# A new approach to formal moduli problems

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.

The main goal of this paper is to introduce a framework for formal moduli
problems, using new methods coming from operadic calculus. We construct an
adjunction between formal moduli problems over some type of algebras and their
Koszul dual algebras, in any characteristic. This adjunction is an equivalence
if and only if some algebras are equivalent to their completions. We give a
concrete homological criterion for it. It gives us a new proof of the
celebrated Lurie–Pridham theorem, as well as of many other generalizations of
it.

###### Key words and phrases:

Formal moduli problems, algebraic operads, deformation theory, homotopical
algebra.

###### 2020 Mathematics Subject Classification:

18M70, 18N40, 14D15, 14D23

###### Contents

  1. 1 Equivalent definitions of formal moduli problems of algebras over an operad
  2. 2 Divided power algebras
  3. 3 Cellular algebras and formal moduli problems
  4. 4 Tempered cooperads
  5. 5 Examples

## Introduction

The idea that every infinitesimal deformation problem is encoded by a
differential graded Lie algebra has a long history. It dates back to the works
of K. Kodaira and D. Spenser on the deformation of complex structure on a
manifold [KS58], and slowly became a guiding principle with the insights of P.
Deligne, V. Drinfeld and many others. We refer to [Toë17, CG21] for
comprehensive surveys. Nevertheless, it was only recently that J. Lurie and J.
Pridham (independently) were able to translate it into a precise mathematical
statement.

######  Theorem ([Lur11, Pri10]).

Let $\Bbbk$ be a field of characteristic zero. There is an equivalence of
$\infty$-categories between the category of formal moduli problems and the
category of dg Lie algebras over $\Bbbk$.

Here the notion of a formal moduli problem corresponds to a formalization of
the idea of an "infinitesimal deformation problem". A formal moduli problem is
a functor from $\Bbbk$-augmented dg Artinian $\Bbbk$-algebras with residue
field $\Bbbk$ to spaces, which preserves certain homotopy pullbacks.
Informally speaking, these dg Artinian $\Bbbk$-algebras correspond to
infinitesimal neighbourhoods of a $\Bbbk$-point of an algebro-geometric
object, for instance a moduli space. A formal moduli problem therefore encodes
the formal neighbourhood of this $\Bbbk$-point via a "functor of points"
perspective.

The Lurie–Pridham theorem has been since then generalized in many directions.
For example, B. Hennion generalized it to the case where the base field is a
connective dg unital commutative $\Bbbk$-algebra $A$ with finite dimensional
homology, see [Hen18]. Formal moduli problems then correspond to formal
neighbourhoods of a $\Bbbk$-point in a algebro-geometric object over $A$, and
these are encoded by dg Lie algebras in dg $A$-modules. More generally, J.
Nuiten showed in [Nui19] that formal neighbourhoods of more general geometric
objects inside a moduli space are encoded by dg Lie algebroids.

D. Calaque, R. Campos and J. Nuiten showed in [CCN19] that the fact that dg
Lie algebras are the ones appearing on the other side of the correspondence is
a particular instance of Koszul duality for operads. This idea can be traced
back to the original approach of J. Lurie in [Lur11, Sections 3 and 4]. One
can define formal moduli problems in a more general setting than just dg
Artinian commutative algebras, defining it over Artinian algebras over any
operad $\mathcal{P}$. They showed that for certain operads satisfying a
technical condition, called splendid operads, the $\infty$-category of formal
moduli problems over $\mathcal{P}$-algebras is equivalent to the
$\infty$-category of dg algebras over a Koszul dual operad.

Another direction of generalizations is given when one considers positive
characteristic base fields. In this case, the naive analogue of the
Lurie–Pridham theorem is false, as dg Lie algebras a notoriously badly behaved
in positive characteristic. Nevertheless, L. Brantner and A. Mathew showed
[BM19] that in this case the $\infty$-category of formal moduli problems is
still equivalence to the $\infty$-category of algebras over a monad (defined
$\infty$-categorically). Algebras over this monad are called partition Lie
algebras. L. Brantner, R. Campos and J. Nuiten later showed in [BCN21] that
this $\infty$-category can be obtained as the localization of a semi-model
category of divided power algebras over an operad. This in particular gives
point-set models for these partitions Lie algebras.

Let us say a few words about the proofs of these theorems. First, it is quite
noticeable that all of these proofs rely on the same underlying machinery
developed in [Lur11]. We will give a brief overview of the key elements in the
original case, but all the other cases are to some extent similar. The first
idea is to relate dg commutative algebras and dg Lie algebra using some kind
of Koszul duality. In this case, it can be done by considering the Chevalley-
Eilenberg complex (which is a dg cocommutative coalgebra) and then taking the
linear dual. This gives a functor

${\widehat{\mathrm{CE}}:\mathsf{dg}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}^{\mathsf{op}}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Com}\text{-}\mathsf{coalg}^{\mathsf{op}}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{Com}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$$\scriptstyle{\mathrm{CE}^{c}}$$\scriptstyle{(-)^{*}}$

between the $\infty$-category of dg Lie algebras and the $\infty$-category of
dg commutative algebras. Using that $\widehat{\mathrm{CE}}$ commutes with
(homotopy) limits, one can then deduce that it must have an
$\infty$-categorical left adjoint, which is often denoted by $\mathfrak{D}$.
This allows one to construct a functor

${\Psi:\mathsf{dg}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{Fun}\left(\mathsf{dg}\leavevmode\nobreak\
\mathsf{Art}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}],\mathcal{S}\right)}$${\mathfrak{g}}$${\left[A\mapsto\mathrm{Map}(\mathfrak{D}(A),\mathfrak{g})\right]\leavevmode\nobreak\
,}$

where $\mathcal{S}$ denotes the $\infty$-category of spaces. Here is where one
needs to prove various key assertions:

  1. (1)

The functor $\mathfrak{D}(-)$ preserves (homotopy) pullback, and therefore the
image of $\Psi$ lands in the $\infty$-category of formal moduli problems.

  2. (2)

The functor $\Psi$ is conservative and preserves sifted (homotopy) colimits.
Since it is accessible and preserves all limits, it admits a left adjoint
$\Phi$.

  3. (3)

The functor $\Phi$ is fully faithful, that is, the unit
$\mathrm{id}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\Psi\circ\Phi$
is a weak-equivalence.

If these assertions are true, then the adjunction $\Phi\dashv\Psi$ is an
equivalence of $\infty$-categories. Indeed, using the fact that the unit of
adjunction is a weak-equivalence and that $\Phi$ is conservative, we can
deduce that the counit is also a weak-equivalence, and therefore that the
adjunction is an equivalence.

The first point is in general non-trivial. The second point is somewhat
easier, as it can be directly checked on $\mathrm{T}\circ\Psi$, where
$\mathrm{T}$ is the tangent space of a formal moduli problem. The hardest
assertion to show is the third one. It can be shown, using hypercoverings,
that any formal moduli problem $F$ is weakly-equivalent to a the geometric
realization of a simplicial object $X_{\bullet}$, where $X_{n}$ is pro-
representable. Using the fact that $\Psi$ commutes with sifted (homotopy)
colimits, one can then reduce to the pro-representable case. Using the fact
that $\Psi$ commutes with filtered (homotopy) colimits, one can further reduce
to the representable case. So the third point can be reduced to showing that
for any dg Artinian algebra $A$,

$\eta_{A}:A\longrightarrow\widehat{\mathrm{CE}}\leavevmode\nobreak\
\mathfrak{D}(A)$

is a quasi-isomorphism. Even then, there are still quite non-trivial
computations to be done. In fact, it is rather the counit of this adjunction
which is shown to be a quasi-isomorphism for dg Lie algebras which are called
good (which we call cellular). And good dg Lie algebras are shown to be
equivalent to dg Artinian algebras and thus the above unit of adjunction is
also an equivalence.

Explaining the method. The main goal of this paper is to propose a new
approach to formal moduli problems. Before stating the general results
obtained in this paper, let us illustrate our new method in the case of the
Lurie–Pridham. The first ingredient is using the duality square constructed in
[RiL22], which contains the operadic calculus methods that will be used in the
proof. In order to study formal moduli problems, it does not matter if we
define them in terms of the $\infty$-category of dg commutative Artinian
algebras or in terms the $\infty$-category of Artinian
$\mathcal{C}_{\infty}$-algebras (algebras commutative up to homotopy). We
consider the following commutative square of Quillen adjunctions

${\left(\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}_{\infty}\text{-}\mathsf{alg}\right)^{\mathsf{op}}}$${\left(\mathsf{dg}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{coalg}^{\mathsf{conil}}\right)^{\mathsf{op}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{abs}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{B}^{\mathsf{op}}}$$\scriptstyle{(-)^{\circ}}$$\scriptstyle{(-)^{*}}$$\scriptstyle{\Omega^{\mathsf{op}}}$$\scriptstyle{\widehat{\Omega}}$$\scriptstyle{(-)^{*}}$$\scriptstyle{\widehat{\mathrm{B}}}$$\scriptstyle{(-)^{\vee}}$${\dashv}$${\dashv}$${\dashv}$${\dashv}$

Here the Bar-Cobar adjunction $\Omega\dashv\mathrm{B}$ and the complete Bar-
Cobar adjunction $\widehat{\Omega}\dashv\widehat{\mathrm{B}}$ are both
instance of the Koszul duality which generalize the Chevalley–Eilenberg
complex. Vertical adjunctions are given in each case by the linear dual
functor $(-)^{*}$ and its adjoint. These adjunctions become Quillen
adjunctions when one considers the left (resp. right) transferred model
structure on dg $\mathcal{C}_{\infty}$-algebras (resp. coalgebras) from dg
modules, and then further transfers it along the Bar-Cobar adjunctions. Notice
that on the bottom right corner, the category of dg absolute Lie algebras
appears. This is a new type of algebraic structure introduced in [GL18], where
infinite sums of operations have a well-defined image without assuming an
underlying topology. These objects are encoded as algebras over a cooperad
instead of an operad.

Now, using the vertical left linear duality adjunction, it is quite simple to
show that Artinian $\mathcal{C}_{\infty}$-algebras are equivalent to
coArtinian $\mathcal{C}_{\infty}$-coalgebras, since the linear duality
adjunction induces an equivalence on objects with finite dimensional homology.
This first step allows us to define formal moduli problems purely in terms of
coalgebras, which coincides with the philosophy of [Hin01] that states that
coalgebras should be regarded as formal derived spaces. Then, using the
complete Bar-Cobar adjunction, which is a Quillen equivalence, it is
straightforward to show that coArtinian $\mathcal{C}_{\infty}$-coalgebras
correspond to cellular dg absolute Lie algebras. This gives equivalences of
$\infty$-categories

$\mathsf{Art}\leavevmode\nobreak\
\mathcal{C}_{\infty}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\simeq\mathsf{coArt}\leavevmode\nobreak\
\mathcal{C}_{\infty}\text{-}\mathsf{coalg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\simeq\mathsf{Cell}\leavevmode\nobreak\
\mathsf{abs}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\ .$

The $\infty$-category of formal moduli problems is given by functors on
Artinian $\mathcal{C}_{\infty}$-algebras that respect certain pullbacks.
Therefore it is equivalent to the $\infty$-category of pre-sheaves on cellular
dg absolute Lie algebras which respect certain pushouts. This highlights the
important role that absolute algebras play in deformation theory.

Finally, we use the presentation of the $\infty$-category of algebras over a
monad in dg modules obtained in [Gri20] to further reduce the initial
question. In op.cit, the first author showed that the $\infty$-category of
algebras over a monad which preserves sifted (homotopy) colimits is equivalent
to the $\infty$-category of functors from cellular algebras to spaces which
preserve certain pushouts. In our case, it gives that

$\mathsf{dg}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\simeq\mathsf{Fun}^{\amalg}\left(\mathsf{Cell}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]^{\mathsf{op}},\mathcal{S}\right)\leavevmode\nobreak\ ,$

that is, the $\infty$-category of dg Lie algebras up to quasi-isomorphisms is
equivalent to the $\infty$-category of pre-sheaves on cellular Lie algebras
that preserve certain pushouts. This entails that formal moduli problems are
equivalent to dg Lie algebras if and only if the absolute envelope functor

$\mathrm{Ab}:\mathsf{dg}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}\longrightarrow\mathsf{dg}\leavevmode\nobreak\
\mathsf{abs}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}^{\mathsf{comp}}$

from dg Lie algebras to dg absolute Lie algebras induces an equivalence
between the $\infty$-categories of cellular Lie algebras and of cellular
absolute Lie algebras. This amounts to the natural inclusion

$\eta:\bigoplus_{n\geq
0}\mathcal{L}\mathrm{ie}(n)\otimes_{\mathbb{S}_{n}}V^{\otimes
n}\rightarrowtail\prod_{n\geq
0}\mathcal{L}\mathrm{ie}(n)\otimes_{\mathbb{S}_{n}}V^{\otimes
n}\leavevmode\nobreak\ ,$

of the free dg Lie algebra into the free completed dg Lie algebra being a
quasi-isomorphism for any finite dimensional dg module $V$ concentrated in
degrees $\leq-1$. It follows from a straightforward observation: since there
are only a finite number of non-zero terms in each degree, the direct sum
coincides with the product and $\eta$ is an isomorphism. This straightforward
observation is the only non-formal step in our approach to formal moduli
problem statements.

