# Homotopical operadic calculus in positive characteristic

Brice Le Grignou Victor Roca i Lucio  Brice Le Grignou,
<EMAIL_ADDRESS>Victor Roca i Lucio, Ecole Polytechnique Fédérale de
Lausanne, EPFL, CH-1015 Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract.

Algebraic operads provide a powerful tool to understand the homotopy theory of
the types of (co)algebras they encode. So far, the principal results and
methods that this theory provides were only available in characteristic zero.
The reason is that operads carry an action of all the symmetric groups, whose
representation theory becomes much more involved in positive characteristic.
The goal of this paper is to extend these results and methods to a positive
characteristic setting. We solve the main problems that appear in this new
setting by using the notion of a quasi-planar cooperad as the building block
of the theory.

###### Key words and phrases:

Homotopical operadic calculus, algebraic operads, Koszul duality, bar-cobar
adjunctions, positive characteristic.

###### 2020 Mathematics Subject Classification:

18M70,18N40,18N55,18N60

###### Contents

  1. 1 Differential graded modules, $\mathbb{S}$-modules, operads and cooperads
  2. 2 Homotopy theory of operads and quasi-planar cooperads
  3. 3 Algebras, coalgebras, and Bar-Cobar adjunctions
  4. 4 Model structure on coalgebras over a cooperad
  5. 5 A Quillen equivalence, $\infty$-morphisms and homotopy transfer theorems for algebras
  6. 6 Model structure on complete algebras over a cooperad
  7. 7 A Quillen equivalence, $\infty$-morphisms and homotopy transfer theorems for coalgebras
  8. 8 Linear duality
  9. A Adjoint lifting theorems, right and left transferred structures

###### Contents

  1. 1 Differential graded modules, $\mathbb{S}$-modules, operads and cooperads
  2. 2 Homotopy theory of operads and quasi-planar cooperads
  3. 3 Algebras, coalgebras, and Bar-Cobar adjunctions
  4. 4 Model structure on coalgebras over a cooperad
  5. 5 A Quillen equivalence, $\infty$-morphisms and homotopy transfer theorems for algebras
  6. 6 Model structure on complete algebras over a cooperad
  7. 7 A Quillen equivalence, $\infty$-morphisms and homotopy transfer theorems for coalgebras
  8. 8 Linear duality
  9. A Adjoint lifting theorems, right and left transferred structures

## Introduction

Operads are algebraic objects which encode other types of algebraic
structures: Lie algebras, associative algebras, Batalin–Vilkovisky algebras,
Gerstenhaber algebras, $\mathcal{A}_{\infty}$-algebras,
$\mathcal{L}_{\infty}$-algebras, and many more. The theory of algebraic
operads provides us with a large set of methods which allow us to study the
homotopy theory of differential graded (dg) $\mathcal{P}$-algebras if they are
encoded by a dg operad $\mathcal{P}$. We will call this set of methods
homotopical operadic calculus. We refer to [LV12] for a detailed account of
the theory of algebraic operads.

Let us explain these methods. Over a characteristic zero field, the category
of dg $\mathcal{P}$-algebras always admits a model category structure where
weak-equivalences are given by quasi-isomorphisms and fibrations are given by
degree-wise epimorphisms. This model structure can be obtained by transfer
along the free-forgetful adjunction and presents the homotopy theory of dg
$\mathcal{P}$-algebras. Cofibrant objects in this category do not admit an
easy description, and the homotopy category is in general quite hard to
understand. One way to get an easier description of this homotopy category is
via bar-cobar adjunctions. To any operad $\mathcal{P}$ one can always
associate a cooperad $\mathcal{C}$ which is Koszul dual in some sense. There
are in fact two canonical choices for $\mathcal{C}$: in certain cases, a
minimal choice given by classical Koszul duality, or in general the one given
by the operadic bar construction on $\mathcal{P}$. For any choice of Koszul
dual cooperad $\mathcal{C}$, this duality is instantiated in the data of a
twisting morphism $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$. And for any
such twisting morphism $\alpha$, there exists a bar-cobar adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg},}$$\scriptstyle{\Omega_{\alpha}}$$\scriptstyle{\mathrm{B}_{\alpha}}$${\dashv}$

between the category of dg $\mathcal{P}$-algebras and the category of dg
$\mathcal{C}$-coalgebras. Notice that these constructions are generalizations
of the classical bar construction of [CE56] and of the classical cobar of
[Ada56] for associative (co)algebras to other types of algebras. This general
construction also recovers the adjunction between dg Lie algebras and
conilpotent dg cocommutative coalgebras of [Qui69]. Let us mention that having
a bar-cobar adjunction is essential for Koszul duality in the sense of
[Pri70], which can be generalized to many types of algebras, see [Mil12,
Idr23, BB20]. Furthermore, it is crucial in order to generalize André–Quillen
cohomology [And71, Qui70] to other types of algebras, see [Mil11].

The natural model category structure on dg $\mathcal{P}$-algebras can always
be transferred along the bar-cobar adjunction, and if $\alpha$ satisfies a
"Koszulness" property, this adjunction becomes a Quillen equivalence. This
type of arguments was first developed for dg Lie algebras by V. Hinich in
[Hin01] using the adjunction of [Qui69]. Note that these are the constructions
that bridge between Quillen and Sullivan’s approaches of [Qui69, Sul77]. Then
K. Lefèvre-Hasegawa developed these methods in [LH03] for non-unital dg
associative algebras and conilpotent dg coassociative coalgebras. The analogue
for unital dg associative algebras was developed by L. Positselski in [Pos11],
where one needs to consider curved conilpotent dg coalgebras on the other
side. The general non-unital case was proven by B. Vallette in [Val20].
Finally, we refer to [LG19] for the general constructions in the unital case.

A great advantage of looking at the transferred model category structure on dg
$\mathcal{C}$-coalgebras is that fibrant-cofibrant objects admit a very easy
description: they are given by dg $\mathcal{C}$-coalgebras whose underlying
graded $\mathcal{C}$-coalgebra structure is cofree (usually called quasi-free
objects). This naturally leads to the notion of an $\infty$-morphism between
two dg $\mathcal{P}$-algebras: it is given by a morphism of dg
$\mathcal{C}$-coalgebras between their respective bar constructions. This
relaxed notion of morphisms has a great advantage: any $\infty$-quasi-
isomorphism admits an homotopy inverse, and it can be shown that two dg
$\mathcal{P}$-algebras are linked by a zig-zag of quasi-isomorphisms if and
only if there exists an $\infty$-quasi-isomorphism between them. For instance,
this fact was crucially exploited by M. Kontsevich in order to prove the
deformation-quantization of Poisson manifolds in [Kon03].

A cofibrant replacement $\mathcal{P}_{\infty}$ of an operad $\mathcal{P}$
encodes a notion of a $\mathcal{P}$-algebra up to homotopy. Any quasi-
isomorphism of operads induces a Quillen equivalence between their respective
categories of algebras, therefore considering algebraic structures up to
homotopy does not change the homotopy category. A key feature of algebraic
structures up to homotopy is the homotopy transfer theorem: given a
contraction of dg modules

${V}$${H\leavevmode\nobreak\
,}$$\scriptstyle{p}$$\scriptstyle{h}$$\scriptstyle{i}$

and a dg $\mathcal{P}_{\infty}$-algebra structure on $V$, there exists a dg
$\mathcal{P}_{\infty}$-algebra structure on $H$ such that $p$ and $i$ can also
be extended into "inverse" $\infty$-quasi-isomorphisms. This is particularly
useful when one chooses a contraction between a dg module and its homology.
Since the transferred structure onto the homology does not lose any
homotopical information, one can always replace a dg
$\mathcal{P}_{\infty}$-algebra by the transferred dg
$\mathcal{P}_{\infty}$-algebra structure on its homology. We refer to [LV12]
for an account of these methods and their history.

Let briefly mention some of the applications of these methods: in rational
homotopy theory, one can use them to study formality questions [Ber14, DCH21]
or to construct rational models [BFMT20, RNV20a]. In symplectic geometry,
$\infty$-morphisms and the homotopy transfer theorem play a key role in
[Fuk02], and more general operadic methods can be applied in Morse theory, see
[Maz22a, Maz22b]. See [Val14] and [LV12] for more detailed instances of
applications.

Dual results were recently proven for coalgebras by D. Lejay and the first
author in [GL22]. Coalgebras over cooperads only encode a certain type of
coalgebras which are called conilpotent. If one wants to encode all coalgebras
(say, coassociative, cocommutative, etc) without a conilpotency condition, one
cannot use cooperads. We use here (reversed) operads. In this case, given a
twisting morphism $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$, they
constructed a complete bar-cobar adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}},}$$\scriptstyle{\widehat{\Omega}_{\alpha}}$$\scriptstyle{\widehat{\mathrm{B}}_{\alpha}}$${\dashv}$

between dg $\mathcal{P}$-coalgebras and complete dg $\mathcal{C}$-algebras.
The notion of an algebra over a cooperad gives a new type of algebraic
structures called absolute algebras. These are algebraic structures endowed
with meaningful notion of infinite sums of operations without presupposing an
underlying topology. Here complete means that the canonical topology induced
by the absolute structure is separated. Most of the classical algebraic
structures encoded by an operad (Lie algebras, associative algebras,
$\mathcal{L}_{\infty}$-algebras, etc.) have an absolute analogue. Once again,
one can transfer (when it exists) the model category structure of dg
$\mathcal{P}$-coalgebras to dg $\mathcal{C}$-algebras using the complete bar-
cobar adjunction. And if the operad $\mathcal{P}$ considered is given by the
cobar construction $\Omega\mathcal{C}$ of $\mathcal{C}$, then the complete
bar-cobar adjunction becomes a Quillen equivalence. The aforementioned
methods, as well as the key notion of an absolute algebra were used by the
second author in [RiL22c] to develop the integration theory of curved absolute
$\mathcal{L}_{\infty}$-algebras, extending the work of [Get09] and [RNV20b].

Fibrant-cofibrant dg $\mathcal{C}$-algebras are also given by quasi-free
objects: one can then define $\infty$-morphisms and $\infty$-quasi-
isomorphisms for coalgebras. They were shown to be "invertible" as well, which
makes them a useful tool to study the homotopy theory of
$\mathcal{P}$-coalgebras. Finally, an analogue version of the homotopy
transfer theorem also holds for dg $\Omega\mathcal{C}$-coalgebras. Note that
coalgebras over an operad can also be interpreted from a properadic
perspective, and $\infty$-morphisms coincide with those introduced by E.
Hoffbeck, J. Leray and B. Vallette in [HLV21].

Linear duality is a bridge between these two bar-cobar adjunctions, and thus
between the homotopy theory of algebras and coalgebras. Indeed, the linear
dual of a dg $\mathcal{P}$-coalgebra is always a dg $\mathcal{P}$-algebra, and
the linear dual of a dg $\mathcal{C}$-coalgebra is a dg $\mathcal{C}$-algebra.
These functors admit adjoints, and one obtains a duality square made of four
commuting adjunctions:

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}^{\mathsf{op}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}^{\mathsf{op}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{B}_{\alpha}^{\mathsf{op}}}$$\scriptstyle{(-)^{\circ}}$$\scriptstyle{(-)^{*}}$$\scriptstyle{\Omega_{\alpha}^{\mathsf{op}}}$$\scriptstyle{\widehat{\Omega}_{\alpha}}$$\scriptstyle{(-)^{*}}$$\scriptstyle{\widehat{\mathrm{B}}_{\alpha}}$$\scriptstyle{(-)^{\vee}}$${\dashv}$${\dashv}$${\dashv}$${\dashv}$

This square can be promoted (under some conditions) into a square of Quillen
adjunctions, using the model category structures constructed so far. This
duality square allowed the second author to show that the homotopy theory of
algebras and coalgebras are equivalent on objects with finite dimensional
homology, see [RiL22a] for more details on these constructions. Finally, let
us mention that all these results also extend to non-augmented operads, which
encode counital types of coalgebras, where on the other side one needs to
consider curved algebras over a curved cooperad.

An overview of the positive characteristic case. A (dg) operad is a collection
$\\{\mathcal{P}(n)\\}_{n\in\mathbb{N}}$ of (dg)
$\Bbbk[\mathbb{S}_{n}]$-modules, for all $n\geq 0$, together with an
additional structure that allows one to compose elements. The category of
$\Bbbk[\mathbb{S}_{n}]$-modules is fairly well-behaved when $\Bbbk$ is a
characteristic zero field: it is semi-simple, there is a combinatorial
description of simple objects, and much more. In particular, the homotopy
theory of dg $\Bbbk[\mathbb{S}_{n}]$-modules behaves in a similar fashion to
the homotopy theory of dg $\Bbbk$-modules.

Things change drastically over a positive characteristic field. Understanding
the representation theory of the symmetric groups is an active subject of
research, with many open questions, see for instance [Kle14]. The homotopy
category of dg $\Bbbk[\mathbb{S}_{n}]$-modules becomes very rich and hard to
understand. A concrete consequence of these facts is that many statements in
fail or at least, require extra hypothesis, over a positive characteristic
field.

The first obstruction is given by the fact that, in general, dg
$\mathcal{P}$-algebras do not admit a model category structure transferred
from dg modules. Under the assumption that $\mathcal{P}(n)$ is a projective dg
$\Bbbk[\mathbb{S}_{n}]$-module for all $n\geq 0$, M. Spitzweck constructed in
[Spi01] a semi-model category structure on dg $\mathcal{P}$-algebras, which is
a weaker notion of a model category structure. B. Fresse then showed in
[Fre09] that the category of dg operad themselves admits a semi-model category
structure and that any quasi-isomorphism between $\mathbb{S}$-projective dg
operads induces an Quillen equivalence of semi-model categories. Parallel to
these results, there is also the axiomatic point of view developed by C.
Berger and I. Moerdijk in [BM03]. They consider admissible operads, those that
admit a transferred model structure from dg modules, and give conditions under
which an operad is admissible (for instance, if it is cofibrant). These
results already have breakthrough applications. As an example, understanding
the homotopy theory of $\mathcal{E}_{\infty}$-algebras in positive
characteristic allowed M. Mandell to construct algebraic models for $p$-adic
homotopy types, see [Man01, Man06].

The homotopy theory of operads themselves plays a crucial role in homotopical
operadic calculus. In this direction, M. Dehling and B. Vallette developed in
[DV21] a general method that gives cofibrant resolutions of the operadic
structure and of the action of the symmetric groups at the same time.
Furthermore, they show that their cofibrant resolution, applied to an operad
$\mathcal{P}$, is in fact isomorphic to
$\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$: here $\Omega,\mathrm{B}$
are respectively the cobar and bar constructions at the operadic level, and
$\mathcal{E}$ is the Barratt-Eccles of [BF04]. This gives a clear indication
that the Barratt-Eccles operad provides a universal way to construct
resolutions that also take into account the action of the symmetric groups.

In a different direction, L. Brantner, R. Campos and J. Nuiten develop in
[BCN21] a theory of divided power operads, in order to encode divided power
algebras. In particular, they endow these categories with semi-model category
structures. This allows them to give point-set models of partition Lie
algebras as defined in [BM19], where they were shown to encode formal moduli
problems over a positive characteristic field. Finally, let us mention several
results in the direction of classical Koszul duality which are valid over any
ring [Fre04, CV06, Fre11].

Main results. The purpose of this paper is to generalize all the
aforementioned methods of homotopical operadic calculus over any field. Let us
mention first that the main motivation for developing these methods in
positive characteristic is the article [GRiL23b]. In op.cit, we provide a new
point of view on formal moduli problems via the duality square constructed
here together with the main results of [Gri20]. Putting these two results
together allows us to give a new proof of many of the main results concerning
formal moduli problems, while generalizing the Koszul duality inherent to
these types of results over a field of any characteristic. Another application
is the paper by the second author about the integration theory of partition
$\mathcal{L}_{\infty}$-algebras [RiL23], which also extensively uses the
results of this paper. We expect that they will find many more applications,
as their analogues in characteristic zero have so far.

From now on, let us assume we work over a field $\Bbbk$. The main idea to
bypass all the major difficulties that arise over a positive characteristic
field is to introduce the notion of a quasi-planar dg cooperad. Informally
speaking, it amounts to a dg cooperad whose underlying graded cooperad is non-
symmetric/planar, and even if its differential can interact in non-trivial
ways with the underlying action of the symmetric groups, this interaction is
still "controlled" by a suitable filtration on the cooperad. Note that even if
our results are valid in the more general unital/curved setting, we will state
them in the augmented/dg setting for simplicity.

######  Definition (Quasi-planar dg cooperad).

Let $(\mathcal{C},d_{\mathcal{C}})$ be a conilpotent dg cooperad. It is quasi-
planar if there exists a ladder of conilpotent dg cooperads

$\mathcal{C}^{(0)}\longrightarrow\mathcal{C}^{(1)}\longrightarrow\cdots\longrightarrow\mathcal{C}^{(i)}\longrightarrow\cdots$

such that $\mathcal{C}$ is the colimit of the diagram and such that the
following conditions are satisfied.

  1. (1)

For all $i$, the underlying conilpotent graded cooperad
$\mathcal{C}^{(i)}_{\mathrm{gr}}$ is planar, meaning that there exists a
conilpotent graded non-symmetric cooperad $\mathcal{C}_{\mathrm{pl}}^{(i)}$
and a given isomorphism

$\mathcal{C}_{\mathrm{pl}}^{(i)}\otimes\mathbb{S}\cong\mathcal{C}^{(i)}_{\mathrm{gr}}\leavevmode\nobreak\
,$

where $(-\otimes\mathbb{S})$ denotes the left adjoint to the forgetful functor
from conilpotent graded cooperads to conilpotent graded non-symmetric
cooperads. Moreover, the filtration preserves this planar structure in the
sense that the graded map $\mathcal{C}^{(i)}\to\mathcal{C}^{(j)}$ (for $i<j$)
is the image through the functor $-\otimes\mathbb{S}$ of the a map
$\mathcal{C}^{(i)}_{\mathrm{pl}}\to\mathcal{C}^{(j)}_{\mathrm{pl}}$.

