quasi-planar ladder $\\{\mathcal{C}^{(i)}\\}_{i\in\alpha}$. Let us prove by an
ordinal induction that $\mathcal{C}^{(i)}$ satisfies the property of the lemma
for every $i<\alpha+1$.

The conilpotent curved cooperad $\mathcal{C}^{(0)}$ is planar, therefore
$d_{\sigma}$ is zero whenever the permutation $\sigma$ is non-trivial. So we
only need to worry about the planar pre-differential $D_{\mathrm{Id}}$. Now
notice that by definition $\mathcal{C}^{(0)}$ admits no non-trivial
decomposition, therefore its pre-differential squares to zero. This proves the
property for $\mathcal{C}^{(0)}$.

Now, let us suppose that $\mathcal{C}^{(j)}$ satisfies the property for every
$j<i$. If $i$ is a limit ordinal, $\mathcal{C}^{(i)}$ also satisfies the
property. Otherwise, $i$ has the form $i=l+1$. One can notice that on
$\mathrm{gr}_{i}\mathcal{C}$, the maps $D_{\sigma}$ is zero whenever the
permutation $\sigma$ is non-trivial and that $d^{2}=D_{\mathrm{Id}}^{2}=0$.
So, for every $x\in\mathcal{C}^{(i)}(n)$, and for every two permutations
$\sigma_{1},\sigma_{2}$, $D_{(\sigma_{1},\sigma_{2})}(x)$ is in
$\mathcal{C}^{(l)}(n)$. This proves the property for $\mathcal{C}^{(i)}$, and
allows us to conclude. Finally, the result for $s^{-1}\overline{\mathcal{C}}$
is a direct consequence of the previous one. ∎

The comodule map. Let us consider the morphism of graded $\mathbb{N}$-modules

${\Delta_{\mathcal{E},\mathcal{C}}:s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}}$${\mathcal{E}_{\mathrm{pl}}\otimes
s^{-1}\overline{\mathcal{C}}}$${s^{-1}x\otimes\\{\mathrm{Id}\\}}$${\displaystyle\sum_{k\geq
0}\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho(\underline{\sigma})\otimes
D_{\underline{\sigma}}(s^{-1}x\otimes\\{\mathrm{Id}\\})\leavevmode\nobreak\
,}$

where $s^{-1}x\in s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(n)$ is a planar
generator and where the sum is taken over all sequences of permutations
$\underline{\sigma}=(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$ for
all $k\geq 0$. The formula is well-defined by Lemma 10. Moreover, notice that
$\Delta_{\mathcal{E},\mathcal{C}}$ is the restriction of the morphism of
operads

$\Omega\mathcal{C}=\mathbb{T}(s^{-1}\overline{\mathcal{C}})\xrightarrow{\Delta_{\mathcal{E},\mathcal{C}}}\mathbb{T}(\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}})\longrightarrow\mathbb{T}(\mathcal{E})\otimes\mathbb{T}(s^{-1}\overline{\mathcal{C}})\longrightarrow\mathcal{E}\otimes\Omega\mathcal{C}\leavevmode\nobreak\
,$

which we also denote by $\Delta_{\mathcal{E},\mathcal{C}}$.

######  Theorem 2.

The map

${\Delta_{\mathcal{E},\mathcal{C}}:s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}}$${\mathcal{E}_{\mathrm{pl}}\otimes
s^{-1}\overline{\mathcal{C}}}$${s^{-1}x\otimes\\{\mathrm{Id}\\}}$${\displaystyle\sum_{k\geq
0}\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho(\underline{\sigma})\otimes
D_{\underline{\sigma}}(s^{-1}x\otimes\\{\mathrm{Id}\\})\leavevmode\nobreak\
,}$

where $s^{-1}x\in s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(n)$ is a planar
generator and where the sum is taken over all sequences of permutations
$\underline{\sigma}=(\sigma_{1},\ldots,\sigma_{k})\in\mathbb{S}_{n}^{k}$ for
all $k\geq 0$, induces a morphism of dg operads

$\Omega\mathcal{C}\longrightarrow\mathcal{E}\otimes\Omega\mathcal{C}$

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

######  Remark 27.

There is an analogue right $\mathcal{E}$-comodule structure, induced by the
same formula.

###### Proof.

By definition, it induces a morphism of graded operads. By Lemma 11, Lemma 13
and Lemma 17, this induced morphism commutes with the differentials. Thus, it
is a morphism of dg operads. By Lemma 18, it defines a left
$\mathcal{E}$-comodule structure. ∎

The rest of this subsection is devoted to proving Theorem 2.