General results. Let $\Bbbk$ be a field of any characteristic. Our first
ingredient will be the generalization of the homotopical operadic calculus to
the positive characteristic case obtained in [GRiL23], and in particular of
the duality square. The second ingredient is again the presentation of
$\infty$-category of algebras over a monad in terms of cellular objects
obtained in [Gri20].

Given a dg operad $\mathcal{P}$ (augmented, $0$-reduced), our goal is to study
formal moduli problems of $\mathcal{P}$-algebras. Without any loss of
generality, we can replace $\mathcal{P}$ with a quasi-isomorphic dg operad of
the form $\Omega\mathcal{C}$, where $\mathcal{C}$ is a particularly well-
behaved dg cooperad which we call quasi-planar.

######  Theorem A (Theorem 4).

There is a canonical adjunction

${\mathsf{FMP}_{\Omega\mathcal{C}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
,}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$${\dashv}$

between the $\infty$-categories of formal moduli problems over
$\Omega\mathcal{C}$-algebras and the $\infty$-category of dg
$\mathcal{C}^{*}$-algebras. Furthermore, this adjunction is an equivalence if
and only if the functor

${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\ .}$$\scriptstyle{\mathrm{Ab}}$

is fully faithful.

Here the functor $\mathrm{Ab}$ compares cellular $\mathcal{C}^{*}$-algebras
(divided power algebras over an operad) with cellular $\mathcal{C}$-algebras
(over a cooperad), therefore it compares the non-absolute and the absolute
versions of the same algebraic structure. We show that, as $\mathcal{C}$ is
$0$-reduced, this functor is fully faithful if and only if the natural
inclusion

$\eta:\bigoplus_{n\geq 0}\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}}\rightarrowtail\prod_{n\geq
0}\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}}\leavevmode\nobreak\ .$

is a quasi-isomorphism, where $V$ is a finite dimensional dg module in degrees
$\leq 0$. Let us point out that the map $\eta$ being a quasi-isomorphisms is a
homotopy completeness condition, analogous to the one introduced by J.Harper
and K. Hess in [HH13]. The condition in Theorem 4 can be reformulated as:
cellular $\mathcal{C}^{*}$-algebras are homotopy complete. Over a
characteristic zero field, it can also be reformulated as: the Goodwillie
tower of the identity functor of dg $\mathcal{C}^{*}$-algebras converges on
cellular objects.

We give a concrete homological criterion on $\mathcal{C}$ called temperedness
that implies that $\eta$ is indeed a quasi-isomorphism.

######  Theorem B (Theorem 6).

Let $\mathcal{C}$ be a tempered quasi-planar conilpotent dg cooperad. The
canonical adjunction

${\mathsf{FMP}_{\Omega\mathcal{C}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
,}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$${\dashv}$

is an equivalence of $\infty$-categories.

Let us point out that the condition of $\mathcal{C}$ being tempered is
equivalent over a zero characteristic field to that of $\Omega\mathcal{C}$
being splendid in the sense of [CCN19]. This gives a new interpretation of the
splendid condition, which is not just a technical condition, but in fact comes
from the comparison between non-absolute and absolute cellular algebras.

We also explain how to recover the formal moduli problem associated to a
homotopy complete dg $\mathcal{C}^{*}$-algebras via an explicit functor with
values in simplicial sets, using the theory of mapping of coalgebras of
[Gri22] and the integration functor for absolute partition
$\mathcal{L}_{\infty}$-algebras constructed in [RiL23].

Finally, we give examples in Section 5. In characteristic zero, our results
recover those of Calaque–Campos–Nuiten in [CCN19], and in particular the
Lurie–Pridham theorem. Over a positive characteristic field, we obtain a
direct proof that formal moduli problems are encoded by divided power algebras
over an operad (which presents the $\infty$-category of partition Lie
algebras). We also recover the fact that formal moduli problems over
$\mathbb{E}_{k}$-algebras are encoded by $\mathbb{E}_{k}$-algebras, where
$\mathbb{E}_{k}$ is a model for the chains of the little $k$-disks operad.
This was also shown by Lurie in [Lur11, Section 3 and 4]. Another example is
given by permutative deformation problems in positive characteristic: we show
that these are encoded by partition pre-Lie algebras. These are divided power
algebras over the pre-Lie operad tensored with the linear dual of the
surjections cooperad of [BCN21]. This indicates that operadic deformation
complexes possess a richer structure than just the pre-Lie product when
working over a positive characteristic field.

### Acknowledgements

We would like to thank Lukas Brantner, Damien Calaque, Ricardo Campos, Joost
Nuiten, Geoffroy Horel and Jérôme Scherer for stimulating discussions and
helpful comments. We also thank Damien Calaque for helpful suggestions on the
draft version.

### Conventions

The base category we consider is the category of differential graded modules
over a field $\Bbbk$ of any characteristic, choosing the homological
convention. It forms a symmetric monoidal category, where one can consider
operads and cooperads. We refer to [GRiL23, Section 1] for more precise
recollections.

Let $\mathcal{C}$ be a category and let $\mathrm{W}$ be a class of arrows in
$\mathcal{C}$. We will denote $\mathcal{C}[\mathrm{W}^{-1}]$ the
$\infty$-category obtained by localizing $\mathcal{C}$ at $\mathrm{W}$. When
working at the $\infty$-categorical level, limits and colimits should be
understood as meaning homotopy limits and colimits. We will denote in the same
way a functor between model categories which preserves weak-equivalences and
the induced functor at the $\infty$-categorical level, making the context
clear.

##  1\. Equivalent definitions of formal moduli problems of algebras over an
operad

###  1.1. First definitions

Let us fix a $0$-reduced dg operad $\mathcal{P}$ for the rest of this section,
i.e: an augmented dg operad such that $\mathcal{P}(0)=0$. Since we are working
over a field $\Bbbk$ of any characteristic, we will furthermore assume that
$\mathcal{P}$ is $\mathbb{S}$-projective, that is, $\mathcal{P}(n)$ is a
cofibrant dg $\mathbb{S}_{n}$-module in the category of dg
$\mathbb{S}_{n}$-modules endowed with the projective model structure. This
hypothesis is necessary in order to make sense of the homotopy theory of dg
$\mathcal{P}$-algebras. Let us point out that all the results of this paper
extend mutatis mutandis to set-colored operads.

######  Notation.

Let $n$ be in $\mathbb{N}$. We denote by $S^{n}$ the dg module given by

$(S^{n})_{m}=\begin{cases}\Bbbk\text{ if }\text{ and }n=m;\\\ 0\text{
otherwise.}\end{cases}$

Likewise, we will denote $D^{n}$ the dg module which is given by $\Bbbk$ is
degrees $n$ and $n-1$, where the differential is given by the identity map.

We assume that $\mathcal{P}$ is augmented that is, equipped with a morphism of
dg operads $\eta:\mathcal{P}\longrightarrow\mathcal{I}$ called the
augmentation. It induces a Quillen adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{mod}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{indec}}$$\scriptstyle{\mathrm{triv}}$${\dashv}$

where the functor $\mathrm{indec}(-)$ is given by indecomposable elements of a
dg $\mathcal{P}$-algebra and the functor $\mathrm{triv}(-)$ endows any dg
module with the structure of a trivial $\mathcal{P}$-algebra. In particular,
the dg modules $S^{n},D^{n}$ can be endowed with a trivial dg
$\mathcal{P}$-algebra structure for all $n$ in $\mathbb{Z}$.

Since $\mathcal{P}$ is a $\mathbb{S}$-projective dg operad, the category of dg
$\mathcal{P}$-algebras can be endowed with a semi-model structure, where weak-
equivalences are given by quasi-isomorphisms and where fibrations are given by
degree-wise epimorphisms. See [Spi01] for more details.

This semi-model structure presents the $\infty$-category of dg
$\mathcal{P}$-algebras localized at the class of quasi-isomorphisms.

######  Definition 1 (Artinian $\mathcal{P}$-algebras).

The $\infty$-category of Artinian $\mathcal{P}$-algebras is the smallest full
subcategory of the $\infty$-category of dg $\mathcal{P}$-algebras such that

  1. (1)

it contains $S^{n}$ for any $n\geq 0$,

  2. (2)

for any Artinian $\mathcal{P}$-algebra $A$ and any morphism $A\longrightarrow
S^{n}$, where $n\geq 1$, the pullback $A\times_{S^{n}}0$ is also Artinian.

######  Remark 1.

Artinian $\mathcal{P}$-algebras are the smallest full sub-$\infty$-category
generated by $\\{S^{n}\\}_{n\geq 0}$ under pullbacks.

######  Remark 2.

Notice that, since $\mathcal{P}$ is $0$-reduced, $0$ is the initial and
terminal object in the category of dg $\mathcal{P}$-algebras.

######  Definition 2 (Formal moduli problems over $\mathcal{P}$-algebras).

The $\infty$-category of formal moduli problem over $\mathcal{P}$-algebras is
defined as the $\infty$-category of functors

$F:\mathsf{Art}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\longrightarrow\mathcal{S}$

from the $\infty$-category of Artinian $\mathcal{P}$-algebras to the
$\infty$-category of spaces satisfying the following conditions:

  1. (1)

$F(0)\simeq\\{*\\}$, where $0$ is the zero $\mathcal{P}$-algebra.

  2. (2)

$F$ preserves pullback diagrams of the form

${A^{\prime}}$${0}$${A}$${S^{n}\leavevmode\nobreak\ ,}$${\lrcorner}$

where $n\geq 1$.

We denote this $\infty$-category by $\mathsf{FMP}_{\mathcal{P}}$.

###  1.2. Changing the operad and strictly Artinian algebras

The definition of the $\infty$-category of formal moduli problems does not
depend on a specific choice of $\mathbb{S}$-projective dg operad.

######  Lemma 1.

Let
$f:\mathcal{Q}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$
be a quasi-isomorphism between two $\mathbb{S}$-projective dg operads. It
induces an equivalence of $\infty$-categories

$\mathsf{FMP}_{\mathcal{Q}}\simeq\mathsf{FMP}_{\mathcal{P}}\leavevmode\nobreak\
.$

###### Proof.

The quasi-isomorphism
$\mathcal{Q}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$
induces a Quillen equivalence of semi-model structures

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{f^{*}}$$\scriptstyle{f_{!}}$${\dashv}$

since both dg operads are $\mathbb{S}$-projective. The restriction functor
$\mathrm{Res}(-)$ clearly preserves the objects $S^{n}$ for any $n$, thus it
induces an equivalence between the $\infty$-categories of Artinian
$\mathcal{P}$-algebras and Artinian $\mathcal{Q}$-algebras. ∎

This implies in particular that one can choose the operad $\mathcal{P}$ to be
cofibrant in the category of dg operads endowed with the semi-model structure
of [Fre09, Chapter 12]. Furthermore, in order to apply the operadic calculus
developed in [GRiL23], we can choose $\mathcal{P}$ to be of the form
$\Omega\mathcal{C}$, where $\mathcal{C}$ is a quasi-planar conilpotent dg
cooperad.

######  Definition 3 (Quasi-planar dg cooperad).

Let $(\mathcal{C},d_{\mathcal{C}})$ be a conilpotent dg cooperad. It is quasi-
planar if there exists a ladder to 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 $\mathcal{C}_{\mathrm{pl}}(i)$ such
that

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

where $(-\otimes\mathbb{S})$ denotes the left adjoint to the forgetful functor
from conilpotent graded cooperads to conilpotent graded non-symmetric
cooperads.

  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

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

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

Notice that $\mathcal{C}$ will be $0$-reduced since we assumed that
$\mathcal{P}$ is $0$-reduced.

######  Remark 3.

There is a specific choice for $\mathcal{C}$, given by the Bar construction
$\mathrm{B}(\mathcal{P}\otimes\mathcal{E}^{\mathrm{nu}})$, where
$\mathcal{E}^{\mathrm{nu}}$ denotes the non-unital Barratt-Eccles operad of
[BF04]. The ladder on
$\mathrm{B}(\mathcal{P}\otimes\mathcal{E}^{\mathrm{nu}})$ is constructed by
combining the coradical filtration with a filtration induced by the
homological degree of $\mathcal{E}^{\mathrm{nu}}$. We refer to [GRiL23] for
more details.

######  Remark 4.

Over a field of characteristic zero, $\mathcal{C}$ can be assumed any
$0$-reduced conilpotent dg cooperad.