  2. (2)

For all $i$, the restriction of the coderivation of $d_{\mathcal{C}(i+1)}$ to
$\mathcal{C}_{\mathrm{pl}}(i+1)\otimes 1$ factors through

$\mathcal{C}_{\mathrm{pl}}^{(i+1)}\otimes
1\longrightarrow\left(\mathcal{C}_{\mathrm{pl}}^{(i+1)}\otimes
1\right)+\mathcal{C}^{(i)}\hookrightarrow\mathcal{C}^{(i+1)};$

in other words, the differential of
$\mathrm{gr}_{i+1}\mathcal{C}=(\mathcal{C}_{\mathrm{pl}}^{(i+1)}/\mathcal{C}_{\mathrm{pl}}^{(i)})\otimes\mathbb{S}$
has the form $d_{\mathrm{pl}}\otimes\mathrm{Id}_{\mathbb{S}}$.

A first version of this notion is already present in [GL22]. Let us point out
that the ladder can be indexed by any small ordinal $\alpha$, not only by
$\omega$. Nevertheless, using Theorem A, we will show that any quasi-planar
cooperad $\mathcal{C}$ admits a canonical quasi-planar ladder indexed by
$\omega$.

Two key observations. Firstly, for any dg operad $\mathcal{P}$, the dg
cooperad $\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ is a quasi-planar
cooperad. Secondly, any operad $\mathcal{P}$ can be replaced by
$\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ up to quasi-isomorphism. If
$\mathcal{P}$ is $\mathbb{S}$-projective, this replacement does not change the
underlying homotopy category. And if it is not, there was no meaningful
homotopy category for dg $\mathcal{P}$-algebras to begin with. So, for all
intents and purposes, one can restrict to operads of the form
$\Omega\mathcal{C}$, where $\mathcal{C}$ is quasi-planar conilpotent dg
cooperad.

Therefore, let us fix a quasi-planar dg cooperad $\mathcal{C}$. We show that
its cobar construction $\Omega\mathcal{C}$ is a cofibrant object in the semi-
model category of dg operad of [Fre09]. Not only that, we also show that any
dg operad of the form $\Omega\mathcal{C}$, with $\mathcal{C}$ quasi-planar,
admits a canonical $\mathcal{E}$-comonoid structure, given by explicit
formulas.

######  Theorem A (Theorem 2).

Let $\mathcal{C}$ be a quasi-planar dg cooperad. There is a canonical morphism
of dg operads

${\Delta_{\mathcal{E},\mathcal{C}}:\Omega\mathcal{C}}$${\mathcal{E}\otimes\Omega\mathcal{C}}$

which endows the dg operad $\Omega\mathcal{C}$ with a left
$\mathcal{E}$-comodule structure.

This theorem can be seen as the positive characteristic analogue of the
following fact: any dg operad $\mathcal{P}$ admits a canonical
$\mathrm{u}\mathcal{C}\mathrm{om}$-comonoid structure. Notice that is is
precisely this $\mathrm{u}\mathcal{C}\mathrm{om}$-comonoid structure that
gives the canonical convolution curved $\mathcal{L}_{\infty}$-algebra
structure on the hom-graded modules between types of coalgebras and types of
algebras which are linked by Koszul duality. Here, it will produce a canonical
convolution (curved absolute) partition $\mathcal{L}_{\infty}$-algebra
structures on the hom-graded modules between types of coalgebras and types of
algebras, in the sense of [BCN21], see also [GRiL23b]. Exploring these
convolution structure will be the subject of future work; we refer to
[GRiL23a].

By standard axiomatic arguments in the same spirit as in [BM03], the dg operad
$\Omega\mathcal{C}$ is both admissible and coadmissible, meaning both dg
$\Omega\mathcal{C}$-algebras and dg $\Omega\mathcal{C}$-coalgebras admit a
transferred structures from dg modules. Then we consider the bar-cobar
adjunction relative to the universal twisting morphism
$\iota:\mathcal{C}\longrightarrow\Omega\mathcal{C}$

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
.}$$\scriptstyle{\Omega_{\iota}}$$\scriptstyle{\mathrm{B}_{\iota}}$${\dashv}$

######  Theorem B (Theorems 6 and 7).

The exists a combinatorial model category structure on the category of dg
$\mathcal{C}$-coalgebras given by the following sets of maps:

  1. (1)

the set of weak-equivalences is given by morphisms $f$ such that
$\Omega_{\iota}(f)$ is a quasi-isomorphism,

  2. (2)

the set of cofibrations is given by degree-wise monomorphisms,

  3. (3)

the set of fibrations is given by morphisms with the right lifting property
with respect to acyclic cofibrations.

Furthermore, considering this model structure promotes the bar-cobar
adjunction into Quillen equivalence.

This means that the homotopy theory of dg $\Omega\mathcal{C}$-algebras can be
read using their Koszul dual dg $\mathcal{C}$-coalgebras. Here again, a great
advantage is that fibrant-cofibrant dg $\mathcal{C}$-coalgebras are exactly
quasi-free dg $\mathcal{C}$-coalgebras which are also images of dg
$\Omega\mathcal{C}$-algebras. Notice that, in principle, the notion of a
coalgebra over a cooperad in positive characteristic encodes types of divided
power conilpotent coalgebras. Nevertheless, since $\mathcal{C}$ is quasi-
planar, there are no divided power operations that appear at the level of
algebraic structures.

We define $\infty$-morphisms and $\infty$-quasi-isomorphisms of dg
$\Omega\mathcal{C}$-algebras: we show that they are "invertible", therefore
two dg $\Omega\mathcal{C}$-algebras are linked by a zig-zag of quasi-
isomorphisms if and only if there exists an $\infty$-quasi-isomorphism between
them. We also prove a version of the homotopy transfer theorem: given a dg
$\Omega\mathcal{C}$-algebra $A$, we construct an $\infty$-quasi-isomorphic dg
$\Omega\mathcal{C}$-algebra structure on the homology of $A$. This means that
$A$ can be replaced by its homology without losing any homotopical data. These
methods should lead to applications in the study of formality questions over
positive characteristic field, in the same spirit as in [DCH21].

Then we turn to dg $\Omega\mathcal{C}$-coalgebras, and generalize the results
of [GL22] in this case. Consider the complete bar-cobar adjunction relative to
$\iota$

${\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}\leavevmode\nobreak\
,}$$\scriptstyle{\widehat{\Omega}_{\iota}}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota}}$${\dashv}$

between the category of dg $\Omega\mathcal{C}$-coalgebras and the category of
qp-complete dg $\mathcal{C}$-algebras. Here qp-complete algebra refer to
algebras which are complete for the canonical quasi-planar ladder of the
quasi-planar cooperad $\mathcal{C}$.

######  Theorem C (Theorems 10 et 11).

The exists a combinatorial model category structure on the category of qp-
complete dg $\mathcal{C}$-algebras given by the following sets of maps:

  1. (1)

the set of weak-equivalences is given by morphisms $f$ such that
$\widehat{\mathrm{B}}_{\iota}(f)$ is a quasi-isomorphism,

  2. (2)

the set of fibrations is given by degree-wise epimorphisms,

  3. (3)

the set of cofibrations is given by morphisms with the left lifting property
with respect to acyclic fibrations.

Furthermore, considering this model structure promotes the bar-cobar
adjunction into Quillen equivalence.

Again, this means that one can read the homotopy theory of dg
$\Omega\mathcal{C}$-coalgebras by using dg $\mathcal{C}$-algebras. Here again,
fibrant-cofibrant dg $\mathcal{C}$-algebras are much simpler as they are
exactly given by quasi-free dg $\mathcal{C}$-algebras which are essentially
the images of the dg $\Omega\mathcal{C}$-coalgebras under the complete cobar
functor. The notion of a dg $\mathcal{C}$-algebra corresponds to the absolute
version of divided powers dg $\mathcal{C}^{*}$-algebras, and is also naturally
endowed with divided power operations.

We define $\infty$-morphisms and $\infty$-quasi-isomorphisms of dg
$\Omega\mathcal{C}$-coalgebras too: we show that they are "invertible",
therefore two dg $\Omega\mathcal{C}$-coalgebras are linked by a zig-zag of
quasi-isomorphisms if and only if there exists an $\infty$-quasi-isomorphism
between them. We also prove a similar version of the homotopy transfer
theorem: given a dg $\Omega\mathcal{C}$-algebra $V$, we construct a
$\infty$-quasi-isomorphic dg $\Omega\mathcal{C}$-coalgebra structure on the
homology of $V$, which allows one to pass from a coalgebra $V$ to its homology
without loosing homotopical data.

Note that once the theory is established for operads of the form
$\Omega\mathcal{C}$, with $\mathcal{C}$ a quasi-planar dg cooperad, then we
can easily extend all the above results to any cofibrant dg operad
$\mathcal{P}$. Indeed, given a dg operad $\mathcal{P}$, we construct what we
call the quasi-planar bar-cobar adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{curv}\leavevmode\nobreak\
\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\text{-}\mathsf{cog}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{B}^{\mathrm{q.p}}_{\pi}}$$\scriptstyle{\Omega^{\mathrm{q.p}}_{\pi}}$${\dashv}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{curv}\leavevmode\nobreak\
\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}\leavevmode\nobreak\
,}$$\scriptstyle{\widehat{\Omega}^{\mathrm{q.p}}_{\pi}}$$\scriptstyle{\widehat{\mathrm{B}}^{\mathrm{q.p}}_{\pi}}$${\dashv}$

and in the case when $\mathcal{P}$ is cofibrant, all the above results can be
translated mutatis mutandis to this setting.

Finally, we extend the duality square of [RiL22a] to this new setting. Recall
that it intertwines the bar-cobar adjunction with the complete bar-cobar
adjunction via a pair of duality adjunctions. Using the model category
structures constructed so far, we prove that all the functors in this duality
square are Quillen adjunctions. As in the characteristic zero case, this
allows us to show that the homotopy category of dg $\mathcal{P}$-algebras with
finite dimensional homology is equivalent to the homotopy category of dg
$\mathcal{P}$-coalgebras with finite dimensional homology, where $\mathcal{P}$
is a cofibrant dg operad.

These results also open the door to a classical theory of Koszul duality in
the positive characteristic setting, both at the operadic level and at the
algebra/coalgebra level. Since quasi-planar dg cooperads seem to be the right
notion, it would be interesting to understand, given an operad $\mathcal{P}$,
what is (and when does it exist) the minimal quasi-planar dg cooperad
$\mathcal{C}$ such that there exists a quasi-isomorphism
$\Omega\mathcal{C}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$.
In this case, one could think of $\mathcal{C}$ as the Koszul dual of
$\mathcal{P}$, in the original sense of [GK94, GJ94]. Very similar ideas were
already explained in [DV21, Section 2.7], where the authors also suggest a
similar approach using their theory of higher cooperads. Finally, at the
algebra level, one should be able to ask similar questions using the bar-cobar
constructions of this paper, knowing that they have a reasonable homotopical
behaviour. These questions are all beyond the reach of this paper.
Nevertheless, let us mention that comparing the approach carried out in this
paper with [DV21], and specially comparing quasi-planar cooperads with higher
cooperads, shall be the subject of future work in [GRiL23a].

### Acknowledgements

We would like to thank Guille Carrión, Aras Ergus, Geoffroy Horel, Joost
Nuiten and Bruno Vallette for interesting discussions. We would also like to
thank Bruno Vallette for useful comments on the draft version.

### Notations and conventions

  1. (1)

Universe. We fix a universe $\mathscr{U}$. A small set is an element of this
universe. A large set is a subset of it.

  2. (2)

Ordinal products. Let $\omega$ be the first infinite ordinal which is the
poset of natural numbers. Let $\omega\cdot\omega$ be the ordinal whose
underlying poset is the two times product of $\omega$ equipped with the
lexicographic order. In general, if $\alpha$ and $\beta$ are two small
ordinals, we will denote $\alpha\cdot\beta$ the small ordinal whose underlying
poset is the product poset equipped with the lexicographic order.

Given a small ordinal $\alpha$ (and more generally a poset), one can view it
as a category. Objects are indexed by the ordinal $\alpha$, and there is only
one non-trivial arrow between two objects $i$ and $j$ in $\alpha$, if and only
if $i<j$.

  3. (3)

Ladders. Let $\mathsf{C}$ be a cocomplete category and let $\alpha$ be a small
ordinal. The data of an $\alpha$-ladder amounts to the data of a functor
cocontinuous functor

$c:1+\alpha\longrightarrow\mathsf{C}\leavevmode\nobreak\ .$

This corresponds to a "ladder" diagram of objects in $\mathsf{C}$

$0\longrightarrow c(0)\longrightarrow c(1)\longrightarrow\cdots\longrightarrow
c(i)\longrightarrow\cdots$

indexed by $\alpha$, such that for every limit ordinal $k\in\alpha$, we have

$c(k)\cong\underset{i<k}{\mathrm{colim}}\leavevmode\nobreak\
c(i)\leavevmode\nobreak\ .$

The colimit of the ladder diagram will be denoted by

$c(\alpha)\coloneqq\underset{i\in\alpha}{\mathrm{colim}}\leavevmode\nobreak\
c(i)\leavevmode\nobreak\ .$

Usually and depending on the category $\mathsf{C}$, additional requirements
will be made on the transition maps $c(i)\longrightarrow c(i+1)$. For
instance, they will be required to be monomorphisms or some kind of
cofibrations.

We considering ladder diagrams in categories of operads and cooperads, the
notation $c(i)$ for the image of $i\in\alpha$ will be replaces by $c^{(i)}$ in
order to avoid confusions with the arity.

  4. (4)

Coladders. Let $\mathsf{D}$ be a complete category and let $\alpha$ be a small
ordinal. The data of an $\alpha$-coladder amounts to the data of a functor
continuous functor

$d:(1+\alpha)^{\mathrm{op}}\longrightarrow\mathsf{D}\leavevmode\nobreak\ .$

This corresponds to a "coladder" diagram of objects in $\mathsf{D}$

$0\longleftarrow d(0)\longleftarrow d(1)\longleftarrow\cdots\longleftarrow
d(i)\longleftarrow\cdots$

indexed by $\alpha$, such that for every limit ordinal $k\in\alpha$, we have

$d(k)\cong\lim_{i<k}d(i)\leavevmode\nobreak\ .$

The limit of the coladder diagram will be denoted by

$d(\alpha)\coloneqq\lim_{i\in\alpha}d(i)\leavevmode\nobreak\ .$

Usually and depending on the category $\mathsf{D}$, additional requirements
will be made on the transition maps $d(i+1)\longrightarrow d(i)$. For
instance, they will be required to be epimorphisms or some kind of fibrations.

  5. (5)

Permutations and shuffles. The symmetric group on $n$ elements, given by the
set of bijections between $\\{1,\cdots,n\\}$ and itself, will be denoted by
$\mathbb{S}_{n}$. Elements in $\mathbb{S}_{n}$ are called permutations. A
permutation $\sigma$ in $\mathbb{S}_{n}$ is determined by its values
$\sigma(1),\cdots,\sigma(n)$.

Let $\sigma$ be a permutation in $\mathbb{S}_{a+b}$. It is a $(a,b)$-shuffle
if it satisfies that

$\sigma(1)<\cdots<\sigma(a)\quad\text{and}\quad\sigma(a+1)<\cdots<\sigma(a+b)\leavevmode\nobreak\
.$

The set of all $(a,b)$-shuffles in $\mathbb{S}_{a+b}$ will be denoted by
$\mathrm{Sh}(a,b)$.

##  1\. Differential graded modules, $\mathbb{S}$-modules, operads and
cooperads

In this section, we introduce the underlying categories of graded, pre-
differential and differential modules over a field $\Bbbk$ of any
characteristic. We recall some results about the homotopy theory of dg
$\Bbbk[G]$-modules, where $G$ is a finite group. These results will be useful
when considering dg $\mathbb{S}$-modules. Finally, we recall the definitions
of planar and symmetric (co)operads.

###  1.1. Graded modules, differential graded modules and pre-differential
graded modules

######  Definition 1 (Graded modules).

Let $\mathsf{gr}\leavevmode\nobreak\ \Bbbk\text{-}\mathsf{mod}$ be the
category of graded $\Bbbk$-modules. Objects are given by families
$X=\\{X_{n}\\}_{n\in\mathbb{Z}}$ of $\Bbbk$-modules indexed by the set of
integers $\mathbb{Z}$, and morphisms by collections of linear maps indexed by
$\mathbb{Z}$ which respect the grading.

It forms a closed symmetric monoidal category endowed with the tensor product
$X\otimes Y$ given by

$(X\otimes Y)_{n}\coloneqq\bigoplus_{i+j=n}X_{i}\otimes
Y_{j}\leavevmode\nobreak\ ,$

where the internal hom $[X,Y]$ is given by

$[X,Y]_{n}\coloneqq\prod_{k}[X_{k},Y_{n+k}],$

for every two graded $\Bbbk$-modules $X,Y$ and every integer $n$ in
$\mathbb{Z}$. An element $x$ of $X_{n}$ will be called an homogeneous element
of degree $n$ of $X$. We denote the degree of a homogeneous element by $|x|$.
A degree $n$ map from $X$ to $Y$ is just a homogeneous element of degree $n$
of $[X,Y]$.

