# Higher Lie theory in positive characteristic

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

The main goal of this article is to develop integration theory over a positive
characteristic field, and as a by-product, construct Lie-type models for the
$p$-adic homotopy types of spaces. We introduce a new type of algebraic
structure called curved absolute partition $\mathcal{L}_{\infty}$-algebra,
which always admits a well-defined Maurer-Cartan equation. We construct a
Quillen adjunction between curved absolute partition
$\mathcal{L}_{\infty}$-algebras and simplicial sets, where the right adjoint
is the integration functor. We show that it satisfies the same properties that
its characteristic zero analogues. We also give an explicit combinatorial
description of gauge equivalences. And using our constructions, we show how
one can recover the $\mathbb{E}_{\infty}$-formal moduli problem associated to
a (spectral) partition $\mathcal{L}_{\infty}$-algebra, under a homotopy
completeness assumption. Finally, we use the left adjoint of the integration
functor to construct Quillen-like models for the $p$-adic homotopy types of
spaces and of their mapping spaces.

###### Key words and phrases:

Algebraic operads, deformation theory, formal moduli problems, homotopical
algebra, Lie theory, $p$-adic homotopy theory.

###### 2020 Mathematics Subject Classification:

18M70, 18N40, 22E60, 55P62, 55U10, 14D15, 14D23

###### Contents

  1. 1 Curved absolute partition $\mathcal{L}_{\infty}$-algebras
  2. 2 The integration theory
  3. 3 Lie-type models in $p$-adic homotopy theory
  4. A Structural formulas

## Introduction

The characteristic zero case. Over a field of characteristic zero, there is an
equivalence of between the $\infty$-category of formal moduli problems and the
$\infty$-category of dg Lie algebras. This became a theorem by the work J.
Lurie and J. P. Pridham [Lur11a, Pri10]. This statement can be roughly
translated as "every infinitesimal deformation problem is encoded by a dg Lie
algebra". In the classical setting, one of the purposes of integration theory
is to understand the "infinitesimal deformation problem" encoded by a
particular dg Lie algebra.

More precisely, the goal is to construct a functor from the category of dg Lie
algebras to the category of simplicial sets which recovers the deformation
$\infty$-groupoid encoded by this dg Lie algebra. Its most basic example can
be traced back to Lie theory and the Baker–Campbell–Hausdorff formula which
produces a group out of a nilpotent Lie algebra. A first general approach to
the integration of dg Lie algebras is given by the seminal work of V. Hinich
in [Hin01], using methods from D. Sullivan [Sul77]. A refined version of the
integration procedure was constructed by E. Getzler in [Get09], where he
constructs the integration procedure to nilpotent
$\mathcal{L}_{\infty}$-algebras (homotopy Lie algebras). More modern
approaches include the new characterization of Getzler’s functor given by D.
Robert-Nicoud and B. Vallette in [RNV20] using operadic calculus. Finally, the
author generalized these results to the case of curved absolute
$\mathcal{L}_{\infty}$-algebras in [RiL22b]. By considering curved absolute
$\mathcal{L}_{\infty}$-algebras, a new type of algebraic structures where all
formal sums of operations are well-defined by definition, the author was also
able to introduce a new set of methods into integration theory: mainly the
complete bar construction and new model category structures transferred using
the complete bar-cobar adjunction.

There is a parallel story concerning rational models for spaces. On the one
hand, one can construct rational models using dg commutative algebras as done
by D. Sullivan in [Sul77]. And on the other, one can construct rational models
using dg Lie algebras, as done by D. Quillen in [Qui69]. There is a duality
between these two types of algebraic structures called Koszul duality. This
duality explains why dg Lie algebras encode infinitesimal deformation problems
and also links Quillen’s and Sullivan’s approaches to rational homotopy
theory. It is a modern insight of [BFMT20], and [RNV20] in the
$\mathcal{L}_{\infty}$ case, that integration theory can help simply Quillen’s
original approach and give a direct construction of Lie models for spaces.
This was also generalized to curved absolute $\mathcal{L}_{\infty}$-algebras
in [RiL22b], where they were shown to provide rational models for non-
necessarily pointed nor connected finite type nilpotent spaces.

The positive characteristic case. Over a field of positive characteristic,
there is an equivalence between the $\infty$-category of
$\mathbb{E}_{\infty}$-formal moduli problems and the $\infty$-category of
(spectral) partition Lie algebras. This was proven by L. Brantner and A.
Mathew in [BM19]. In their work, they define the $\infty$-category of
partition Lie algebras as algebras over an $\infty$-categorical monad. Later,
L. Brantner, R. Campos and J. Nuiten gave in [BCN21] an explicit point-set
model for the $\infty$-category of partition Lie algebras: it can be obtained
by localizing at quasi-isomorphisms the category of divided powers dg
$\Omega(\mathbb{E}^{\mathrm{nu}}_{\infty})^{*}$-algebras, where
$(\mathbb{E}^{\mathrm{nu}}_{\infty})^{*}$ is the linear dual of any non-unital
$\mathbb{S}$-projective resolution of the commutative operad. We will refer to
these algebras as partition $\mathcal{L}_{\infty}$-algebras.

Nevertheless, even if we have point-set algebraic models for
$\mathbb{E}_{\infty}$-formal moduli problems, there are no analogues of
integration theory in positive characteristic. One of the main goals of this
paper is to fill this gap by constructing a well-behaved integration functor.
In order to do so, we build on the ideas introduced in [RiL22b] and heavily
rely on the positive characteristic homotopical operadic calculus developed by
B. Le Grignou and the author in [GRiL23a].

The other main goal is the construct Lie-type models for the $p$-adic homotopy
type of spaces using these methods. M. Mandell proved in [Man01] that
$\mathbb{E}_{\infty}$-algebras over the algebraic closure of $\mathbb{F}_{p}$
provide us with algebraic models for the $p$-adic homotopy type of spaces.
This result can be thought as the positive characteristic analogue of
Sullivan’s results. The question is then whether one can Koszul-dualize
Mandell’s approach to obtain Lie-type models for spaces like in [Qui69], given
this time in terms of partition Lie or partition
$\mathcal{L}_{\infty}$-algebras. One of the main difficulties is that cochains
on spaces are formally étale over a positive characteristic field as shown in
[Lur11b], and therefore the bar constructions on these algebras are always
acyclic. This suggests that constructing these Lie-type models directly at the
$\infty$-categorical level (or with model categories where weak-equivalences
are quasi-isomorphisms) might not be possible. We overcome this difficulty by
working with absolute partition $\mathcal{L}_{\infty}$-algebras instead, which
have a model structure transferred from coalgebras. In this case, this finer
notion of weak equivalence allows us to encode the $p$-adic homotopy type of
connected finite type nilpotent spaces. A possible explanation for this
phenomenon is the following: in characteristic zero, the dg Lie or
$\mathcal{L}_{\infty}$-algebras that give rational models are homotopy
complete, meaning that they are fully faithfully included into the absolute
counterparts. Hence one can consider rational models both in terms of absolute
or classical dg Lie algebras. Whereas in positive characteristic, it could be
that one is forced to consider absolute partition
$\mathcal{L}_{\infty}$-algebras.

Main results. We first introduce curved absolute partition
$\mathcal{L}_{\infty}$-algebras, where we consider curved objects in order to
have the maximum degree of generality. Absolute partition
$\mathcal{L}_{\infty}$-algebras are included into curved ones, as they are
precisely those with zero curvature. On the other side of Koszul duality, this
degree of generality amounts to working with all unital
$\mathbb{E}_{\infty}$-algebras instead of only with augmented
$\mathbb{E}_{\infty}$-algebras.

The idea is to consider the Barratt-Eccles dg operad $\mathcal{E}$ introduced
by C. Berger and B. Fresse in [BF04] as a particular model for unital
$\mathbb{E}_{\infty}$-(co)algebras. Curved algebras over the curved cooperad
$\mathrm{B}\mathcal{E}$, where $\mathrm{B}$ stands for a particular bar
construction, are the positive characteristic analogues of curved absolute
$\mathcal{L}_{\infty}$-algebras. The cooperad $\mathrm{B}\mathcal{E}$ is an
example of a quasi-planar in the sense of [GRiL23a]. This allows us to apply
all the tools from the homotopical operadic calculus of op.cit. to this
particular situation, and therefore to generalize the methods of [RiL22b] to
this new setting.

The cellular chains functor $C^{c}_{*}(-)$ on spaces can be endowed with a
functorial dg $\mathcal{E}$-coalgebra structure as shown in [BF04]. By pulling
back this structure along a cofibrant resolution (as an operad) of
$\mathcal{E}$ which we call $u\mathcal{EE}_{\infty}$, we obtain the following
commuting triangle of Quillen adjunctions:

######  Theorem A (Theorem 2.5).

There is a commuting triangle

${\hskip
12.0ptu\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{sSet}}$${\hskip
36.0pt\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\textsf{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}~{},}$$\scriptstyle{\widehat{\Omega}_{\iota}}$$\scriptstyle{\overline{\mathcal{R}}}$$\scriptstyle{C^{c}_{*}(-)}$$\scriptstyle{\mathcal{L}}$${\dashv}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota}}$$\scriptstyle{\mathcal{R}}$${\dashv}$${\dashv}$

made of Quillen adjunctions between the categories of simplicial sets with the
Kan-Quillen model structure, of $u\mathcal{EE}_{\infty}$-coalgebras endowed
with a model category structure where weak-equivalences are given by quasi-
isomorphisms, and of curved absolute partition $\mathcal{L}_{\infty}$-algebras
with a transferred model category structure along the complete bar-cobar
adjunction.

As in [RiL22b], the complete bar-cobar adjunction proves to be an essential
tool in integration theory, as we can also transfer model category structures
along this adjunction using the results of [GRiL23a]. The integration functor
$\mathcal{R}$ is, by definition, a right Quillen functor. This simple fact
directly implies the main properties that a well-behaved integration functor
needs to satisfy.

######  Theorem B (Theorem 2.6).

  1. (1)

For any complete curved absolute partition $\mathcal{L}_{\infty}$-algebra
$\mathfrak{g}$, the simplicial set $\mathcal{R}(\mathfrak{g})$ is a Kan
complex.

  2. (2)

Let $f:\mathfrak{g}\twoheadrightarrow\mathfrak{h}$ be an epimorphism of
complete curved absolute partition $\mathcal{L}_{\infty}$-algebras. Then

$\mathcal{R}(f):\mathcal{R}(\mathfrak{g})\twoheadrightarrow\mathcal{R}(\mathfrak{h})$

is a fibrations of simplicial sets.

  3. (3)

The functor $\mathcal{R}$ preserves weak-equivalences. In particular, it sends
any graded quasi-isomorphism
$f:\mathfrak{g}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{h}$
of complete curved absolute partition $\mathcal{L}_{\infty}$-algebras to a
weak homotopy equivalence of simplicial sets.

Furthermore, the functor $\mathcal{R}$ admits a combinatorial description. The
$0$-simplicies of $\mathcal{R}(\mathfrak{g})$ are given by Maurer-Cartan
elements: not only the equation makes sense in this context (without any
$1/n!$ terms), but it is always well-defined for any curved absolute partition
$\mathcal{L}_{\infty}$-algebra, since these are absolute algebras and any
formal sum of operations admits a well-defined image in the absolute setting.
We also completely characterize $1$-simplicies in $\mathcal{R}(\mathfrak{g})$
by explicit formulas, giving a combinatorial notion of gauge equivalences.
This allows us to describe $\pi_{0}(\mathcal{R}(\mathfrak{g}))$, that is, the
set of Maurer–Cartan elements up to gauge equivalence. This set is the
underlying classical deformation problem of a homotopical deformation problem.
Notice that one can always restrict the integration functor $\mathcal{R}$ to
the full sub-category of nilpotent partition $\mathcal{L}_{\infty}$-algebras,
which are the positive characteristic analogues of the nilpotent
$\mathcal{L}_{\infty}$-algebras studied in [Get09]. In particular, for abelian
partition $\mathcal{L}_{\infty}$-algebras, the functor $\mathcal{R}$ is
isomorphic to the right adjoint of the Dold–Kan correspondence. Furthermore,
under a natural homotopy completeness assumption, we show that our functor
presents the formal moduli problem associated to a partition
$\mathcal{L}_{\infty}$-algebra. We work over a field $\Bbbk$ of any
characteristic: as a sanity check, when $\Bbbk$ is of characteristic zero, we
show that our integration functor $\mathcal{R}$ is naturally weakly-equivalent
to the previously known functors.

Now, using curved absolute partition $\mathcal{L}_{\infty}$-algebras, we
construct Quillen-like models for the $p$-adic homotopy theory of spaces, by
applyng the left adjoint functor $\mathcal{L}$. This approach is Koszul dual
to M. Mandell’s approach of [Man01], much in the same way as Quillen’s
approach of [Qui69] to rational homotopy theory is Koszul dual to Sullivan’s
approach.

######  Theorem C (Theorem 3.5).

Let $X$ be a connected finite type nilpotent simplicial set. The unit of
adjunction

$\eta_{X}:X\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{R}\mathcal{L}(X)$

is an $\mathbb{F}_{p}$-equivalence.