From now on, let us fix a $0$-reduced quasi-planar conilpotent dg cooperad
$\mathcal{C}$. Our goal will be to describe the $\infty$-category of formal
moduli problems over $\Omega\mathcal{C}$. Any dg operad of the form
$\Omega\mathcal{C}$ is admissible, thus the semi-model category structure on
dg $\Omega\mathcal{C}$-algebras actually forms a model category structure.
Moreover, there is a Bar-Cobar adjunction relative to the universal twisting
morphism $\iota:\mathcal{C}\longrightarrow\Omega\mathcal{C}$

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

which becomes a Quillen equivalence when one considers the transferred model
category structure along this adjunction on the category of dg
$\mathcal{C}$-coalgebras. See [GRiL23, Section 3].

######  Definition 4 (Square-zero extension).

Let $f:B\longrightarrow A$ be a morphism of dg $\Omega\mathcal{C}$-algebras.
It is a square-zero extension if $f$ fits in a homotopy pullback

${S^{n}}$${B}$${0}$${A\leavevmode\nobreak\ ,}$${\lrcorner}$$\scriptstyle{f}$

for some $n\geq 0$.

######  Definition 5 (Strictly Artinian algebra).

Let $A$ be a dg $\Omega\mathcal{C}$-algebra. It is strictly Artinian if the
terminal morphism $A\longrightarrow 0$ admits a factorization

$A=A^{(n)}\longrightarrow\cdots\longrightarrow A^{(i)}\longrightarrow
A^{(i-1)}\longrightarrow\cdots\longrightarrow A^{(0)}=0\leavevmode\nobreak\ ,$

where each morphism $A^{(i)}\longrightarrow A^{(i-1)}$ is a square-zero
extension.

######  Proposition 1.

There is an equivalence of $\infty$-categories between the $\infty$-category
of Artinian $\Omega\mathcal{C}$-algebras and the $\infty$-category of strictly
Artinian $\Omega\mathcal{C}$-algebras.

###### Proof.

It can easily shown that any strictly Artinian $\Omega\mathcal{C}$-algebra is
Artinian, see [CCN19, Example 2.3]. In order to show that any Artinian
$\Omega\mathcal{C}$-algebra is quasi-isomorphic to a strictly Artinian
$\Omega\mathcal{C}$-algebra, we follow the same steps as in [CCN19, Lemma
5.12]. It suffices to show that strictly Artinian $\Omega\mathcal{C}$-algebras
are also closed under homotopy pullbacks of the form $0\longrightarrow S^{n}$
for $n\geq 1$. Let $A$ be a strictly Artinian $\Omega\mathcal{C}$-algebra, we
consider the following (homotopy) pullback

${Y}$${D^{n}}$${\Omega_{\iota}\mathrm{B}_{\iota}A}$${S^{n}\leavevmode\nobreak\
.}$${\lrcorner}$

We want to show that the map
$Y\longrightarrow\Omega_{\iota}\mathrm{B}_{\iota}A$ is naturally quasi-
isomorphic to a square-zero extension $B\longrightarrow A$ with kernel
$S^{n}$. Using the Bar-Cobar adjunction relative to $\iota$, we can replace
the above pullback by the following pullback

${X}$${\mathcal{C}\circ(D^{n})}$${\mathrm{B}_{\iota}A}$${\mathcal{C}\circ(S^{n})\leavevmode\nobreak\
,}$${\lrcorner}$

in the category of dg $\mathcal{C}$-coalgebras. We conclude by the fact that
the map $X\longrightarrow\mathrm{B}_{\iota}A$ is naturally isomorphic to a map
$\mathrm{B}_{\iota}B\longrightarrow\mathrm{B}_{\iota}A$ where $B$ is the
desired square-zero extension. ∎

######  Remark 5.

The original statement concerning Artinian $\mathbb{E}_{k}$-algebras can be
found in [Lur17, Theorem 7.4.1.26].

######  Corollary 1.

Let $A$ be an Artinian dg $\Omega\mathcal{C}$-algebra. The homology of $A$ is
degree-wise finite dimensional and concentrated in finitely many non-negative
homological degrees.

###### Proof.

It follows from Proposition 1, since the homology of a strictly Artinian
algebra is necessarily a perfect dg module concentrated in non-positive
degrees. ∎

######  Remark 6.

The $\infty$-category of Artinian $\Omega\mathcal{C}$-algebras is a full
sub-$\infty$-category of the $\infty$-category of dg
$\Omega\mathcal{C}$-algebras with degree-wise finite dimensional homology.

###  1.3. The coalgebraic point of view of formal moduli problems

The goal of this section is to define formal moduli problems exclusively in
terms of coalgebras. When one considers formal moduli problems over
$\mathbb{E}_{\infty}$-algebras, this point of view is very close to the one
expressed in [Hin01]: indeed, it shows that $\mathbb{E}_{\infty}$-coalgebras
can be thought as derived formal stacks. From this point of view, formal
moduli problems are defined as functors that assign to some "elementary"
derived formal stacks a space, together with a "gluing condition".

The dg operad $\Omega\mathcal{C}$ is cofibrant, thus it is coadmissible. This
means that the category of dg $\Omega\mathcal{C}$-coalgebras admits a left-
transferred model structure from dg modules, where weak-equivalences are given
by quasi-isomorphisms and where cofibrations are given by degree-wise
monomorphisms.

Let $\eta:\Omega\mathcal{C}\longrightarrow\mathcal{I}$ be the augmentation
morphisms. It induces a Quillen adjunction

${\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathsf{mod}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{prim}}$$\scriptstyle{\mathrm{triv}}$${\dashv}$

where the functor $\mathrm{prim}(-)$ is given by primitive elements of a dg
$\Omega\mathcal{C}$-coalgebra and the functor $\mathrm{triv}(-)$ endows any dg
module with the structure of a trivial $\Omega\mathcal{C}$-coalgebra
structure. In particular, the dg modules $S^{n},D^{n}$ can be endowed with a
trivial $\Omega\mathcal{C}$-coalgebra structure for all $n$ in $\mathbb{Z}$.

######  Definition 6 (coArtinian $\Omega\mathcal{C}$-coalgebras).

The $\infty$-category of coArtinian $\Omega\mathcal{C}$-coalgebras is the
smallest full subcategory of the $\infty$-category of dg
$\Omega\mathcal{C}$-coalgebras such that

  1. (1)

it contains $S^{n}$ for any $n\leq 0$,

  2. (2)

for any coArtinian $\Omega\mathcal{C}$-coalgebra $C$ and any morphism
$S^{n}\longrightarrow C$, where $n\leq-1$, the pushout $C\amalg_{S^{n}}0$ is
also coArtinian.

######  Remark 7.

The $\infty$-category of coArtinian coalgebras is obtained by iteratively
gluing spheres (viewed as a formal derived space) endowed with the trivial
coalgebra structure.

######  Definition 7 (Formal moduli problems over
$\Omega\mathcal{C}$-coalgebras).

The $\infty$-category of formal moduli problem over
$\Omega\mathcal{C}$-coalgebras is defined as the $\infty$-category of functors

$F:\mathsf{coArt}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]^{\mathsf{op}}\longrightarrow\mathcal{S}$

from the $\infty$-category of coArtinian $\Omega\mathcal{C}$-coalgebras to the
$\infty$-category of spaces satisfying the following conditions:

  1. (1)

$F(0)\simeq\\{*\\}$, where $0$ is the zero $\Omega\mathcal{C}$-coalgebra.

  2. (2)

$F$ sends pushouts diagrams of the form

${S^{n}}$${0}$${C}$${C^{\prime}\leavevmode\nobreak\ ,}$${\ulcorner}$

to pullbacks in spaces, where $n\leq-1$.

We denote this $\infty$-category by $\mathsf{FMP}_{\Omega\mathcal{C}}^{c}$.

Our goal is to show that the above $\infty$-category is equivalent the
$\infty$-category of formal moduli problems over $\Omega\mathcal{C}$-algebras.

######  Lemma 2.

Let $C$ be an coArtinian $\Omega\mathcal{C}$-coalgebra. The homology of $C$ is
degree-wise finite dimensional and concentrated in finitely many non-positive
homological degrees.

###### Proof.

Any coArtinian $\Omega\mathcal{C}$-coalgebra is quasi-isomorphic to a strictly
coArtinian $\Omega\mathcal{C}$-coalgebra, that is, to a
$\Omega\mathcal{C}$-coalgebra $C$ whose initial morphism $0\longrightarrow C$
can be factored as

$0=C^{(n)}\longrightarrow\cdots\longrightarrow C^{(i)}\longrightarrow
C^{(i-1)}\longrightarrow\cdots\longrightarrow C^{(0)}=C\leavevmode\nobreak\ ,$

where each morphism $C^{(i)}\longrightarrow C^{(i-1)}$ fits into a homotopy
pushout of the form

${C^{(i)}}$${C^{(i-1)}}$${0}$${S^{n}\leavevmode\nobreak\ ,}$${\ulcorner}$

for some $n\leq 0$. This can be shown using the same arguments as in
Proposition 1, where one considers this time the complete Bar-Cobar adjunction
relative to $\iota$

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

which is again a Quillen equivalence when one considers the transferred
structure along this adjunction on dg $\mathcal{C}$-algebras. Finally, it is
straightforward to see that any strictly coArtinian
$\Omega\mathcal{C}$-coalgebra satisfies the desired property. ∎

######  Theorem 1.

There is a canonical equivalence of $\infty$-categories

$\mathsf{FMP}_{\Omega\mathcal{C}}\simeq\mathsf{FMP}_{\Omega\mathcal{C}}^{c}$

between the $\infty$-category of formal moduli problems over
$\Omega\mathcal{C}$-algebras and the $\infty$-category of formal moduli
problems over $\Omega\mathcal{C}$-coalgebras.

###### Proof.

We showed in [GRiL23, Section 5] that the Sweedler dual-linear dual adjunction
induces an equivalence of $\infty$-categories

${\left(\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{alg}^{\mathbf{f.d}}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\right)^{\mathsf{op}}}$${\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}^{\mathbf{f.d}}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
.}$$\scriptstyle{(-)^{\circ}}$$\scriptstyle{(-)^{*}}$${\dashv}$

between the $\infty$-category of dg $\Omega\mathcal{C}$-algebras with degree-
wise finite dimensional homology and the $\infty$-category of
$\Omega\mathcal{C}$-coalgebras with degree-wise finite dimensional homology.

It is straightforward to show that the above equivalence restricts to an
equivalence of $\infty$-categories between Artinian
$\Omega\mathcal{C}$-algebras and coArtinian $\Omega\mathcal{C}$-coalgebras,
using Corollary 1 and Lemma 2. The equivalence between the two types of formal
moduli problems follows by identifying the conditions imposed on these functor
categories via the aforementioned equivalence. ∎

###  1.4. Description in terms of cellular algebras over a cooperad

We identify via Koszul duality coArtinian $\Omega\mathcal{C}$-coalgebras with
cellular $\mathcal{C}$-algebras, giving another description of formal moduli
problems over $\Omega\mathcal{C}$-algebras.

Recall that the twisting morphism
$\iota:\mathcal{C}\longrightarrow\Omega\mathcal{C}$ induces a complete Bar-
Cobar adjunction

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

which is a Quillen equivalence when one considers the transferred model
category structure along this adjunction on the category of complete dg
$\mathcal{C}$-algebras. Let us make this transferred model category more
explicit:

  1. (1)

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

  2. (2)

fibrations are given by degree-wise epimorphisms,

  3. (3)

cofibrations are determined by the left-lifting property.

The $\infty$-category of $\mathcal{C}$-algebras that we consider is given by
localizing this model category at the aforementioned class of weak-
equivalences, which is a sub-class of quasi-isomorphisms.

######  Proposition 2.

The class of weak-equivalences of dg $\mathcal{C}$-algebras is strictly
included in the class of quasi-isomorphisms of dg $\mathcal{C}$-algebras.

###### Proof.

Follows from [GRiL23, Section 4.10]. ∎

######  Definition 8 (Cellular $\mathcal{C}$-algebras).

The $\infty$-category of cellular $\mathcal{C}$-algebras is the smallest full
sub-category of the $\infty$-category of $\mathcal{C}$-algebras

  1. (1)

it contains $(S^{n})^{\mathcal{C}}$ for any $n\leq 0$,

  2. (2)

for any cellular $\mathcal{C}$-algebra $\mathfrak{g}$ and any morphism
$(S^{n})^{\mathcal{C}}\longrightarrow C$, where $n\leq-1$, the pushout
$\mathfrak{g}\amalg_{(S^{n})^{\mathcal{C}}}0$ is also cellular.

######  Remark 8.

Following the terminology of [Gri20], these objects should be referred to as
$0$-cellular objects.

######  Definition 9 (Cocartesian presheaves).

A cocartesian presheaf $F$ on the $\infty$-category of cellular
$\mathcal{C}$-algebras amounts to the data of a functor

$F:\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]^{\mathsf{op}}\longrightarrow\mathcal{S},$

satisfying the following conditions:

  1. (1)

$F(0)\simeq\\{*\\}$, where $0$ is the zero $\mathcal{C}$-algebra.