Moreover, let $f\in[X,X^{\prime}]_{n}$ and $g\in[Y,Y^{\prime}]_{m}$. We denote
$f\otimes g$ the element of $[X\otimes Y,X^{\prime}\otimes Y^{\prime}]_{n+m}$
defined as

$(f\otimes g)(x\otimes y)\coloneqq(-1)^{|g||x|}f(x)\otimes g(y)$

and $[f,g]$ the element of $[[X^{\prime},Y],[X,Y^{\prime}]]_{n+m}$ defined as

$[f,g](h)\coloneqq(-1)^{|f||h|}g\circ h\circ f.$

The category of graded $\Bbbk$-modules enjoys several categorical properties.

  1. (1)

The (monadic) forgetful functor towards graded sets commutes with sifted
colimits (it commutes with filtered colimits and reflexive coequalisers).

  2. (2)

The tensor product commutes with colimits and with finite limits.

  3. (3)

Subsequently the functor $X\mapsto X^{\otimes n}$ commutes with sifted
colimits and with finite cosifted limits.

  4. (4)

Filtered colimits commute with finite limits (since both are computed in
graded sets).

  5. (5)

Coproducts commute with finite limits.

  6. (6)

Products commute with finite colimits.

######  Example 1.

However, cofiltered limits do not commute in general with finite colimits.
Indeed, in the context where $\Bbbk=\mathbb{R}$, let $X$ be the sub
$\mathbb{R}$-module of $\mathbb{R}^{\mathbb{N}}$ spanned by sequence
$(x_{0},x_{1},\ldots)$ so that $\sum_{n}|x_{n}|<+\infty$ and let
$s:X\longrightarrow X$ be the shift endomorphism

$s(x_{0},x_{1},x_{2},\ldots)=(0,x_{0},x_{1},x_{2},\ldots).$

Moreover, let $Y\subseteq X$ the sub $\mathbb{R}$-module spanned by sequence
whose sum is zero. It is stable through $s$. Moreover the following sequence
is exact

$0\longrightarrow Y\hookrightarrow
X\xrightarrow{\Sigma}\mathbb{R}\longrightarrow 0$

where $\Sigma$ denotes the sum. The following diagram is commutative

${\cdots}$${Y}$${\cdots}$${Y}$${Y}$${\cdots}$${X}$${\cdots}$${X}$${X}$${\cdots}$${\mathbb{R}}$${\cdots}$${\mathbb{R}}$${\mathbb{R}.}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{\Sigma}$$\scriptstyle{s}$$\scriptstyle{s}$$\scriptstyle{\Sigma}$$\scriptstyle{\Sigma}$

The limit of the two first row are $0$ while the limit of the last row is
$\mathbb{R}$. Thus, it cannot be the quotient of the second limit by the first
one.

######  Definition 2 (Pre-differential graded modules).

We denote $\mathsf{pdg}\leavevmode\nobreak\ \Bbbk\text{-}\mathsf{mod}$ the
category of pre-differential graded (pdg) $\Bbbk$-modules. Objects are given
by graded $\Bbbk$-modules $\\{X_{n}\\}_{n\in\mathbb{Z}}$ equipped with a
degree $-1$ endomorphism $d:X_{(-)}\longrightarrow X_{(-)-1}$, and morphisms
by morphisms of graded modules which commute with the pre-differentials.

The closed symmetric monoidal category structure on graded $\Bbbk$-modules
induces a closed symmetric monoidal category structure on pdg $\Bbbk$-modules.
For any two pdg $\Bbbk$-modules $X,Y$, the graded tensor product $X\otimes Y$
is equipped with the pre-differential

$d_{X\otimes Y}\coloneqq
d_{x}\otimes\mathrm{Id}_{Y}+d_{X}\otimes\mathrm{Id}_{Y}\leavevmode\nobreak\ ,$

and the internal hom $[X,Y]$ with the pre-differential

$d_{[X,Y]}\coloneqq[\mathrm{Id},d_{Y}]-[d_{x},\mathrm{Id}].$

######  Definition 3 (Differential graded modules).

We denote $\mathsf{dg}\leavevmode\nobreak\ \Bbbk\text{-}\mathsf{mod}$ the
category of differential graded (dg) $\Bbbk$-modules. It is the full
subcategory of pdg $\Bbbk$-modules $(X,d)$ that $d^{2}=d\circ d=2$.

######  Remark 1.

A differential graded module is also referred as a chain complex.

It inherits from pdg $\Bbbk$-modules the structure of a closed symmetric
monoidal category. One can check that if $X$ and $Y$ are dg modules, their
tensor product $X\otimes Y$ and their internal hom $[X,Y]$ is again a dg
module.

######  Definition 4 (Spheres and disks).

Let $n\in\mathbb{Z}$ be an integer. The $n$-th disk, denoted by $D^{n}$, is
the dg module given by

$D^{n}_{m}\coloneqq\begin{cases}\Bbbk\text{ if }m=n-1,n\leavevmode\nobreak\
,\\\ 0\text{ otherwise}\leavevmode\nobreak\ ,\end{cases}$

where the differential $D^{n}_{n}\longrightarrow D^{n}_{n-1}$ is the identity
of map of $\Bbbk$.

The $n$-th sphere, denoted by $S^{n}$, is the dg module given by

$S^{n}_{m}\coloneqq\begin{cases}\Bbbk\text{ if }m=n\leavevmode\nobreak\ ,\\\
0\text{ otherwise}.\end{cases}$

######  Definition 5 (Suspensions).

Let $k\in\mathbb{Z}$ be an integer and let $X$ is a graded (resp. pdg, dg)
$\Bbbk$-module. The tensor product $S^{k}\otimes X$ is referred as the $k$-th
suspension of $X$ and will be denoted by $s^{k}X$.

The category of dg $\Bbbk$-modules can be endowed with a combinatorial model
category structure, determined by the following sets of maps:

  1. (1)

the set of weak equivalences is given by quasi-isomorphisms;

  2. (2)

the set of fibrations is given by degree-wise epimorphisms;

  3. (3)

the set of cofibrations is given by degree-wise injections;

  4. (4)

the set of generating cofibrations is given by the canonical inclusion maps
$S^{n}\longrightarrow D^{n+1}$ for all $n$ in $\mathbb{Z}$;

  5. (5)

the set of generating acyclic cofibrations is given by the inclusion maps
$0\longrightarrow D^{n+1}$ for all $n$ in $\mathbb{Z}$.

######  Remark 2.

Since $\Bbbk$ is a field, this model category structure is both the injective
model structure and the projective model structure.

This model category structure can be left-transferred along the forgetful-
truncation adjunction

${\mathsf{dg}\leavevmode\nobreak\ \Bbbk\text{-}\mathsf{mod}_{\geq
0}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod},}$$\scriptstyle{\mathrm{U}}$$\scriptstyle{\tau_{\geq
0}}$${\dashv}$

to dg $\Bbbk$-modules in non-negative degrees. It can also be right-
transferred via the truncation-forgetful adjunction

${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}_{\leq 0},}$$\scriptstyle{\tau_{\leq
0}}$$\scriptstyle{\mathrm{U}}$${\dashv}$

to dg $\Bbbk$-modules in non-positive degrees.

###  1.2. Finite group action on dg modules

Let $G$ be a finite group. The data of an action of $G$ on a dg $\Bbbk$-module
is equivalent to the data of a dg $\Bbbk[G]$-module, where $\Bbbk[G]$ denotes
the group algebra of $G$.

The functor

${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk[G]\text{-}\mathsf{mod}}$${X}$${\Bbbk[G]\otimes X}$

is both left and right adjoint to the forgetful functor $U_{G}$, and fits in
the following adjunction diagram

${\mathsf{dg}\leavevmode\nobreak\
\Bbbk[G]\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\leavevmode\nobreak\
.}$$\scriptstyle{U_{G}}$$\scriptstyle{\Bbbk[G]\leavevmode\nobreak\
\otimes-}$$\scriptstyle{\Bbbk[G]\leavevmode\nobreak\
\otimes-}$${\dashv}$${\dashv}$

There is a bialgebra structure on $\Bbbk[G]$, where the coproduct $\Delta$ is
induced by the set-theoretical diagonal of $G$. It allows us to endow the
category of dg $\Bbbk[G]$-modules with with a monoidal category structure. The
tensor product of two dg $\Bbbk[G]$-modules $X,Y$ is given by the underlying
dg module $X\otimes Y$ endowed with the following dg $\Bbbk[G]$-module
structure:

${\Bbbk[G]\otimes(X\otimes Y)}$${\Bbbk[G\times G]\otimes(X\otimes
Y)}$${(\Bbbk[G]\otimes X)\otimes(\Bbbk[G]\otimes Y)}$${X\otimes
Y\leavevmode\nobreak\ ,}$$\scriptstyle{\Delta\leavevmode\nobreak\
\otimes\leavevmode\nobreak\
\mathrm{id}}$$\scriptstyle{\cong}$$\scriptstyle{\gamma_{X}\leavevmode\nobreak\
\otimes\leavevmode\nobreak\ \gamma_{Y}}$

where $\gamma_{X},\gamma_{Y}$ denote, respectively, the dg $\Bbbk[G]$-module
structures of $X$ and $Y$.

One can either right-transfer or left-transfer the model category structure on
dg modules to dg $\Bbbk[G]$-modules. This gives two different combinatorial
model structures.

  1. (1)

The right-transferred structure is called the projective model structure.
Fibrations are given by degree-wise epimorphisms and weak-equivalences by
quasi-isomorphisms.

  2. (2)

The left-transferred structure is called the injective model structure.
Cofibrations are given by degree-wise monomorphisms and weak-equivalences by
quasi-isomorphisms.

######  Definition 6 (Projective and injective modules).

Let $X$ be a dg $\Bbbk[G]$-module.

  1. (1)

$X$ is said to be projective if it is a fibrant-cofibrant object in the
projective model structure.

  2. (2)

$X$ is said to be injective if it is a fibrant-cofibrant object in the
injective model structure.

######  Lemma 1.

Let $X$ be dg $\Bbbk[G]$-module.

  1. (1)

If $X$ is injective, then its linear dual $X^{*}$ is projective.

  2. (2)

If $X$ is projective, then its linear dual $X^{*}$ is injective.

###### Proof.

Linear duality defines a Quillen adjunction

${\mathsf{dg}\leavevmode\nobreak\
\Bbbk[G]\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk[G]\text{-}\mathsf{mod}^{\mathsf{op}},}$$\scriptstyle{(-)^{*}}$$\scriptstyle{(-)^{*}}$${\dashv}$

when the category on the left is endowed with the projective model structure
and the category on the right with the opposite of the injective model
structure. ∎

The two adjunctions $\Bbbk[G]\otimes-\dashv U_{G}\dashv\Bbbk[G]\otimes-$
restrict to dg $\Bbbk$-modules and dg $\Bbbk[G]$-modules which are either in
non-negative degrees or in non-positive degrees.

  1. (1)

The category of dg $\Bbbk[G]$-modules in non-negative degrees can be endowed
with the projective model structure. Fibrations are given by degree-wise
epimorphisms and weak-equivalences by quasi-isomorphisms. Moreover,
cofibrations are degree-wise injections whose cokernel is degre-ewise
projective.

  2. (2)

The category of dg $\Bbbk[G]$-modules in non-positive degrees can be endowed
with the injective model structure. Cofibrations are given by degree-wise
injections and weak-equivalences by quasi-isomorphisms. Moreover, fibrations
are degree-wise epimorphisms whose kernel is degree-wise injective.

######  Definition 7 (Quasi-free modules).

A dg $\Bbbk[G]$-module is quasi-free if its underlying graded
$\Bbbk[G]$-module is free, that is, in the essential image of the functor
$\Bbbk[G]\otimes-$ from graded $\Bbbk$-module to graded $\Bbbk[G]$-modules.

A quasi-free dg $\Bbbk[G]$-module $X$ is degree-wise projective and degree-
wise injective. Nevertheless, it might not be cofibrant in the projective
model structure nor fibrant in the injective model structure.

  1. (1)

If $X$ is bounded below, it is cofibrant in the projective model structure
since, up to some a finite degree translation, it is the image of a cofibrant
object of $\mathsf{dg}\leavevmode\nobreak\ \Bbbk[G]\text{-}\mathsf{mod}_{\geq
0}$.

  2. (2)

If $X$ is bounded above, it is fibrant in the injective model structure since,
up to some a finite degree translation, it is the image of a fibrant object of
$\mathsf{dg}\leavevmode\nobreak\ \Bbbk[G]\text{-}\mathsf{mod}_{\leq 0}$.

######  Remark 3.

To the best of our knowledge, these results are particular instances of
results in [AFH03]. As we were unable to find it, we refer to [Abb14] for more
details.

######  Proposition 1.

Quasi-free dg $\Bbbk[G]$-modules are stable through tensor product in the
following sense: $X\otimes Y$ is quasi-free whenever $X$ or $Y$ is quasi-free.

###### Proof.

Let $X$ and $Y$ be dg $\Bbbk[G]$-modules, and assume that
$Y\cong\Bbbk[G]\otimes Z$ as graded $\Bbbk[G]$-modules. The canonical morphism
of dg $\Bbbk[G]$-modules

$\Bbbk[G]\otimes(U_{G}(X)\otimes Z)\longrightarrow X\otimes Y$

is an isomorphism, with inverse

$x\otimes g\otimes z\mapsto g\otimes(g^{-1}(x)\otimes z).$

∎

######  Remark 4.

The above proposition holds if one replaces $\Bbbk[G]$ by any Hopf algebra
$H$.

######  Proposition 2.

Let $X$ be a quasi-free dg $\Bbbk[G]$-module. The canonical norm map

$\mathbb{N}_{X}:X_{G}\longrightarrow X^{G}$

from the coinvariants to the invariants is an isomorphism of dg modules.

###### Proof.

Follows from the fact that $X$ is quasi-free and that limits and colimits are
computed degree-wise. ∎

Finally, the projective and injective model category structures are compatible
in the following sense.

######  Proposition 3.

The category of dg $\Bbbk[G]$-modules together with the projective model
structure is homotopically enriched-tensored-cotensored over the category dg
$\Bbbk[G]$-module together with the injective model structure. For every
injective cofibration (i.e: a degree-wise injection) $f:A\rightarrowtail B$
and every projective cofibration $g:X\rightarrowtail Y$, the morphism

$f\diamond g:(A\otimes Y)\coprod_{A\leavevmode\nobreak\
\otimes\leavevmode\nobreak\ X}(B\otimes X)\rightarrowtail B\otimes Y$

is a projective cofibration. Furthermore, it is acyclic whenever $f$ or $g$
is.

###### Proof.

Let us suppose that $g$ is a generating projective cofibration, given by the
inclusion $S^{k}\otimes\Bbbk[G]\longrightarrow D^{k+1}\otimes\Bbbk[G]$. In
that case, one has a canonical isomorphism of diagrams between the commutative
square

${A\otimes(S^{k}\otimes\Bbbk[G])}$${A\otimes(D^{k+1}\otimes\Bbbk[G])}$${B\otimes(S^{k}\otimes\Bbbk[G])}$${B\otimes(D^{k+1}\otimes\Bbbk[G])}$

and the commutative square

${(U_{G}(A)\otimes S^{k})\otimes\Bbbk[G]}$${(U_{G}(A)\otimes
D^{k+1})\otimes\Bbbk[G]}$${(U_{G}(B)\otimes
S^{k})\otimes\Bbbk[G]}$${(U_{G}(B)\otimes
D^{k+1})\otimes\Bbbk[G]\leavevmode\nobreak\ .}$

Thus, the map $f\diamond g$ is isomorphic to the image through the functor
$-\otimes\Bbbk[G]$ of the map

$(U_{G}(A)\otimes D^{k+1})\coprod_{(U_{G}(A)\leavevmode\nobreak\
\otimes\leavevmode\nobreak\ S^{k})}((U_{G}(B)\otimes
S^{k}))\to(U_{G}(B)\otimes D^{k+1})\leavevmode\nobreak\ ,$

which is a cofibration of dg $\Bbbk$-modules. Hence $f\diamond g$ is a
projective cofibration. Now the set of morphisms $g$ so that $f\diamond g$ is
a projective cofibration is stable through pushouts, transfinite composition
and retracts. Hence it contains all the projective cofibrations.

Finally, if $f$ is acyclic or $g$ are acyclic, then $f\diamond g$ is acyclic
too, as it is an acylic cofibration of dg $\Bbbk$-modules and since the model
category on dg $\Bbbk$-modules is a monoidal model category. ∎

###  1.3. $\mathbb{N}$-modules, planar operads and planar cooperads

We consider $\mathbb{N}$ as a category where objects are natural integers and
where there are only identity morphisms.

######  Definition 8 (dg $\mathbb{N}$-modules).

A dg $\mathbb{N}$-modules amounts to the data of a functor

$X:\mathbb{N}\longrightarrow\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}.$

The object $X(n)$ is called the arity $n$ part of $X$. We denote
$\mathsf{dg}\leavevmode\nobreak\ \mathbb{N}\text{-}\mathsf{mod}$ the category
of dg $\mathbb{N}$-modules.

######  Remark 5.

The category of dg $\mathbb{N}$-modules admits a canonical combinatorial model
category structure, determined by the following sets of maps:

  1. (1)

the set of weak equivalences
$f:X\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}Y$ is
given by arity-wise quasi-isomorphisms
$f(n):X(n)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}Y(n)$
for all $n\geq 0$;

  2. (2)

the set of fibrations $f:X\twoheadrightarrow Y$ is given by arity-wise degree-
wise epimorphisms $f(n):X(n)\twoheadrightarrow Y(n)$ for all $n\geq 0$;

  3. (3)

the set of cofibrations $f:X\rightarrowtail Y$ is given by arity-wise degree-
wise injections $f(n):X(n)\rightarrowtail Y(n)$ for all $n\geq 0$.

The planar horizontal product on dg $\mathbb{N}$-modules $X,Y$ is given by the
Day convolution product

$(X\circledast_{\mathrm{pl}}Y)(n)\coloneqq\bigoplus_{k+l=n}X(k)\otimes
Y(l)\leavevmode\nobreak\ .$

This endows dg $\mathbb{N}$ with a symmetric monoidal category structure,
where the unit is given by $\Bbbk$ concentrated in arity $0$.

