero for
all $k\leqslant n$. Let us prove by induction that $K^{n}_{\varphi_{t}}$ is
also zero. The canonical map

$H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)\hookrightarrow
H_{-1}\left(\mathfrak{g}^{\varphi_{*}}\right)$

is injective and its image contains the class $K^{1}_{\varphi_{t}}$. The same
applies for $\tilde{\mathfrak{g}}_{H(A_{S})}$ and $K^{1}_{\varphi_{t}\otimes