######  Lemma 11.

The diagram

${s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}}$${\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\otimes\mathbb{S}}$${\mathcal{I}}$${\mathcal{E}\otimes\mathcal{I}\otimes\mathbb{S}}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$$\scriptstyle{d_{\theta}}$$\scriptstyle{\mathrm{Id}\otimes
d_{\theta}}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$

is commutative, where $d_{\theta}$ is the term of the differential in
$\Omega\mathcal{C}$ induced by the curvature $\theta$ of $\mathcal{C}$.

###### Proof.

The diagram trivially commutes at arity $n\neq 1$. In arity $1$, the
commutation is a direct consequence of the fact that
$\Delta_{\mathcal{E},\mathcal{C}}(x\otimes\mathrm{Id})=(\mathrm{Id})\otimes
x\otimes\\{\mathrm{Id}\\}$ for any $x\in
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(1)$. ∎

######  Lemma 12.

For every natural integers, $n\geq 2,k\geq 1$, we have an equality of degree
$-1$ maps from $s^{-1}\overline{\mathcal{C}}(n)$ to $\mathcal{E}(n)\otimes
s^{-1}\overline{\mathcal{C}}(n)$:

$\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}d_{\mathcal{E}}\rho(\underline{\sigma})\otimes
D_{\underline{\sigma}}(-)=\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho((\sigma_{1},\ldots,\sigma_{k-1}))^{\sigma_{k}}\otimes
D_{\underline{\sigma}}(-)+(-1)^{k}\rho((\sigma_{2},\ldots,\sigma_{k}))\otimes
D_{\underline{\sigma}}(-)\leavevmode\nobreak\ .$

###### Proof.

We have

$d_{\mathcal{E}}(\rho(\underline{\sigma}))=\rho((\sigma_{1},\ldots,\sigma_{k-1}))^{\sigma_{k}}+\sum_{0<i<k}(-1)^{k-i}\delta_{i}(\rho(\underline{\sigma}))+(-1)^{k}\rho((\sigma_{2},\ldots,\sigma_{k}))\leavevmode\nobreak\
,$

where

$\delta_{i}(\rho(\underline{\sigma}))=\rho(\sigma_{1}\ldots,\sigma_{i}\sigma_{i+1},\ldots,\sigma_{k})\leavevmode\nobreak\
.$

This gives us

$\displaystyle\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}d_{\mathcal{E}}\rho(\overline{\sigma})\otimes
D_{\underline{\sigma}}(-)=$
$\displaystyle\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\Big{(}\rho((\sigma_{1},\ldots,\sigma_{k-1}))^{\sigma_{k}}\otimes
D_{\underline{\sigma}}(-)$
$\displaystyle+\sum_{0<i<k}(-1)^{k-i}\sum_{\underline{\sigma}}\delta_{i}(\rho(\underline{\sigma}))\otimes
D_{\underline{\sigma}}(-)\Big{)}$
$\displaystyle+(-1)^{k}\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho((\sigma_{2},\ldots,\sigma_{k}))\otimes
D_{\underline{\sigma}}(-)\leavevmode\nobreak\ .$

Since the underlying graded conilpotent cooperad of $\mathcal{C}$ is planar,
the curvature equation tells us that $d_{s^{-1}\overline{\mathcal{C}}}^{2}$ is
planar and that, for every permutation $\sigma\in\mathbb{S}_{n}$,

$\sum_{\mu\xi=\sigma}D_{\mu}D_{\xi}=\begin{cases}d_{s^{-1}\overline{\mathcal{C}}}^{2}\text{
if }\sigma=\mathrm{Id}\leavevmode\nobreak\ ,\\\ 0\text{
otherwise.}\end{cases}$

Noticing that $\delta_{i}(\rho(\underline{\sigma}))$ depends only on the
product $\sigma_{i}\sigma_{i+1}$ and not on the particular values of
$\sigma_{i}$ and $\sigma_{i+1}$, we have

$\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\delta_{i}(\rho(\underline{\sigma}))\otimes
D_{\underline{\sigma}}(-)=\sum_{\underline{\mu}=(\mu_{1},\ldots,\mu_{k-1})\in\mathbb{S}_{n}^{k-1}}\sum_{\sigma\sigma^{\prime}=\mu_{i}}\rho(\underline{\mu})\otimes
D_{(\mu_{1},\ldots,\sigma\sigma^{\prime},\ldots,\mu_{k-1})}(-)\leavevmode\nobreak\
.$