There is another monoidal structure given by the planar composition product

$(X\circ_{\mathrm{pl}}Y)(n)\coloneqq\bigoplus_{k\geq 0}X(k)\otimes
Y^{\circledast_{\mathrm{pl}}k}(n)\leavevmode\nobreak\ .$

The unit for the composition is given by $\mathcal{I}$, defined as follows

$\mathbb{1}(n)\coloneqq\begin{cases}0\text{ if }n\neq 1\leavevmode\nobreak\
,\\\ \Bbbk\text{ if }n=1.\end{cases}$

######  Definition 9 (Planar dg operad).

A planar dg operad $\mathcal{P}$ amounts to the data of a monoid
$(\mathcal{P},\gamma,\eta)$ in the category of dg $\mathbb{N}$-modules with
respect to the composition product.

######  Definition 10 (Augmented planar dg operad).

An augmented planar dg operad $\mathcal{P}$ amounts to the data of a planar dg
operad $(\mathcal{P},\gamma,\eta)$ equipped with a morphism of planar dg
operads $\nu:\mathcal{P}\longrightarrow\mathcal{I}$ such that
$\nu\circ\eta=\mathrm{id}.$

Given an augmented planar dg operad $\mathcal{P}$, we will denote by
$\overline{\mathcal{P}}$ the kernel of the augmentation map.

######  Definition 11 (Planar dg cooperad).

A planar dg cooperad $\mathcal{C}$ amounts to the data of a comonoid
$(\mathcal{C},\Delta,\epsilon)$ in the category of dg $\mathbb{N}$-modules
with respect to the composition product.

Given a planar dg cooperad $\mathcal{C}$, we will denote by
$\overline{\mathcal{C}}$ the kernel of the counit map.

######  Definition 12 (Coaugmented planar dg cooperad).

A coaugmented planar dg cooperad $\mathcal{C}$ amounts to the data of a planar
dg cooperad $(\mathcal{C},\Delta,\epsilon)$ equipped with a morphism of planar
dg cooperads $\mu:\mathcal{I}\longrightarrow\mathcal{C}$ such that
$\epsilon\circ\mu=\mathrm{id}$.

######  Remark 6.

All the definitions of this subsection make sense in the graded or the pre-
differential setting.

###  1.4. $\mathbb{S}$-modules, operads and cooperads

In this subsection, we deal with dg (resp. graded or pdg)
$\mathbb{S}$-modules.

######  Definition 13 (dg $\mathbb{S}$-module).

Let $\mathbb{S}$ be the groupoid whose objects are natural integers and whose
morphisms are given by

$\hom_{\mathbb{S}}(n,m)=\begin{cases}\emptyset\text{ if }n\neq
m\leavevmode\nobreak\ ,\\\ \mathbb{S}_{n}\text{ if }n=m\leavevmode\nobreak\
.\end{cases}$

A dg $\mathbb{S}$-module $M$ amounts to the data of a functor

$M:\mathbb{S}^{\mathrm{op}}\longrightarrow\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}$

from $\mathbb{S}^{\mathrm{op}}$ to dg modules. It corresponds to collection of
dg modules $\\{M(n)\\}$ for $n\geq 0$, where each $M(n)$ is endowed with a
(right) action of $\mathbb{S}_{n}$. We denote by
$\mathsf{dg}\leavevmode\nobreak\ \mathbb{S}\text{-}\mathsf{mod}$ the category
of dg $\mathbb{S}$-modules.

######  Remark 7.

We define analogously the categories of graded or pdg $\mathbb{S}$-modules.

There is a diagram of adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathbb{S}\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\mathbb{N}\text{-}\mathsf{mod}\leavevmode\nobreak\
,}$$\scriptstyle{U}$$\scriptstyle{-\otimes\leavevmode\nobreak\
\mathbb{S}}$$\scriptstyle{-\otimes\leavevmode\nobreak\
\mathbb{S}}$${\dashv}$${\dashv}$

between the categories of dg $\mathbb{N}$-modules and of dg
$\mathbb{S}$-modules. The functor $-\otimes\mathbb{S}$ is given by

$(X\otimes\mathbb{S})(n)\coloneqq
X(n)\otimes\Bbbk[\mathbb{S}_{n}]\leavevmode\nobreak\ ,$

for all $n\geq 0$. We will also denote by $-\otimes\mathbb{S}$ the endofunctor
of dg $\mathbb{N}$-modules that is given by the (co)free dg
$\mathbb{S}$-module functor composed with the forgetful functor.

One can either right-transfer or left-transfer the model category structure on
dg $\mathbb{N}$-modules to dg $\mathbb{S}$-modules. This gives two different
combinatorial model structures.

  1. (1)

The right-transferred structure is called the projective model structure.
Fibrations are given by degree-wise arity-wise epimorphisms and weak-
equivalences by arity-wise quasi-isomorphisms.

  2. (2)

The left-transferred structure is called the injective model structure.
Cofibrations are given by degree-wise arity-wise monomorphisms and weak-
equivalences by arity-wise quasi-isomorphisms.

######  Remark 8.

All the results of Subsection 1.2 can be translated to dg $\mathbb{S}$-modules
as the hold for any finite group $G$ and since the homotopy theory of dg
$\mathbb{S}$-modules is determined arity-wise.

The composition product $\circ$ of two dg $\mathbb{S}$-modules $M,N$ is
defined as follows

$M\circ N(n)\coloneqq\bigoplus_{k\geq
0}M(k)\otimes_{\mathbb{S}_{k}}\left(\bigoplus_{i_{1}+\cdots+i_{k}=n}\mathrm{Ind}_{\mathbb{S}_{i_{1}}\times\cdots\times\mathbb{S}_{i_{k}}}^{\mathbb{S}_{n}}(N(i_{1})\otimes\cdots\otimes
N(i_{k}))\right)\leavevmode\nobreak\ .$

The unit for the composition is given by $\mathcal{I}$, defined as follows

$\mathbb{1}(n)\coloneqq\begin{cases}0\text{ if }n\neq 1\leavevmode\nobreak\
,\\\ \Bbbk\text{ if }n=1.\end{cases}$

They endow the category of dg $\mathbb{S}$-modules with a monoidal category
structure.

######  Definition 14 (dg operad).

A dg operad $\mathcal{P}$ amounts to the data of a monoid
$(\mathcal{P},\gamma,\eta)$ in the category of dg $\mathbb{S}$-modules with
respect to the composition product.

######  Definition 15 (augmented dg operad).

An augmented dg operad $\mathcal{P}$ amounts to the data of a dg operad
$(\mathcal{P},\gamma,\eta)$ equipped with a morphism of dg operads
$\nu:\mathcal{P}\longrightarrow\mathcal{I}$ such that
$\nu\circ\eta=\mathrm{id}.$

Given an augmented planar dg operad $\mathcal{P}$, we will denote by
$\overline{\mathcal{P}}$ the kernel of the augmentation map.

######  Definition 16 ($\mathbb{S}$-something dg operad).

Let $\mathcal{P}$ be a dg operad.

  1. (1)

It is called $\mathbb{S}$-projective if its underlying dg $\mathbb{S}$-module
is cofibrant for the projective model structure.

  2. (2)

It is called $\mathbb{S}$-injective if its underlying dg $\mathbb{S}$-module
is fibrant for the injective model structure.

  3. (3)

It is called $\mathbb{S}$-quasi-free if its underlying dg $\mathbb{S}$-module
is quasi-free.

######  Remark 9.

An $\mathbb{S}$-projective dg operad is usually called an
$\mathbb{S}$-cofibrant dg operad in the literature. We adopt this non-standard
terminology in order to be able to differentiate between
$\mathbb{S}$-projective and $\mathbb{S}$-injective dg operads, which both have
cofibrant underlying dg $\mathbb{S}$-modules, but in different model category
structures.

######  Definition 17 (dg cooperad).

A dg cooperad $\mathcal{C}$ amounts to the data of a comonoid
$(\mathcal{C},\Delta,\epsilon)$ in the category of dg $\mathbb{S}$-modules
with respect to the composition product.

Given a dg cooperad $\mathcal{C}$, we will denote by $\overline{\mathcal{C}}$
the kernel of the counit map.

######  Definition 18 (coaugmented dg cooperad).

A coaugmented dg cooperad $\mathcal{C}$ amounts to the data of a dg cooperad
$(\mathcal{C},\Delta,\epsilon)$ equipped together with a morphism of planar dg
cooperads $\mu:\mathcal{I}\longrightarrow\mathcal{C}$ such that
$\epsilon\circ\mu=\mathrm{id}$.

There is a strong monoidal structure on the functor $-\otimes\mathbb{S}$ which
yields two adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Operads}_{\mathrm{pl}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Operads},}$$\scriptstyle{-\otimes\mathbb{S}}$$\scriptstyle{\mathrm{U}_{\mathbb{S}}}$${\dashv}$
${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads}_{\mathrm{pl}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads},}$$\scriptstyle{-\otimes\mathbb{S}}$$\scriptstyle{\mathrm{U}_{\mathbb{S}}}$${\dashv}$

that lift, respectively, the adjunction $-\otimes\mathbb{S}\dashv
U_{\mathbb{S}}$ and the adjunction $U_{\mathbb{S}}\dashv-\otimes\mathbb{S}$
that relate dg $\mathbb{S}$-modules to dg $\mathbb{N}$-modules.

######  Remark 10.

The adjunction $-\otimes\mathbb{S}\dashv U_{\mathbb{S}}$ relating dg operads
to planar dg operads is monadic since its right adjoint preserves coreflexive
equalisers and is conservative. However, the other adjunction
$U_{\mathbb{S}}\dashv-\otimes\mathbb{S}$ is not a priori comonadic.

######  Remark 11.

All the definitions of this subsection make sense in the graded or the pre-
differential setting.

###  1.5. Tree modules and conilpotent cooperads

In this subsection, we briefly recall how operads are algebras over the tree
monad and how conilpotent cooperads are exactly coalgebras over the tree
comonad, both in the planar and in the symmetric case. These constructions can
all be found in [LV12]. For a more detailed discussion about the point of view
adopted here, see the forthcoming note [GRiL23c].

Planar tree endofunctor. For every dg $\mathbb{N}$-module $X$, one can define
the planar tree module $\mathbb{T}_{\mathrm{pl}}(X)$ of $X$, which is the dg
$\mathbb{N}$-module given, for $m\geq 0$, by

$\mathbb{T}_{\mathrm{pl}}(X)(m)=\bigoplus_{t}t(X)\leavevmode\nobreak\ ,$

where the sum is taken over the isomorphism classes of planar trees with $m$
leaves.

Let $n$ be in $\mathbb{N}$. We define the following sub-functors of the planar
tree module:

  * •

The reduced planar tree endofunctor $\overline{\mathbb{T}}_{\mathrm{pl}}(X)$,
given by the sum over all non-trivial planar trees;

  * •

The $n$-levelled planar tree endofunctor $\mathbb{T}_{\mathrm{pl},\leq n}(X)$,
given by the sum over planar trees whose height is equal or lower than $n$
(recall that the height of the trivial tree with no node is $0$);

  * •

The $n$-weight planar tree endofunctor $\mathbb{T}_{\mathrm{pl}}^{(\leq
n)}(X)$, given by the sum over planar trees with $n$ nodes or less.

All these constructions are natural in $X$ and define endofunctors of the
category of dg $\mathbb{N}$-modules.

Planar operads. The planar tree module endofunctor $\mathbb{T}_{\mathrm{pl}}$
admits a monad structure induced by the grafting of planar trees.

######  Proposition 4.

The category of algebras over the monad $\mathbb{T}_{\mathrm{pl}}$ is
canonically isomorphic to the category of planar dg operads.

Conilpotent planar cooperads. The reduced planar tree module endofunctor
$\overline{\mathbb{T}}_{\mathrm{pl}}$ admits a comonad structure induced by
partitioning planar trees. Furthermore, there is a fully faithful functor

$\mathrm{Conil}:\mathsf{dg}\leavevmode\nobreak\
\overline{\mathbb{T}}_{\mathrm{pl}}\text{-}\mathsf{cog}\longrightarrow(\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads}_{\mathrm{pl}})_{\mathcal{I}/}$

from dg $\overline{\mathbb{T}}_{\mathrm{pl}}$-coalgebras to coaugmented planar
dg cooperads.

######  Definition 19 (Conilpotent planar dg cooperad).

Let $\mathcal{C}$ be a coaugmented planar dg cooperad. It is conilpotent if it
belongs to the essential image of the functor $\mathrm{Conil}$ from dg
$\overline{\mathbb{T}}_{\mathrm{pl}}$-coalgebras to planar dg cooperads. We
denote $\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}}_{\mathrm{pl}}$ the full sub-category of
coaugmented planar dg cooperads spanned by conilpotent ones.

######  Remark 12.

The idea behind this definition is the following: a (non-counital) cooperad
can also be described in terms of partial decomposition maps

$\Delta_{i}:\mathcal{C}(n+k-1)\longrightarrow\mathcal{C}(n)\otimes\mathcal{C}(k)\leavevmode\nobreak\
,$

and it is conilpotent if and only if any iteration of these partial
decompositions is eventually trivial. If this is the case, then the data of
all the possible iterations is exactly encoded by the
$\overline{\mathbb{T}}_{\mathrm{pl}}$-coalgebra structure.

For every natural integer $n\geq 1$, the comonad structure on
$\overline{\mathbb{T}}_{\mathrm{pl}}$ restricts to
$\overline{\mathbb{T}}_{\mathrm{pl}}^{(\leq n)}$. The inclusion of comonads
$\overline{\mathbb{T}}_{\mathrm{pl}}^{(\leq
n)}\rightarrowtail\overline{\mathbb{T}}_{\mathrm{pl}}$ induces an endofunctor
$\mathrm{F}^{\mathrm{rad}}_{n}$ in the category of conilpotent planar dg
cooperads.

######  Definition 20 (Coradical filtration).

Let $\mathcal{C}$ be a conilpotent planar dg cooperad. Its $n$-coradical
filtration is given by the conilpotent planar dg cooperad
$\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}$. It induces a ladder diagram

$\mathrm{F}^{\mathrm{rad}}_{0}\mathcal{C}\rightarrowtail\mathrm{F}^{\mathrm{rad}}_{1}\mathcal{C}\rightarrowtail\cdots\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}\rightarrowtail\cdots\leavevmode\nobreak\
,$

indexed by $\mathbb{N}$, where all the arrows are monomorphisms.

For any conilpotent planar dg cooperad $\mathcal{C}$, there is a canonical
isomorphism between $\mathcal{C}$ and the colimit of the following ladder
diagram

$\mathrm{F}^{\mathrm{rad}}_{0}\mathcal{C}\rightarrowtail\mathrm{F}^{\mathrm{rad}}_{1}\mathcal{C}\rightarrowtail\cdots\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}\rightarrowtail\cdots$

in the category of conilpotent planar dg cooperads.

######  Proposition 5.

Let $n\geq 0$ and $\mathcal{C}$ be a conilpotent planar dg cooperad. Then
$\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}$ fits in the following pullback

${\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}}$${\overline{\mathbb{T}}_{\mathrm{pl}}^{(\leq
n)}\mathcal{C}}$${\mathcal{C}}$${\overline{\mathbb{T}}_{\mathrm{pl}}\mathcal{C}.}$${\lrcorner}$$\scriptstyle{\delta_{\mathcal{C}}}$

in the category of dg $\mathbb{N}$-modules, where $\delta_{\mathcal{C}}$
denotes the dg $\overline{\mathbb{T}}_{\mathrm{pl}}$-coalgebra structure of
$\mathcal{C}$.

###### Proof.

This is a direct application of Proposition 71. ∎

Tree endofunctor. Let $M$ be a dg $\mathbb{S}$-module, we can define the tree
endofunctor $\mathbb{T}(M)$ as the following reflexive coequalizer

${\mathrm{Coeq}\Bigg{(}\displaystyle(\mathbb{T}_{\mathrm{pl}}((\mathrm{U}(M)\otimes\mathbb{S})\otimes\mathbb{S}}$${\mathbb{T}_{\mathrm{pl}}(\mathrm{U}(M))\otimes\mathbb{S}\Bigg{)}}$${\mathbb{T}(M)\leavevmode\nobreak\
,}$

where one of the maps is build from the dg $\mathbb{S}$-module structure of
$M$ and the other using the monad structures $-\otimes\mathbb{S}$. Note that
this definition is equivalent to the more standard one in [LV12].

######  Notation.

Let $n$ be a natural integer. We define analogously variants of the tree
endofunctor by replacing the planar tree endofunctor
$\mathbb{T}_{\mathrm{pl}}$ in the above coequalizer.

  * •

The reduced tree endofunctor $\overline{\mathbb{T}}$ is given by replacing
$\mathbb{T}_{\mathrm{pl}}$ with $\overline{\mathbb{T}}_{\mathrm{pl}}$.

  * •

We denote by $\mathbb{T}_{\leq n}$ the endofunctor obtained by replacing
$\mathbb{T}_{\mathrm{pl}}$ with $\overline{\mathbb{T}}_{\mathrm{pl},\leq n}$.

  * •

We denoted by $\mathbb{T}^{(n)}$ the endofunctor obtained by replacing
$\mathbb{T}_{\mathrm{pl}}$ with $\overline{\mathbb{T}}_{\mathrm{pl}}^{(n)}$.

  * •

We denoted by $\mathbb{T}^{(\leq n)}$ the endofunctor obtained by replacing
$\mathbb{T}_{\mathrm{pl}}$ with $\overline{\mathbb{T}}_{\mathrm{pl}}^{(\leq
n)}$.