  2. (2)

$F$ sends pushouts diagrams of the form

${(S^{n})^{\mathcal{C}}}$${0}$${\mathfrak{g}}$${\mathfrak{g}^{\prime}\leavevmode\nobreak\
,}$${\ulcorner}$

to pullbacks in spaces, where $n\leq-1$.

We denote $\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}])$ the full reflective sub $\infty$-category of presheaves on
cellular $\mathcal{C}$-algebras.

######  Proposition 3.

The $\infty$-category of formal moduli problem over
$\Omega\mathcal{C}$-algebras is canonically equivalent to the
$\infty$-category of cocartesian presheaves on cellular
$\mathcal{C}$-algebras.

###### Proof.

Follows directly from Theorem 1, using the fact that the complete Bar-Cobar
adjunction with respect to $\iota$ is a Quillen equivalence, simply by
noticing that the image of $S^{n}$ under the complete Cobar functor is given
by $(S^{n})^{\mathcal{C}}$. ∎

######  Remark 9.

One can not apply the methods of [Gri20] in order to conclude that formal
moduli problems over $\Omega\mathcal{C}$-algebras are always encoded by
complete dg $\mathcal{C}$-algebras. The obstruction is that the monad

$(-)^{\mathcal{C}}\coloneqq\displaystyle\prod_{n\geq
0}\mathrm{Hom}_{\mathbb{S}_{n}}(\mathcal{C}(n),(-)^{\otimes n})$

clearly does not preserve sifted homotopy colimits. Therefore one can not
conclude that the category of complete dg $\mathcal{C}$-algebras admits a
presentation in terms of cellular $\mathcal{C}$-algebras. This fact is what
forces us to consider instead the category of dg $\mathcal{C}^{*}$-algebras,
which admits such a presentation.

##  2\. Divided power algebras

Our goal in this section is to define the $\infty$-category of
$\mathcal{C}^{*}$-algebras, which, under certain hypothesis, will encode
formal moduli problems over $\Omega\mathcal{C}$-algebras. Since we are working
over a field $\Bbbk$ of any characteristic, we first give a series of
recollections on tame model structures and semi-model structures on
$\mathbb{S}$-tame operads, which mostly come from [BCN21].

###  2.1. The tame model structure

Let $R$ be a dg unital associative algebra. It this section we briefly recall
the tame or contraderived model structure on the category of dg $R$-modules.
It is given by the following classes of maps:

  1. (1)

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

  2. (2)

the class of cofibrations is given by monomorphisms whose cokernel is a
graded-projective $R$-module,

  3. (3)

and the class of weak-equivalences is given by morphisms $f:A\longrightarrow
B$ which induce quasi-isomorphisms

$f_{*}:\mathrm{hom}(P,A)\longrightarrow\mathrm{hom}(P,B)\leavevmode\nobreak\
,$

for all graded-projective dg $R$-modules $P$, that is, for all dg $R$-modules
such that $P_{k}$ is a projective $R$-module for all $k$ in $\mathbb{Z}$.

See [Pos11] where contraderived categories are introduced. The projective
model structure can be obtained as a right Bousfield localization of the tame
model structure. Both model structures actually agree on bounded below dg
$R$-modules, meaning that any quasi-isomorphism of bounded below dg
$R$-modules is a tame equivalence.

######  Example 1.

When $R$ is a field $\Bbbk$, the tame model structure on dg $\Bbbk$-modules
coincides with the projective model structure.

######  Remark 10.

When $R$ is a commutative dga which homologically bounded and coherent, the
$\infty$-category presented by this model structure can be identified with the
$\infty$-category of pro-coherent $R$-modules. See [Nui19, Section 8].

In particular, one can consider the tame model structure on dg
$\Bbbk[\mathbb{S}_{n}]$-modules for any $n\geq 0$. This in turn induces a tame
model structure on the category of dg $\mathbb{S}$-modules, where weak-
equivalences are tame equivalences arity-wise.

######  Definition 10 ($\mathbb{S}$-tame).

A dg $\mathbb{S}$-module $M$ is said to be $\mathbb{S}$-tame if $M_{n}$ it is
a cofibrant object in the tame model structure of dg $\mathbb{S}_{n}$-modules
for all $n$.

######  Example 2.

A quasi-free dg $\mathbb{S}$-module, although it might not be
$\mathbb{S}$-projective nor $\mathbb{S}$-injective, it is always
$\mathbb{S}$-tame.

######  Proposition 4.

[BCN21, Proposition 4.31] Let $\mathcal{P}$ be a dg operad whose underlying dg
$\mathbb{S}$-module is $\mathbb{S}$-tame. The category of dg
$\mathcal{P}$-algebras admits a cofibrantly generated semi-model category
structure given by

  1. (1)

weak-equivalences are given by quasi-isomorphisms,

  2. (2)

fibrations are given by degree-wise epimorphisms,

  3. (3)

cofibrations are determined by the left-lifting property.

######  Remark 11.

The $\infty$-category presented by this model structure does not depend on the
choice of the dg operad $\mathcal{P}$ in the following sense: any tame
equivalence of dg operads induces a Quillen equivalence between the
corresponding semi-model category structures.

L. Brantner, R. Campos and J. Nuiten work in a more general setting where
$\Bbbk$ is any coherent ring, therefore the semi-model structure on dg
$\mathcal{P}$-algebras is also given by tame equivalences. The same phenomenon
as in Example 1 reduces tame equivalences of dg $\mathcal{P}$-algebras to
quasi-isomorphisms when the base category is that of dg modules over a field.
We refer to [BCN21, Section 4] for more details.

###  2.2. The linear dual operad

For the rest of this section, we fix a $0$-reduced quasi-planar conilpotent dg
cooperad $\mathcal{C}$.

######  Lemma 3.

The dg $\mathbb{S}_{n}$-module $\mathcal{C}(n)^{*}$ is
$\mathbb{S}_{n}$-injective module.

###### Proof.

The dg $\mathbb{S}_{n}$-module $\mathcal{C}(n)$ is projective since
$\mathcal{C}$ is quasi-planar. The linear dual of a
$\mathbb{S}_{n}$-projective module is a $\mathbb{S}_{n}$-injective module, see
[GRiL23, Section 1] for more details. ∎

######  Lemma 4.

There is a canonical dg operad structure on the dg $\mathbb{S}$-module
$\\{\mathcal{C}(n)^{*}\\}$.

###### Proof.

The decomposition map
$\Delta_{\mathcal{C}}:\mathcal{C}\longrightarrow\mathcal{C}\circ\mathcal{C}$
induces a map $\Delta_{\mathcal{C}}^{*}:\mathcal{C}^{*}\leavevmode\nobreak\
\tilde{\circ}\leavevmode\nobreak\
\mathcal{C}^{*}\longrightarrow\mathcal{C}^{*}$ for the composition product
$\tilde{\circ}$, which is given by

$\mathcal{C}^{*}\leavevmode\nobreak\ \tilde{\circ}\leavevmode\nobreak\
\mathcal{C}^{*}\coloneqq\bigoplus_{k\geq
0}\left(\mathcal{C}^{*}(k)\otimes\left(\bigoplus_{i_{1}+\cdots+i_{k}=n}\mathsf{Ind}_{\mathbb{S}_{i_{1}}\times\cdots\times\mathbb{S}_{i_{k}}}^{\mathbb{S}_{n}}(\mathcal{C}^{*}(i_{1})\otimes\cdots\otimes\mathcal{C}^{*}\mathcal{C}^{*}(i_{k}))\right)\right)^{\mathbb{S}_{k}}\leavevmode\nobreak\
.$

Here the first direct sum coincides with the product since
$\mathcal{C}^{*}(0)\cong 0$. The map $\Delta_{\mathcal{C}}^{*}$ endows
$\mathcal{C}^{*}$ with a monoid structure with respect to the composition
product $\tilde{\circ}$. Finally, there is a canonical isomorphism
$\mathcal{C}^{*}\leavevmode\nobreak\ \tilde{\circ}\leavevmode\nobreak\
\mathcal{C}^{*}\cong\mathcal{C}^{*}\circ\mathcal{C}^{*}$, thus
$\mathcal{C}^{*}$ can be endowed with a canonical operad structure. ∎

###  2.3. Divided power algebras

The dg operad $\mathcal{C}^{*}$ induces a $1$-categorical monad on the
category of dg module via its Schur functor:

$\displaystyle\bigoplus_{n\geq
0}\mathcal{C}^{*}(n)\otimes_{\mathbb{S}_{n}}(-)^{\otimes
n}:\mathsf{dg}\leavevmode\nobreak\
\mathsf{mod}\longrightarrow\mathsf{dg}\leavevmode\nobreak\
\mathsf{mod}\leavevmode\nobreak\ .$

######  Lemma 5.

There is a canonical natural isomorphism

$\displaystyle\bigoplus_{n\geq
0}\mathcal{C}^{*}(n)\otimes_{\mathbb{S}_{n}}(-)^{\otimes
n}\cong\bigoplus_{n\geq 0}\left(\mathcal{C}^{*}(n)\otimes(-)^{\otimes
n}\right)^{\mathbb{S}_{n}}$

which is given by the norm map.

###### Proof.

The dg $\mathbb{S}_{n}$-module $\mathcal{C}^{*}(n)$ is quasi-free, thus the
norm map is an isomorphism. ∎

In general, the linear dual $\mathcal{C}^{*}$ should be a divided power dg
operad, thus dg $\mathcal{C}^{*}$-algebras should be endowed with divided
power operations in the sense of [Fre00]. This does not happen because of the
quasi-planar hypothesis on $\mathcal{C}$, which allows us identify invariants
with coinvariants all the way through.

This is no longer true when we pass to the $\infty$-categorical level. Since
$\mathcal{C}^{*}(n)$ is not $\mathbb{S}_{n}$-projective, we don’t have a
natural quasi-isomorphism of dg modules

$\displaystyle\bigoplus_{n\geq
0}\mathcal{C}^{*}(n)\otimes_{\mathbb{S}_{n}}(-)^{\otimes
n}\not\simeq\bigoplus_{n\geq 0}\left(\mathcal{C}^{*}(n)\otimes(-)^{\otimes
n}\right)_{h\mathbb{S}_{n}}\leavevmode\nobreak\ ,$

where on the right we consider homotopy coinvariants. This illustrates the
fact that although divided power operations do not appear $1$-categorically on
a dg $\mathcal{C}^{*}$-algebra, the $\infty$-category of
$\mathcal{C}^{*}$-algebras will still behaves like an $\infty$-category of
divided power algebras and not like an $\infty$-category of usual algebras.

The $\infty$-category of $\mathcal{C}^{*}$-algebras can be presented by a
semi-model structure.

######  Proposition 5.

The category of dg $\mathcal{C}^{*}$-algebras admits a cofibrantly generated
semi-model category structure given by

  1. (1)

weak-equivalences are given by quasi-isomorphisms,

  2. (2)

fibrations are given by degree-wise epimorphisms,

  3. (3)

cofibrations are determined by the left-lifting property.

###### Proof.

Follows from Proposition 4 since $\mathcal{C}^{*}$ is $\mathbb{S}$-tame. ∎

Since this semi-model category structure is transferred from the category of
dg modules, the free-forget adjunction induces a monadic adjunction at the
level of $\infty$-categories.

######  Proposition 6.

[BCN21, Section 4] The adjunction induced at the level of $\infty$-categories
by the free-forgetful Quillen adjunction

${\mathsf{dg}\leavevmode\nobreak\ \mathsf{mod}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
,}$$\scriptstyle{\mathcal{C}^{*}\circ\leavevmode\nobreak\
(-)}$$\scriptstyle{\mathrm{U}}$${\dashv}$

is monadic and the forgetful functor preserves sifted colimits.

###### Proof.

The right adjoint $\infty$-functor preserves sifted colimits by [BCN21,
Proposition 4.32] combined with [BCN21, Lemma 4.26, Remark 4.30]. Moreover, it
is clearly conservative. Thus, it is also monadic by the monadicity theorem
for $\infty$-categories [Lur17, Theorem 4.7.0.3]. ∎

##  3\. Cellular algebras and formal moduli problems

The goal of this section is to construct the $\infty$-category of cellular
$\mathcal{C}^{*}$-algebras, and show that the $\infty$-category of
$\mathcal{C}^{*}$-algebras admits a presentation in terms of cocartesian
presheaves on cellular $\mathcal{C}^{*}$-algebras. This will allow to reduce
any formal moduli problem type of statement to a comparison result between
cellular $\mathcal{C}$-algebras and cellular $\mathcal{C}^{*}$-algebras. That
is, between the absolute and the non-absolute versions of the same algebraic
structure. For the rest of this section, $\mathcal{C}$ is a $0$-reduced quasi-
planar conilpotent dg cooperad.

###  3.1. A presentation of algebras over a monad

We use the main theorem of [Gri20] in order to give a presentation of the
$\infty$-category of $\mathcal{C}^{*}$-algebras purely in terms of cocartesian
presheaves on cellular $\mathcal{C}^{*}$-algebras. Let

${\mathsf{dg}\leavevmode\nobreak\ \mathsf{mod}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathrm{T}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{T}(-)}$$\scriptstyle{\mathrm{U}}$${\dashv}$

be a monadic adjunction of $\infty$-categories.