If $\mu_{i}=\mathrm{Id}$ then $\rho(\underline{\mu})=0$ and if
$\mu_{i}\neq\mathrm{Id}$ then
$D_{(\mu_{1},\ldots,\sigma\sigma^{\prime},\ldots,\mu_{k-1})}=0$. Thus

$\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\delta_{i}(\rho(\underline{\sigma}))\otimes
D_{\underline{\sigma}}(-)=0\leavevmode\nobreak\ ,$

which concludes the proof. ∎

######  Lemma 13.

The following diagram

${s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}}$${\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\otimes\mathbb{S}}$${s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\otimes\mathbb{S}}$${\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\otimes\mathbb{S}}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$$\scriptstyle{d_{s^{-1}\overline{\mathcal{C}}}}$$\scriptstyle{d_{\mathcal{E}}\otimes\mathrm{Id}+\mathrm{Id}\otimes
d_{s^{-1}\overline{\mathcal{C}}}}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$

is commutative.

###### Proof.

In arity $0$ and $1$, the commutation is straightforward to check. In arity
$n\geq 2$, we have
$d_{s^{-1}\overline{\mathcal{C}}}=\sum_{\sigma\in\mathbb{S}_{n}}D_{\sigma}$.
This gives the following equalities of maps on
$s^{-1}\overline{\mathcal{C}}(n)$

$\displaystyle\Delta_{\mathcal{E},\mathcal{C}}d_{s^{-1}\overline{\mathcal{C}}}(-)$
$\displaystyle=\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k},\leavevmode\nobreak\
\sigma\in\mathbb{S}_{n}}\rho(\underline{\sigma})^{\sigma}\otimes
D_{\underline{\sigma}}D_{\sigma}(-)\leavevmode\nobreak\ ,$
$\displaystyle(\mathrm{Id}\otimes
d_{s^{-1}\overline{\mathcal{C}}})\Delta_{\mathcal{E},\mathcal{C}}(-)$
$\displaystyle=\sum_{k\geq
0}\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k},\leavevmode\nobreak\
\sigma\in\mathbb{S}_{n}}(-1)^{k}\rho(\underline{\sigma})\otimes
D_{\sigma}D_{\underline{\sigma}}(-)\leavevmode\nobreak\ .$

It follows from Lemma 12 that

$\Delta_{\mathcal{E},\mathcal{C}}d_{s^{-1}\overline{\mathcal{C}}}(-)=(d_{\mathcal{E}}\otimes\mathrm{Id})\Delta_{\mathcal{E},\mathcal{C}}(-)+(\mathrm{Id}\otimes
d_{s^{-1}\overline{\mathcal{C}}})\Delta_{\mathcal{E},\mathcal{C}}(-)\leavevmode\nobreak\
,$

and thus that the square also commutes in arity $n\geq 2$. ∎

######  Lemma 14.

Let $p,q\geq 1$, $1\leq i\leq p$,$n=p+q-1$ and let $\sigma\in\mathbb{S}_{n}$
be a non-trivial permutation. One has the following egalites between maps from
$s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}})(n)$ to
$(\mathbb{T}_{\mathrm{pl}}^{(2)}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}))(n)\otimes\mathbb{S}_{n}$
depending on the $(p,q,i)$-admissibility of $\sigma$.

  1. (1)

If $\sigma$ is $(p,q,i)$-top admissible, that is, if there exists a unique
non-trivial permutation $\psi\in\mathbb{S}_{q}$ such that
$\sigma=\mathrm{Id}_{p}\circ_{i}\psi$, then

$\Delta_{i}^{p,q}D_{\sigma}=-(\mathrm{Id}\otimes
D_{\psi})\Delta_{i}^{p,q}\leavevmode\nobreak\ .$

  2. (2)

If $\sigma$ is $(p,q,i)$-bottom admissible, that is, if there exists a unique
non-trivial permutation $\mu\in\mathbb{S}_{p}$ such that
$\sigma=\mu\circ_{\mu^{-1}(i)}\mathrm{Id}_{q}$, then

$\Delta_{i}^{p,q}D_{\sigma}=-(D_{\mu}\otimes\mathrm{Id})\Delta_{\mu^{-1}(i)}^{p,q}\leavevmode\nobreak\
.$

  3. (3)