######  Proposition 6.

Let $X$ be a dg $\mathbb{N}$-module. The canonical map

$\nu_{X}:\mathbb{T}(X\otimes\mathbb{S})\longrightarrow\mathbb{T}_{\mathrm{pl}}(X)\otimes\mathbb{S}$

is a isomorphism of dg $\mathbb{S}$-modules, natural in $X$.

Operads. There is a monad structure on the tree endofunctor $\mathbb{T}$ which
can be constructed using the monad structure on the planar tree endofunctor.
We refer to [GRiL23c] for more details.

######  Proposition 7.

The category of algebras over the monad $\mathbb{T}$ is canonically isomorphic
to the category of dg operads.

Conilpotent cooperads. There is a comonad structure on the reduced tree
endofunctor $\overline{\mathbb{T}}$, which again can be constructed from the
comonad structure on the reduced planar tree endofunctor
$\overline{\mathbb{T}}_{\mathrm{pl}}$. We refer to [GRiL23c] for more details.
There is a fully faithful functor

$\mathrm{Conil}:\mathsf{dg}\leavevmode\nobreak\
\overline{\mathbb{T}}\text{-}\mathsf{cog}\longrightarrow(\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads})_{\mathcal{I}/}$

from dg $\overline{\mathbb{T}}$-coalgebras to coaugmented dg cooperads.

######  Definition 21 (Conilpotent dg cooperad).

Let $\mathcal{C}$ be a coaugmented dg cooperad. It is conilpotent if it
belongs to the essential image of the functor $\mathrm{Conil}$ from dg
$\overline{\mathbb{T}}$-coalgebras to dg cooperads. We denote
$\mathsf{dg}\leavevmode\nobreak\ \mathsf{Cooperads}^{\mathsf{conil}}$ the full
sub-category of coaugmented dg cooperads spanned by conilpotent ones.

The adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads}_{\mathrm{pl}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads},}$$\scriptstyle{-\otimes\mathbb{S}}$$\scriptstyle{\mathrm{U}_{\mathbb{S}}}$${\dashv}$

restricts to an adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads}_{\mathrm{pl}}^{\mathsf{conil}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}},}$$\scriptstyle{-\otimes\mathbb{S}}$$\scriptstyle{\mathrm{U}_{\mathbb{S}}}$${\dashv}$

between conilpotent planar dg cooperads and conilpotent dg cooperads.

For every natural integer $n\geq 1$, the comonad structure on
$\overline{\mathbb{T}}$ restricts to $\overline{\mathbb{T}}^{(\leq n)}$. The
inclusion of comonads $\overline{\mathbb{T}}^{(\leq
n)}\rightarrowtail\overline{\mathbb{T}}$ induces an endofunctor
$\mathrm{F}^{\mathrm{rad}}_{n}$ in the category of conilpotent dg cooperads.

######  Definition 22 (Coradical filtration).

Let $\mathcal{C}$ be a conilpotent dg cooperad. Its $n$-coradical filtration
is given by the conilpotent dg cooperad
$\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}$. It induces a ladder diagram

$\mathrm{F}^{\mathrm{rad}}_{0}\mathcal{C}\rightarrowtail\mathrm{F}^{\mathrm{rad}}_{1}\mathcal{C}\rightarrowtail\cdots\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}\rightarrowtail\cdots\leavevmode\nobreak\
,$

indexed by $\mathbb{N}$, where all the arrows are monomorphisms.

For any conilpotent dg cooperad $\mathcal{C}$, there is a canonical
isomorphism between $\mathcal{C}$ and the colimit of the following ladder
diagram

$\mathrm{F}^{\mathrm{rad}}_{0}\mathcal{C}\rightarrowtail\mathrm{F}^{\mathrm{rad}}_{1}\mathcal{C}\rightarrowtail\cdots\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}\rightarrowtail\cdots$

in the category of conilpotent dg cooperads.

All the construction we have performed in this section commutes with the
forgetful functors

$\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\longrightarrow\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\leavevmode\nobreak\ .$

This implies that if a dg (or pdg) conilpotent cooperad $\mathcal{C}$
proceeds, as a graded conilpotent cooperad, from a planar one
$\mathcal{C}_{\mathrm{pl}}$ (in the sense that there is an isomorphism
$\mathcal{C}\cong\mathcal{C}_{\mathrm{pl}}\otimes\mathbb{S}$ of graded
cooperads), then the conilpotent dg cooperad
$\overline{\mathrm{F}}_{n}\mathcal{C}$ fits in the following pullback diagram

${\mathrm{F}^{\mathrm{rad}}_{n}\mathcal{C}}$${\overline{\mathbb{T}}^{(\leq
n)}\mathcal{C}}$${\mathcal{C}}$${\overline{\mathbb{T}}\mathcal{C}\leavevmode\nobreak\
,}$${\lrcorner}$$\scriptstyle{\delta_{\mathcal{C}}}$

in the category of dg $\mathbb{S}$-modules, where $\delta_{\mathcal{C}}$
denotes the dg $\overline{\mathbb{T}}$-coalgebra structure of $\mathcal{C}$.

######  Remark 13.

If the characteristic of the base field $\Bbbk$ is zero, both the tree module
$\mathbb{T}(-)$ and the composition product $\circ$ preserve finite cosifted
limits. Therefore the forgetful functors from (conilpotent) dg cooperads to dg
$\mathbb{S}$-modules preserve these limits. Thus, as in the planar case, given
a dg cooperad $\mathcal{C}$, the conilpotent dg cooperad
$\overline{\mathrm{F}}_{n}\mathcal{C}$ fits in a pullback diagram as above.

##  2\. Homotopy theory of operads and quasi-planar cooperads

The main goal of this section is to introduce the notion of a quasi-planar
cooperad. First, we recall the bar-cobar constructions at the operadic level
(the unital/curved version of [LG19]). Then, we recall the semi-model category
structure of [Fre09], which encodes the homotopy theory of dg operads.
Finally, we introduce the notion of a quasi-planar cooperad, construct basic
examples, and show their cobar constructions provide us with cofibrant
resolutions which are particularly well-behaved when working over a field of
any characteristic. Furthermore, for any quasi-planar conilpotent curved
cooperad $\mathcal{C}$, where construct a canonical $\mathcal{E}$-comonoid
structure over $\Omega\mathcal{C}$, where $\mathcal{E}$ is the Barratt-Eccles
operad. This allows us to define a canonical quasi-planar ladder for any
quasi-planar conilpotent curved cooperad $\mathcal{C}$, which can be viewed as
the positive characteristic analogue of the coradical filtration.

###  2.1. Conilpotent curved cooperads and the operadic bar-cobar adjunction

The Koszul dual notion of a non-necessarily augmented dg operad is a
conilpotent curved cooperad. We recall the bar-cobar adjunction at the
operadic level that links these two notions constructed in [LG19].

######  Definition 23 (Curved cooperad).

A curved cooperad $\mathcal{C}$ amounts to the data of a pdg cooperad
$(\mathcal{C},\Delta,\epsilon,d)$ equipped with a degree $-2$ map
$\theta:\mathcal{C}\longrightarrow\mathcal{I}$ called the curvature such that
$\theta\circ d=0$, and such that the following diagram commutes:

${\mathcal{C}}$${\mathbb{T}^{(2)}\overline{\mathcal{C}}}$${\mathcal{C}\leavevmode\nobreak\
.}$$\scriptstyle{\Delta_{(1)}}$$\scriptstyle{d^{2}}$$\scriptstyle{(\mathrm{Id}\leavevmode\nobreak\
\otimes\leavevmode\nobreak\ \theta)\leavevmode\nobreak\ -\leavevmode\nobreak\
(\theta\leavevmode\nobreak\ \otimes\leavevmode\nobreak\
\mathrm{Id})\leavevmode\nobreak\ }$

Here $\Delta_{(1)}$ is given by the sum over all possible decompostions into
pairs.

######  Remark 14.

A curved cooperad with zero curvature is a dg cooperad.

######  Remark 15.

A curved cooperad is said to be coaugmented (resp. conilpotent) if its
underlying pdg cooperad is.

The operadic bar construction. Given a dg operad $\mathcal{P}$, one can build
a conilpotent curved cooperad $\mathrm{B}\mathcal{P}$ whose underlying
conilpotent graded cooperad is given by $\mathbb{T}(s\mathcal{P}\oplus
s^{2}\mathcal{I})$. We endow it with the unique coderivation whose projection
onto the generators is the sum of the following maps:

  1. (1)

the map

$\displaystyle\mathbb{T}(s\mathcal{P}\oplus
s^{2}\mathcal{I})\twoheadrightarrow\mathbb{T}^{(2)}(s\mathcal{P})$
$\displaystyle\longrightarrow s\mathcal{P}$
$\displaystyle(sp\otimes_{i}sp^{\prime})\otimes\\{\sigma\\}$
$\displaystyle\mapsto(-1)^{|p|}s(p\circ_{i}p^{\prime})^{\sigma}$

where $sp\otimes_{i}sp^{\prime}$ labels a 2 nodes tree whose second node is
plugged at the $i^{th}$ leaf of the root node,

  2. (2)

the map

$\displaystyle\mathbb{T}(s\mathcal{P}\oplus
s^{2}\mathcal{I})\twoheadrightarrow s\mathcal{P}$
$\displaystyle\longrightarrow s\mathcal{P}$ $\displaystyle sp$
$\displaystyle\mapsto-sd(p)\leavevmode\nobreak\ ,$

  3. (3)

the map

$\displaystyle\mathbb{T}(s\mathcal{P}\oplus
s^{2}\mathcal{I})\twoheadrightarrow s^{2}\mathcal{I}$
$\displaystyle\longrightarrow s\mathcal{P}$ $\displaystyle s^{2}1$
$\displaystyle\mapsto-s1_{\mathcal{P}}.$

We denote by $d_{\gamma}$, $d_{\mathcal{P}}$ and $d_{u}$ the respective
coderivations induced by these maps. The curvature is given by the following
map

$\displaystyle\Theta:\mathbb{T}(s\mathcal{P}\oplus
s^{2}\mathcal{I})\twoheadrightarrow s^{2}\mathcal{I}$
$\displaystyle\longrightarrow\mathcal{I}$ $\displaystyle s^{2}1$
$\displaystyle\mapsto 1.$

One can check that $(\mathbb{T}(s\mathcal{P}\oplus
s^{2}\mathcal{I}),d_{\gamma}+d_{\mathcal{P}}+d_{u},\Theta)$ forms a
conilpotent curved cooperad.

The operadic cobar construction. Given a coaugmented curved cooperad
$\mathcal{C}$, one can build a dg operad $\Omega\mathcal{C}$ whose underlying
graded operad is $\mathbb{T}(s^{-1}\overline{\mathcal{C}})$. We endow it with
the unique derivation whose restriction to the generators is the sum of the
following maps:

  1. (1)

the map

$\displaystyle s^{-1}\overline{\mathcal{C}}$
$\displaystyle\longrightarrow\mathbb{T}^{(2)}(s^{-1}\overline{\mathcal{C}})\hookrightarrow\mathbb{T}(s^{-1}\overline{\mathcal{C}})$
$\displaystyle s^{-1}c$
$\displaystyle\mapsto-\sum(-1)^{|c_{(1)}|}sc_{(1)}\otimes
sc_{(2)}\otimes\\{\sigma\\}$

where $\Delta_{(1)}(c)=\sum c_{(1)}\otimes c_{(2)}\otimes\\{\sigma\\}$ denotes
the sum of all possible decompositions of $c$ into a pairs,

  2. (2)

the map

$\displaystyle s^{-1}\overline{\mathcal{C}}$ $\displaystyle\longrightarrow
s^{-1}\overline{\mathcal{C}}\hookrightarrow\mathbb{T}(s^{-1}\overline{\mathcal{C}})$
$\displaystyle s^{-1}c$ $\displaystyle\mapsto-s^{-1}d(c)$

  3. (3)

the map

$\displaystyle s^{-1}\overline{\mathcal{C}}$
$\displaystyle\longrightarrow\mathcal{I}\hookrightarrow\mathbb{T}(s^{-1}\overline{\mathcal{C}})$
$\displaystyle s^{-1}c$ $\displaystyle\mapsto\theta(c).$

We denote by $d_{\Delta},d_{\mathcal{C}}$ and $d_{\theta}$ the respective
derivations induced by these maps. One can check
$(\mathbb{T}(s^{-1}\overline{\mathcal{C}}),d_{\Delta}+d_{\mathcal{C}}+d_{\theta})$
forms a dg operad.

The operadic bar-cobar adjunction. This two functors form an adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Operads}}$${\mathsf{curv}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}},}$$\scriptstyle{\mathrm{B}}$$\scriptstyle{\Omega}$${\dashv}$

between dg operads and conilpotent curved cooperads.

######  Remark 16.

If a dg operad $\mathcal{P}$ is augmented, then its bar construction is in
fact a conilpotent dg cooperad. Up to natural weak-equivalences, this
adjunction coincides with the bar-cobar adjunction of [LV12] between augmented
dg operads and conilpotent dg cooperads.

###  2.2. The Barratt–Eccles operad and the Hadamard tensor product

Let us review the construction of the Barratt–Eccles operad of C. Berger and
B. Fresse in [BF04].

######  Definition 24 (Barratt–Eccles dg operad).

The unital barrat-Eccles dg operad, denoted by $\mathcal{E}$, is defined as
follows. For $n\geq 2$, the arity $n$ component $\mathcal{E}(n)$ is given, in
degree $m$, by the free $\Bbbk$-module generated by the sequences of distinct
permutations

$(\sigma_{0},\sigma_{1},\ldots,\sigma_{m})\in\mathbb{S}_{n}^{m+1}$

where $\sigma_{i}\neq\sigma_{i+1}$ for $0\leq i\leq m-1$. The right
$\mathbb{S}_{n}$-action is given by

$(\sigma_{0},\sigma_{1},\ldots,\sigma_{m})^{\sigma}=(\sigma_{0}\sigma,\sigma_{1}\sigma,\ldots,\sigma_{m}\sigma).$

The differential of $\mathcal{E}(n)$ is given as follows:

$d((\sigma_{0},\sigma_{1},\ldots,\sigma_{m}))=(\sigma_{1},\ldots,\sigma_{m})-(\sigma_{0},\sigma_{2},\ldots,\sigma_{m})+\cdots+(-1)^{m}(\sigma_{0},\sigma_{1},\ldots,\sigma_{m-1})\leavevmode\nobreak\
.$

For $n=0,1$, $\mathcal{E}(0)=\mathcal{E}(1)=\Bbbk$ endowed with the trivial
action and the zero differential. We refer to [BF04, Section 1.13] for the
specific formulae of the operadic compositions.

######  Remark 17.

Notice that, for all $n\geq 0$, the dg $\Bbbk[S_{n}]$-module $\mathcal{E}(n)$
is quasi-free and concentrated in positive degrees, therefore it is also
projective. Therefore $\mathcal{E}$ is both an $\mathbb{S}$-quasi-free and an
$\mathbb{S}$-projective dg operad.

The canonical morphism of operads
$\mathcal{E}\longrightarrow\mathrm{u}\mathcal{C}\mathrm{om}$ is arity-wise a
quasi-isomorphism. To see this, it suffices to notice it admits a section and
that the degree $1$ endomorphism of $\mathcal{E}$

$h:(\sigma_{0},\sigma_{1},\ldots,\sigma_{m})\mapsto(1,\sigma_{0},\sigma_{1},\ldots,\sigma_{m})$

satisfies $\partial(h)=\pi_{\mathrm{u}\mathcal{C}\mathrm{om}}$, where
$\pi_{\mathrm{u}\mathcal{C}\mathrm{om}}$ is the projection onto the image of
this section.

######  Definition 25.

We define $\mathcal{E}_{\mathrm{pl}}$ as the sub-graded $\mathbb{N}$-module of
$\mathcal{E}$ given, in arity $n\geq 2$ and degree $m$, by the free
$\Bbbk$-module generated by the sequences

$(\sigma_{0},\sigma_{1},\ldots,\sigma_{m})\in\mathbb{S}_{n}^{m+1}$

where $\sigma_{i}\neq\sigma_{i+1}$ for $0\leq i\leq m-1$ and such that
$\sigma_{0}=1$. In arities $n=0,1$, we have
$\mathcal{E}_{\mathrm{pl}}(0)=\mathcal{E}_{\mathrm{pl}}(1)=\Bbbk$.

The canonical morphism of graded $\mathbb{S}$-modules

$\mathcal{E}_{\mathrm{pl}}\otimes\mathbb{S}\longrightarrow\mathcal{E}$

is an isomorphism.

######  Definition 26 (The Hadamard tensor product).

Let $\mathcal{P}$ be a dg operad. We denote by $\mathcal{E}\otimes\mathcal{P}$
the Hadamard tensor product of $\mathcal{P}$ with $\mathcal{E}$. It is the dg
operad whose underlying dg $\mathbb{S}$-module is

$(\mathcal{E}\otimes\mathcal{P})(n)=\mathcal{E}(n)\otimes\mathcal{P}(n)$

equipped with the diagonal action of $\mathbb{S}_{n}$. The operad structure is
given by the map

$(\mathcal{E}\otimes\mathcal{P})\circ(\mathcal{E}\otimes\mathcal{P})\longrightarrow(\mathcal{E}\otimes\mathcal{E})\otimes(\mathcal{P}\circ\mathcal{P})\longrightarrow\mathcal{E}\otimes\mathcal{P}\leavevmode\nobreak\
.$

######  Proposition 8.

Let $\mathcal{P}$ be a dg operad. The dg $\mathbb{S}$-module of the Hadamard
tensor product $\mathcal{E}\otimes\mathcal{P}$ is quasi-free with generators
$\mathcal{E}_{\mathrm{pl}}\otimes U_{\mathbb{S}}(\mathcal{P})$.

###### Proof.

Follows from Proposition 1. ∎

Notice that the canonical map
$\mathcal{E}\otimes\mathcal{P}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$
is also an arity-wise quasi-isomorphims.