######  Definition 11 ($m$-cellular $\mathrm{T}$-algebras).

Let $m$ be in $\mathbb{Z}\cup\\{\infty\\}$. The $\infty$-category of
$m$-cellular $\mathrm{T}$-algebras is the smallest full sub-category of
$\mathrm{T}$-algebras

  1. (1)

it contains $\mathrm{T}(S^{n})$ for any $n\leq m$,

  2. (2)

for any $m$-cellular $\mathrm{T}$-algebra $A$ and any morphism
$\mathrm{T}(S^{n})\longrightarrow A$, where $n\leq m-1$, the pushout
$A\amalg_{\mathrm{T}(S^{n})}\mathrm{T}(0)$ is also cellular.

######  Definition 12 (Cocartesian presheaves).

A cocartesian presheaf $F$ on the $\infty$-category on $m$-cellular
$\mathrm{T}$-algebras amounts to the data of a functor

$F:\mathsf{Cell}^{\leq m}\leavevmode\nobreak\
\mathrm{T}\text{-}\mathsf{alg}^{\mathsf{op}}\longrightarrow\mathcal{S},$

satisfying the following conditions:

  1. (1)

$F(\mathrm{T}(0))\simeq\\{*\\}$, where $\mathrm{T}(0)$ is the initial
$\mathrm{T}$-algebra,

  2. (2)

$F$ sends pushouts diagrams of the form

${\mathrm{T}(S^{n})}$${\mathrm{T}(0)}$${A}$${A^{\prime}\leavevmode\nobreak\
,}$${\ulcorner}$

to pullbacks in spaces, where $n\leq m-1$.

We denote $\mathsf{Psh}_{\amalg}\left(\mathsf{Cell}^{\leq
m}\leavevmode\nobreak\ \mathrm{T}\text{-}\mathsf{alg}\right)$ the full
reflective sub $\infty$-category of presheaves on cellular
$\mathrm{T}$-algebras.

######  Theorem 2 ([Gri20, Theorem 3]).

Let $m$ be in $\mathbb{Z}\cup\\{\infty\\}$ and let

${\mathsf{dg}\leavevmode\nobreak\ \mathsf{mod}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathrm{T}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{T}(-)}$$\scriptstyle{\mathrm{U}}$${\dashv}$

be a monadic adjunction of $\infty$-categories such that the forgetful functor
$\mathrm{U}$ preserves sifted colimits. There is a canonical equivalence of
$\infty$-categories

${\mathsf{Psh}_{\amalg}\left(\mathsf{Cell}^{\leq m}\leavevmode\nobreak\
\mathrm{T}\text{-}\mathsf{alg}\right)}$${\mathrm{T}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{Lan}(i_{\mathrm{T}})}$$\scriptstyle{\mathrm{Y}}$${\dashv}$

where $\mathrm{Y}$ is the Yoneda nerve $A\mapsto\mathrm{Map}(-,A)$ and
$\mathrm{Lan}(i)$ is the left Kan extension of the inclusion functor
$i_{\mathrm{T}}$ from cellular $\mathrm{T}$-algebras into all
$\mathrm{T}$-algebras.

######  Notation.

The $\infty$-category of $0$-cellular $\mathcal{C}^{*}$-algebras will be
referred simply as the $\infty$-category of cellular
$\mathcal{C}^{*}$-algebras.

######  Theorem 3.

There is a canonical equivalence of $\infty$-categories

${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}])}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{Lan}(i_{\mathcal{C}^{*}})}$$\scriptstyle{\mathrm{Y}}$${\dashv}$

between the $\infty$-category of $\mathcal{C}^{*}$-algebras and the
$\infty$-category of cocartesian presheaves on cellular
$\mathcal{C}^{*}$-algebras.

###### Proof.

Follows from Propositions 6 by applying Theorem 2 with $m=0$. ∎

Every cellular dg $\mathcal{C}^{*}$-algebra is quasi-isomorphic strictly
cellular dg $\mathcal{C}^{*}$-algebra, which is cofibrant in the semi-model
category of dg $\mathcal{C}^{*}$-algebras.

######  Definition 13 (Strictly cellular dg $\mathcal{C}^{*}$-algebra).

A strictly cellular dg $\mathcal{C}^{*}$-algebra is an quasi-free dg
$\mathcal{C}^{*}$-algebra of the form

$\left(\mathcal{C}^{*}\circ V,d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V}+d_{2}\right)\leavevmode\nobreak\ ,$

where

  1. (1)

$V$ is a dg module concentrated in degrees $\leq 0$ of total finite dimension,
endowed with a exhaustive filtration

$0=V(-1)\subseteq V(0)\subseteq V(1)\subseteq\cdots\subseteq
V\leavevmode\nobreak\ ,$

  2. (2)

the differential $d_{2}$ satisfies that

$d_{2}(V(i))\subseteq\mathcal{C}^{*}\circ V(i-1)\leavevmode\nobreak\ .$

######  Proposition 7.

Any cellular $\mathcal{C}^{*}$-algebra is quasi-isomorphic to a strictly
cellular dg $\mathcal{C}^{*}$-algebra.

###### Proof.

Follows from the definition of a cellular $\mathcal{C}$-algebra. Indeed, they
are given by a finite number of (homotopy) pushouts of the form

${\mathcal{C}^{*}\circ S^{n}}$${\mathcal{C}^{*}\circ
D^{n}}$${\mathcal{C}^{*}\circ V(i-1)}$${\mathcal{C}^{*}\circ
V(i)\leavevmode\nobreak\ ,}$${\ulcorner}$

so the generators are a finite dimensional dg module in degrees $\leq 0$, and
it clearly follows that $d_{2}(V(i))\subseteq\mathcal{C}^{*}\circ
V(i-1)\leavevmode\nobreak\ .$ ∎

###  3.2. The canonical adjunction

There is a canonical adjunction between the $\infty$-category of
$\mathcal{C}$-algebras and the $\infty$-category of
$\mathcal{C}^{*}$-algebras. This adjunction induces an adjunction between the
$\infty$-categories of formal moduli problems over
$\Omega\mathcal{C}$-algebras and of $\mathcal{C}^{*}$-algebras.

######  Lemma 6.

There is a Quillen adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{Res}}$$\scriptstyle{\mathrm{Ab}}$${\dashv}$

between the category of complete dg $\mathcal{C}$-algebras and the category of
dg $\mathcal{C}^{*}$-algebras.

###### Proof.

There is a morphism of monads

$\bigoplus_{n\geq 0}\mathcal{C}(n)^{*}\otimes_{\mathbb{S}_{n}}(-)^{\otimes
n}\rightarrowtail\prod_{n\geq
0}\mathrm{Hom}_{\mathbb{S}_{n}}(\mathcal{C}(n),(-)^{\otimes n})$

which induces the adjunction $\mathrm{Ab}\dashv\mathrm{Res}$, given by the
composition of the norm map from coinvariants to invariants with the natural
inclusion $\mathcal{C}(n)^{*}\otimes(-)^{\otimes
n}\longrightarrow\mathrm{Hom}(\mathcal{C}(n),(-)^{\otimes n})$. The functor
$\mathrm{Res}$ is a right Quillen functor since it preserves fibrations and
since any weak-equivalence of dg $\mathcal{C}$-algebras is in particular a
quasi-isomorphism by Proposition 2. ∎

######  Lemma 7.

The functor $\mathrm{Ab}$ restricts to a functor

${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\ ,}$$\scriptstyle{\mathrm{Ab}}$

between the sub-$\infty$-categories of cellular $\mathcal{C}^{*}$-algebras and
cellular $\mathcal{C}$-algebras.

###### Proof.

The adjunction of Lemma 6 induces an adjunction between the
$\infty$-categories of $\mathcal{C}$-algebras and of
$\mathcal{C}^{*}$-algebras. Clearly, for any $n\leq 0$, there is a weak-
equivalence (in fact, an isomorphism) of $\mathcal{C}$-algebras

$\mathrm{Ab}(\mathcal{C}^{*}\circ S^{n})\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ (S^{n})^{\mathcal{C}}\leavevmode\nobreak\ ,$

and moreover, it preserves homotopy pushouts, therefore it sends any cellular
$\mathcal{C}^{*}$-algebra to a cellular $\mathcal{C}$-algebra. ∎

######  Remark 12.

The functor $\mathrm{Ab}$ preserves (quasi)-free objects in general.

Since the left adjoint functor $\mathrm{Ab}$ restricts to cellular objects on
both sides, it induces the following commutative square of left adjoint
functors

${\mathsf{Psh}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}])}$${\mathsf{Psh}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}])}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\
.}$$\scriptstyle{\mathrm{Ab}_{!}}$$\scriptstyle{\mathrm{Lan}(i_{\mathcal{C}^{*}})}$$\scriptstyle{\mathrm{Lan}(i_{\mathcal{C}})}$$\scriptstyle{\mathrm{Ab}}$

The left Kan extensions $\mathrm{Lan}(i)$ of the inclusion of cellular
algebras into all algebras factor on both sides through cocartesian
presheaves. This gives the following commutative square

${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}])}$${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}])}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\
.}$$\scriptstyle{\mathrm{Ab}_{!}}$$\scriptstyle{\mathrm{Lan}(i_{\mathcal{C}^{*}})}$$\scriptstyle{\mathrm{Lan}(i_{\mathcal{C}})}$$\scriptstyle{\mathrm{Ab}}$

of left adjoint functors. By Theorem 3, the functor
$\mathrm{Lan}(i_{\mathcal{C}^{*}})$ is an equivalence. By Proposition 3, there
is also an equivalence of categories

$\widehat{\mathrm{B}}_{\iota,!}:\mathsf{FMP}^{c}\longrightarrow\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}])\leavevmode\nobreak\ .$

Therefore pre and post composing the adjunction

${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}])\leavevmode\nobreak\
}$${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]),}$$\scriptstyle{\mathrm{Ab}_{!}}$$\scriptstyle{\mathrm{Ab}^{*}}$${\dashv}$

with the aforementioned two equivalences, we get an adjunction

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
}$${\mathsf{FMP}_{\Omega\mathcal{C}}^{c},}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$${\dashv}$

relating the $\infty$-category of divided power dg $\mathcal{C}^{*}$-algebras
to the $\infty$-category of formal moduli problems over
$\Omega\mathcal{C}$-algebras.

######  Lemma 8.

The following assertions about the the functor

${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\ .}$$\scriptstyle{\mathrm{Ab}}$

are equivalent

  1. (1)

it is fully faithful,

  2. (2)

it is an equivalence of $\infty$-categories.

###### Proof.

Suppose the functor $\mathrm{Ab}$ is homotopically fully faithful, then it
induces an equivalence between the $\infty$-categories of cellular
$\mathcal{C}^{*}$-algebras and of cellular $\mathcal{C}$-algebras. Indeed,
since cellular $\mathcal{C}$-algebras are the smallest sub-$\infty$-category
of $\mathcal{C}$-algebras generated by $(S^{n})^{\mathcal{C}}$ under pushouts,
$\mathrm{Ab}$ is also essentially surjective. ∎

######  Theorem 4.

The following assertions are equivalent

  1. (1)

The adjunction

${\mathsf{FMP}_{\Omega\mathcal{C}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
,}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$${\dashv}$

is equivalence of $\infty$-categories.

  2. (2)

The functor

${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{W}^{-1}]\leavevmode\nobreak\ .}$$\scriptstyle{\mathrm{Ab}}$

is fully faithful.

  3. (3)

For every strictly cellular dg $\mathcal{C}^{\ast}$-algebra $A$, the derived
unit map of the $\mathrm{Ab}\dashv\mathrm{Res}$ adjunction is a quasi-
isomorphism.

###### Proof.

The implication $(3)\Rightarrow(2)$ follows from Proposition 7. The
implication $(2)\Rightarrow(1)$ follows from Lemma 8.

The implication $(1)\Rightarrow(2)$ is a consequence of the fact that cellular
$\mathcal{C}$-algebras and cellular $\mathcal{C}^{\ast}$-algebras are full
subcategories of, respectively, formal moduli problems over
$\Omega\mathcal{C}$-algebras and $\mathcal{C}^{*}$-algebras.