We can recover information about $X$ using these models. If a curved absolute
partition $\mathcal{L}_{\infty}$-algebra $\mathfrak{g}_{X}$ is a $p$-adic
model for a space $X$, then we can recover the homology of $X$ by computing
the homology of the complete bar construction of $\mathfrak{g}_{X}$.

We also develop a notion of minimal model in this context, analogue to the one
introduced in [RiL22b]. The generators of the minimal model of a $p$-adic
model will correspond to the homology of the space. Finally, using the theory
of mapping coalgebras developed by B. Le Grignou in a forthcoming paper
[Gri23], we construct models for mapping spaces.

######  Theorem D (Theorem 3.17).

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}_{\infty}$-algebra and let $X$ be a simplicial set. There is a
weak equivalence of Kan complexes

$\mathrm{Map}(X,\mathcal{R}(\mathfrak{g}))\simeq\mathcal{R}\left(\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})\right)~{},$

which is natural in $X$ and in $\mathfrak{g}$, where
$\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})$ denotes the convolution curved
absolute partition $\mathcal{L}_{\infty}$-algebra.

Finally, let us point out that the integration theory introduced in this paper
lies at the crossroad of three domains. First, Lie theory, where partition
$\mathcal{L}_{\infty}$-algebras are explicit, well-behaved substitutes for Lie
algebras in positive characteristic (different from $p$-restricted Lie
algebras). Extending classical results of Lie theory to this setting will be
the subject of future work. Secondly, and as already explained, it is deeply
related to deformation theory via formal moduli problems. Finally, as in the
characteristic zero case, the adjoint of the integration functor allows us to
construct models for homotopy types.

### Acknowledgments

I would like to thank Lukas Brantner, Ricardo Campos, Brice Le Grignou, Joost
Nuiten, Daniel Robert-Nicoud, Bruno Vallette and Felix Wierstra for
stimulating conversations about these and related topics.

### Conventions

Let $\Bbbk$ be a field. The base category will be the symmetric monoidal
category $\left(\mathsf{pdg}\textsf{-}\mathsf{mod},\otimes,\Bbbk\right)$ of
pre-differential graded $\Bbbk$-modules with the tensor product. This
framework as well as the basic notion used in this paper are explained, for
instance, in [RiL22a].

##  1\. Curved absolute partition $\mathcal{L}_{\infty}$-algebras

The goal of this section is to introduce the positive characteristic analogue
of the curved absolute $\mathcal{L}_{\infty}$-algebras that were introduced in
[RiL22b], and to give a description of this algebraic structure.

###  1.1. Explicit description

Let $\mathcal{E}$ be the unital version of the Barratt-Eccles operad. We
consider the conilpotent curved cooperad
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$, where $\mathrm{B}^{\mathrm{s.a}}$
stands for the semi-augmented bar construction of [HM12], see also [RiL22b,
Appendix]. As we will later show, this cooperad is quasi-planar cooperad in
the sense of [GRiL23a]. This will allow us to apply the homotopical operadic
calculus developed in op.cit. to this particular context.

######  Definition 1.1 (Curved absolute partition
$\mathcal{L}_{\infty}$-algebra).

A curved absolute partition $\mathcal{L}_{\infty}$-algebra $\mathfrak{g}$
amounts the data $(\mathfrak{g},\gamma_{\mathfrak{g}},d_{\mathfrak{g}})$ of a
curved $\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$-algebra.

The name curved absolute partition $\mathcal{L}_{\infty}$-algebra is justified
by the fact that these algebras are the curved absolute analogues of algebras
over the operad $\Omega(\mathcal{E}^{\mathrm{nu}})^{*}$, where
$\mathcal{E}^{\mathrm{nu}}$ is a non-unital version of the Barratt-Eccles
operad. The category of dg $\Omega(\mathcal{E}^{\mathrm{nu}})^{*}$-algebras
together with quasi-isomorphisms is a model for the $\infty$-category of
spectral partition Lie algebras of [BM19], as it was shown in [BCN21, Theorem
4.44]. We refer to Subsection 2.4 for more details on this.

######  Notation 1.2.

A curved absolute partition $\mathcal{L}_{\infty}$-algebra will also be
referred to as a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra, in a
somewhat vain effort to make notation less heavy.

Let us make this definition more explicit. A curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra structure on a pdg module
$(\mathfrak{g},d_{\mathfrak{g}})$ amounts to the data of a structural morphism

$\gamma_{\mathfrak{g}}:\prod_{n\geq
0}\mathrm{Hom}_{\mathbb{S}_{n}}\left(\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n),\mathfrak{g}^{\otimes
n}\right)\longrightarrow\mathfrak{g}~{},$

which satisfies the conditions of a curved algebra over a curved cooperad.
These conditions consist of a compatibility condition with the pre-
differentials, the associativity condition of an algebra over a monad and
finally a compatibility condition with the curvature of the cooperad. See
[GRiL23a, Section 3] for more details. Using the constructions of [RiL22b,
Appendix], let us give a more explicit description of the structural map.

######  Lemma 1.3.

Let $(\mathfrak{g},d_{\mathfrak{g}})$ be a pdg module. There is an isomorphism
of pdg modules

$\prod_{n\geq
0}\mathrm{Hom}_{\mathbb{S}_{n}}\left(\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n),\mathfrak{g}^{\otimes
n}\right)\cong\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}~{},$

natural in $\mathfrak{g}$, where $\widehat{\Omega}^{\mathrm{s.a}}$ denotes the
semi-augmented complete cobar construction and where $\widehat{\otimes}$
denotes the completed tensor product with respect to the canonical filtration
on the semi-augmented complete cobar construction.

###### Proof.

There is an isomorphism of pdg modules

$\mathrm{Hom}_{\mathbb{S}_{n}}\left(\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n),\mathfrak{g}^{\otimes
n}\right)\cong\left(\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}~{}\mathfrak{g}^{\otimes
n}\right)^{\mathbb{S}_{n}}~{},$

since $\omega$-stage of the coradical filtration
$\mathscr{R}_{\omega}\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n)$ is degree-wise
finite dimensional over $\Bbbk$. Furthermore, there is an isomorphism

$\left(\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}~{}\mathfrak{g}^{\otimes
n}\right)^{\mathbb{S}_{n}}\cong\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}~{},$

given by the norm map, since
$\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)$ is a quasi-free dg
$\mathbb{S}_{n}$-module for all $n\geq 0$. See [GRiL23a, Section 1.2] for more
details on quasi-freeness. ∎

Let $\mathrm{SCRT}_{n}^{\omega}$ denote the set of symmetric corked rooted
trees of arity $n$ and with $\omega$ internal edges. A symmetric corked rooted
tree is a rooted tree where vertices either have at least two incoming edges
or zero incoming edges (called corks). In a symmetric corked rooted tree, each
vertex $v$ with at least two incoming edges is labelled by a $r_{v}+1$-uple
$(\sigma_{0},\cdots,\sigma_{r_{v}})$, where $\sigma_{j}$ is an element of
$\mathbb{S}_{\mathrm{In}(v)}$, where $r_{v}\geq 0$, and where
$\sigma_{i}\neq\sigma_{j}$ for $i\neq j$. If $v$ has no incoming edges then it
is not labelled. The degree $\mathrm{deg}(\tau)$ of a symmetric corked rooted
tree $\tau$ is given by

$\mathrm{deg}(\tau)\coloneqq\mathrm{Cork}(\tau)+\sum_{v\in\mathrm{V}(\tau)}r_{v}+1~{},$

where $\mathrm{Cork}(\tau)$ is the number of corks of $\tau$ and where
$\mathrm{V}(\tau)$ is the set of all vertices of $\tau$. We denote
$\mathrm{SCRT}(\delta)$ the finite set of symmetric corked rooted trees of
degree $\delta\geq 0$. The arity of a symmetric corked rooted tree is the
number of non-corked leaves. The weight is the total number of vertices. In
general, we denote by $\mathrm{SCRT}_{n}^{\omega}$ the set of symmetric corked
rooted trees of weight $\omega$ and arity $n$. In summary, these are rooted
trees labelled by elements in the Barratt–Eccles operad, and they form a basis
of $\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$.

We will denote by $c_{n}^{(\sigma_{0},\cdots,\sigma_{r})}$ the $n$-corolla
labelled by $(\sigma_{0},\cdots,\sigma_{r})$. Weight one symmetric corked
rooted trees are given by $n$-corollas. The only symmetric corked rooted tree
of weight (and of degree) $0$ is the trivial tree of arity one with zero
vertices.

######  Lemma 1.4.

Let $\mathfrak{g}$ be a graded module with a basis $\left\\{g_{b}~{}|~{}b\in
B\right\\}~{}.$ The graded module

$\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}$

admits a basis given by formal power series of linear combinations of
symmetric corked rooted trees labelled by the basis elements of
$\mathfrak{g}$. That is, elements of the form

$\sum_{n\geq 0}\sum_{\omega\geq
0}\sum_{\tau\in\mathrm{SCRT}_{n}^{\omega}}\sum_{i\in\mathrm{I}_{\tau}}\lambda_{\tau}^{(i)}\tau\left(g_{i_{1}}^{(i)},\cdots,g_{i_{n}}^{(i)}\right)~{},$

where $\mathrm{I}_{\tau}$ is a finite set, $\lambda_{\tau}$ is a scalar in
$\Bbbk$ and $(i_{1},\cdots,i_{n})$ is in $B^{n}$. Here
$\tau(g_{i_{1}},\cdots,g_{i_{n}})$ refers to the symmetric corked rooted tree
$\tau$ with input leaves decorated by the elements
$\tau(g_{i_{1}},\cdots,g_{i_{n}})$. The homological degree of
$\tau(g_{i_{1}},\cdots,g_{i_{n}})$ is given by
$\sum_{j}\mathrm{deg}(g_{i_{j}})-\mathrm{deg}(\tau)$, where
$\mathrm{deg}(\tau)$ is the degree of $\tau$ defined above.

###### Proof.

It follows by direct inspection from the description of
$\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)$ in terms of symmetric
corked rooted trees, since it is the linear dual of
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$. ∎

######  Remark 1.5.

Notice that instead of considering formal power series over the arity and the
weight of the symmetric corked rooted trees, we could also consider formal
power series indexed by the degree of the symmetric corked rooted trees

$\sum_{\delta\geq
0}\sum_{\tau\in\mathrm{SCRT}(\delta)}\sum_{i\in\mathrm{I}_{\tau}}\lambda_{\tau}^{(i)}\tau\left(g_{i_{1}}^{(i)},\cdots,g_{i_{n}}^{(i)}\right)~{},$

as the basis elements in Lemma 1.4, as this sum also ranges over every
symmetric corked rooted tree.

Let $(\mathfrak{g},d_{\mathfrak{g}})$ be a pdg module. Let us describe the
pre-differential on the pdg module

$\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}~{}.$

It is given by the sum of three terms $d_{1}$, $d_{2}$ and $d_{3}$.

  1. (1)

The first term $d_{1}$ is the sum of the pre-differential $d_{\mathfrak{g}}$
applied on each of the elements $g_{i_{j}}$ that decorate the leaves of a
symmetric corked rooted tree $\tau$.

  2. (2)

The second term $d_{2}$ is the sum over all vertices of $\tau$ of the
following action on their labels:

$d_{2}((\sigma_{0},\cdots,\sigma_{r}))=\sum_{\begin{subarray}{c}\sigma\in\mathbb{S}_{n}\\\
\sigma\neq\sigma_{j}\end{subarray}}\sum_{i=0}^{r}(-1)^{i}(\sigma_{0},\cdots,\underbrace{\sigma}_{i\text{-th
spot}},\cdots,\sigma_{r})~{},$

where the first sum is taken over all $\sigma$ in $\mathbb{S}_{n}$ such that
$\sigma\neq\sigma_{j}$ for $0\leq j\leq r$ and where $\sigma$ is added in the
$i$-th spot. This term corresponds to the differential on $\mathcal{E}^{*}$.

  3. (3)

The third term $d_{3}$ is induced by the partial decompositions maps of
counital partial cooperad $\mathcal{E}^{*}$. Applied to a corolla with a
vertex $v$ labelled by $(\sigma_{0},\cdots,\sigma_{r_{v}})$, it gives a sum of
symmetric corked rooted trees with two vertices $v^{(1)}$ and $v^{(2)}$,
labelled by all the possible partial decompositions of
$(\sigma_{0},\cdots,\sigma_{r_{v}})$. The corolla $v^{(2)}$ lies on the $i$-th
input of $v^{(1)}$ precisely when the partial decomposition $\Delta_{i}$ has
been applied to $(\sigma_{0},\cdots,\sigma_{r_{v}})$.

In general, the term $d_{3}$ is the sum over all the vertices of a symmetric
corked tree of the previous rule on corollas. For more concrete formulas of
the partial decompositions of $\mathcal{E}^{*}$, see the Appendix A.