######  Remark 18.

We choose, as a convention, to systematically consider $\mathcal{E}$ on the
left hand side of the tensor products $\mathcal{E}\otimes\mathcal{P}$ for
consistency reasons that will become more apparent later. One could have
chosen the opposite convention, all the results also hold.

###  2.3. Another presentation of the Barratt–Eccles operad

In this subsection, we give a new presentation of the Barratt-Eccles operad
and its composition. The computations introduced here will be used in
Subsections 2.8 and 2.9.

Let us describe the composition in the Barratt-Eccles operad in terms of
shuffle permutations. Throughout this subsection and the next one, given two
permutations $\mu\in\mathbb{S}_{p},\psi\in\mathbb{S}_{q}$ and $1\leq i\leq p$,
$\mu\circ_{i}\psi$ refers to the composition of $\mu$ with $\psi$ at the
$i$-th leaf. This composition is given by the partial compositions of the
associative operad in the category of sets.

######  Definition 27 (Admissible permutations).

Let $p,q$ be two natural integers with $p\geq 1$, let $1\leq i\leq p$ and let
$n=p+q-1$. We say that a permutation $\sigma\in\mathbb{S}_{n}$ is

  1. (1)

$(p,q,i)$-bottom admissible if there exists a permutation
$\mu\in\mathbb{S}_{p}$ such that

$\sigma=\mu\circ_{\mu^{-1}(i)}\mathrm{Id}_{q}\leavevmode\nobreak\ ;$

  2. (2)

$(p,q,i)$-top admissible if there exists a permutation $\psi\in\mathbb{S}_{q}$
such that

$\sigma=\mathrm{Id}_{p}\circ_{i}\psi\leavevmode\nobreak\ ;$

  3. (3)

$(p,q,i)$-admissible if it is $(p,q,i)$-top admissible or $(p,q,i)$-bottom
admissible;

  4. (4)

$(p,q,i)$-non admissible if it is not $(p,q,i)$-admissible.

######  Lemma 2.

Given $p,q\geq 1$ and $1\leq i\leq p$, the function

$\displaystyle\mathbb{S}_{p}\times\mathbb{S}_{q}$
$\displaystyle\longrightarrow\mathbb{S}_{n}$ $\displaystyle(\mu,\psi)$
$\displaystyle\mapsto\mu\circ_{\mu^{-1}(i)}\psi$

is injective. In particular, the intersection of $(p,q,i)$-bottom admissible
permutations with $(p,q,i)$-top admissible permutations only contains the
trivial permutation.

###### Proof.

A permutation $\sigma$ in $\mathbb{S}_{n}$ can be written as a composition
$\mu\circ_{\mu^{-1}(i)}\psi$, with $\mu\in\mathbb{S}_{p}$ and
$\psi\in\mathbb{S}_{q}$ if and only if $\sigma^{-1}$ sends the segment
$\\{i,\cdots,i+q-1\\}$ to a segment. Then the permutation $\psi$ is determined
by the induced function between these two segments and $\mu$ is determined by
the function induced by collapsing these each of these segments into one
element. ∎

Notice that for every $\sigma_{1},\sigma_{2}\in\mathbb{S}_{n}$ and
$\mu_{1},\mu_{2}\in\mathbb{S}_{m}$ and $1\leq 1\leq n$ we have

$(\sigma_{2}\sigma_{1})\circ_{i}(\mu_{2}\mu_{1})=(\sigma_{2}\circ_{\sigma_{1}(i)}\mu_{2})(\sigma_{1}\circ_{i}\mu_{1})\leavevmode\nobreak\
.$

######  Definition 28.

Let $k,p\geq 1,q\geq 0$, $1\leq i\leq p$ and $n=p+q-1$ be natural integers and
let us consider two sequences of permutations
$\underline{\mu}\in\mathbb{S}_{p}^{k}$ and
$\underline{\psi}\in\mathbb{S}_{q}^{k}$. We define
$\underline{\mu}\ltimes_{i}\underline{\psi}$ as the sequence
$\underline{\sigma}=(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$ given
by

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One can notice that

$(\mu_{1},\ldots,\mu_{k-1},\mu_{k})\ltimes_{i}(\psi_{1},\ldots,\psi_{k-1},\psi_{k})=(\mu_{1}\circ_{\mu_{1}^{-1}(i)}\psi_{1})\sqcup((\mu_{2},\ldots,\mu_{k})\ltimes_{\mu_{1}^{-1}(i)}(\psi_{2},\ldots,\psi_{k}))\leavevmode\nobreak\
,$

where $\sqcup$ stands for the concatenation of sequences of permutations.

######  Lemma 3.

If $q\geq 1$, the function
$-\ltimes_{i}-:\mathbb{S}_{p}^{k}\times\mathbb{S}_{q}^{k}\longrightarrow\mathbb{S}_{n}^{k}$
is injective.

###### Proof.

This follows from a straightforward induction on $k$ using Lemma 2. ∎

######  Definition 29 (Admissible sequence).

Let $p,q,k$ be three natural integers such that $p,k\geq 1$, let $1\leq i\leq
p$ and let $n=p+q-1$. A sequence of non-trivial permutations
$\underline{\sigma}\in\mathbb{S}_{n}^{k}$ is

  1. (1)

$(p,q,i)$-admissible if there exists two sequences of permutations
$\underline{\mu}\in\mathbb{S}_{p}^{k}$ and
$\underline{\psi}\in\mathbb{S}_{q}^{k}$ such that
$\underline{\sigma}=\underline{\mu}\ltimes_{i}\underline{\psi}$, and if for
every $1\leq j\leq k$ at least (and necessarily at most) one of the two
permuations $\mu_{j},\psi_{j}$ is trivial;

  2. (2)

$(p,q,i)$-non admissible if it is not $(p,q,i)$-admissible.

######  Definition 30.

Let $\underline{\mu}\in\mathbb{S}_{p}^{a}$ and
$\underline{\psi}\in\mathbb{S}_{q}^{b}$ be two sequences of permutations with
$p,a,b\geq 1$ and let $\phi$ be a $(a,b)$-shuffle. We denote
$\underline{\mu}\circ_{i,\phi}\underline{\psi}$ the sequence of permutation in
$\mathbb{S}_{n}^{k}$ (where $k=a+b$ and $n=p+q-1$) given by

$\underline{\mu}\circ_{i,\phi}\underline{\psi}\coloneqq\phi(\underline{\mu}\sqcup\mathrm{Id}_{p}^{b})\ltimes_{i}\phi(\mathrm{Id}_{q}^{a}\sqcup\underline{\psi})\leavevmode\nobreak\
,$

where

  1. (1)

$\phi(\underline{\mu}\sqcup\mathrm{Id}_{p}^{b})\in\mathbb{S}_{p}^{k}$ is the
sequence of permutations $(\mu^{\prime}_{1},\ldots,\mu^{\prime}_{k})$ such
that $\mu^{\prime}_{j}=\mu_{\phi^{-1}(j)}$ if $\phi^{-1}(j)\leq a$ and
$\mu^{\prime}_{j}=\mathrm{Id}_{p}$ otherwise;

  2. (2)

$\phi(\mathrm{Id}_{q}^{a}\sqcup\underline{\psi})\in\mathbb{S}_{q}^{k}$ is the
sequence of permutations $(\psi^{\prime}_{1},\ldots,\psi^{\prime}_{k})$ such
that $\psi^{\prime}_{j}=\psi_{\phi^{-1}(j)-a}$ if $\phi^{-1}(j)\geq a+1$ and
$\psi^{\prime}_{j}=\mathrm{Id}_{q}$ otherwise.

######  Lemma 4.

The function

${\displaystyle\coprod_{a+b=k}(\mathbb{S}_{p}-\\{\mathrm{Id}_{p}\\})^{a}\times(\mathbb{S}_{q}-\\{\mathrm{Id}_{p}\\})^{b}\times\mathrm{Sh}(a,b)}$${(\mathbb{S}_{n}-\\{\mathrm{Id}_{p}\\})^{k}}$${(\underline{\mu},\underline{\psi},\phi)}$${\underline{\mu}\circ_{i,\phi}\underline{\psi}}$

is injective, and its image is given by $(p,q,i)$-admissible sequences.

###### Proof.

This is a direct consequence of the definition of a $(p,q,i)$-admissible
sequence combined with Lemma 3. ∎

Let again $\mathcal{E}$ denote the Barratt-Eccles dg operad of [BF04].

######  Definition 31.

Let
$\underline{\sigma}\coloneqq(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$
be a sequence of permutations, we define $\rho(\underline{\sigma})$, an
element of $\mathcal{E}(n)_{k}$, as follows as follows

  1. (1)

if $k=0$, then $\rho(\underline{\sigma})\coloneqq\rho(*)=(\mathrm{Id}_{n})$,

  2. (2)

if $k\geq 1$ and $\underline{\sigma}=(\sigma_{1},\ldots,\sigma_{k})$,

$\rho(\underline{\sigma})\coloneqq(\mathrm{Id}_{n},\sigma_{k},\sigma_{k-1}\sigma_{k},\ldots,\sigma_{1}\cdots\sigma_{k})\leavevmode\nobreak\
.$

######  Remark 19.

A direct computation shows that
$\\{\rho(\underline{\sigma})\\}_{\underline{\sigma}}$ for
$\underline{\sigma}\coloneqq(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$
forms a basis of $(\mathcal{E}(n)_{\mathrm{pl}})_{k}$ for all $n,k\geq 0$ as a
graded $\mathbb{N}$-module. Thus it is a basis of $\mathcal{E}(n)_{k}$ as a
graded $\mathbb{S}$-module.

Partial compositions. Let us recall the partial compositions of the Barratt-
Eccles dg operad, constructed in [BF04, Section 1.1.3]. For two elements
$x=(\mu_{0},\ldots,\mu_{a})\in\mathcal{E}(p)_{a}$ and
$y=(\psi_{0},\ldots,\psi_{b})\in\mathcal{E}(p)_{b}$, given $1\leq i\leq a$ and
an $(a,b)$-shuffle $\phi$, let us consider

$x\circ_{i,\phi}^{\mathcal{E}}y\in\mathcal{E}(p+q-1)_{a+b}\leavevmode\nobreak\
.$

It is the sequence of permutations $(\sigma_{0},\ldots,\sigma_{a+b})$ defined
by

$\sigma_{j}=\mu_{j_{d}}\circ_{i}\psi_{j_{u}}\leavevmode\nobreak\ ,$

where

$\displaystyle j_{d}$ $\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\
\phi(k)\leq j\\}\leavevmode\nobreak\ ,$ $\displaystyle j_{u}$
$\displaystyle=\\#\\{k\in\mathbb{N}|a+1\leq k\leq a+b,\ \phi(k)\leq j\\}.$

The partial compositions in $\mathcal{E}$ are given by

$x\circ_{i}^{\mathcal{E}}y=\sum_{\phi\in\mathrm{Sh}(a,b)}(-1)^{\epsilon(\phi)}x\circ_{i,\phi}^{\mathcal{E}}y\leavevmode\nobreak\
,$

where the sum is taken over all $(a,b)$-shuffles and where $\epsilon(\phi)$ is
the sign of the permutation $\phi$.

######  Notation.

For every $(\sigma_{1},\ldots,\sigma_{n})\in\mathcal{E}(k)_{n+1}$ we denote

$\mathrm{inv}((\sigma_{1},\ldots,\sigma_{n}))=(\sigma_{n},\ldots,\sigma_{1})\in\mathcal{E}(k)_{n+1}.$

######  Lemma 5.

For every
$\underline{\mu}\in\mathbb{S}_{p}^{a},\underline{\psi}\in\mathbb{S}_{q}^{b}$
and every $(a,b)$-shuffle $\phi$, we have

$(\mathrm{inv}\rho(\underline{\mu}))\circ_{\underline{\mu}^{-1}(i),\phi}^{\mathcal{E}}(\mathrm{inv}\rho(\underline{\psi}))=\mathrm{inv}\rho(\underline{\mu}\circ_{i,\phi}\underline{\psi})\leavevmode\nobreak\
,$

where $\underline{\mu}^{-1}(i)=\mu_{a}^{-1}\cdots\mu_{1}^{-1}(i)$.
Subsequently,

$(\mathrm{inv}\rho(\underline{\mu}))\circ_{\underline{\mu}^{-1}(i)}(\mathrm{inv}\rho(\underline{\psi}))=\sum_{\phi\in\mathrm{Sh}(a,b)}(-1)^{\epsilon({\phi})}\mathrm{inv}\rho(\underline{\mu}\circ_{i,\phi}\underline{\psi})\leavevmode\nobreak\
,$

where the sum is taken over the $(a,b)$-shuffles and $\epsilon({\phi})$ is the
sign of the shuffle ${\phi}$.

###### Proof.

For $1\leq j\leq a+b$ we denote $\mu^{\prime}_{j}=\mu_{\phi^{-1}(j)}$ if
$\phi^{-1}(j)\leq a$ and $\mu^{\prime}_{j}=\mathrm{Id}_{p}$ otherwise;
similarly $\psi^{\prime}_{j}=\psi_{\phi^{-1}(j)-a}$ if $\phi^{-1}(j)\geq a+1$
and $\mu^{\prime}_{j}=\mathrm{Id}_{q}$ otherwise. One can check that both

$(\mathrm{inv}\rho(\underline{\mu}))\circ_{\underline{\mu}^{-1}(i),\phi}^{\mathcal{E}}(\mathrm{inv}\rho(\underline{\psi}))\quad\text{and}\quad\mathrm{inv}\rho(\underline{\mu}\circ_{i,\phi}\underline{\psi})$

are equal to

$(\mathrm{Id}_{a+b},\mu^{\prime}_{a+b}\circ_{\underline{\mu}^{-1}(i)}\psi^{\prime}_{a+b},\cdots,(\mu^{\prime}_{1}\cdots\mu^{\prime}_{a+b})\circ_{\underline{\mu}^{-1}(i)}(\psi^{\prime}_{1}\cdots\psi^{\prime}_{a+b}))\leavevmode\nobreak\
.$

∎

######  Definition 32 (Opposite shuffle).

Let $\phi$ be a $(a,b)$-shuffle. Its opposite $(a,b)$-shuffle
$\overline{\phi}$ is given by

$\overline{\phi}(j)=\begin{cases}a+b+1-\phi(a+1-j)\text{ if }j\leq
a\leavevmode\nobreak\ ;\\\ a+b+1-\phi(2a+b1-j)\text{ if }j\geq
a+1\leavevmode\nobreak\ .\end{cases}$

In other words, the opposite $(a,b)$-shuffle $\overline{\phi}$ is given by
inverting indepedently the segments $\\{1,\ldots,a\\}$ and
$\\{a+1,\ldots,a+b\\}$, applying $\phi$ to them, and finally inverting the
whole segment $\\{1,\ldots,a+b\\}$. By inverting, we mean applying the unique
permutation that yields the maximum number of inversions. Notice that the
signatures of a $(a,b)$-shuffle and its opposite are related by the following
formula

$(-1)^{\epsilon(\overline{\phi})}=(-1)^{ab+\epsilon({\phi})}\leavevmode\nobreak\
.$

######  Lemma 6.

For every
$(\mu_{0},\ldots,\mu_{a})\in\mathcal{E}(p)_{a},(\psi_{0},\ldots,\psi_{b})\in\mathcal{E}(q)_{b}$,
$1\leq i\leq p$ and every $(a,b)$-shuffle $\phi$ we have

$\mathrm{inv}(\mu_{0},\ldots,\mu_{a})\circ_{i,\phi}^{\mathcal{E}}\mathrm{inv}(\psi_{0},\ldots,\psi_{b})=\mathrm{inv}((\mu_{0},\ldots,\mu_{a})\circ_{i,\overline{\phi}}^{\mathcal{E}}(\psi_{0},\ldots,\psi_{b}))\leavevmode\nobreak\
.$

Subsequently

$\mathrm{inv}(\mu_{0},\ldots,\mu_{a})\circ_{i}\mathrm{inv}(\psi_{0},\ldots,\psi_{b})=(-1)^{ab}\mathrm{inv}((\mu_{0},\ldots,\mu_{a})\circ_{i}(\psi_{0},\ldots,\psi_{b}))\leavevmode\nobreak\
.$

###### Proof.

Let us denote

$(\sigma_{0},\ldots,\sigma_{a+b})=\mathrm{inv}(\mathrm{inv}(\mu_{0},\ldots,\mu_{a})\circ_{i,\phi}^{\mathcal{E}}\mathrm{inv}(\psi_{0},\ldots,\psi_{b}))\leavevmode\nobreak\
.$

By definition, one has

$\sigma_{a+b-j}=\mu_{a-j_{d}}\circ_{i}\psi_{b-j_{u}}$

for every $0\leq j\leq a+b$, where as above

$\displaystyle j_{d}$ $\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\
\phi(k)\leq j\\}$ $\displaystyle j_{u}$
$\displaystyle=\\#\\{k\in\mathbb{N}|a+1\leq k\leq a+b,\ \phi(k)\leq j\\}.$

By denoting $\overline{j}=a+b-j$, $\overline{j}_{d}=a-j_{d}$ and
$\overline{j}_{u}=b-j_{u}$, this rewrites as

$\sigma_{\overline{j}}=\mu_{\overline{j}_{d}}\circ_{i}\psi_{\overline{j}_{u}}$

for every $0\leq\overline{j}\leq a+b$. Combined with the fact that

$\displaystyle\overline{j}_{d}=a-j_{d}$
$\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\ \overline{\phi}(k)>j\\}$
$\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\ \overline{\phi}(k)\geq
j+1\\}$ $\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\
a+b+1-\overline{\phi}(k)\leq a+b+1-(j+1)\\}$
$\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\
a+b+1-\overline{\phi}(a+1-k)\leq\overline{j}\\}$
$\displaystyle=\\#\\{k\in\mathbb{N}|1\leq k\leq a,\
\overline{\phi}(k)\leq\overline{j}\\}\leavevmode\nobreak\ ,$

and similarly

$\overline{j}_{u}=\\#\\{k\in\mathbb{N}|a+1\leq k\leq a+b,\
\overline{\phi}(k)\leq\overline{j}\\}\leavevmode\nobreak\ ,$

we get that

$(\sigma_{0},\ldots,\sigma_{a+b})=(\mu_{0},\ldots,\mu_{a})\circ_{i,\overline{\phi}}^{\mathcal{E}}(\psi_{0},\ldots,\psi_{b})\leavevmode\nobreak\
.$

∎

######  Lemma 7.

For every
$\underline{\mu}\in\mathbb{S}_{p}^{a},\underline{\psi}\in\mathbb{S}_{q}^{b}$,
we have

$\rho(\underline{\mu})\circ_{\underline{\mu}^{-1}(i)}\rho(\underline{\psi})=(-1)^{ab}\sum_{\phi\in\mathrm{Sh}(a,b)}{\epsilon({\phi})}\rho(\underline{\mu}\circ_{i,\phi}\underline{\psi})\leavevmode\nobreak\
,$

where $\underline{\mu}^{-1}(i)=\mu_{a}^{-1}\cdots\mu_{1}^{-1}(i)$, where the
sum is taken over the $(a,b)$-shuffles and where $\epsilon({\phi})$ is the
sign of the $(a,b)$-shuffle ${\phi}$.