Finally, let us suppose $(2)$. Let $\mathfrak{g},\mathfrak{g}^{\prime}$ be two
cellular dg $\mathcal{C}^{\ast}$ algebra. Without any loss of generality, we
can suppose that $\mathfrak{g}$ is strictly cellular by Proposition 7. The map

$\mathrm{Map}_{\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}\left(\mathfrak{g}^{\prime},\mathfrak{g}\right)\simeq\mathrm{Map}_{\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}\left(\mathfrak{g}^{\prime},\mathrm{Res}\leavevmode\nobreak\
\mathrm{Ab}(\mathfrak{g})\right)$

is a equivalence since $\mathrm{Ab}$ is fully faithful. This means that the
derived unit map

$\mathbb{L}\eta_{\mathfrak{g}}:\mathfrak{g}\longrightarrow\mathrm{Res}\leavevmode\nobreak\
\mathrm{Ab}(\mathfrak{g})$

is local with respect to every cellular algebra $\mathfrak{g}^{\prime}$. In
turn, this implies that it is a weak-equivalence, since every dg
$\mathcal{C}^{\ast}$ algebra is a colimit of cellular algebras. ∎

###  3.3. Formal moduli problem associated to an algebra

For the rest of this subsection, we will suppose that there is an equivalence
between the $\infty$-category of formal moduli problems of
$\Omega\mathcal{C}$-algebras and the $\infty$-category of dg
$\mathcal{C}^{*}$-algebras. This will for instance be the case if
$\mathcal{C}$ is tempered, as explained in Section 4. Our goal here of this
subsection is to describe $\Psi$ for different kinds of dg
$\mathcal{C}^{*}$-algebras.

######  Lemma 9.

The functor which associates to a dg $\mathcal{C}^{*}$-algebra the formal
moduli problem it encodes is given by

${\Psi:\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{Psh}\left(\mathsf{coArt}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\right)}$${\mathfrak{g}}$${\mathrm{Map}_{\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}[\mathrm{Q.iso}^{-1}]}(\mathrm{Res}\leavevmode\nobreak\
\widehat{\Omega}_{\iota}(-),\mathfrak{g})\leavevmode\nobreak\ ,}$

at the $\infty$-categorical level.

###### Proof.

Since $\mathrm{Ab}$ is an equivalence, $\mathrm{Res}$ preserves pushouts of
cellular dg $\mathcal{C}$-algebras. So $\Psi(\mathfrak{g})$ is indeed a formal
moduli problem. The result follows from direct inspection. ∎

######  Remark 13.

We chose to describe the formal moduli problem $\Psi(\mathfrak{g})$ using the
test category of coArtinian $\Omega\mathcal{C}$-coalgebras as it is more
natural to do so. A description in terms of Artinian
$\Omega\mathcal{C}$-algebras can by obtained directly by pre-composing
$\Psi(\mathfrak{g})$ with the derived Sweedler dual functor $(-)^{\circ}$.

######  Theorem 5.

Let $\mathfrak{g}$ be a dg $\mathcal{C}^{*}$-algebra. The following assertions
are equivalent.

  1. (1)

The dg $\mathcal{C}^{*}$-algebra is in the homotopy essential image of the
functor

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\ .}$$\scriptstyle{\mathrm{Res}}$

  2. (2)

The formal moduli problem $\Psi(\mathfrak{g})$ is corepresentable by a dg
$\Omega\mathcal{C}$-coalgebra $V_{\mathfrak{g}}$, meaning it is given by

${\Psi(\mathfrak{g}):\mathsf{coArt}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathcal{S}}$${C}$${\mathrm{Map}_{\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}[\mathrm{Q.iso}^{-1}]}(C,V_{\mathfrak{g}})\leavevmode\nobreak\
.}$

###### Proof.

Follows from the commutativity of the following diagram of functors

${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}])}$${\mathsf{Psh}_{\amalg}(\mathsf{Cell}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{W}^{-1}])}$${\mathsf{FMP}^{c}_{\Omega\mathcal{C}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{comp}}\leavevmode\nobreak\
[\mathrm{W}^{-1}]}$${\mathsf{dg}\leavevmode\nobreak\
{\Omega\mathcal{C}}\text{-}\mathsf{cog}\leavevmode\nobreak\
{\left[\mathrm{Q.iso}^{-1}\right]}\leavevmode\nobreak\
.}$$\scriptstyle{\mathrm{Ab}_{!}}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota,!}}$$\scriptstyle{\mathrm{Y}}$$\scriptstyle{\mathrm{Y}}$$\scriptstyle{\mathrm{Res}}$$\scriptstyle{\mathrm{Y}}$$\scriptstyle{\widehat{\Omega}_{\iota}}$

∎

######  Definition 14 (Homotopy complete algebra).

Let $\mathfrak{g}$ be a dg $\mathcal{C}^{*}$-algebra. It is homotopy complete
if the derived unit of adjunction

$\mathbb{L}\eta_{\mathfrak{g}}:\mathfrak{g}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{Res}\leavevmode\nobreak\
\mathbb{L}\mathrm{Ab}\leavevmode\nobreak\ \mathfrak{g}$

is a quasi-isomorphism, where $\mathbb{L}\mathrm{Ab}$ denotes the derived
functor of $\mathrm{Ab}$.

######  Remark 14.

The definition of homotopy completeness is analogous to the one of Harper and
Hess in [HH13]. Over a characteristic zero field, homotopy complete algebras
are also the convergence radius of the Goodwillie tower of the identity
functor in the category of dg $\mathcal{C}^{*}$-algebras. One can then
reinterpret the condition $(2)$ in Theorem 4 as: cellular
$\mathcal{C}^{*}$-algebras are homotopy complete.

The notion of homotopy completeness is deeply linked to nilpotency. For
example, nilpotent dg Lie algebras in non-negative degrees are homotopy
complete as it was shown in [CPRNW20, Section 5]. The other way around, if a
quasi-free dg Lie algebra generated in non-negative degrees is homotopy
complete, its homology is a pro-nilpotent graded Lie algebra, see [FMFT19].

######  Corollary 2.

Let $\mathfrak{g}$ be a homotopy complete dg $\mathcal{C}^{*}$-algebra. The
formal moduli problem $\Psi(\mathfrak{g})$ associated to $\mathfrak{g}$ is
given by the functor

${\Psi(\mathfrak{g}):\mathsf{coArt}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]}$${\mathcal{S}}$${C}$${\mathrm{Map}_{\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{W}^{-1}]}\left(\widehat{\Omega}_{\iota}C,\mathbb{L}\mathrm{Ab}(\mathfrak{g})\right)\leavevmode\nobreak\
.}$

###### Proof.

We have that $\mathfrak{g}\simeq\mathrm{Res}\leavevmode\nobreak\
\mathbb{L}\mathrm{Ab}(\mathfrak{g})$ since it is homotopy complete.
Furthermore, $\mathbb{L}\mathrm{Ab}(\mathfrak{g})$ is always weakly-equivalent
to the complete Cobar construction of a dg $\Omega\mathcal{C}$-coalgebra since
the complete Bar-Cobar adjunction is a Quillen equivalence. ∎

The bright point about Theorem 5 is that it is possible to give explicit
models for the mapping spaces that describe the formal moduli problem
$\Psi(\mathfrak{g})$ when it is corepresentable by a dg
$\Omega\mathcal{C}$-coalgebra. The simplicial sets that model these mapping
spaces will have as $0$-simplices solutions to a Maurer-Cartan equation, and
the higher simplicies will be given by combinatorial formulae. In particular,
we obtain that such models exist for any formal moduli problem encoded by a
homotopy complete dg $\mathcal{C}^{*}$-algebra. This is analogous to the
various (pro)-nilpotent conditions that are present in integration theory in
the literature, see for instance [Get09].

Let us describe how to construct such models. The dg operad
$\Omega\mathcal{C}$ is cofibrant, therefore it is $\mathcal{E}$-split: there
exists a morphism of dg operad
$\Omega\mathcal{C}\longrightarrow\Omega\mathcal{C}\otimes\mathcal{E}$, where
$\mathcal{E}$ is the Barratt-Eccles dg operad. This implies that given a dg
$\Omega\mathcal{C}$-coalgebra $C$ and a dg $\mathcal{C}$-algebra $A$, there
exists a convolution curved $\mathrm{B}\mathcal{E}$-algebra structure on the
graded module of graded maps $\mathrm{hom}(C,A)$. We refer to [Gri22] for the
general theory of mapping coalgebras and convolution algebras.

A curved $\mathrm{B}\mathcal{E}$-algebra can also be called a curved absolute
partition $\mathcal{L}_{\infty}$-algebras. Using the integration functor
$\mathcal{R}$ for curved absolute partition $\mathcal{L}_{\infty}$-algebras
constructed in [RiL23], we get a rectification of the functor
$\Psi(\mathfrak{g})$. It is induced by the following functor

${\psi(\mathfrak{g}):\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{coalg}}$${\mathsf{sSets}}$${C}$${\mathcal{R}(\mathrm{hom}(C,\mathbb{L}\mathrm{Ab}(\mathfrak{g})))\leavevmode\nobreak\
,}$

when restricted to models for coArtinian dg $\Omega\mathcal{C}$-coalgebras.
Furthermore, the assignment is functorial in homotopy complete dg
$\mathcal{C}^{*}$-algebras: any quasi-isomorphisms induces a natural weak-
equivalence of functors. See also [RiL23, Section 2.5].

##  4\. Tempered cooperads

For the rest of this section let $\mathcal{C}$ be a $0$-reduced quasi-planar
conilpotent dg cooperads. We give a powerful criterion that turns the functor
relating the $\infty$-categories of cellular $\mathcal{C}$-algebras and
cellular $\mathcal{C}^{*}$-algebras an equivalence. In its essence, it holds
if the homology of $\mathcal{C}$ is concentrated in increasingly positive
homological degrees. The reason behind is that cellular $\mathcal{C}$-algebras
and cellular $\mathcal{C}^{*}$-algebras are equivalent if a graded direct sum
is equivalent to a graded direct product, which will be true if there are
finitely many non-trivial elements in each homological degree.

###  4.1. Reduction to the free case

Recall that by Theorem 4, in order to have an equivalence of
$\infty$-categories between formal moduli problems over
$\Omega\mathcal{C}$-algebras and dg $\mathcal{C}^{*}$-algebras, it is enough
to check that the (derived) unit of the $\mathrm{Ab}\dashv\mathrm{Res}$ is a
quasi-isomorphism for every strictly cellular dg $\mathcal{C}^{*}$-algebra. It
is in fact possible to reduce even further to the case of free dg
$\mathcal{C}^{*}$-algebras.

######  Proposition 8.

As the conilpotent dg cooperad $\mathcal{C}$ is quasi-planar and $0$-reduced,
the following assertions are equivalent.

  1. (1)

For every strictly cellular dg $\mathcal{C}^{\ast}$-algebra the unit of the
$\mathrm{Ab}\dashv\mathrm{Res}$ adjunction is a quasi-isomorphism.

  2. (2)

For every free dg $\mathcal{C}^{*}$-algebra generated by a finite dimensional
dg module in non-positve degrees, the unit of the
$\mathrm{Ab}\dashv\mathrm{Res}$ adjunction is a quasi-isomorphism.

###### Proof.

The first point $(1)$ clearly implies the second point $(2)$. Let
$\left(\mathcal{C}^{*}\circ V,d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V}+d_{2}\right)$ be a strictly cellular dg $\mathcal{C}^{*}$-algebra. We will
first reduce to the case where $d_{2}$ is concentrated in arity one, that is
$d_{2}(V)\subseteq\mathcal{C}^{*}(1)\otimes V$, before reducing to the case
where $d_{2}$ is trivial, that is, when the stricly cellular algebra is in
fact free.

We consider the following decreasing exhaustive filtrations

$\mathrm{F}_{m}\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V\coloneqq\bigoplus_{n\geq m}\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}}\leavevmode\nobreak\
,\quad\quad\mathrm{F}_{m}\leavevmode\nobreak\
(V)^{\mathcal{C}}\coloneqq\prod_{n\geq
m}\mathrm{Hom}\left(\mathcal{C}(n),V^{\otimes
n}\right)^{{\mathbb{S}_{n}}}\leavevmode\nobreak\ ,$

for all $m\geq 0$. Since $\mathcal{C}$ is $0$-reduced, both filtrations are
stable by the differentials. The unit map is given by the natural inclusion

$\eta:\bigoplus_{n\geq m}\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}}\rightarrowtail\prod_{n\geq
m}\mathrm{Hom}\left(\mathcal{C}(n),V^{\otimes n}\right)^{{\mathbb{S}_{n}}}$

preserves these filtrations. Notice that $\eta$ induces an isomorphism at the
first page of the induced spectral sequences. Indeed, since $V$ is a finite
dimensional dg module, there is an isomorphism

$\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}}\cong\mathrm{Hom}\left(\mathcal{C}(n),V^{\otimes
n}\right)^{{\mathbb{S}_{n}}}\leavevmode\nobreak\ .$

So in order to prove that $\eta$ is a quasi-isomorphism, it is enough to show
that both spectral sequences strongly converge. The filtration on the infinite
product is complete and Hausdorff, therefore it strongly converges.