######  Remark 1.6 (The norm map).

Both $\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}$ and
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ can be fully described in terms of
symmetric corked rooted trees. The isomorphism induced by the norm map in
Lemma 1.3 is given explicitly by

${\mathbb{N}:\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}}$${\mathrm{Hom}_{\mathbb{S}_{n}}\left(\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n),\mathfrak{g}^{\otimes
n}\right)}$${\tau(g_{1},\cdots,g_{n})}$${\displaystyle\sum_{\sigma\in\mathbb{S}_{n}}\left[\mathrm{ev}_{(g_{\sigma^{-1}(1)},\cdots,g_{\sigma^{-1}(n)})}(\sigma\bullet\tau):\sigma\bullet\tau\mapsto
g_{\sigma^{-1}(1)}\otimes\cdots\otimes g_{\sigma^{-1}(n)}\right]~{}.}$

where $\mathrm{ev}_{(g_{\sigma(1)},\cdots,g_{\sigma(n)})}(\sigma\bullet\tau)$
is a Dirac function which is zero everywhere outside of the prescribed value
and where $\sigma\bullet\tau$ is given by the action of $\mathbb{S}_{n}$ on
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n)$. This allows us to pass from one
presentation to the other. For example, this morphism is given on corollas
labelled by $w=(\sigma_{0},\cdots,\sigma_{r})$ as:

$\mathbb{N}\left(c_{n}^{w}(g_{1},\cdots,g_{n})\right)=\sum_{\sigma\in\mathbb{S}_{n}}\left[\mathrm{ev}_{(g_{\sigma^{-1}(1)},\cdots,g_{\sigma^{-1}(n)})}\left(c_{n}^{\sigma.w}\right):c_{n}^{\sigma.w}\mapsto(g_{\sigma^{-1}(1)},\cdots,g_{\sigma^{-1}(n)})\right]~{},$

where $\sigma.w$ is given by $(\sigma.\sigma_{0},\cdots,\sigma.\sigma_{r})$.
Notice that a basis of the $n$-corollas, as a free graded
$\mathbb{S}_{n}$-module, is given by $n$-corollas labelled by
$(\mathrm{id},\sigma_{1},\cdots,\sigma_{r})$.

Structural data. The structural data of a curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra structure on a pdg module
$(\mathfrak{g},d_{\mathfrak{g}})$ thus amounts to the data of a map of pdg
modules

$\gamma_{\mathfrak{g}}:\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}\longrightarrow\mathfrak{g}~{},$

which sends any formal power series of the form

$\sum_{n\geq 0}\sum_{\omega\geq
0}\sum_{\tau\in\mathrm{SCRT}_{n}^{\omega}}\sum_{i\in\mathrm{I}_{\tau}}\lambda_{\tau}^{(i)}\tau\left(g_{i_{1}}^{(i)},\cdots,g_{i_{n}}^{(i)}\right)$

to a well-defined image

$\gamma_{\mathfrak{g}}\left(\sum_{n\geq 0}\sum_{\omega\geq
0}\sum_{\tau\in\mathrm{SCRT}_{n}^{\omega}}\sum_{i\in\mathrm{I}_{\tau}}\lambda_{\tau}^{(i)}\tau\left(g_{i_{1}}^{(i)},\cdots,g_{i_{n}}^{(i)}\right)\right)~{},$

in the pdg module $\mathfrak{g}$. This does not presuppose an underlying
topology on the pdg module $\mathfrak{g}$.

The elementary operations induced by the structural map
$\gamma_{\mathfrak{g}}$ are the family of operations of degree $-r-1$ given by

$\left\\{l_{n}^{(\sigma_{0},\cdots,\sigma_{r})}\coloneqq\gamma_{\mathfrak{g}}\left(c_{n}^{(\sigma_{0},\cdots,\sigma_{r})}(-,\cdots,-)\right):\mathfrak{g}^{\otimes
n}\longrightarrow\mathfrak{g}~{}\right\\}$

for all $n\neq 1$ and all $r$-tuples $(\sigma_{1},\cdots,\sigma_{r})$, where
each $\sigma_{j}$ is in $\mathbb{S}_{n}$ and where $r\geq 1$. Recall that
$c_{n}$ stands for the $n$-corolla.

Conditions on the structural data. The structural map $\gamma_{\mathfrak{g}}$
needs to satisfy associativity and a compatibility with the pre-differential
conditions that are analogous to those of [RiL22b, Section 1]. Furthemore, in
order to form a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra
structure, the following equation on the elementary operations

(1) $d_{\mathfrak{g}}^{2}(g)=l_{2}^{(12)}(l_{0},g)+l_{2}^{(21)}(l_{0},g)~{},$

needs to hold. We will refer to the term
$l_{0}:\Bbbk\longrightarrow\mathfrak{g}$ as the curvature.

Quasi-planar completeness. As mentioned before, the conilpotent curved
cooperad $\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ is quasi-planar, and hence
comes equipped with a canonical quasi-planar filtration, see [GRiL23a, Section
2]. This filtration can be thought as the analogue of the coradical filtration
in the quasi-planar setting. In this particular case, the quasi-planar
filtration on $\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ is precisely the one
induced by the degree of symmetric corked rooted trees, see Lemma 1.12. It
induces a canonical filtration on any curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra, which we call the qp-filtration.

Let $\mathfrak{g}$ be a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra,
its qp-filtration is given by

$\mathrm{W}_{\delta}\mathfrak{g}\coloneqq\mathrm{Im}\left(\gamma_{\mathfrak{g}}|_{\mathscr{F}_{\delta}}:\prod_{n\geq
0}\mathscr{F}_{\delta}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}\longrightarrow\mathfrak{g}\right)~{},$

for $\delta\geq 0$, where
$\mathscr{F}_{\delta}\widehat{\Omega}^{\mathrm{s.a}}u\mathcal{C}om^{*}(n)$ is
spanned by symmetric corked rooted trees of degree greater or equal to
$\delta$. Therefore an element $g$ is in
$\mathrm{W}_{\delta_{0}}\mathfrak{g}$, and therefore of weight $\delta$, if
and only if it can be written as

$g=\gamma_{\mathfrak{g}}\left(\sum_{\delta\geq\delta_{0}}\sum_{\tau\in\mathrm{SCRT}(\delta)}\lambda_{\tau}\tau(g_{i_{1}},\cdots,g_{i_{n}})\right)~{}.$

One can check that $\mathfrak{g}/\mathrm{W}_{\delta}\mathfrak{g}$ has an
unique curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra structure given by
the quotient structural map. For instance, the trivial algebra
$\mathfrak{g}/\mathrm{W}_{1}\mathfrak{g}$ is given by the generators of the
algebra $\mathfrak{g}$. See [GRiL23a, Section 3.6] for more details on this
type of filtrations.

######  Definition 1.7 (Qp-complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra).

Let $(\mathfrak{g},\gamma_{\mathfrak{g}},d_{\mathfrak{g}})$ be a curved
absolute $\mathcal{L}^{\pi}_{\infty}$-algebra. It is qp-complete if the
canonical epimorphism

$\varphi_{\mathfrak{g}}:\mathfrak{g}\twoheadrightarrow\lim_{\delta}\mathfrak{g}/\mathrm{W}_{\delta}\mathfrak{g}$

is an isomorphism of curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras.

######  Remark 1.8.

We have that $\varphi_{\mathfrak{g}}$ is always an epimorphism by [GRiL23a,
Section 3.6].

######  Definition 1.9 (Associated graded).

Let $\mathfrak{g}$ be a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra.
Its associated graded of weight $\delta$ is given by

$\mathrm{gr}_{\delta}\mathfrak{g}\coloneqq\mathrm{W}_{\delta}\mathfrak{g}/\mathrm{W}_{\delta+1}\mathfrak{g}~{},$

for any $\delta\geq 0$.

######  Remark 1.10.

Notice that the associated graded is a dg module for all $\delta\geq 0$, as
the curvature lies in weight one.

######  Definition 1.11 (Maurer–Cartan equation).

Let $(\mathfrak{g},\gamma_{\mathfrak{g}},d_{\mathfrak{g}})$ be a curved
absolute $\mathcal{L}^{\pi}_{\infty}$-algebra. A Maurer–Cartan element
$\alpha$ is an element in $\mathfrak{g}$ of degree $0$ which satisfies the
following equation

$\gamma_{\mathfrak{g}}\left(\sum_{n\geq 0,~{}n\neq
1}c_{n}^{\mathrm{id}}(\alpha,\cdots,\alpha)\right)+d_{\mathfrak{g}}(\alpha)=0~{}.$

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

###  1.2. Model structures

We endow the category of curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras
with a transferred model structure from the category of
$\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$-coalgebras along the complete
bar-cobar adjunction, which becomes Quillen equivalence. From now on, we
denote $\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ by
$u\mathcal{EE}_{\infty}$.

######  Lemma 1.12.

  1. (1)

The conilpotent curved cooperad $\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ is
quasi-planar.

  2. (2)

Its canonical quasi-planar filtration is given by the degree of symmetric
corked rooted trees.

###### Proof.

The analogue of this result for $\mathrm{B}\mathcal{E}$ is proven in [GRiL23a,
Section 2], and it holds mutatis mutandis for the semi-augmented bar
construction as well. Let us explain why it is quasi-planar: the filtration
induced by considering only symmetric corked rooted trees of degree $\leq i$
induces an $\omega$-ladder of conilpotent curved cooperads whose colimit is
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$. The underlying conilpotent graded
cooperads of each step of the filtration is clearly planar and the
differential simply vanishes on the associated graded, hence it is a quasi-
planar filtration. The canonical quasi-planar filtration is induced by the
$\mathcal{E}$-comodule structure of
$\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$, which can be explicitly computed
like in [GRiL23a, Section 2.9] and coincides with the above filtration. ∎

######  Proposition 1.13.

There is a model structure on the category of
$u\mathcal{EE}_{\infty}$-coalgebras left-transferred along the cofree-
forgetful adjunction

${\mathsf{dg}~{}\mathsf{mod}}$${u\mathcal{EE}_{\infty}\textsf{-}\mathsf{coalg}~{},}$$\scriptstyle{\mathcal{C}(u\mathcal{EE}_{\infty})(-)}$$\scriptstyle{U}$${\dashv}$

where

  1. (1)

the class of weak equivalences is given by quasi-isomorphisms,

  2. (2)

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

  3. (3)

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

###### Proof.

By Lemma 1.12, $\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ is a quasi-planar curved
conilpotent cooperad. Therefore the dg operad
$\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ is cofibrant in the semi-model
category of [Fre09]. Finally, any cofibrant dg operad is coadmissible. See
[GRiL23a] for more details on this. ∎

Consider the curved twisting morphism
$\iota:\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\longrightarrow\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$.
It induces a complete bar-cobar adjunction relating
$u\mathcal{EE}_{\infty}$-coalgebras and qp-complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras:

${u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}~{}.}$$\scriptstyle{\widehat{\Omega}_{\iota}}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota}}$${\dashv}$

Using this adjunction, one can transfer the model structure on
$u\mathcal{EE}_{\infty}$-coalgebras to the category of qp-complete curved
absolute $\mathcal{L}^{\pi}_{\infty}$-algebras.

######  Theorem 1.14.

There is a model structure on the category of qp-complete curved absolute
partition $\mathcal{L}^{\pi}_{\infty}$-algebras right-transferred along the
complete bar-cobar adjunction

${u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}}$$\scriptstyle{\widehat{\Omega}_{\iota}}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota}}$${\dashv}$

where

  1. (1)

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

  2. (2)

the class of fibrations is given is given by morphisms $f$ which are degree-
wise epimorphisms,

  3. (3)

and the class of cofibrations is given by left lifting property with respect
to acyclic fibrations.

Furthermore, this adjunction is a Quillen equivalence.

###### Proof.

Follows from [GRiL23a, Section 6 and 7], using the quasi-planarity of
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$. ∎

There is a particular kind of weak-equivalences between qp-complete curved
absolute partition $\mathcal{L}^{\pi}_{\infty}$-algebras which admits an easy
description.

######  Definition 1.15 (Graded quasi-isomorphisms).

Let $f:\mathfrak{g}\longrightarrow\mathfrak{h}$ be a morphism of qp-complete
curved absolute partition $\mathcal{L}^{\pi}_{\infty}$-algebras. It is a
graded quasi-isomorphism if in induces a quasi-isomorphism

$\mathrm{gr}_{\delta}(f):\mathrm{gr}_{\delta}\mathfrak{g}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{gr}_{\delta}\mathfrak{h}$

between the associated graded, for all $\delta\geq 0$.

######  Proposition 1.16.

Let $f:\mathfrak{g}\longrightarrow\mathfrak{h}$ be a graded quasi-isomorphism
between two qp-complete curved absolute partition
$\mathcal{L}^{\pi}_{\infty}$-algebras. Then

$\widehat{\mathrm{B}}_{\iota}(f):\widehat{\mathrm{B}}_{\iota}(\mathfrak{g})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\widehat{\mathrm{B}}_{\iota}(\mathfrak{h})$

is a quasi-isomorphism and $f$ is a weak-equivalence of qp-complete curved
absolute partition $\mathcal{L}^{\pi}_{\infty}$-algebras.

###### Proof.

Follows from [GRiL23a, Section 6]. ∎

###  1.3. Monoidal structures and convolution algebras

The category of $u\mathcal{EE}_{\infty}$-coalgebras is admits a monoidal
structure, whose product is given by the underlying tensor product of dg
modules. This monoidal structure is compatible with the model structure and is
in fact closed. The enrichment of $u\mathcal{EE}_{\infty}$-coalgebras over
themselves is compatible with the construction of convolution curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras. These constructions are based on the
results of [Gri22, Gri23], and will be treated with more detail in the
positive characteristic case in [GRiL24]. See [RiL22b, Section 1] for their
analogue in characteristic zero.