If $\sigma$ is $(p,q,i)$-non admissible, then

$\Delta_{i}^{p,q}D_{\sigma}=0\leavevmode\nobreak\ .$

###### Proof.

Since the derivation of $\Omega\mathcal{C}$ squares to zero, one has the
following equation

$d_{\Delta}d_{s^{-1}\overline{\mathcal{C}}}+(d_{s^{-1}\overline{\mathcal{C}}}\otimes\mathrm{Id}+\mathrm{Id}\otimes
d_{s^{-1}\overline{\mathcal{C}}})d_{\Delta}=0$

between maps from $s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}$ to
$(\mathbb{T}^{(2)}(s^{-1}\overline{\mathcal{C}}))\simeq(\mathbb{T}_{\mathrm{pl}}^{(2)}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}))\otimes\mathbb{S}$.
In arity $n$, we denote $\pi_{p,q,i,\sigma}$ the projection

$(\mathbb{T}_{\mathrm{pl}}^{(2)}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}))(n)\otimes\mathbb{S}_{n}\twoheadrightarrow\mathbb{T}^{(2)}_{\mathrm{pl},p,q,i}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}})\otimes\\{\sigma\\}\leavevmode\nobreak\
,$

where
$\mathbb{T}^{(2)}_{\mathrm{pl},p,q,i}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}})$
denotes the planar trees labelled by
$s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}$ that have two nodes: the root
node with $p$ input and another node with $q$ inputs, where the top node is
attached to the $i^{th}$ input of the root node. One has

$\pi_{p,q,i,\sigma}d_{\Delta}d_{s^{-1}\overline{\mathcal{C}}}=\Delta_{i}^{p,q}D_{\sigma}.$

Moreover,

$\pi_{p,q,i,\sigma}(d_{s^{-1}\overline{\mathcal{C}}}\otimes\mathrm{Id})d_{\Delta}=\begin{cases}(D_{\mu}\otimes\mathrm{Id})\Delta_{\mu^{-1}(i)}^{p,q}\text{
if $\sigma$ is bottom admissible;}\\\ 0\text{ otherwise.}\end{cases}$

and

$\pi_{p,q,i,\sigma}(\mathrm{Id}\otimes
d_{s^{-1}\overline{\mathcal{C}}})d_{\Delta}=\begin{cases}(\mathrm{Id}\otimes
D_{\psi})\Delta_{i}^{p,q}\text{ if $\sigma$ is top admissible;}\\\ 0\text{
otherwise.}\end{cases}$

Then, the three equations of the proposition are just the three possible cases
(in terms of $(p,q,i)$-admissibility of $\sigma$) of the equation

$\pi_{p,q,i,\sigma}(d_{\Delta}d_{s^{-1}\overline{\mathcal{C}}}+(d_{s^{-1}\overline{\mathcal{C}}}\otimes\mathrm{Id}+\mathrm{Id}\otimes
d_{s^{-1}\overline{\mathcal{C}}})d_{\Delta})=0\leavevmode\nobreak\ .$

∎

######  Lemma 15.

Let us denote by $u$ the generator element of
$u\mathcal{A}ss(0)=\mathcal{E}(0)$. Let $p\geq 1$, let
$\sigma\in\mathbb{S}_{p}$ be a non-trivial permutation and let $1\leq i\leq
p+1$. Then,

$\sum_{\mu\leavevmode\nobreak\
\circ_{\mu^{-1}(i)}u=\sigma}(D_{\mu}\otimes\mathrm{Id})\Delta_{\mu^{-1}(i)}^{p,0}=-\Delta_{i}^{p,0}D_{\sigma}\leavevmode\nobreak\
,$

where the sum ranges over all $\mu$ in $\mathbb{S}_{p+1}$ such that
$\mu\leavevmode\nobreak\ \circ_{\mu^{-1}(i)}u=\sigma$.

###### Proof.

It follows from the same arguments as those used in the proof of Lemma 14. ∎

######  Lemma 16.

Let $k\geq 0$, $p\geq 1$, $q\geq 0$, $1\leq i\leq p$, $n=p+q-1$ and let
$\underline{\sigma}\in\mathbb{S}_{n}^{k}$ be a sequence of non-trivial
permutations. Then

$\Delta_{i}^{p,q}D_{\underline{\sigma}}(-)=\sum_{\underline{\mu}\circ_{i,\phi}\underline{\psi}=\underline{\sigma}}(-1)^{k}\epsilon(\phi)(D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta_{\underline{\mu}^{-1}(i)}^{p,q}(-)\leavevmode\nobreak\
,$

where the sum ranges over all $\underline{\mu}\in\mathbb{S}_{p}^{a}$ and all
$\underline{\psi}\in\mathbb{S}_{q}^{b}$, with $a\geq 1$, $b\geq 1$ and
$a+b=k$, such that there exists a $(a,b)$-shuffle $\phi$ such that
$\underline{\mu}\circ_{i,\phi}\underline{\psi}=\underline{\sigma}$, and where
$\underline{\mu}^{-1}(i)=\mu_{a}^{-1}\cdots\mu_{1}^{-1}(i)$.