###### Proof.

This is a combination of Lemma 6 and Lemma 5. ∎

Comonoid structure. As described in [BF04, Section 1.1], the Barratt-Eccles dg
operad $\mathcal{E}$ has the structure of a comonoid for the Hadamard tensor
product. This structure is given by the map

${\Delta_{\mathcal{E}}:\mathcal{E}}$${\mathcal{E}\otimes\mathcal{E}}$${(\sigma_{0},\ldots,\sigma_{n})}$${\displaystyle\sum_{i=0}^{k}(\sigma_{0},\ldots,\sigma_{i})\otimes(\sigma_{i},\ldots,\sigma_{n})\leavevmode\nobreak\
.}$

######  Lemma 8.

For every $(\sigma_{0},\ldots,\sigma_{k})\in\mathcal{E}(n)_{k}$, one has that

$\Delta_{\mathcal{E}}(\rho((\sigma_{0},\ldots,\sigma_{k})))=\sum_{i=1}^{k+1}\rho((\sigma_{i},\ldots,\sigma_{k}))\otimes\rho((\sigma_{1},\ldots,\sigma_{i-1}))^{\sigma_{i}\cdots\sigma_{k}}\leavevmode\nobreak\
,$

where $\Delta_{\mathcal{E}}$ is the structure of a comonoid of the Barratt-
Eccles dg operad $\mathcal{E}$ for the Hadamard tensor product.

###### Proof.

This follows from direct inspection. ∎

###  2.4. Homotopy theory of operads

There is a semi-model category structure on the category of dg operads
constructed by B. Fresse in [Fre09].

######  Theorem 1 ([Fre09, Chapter 12]).

The category of dg operads admits a semi-model structure, determined by the
following sets of maps:

  * •

the set of fibrations is given by morphisms of dg operads which are arity-wise
degree-wise epimorphisms,

  * •

the set of weak-equivalences is given by morphisms of dg operads which are
arity-wise quasi-isomorphism,

  * •

the set of cofibrations is given by morphisms of dg operads which have the
left lifting property with respect to maps that are both fibrations and weak-
equivalences.

Let us denote by $[1]$ the category $0\longrightarrow 1$ with two objects and
a single non-trivial arrow. A functor from this category is equivalent to the
data of two objects and a morphism between them.

######  Lemma 9.

Let us consider a morphism of dg operad
$\mathcal{P}^{(0)}\longrightarrow\mathcal{P}^{(1)}$. Suppose there exists a
morphism of graded $\mathbb{N}$-modules $X^{(0)}\longrightarrow X^{(1)}$ such
that the following diagram of functors

${{[1]}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Operads}}$${\mathsf{gr}\leavevmode\nobreak\
\mathbb{N}\text{-}\mathsf{mod}}$${\mathsf{gr}\leavevmode\nobreak\
\mathsf{Operads}_{\mathrm{pl}}}$${\mathsf{gr}\leavevmode\nobreak\
\mathsf{Operads}}$$\scriptstyle{\mathcal{P}}$$\scriptstyle{X}$$\scriptstyle{\mathrm{U}}$$\scriptstyle{\mathbb{T}_{\mathrm{pl}}}$$\scriptstyle{\mathbb{S}\otimes-}$

commutes. Furthermore, suppose that

  1. (1)

the map $X^{(0)}\longrightarrow X^{(1)}$ is a degree-wise injection;

  2. (2)

the restriction of the derivation of $\mathcal{P}^{(1)}$ to the generators
$X^{(1)}$ factors as

$X^{(1)}\longrightarrow\mathcal{P}^{(0)}+X^{(1)}\hookrightarrow\mathcal{P}^{(1)}\leavevmode\nobreak\
.$

Then the map $\mathcal{P}^{(0)}\longrightarrow\mathcal{P}^{(1)}$ is a
cofibration.

###### Proof.

Let us decompose $X^{(1)}\cong X^{(0)}\oplus Y$ as graded
$\mathbb{N}$-modules. The restriction to $Y$ of the derivation of
$\mathcal{P}^{(1)}$ is a degree $-1$ map $(d_{Y},\phi):Y\longrightarrow
Y\oplus\mathcal{P}^{(0)}$. The fact that the whole derivation of
$\mathcal{P}^{(1)}$ squares to zero amounts to the facts that

  1. (1)

the map $d_{Y}$ squares to zero and thus $Y$ gets the structure of a dg module
;

  2. (2)

the map $\psi:S^{-1}\otimes Y\longrightarrow\mathcal{P}^{(0)}$, that sends
$s\otimes y$ to $\phi(y)$ is a morphism of dg modules.

Thus the following diagram

${\mathbb{T}((S^{-1}\otimes
Y)\otimes\mathbb{S})}$${\mathcal{P}^{(0)}}$${\mathbb{T}((D^{0}\otimes
Y)\otimes\mathbb{S})}$${\mathcal{P}^{(1)}.}$$\scriptstyle{\psi}$${\ulcorner}$

is a pushout square of dg operads. Since the left vertical map is cofibration,
so is the right vertical one. ∎

Given a small ordinal $\alpha$, one can view it as a category. Objects are
indexed by the ordinal $\alpha$, and there is only one non-trivial arrow
between two objects $i$ and $j$ in $\alpha$, if and only if $i<j$.
Cocontinuous functors from $\alpha$ to a category give what we call
$\alpha$-indexed ladders. We refer to Subsection Notations and conventions for
more details.

######  Proposition 9.

Let $\alpha$ be a small ordinal and let

$\mathcal{P}^{(0)}\longrightarrow\mathcal{P}^{(1)}\longrightarrow\mathcal{P}^{(2)}\longrightarrow\cdots\longrightarrow\mathcal{P}^{(i)}\longrightarrow\cdots\leavevmode\nobreak\
,$

be an $\alpha$-indexed ladder of dg operads. Suppose there exists an
$\alpha$-indexed ladder

$X^{(0)}\longrightarrow X^{(1)}\longrightarrow
X^{(2)}\longrightarrow\cdots\longrightarrow
X^{(i)}\longrightarrow\cdots\leavevmode\nobreak\ ,$

of graded $\mathbb{N}$-modules such that the following diagram of functors

${\alpha}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Operads}}$${\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{gr}\leavevmode\nobreak\
\mathsf{Operads}_{\mathrm{pl}}}$${\mathsf{gr}\leavevmode\nobreak\
\mathsf{Operads}\leavevmode\nobreak\
.}$$\scriptstyle{\mathcal{P}}$$\scriptstyle{X}$$\scriptstyle{\mathrm{U}}$$\scriptstyle{\mathbb{T}_{\mathrm{pl}}}$$\scriptstyle{\mathbb{S}\otimes-}$

commutes. Furthermore, suppose that

  1. (1)

for every $i,i+1\in\alpha$, the map $X^{(i)}\longrightarrow X^{(i+1)}$ is a
degree-wise injection;

  2. (2)

for every $i\in\alpha$, the restriction of the derivation of
$\mathcal{P}^{(i+1)}$ to the generators $X^{(i+1)}$ factors as

$X^{(i+1)}\longrightarrow\mathcal{P}^{(i)}+X^{(i+1)}\hookrightarrow\mathcal{P}^{(i+1)}.$

Then the colimit dg operad

$\mathcal{P}^{(\alpha)}\coloneqq\underset{i\in\alpha}{\mathrm{colim}}\leavevmode\nobreak\
\mathcal{P}^{(i)}$

is cofibrant.

###### Proof.

Let $v$ be the subset of $\alpha+1$ of elements $i$ so that
$\mathcal{P}^{(i)}$ is cofibrant. It satisfies the following properties:

  1. (1)

it contains the first element $0\in\alpha+1$ since
$\mathcal{P}^{(0)}\cong\mathcal{I}$;

  2. (2)

for $i,i+1\in\alpha+1$, if $i\in v$, then $i+1\in v$ by Lemma 9;

  3. (3)

if $i\in\alpha+1$ is a limit ordinal and if $v\sqcap\\{j<i\\}=\\{j<i\\}$, then
$i\in v$ since cofibrations are stable through transfinite composition.

Therefore $v=\alpha$, which implies that $P^{(\alpha)}$ is cofibrant. ∎

###  2.5. Conilpotent cooperad ladders

######  Definition 33 (Cooperad ladder).

Let $\alpha$ be a small ordinal. An $\alpha$-cooperad ladder amounts to the
data of a cocontinuous functor

$\mathcal{C}:\alpha\longrightarrow\mathsf{curv}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}}\leavevmode\nobreak\ ,$

which satisfies the following conditions:

  1. (1)

for every $i,i+1\in\alpha$, the map
$\mathcal{C}^{(i)}\longrightarrow\mathcal{C}^{(i+1)}$ is an arity-wise degree-
wise injection;

  2. (2)

the decomposition map of the underlying pdg conilpotent cooperad

${\mathcal{C}^{(i+1)}}$${\mathbb{T}\mathcal{C}^{(i+1)}}$${\overline{\mathbb{T}}\mathcal{C}^{(i+1)}}$

factors through $\overline{\mathbb{T}}\mathcal{C}^{(i)}$.

We denote

$\mathcal{C}^{(\alpha)}\coloneqq\underset{i\in\alpha}{\mathrm{colim}}\leavevmode\nobreak\
\mathcal{C}^{(i)}\leavevmode\nobreak\ ,$

the colimit of the cooperad ladder.

The associated graded of this ladder is given by

$\mathrm{gr}_{i}\leavevmode\nobreak\
\mathcal{C}\coloneqq\mathcal{C}^{(i)}/\underset{j<i}{\mathrm{colim}}\leavevmode\nobreak\
\mathcal{C}^{(j)}\leavevmode\nobreak\ .$

In fact, we have that $\mathrm{gr}_{i}\leavevmode\nobreak\
\mathcal{C}\cong\mathcal{C}^{(i)}/\mathcal{C}^{(i-1)}$ and that
$\mathrm{gr}_{0}\leavevmode\nobreak\ \mathcal{C}=\mathcal{C}^{(0)}$.
Furthermore, $\mathrm{gr}_{i}\leavevmode\nobreak\ \mathcal{C}=0$ when $i$ is a
limit ordinal.

######  Remark 20.

For all $i\in\alpha$, the associated graded object
$\mathrm{gr}_{i}\mathcal{C}$ is a dg module. Indeed, the induced coderivation
on it squares to zero because of the condition in Definition 23.

######  Example 2 (Coradical ladder).

Let $\mathcal{C}$ be a conilpotent curved cooperad. The coradical filtration
of $\mathcal{C}$

$\mathrm{F}^{\mathrm{rad}}_{0}\leavevmode\nobreak\
\mathcal{C}\rightarrowtail\mathrm{F}^{\mathrm{rad}}_{1}\leavevmode\nobreak\
\mathcal{C}\rightarrowtail\cdots\rightarrowtail\mathrm{F}^{\mathrm{rad}}_{n}\leavevmode\nobreak\
\mathcal{C}\rightarrowtail\cdots\rightarrowtail\underset{i\in\omega}{\mathrm{colim}}\leavevmode\nobreak\
\mathrm{F}^{\mathrm{rad}}_{i}\leavevmode\nobreak\ \mathcal{C}\cong\mathcal{C}$

is a cooperad ladder called the coradical ladder. Notice that it is also the
coradical development of the constant ladder.

###  2.6. Quasi-planar cooperad ladders and quasi-planar cooperads

######  Definition 34 (Quasi-planar cooperad ladder).

A quasi-planar cooperad ladder amounts to the data of a small ordinal $\alpha$
and a commutative diagram of functors

${\alpha}$${\mathsf{curv}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}}}$${\mathsf{gr}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}}_{\mathrm{pl}}}$${\mathsf{gr}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}}\leavevmode\nobreak\
,}$$\scriptstyle{\mathcal{C}}$$\scriptstyle{\mathcal{C}_{\mathrm{pl}}}$$\scriptstyle{\mathrm{U}}$$\scriptstyle{-\otimes\mathbb{S}}$

where the natural isomorphism $\varphi$ that makes the diagram commute is
specified, meaning there is a given family

$\varphi^{(i)}:\mathcal{C}_{\mathrm{pl}}^{(i)}\otimes\mathbb{S}\longrightarrow\mathcal{C}^{(i)}$

of isomorphisms of graded conilpotent cooperads for all $i\in\alpha$, called
the quasi-planar isomorphisms. This data satisfies the following conditions:

  1. (1)

the functor $\mathcal{C}$ is a cooperad ladder (thus
$\mathcal{C}_{\mathrm{pl}}$ is also cocontinuous);

  2. (2)

for every $i,i+1\in\alpha$, the restriction of the coderivation of
$\mathcal{C}^{(i+1)}$ to $\mathcal{C}_{\mathrm{pl}}^{(i+1)}\otimes 1$ factors
through

$\mathcal{C}_{\mathrm{pl}}^{(i+1)}\otimes
1\longrightarrow\mathcal{C}_{\mathrm{pl}}^{(i+1)}\otimes
1+\mathcal{C}^{(i)}\hookrightarrow\mathcal{C}^{(i+1)};$

in other words, the differential of $\mathrm{gr}_{i+1}\leavevmode\nobreak\
\mathcal{C}=\left(\mathcal{C}_{\mathrm{pl}}^{(i+1)}/\mathcal{C}_{\mathrm{pl}}^{(i)}\right)\otimes\mathbb{S}$
has the form $d_{\mathrm{pl}}\otimes\mathrm{Id}_{\mathbb{S}}$.

######  Definition 35 (Quasi-planar cooperad).

A conilpotent curved cooperad $\mathcal{C}$ is quasi-planar if it is
isomorphic to the colimit of a quasi-planar cooperad ladder.

######  Remark 21.

The main ideas for this definition where already in [GL22, Definition 11.2].

######  Remark 22.

It would be very interesting to compare the notion of a quasi-planar cooperad
with the notion of a higher cooperad introduced in [DV21].

######  Definition 36 (Quasi-planar morphism).

Let $f:\mathcal{C}\longrightarrow\mathcal{D}$ be a morphism of curved
cooperads between two quasi-planar conilpotent curved cooperads. It is quasi-
planar if it restricts to a morphism
$f_{\mathrm{pl}}:\mathcal{C}_{\mathrm{pl}}\longrightarrow\mathcal{D}_{\mathrm{pl}}$
such that the following diagram commutes

${\mathcal{C}_{\mathrm{pl}}\otimes\mathbb{S}}$${\mathcal{D}_{\mathrm{pl}}\otimes\mathbb{S}}$${\mathcal{C}^{(i)}}$${\mathcal{D}\leavevmode\nobreak\
,}$$\scriptstyle{f_{\mathrm{pl}}\otimes\mathbb{S}}$$\scriptstyle{\varphi_{\mathcal{C}}}$$\scriptstyle{\varphi_{\mathcal{D}}}$$\scriptstyle{f}$

where $\varphi_{\mathcal{C}}$ and $\varphi_{\mathcal{D}}$ are the quasi-planar
isomorphisms.

There are many examples of quasi-planar conilpotent curved cooperads. The main
example is the following.

######  Proposition 10.

Let $\mathcal{P}$ be a dg operad. The conilpotent curved cooperad
$\mathrm{B}(\mathcal{P}\otimes\mathcal{E})$ is quasi-planar.

######  Remark 23.

A first version of this proposition was proven in [GL22, Proposition 11.31].
We will give a new proof in Subsection 2.9, Proposition 15.

######  Example 3.

Let $f:\mathcal{P}\longrightarrow\mathcal{Q}$ be a morphism of dg operads. It
induces a quasi-planar morphism
$\mathrm{B}(f\otimes\mathcal{E}):\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\longrightarrow\mathrm{B}(\mathcal{E}\otimes\mathcal{Q})$
of quasi-planar conilpotent curved cooperads.

Over a characteristic zero field, dg operads admit a model category structure
where weak-equivalences are given by arity-wise quasi-isomorphisms and where
fibrations are given by arity-wise degree-wise epimorphisms. One can then
consider the transferred model category structure, along the operadic bar-
cobar adjunction, on conilpotent curved cooperads constructed in [Gri21].
Every conilpotent curved cooperad is then shown to be weak-equivalent to a
quasi-planar conilpotent curved cooperad.