######  Proposition 1.17.

The category of $u\mathcal{EE}_{\infty}$-coalgebras forms a monoidal model
category, where the monoidal product is given by tensor product.

###### Proof.

The dg operad $\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ is isomorphic to
the cellular chains of the Boardman-Vogt construction of the simplicial
Barratt-Eccles operad. This implies that it inherits a Hopf structure.
Finally, the tensor product trivially satisfies the compatibility conditions
of [Hov99, Chapter 4], that is, the pushout-product and the unit axioms. ∎

This monoidal category is biclosed, where the right adjoint to the tensor
product is given by an internal hom bifunctor

$\\{-,-\\}:\left(u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}\right)^{\mathsf{op}}\times
u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}\longrightarrow
u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}~{}.$

For any triple of $u\mathcal{EE}_{\infty}$-coalgebras $C,D,E$, there are
isomorphisms

$\mathrm{Hom}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}(C\otimes
D,E)\cong\mathrm{Hom}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}(C,\\{D,E\\})\cong\mathrm{Hom}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}(D,\\{C,E\\})~{},$

which are natural in $C,D$ and $E$. Finally, given a
$u\mathcal{EE}_{\infty}$-coalgebra $C$ and a curved absolute partition
$\mathcal{L}^{\pi}_{\infty}$-algebra $\mathfrak{g}$, there is a convolution
curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra structure on the pdg
module of graded morphisms $(\mathrm{hom}(C,\mathfrak{g}),\partial)$. The
internal hom and the convolution algebra are compatible in the following
sense.

######  Theorem 1.18 ([Gri23]).

Let $C$ be a $u\mathcal{EE}_{\infty}$-coalgebra and let $\mathfrak{g}$ be a
curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra. There is a natural
isomorphism of $u\mathcal{EE}_{\infty}$-coalgebras

$\left\\{C,\widehat{\mathrm{B}}_{\iota}(\mathfrak{g})\right\\}\cong\widehat{\mathrm{B}}_{\iota}\left(\mathrm{hom}(C,\mathfrak{g})\right)~{},$

where $\mathrm{hom}(C,\mathfrak{g})$ denotes the convolution curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra of $C$ and $\mathfrak{g}$.

##  2\. The integration theory

In section, we develop the integration theory of curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras. We show that the integration functor is
well-behaved in an appropriate sense and describe it combinatorially. Finally,
we explain how it allows us to present formal moduli problems in certain
cases.

###  2.1. Cellular chain functor

Consider the $\mathcal{E}$-coalgebra structure on the cellular chain functor
given by the formulae of [BF04].

######  Remark 2.1.

Since $C^{c}(\Delta^{\bullet})$ is a non-conilpotent $\mathcal{E}$-coalgebra,
this structure cannot be encoded by a cooperad.

######  Theorem 2.2 ([BF04]).

There is a Quillen adjunction

${\mathsf{sSet}}$${\mathcal{E}\text{-}\mathsf{coalg}~{},}$$\scriptstyle{C^{c}(-)}$$\scriptstyle{\overline{\mathrm{R}}}$${\dashv}$

where $C^{c}(-)$ denotes the cellular chain functor endowed with a
$\mathcal{E}$-coalgebra structure and where the right adjoint is given by

$\overline{\mathrm{R}}(C)_{\bullet}\coloneqq\mathrm{Hom}_{\mathcal{E}_{\infty}\text{-}\mathsf{cog}}(C^{c}(\Delta^{\bullet}),C)~{}.$

###### Proof.

The key point is constructing the functorial $\mathcal{E}$-coalgebra structure
on $C^{c}(-)$, which is done in [BF04]. Using the cylinder object argument, it
is possible to show that $\mathcal{E}$-coalgebras admit a left transferred
model category structure, analogous to the one in Proposition 1.13. Then it is
immediate that $C^{c}(-)$ is a left Quillen functor. ∎

The counit morphism
$\epsilon:\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{E}$
induces a Quillen adjunction:

${\mathcal{E}\text{-}\mathsf{coalg}}$${u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}~{}.}$$\scriptstyle{\mathrm{Res}_{\epsilon}}$$\scriptstyle{\mathrm{Coind}_{\epsilon}}$${\dashv}$

By composing both Quillen adjunctions, we get the following Quillen
adjunction:

${\mathsf{sSet}}$${u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}~{},}$$\scriptstyle{\mathrm{Res}_{\epsilon}~{}\circ~{}C^{c}(-)}$$\scriptstyle{\overline{\mathcal{R}}}$${\dashv}$

The right adjoint is given, for a $u\mathcal{EE}_{\infty}$-coalgebra $C$, by

$\overline{\mathcal{R}}(C)_{\bullet}\coloneqq\mathrm{Hom}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{cog}}\left(\mathrm{Res}_{\epsilon}C^{c}(\Delta^{\bullet}),C\right)\cong\mathrm{Hom}_{\mathcal{E}\text{-}\mathsf{cog}}\left(C^{c}(\Delta^{\bullet}),\mathrm{Coind}_{\epsilon}C\right)~{}.$

######  Remark 2.3.

We embed the cellular chains functor $C^{c}(-)$ inside the bigger category of
$u\mathcal{EE}_{\infty}$-coalgebras since their homotopy theory is better
behaved than the homotopy theory of $\mathcal{E}$-coalgebras. It is not known
to us if the quasi-isomorphism
$\epsilon:\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{E}$
induces a Quillen equivalence since $\mathcal{E}$ is not cofibrant as a dg
operad.

######  Notation 2.4.

Since the inclusion functor $\mathrm{Res}_{\epsilon}$ do not change the
underlying dg module nor the coalgebraic structure of the cellular chain
functor $C^{c}(-)$, we denote the composition
$\mathrm{Res}_{\epsilon}C^{c}(-)$ simply by $C^{c}(-)$ from now on.

######  Theorem 2.5.

There is a commuting triangle

${\hskip
12.0ptu\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{sSet}}$${\hskip
36.0pt\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\textsf{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}~{},}$$\scriptstyle{\widehat{\Omega}_{\iota}}$$\scriptstyle{\overline{\mathcal{R}}}$$\scriptstyle{C^{c}_{*}(-)}$$\scriptstyle{\mathcal{L}}$${\dashv}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota}}$$\scriptstyle{\mathcal{R}}$${\dashv}$${\dashv}$

made of Quillen adjunctions.

###### Proof.

Follows directly from the above, together with Theorem 1.14. ∎

###  2.2. The integration functor

The functor $\mathcal{R}$ in the triangle above is the integration functor we
were looking for. It is the positive characteristic analogue of the functor
defined in [RiL22b], which generalizes to the curved absolute case the
integration functor defined by E. Getzler in [Get09]. See also [RNV20] for an
operadic construction of Getzler’s functor.

######  Theorem 2.6.

  1. (1)

For any qp-complete curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra
$\mathfrak{g}$, the simplicial set $\mathcal{R}(\mathfrak{g})$ is a Kan
complex.

  2. (2)

Let $f:\mathfrak{g}\twoheadrightarrow\mathfrak{h}$ be an epimorphism of qp-
complete curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras. Then

$\mathcal{R}(f):\mathcal{R}(\mathfrak{g})\twoheadrightarrow\mathcal{R}(\mathfrak{h})$

is a fibrations of simplicial sets.

  3. (3)

The functor $\mathcal{R}$ preserves weak-equivalences. In particular, it sends
any graded quasi-isomorphism
$f:\mathfrak{g}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{h}$
of qp-complete curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras to a weak
homotopy equivalence of simplicial sets.

###### Proof.

The functor $\mathcal{R}$ is a right Quillen functor. ∎

######  Remark 2.7.

The above theorem provides us with the positive characteristic analogues of
the main properties that integration functors satisfy. In particular, the last
point can be interpreted as a Goldman–Millson type of theorem. See [RiL22b,
Remark 2.13] for further explanations.

######  Corollary 2.8.

The integration functor commutes with limits of curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras and homotopy limits of qp-complete
curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras.

###### Proof.

The functor $\mathcal{R}$ is a right Quillen functor, and since every qp-
complete curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra is fibrant, it
is also equal to its derived functor. ∎

######  Remark 2.9.

Let $\mathfrak{g}$ be a qp-complete curved absolute partition
$\mathcal{L}^{\pi}_{\infty}$-algebra. There is an isomorphism

$\mathcal{R}(\mathfrak{g})_{\bullet}\cong\lim_{\delta}\mathcal{R}(\mathfrak{g}/\mathrm{W}_{\delta}\mathfrak{g})_{\bullet}~{}.$

of Kan complexes. Thus $\mathcal{R}(\mathfrak{g})$ is the (homotopy) limit of
a tower of simplicial sets obtained by integrating nilpotent curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras, see Subsection 2.4 for more on this
notion.

Non-abelian Dold-Kan correspondence. The adjunction

${\mathsf{sSet}}$${\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\textsf{-}\mathsf{alg}}$$\scriptstyle{\mathcal{L}}$$\scriptstyle{\mathcal{R}}$${\dashv}$

is a generalization of the Dold-Kan correspondence over a field of positive
characteristic. Notice that a dg module $(V,d_{V})$ is a particular example of
a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra, where the structural
morphism

$\gamma_{V}:\displaystyle\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}V^{\otimes
n}\longrightarrow V$

is the zero morphism on any non-trivial symmetric corked rooted tree. We refer
to these as abelian curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras.

######  Proposition 2.10.

Let $(V,d_{V})$ be an abelian curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra. There is an isomorphism

$\mathcal{R}(V)\cong\Gamma(V)~{},$

where $\Gamma(-)$ is the Dold–Kan functor.

###### Proof.

Notice that

$\mathcal{R}(V)\cong\mathrm{Hom}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{cog}}\left(C^{c}(\Delta^{\bullet}),\mathscr{C}(u\mathcal{EE}_{\infty})(V)\right)\cong\mathrm{Hom}_{\mathsf{dg}\text{-}\mathsf{mod}}\left(C^{c}(\Delta^{\bullet}),V\right)~{},$

simply by cofreeness, since the term of the differential of the complete bar
construction induced by the structure of $V$ is trivial. ∎

###  2.3. Maurer–Cartan elements and gauge equivalences

The $0$-simplicies of $\mathcal{R}(\mathfrak{g})$ correspond to Maurer–Cartan
elements in $\mathfrak{g}$. We give explicit formulae for the $1$-simplicies,
which correspond to gauge equivalences between Maurer–Cartan elements.

######  Proposition 2.11.

Let $\mathfrak{g}$ be a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra.
There is a bijection

$\mathcal{R}(\mathfrak{g})_{0}\cong\mathcal{MC}(\mathfrak{g})~{},$

between the $0$-simplicies of $\mathcal{R}(\mathfrak{g})$ and the
Maurer–Cartan elements of $\mathfrak{g}$.

###### Proof.

The $0$-simplicies of $\mathcal{R}(\mathfrak{g})$ are determined by the
$\mathcal{E}$-coalgebra structure of $C_{*}^{c}(\\{*\\})\cong\Bbbk$. One can
check that this coalgebra structure is given by maps

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where the other decomposition maps are zero for degree reasons. Notice that

$\mathbb{N}(c_{n}^{\mathrm{id}}(a_{0},\cdots,a_{0}))=\sum_{\sigma_{0}\in\mathbb{S}_{n}}\left[\mathrm{ev}_{a_{0}}:c_{n}^{\sigma_{0}}\mapsto
a_{0}\otimes\cdots\otimes a_{0}\right]$

under the norm map isomorphism

$\mathbb{N}:\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}\longrightarrow\mathrm{Hom}_{\mathbb{S}_{n}}\left(\mathrm{B}^{\mathrm{s.a}}\mathcal{E}(n),\mathfrak{g}^{\otimes
n}\right)~{}.$

Thus

$\widehat{\Omega}_{\iota}(\Bbbk.a_{0})\cong\left(\displaystyle\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}(\Bbbk.a_{0})^{\otimes
n},d_{\mathrm{cobar}}(a_{0})=-\sum_{n\geq 0~{},~{}n\neq
1}c_{n}^{\mathrm{id}}(a_{0},\cdots,a_{0})\right)~{}.$

Therefore the data of a morphism of curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras
$\widehat{\Omega}_{\iota}(\Bbbk.a_{0})\longrightarrow\mathfrak{g}$ is
equivalent to the data of an element $\alpha$ in $\mathfrak{g}$ of degree $0$
such that

$d_{\mathfrak{g}}(\alpha)=-\gamma_{\mathfrak{g}}\left(\sum_{n\geq 0,n\neq
1}c_{n}^{\mathrm{id}}(\alpha,\cdots,\alpha)\right)~{}.$

∎

Gauge equivalences and the coalgebra structure on the interval. We define
gauge equivalences and show that they are equivalent to paths in
$\mathcal{R}(\mathfrak{g})$. This allows for a combinatorial description of
$\pi_{0}(\mathcal{R}(\mathfrak{g}))$. From the point of view of deformation
theory, $\pi_{0}(\mathcal{R}(\mathfrak{g}))$ is the set of (infinitesimal)
deformations up to equivalence. Thus if $\mathfrak{g}$ encodes a spectral
deformation problem, $\pi_{0}(\mathcal{R}(\mathfrak{g}))$ gives the classical
deformation problem associated to $\mathfrak{g}$.