###### Proof.

First, in the case where $q\geq 1$, a straightforward induction using Lemma 14
shows that:

  1. (1)

if $\underline{\sigma}$ is $(p,q,i)$-admissible, that is, if there exist an
unique shuffle permutation $\phi\in\mathbb{S}_{k}$ (which is an
$(a,b)$-shuffle) and an unique pair of sequences of permutations
$\underline{\mu}\in\mathbb{S}_{p}^{a},\underline{\psi}\in\mathbb{S}_{q}^{b}$
such that

$\underline{\sigma}=\underline{\mu}\circ_{i,\phi}\underline{\psi}\leavevmode\nobreak\
,$

then

$\Delta_{i}^{p,q}D_{\underline{\sigma}}(-)=(-1)^{k}\epsilon(\phi)(D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta_{j}^{p,q}(-)\leavevmode\nobreak\ ,$

where $j=\mu_{a}^{-1}\cdots\mu_{1}^{-1}(i)$.

  2. (2)

If $\underline{\sigma}$ is not $(p,q,i)$-admissible, then

$\Delta_{i}^{p,q}D_{\underline{\sigma}}(-)=0\leavevmode\nobreak\ .$

In the case where $q=0$, the result follows from a similar induction using
Lemma 15. ∎

######  Lemma 17.

The following diagram

${s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}}$${\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\otimes\mathbb{S}}$${\mathbb{T}^{(2)}_{\mathrm{pl}}(s^{-1}\mathcal{C}_{\mathrm{pl}})}$${\mathbb{T}^{(2)}_{\mathrm{pl}}(\mathcal{E}\otimes
s^{-1}\mathcal{C})}$${\mathcal{E}\otimes\mathbb{T}^{(2)}_{\mathrm{pl}}(s^{-1}\mathcal{C}_{\mathrm{pl}})\otimes\mathbb{S}.}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$$\scriptstyle{d_{\Delta}}$$\scriptstyle{\mathrm{Id}\otimes
d_{\Delta}}$$\scriptstyle{\mathbb{T}^{(2)}_{\mathrm{pl}}(\Delta_{\mathcal{E},\mathcal{C}})}$$\scriptstyle{\gamma}$

is commutative, where $\gamma$ is the composition within the operad
$\mathcal{E}\otimes\Omega\mathcal{C}$.

###### Proof.

First let $p\geq 1,q\geq 0$, $1\leq i\leq p$ and $n=p+q-1$. As in the proof of
Lemma 14, we denote
$\mathbb{T}^{(2)}_{\mathrm{pl},p,q,i}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}})$
the planar trees labelled by $s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}$ that
have two nodes: the root node with $p$ inputs and another node with $q$
inputs, which is attached to the $i^{th}$ input of the root node. Let us prove
that, in arity $n$, the composition of the two maps

$s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}(n)\rightrightarrows\mathcal{E}(n)\otimes\mathbb{T}^{(2)}_{\mathrm{pl}}(s^{-1}\mathcal{C}_{\mathrm{pl}})(n)\otimes\mathbb{S}_{n}$

with the projection

$\mathrm{Id}\otimes\pi_{p,q,i}\otimes\mathrm{Id}:\mathcal{E}(n)\otimes\mathbb{T}^{(2)}_{\mathrm{pl}}(s^{-1}\mathcal{C}_{\mathrm{pl}})(n)\otimes\mathbb{S}_{n}\twoheadrightarrow\mathcal{E}(n)\otimes\mathbb{T}^{(2)}_{\mathrm{pl},p,q,i}(s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}})\otimes\mathbb{S}_{n}$

are equal. This amounts to show that

$\sum_{\underline{\mu}\in\mathbb{S}_{p}^{a},\underline{\psi}\in\mathbb{S}_{q}^{b}}(-1)^{|\mu||\psi|}(\rho(\underline{(}\mu))\circ_{\underline{\mu}^{-1}(i)}\rho(\underline{(}\psi)))\otimes((D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta^{p,q}_{\underline{\mu}^{-1}(i)})=(\mathrm{Id}\otimes\Delta^{p,q}_{i})\left(\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho(\underline{\sigma})\otimes
D_{\underline{\sigma}}\right)\leavevmode\nobreak\ ,$