Let us show that the spectral sequence induced by the filtration
$\mathrm{F}_{m}\leavevmode\nobreak\ \mathcal{C}^{*}\circ V$ is bounded at page
two. We suppose that the unit of adjunction is a quasi-isomorphism for
$(\mathcal{C}^{*}\circ V,d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V}+d_{2}^{\leq 1})$, where $d_{2}^{\leq 1}$ is the differential $d_{2}$
truncated at arity one. This unit is given in homology by

$\bigoplus_{n\geq 0}\mathrm{H}_{*}\left(\left(\mathcal{C}(n)^{*}\otimes
V^{\otimes n}\right)^{\mathbb{S}_{n}},d_{\leavevmode\nobreak\
\mathcal{C}^{*}\circ V}+d_{2}^{\leq 1}\right)\rightarrowtail\prod_{n\geq
0}\mathrm{H}_{*}\left(\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}},d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V}+d_{2}^{\leq 1}\right)\leavevmode\nobreak\ .$

It is an isomorphism if and only if for every $k$ in $\mathbb{Z}$, there
exists a $m_{k}$ such that

$\mathrm{H}_{k}\left(\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}},d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V}+d_{2}^{\leq 1}\right)=0\leavevmode\nobreak\ ,$

for all $n\geq m_{k}$. This implies that the spectral sequence induced by the
filtration $\mathrm{F}_{m}\leavevmode\nobreak\ \mathcal{C}^{*}\circ V$ is
bounded at page two.

We are left to show that if the unit map is a quasi-isomorphism for
$(\mathcal{C}^{*}\circ V,d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ V})$,
then it implies that is a quasi-isomorphism for $(\mathcal{C}^{*}\circ
V,d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ V}+d_{2}^{\leq 1})$. It
suffices to show that if

$\mathrm{H}_{k}\left(\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}},d_{\leavevmode\nobreak\ \mathcal{C}^{*}\circ
V}\right)=0\quad\text{then}\quad\mathrm{H}_{k}\left(\left(\mathcal{C}(n)^{*}\otimes
V^{\otimes n}\right)^{\mathbb{S}_{n}},d_{\leavevmode\nobreak\
\mathcal{C}^{*}\circ V}+d_{2}^{\leq 1}\right)=0\leavevmode\nobreak\ .$

Recall that $V$ is endowed with an exhaustive finite filtration $V(i)$ since
$\mathcal{C}^{*}\circ V$ is strictly cellular. We consider the following
increasing filtration

$\mathrm{F}_{i}\left(\mathcal{C}(n)^{*}\otimes V^{\otimes
n}\right)^{\mathbb{S}_{n}}\coloneqq\left(\mathcal{C}(n)^{*}\otimes\left(\sum_{i_{1}+\cdots+i_{l}=i}V(i_{1})\otimes\cdots\otimes
V(i_{l})\right)\right)^{\mathbb{S}_{n}}\leavevmode\nobreak\ .$

The filtration is finite, therefore the spectral sequence associated it is
strongly convergent. Using this spectral sequence, it is straightforward to
obtain the desired vanishing result for $\mathrm{H}_{k}$. ∎

###  4.2. Tempered cooperads

######  Definition 15 (Tempered cooperad).

Let $\mathcal{C}$ be a quasi-planar $0$-reduced conilpotent dg cooperad. It is
tempered if for all $\kappa\geq 0$, there exist an $n_{\kappa}$ such that, for
all $n\geq n_{\kappa}$ and $k\leq\kappa$, $\mathrm{H}_{k}(\mathcal{C}(n))=0$.

######  Remark 15.

Equivalently, for all $\kappa\geq 0$, there exist an $n_{\kappa}$ such that,
for all $n\geq n_{\kappa}$ and $k\geq\kappa$,
$\mathrm{H}_{k}(\mathcal{C}^{\ast}(n))=0$. Essentially, it means that the
homology of $\mathcal{C}$ is concentrated in increasingly bigger degrees as
the arity goes to infinity.

######  Remark 16 (Comparison with splendid operads).

Let $\Bbbk$ be a field of characteristic zero and let $\mathcal{P}$ be a
connective splendid dg operad in the sense of [CCN19]. One can always transfer
an $\infty$-operad structure onto $\mathrm{H}\mathcal{P}$, and there is a
weak-equivalence of conilpotent dg cooperads
$\mathrm{B}\mathcal{P}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{B}\mathrm{H}\mathcal{P}$.
Here $\mathrm{B}\mathrm{H}\mathcal{P}$ is a tempered conilpotent dg cooperad
concentrated in degrees $\geq 1$. This gives a tempered model for any splendid
dg operad.

On the other hand, if a conilpotent dg cooperad $\mathcal{C}$ is tempered,
then one can check that the dg operad $\Omega\mathcal{C}$ is splendid. See
[CCN19, Variant 5.18] for more details. Notice that op.cit. uses a
cohomological convention.

######  Lemma 10.

Let $\mathcal{C}$ be a tempered $0$-reduced quasi-planar conilpotent dg
cooperad and let $n\geq 0$. Suppose that $\mathrm{H}_{*}(\mathcal{C}(n))$ is
concentrated in degrees $>\kappa$. Let $V$ be a finite dimensional dg module
concentrated in degrees $\leq 0$. The homology

$\mathrm{H}_{*}\left(\mathrm{Hom}\left(\mathcal{C}(n),V^{\otimes
n}\right)^{\mathbb{S}_{n}}\right)$

is concentrated in degrees $<-\kappa$.

###### Proof.

Let us denote $\tau_{\kappa+1}(\mathcal{C}(n))$ the "intelligent" truncation
of $\mathcal{C}(n)$ at degree $\kappa+1$ given by

$\tau_{\kappa+1}(\mathcal{C}(n))_{k}=\begin{cases}\mathcal{C}(n)_{k}\text{ if
}k\geq\kappa+1\leavevmode\nobreak\ ,\\\ 0\text{ if
}k\leq\kappa\leavevmode\nobreak\ ,\\\
\mathrm{Ker}(d:\mathcal{C}(n)_{\kappa+1}\to\mathcal{C}(n)_{\kappa+1})\text{ if
}k=\kappa+1\leavevmode\nobreak\ .\end{cases}$

There is a quasi-isomorphism

$\tau_{\kappa+1}(\mathcal{C}(n))\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{C}(n)\leavevmode\nobreak\
.$

The $\mathbb{S}_{n}$-module $\mathcal{C}(n)$ is $\mathbb{S}_{n}$-projective
since $\mathcal{C}$ is quasi-planar. In general
$\tau_{\kappa+1}\mathcal{C}(n)$ is not a $\mathbb{S}_{n}$-projective any more.
Nevertheless, we can take a cofibrant resolution $P$ of it in the model
category of dg $\mathbb{S}_{n}$-module in degrees strictly above $\kappa$
endowed with the projective model structure. It is also cofibrant when seen as
an unbounded dg $\mathbb{S}_{n}$-module for the projective model strucutre.
Thus we get a composite quasi-isomorphism of $\mathbb{S}_{n}$-projective dg
$\mathbb{S}_{n}$-modules
$P\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{C}(n)$.

The model category of dg $\mathbb{S}_{n}$-module with the projective model
structure is homotopically enriched-tensored-cotensored over the dg
$\mathbb{S}_{n}$-modules with the injective model structure. By applying the
functor $\mathrm{Hom}\left(-,V^{\otimes n}\right)$ we get a quasi-isomorphism

$\mathrm{Hom}\left(\mathcal{C}(n),V^{\otimes
n}\right)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{Hom}\left(P,V^{\otimes
n}\right)$

of $\mathbb{S}_{n}$-injective dg $\mathbb{S}_{n}$-modules. This give a quasi-
isomorphism between their respective $\mathbb{S}_{n}$-invariants since these
also compute the homotopy invariants. But in degree $-\kappa$ and above, the
dg module $\mathrm{Hom}_{\mathbb{S}_{n}}\left(P,V^{\otimes n}\right)$ is zero,
therefore so is its homology. ∎

######  Theorem 6.

Let $\mathcal{C}$ be a tempered $0$-reduced quasi-planar conilpotent dg
cooperad. The canonical adjunction of $\infty$-categories

${\mathsf{FMP}_{\Omega\mathcal{C}}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{C}^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\
,}$$\scriptstyle{\Phi}$$\scriptstyle{\Psi}$${\dashv}$

between the $\infty$-categories of formal moduli problems over
$\Omega\mathcal{C}$-algebras and the $\infty$-category of dg
$\mathcal{C}^{*}$-algebras is an equivalence.

###### Proof.

As a consequence of Lemma 10, the (derived) unit map is a quasi-isomorphism
for every dg $\mathcal{C}^{\ast}$-algebra freely generated by a finite
dimensionsal dg module in non positive degrees. We conclude by Proposition 8.
∎

##  5\. Examples

###  5.1. The characteristic zero case

If $\Bbbk$ is a field of characteristic zero, the constructions performed so
far simplify: we do not need to assume that $\mathcal{C}$ is quasi-planar any
more, since every cooperad $\mathcal{C}$ is equivalent to a quasi-planar one.
In this context, the methods developed so far provide a new proof of the main
results of D. Calaque, R. Campos and J. Nuiten in [CCN19]. See Remark 16 for a
comparison between tempered cooperads and splendid operads.

######  Theorem 7 ([CCN19, Theorem 1.3]).

Let $\mathcal{P}$ be a connective splendid dg operad. There is a canonical
equivalence

$\mathsf{FMP}_{\mathcal{P}}\simeq\mathsf{dg}\leavevmode\nobreak\
(\mathrm{B}\mathcal{P})^{*}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\ .$

between the $\infty$-categories of formal moduli problems over
$\mathcal{P}$-algebras and of dg $(\mathrm{B}\mathcal{P})^{*}$-algebras when
localized at quasi-isomorphisms.

While their results provided inspiration for the ones obtained in this paper,
it should be noted that their arguments are very similar to the original
arguments of J. Lurie [Lur11]. In particular, the condition that $\mathcal{P}$
is splendid (which is equivalent to $(\mathrm{B}\mathcal{P})^{*}$ being
tempered) only appears as a technical side-condition in order to prove the
prerequisites that make the $\infty$-categorical machinery constructed in
[Lur11] run. On the other hand, from the perspective of the new proof given in
this paper, this condition is conceptually explained by the comparison
functors between cellular $\mathrm{B}\mathcal{P}$-algebras and cellular
$(\mathrm{B}\mathcal{P})^{*}$-algebras.

Lurie–Pridham Theorem. Let us consider $\mathcal{P}$ to be the dg operad
$\mathcal{C}\mathrm{om}$, which encodes non-unital commutative algebras. In
this case the conilpotent dg cooperad obtained is
$\mathrm{B}\mathcal{C}\mathrm{om}$, and its linear dual
$\Omega\mathcal{C}\mathrm{om}^{*}$ encodes shifted
$\mathcal{L}_{\infty}$-algebras.

######  Lemma 11.

The dg cooperad $\mathrm{B}\leavevmode\nobreak\ \mathcal{C}\mathrm{om}$ is
tempered.

###### Proof.

It is quasi-isomorphic to the shifted Lie cooperad
$s\mathcal{L}\mathrm{ie}^{*}$, which is tempered by direct inspection. ∎

######  Theorem 8 (Lurie-Pridham).

There is a canonical equivalence

$\mathsf{FMP}_{\mathcal{C}\mathrm{om}}\simeq\mathsf{dg}\leavevmode\nobreak\
\mathsf{Lie}\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\ .$

between the $\infty$-categories of formal moduli problems and of dg Lie
algebras.

###### Proof.

Follows from Theorem 6 and Lemma 11. ∎

The heart of this new approach to the celebrated Lurie–Pridham theorem can be
condensed in the fact that the canonical inclusion of dg Lie algebras

$\mathcal{L}\mathrm{ie}(V)\to\widehat{\mathcal{L}\mathrm{ie}}(V)$

between the free Lie algebra $\mathcal{L}\mathrm{ie}(V)$ and the completed
free Lie algebra $\widehat{\mathcal{L}\mathrm{ie}}(V)$ on a finite dimensional
dg module concentrated in degree $\leq 1$ is an isomorphism.

######  Remark 17.

The results of [CCN19] have an extra level of generality with respect to our
current work. They are valid over coherent homologically bounded dg unital
commutative $\Bbbk$-algebra $A$ (this generalization was already achieved by
B. Hennion in [Hen18] in the context of formal moduli problems of commutative
algebras). Moreover, $A$ can also be a dg category satisfying analogue
conditions, which heuristically corresponds to "adding colours" to a dg operad
in dg $A$-modules. We nevertheless expect that the methods introduced in this
paper can be generalized to a similar setting.

###  5.2. Partition Lie algebras

Let $\Bbbk$ be field of any characteristic. We consider formal moduli problems
over an $\mathbb{E}^{\mathrm{nu}}_{\infty}$ operad, that is, an operad which
is a $\mathbb{S}$-projective resolution of the operad
$\mathcal{C}\mathrm{om}$.

It was shown by L. Brantner and A. Mathew in [BM19] that the $\infty$-category
of these formal moduli problems is equivalent to the $\infty$-category of
partition Lie algebras. The $\infty$-category of partition Lie algebras was
first defined to be the $\infty$-category of algebras over a monad. The
question of whether this $\infty$-category could admit a simple presentation
in terms of an algebraic $1$-category localized at some class of equivalences
was settled by L. Brantner, R. Campos and J. Nuiten in [BCN21].