Paths in $\mathcal{R}(\mathfrak{g})$ are given by the $\mathcal{E}$-coalgebra
structure on the cellular chains of the interval $C_{*}^{c}(\Delta^{1})$. The
non-counital version of this coalgebra structure was explicitly computed in
[BF04, Section 3.2], and the counital version simply amounts to adding another
group-like element. Following [BF04], we define the linear maps
$\epsilon_{s}:\mathcal{E}(n)\longrightarrow\Bbbk$ by the following assignment:

${(\sigma_{0},\cdots,\sigma_{k})}$${\left\\{\begin{array}[]{ll}\epsilon(\sigma_{0}(1),\cdots,\sigma_{s-1}(1))~{},~{}~{}(\sigma_{0}(1),\cdots,\sigma_{s-1}(1))~{}~{}\text{is
a permutation of}~{}~{}\\{1,\cdots,s\\}~{},\\\ \displaystyle
0~{}~{}\mathrm{otherwise}~{},\end{array}\right.}$

for any $s\geq 0$, where $\epsilon(\sigma_{0}(1),\cdots,\sigma_{s-1}(1))$ is
the sign of the permutation in $\mathbb{S}_{s}$.

######  Definition 2.12 (Gauge equivalences).

Let $\mathfrak{g}$ be a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra,
and let $\alpha,\beta$ be two Maurer–Cartan elements. They are gauge
equivalent $\alpha\sim\beta$ if there exists a degree $1$ element $\lambda$ in
$\mathfrak{g}$ such that

$d_{\mathfrak{g}}(\lambda)=\beta-\alpha+\gamma_{\mathfrak{g}}\left(\sum_{n\geq
2}\sum_{s+a+b=n}\sum_{\begin{subarray}{c}w\in\mathcal{E}(n)_{s-1}\\\
w=(\sigma_{0},\cdots,\sigma_{s-1})\end{subarray}}\epsilon_{s}(w)~{}c_{n}^{w}\left(\underbrace{\lambda,\cdots,\lambda}_{s},\underbrace{\alpha,\cdots,\alpha}_{a},\underbrace{\beta,\cdots,\beta}_{b}\right)\right)~{}.$

######  Remark 2.13.

If $\mathfrak{g}$ is qp-complete, the above formula can be split in several
components according to the weight of the operations.

######  Theorem 2.14.

Let $\mathfrak{g}$ be a qp-complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra. There is a canonical bijection

$\pi_{0}(\mathcal{R}(\mathfrak{g}))\cong\mathcal{MC}(\mathfrak{g})/\sim_{\mathrm{gauge}}~{}.$

###### Proof.

There are canonical bijections

$\pi_{0}(\mathcal{R}(\mathfrak{g}))\cong\mathrm{Hom}_{\mathsf{sSets}_{*}}(\\{*\\},\mathcal{R}(\mathfrak{g}))/\sim_{\mathrm{hmt}}\cong\mathrm{Hom}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}\left(C_{*}^{c}(\\{*\\}),\widehat{\mathrm{B}}_{\iota}(\mathfrak{g})\right)/\sim_{\mathrm{hmt}}~{},$

since $C_{*}^{c}\dashv\overline{\mathcal{R}}$ is a Quillen adjuntion. In order
to compute homotopies in $u\mathcal{EE}_{\infty}$-coalgebras, we need an
cylinder object. This cylinder object is provided by tensoring with the
interval object $C_{*}^{c}(\Delta^{1})$. Notice that the
$u\mathcal{EE}_{\infty}$-coalgebra structure of $C_{*}^{c}(\Delta^{1})$ is the
given by its $\mathcal{E}$-coalgebra structure.

Using the formulae of [BF04, Section 3.2], the $\mathcal{E}$-coalgebra
structure on
$C_{*}^{c}(\Delta^{1})\cong\Bbbk.a_{0}\oplus\Bbbk.a_{1}\oplus\Bbbk.a_{01}$ is
given by the following structure map:

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Finally, the data of a morphism
$h:C_{*}^{c}(\Delta^{1})\longrightarrow\widehat{\mathrm{B}}_{\iota}(\mathfrak{g})$
is equivalent to the data of a curved twisting morphism
$\iota_{h}:C_{*}^{c}(\Delta^{1})\longrightarrow\mathfrak{g}$, and this data
exactly coincides with the definition of a gauge equivalence, once we pullback
the coalgebra structure along the norm isomorphism. ∎

######  Remark 2.15.

Following [BF04, Section 3.2], one can also give an explicit description of
the $\mathcal{E}$-coalgebra structure on $C_{*}^{c}(\mathbb{S}^{n})$, where
$\mathbb{S}^{n}$ is the simplicial model for the $n$-sphere given by
$\Delta^{n}/\partial\Delta^{n}$. From this data, it is possible to retrieve an
explicit combinatorial description of the higher homotopy groups
$\pi_{n}(\mathcal{R}(\mathfrak{g}),\alpha)$ at a given Maurer–Cartan element.
Nevertheless, and unlike in the characteristic zero case, this combinatorial
description does not seem to be linked to a analogue of Berglund’s theorem in
[Ber15]. Understanding the homotopy groups of $\mathcal{R}(\mathfrak{g})$ via
$\mathfrak{g}$ will be the subject of future work.

###  2.4. Nilpotent partition $\mathcal{L}^{\pi}_{\infty}$-algebras

The goal of this sections is to give particular examples of specific objects
which can be integrated using the functor $\mathcal{R}$. From the point of
view of higher Lie theory, nilpotent partition
$\mathcal{L}^{\pi}_{\infty}$-algebras are of particular interest. They are the
positive characteristic analogues of the nilpotent
$\mathcal{L}_{\infty}$-algebras considered in [Get09].

Let $\mathfrak{g}$ be a curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebra.
It is an absolute $\mathcal{L}^{\pi}_{\infty}$-algebra if the curvature term
$l_{0}:\Bbbk\longrightarrow\mathfrak{g}$ is zero. Absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras are algebras over the conilpotent dg
cooperad $\mathrm{B}\mathcal{E}^{\mathrm{nu}}$, where
$\mathcal{E}^{\mathrm{nu}}$ is the non-unital version of the Barratt-Eccles
operad. Their structural morphism

$\gamma_{\mathfrak{g}}:\prod_{n\geq
1}\Omega(\mathcal{E}^{\mathrm{nu}})^{*}(n)\otimes_{\mathbb{S}_{n}}~{}\mathfrak{g}^{\otimes
n}\longrightarrow\mathfrak{g}$

can also be made explicit, like in Subsection 1.1. It associates to any formal
power series of symmetric rooted trees (without corks) labelled by elements of
$\mathfrak{g}$ a well-defined image in $\mathfrak{g}$, and satisfies analogue
axioms. Theses objects also admit a canonical quasi-planar filtration, again
given by the degree of the symmetric rooted trees.

######  Definition 2.16 (Partition $\mathcal{L}_{\infty}$-algebra).

A $\mathcal{L}^{\pi}_{\infty}$-algebra
$(\mathfrak{h},\gamma_{\mathfrak{h}},d_{\mathfrak{h}})$ amounts to the data of
a dg module $(\mathfrak{h},d_{\mathfrak{h}})$ together with a dg
$\Omega(\mathcal{E}^{\mathrm{nu}})^{*}$-algebra structure

$\gamma_{\mathfrak{h}}:\bigoplus_{n\geq
1}\Omega(\mathcal{E}^{\mathrm{nu}})^{*}(n)\otimes_{\mathbb{S}_{n}}~{}\mathfrak{h}^{\otimes
n}\longrightarrow\mathfrak{h}~{}.$

The data of the structural morphism $\gamma_{\mathfrak{h}}$ is equivalent to
the data of a family of elementary operations

$\left\\{l_{n}^{(\sigma_{0},\cdots,\sigma_{r})}\coloneqq\gamma_{\mathfrak{h}}\left(c_{n}^{(\sigma_{0},\cdots,\sigma_{r})}(-,\cdots,-)\right):\mathfrak{h}^{\otimes
n}\longrightarrow\mathfrak{h}\right\\}$

for $n\geq 2$, and every $r+1$-tuple of distinct permutations in
$\mathbb{S}_{n}$, for $r\geq 0$. We will give a description of the relations
that these elementary operations satisfy in Appendix A.

######  Remark 2.17.

The category of $\mathcal{L}^{\pi}_{\infty}$-algebras admits a semi-model
category structure transferred from dg modules, where weak-equivalences are
given by quasi-isomorphisms and fibrations by degree-wise surjections. Its
underlying $\infty$-category is equivalent to the $\infty$-category of
$\mathbb{E}_{\infty}$-formal moduli problems. This follows from the results of
[BM19] combined with those of [BCN21]. There is also a direct proof in given
in [GRiL23b].

######  Lemma 2.18.

There is an adjunction

${\mathcal{L}^{\pi}_{\infty}\text{-}\mathsf{alg}}$${~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\text{-}\mathsf{alg}}$$\scriptstyle{\mathrm{Ab}}$$\scriptstyle{\mathrm{Res}}$${\dashv}$

between the category of absolute $\mathcal{L}^{\pi}_{\infty}$-algebras and the
category of $\mathcal{L}^{\pi}_{\infty}$-algebras.

###### Proof.

It is induced by the canonical morphism of monads

$\iota:\bigoplus_{n\geq
1}\Omega(\mathcal{E}^{\mathrm{nu}})^{*}(n)\otimes_{\mathbb{S}_{n}}(-)^{\otimes
n}\longrightarrow\prod_{n\geq
1}\Omega(\mathcal{E}^{\mathrm{nu}})^{*}(n)\otimes_{\mathbb{S}_{n}}~{}(-)^{\otimes
n}~{},$

given by the inclusion. The restriction along this morphism preserves all
limits and both categories are presentable. ∎

######  Definition 2.19 (Nilpotent partition $\mathcal{L}_{\infty}$-algebra).

Let $(\mathfrak{h},\gamma_{\mathfrak{h}},d_{\mathfrak{h}})$ be a
$\mathcal{L}^{\pi}_{\infty}$-algebra. It is nilpotent if the structural
morphism

$\gamma_{\mathfrak{h}}:\bigoplus_{n\geq
1}\Omega(\mathcal{E}^{\mathrm{nu}})^{*}(n)\otimes_{\mathbb{S}_{n}}~{}\mathfrak{h}^{\otimes
n}\longrightarrow\mathfrak{h}~{}.$

factors through

$\gamma_{\mathfrak{h}}:\bigoplus_{n\geq
1}^{k}\left(\Omega(\mathcal{E}^{\mathrm{nu}})^{*}\right)^{\leq\delta}(n)\otimes_{\mathbb{S}_{n}}\mathfrak{h}^{\otimes
n}\longrightarrow\mathfrak{h}$

for some $k\geq 1$ and some $\delta\geq 1$, where
$\left(\Omega(\mathcal{E}^{\mathrm{nu}})^{*}\right)^{\leq\delta}$ is given by
symmetric rooted trees of degree $\leq\delta$.

######  Remark 2.20.

One can define a lower central series for partition
$\mathcal{L}_{\infty}$-algebras, in an analogue way to [Get09, Definition
4.2]. Being arity-nilpotent amounts to this lower central series terminating.
Nevertheless, arity-nilpotency does not imply weight-nilpotency in this case,
as there are operations of arbitrarily high weight in each arity.

######  Proposition 2.21.

Any nilpotent partition $\mathcal{L}_{\infty}$-algebra is a qp-complete
absolute partition $\mathcal{L}_{\infty}$-algebra.

###### Proof.

One can extend the structural map to the infinite product by zero, and the
conditions imposed on an absolute partition $\mathcal{L}_{\infty}$-algebra are
automatically satisfied. One can check that the canonical qp-filtration of a
nilpotent partition $\mathcal{L}_{\infty}$-algebra is eventually zero since it
is nilpotent. Therefore it is qp-complete. ∎

######  Remark 2.22.

One can show that the restriction functor of Lemma 2.18 is fully faithful on
nilpotent partition $\mathcal{L}_{\infty}$-algebra, and that the adjunction is
an equivalence on nilpotent objects.

######  Remark 2.23.

Since qp-complete absolute partition $\mathcal{L}_{\infty}$-algebras admit all
limits, any limit of nilpotent partition $\mathcal{L}_{\infty}$-algebras is
again a qp-complete absolute partition $\mathcal{L}_{\infty}$-algebra. In
particular, any pro-nilpotent partition $\mathcal{L}_{\infty}$-algebras is.
Notice that these limits are computed in the underlying category. Furthermore,
since the functor $\mathcal{R}$ preserves limits, the image of a limit of
nilpotent partition $\mathcal{L}_{\infty}$-algebras by the functor
$\mathcal{R}$ will be the limit of their images.

###  2.5. Formal moduli problems and integration theory

The goal of this section is to show that the integration functor $\mathcal{R}$
allows us to recover the formal moduli problem associated to an algebra, under
certain homotopy completeness hypothesis. We first develop the general case,
and then specify the results for formal moduli problems of
$\mathbb{E}_{\infty}$-algebras.