######  Proposition 11 ([GL22, Lemma 11.32]).

Let $\Bbbk$ be a field of characteristic zero and let $\mathcal{C}$ be a
conilpotent curved cooperad. There exists a quasi-planar conilpotent curved
cooperad $\mathcal{C}^{\prime}$ and an acyclic cofibration
$\mathcal{C}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{C}^{\prime}$.

###### Proof.

The canonical quasi-morphism
$\mathcal{E}\otimes\Omega\mathcal{C}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\Omega\mathcal{C}$
admits a section, since both dg operads are cofibrant. Therefore it suffices
to consider the following composition

$\mathcal{C}\rightarrowtail\mathrm{B}\Omega\mathcal{C}\rightarrowtail\mathrm{B}(\mathcal{E}\otimes\Omega\mathcal{C})\leavevmode\nobreak\
,$

which is an acylic cofibration since it is the composition of two acyclic
cofibrations. ∎

######  Remark 24.

Proposition 11 will imply that when $\Bbbk$ is a field of characteristic zero,
one does recover the previously known homotopical operadic calculus of [LG19]
and [GL22].

######  Proposition 12.

Let $\mathcal{C}$ be a quasi-planar conilpotent dg cooperad. The underlying dg
$\mathbb{S}$-module of $\mathcal{C}$ is projective.

###### Proof.

Let us prove this by induction on the quasi-planar ladder that defines
$\mathcal{C}$. The dg $\mathbb{S}$-module $\mathcal{C}^{(0)}$ is free as a dg
$\mathbb{S}$-module, therefore it is projective.

Let us write $\mathcal{C}^{(i)}$ as the direct sum $\mathcal{C}^{(i-1)}\oplus
Y$ in the category of graded $\mathbb{S}$-modules. Then $\mathcal{C}^{(i)}$ is
obtained by attaching $Y$ to $\mathcal{C}^{(i-1)}$ via a pushout of the form

${(S^{-1}\otimes Y)\otimes\mathbb{S}}$${\mathcal{C}^{(i-1)}}$${(D^{0}\otimes
Y)\otimes\mathbb{S}}$${\mathcal{C}^{(i)}\leavevmode\nobreak\ ,}$${\ulcorner}$

which implies that $\mathcal{C}^{(i)}$ is again cofibrant in the category of
dg $\mathbb{S}$-modules endowed with the projective model structure. We
conclude by the fact that the colimit of cofibrant objects is again a
cofibrant. ∎

######  Remark 25.

When $\mathcal{C}$ is a quasi-planar conilpotent curved cooperad, the pre-
differential of its underlying pdg $\mathbb{S}$-module can still be
constructed by attaching cells at each step.

Quasi-planar conilpotent curved cooperads are sent to cofibrant dg operads in
the semi-model category structure of [Fre09] by the operadic cobar
construction of Subsection 2.1.

######  Proposition 13.

Let $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad. Then the dg
operad $\Omega\mathcal{C}$ is cofibrant.

###### Proof.

We can notice that the composition of the following functors

${\alpha}$${\mathsf{curv}\leavevmode\nobreak\
\mathsf{Cooperads}^{\mathsf{conil}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Operads}}$$\scriptstyle{\mathcal{C}}$$\scriptstyle{\Omega}$

satisfies the conditions of Proposition 9, which allows us to conclude. ∎

######  Corollary 1.

Let $\mathcal{C}$ be a conilpotent curved cooperad. The following assertions
are equivalent

  1. (1)

the operad $\Omega\mathcal{C}$ is cofibrant;

  2. (2)

the morphism of dg operads
$\Omega\mathcal{C}\otimes\mathcal{E}\longrightarrow\Omega\mathcal{C}$ admits a
section.

###### Proof.

If $\Omega\mathcal{C}$ is cofibrant, it is clear that the map
$\Omega\mathcal{C}\otimes\mathcal{E}\longrightarrow\Omega\mathcal{C}$ admits a
section. Conversely, let us suppose that this map admits a section. Then the
projection
$\Omega\mathrm{B}((\Omega\mathcal{C})\otimes\mathcal{E})\longrightarrow\Omega\mathcal{C}$
also admits a section given by

$\Omega\mathcal{C}\longrightarrow\Omega\mathrm{B}\Omega\mathcal{C}\longrightarrow\Omega\mathrm{B}(\mathcal{E}\otimes\Omega\mathcal{C})\leavevmode\nobreak\
.$

Therefore $\Omega\mathcal{C}$ is a retract of
$\Omega\mathrm{B}(\mathcal{E}\otimes\Omega\mathcal{C})$. Since
$\Omega\mathrm{B}(\mathcal{E}\otimes\Omega\mathcal{C})$ is cofibrant (by
Proposition 13 combined with Proposition 10), then its retract
$\Omega\mathcal{C}$ is also cofibrant. ∎

######  Remark 26.

In fact, for any $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad,
the dg operad $\Omega\mathcal{C}$ is not only $\mathcal{E}$-split, but there
exists explicit canonical map
$\Omega\mathcal{C}\longrightarrow\mathcal{E}\otimes\Omega\mathcal{C}$ which
endows $\Omega\mathcal{C}$ with a $\mathcal{E}$-comodule structure. This is
done in Subsection 2.8.

###  2.7. The cofibrant resolution

Finally, for any dg operad $\mathcal{P}$, there is always a cofibrant
replacement of the form $\Omega\mathcal{C}$, where $\mathcal{C}$ is a quasi-
planar conilpotent curved cooperad. This coincides a previous result for
augmented dg operads, see [DV21, Theorem 2, Proposition 17].

######  Proposition 14.

For every operad $\mathcal{P}$,
$\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ is cofibrant and the
canonical morphism

$\psi:\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$

is an arity-wise quasi-isomorphism.

###### Proof.

The fact that $\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ is cofibrant
follows from the fact that $\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ is
quasi-planar and that $\Omega$ sends quasi-planar conilpotent curved cooperads
to cofibrant dg operads.

Let us us show that the map
$\psi:\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\longrightarrow\mathcal{P}$
is an arity-wise quasi-isomorphism. It suffices to show that the counit
morphism
$\epsilon:\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\longrightarrow\mathcal{E}\otimes\mathcal{P}$
is an arity-wise quasi-isomorphism. It admits a canonical section $s$ in the
category of dg $\mathbb{S}$-modules, given by the following composition:

$\mathcal{E}\otimes\mathcal{P}\cong
s^{-1}s(\mathcal{E}\otimes\mathcal{P})\longrightarrow
s^{-1}\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\longrightarrow\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\leavevmode\nobreak\
.$

Let us denote $\pi_{\mathcal{P}}=s\circ\epsilon$. Since $\epsilon\circ
s=\mathrm{Id}$, the counit $\epsilon$ is an arity-wise quasi-isomorphism if
and only if $\pi_{\mathcal{P}}$ is arity-wise quasi-isomorphism. So let us
prove that $\pi_{\mathcal{P}}$ is an arity-wise quasi-isomorphism.

Let us first assume that the unit map
$\eta:\mathcal{I}\longrightarrow\mathcal{P}$ has a left inverse in the
category of graded $\mathbb{S}$-modules. Let us denote $\mathcal{Q}$ the
graded $\mathbb{S}$-module defined as follows

$\mathcal{Q}\coloneqq\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P}\leavevmode\nobreak\
.$

The map $\mathcal{I}\longrightarrow\mathcal{Q}$ also has a left inverse. We
denote $\overline{\mathcal{Q}}$ the kernel of this section. Let us denote
$\mathcal{C}$ the following planar graded conilpotent cooperad

$\mathcal{C}\coloneqq\mathbb{T}_{\mathrm{pl}}(s\mathcal{Q}\oplus
s^{2}\mathcal{I})\cong\mathbb{T}_{\mathrm{pl}}(s\overline{\mathcal{Q}}\oplus
s\mathcal{I}\oplus s^{2}\mathcal{I})\leavevmode\nobreak\ .$

We have a canonical isomorphism of graded operads

$\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\simeq\mathbb{T}_{\mathrm{pl}}(s^{-1}\overline{\mathcal{C}})\otimes\mathbb{S}\leavevmode\nobreak\
.$

Let $h$ be the degree $1$ endomorphism of
$\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ given as follows.

  1. (1)

On the trivial tree (with no node):

$\displaystyle\mathcal{I}$ $\displaystyle\longrightarrow
s^{-1}s^{2}\mathcal{I}\longrightarrow s^{-1}\overline{\mathcal{C}}$
$\displaystyle 1$ $\displaystyle\mapsto s^{-1}s^{2}1\leavevmode\nobreak\ .$

  2. (2)

On planar trees with one node, it is zero.

  3. (3)

On planar trees $t$ with two nodes or more it is

${t(\overline{\mathcal{C}})\otimes\mathbb{S}}$${s^{-1}\overline{\mathcal{C}}(n_{1})\otimes
s^{-1}\overline{\mathcal{C}}(n_{2})\otimes\cdots\otimes\mathbb{S}}$${s^{-1}F_{1}^{\mathrm{rad}}(\overline{\mathcal{C}})(n_{1})\otimes
s^{-1}\overline{\mathcal{C}}(n_{2})\otimes\cdots\otimes\mathbb{S}}$${s^{-2}F_{1}^{\mathrm{rad}}(\overline{\mathcal{C}})(n_{1})\otimes\overline{\mathcal{C}}(n_{2})\otimes\cdots\otimes\mathbb{S}}$${s^{-1}\overline{\mathcal{C}}(n_{1})\otimes\overline{\mathcal{C}}(n_{2})\otimes\cdots\otimes\mathbb{S}}$${s^{-1}\overline{\mathcal{C}}(n_{1}+n_{2}-1)\otimes\cdots\otimes\mathbb{S}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the third and the fourth map are given by

$s^{-1}x\otimes
s^{-1}y\otimes\cdots\otimes\\{\sigma\\}\mapsto-(-1)^{|x|}s^{-2}x\otimes
y\otimes\cdots\otimes\\{\sigma\\}\mapsto-(-1)^{|x|}s^{-1}x\otimes
y\otimes\cdots\otimes\\{\sigma\\}\leavevmode\nobreak\ .$

Let show that $\partial(h)+\pi_{\mathcal{P}}$ is an isomorphism. We filter
$\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$ using the tree filtration
relative to the coradical filtration of
$\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$, with the difference that we
define the fist part not to be the trivial tree but $0$. This gives

$\displaystyle F_{0}\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$
$\displaystyle=0\leavevmode\nobreak\ ,$ $\displaystyle
F_{1}\Omega\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$
$\displaystyle=\mathcal{I}\oplus s^{-1}(s(\mathcal{E}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I})\leavevmode\nobreak\ .$

This filtration is preserved by $h$, $\pi_{\mathcal{P}}$ and the derivation.
Let us show that the map on the associated graded object
$\mathrm{gr}(\partial(h)+\pi_{\mathcal{P}})$ is an isomorphism.

  1. (1)

On $\mathrm{gr}_{1}(\Omega B(\mathcal{E}\otimes\mathcal{P}))$ it is the
identity.

  2. (2)

For $n\geq 2$, we filter $\mathrm{gr}_{n}(\Omega
B(\mathcal{E}\otimes\mathcal{P}))$ by the total sum of the degrees of elements
of $\mathcal{E}$ that appear in the tensors. This filtration is again
preserved by all maps of interest. On the associated graded object, the map
$\mathrm{gr}_{*}\mathrm{gr}_{n}(\partial(h)+\pi_{\mathcal{P}})$ is the
identity.

Thus $\mathrm{gr}(\partial(h)+\pi_{\mathcal{P}})$ is an isomorphism and
therefore so is $\partial(h)+\pi_{\mathcal{P}}$. This means that
$\pi_{\mathcal{P}}$ is an arity-wise quasi-isomorphism as it is chain
homotopic to an arity-wise quasi-isomorphism.

Finally, if the unit $\mathcal{I}\longrightarrow\mathcal{P}$ does not admit a
left inverse in the category of graded $\mathbb{S}$-modules, then
$\mathcal{P}=0$ and the result holds trivially. ∎

###  2.8. The universal structure of a Barratt-Eccles comodule

In this subsection, we equip any dg operad of the form $\Omega\mathcal{C}$
with a $\mathcal{E}$-comodule structure, where $\mathcal{C}$ is again a quasi-
planar conilpotent curved cooperad and where $\mathcal{E}$ denotes the
Barratt-Eccles operad. This comonoid structure can be viewed as the positive
characteristic analogue of the $\mathrm{u}\mathcal{C}\mathrm{om}$-comonoid
structure that exists for any dg operad. By that, we mean that it produces
universal convolution (curved absolute) partition
$\mathcal{L}_{\infty}$-algebra structures on the hom-graded modules between
types of coalgebras and types of algebras. For more details, we refer to
[GRiL23a].

Recall that the restriction of the differential on $\Omega\mathcal{C}$ to the
generators $s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}$ decomposes into three
maps

  * •

$d_{s^{-1}\overline{\mathcal{C}}}:s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\longrightarrow
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\otimes\mathbb{S}\cong
s^{-1}\overline{\mathcal{C}}$, which is given by
$s^{-1}d_{\overline{\mathcal{C}}}$;

  * •

$d_{\Delta}:s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\longrightarrow\mathbb{T}_{\mathrm{pl}}^{(2)}s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}$,
induced by the (planar) partial decompositions;

  * •

$d_{\theta}:s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\longrightarrow\mathcal{I}$,
induced by the curvature of $\mathcal{C}$.

For $p\geq 1,q\geq 0,1\leq i\leq p$, let us denote

$\Delta^{p,q}_{i}:s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(p+q-1)\longrightarrow
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(p)\otimes
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(q)$

the $i$-th (planar) partial decomposition map of the cooperad
$\mathcal{C}_{\mathrm{pl}}$ into arity $p$ and arity $q$ elements. It is the
composition of the map $d_{\Delta}$ at arity $p+q-1$ with the projection onto
planar trees $t$ with two nodes

  * •

the root node with $p$ leaves;

  * •

the second node with $q$ leaves and that is plugged to the root node at its
ith leaf.

Let $M$ be a pdg $\mathbb{S}$-module that is quasi-free, that is, such that
there exists a graded $\mathbb{N}$-module $N$ and an isomorphism of graded
$\mathbb{S}$-module $M\cong N\otimes\mathbb{S}$. We are interested in how the
pre-differential interacts with the action of the symmetric groups. Note for
any given $n\geq 0$, there is an isomorphism

$M(n)\cong\bigoplus_{\sigma\in\mathbb{S}_{n}}N(n)\otimes\\{\sigma\\}\leavevmode\nobreak\
,$

of graded modules.

######  Definition 37 ($\sigma$ pre-differential).

Let $M$ be a pdg $\mathbb{S}$-module that is quasi-free, that is, such that
there exists a graded $\mathbb{N}$-module $N$ and an isomorphism of graded
$\mathbb{S}$-module $M\cong N\otimes\mathbb{S}$. Let $\sigma$ be in
$\mathbb{S}_{n}$, the $\sigma$ pre-differential, denoted by $d_{\sigma}$, is
the degree $-1$ endomorphism of $N(n)$ given by the composite

$N(n)\simeq N(n)\otimes\\{\mathrm{Id}\\}\hookrightarrow
M(n)\xrightarrow{d_{M}}M(n)\twoheadrightarrow N(n)\otimes\\{\sigma\\}\simeq
N(n)\leavevmode\nobreak\ .$

We denote by $D_{\sigma}$ the $\mathbb{S}_{n}$-equivariant degree $-1$
endomorphism of $M(n)$ whose restriction to the generators $N$ is the
composition

$N(n)\otimes\\{\mathrm{Id}\\}\hookrightarrow
M(n)\xrightarrow{d_{M}}M(n)\twoheadrightarrow
N(n)\otimes\\{\sigma\\}\hookrightarrow M(n)\leavevmode\nobreak\ .$

For any $x$ in $N(n)$, we have
$D_{\sigma}(X\otimes\\{\mathrm{Id}\\})=d_{\sigma}(x)\otimes\\{\sigma\\}$.

Similarly, for a sequence of permutations
$\underline{\sigma}\coloneqq(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$,
we set

$d_{\underline{\sigma}}\coloneqq d_{\sigma_{1}}\cdots
d_{\sigma_{k}}\quad\text{and}\quad D_{\underline{\sigma}}\coloneqq
D_{\sigma_{1}}\cdots D_{\sigma_{k}}\leavevmode\nobreak\ .$

Notice that, for any $x$ in $N(n)$, we have

$D_{\emptyset}(x\otimes\\{\mathrm{Id}\\})=x\otimes\\{\mathrm{Id}\\}\leavevmode\nobreak\
;\quad
D_{(\sigma_{1},\ldots,\sigma_{k})}(x\otimes\\{\mathrm{Id}\\})=d_{(\sigma_{1},\ldots,\sigma_{k})}(x)\otimes\\{\sigma_{1}\cdots\sigma_{k}\\}\leavevmode\nobreak\
.$

The case of interest is when $M$ is the underlying pdg $\mathbb{S}$-module of
the quasi-planar curved cooperad $\mathcal{C}$ and $N$ is the graded
$\mathbb{N}$-module $\mathcal{C}_{\mathrm{pl}}$ of planar generators.

######  Lemma 10.

Let $n\geq 0$ and let $\mathcal{C}$ be a quasi-planar conilpotent curved
cooperad. For every planar generator $x\in\mathcal{C}_{\mathrm{pl}}(n)$, there
exists a natural integer $k_{x}$ such that for every sequence of permutations
$(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$ of length $k\geq k_{x}$,
we have

$d_{(\sigma_{1},\ldots,\sigma_{k})}(x)=0\leavevmode\nobreak\ .$

The same result holds, mutatis mutandis, for $s^{-1}\overline{\mathcal{C}}$
equipped with the pre-differential
$d_{s^{-1}\overline{\mathcal{C}}}=s^{-1}d_{\overline{\mathcal{C}}}$.

###### Proof.

Since the cooperad $\mathcal{C}$ is quasi-planar, it is the colimit of a