######  Theorem 9 ([BCN21, Proposition 4.40]).

Let $\mathbb{E}$ be a connected $\mathbb{E}^{\mathrm{nu}}_{\infty}$-operad.
There is a canonical equivalence

$\mathsf{dg}\leavevmode\nobreak\
\Omega(\mathbb{E}^{*})\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\simeq\mathcal{L}ie^{\pi}\text{-}\mathsf{alg}$

between the $\infty$-categories of dg $\Omega(\mathbb{E}^{*})$-algebras and
the $\infty$-category of partition Lie algebras.

Therefore, the semi-model category of dg $\Omega(\mathbb{E}^{*})$-algebras is
a model for partition Lie algebras, and a fortiori, their localized
$\infty$-category is equivalent to the $\infty$-category of formal moduli
problems of $\mathbb{E}^{\mathrm{nu}}_{\infty}$-algebras over a field $\Bbbk$
of any characteristic.

On the other hand, the methods introduced in this paper give a direct proof
that there is an equivalence of $\infty$-categories between the
$\infty$-category of formal moduli problems of
$\mathbb{E}^{\mathrm{nu}}_{\infty}$-algebras and the $\infty$-category of dg
$\Omega(\mathbb{E}^{*})$-algebras. For that, we start by choosing a specific
model, the non-unital Barratt-Eccles dg operad $\mathcal{E}^{\mathrm{nu}}$.

######  Lemma 12.

The conilpotent dg cooperad $\mathrm{B}(\mathcal{E}^{\mathrm{nu}})$ is quasi-
planar and tempered.

###### Proof.

For the fact that $\mathrm{B}(\mathcal{E}^{\mathrm{nu}})$ is quasi-planar, see
[GRiL23]. It follows from [Fre04, Theorem 6.8] that the homology of
$\mathrm{B}(\mathcal{E}^{\mathrm{nu}})$ is the underlying $\mathbb{S}$-module
of the shifted Lie cooperad $s\mathcal{L}\mathrm{ie}^{*}$. ∎

######  Theorem 10.

Let $\mathcal{E}^{\mathrm{nu}}$ be the non-unital Barratt-Eccles dg operad.
There is a canonical equivalence

$\mathsf{FMP}_{\mathcal{E}^{\mathrm{nu}}}\simeq\mathsf{dg}\leavevmode\nobreak\
\Omega((\mathcal{E}^{\mathrm{nu}})^{*})\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\ .$

between the $\infty$-categories of dg
$\Omega((\mathcal{E}^{\mathrm{nu}})^{*})$-algebras and the $\infty$-category
of formal moduli problems of $\mathcal{E}^{\mathrm{nu}}$-algebras.

###### Proof.

Follows from Theorem 6 and Lemma 12. ∎

This proof has the advantage of directly providing point-set models.

######  Remark 18.

The result of Theorem 10 can be subsequently extended to other connected
models for of $\mathbb{E}^{\mathrm{nu}}_{\infty}$-operads by the homotopy-
invariance of the $\infty$-categories considered.

######  Remark 19.

The results of [BM19] (resp. of [BCN21]) have an extra level of generality
with respect to our current work since their results are also valid when
$\Bbbk$ is a complete local ring (resp. a coherent
$\mathbb{E}_{\infty}$-ring). We nevertheless expect that the methods
introduced in this paper can be generalized to a similar setting.

###  5.3. $\mathbb{E}_{k}$ formal moduli problems

Over a field of any characteristic $\Bbbk$, one recovers the correspondence
between formal moduli problems over $\mathbb{E}_{k}^{\mathrm{nu}}$-algebras
and $\mathbb{E}_{k}^{\mathrm{nu}}$-algebras, where
$\mathbb{E}_{k}^{\mathrm{nu}}$ is a model for the non-unital little $k$-cubes
operad. For $k=1$, this gives that formal moduli problems of non-unital
associative algebras are encoded by non-unital dg associative algebras. See
[Lur11, Section 3 and 4].

Let $\mathcal{E}^{\mathrm{nu}}$ be the non-unital Barratt-Eccles dg operad. It
admits a filtration of sub dg operads

$\mathcal{E}^{\mathrm{nu}}_{1}\rightarrowtail\cdots\rightarrowtail\mathcal{E}^{\mathrm{nu}}_{k}\rightarrowtail\cdots\rightarrowtail\mathrm{colim}_{k}\leavevmode\nobreak\
\mathcal{E}^{\mathrm{nu}}_{k}\cong\mathcal{E}^{\mathrm{nu}}\leavevmode\nobreak\
,$

where each $\mathcal{E}_{k}^{\mathrm{nu}}$ is a model for
$\mathbb{E}_{k}^{\mathrm{nu}}$. See [BF04, Section 1.6] for more details on
this.

######  Lemma 13.

Let $k\geq 1$. The conilpotent dg cooperad
$\mathrm{B}\mathcal{E}_{k}^{\mathrm{nu}}$ is tempered and quasi-planar.

###### Proof.

Let us notice that for any $k\geq 1$, every $\mathcal{E}_{k}^{\mathrm{nu}}(n)$
is a quasi-free dg $\mathbb{S}_{n}$-module. Therefore the underlying
conilpotent graded cooperad $\mathrm{B}\mathcal{E}_{k}^{\mathrm{nu}}$ is
planar. Filtering by the homological degree of $\mathcal{E}^{\mathrm{nu}}_{k}$
gives the quasi-planar filtration on
$\mathrm{B}\mathcal{E}^{\mathrm{nu}}_{k}$.

It was shown in [Fre11] that the homology of
$\mathrm{B}\mathcal{E}^{\mathrm{nu}}_{k}$ is given by the underlying
$\mathbb{S}$-module of the $k$-fold suspension of the conilpotent dg cooperad
$(k-1)\text{-}\mathcal{G}erst^{*}$, which is the linear dual cooperad of the
operad encoding non-unital $(k-1)$-Gerstenhaber algebras. It follows directly
that $\mathrm{B}\mathcal{E}^{\mathrm{nu}}_{k}$ is tempered. ∎

######  Theorem 11.

Let $k\geq 1$ and let $\mathcal{E}^{\mathrm{nu}}_{k}$ the sub dg operad of the
non-unital Barratt-Eccles dg operad encoding
$\mathbb{E}^{\mathrm{nu}}_{k}$-algebras. There is a canonical equivalence

$\mathsf{FMP}_{\mathcal{E}^{\mathrm{nu}}_{k}}\simeq\mathsf{dg}\leavevmode\nobreak\
\Omega((\mathcal{E}^{\mathrm{nu}}_{k})^{*})\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\ .$

between the $\infty$-categories of dg
$\Omega((\mathcal{E}^{\mathrm{nu}}_{k})^{*})$-algebras and of formal moduli
problems over $\mathcal{E}^{\mathrm{nu}}_{k}$-algebras.

###### Proof.

Follows from Theorem 6 and Lemma 13. ∎

######  Remark 20.

The result of Theorem 11 can be subsequently extended to other connected
models for of $\mathbb{E}_{k}^{\mathrm{nu}}$-operads by the homotopy-
invariance of the $\infty$-categories considered.

###  5.4. Partition pre-Lie algebras and permutative formal moduli problems

Over a field of characteristic zero $\Bbbk$, the $\infty$-category of formal
moduli problems over permutative algebras is equivalent to the
$\infty$-category of pre-Lie algebras. This was shown in [CCN19], by applying
Theorem 7 to the operad $\mathcal{P}\mathrm{erm}$ which encodes permutative
algebras. See [LV12, Section 13.4] for more information on permutative
algebras and pre-Lie algebras. The main example of formal moduli problems of
permutative algebras is given by operadic deformation theory: given a
coaugmented dg cooperad $\mathcal{C}$ and a dg operad $\mathcal{P}$, one can
define

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where $\mathcal{P}\otimes A$ is again a dg operad, since $A$ is a dg
permutative algebra. This formal moduli problem encodes the deformations of
the trivial morphism $\Omega\mathcal{C}\longrightarrow\mathcal{P}$. When
$\mathcal{P}$ is the endomorphism operad $\mathrm{End}_{V}$ of some dg module
$V$, deformations of this trivial morphism correspond to deformations of the
trivial dg $\Omega\mathcal{C}$-algebra structure on $V$. This permutative
formal moduli problem is encoded by the convolution dg pre-Lie algebra

$\prod_{n\geq
0}\mathrm{Hom}_{\mathbb{S}_{n}}(\mathcal{C}(n),\mathcal{P}(n))\leavevmode\nobreak\
,$

where the pre-Lie bracket is constructed using the partial decomposition of
$\mathcal{C}$ and the partial composition of $\mathcal{P}$.

Over a field of positive characteristic, the operad $\mathcal{P}\mathrm{erm}$
is not $\mathbb{S}$-projective. Therefore in order to make sense of a
permutative formal moduli problem, we can consider
$\mathcal{P}\mathrm{erm}\otimes\mathcal{E}^{\mathrm{nu}}$, where
$\mathcal{E}^{\mathrm{nu}}$ is the non-unital Barratt-Eccles operad.

######  Lemma 14.

The conilpotent dg cooperad
$\mathrm{B}(\mathcal{P}erm\otimes\mathcal{E}^{\mathrm{nu}})$ is tempered.

###### Proof.

It follows from [CV06, Theorem 1.13] that the homology of
$\mathrm{B}(\mathcal{P}\mathrm{erm}\otimes\mathcal{E}^{\mathrm{nu}})$ (which
is quasi-isomorphic to $\mathrm{B}(\mathcal{P}\mathrm{erm})$) is the
underlying $\mathbb{S}$-module of the shifted pre-Lie cooperad
$s\mathrm{pre}\text{-}\mathcal{L}\mathrm{ie}^{*}$. Thus it clearly satisfies
the tempered condition. ∎

######  Theorem 12.

There is a canonical equivalence

$\mathsf{FMP}_{\mathcal{P}\mathrm{erm}\leavevmode\nobreak\
\otimes\leavevmode\nobreak\
\mathcal{E}^{\mathrm{nu}}}\simeq\mathsf{dg}\leavevmode\nobreak\
\Omega(\mathcal{P}\mathrm{erm}^{*}\otimes(\mathcal{E}^{\mathrm{nu}})^{*})\text{-}\mathsf{alg}\leavevmode\nobreak\
[\mathrm{Q.iso}^{-1}]\leavevmode\nobreak\ .$

between the $\infty$-categories of dg
$\Omega(\mathcal{P}\mathrm{erm}^{*}\otimes(\mathcal{E}^{\mathrm{nu}})^{*})$-algebras
and of formal moduli problems over
$\mathcal{P}\mathrm{erm}\leavevmode\nobreak\ \otimes\leavevmode\nobreak\
\mathcal{E}^{\mathrm{nu}}$-algebras.

###### Proof.

Follows from Theorem 6 and Lemma 14. ∎

######  Remark 21.

Adapting the arguments of [BCN21, Proposition 4.51], one can show that
$\mathcal{S}\mathrm{urj}^{*}\otimes
s\mathrm{pre}\text{-}\mathcal{L}\mathrm{ie}$ is also a model for
$\Omega(\mathcal{P}\mathrm{erm}^{*}\otimes(\mathcal{E}^{\mathrm{nu}})^{*})$,
where $\mathcal{S}\mathrm{urj}^{*}$ is the dual of the surjections cooperad
constructed in the appendix of op.cit.

In order to define the analogue of the functor
$\mathrm{Def}_{\Omega\mathcal{C}\rightarrow\mathcal{P}}^{0}$ in positive
characteristic, one can replace dg operads by some kinds of operads up to
homotopy ; for instance dg algebras over the coloured operads
$\mathcal{O}p\otimes\mathcal{E}^{\mathrm{nu}}$ (where $\mathcal{O}p$ is the
coloured operad that encodes non-unital dg operads). These dg operads up to
homotopy are tensored over
$\mathcal{P}\mathrm{erm}\otimes\mathcal{E}^{\mathrm{nu}}$-algebras. For two
operads $\mathcal{P},\mathcal{Q}$, we can then define a formal moduli problem
over $\mathcal{P}\mathrm{erm}\otimes\mathcal{E}^{\mathrm{nu}}$-algebras in an
analogous way

${A}$${\mathrm{Map}_{\mathsf{dg}\leavevmode\nobreak\
{\mathcal{O}p\leavevmode\nobreak\ \otimes\leavevmode\nobreak\
\mathcal{E}^{\mathrm{nu}}}\text{-}\mathsf{alg}}\left(\mathcal{Q},\mathcal{P}\otimes
A\right)\leavevmode\nobreak\ .}$

Replacing $\mathcal{Q}$ by the Cobar construction $\Omega\mathcal{C}$ of a
quasi-planar conilpotent dg cooperad $\mathcal{C}$, the convolution
construction between $\mathcal{C}$ and $\mathcal{P}$ should encode this formal
moduli problem, and therefore should be endowed with a dg partition pre-Lie
algebra structure that extends the classical pre-Lie structure.

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