Let $\mathcal{C}$ be a conilpotent dg cooperad. There is an adjunction

${\mathsf{dg}~{}\mathcal{C}^{*}\text{-}\mathsf{alg}}$${\mathsf{dg}~{}\mathcal{C}\text{-}\mathsf{alg}}$$\scriptstyle{\mathrm{Ab}}$$\scriptstyle{\mathrm{Res}}$${\dashv}$

between the category of dg $\mathcal{C}$-algebras and the category of dg
$\mathcal{C}^{*}$-algebras. This adjunction relates the absolute and the
standard categories of the same algebraic structure. Notice that Lemma 2.18 is
a particular instance of this phenomenon.

If $\mathcal{C}$ is quasi-planar in the sense of [GRiL23a], this adjunction is
a Quillen adjunction. Here the left-hand side is endowed with a transferred
semi-model structure from dg modules where weak-equivalences are given by
quasi-isomorphisms. And the right-hand side is endowed with a model category
structure transferred from dg $\Omega\mathcal{C}$-coalgebras along the
complete bar-cobar adjunction. Transferred weak-equivalences are in particular
quasi-isomorphisms, thus the restriction functor is right Quillen, see
[GRiL23a, Section 7.5].

######  Definition 2.24 (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$}}\,}\mathrm{Res}~{}\mathbb{L}\mathrm{Ab}(\mathfrak{g})$

is a quasi-isomorphism.

######  Remark 2.25.

This definition is extremely similar to the definition of homotopy
completeness given by J. Harper and K. Hess in [HH13] when $\mathcal{C}$ is
$1$-reduced.

In [GRiL23b], Brice Le Grignou and the author showed that if $\mathcal{C}$ is
a $0$-reduced tempered quasi-planar conilpotent dg cooperad, the
$\infty$-category of formal moduli problems over $\Omega\mathcal{C}$-algebras
is equivalent to the $\infty$-category of dg $\mathcal{C}^{*}$-algebras. The
proof boils down to showing, using the tempered condition, that cellular dg
$\mathcal{C}^{*}$-algebras are homotopy complete. This equivalence is then
induced by the following functor

${\Psi:\mathsf{dg}~{}\mathcal{C}^{*}\text{-}\mathsf{alg}~{}[\mathrm{Q.iso}^{-1}]}$${\mathsf{Fun}\left(\mathsf{Art}~{}\Omega\mathcal{C}\text{-}\mathsf{alg}~{}[\mathrm{Q.iso}^{-1}],\mathcal{S}\right)}$${\mathfrak{g}}$${\left[A\mapsto\mathrm{Map}_{\mathsf{dg}~{}\mathcal{C}^{*}\text{-}\mathsf{alg}[\mathrm{Q.iso}^{-1}]}(\mathrm{Res}~{}\widehat{\Omega}_{\iota}(\mathrm{H}_{*}(A)^{*}),\mathfrak{g})\right]~{},}$

at the $\infty$-categorical level, where
$\mathrm{Res}~{}\widehat{\Omega}_{\iota}(A^{\circ})$ is the restriction of
complete cobar construction relative to the twisting morphism
$\iota:\mathcal{C}\longrightarrow\Omega\mathcal{C}$, applied to the linear
dual of the homology of $A$ with its transferred structure. Let us give a
rectification of $\Psi(\mathfrak{g})$ using the integration functor
$\mathcal{R}$.

######  Proposition 2.26.

Let $\mathfrak{g}$ be a homotopy complete dg $\mathcal{C}^{*}$-algebra. The
functor $\Psi(\mathfrak{g})$ is induced by the following functor

${\psi(\mathfrak{g}):\mathsf{dg}~{}\Omega\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{sSets}}$${A}$${\mathcal{R}(\mathrm{hom}(\mathrm{H}_{*}(A)^{*},\mathbb{L}\mathrm{Ab}(\mathfrak{g})))~{},}$

where $\mathrm{hom}(-,-)$ refers to a convolution curved absolute partition
$\mathcal{L}_{\infty}$-algebra construction. Furthermore, the assignment is
functorial in homotopy complete dg $\mathcal{C}^{*}$-algebras: any quasi-
isomorphism induces a natural weak-equivalence of functors.

###### Proof.

By [GRiL23b, Theorem 5], we know that the functor $\Psi(\mathfrak{g})$ is
equivalently given by

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

when $\mathfrak{g}$ is homotopy complete. We can give explicit models for this
mapping space in terms of a convolution curved absolute partition
$\mathcal{L}_{\infty}$-algebra construction.

Indeed, since $\mathcal{C}$ is quasi-planar, the dg operad $\Omega\mathcal{C}$
admits a canonical $\mathcal{E}$-comodule structure
$\Omega\mathcal{C}\longrightarrow\Omega\mathcal{C}\otimes\mathcal{E}$, as
described in [GRiL23a, Section 2]. Applying the constructions of [Gri23], this
gives an enrichment $\left\\{-,-\right\\}$ of dg
$\Omega\mathcal{C}$-coalgebras over dg $\mathcal{E}$-coalgebras. This
enrichment is adjoint to the tensorisation of dg
$\Omega\mathcal{C}$-coalgebras by dg $\mathcal{E}$-coalgebras. Analogue to the
constructions of Subsection 1.3, given a dg $\Omega\mathcal{C}$-coalgebras $C$
and a dg $\mathcal{C}$-algebra $\mathfrak{g}$, one can construction a
convolution curved absolute partition $\mathcal{L}_{\infty}$-algebra
$\mathrm{hom}(C,\mathfrak{g})$ such that there is a natural isomorphism

$\widehat{\mathrm{B}}_{\iota}\mathrm{hom}(C,\mathfrak{g})\cong\left\\{C,\widehat{\mathrm{B}}_{\iota}(\mathfrak{g})\right\\}$

of $u\mathcal{EE}_{\infty}$-coalgebras. We refer to [GRiL24] for a more
detailed description of these convolution algebras. In our case, the mapping
space in dg $\mathcal{C}$-algebras between
$\widehat{\Omega}_{\iota}\mathrm{H}_{*}(A)^{*}$ and $\mathbb{L}\mathrm{Ab}$
can be computed as follows:

$\displaystyle\mathrm{Hom}_{\mathsf{dg}~{}\Omega\mathcal{C}\text{-}\mathsf{coalg}}\left(\mathrm{H}_{*}(A)^{*}\otimes
C_{*}^{c}(\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}\mathbb{L}\mathrm{Ab}(\mathfrak{g})\right)$
$\displaystyle\cong\mathrm{Hom}_{\mathsf{dg}~{}\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{coalg}}\left(C_{*}^{c}(\Delta^{\bullet}),\left\\{\mathrm{H}_{*}(A)^{*},\widehat{\mathrm{B}}_{\iota}\mathbb{L}\mathrm{Ab}(\mathfrak{g})\right\\}\right)$
$\displaystyle\cong\mathrm{Hom}_{\mathsf{dg}~{}\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{coalg}}\left(C_{*}^{c}(\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}\mathrm{hom}(\mathrm{H}_{*}(A)^{*},\mathbb{L}\mathrm{Ab}(\mathfrak{g}))\right)$
$\displaystyle\cong\mathcal{R}(\mathrm{hom}(\mathrm{H}_{*}(A)^{*},\mathbb{L}\mathrm{Ab}(\mathfrak{g})))~{},$

since $\mathrm{H}_{*}(A)^{*}\otimes C_{*}^{c}(\Delta^{\bullet})$ is a Reedy
cofibrant resolution of $\mathrm{H}_{*}(A)^{*}$ as a dg
$\Omega\mathcal{C}$-coalgebra.

Finally, any quasi-isomorphism
$\mathfrak{g}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{g}^{\prime}$
induces a weak-equivalence
$\mathbb{L}\mathrm{Ab}(\mathfrak{g})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathbb{L}\mathrm{Ab}(\mathfrak{g}^{\prime})$
of dg $\mathcal{C}$-algebras. And the convolution algebra construction
$\mathrm{hom}(-,-)$ preserves weak-equivalences of dg $\mathcal{C}$-algebras
in the second variable. ∎

######  Remark 2.27.

The homotopy completeness assumption needed to present the formal moduli
problem associated to a dg $\mathcal{C}^{*}$-algebra is quite natural: it is
related to the usual (pro)-nilpotency conditions present in the literature.
Let us point out that a more general class of formal moduli problems admit a
similar rectification: any formal moduli problem encoded by a dg
$\mathcal{C}^{*}$-algebra in the homotopical essential image of the functor
$\mathrm{Res}$. These correspond to formal moduli problems corepresented by a
dg $\Omega\mathcal{C}$-coalgebra.

$\mathbb{E}_{\infty}$-formal moduli problems. The case where
$\mathcal{C}=\mathrm{B}\mathcal{E}^{\mathrm{nu}}$ gives by the equivalence
between formal moduli problems of $\mathbb{E}_{\infty}$-algebras (encoded by
the particular model $\Omega\mathrm{B}\mathcal{E}^{\mathrm{nu}}$) and dg
$\Omega(\mathcal{E}^{\mathrm{nu}})^{*}$-algebras.

######  Corollary 2.28.

Let $\mathfrak{g}$ be a partition $\mathcal{L}_{\infty}$-algebra which is
homotopy complete. Then $\Psi(\mathfrak{g})$ is induced by the following
functor

${\psi(\mathfrak{g}):\mathsf{dg}~{}\Omega\mathrm{B}\mathcal{E}^{\mathrm{nu}}\text{-}\mathsf{alg}}$${\mathsf{sSets}}$${A}$${\mathcal{R}(\mathrm{hom}(\mathrm{H}_{*}(A)^{*},\mathbb{L}\mathrm{Ab}(\mathfrak{g})))~{}.}$

Furthermore, the assignment is functorial in homotopy complete partition
$\mathcal{L}_{\infty}$-algebras: any quasi-isomorphism induces a natural weak-
equivalence of functors.

###### Proof.

Follows directly from Theorem 2.26. ∎

######  Example 2.29.

Partition $\mathcal{L}_{\infty}$-algebras which admit as a quasi-free model
generated by a finite dimensional complex $V$ in degrees $\leq 0$ are homotopy
complete, as shown in [GRiL23b]. These correspond to representable formal
moduli problems, which are then presented as explained in the above
proposition.

######  Remark 2.30.

Nilpotent partition $\mathcal{L}_{\infty}$-algebras should be prime examples
of homotopy complete partition $\mathcal{L}_{\infty}$-algebras. In fact, any
homotopy (pro)-nilpotent partition $\mathcal{L}_{\infty}$-algebra should be.
This will be the subject of future work.

###  2.6. Comparison with the characteristic zero case

Our constructions work over any field $\Bbbk$. The goal of this subsection is
to show that the simplicial set obtained using the integration functor
$\mathcal{R}$ is weakly-equivalent to the one obtained by applying the
integration functor constructed in [RiL22b], when $\Bbbk$ is a characteristic
zero field. This entails that on nilpotent $\mathcal{L}_{\infty}$-algebras, it
is weakly-equivalent to Getzler’s functor of [Get09].

Over a characteristic zero field, the canonical quasi-isomorphism
$\mathcal{E}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}u\mathcal{C}om$
induces a quasi-isomorphism of cofibrant dg operads
$\varphi:u\mathcal{EE}_{\infty}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}u\mathcal{CC}_{\infty}$,
hence a Quillen equivalence

${u\mathcal{CC}_{\infty}\text{-}\mathsf{coalg}}$${u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}~{},}$$\scriptstyle{\mathrm{Res}_{\varphi}}$$\scriptstyle{\mathrm{Coind}_{\varphi}}$${\dashv}$

where $u\mathcal{CC}_{\infty}$ are precisely the models for counital
cocommutative up to homotopy coalgebras that used for integration theory in
[RiL22b].

######  Remark 2.31.

This implies that the adjunction between curved absolute
$\mathcal{L}_{\infty}$-algebras and curved absolute partition
$\mathcal{L}_{\infty}$-algebras, induced by the morphism of conilpotent curved
cooperads
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\longrightarrow\mathrm{B}^{\mathrm{s.a}}u\mathcal{C}om$,
is also a Quillen equivalence.

Let us denote by $\mathcal{R}^{\mathrm{zero}}$ the integration functor
constructed in [RiL22b].

######  Proposition 2.32.

Let $\mathfrak{g}$ be a curved absolute partition
$\mathcal{L}_{\infty}$-algebra. The following simplicial sets are naturally
weakly-equivalent

$\mathcal{R}(\mathfrak{g})\simeq\mathcal{R}^{\mathrm{zero}}(\mathrm{Res}(\mathfrak{g}))~{},$

where $\mathrm{Res}$ is the restriction functor from curved absolute partition
$\mathcal{L}_{\infty}$-algebras to curved absolute
$\mathcal{L}_{\infty}$-algebras.

###### Proof.

There is a commuting square of Quillen equivalences

${u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\textsf{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}}$${u\mathcal{CC}_{\infty}\text{-}\mathsf{coalg}}$${\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}_{\infty}\textsf{-}\mathsf{alg}^{\mathsf{comp}}~{},}$$\scriptstyle{\widehat{\Omega}^{1}_{\iota}}$$\scriptstyle{\mathrm{Coind}_{\varphi}}$$\scriptstyle{\mathrm{Res}}$$\scriptstyle{\widehat{\mathrm{B}}^{1}_{\iota}}$$\scriptstyle{\widehat{\Omega}^{2}_{\iota}}$$\scriptstyle{\mathrm{Res}_{\varphi}}$$\scriptstyle{\widehat{\mathrm{B}}^{2}_{\iota}}$$\scriptstyle{\mathrm{Ind}}$${\dashv}$${\dashv}$${\dashv}$${\dashv}$

where the right vertical adjunction is the one induced by the morphism of
conilpotent curved cooperads
$\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\longrightarrow\mathrm{B}^{\mathrm{s.a}}u\mathcal{C}om$.
We add the superscripts $1$ and $2$ to distinguish the two complete bar-cobar
adjunctions. These induce adjunctions (and equivalences) at the
$\infty$-categorical level.

It is straightforward to compute that, for the canonical
$u\mathcal{CC}_{\infty}$-coalgebra structure on $\Bbbk$,
$\mathrm{Res}_{\varphi}(\Bbbk)$ is $\Bbbk$ with its canonical
$u\mathcal{EE}_{\infty}$-coalgebra structure. Let $\mathfrak{g}$ be a curved
absolute $\mathcal{L}_{\infty}$-algebra, the simplicial set
$\mathcal{R}^{\mathrm{zero}}(\mathfrak{g})$ has the following homotopy type

$\mathcal{R}(\mathfrak{g})\simeq\mathrm{Map}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}\left(\Bbbk,\widehat{\mathrm{B}}^{2}_{\iota}(\mathfrak{g})\right)\simeq\mathrm{Map}_{u\mathcal{EE}_{\infty}\text{-}\mathsf{coalg}}\left(\mathrm{Res}_{\varphi}(\Bbbk),\widehat{\mathrm{B}}^{2}_{\iota}(\mathfrak{g})\right)$

which is naturally equivalent via adjunction to

$\mathrm{Map}_{u\mathcal{CC}_{\infty}\text{-}\mathsf{coalg}}\left(\Bbbk,\mathrm{Coind}_{\varphi}\widehat{\mathrm{B}}^{2}_{\iota}(\mathfrak{g})\right)\simeq\mathrm{Map}_{u\mathcal{CC}_{\infty}\text{-}\mathsf{coalg}}\left(\Bbbk,\widehat{\mathrm{B}}^{1}_{\iota}(\mathfrak{g})\right)\simeq\mathcal{R}^{\mathrm{zero}}(\mathrm{Res}(\mathfrak{g}))~{}.$

∎

##  3\. Lie-type models in $p$-adic homotopy theory

In this section, we build, for the first time, Quillen-like models for the
$p$-adic homotopy type of connected finite type nilpotent spaces. These models
are Koszul dual to the models constructed by Mandell in [Man01], much in the
same way that Koszul duality relates Quillen’s and Sullivan’s approaches.

###  3.1. Comparing derived units of adjunctions

From now on, we assume the base field $\Bbbk$ to be the algebraic closed field
$\overline{\mathbb{F}_{p}}$, which is the algebraic closure of
$\mathbb{F}_{p}$. Furthermore, we consider the category of simplicial sets
endowed with the $\mathbb{F}_{p}$-local model structure constructed in
[Bou75], where cofibrations are given by monomorphisms and where weak-
equivalences are given by morphisms $f:X\longrightarrow Y$ such that
$\mathrm{H}_{*}(f,\mathbb{F}_{p}):\mathrm{H}_{*}(X,\mathbb{F}_{p})\longrightarrow\mathrm{H}_{*}(Y,\mathbb{F}_{p})$
is an isomorphism.

Cochains adjunction: There is an adjunction

${\mathsf{sSet}}$${\mathsf{dg}~{}\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\textsf{-}\mathsf{alg}^{\mathsf{op}}~{},}$$\scriptstyle{C^{*}_{c}(-)}$$\scriptstyle{\mathrm{U}}$${\dashv}$

where $C^{*}_{c}(-)$ is the normalized cochains functor, endowed with a
$\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$-algebra structure, which is given
by pulling back the $\mathcal{E}$-algebra structure along the morphism
$\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\longrightarrow\mathcal{E}$.

Finite type nilpotent spaces. We recall what finite type nilpotent spaces are
and we state the main theorem of this subsection.

######  Definition 3.1 (Finite type simplicial set).

Let $X$ be a simplicial set. It is said to be of finite type if the homology
groups $\mathrm{H}_{n}(X,\mathbb{F}_{p})$ are finite dimensional for all
$n\geq 0$.

######  Definition 3.2 (Nilpotent simplicial set).

Let $X$ be a simplicial set. It is said to be nilpotent if for every
$0$-simplex $\alpha$ it satisfies the following conditions:

  1. (1)

The group $\pi_{1}(X,\alpha)$ is nilpotent.

  2. (2)

The $\pi_{1}(X,\alpha)$-module $\pi_{n}(X,\alpha)$-module is a nilpotent
$\pi_{1}(X,\alpha)$-module.

######  Theorem 3.3 ([Man01]).

Let $X$ be a connected finite type nilpotent simplicial set. There derived
unit of adjunction

$\eta_{X}:X\longrightarrow\mathbb{R}\mathrm{U}C^{*}_{c}(X)$

is an $\mathbb{F}_{p}$-equivalence.

###### Proof.

The dg operad $\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$ qualifies as an
$\mathcal{E}_{\infty}$-operad, that is, a $\mathbb{S}$-projective resolution
of the commutative operad. Thus the main theorem of [Man01] applies. ∎

######  Proposition 3.4.

Let $X$ be a finite type simplicial set. There is a weak-equivalence of
simplicial sets

$\mathcal{R}\mathcal{L}(X)\simeq\mathbb{R}\mathrm{U}C^{*}_{c}(X)~{},$

which is natural on the subcategory of finite type simplicial sets.

###### Proof.

There is an equivalence

$\mathbb{R}\mathrm{U}C^{*}_{c}(X)\simeq\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{alg}}(\Omega_{\iota}\mathrm{B}_{\iota}C_{c}^{*}(X),C_{c}^{*}(\Delta^{\bullet}))~{}.$

The following square of Quillen adjunctions

${\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{alg}^{\mathsf{op}}}$${\mathsf{curv}~{}\mathrm{B}^{\mathsf{s.a}}\mathcal{E}\text{-}\mathsf{coalg}^{\mathsf{op}}}$${\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{coalg}}$${\mathsf{curv}~{}\mathrm{B}^{\mathsf{s.a}}\mathcal{E}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}~{},}$$\scriptstyle{\mathrm{B}_{\iota}^{\mathsf{op}}}$$\scriptstyle{(-)^{\circ}}$$\scriptstyle{(-)^{*}}$$\scriptstyle{\Omega_{\iota}^{\mathsf{op}}}$$\scriptstyle{\widehat{\Omega}_{\iota}}$$\scriptstyle{(-)^{*}}$$\scriptstyle{\widehat{\mathrm{B}}_{\iota}}$$\scriptstyle{(-)^{\vee}}$${\dashv}$${\dashv}$${\dashv}$${\dashv}$

commutes. The left-hand side categories are endowed with a transferred model
structure from dg modules and the right-hand side with a transferred model
structure using each of these bar-cobar adjunctions. This follows from
[GRiL23a, Section 8], using the fact that
$\mathrm{B}^{\mathsf{s.a}}\mathcal{E}$ is a quasi-planar curved conilpotent
cooperad as shown in Lemma 1.12. Since $X$ is a finite type simplicial set,
the homology of $\Omega_{\iota}\mathrm{B}_{\iota}C_{c}^{*}(X)$ is degree-wise
finite dimensional and bounded above. Therefore the generalized Sweedler dual
$(-)^{\circ}$ is homotopically fully-faithful and we get

$\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{alg}}(\Omega_{\iota}\mathrm{B}_{\iota}C_{c}^{*}(X),C_{c}^{*}(\Delta^{\bullet}))\simeq\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{coalg}}(C_{*}^{c}(\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}\widehat{\Omega}_{\iota}C_{*}^{c}(X))~{},$

which concludes the proof. See [RiL22b, Section 3] for an analogue of this
proof in characteristic zero. ∎

######  Theorem 3.5.

Let $X$ be a connected finite type nilpotent simplicial set. The unit of
adjunction

$\eta_{X}:X\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{R}\mathcal{L}(X)$

is an $\mathbb{F}_{p}$-equivalence.

###### Proof.

Follows directly from Proposition 3.4. ∎

######  Remark 3.6.

Let $X$ connected finite type simplicial set. The unit of adjunction

$\eta_{X}:X\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{R}\mathcal{L}(X)$

is weakly-equivalent to the Bousfield-Kan $p$-completion.

######  Remark 3.7.

The above theorem should hold without the connected hypothesis as it does in
characteristic zero, since we are working with curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras.

###  3.2. Minimal models

In this subsection, we show that complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras always admit a minimal resolution, which
is unique up to isomorphism. Furthermore, as a corollary of our constructions,
we show that one can also recover the homology of a space $X$ via the homology
of the complete bar construction of its models.

######  Definition 3.8 (Minimal model).

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra. A minimal model
$(V,\varphi_{d_{V}},\psi_{V})$ amounts to the data of

  1. (1)

A graded module $V$ together with a map

$\varphi_{d_{V}}:V\longrightarrow\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)^{(\geq
1)}~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}V^{\otimes n}~{}$

which lands on elements of weight greater or equal to $1$, such that the
induced derivation $d_{V}$ satisfies

$d_{V}^{2}=l_{2}^{(12)}(l_{0},g)+l_{2}^{(21)}(l_{0},g)~{}.$

  2. (2)

A weak equivalence of curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras

$\psi_{V}:\left(\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}V^{\otimes
n},d_{V}\right)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{g}~{}.$

######  Remark 3.9.

Given a graded module $V$, the data of the derivation $d_{V}$ amounts to a
$u\mathcal{EE}_{\infty}$-coalgebra structure on $V$. Thus the data
$(V,\varphi_{d_{V}})$ above amounts to the data of a minimal
$u\mathcal{EE}_{\infty}$-coalgebra, that is, a
$u\mathcal{EE}_{\infty}$-coalgebra whose underlying differential is zero.

######  Proposition 3.10.

Let $(V,\varphi_{d_{V}},\psi_{V})$ and $(W,\varphi_{d_{W}},\psi_{W})$ be two
minimal models of a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra $\mathfrak{g}$. There is an isomorphism
of graded modules $V\cong W$.

###### Proof.

If $(V,\varphi_{d_{V}},\psi_{V})$ and $(W,\varphi_{d_{W}},\psi_{W})$ are two
minimal models of $\mathfrak{g}$, then, by definition, there exists a weak
equivalence of curved absolute $\mathcal{L}^{\pi}_{\infty}$-algebras

$\left(\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}V^{\otimes
n},d_{V}\right)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\left(\prod_{n\geq
0}\widehat{\Omega}^{\mathrm{s.a}}\mathcal{E}^{*}(n)~{}\widehat{\otimes}_{\mathbb{S}_{n}}~{}W^{\otimes
n},d_{W}\right)~{}.$

This data amounts to an $\infty$-quasi-isomorphism between the
$u\mathcal{EE}_{\infty}$-coalgebras $V$ and $W$. Since their differentials are
trivial, this data is an $\infty$-isomorphism, and in particular, an
isomorphism of graded modules between $V$ and $W$. See [GRiL23a, Section 7]
for the statements related to $\infty$-morphisms of coalgebras in positive
characteristic. ∎

######  Proposition 3.11.

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra. It admits a minimal model where the
graded module of generators is given by the homology of its complete bar
construction.

###### Proof.

Since we are working over a field $\Bbbk$, we can choose a contraction between
$\widehat{\mathrm{B}}_{\iota}\mathfrak{g}$ and its homology. We can then apply
the homotopy transfer theorem of [GRiL23a, Section 7] in order to transfer an
$\infty$-quasi-isomorphic $u\mathcal{EE}_{\infty}$-coalgebra structure onto
its homology. This gives a weak-equivalence of complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras

$\widehat{\Omega}_{\iota}\mathrm{H}_{*}\left(\widehat{\mathrm{B}}_{\iota}\mathfrak{g}\right)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\widehat{\Omega}_{\iota}\widehat{\mathrm{B}}_{\iota}\mathfrak{g}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{g}~{},$

by composing the $\infty$-quasi-isomorphism with the counit of the complete
bar-cobar adjunction. ∎

######  Definition 3.12 ($p$-adic model).

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra. It is a $p$-adic model if there exists a
simplicial set $X$ and a zig-zag of weak-equivalences of curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras

$\mathcal{L}(X)\xleftarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\cdot\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\cdots\xleftarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\cdot\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{g}~{},$

and if, furthermore, $\mathcal{R}(\mathfrak{g})$ is $\mathbb{F}_{p}$-weakly
equivalent to $X$.

######  Example 3.13.

If $\mathfrak{g}$ is weakly equivalent to $\mathcal{L}(X)$, where $X$ is a
connected finite type nilpotent simplicial set, then $\mathfrak{g}$ is a
$p$-adic model.

######  Proposition 3.14.

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra that is a $p$-adic model. The canonical
morphism of $u\mathcal{EE}_{\infty}$-algebras

$C_{*}^{c}(\mathcal{R}(\mathfrak{g}))\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\widehat{\mathrm{B}}_{\iota}\mathfrak{g}$

is a quasi-isomorphism.

###### Proof.

In this situation,
$\mathcal{L}(\mathcal{R}(\mathfrak{g}))\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathfrak{g}$
is a weak-equivalence of curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebras. Hence its transpose
$C_{*}^{c}(\mathcal{R}(\mathfrak{g}))\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\widehat{\mathrm{B}}_{\iota}\mathfrak{g}$
is a quasi-isomorphism of $u\mathcal{EE}_{\infty}$-coalgebras, since the
complete bar-cobar adjunction is a Quillen equivalence. ∎

######  Corollary 3.15.

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra which is a $p$-adic model for a
simplicial set $X$. Then its minimal model is generated by
$\mathrm{H}_{*}(X)$.

###### Proof.

By Proposition 3.14, we have that

$\mathrm{H}_{*}\left(\widehat{\mathrm{B}}_{\iota}\mathfrak{g}\right)\cong\mathrm{H}_{*}(X)~{}.$

And by Proposition 3.11, the graded module $\mathrm{H}_{*}(X)$ is the
generator of the minimal model. ∎

###  3.3. Mapping spaces

In this subsection, we construct explicit $p$-adic models for mapping spaces,
without any assumption on the source simplicial set.

Recall that, if $X$ and $Y$ are simplicial sets, there is an explicit model
for their mapping space given by

$\mathrm{Map}(X,Y)_{\bullet}\coloneqq\mathrm{Hom}_{\mathsf{sSet}}(X\times\Delta^{\bullet},Y)~{},$

which forms a Kan complex when $Y$ is so.

######  Lemma 3.16.

Let $X$ and $Y$ be two simplicial sets. There is a quasi-isomorphism

$\mathrm{EZ}:C^{c}_{*}(X)\otimes
C_{*}^{c}(Y)\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}C^{c}_{*}(X\times
Y)~{},$

of $u\mathcal{EE}_{\infty}$-coalgebras given by the Eilenberg-Zilber map,
which furthermore is natural in $X$ and $Y$.

###### Proof.

The Eilenberg-Zilber map is know to be a quasi-isomorphism of
$\mathcal{E}$-coalgebras, see [BF04], since all the structures are given by
pullback, it is a quasi-isomorphism of
$\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}$-coalgebras. ∎

######  Theorem 3.17.

Let $\mathfrak{g}$ be a complete curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra and let $X$ be a simplicial set. There is
a weak-equivalence of Kan complexes

$\mathrm{Map}(X,\mathcal{R}(\mathfrak{g}))\simeq\mathcal{R}\left(\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})\right)~{},$

which is natural in $X$ and in $\mathfrak{g}$, where
$\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})$ denotes the convolution curved
absolute $\mathcal{L}^{\pi}_{\infty}$-algebra.

###### Proof.

There is an isomorphism of simplicial sets

$\mathrm{Map}(X,\mathcal{R}(\mathfrak{g}))_{\bullet}\coloneqq\mathrm{Hom}_{\mathsf{sSet}}(X\times\Delta^{\bullet},\mathcal{R}(\mathfrak{g}))\cong\mathrm{Hom}_{\mathsf{sSet}}(C^{c}_{*}(X\times\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}(\mathfrak{g}))~{}.$

We can pre-compose by the Eilenberg-Zilber map of Lemma 3.16, giving a weak-
equivalence of simplicial sets

$\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{cog}}(C^{c}_{*}(X\times\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}(\mathfrak{g}))\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{cog}}(C^{c}_{*}(X)\otimes
C^{c}_{*}(\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}(\mathfrak{g}))~{},$

since both $C^{c}_{*}(X\times\Delta^{\bullet})$ and $C^{c}_{*}(X)\otimes
C^{c}_{*}(\Delta^{\bullet})$ are Reedy cofibrant. Let’s compute this last
simplicial set:

$\displaystyle\mathrm{Hom}_{u\mathcal{CC}_{\infty}\text{-}\mathsf{cog}}(C^{c}_{*}(X)\otimes
C^{c}_{*}(\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}(\mathfrak{g}))$
$\displaystyle\cong\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{cog}}\left(C^{c}_{*}(\Delta^{\bullet}),\left\\{C^{c}_{*}(X),\widehat{\mathrm{B}}_{\iota}(\mathfrak{g})\right\\}\right)$
$\displaystyle\cong\mathrm{Hom}_{\Omega\mathrm{B}^{\mathrm{s.a}}\mathcal{E}\text{-}\mathsf{cog}}\left(C^{c}_{*}(\Delta^{\bullet}),\widehat{\mathrm{B}}_{\iota}\left(\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})\right)\right)$
$\displaystyle\cong\mathrm{Hom}_{\mathsf{curv}~{}\mathsf{abs}~{}\mathcal{L}^{\pi}_{\infty}\text{-}\mathsf{alg}}\left(\widehat{\Omega}_{\iota}(C^{c}_{*}(\Delta^{\bullet})),\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})\right)$
$\displaystyle\cong\mathcal{R}\left(\mathrm{hom}(C^{c}_{*}(X),\mathfrak{g})\right)~{}.$

∎

######  Corollary 3.18.

Let $X$ be a simplicial set and let $Y$ be a connected finite type nilpotent
simplicial set. There is a weak-equivalence of simplicial sets

$\mathrm{Map}(X,Y_{\mathbb{F}_{p}})\simeq\mathcal{R}\left(\mathrm{hom}(C^{c}_{*}(X),\mathcal{L}(Y))\right)~{},$

where $Y_{\mathbb{F}_{p}}$ denotes the Bousfield-Kan p-completion of $Y$.

###### Proof.

There is a weak-equivalence of simplicial sets

$Y_{\mathbb{F}_{p}}\simeq\mathcal{RL}(Y)~{},$

which induces the following weak-equivalences

$\mathrm{Map}(X,Y_{\mathbb{F}_{p}})\simeq\mathrm{Map}(X,\mathcal{RL}(Y))\simeq\mathcal{R}\left(\mathrm{hom}(C^{c}_{*}(X),\mathcal{L}(Y))\right)~{}.$

∎

##  Appendix A Structural formulas

The goal of this appendix is to give formulas for the relations that the
elementary operations of a curved absolute
$\mathcal{L}^{\pi}_{\infty}$-algebra satisfy. These relations are imposed by
the partial decomposition maps of the counital partial cooperad
$\mathcal{E}^{*}$, which are hard to understand. Thus, our goal is to describe
the linear dual of the structure maps defined in [BF04, Section 1.1.3].

######  Lemma A.1.

Let $\sigma$ be an element in $\mathbb{S}_{n+k-1}$, with $n,k\geq 1$. For any
$1\leq i\leq n$, there is at most one pair $(\tau,\nu)$ with
$\tau\in\mathbb{S}_{n}$ and $\nu\in\mathbb{S}_{k}$ such that
$\tau\circ_{i}\nu=\sigma~{}.$

###### Proof.

The decomposition exists if and only if $\sigma$ stabilizes
$\\{1,\cdots,n+k-1\\}-\\{i,\cdots,i+k-1\\}$. Then the permutation $\tau$ is
determined by how $\sigma$ restricts to this subset, and therefore $\nu$ is
also completely determined. ∎

Let $\sigma$ be in $\mathbb{S}_{n+k-1}$. We say it is $(n,k,i)$-admissible if
it admits a decomposition $\Delta^{n,k}_{i}(\sigma)$, and we denote
$\sigma^{(1)}$ and $\sigma^{(2)}$ the unique pair in
$\mathbb{S}_{n}\times\mathbb{S}_{k}$ such that
$\sigma^{(1)}\circ_{i}\sigma^{(2)}=\sigma$.

Let $(\sigma_{0},\cdots,\sigma_{r})$ be an $r$-tuple of elements in
$\mathbb{S}_{n+k-1}$. Suppose that $\sigma_{j}$ is $(n,k,i)$-admissible for
all $1\leq j\leq r$, and denote $(\sigma_{0}^{(1)},\cdots,\sigma_{r}^{(1)})$
and $(\sigma_{0}^{(2)},\cdots,\sigma_{r}^{(2)})$ the $r$-tuples of
decompositions.

Let $a,b$ be two natural numbers such that $a+b=r$ and let
$\varphi:(0,0)\longrightarrow(a,b)$ be a lattice path. The $r$-tuple
$(\sigma_{0},\cdots,\sigma_{r})$ is $(\varphi,i)$-admissible if for any step
$(j,k)\longrightarrow(j+1,k)$ in $\varphi$

$\sigma_{j}^{(2)}=\sigma_{j+1}^{(2)}~{},$

and for any step $(j,k)\longrightarrow(j,k+1)$,

$\sigma_{k}^{(1)}=\sigma_{k+1}^{(1)}~{}.$

One can write $(\sigma_{0}^{(1)},\cdots,\sigma_{a}^{(1)})$ and
$(\sigma_{0}^{(2)},\cdots,\sigma_{b}^{(2)})$ by suppressing the repetitions
when $(\sigma_{0},\cdots,\sigma_{r})$ is $(\varphi,i)$-admissible for a given
lattice path $\varphi:(0,0)\longrightarrow(a,b)$.

For any lattice path $\varphi:(0,0)\longrightarrow(a,b)$, the lattice sign
$\epsilon(\varphi)$ is the sign of the shuffle permutation which takes
horizontal segments to the first place and vertical segments to the last
place, like in [BF04, Section 1.1.3].

######  Proposition A.2.

The partial decomposition map

$\Delta^{n,k}_{i}:\mathcal{E}^{*}(n+k-1)\longrightarrow\mathcal{E}^{*}(n)\otimes\mathcal{E}^{*}(k)$

is given by as follows: let $(\sigma_{0},\cdots,\sigma_{r})$ be an element in
$\mathcal{E}^{*}(n+k-1)$.

  1. (1)

If $\sigma_{j}$ is $(n,k,i)$-admissible for all $1\leq j\leq r$, then

$\Delta^{n,k}_{i}(\sigma_{0},\cdots,\sigma_{r})=\sum_{a+b=r}\sum_{\begin{subarray}{c}\varphi:(0,0)\rightarrow(a,b),\\\
(\varphi,i)\text{-}\mathrm{admissible}\end{subarray}}(-1)^{\epsilon(\varphi)}(\sigma_{0}^{(1)},\cdots,\sigma_{a}^{(1)})\otimes(\sigma_{0}^{(2)},\cdots,\sigma_{b}^{(2)})~{},$

where the second sum is taken over all lattice paths
$\varphi:(0,0)\longrightarrow(a,b)$ for which $(\sigma_{0},\cdots,\sigma_{r})$
is $(\varphi,i)$-admissible.

  2. (2)

If there exists a $\sigma_{j}$ which is not $(n,k,i)$-admissible, then the
image of $(\sigma_{0},\cdots,\sigma_{r})$ by the decomposition
$\Delta^{n,k}_{i}$ is zero.

###### Proof.

If one of the permutations $\sigma_{j}$ in $\mathbb{S}_{n+k-1}$ cannot be
obtained as the $i$-th composition of a permutation in $\mathbb{S}_{n}$ and a
permutation in $\mathbb{S}_{k}$, then it not reached by

$\circ^{n,k}_{i}:\mathcal{E}(n)\otimes\mathcal{E}(k)\longrightarrow\mathcal{E}(n+k-1)$

as defined in [BF04, Section 1.1.3]. Therefore its image by
$\Delta^{n,k}_{i}=(\circ_{i}^{n,k})^{*}$ is zero. The other part of the
formula follows from direct inspection. ∎

######  Corollary A.3.

Let $\mathfrak{g}$ be a curved absolute partition
$\mathcal{L}_{\infty}$-algebra. Its elementary operations
$\\{l_{n}^{(\sigma_{0},\cdots,\sigma_{r})}\\}$ satisfy the following
relations:

$\displaystyle\partial\left(l_{n}^{(\sigma_{0},\cdots,\sigma_{r})}\right)=\sum_{\begin{subarray}{c}\bar{\sigma}\in\mathbb{S}_{n}\\\
\bar{\sigma}\neq\sigma_{e}\end{subarray}}\sum_{j=0}^{r}(-1)^{j}l_{n}^{(\sigma_{0},\cdots,\bar{\sigma},\cdots,\sigma_{r})}+$
$\displaystyle+\sum_{\begin{subarray}{c}p+q=n+1,\\\
a+b=r\end{subarray}}\sum_{i=0}^{p}\sum_{\begin{subarray}{c}\varphi:(0,0)\rightarrow(a,b),\\\
(\varphi,i)\text{-}\mathrm{admissible}\end{subarray}}(-1)^{\epsilon(\varphi)}l_{p}^{(\sigma_{0}^{(1)},\cdots,\sigma_{a}^{(1)})}\circ_{i}l_{q}^{(\sigma_{0}^{(2)},\cdots,\sigma_{b}^{(2)})}~{},$

where $\bar{\sigma}$ is added in the $j$-spot in
$(\sigma_{0},\cdots,\bar{\sigma},\cdots,\sigma_{r})$.

###### Proof.

Follows directly from Proposition A.2. ∎

######  Remark A.4.

If $\mathfrak{g}$ is a partition $\mathcal{L}_{\infty}$-algebra, these are
precisely the relations satisfied by the structural operations.

######  Remark A.5.

There is a Python package written by A. M. Medina-Mardones where elements of
the Barratt-Eccles operad and their partial compositions are coded, see
[MM21]. These formulas can also be implemented using the aforementioned
package.

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