which follows from the sequence of equalities

$\displaystyle\sum_{\underline{\mu}\in\mathbb{S}_{p}^{a},\underline{\psi}\in\mathbb{S}_{q}^{b}}(-1)^{|\mu||\psi|}(\rho(\underline{(}\mu))\circ_{\underline{\mu}^{-1}(i)}\rho(\underline{(}\psi)))\otimes((D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta^{p,q}_{\underline{\mu}^{-1}(i)})$ $\displaystyle=$
$\displaystyle\sum_{\underline{\mu}\in\mathbb{S}_{p}^{a},\underline{\psi}\in\mathbb{S}_{q}^{b},\phi\in\mathrm{Sh}(a,b)}(-1)^{\epsilon(\phi)}\rho(\underline{\mu}\circ_{i,\phi}\underline{\psi})\otimes((D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta^{p,q}_{\underline{\mu}^{-1}(i)})\quad\text{ by
Lemma \ref{lemma:inversecompinbe}}$ $\displaystyle=$
$\displaystyle\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\sum_{\underline{\mu}\circ_{i,\phi}\underline{\psi}=\underline{\sigma}}(-1)^{\epsilon(\phi)}\rho(\underline{\sigma})\otimes((D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta^{p,q}_{\underline{\mu}^{-1}(i)})$ $\displaystyle=$
$\displaystyle\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho(\underline{\sigma})\otimes\left(\sum_{\underline{\mu}\circ_{i,\phi}\underline{\psi}=\underline{\sigma}}(-1)^{\epsilon(\phi)}((D_{\underline{\mu}}\otimes
D_{\underline{\psi}})\Delta^{p,q}_{\underline{\mu}^{-1}(i)})\right)$
$\displaystyle=$
$\displaystyle\sum_{\underline{\sigma}\in\mathbb{S}_{n}^{k}}\rho(\underline{\sigma})\otimes\left((-1)^{|\sigma|}(\Delta^{p,q}_{i}D_{\underline{\sigma}})\right)\quad\text{
by Lemma \ref{propositionbeecomposition}}$ $\displaystyle=$
$\displaystyle(\mathrm{Id}\otimes\Delta^{p,q}_{i})(\sum_{\underline{\sigma}}\rho(\underline{\sigma})\otimes
D_{\underline{\sigma}})\leavevmode\nobreak\ .$

This proves the lemma. ∎

######  Lemma 18.

The following diagram commutes

${s^{-1}\overline{\mathcal{C}}}$${\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}}$${\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}}$${\mathcal{E}\otimes\mathcal{E}\otimes
s^{-1}\overline{\mathcal{C}}\leavevmode\nobreak\
,}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$$\scriptstyle{\Delta_{\mathcal{E},\mathcal{C}}}$$\scriptstyle{\Delta_{\mathcal{E}}\otimes\mathrm{Id}}$$\scriptstyle{\mathrm{Id}\otimes\Delta_{\mathcal{E},\mathcal{C}}}$

###### Proof.

This is direct consequence of Lemma 8 ∎

###  2.9. The quasi-planar filtration of quasi-planar cooperads

Let $\mathcal{P}$ be a dg operad. For all arities $m\geq 0$, remember that
there is a canonical isomorphism of graded $\mathbb{S}$-modules

${\mathcal{E}(m)\otimes\mathcal{P}(m)\cong(\mathcal{E}_{\mathrm{pl}}(m)\otimes\mathbb{S}_{m})\otimes\mathcal{P}(m)}$${(\mathcal{E}_{\mathrm{pl}}(m)\otimes\mathcal{P}(m))\otimes\mathbb{S}_{m}}$${e\otimes(p\otimes\\{\sigma\\})}$${\displaystyle(e\otimes
p^{\sigma^{-1}})\otimes\\{\sigma\\}\leavevmode\nobreak\ ,}$

where the right action of $\mathbb{S}_{m}$ on
$\mathcal{E}(m)\otimes\mathcal{P}(m)$ is diagonal, given by $(e\otimes
p)^{\sigma}=e^{\sigma}\otimes p^{\sigma}$, and where
$(\mathcal{E}_{\mathrm{pl}}(m)\otimes\mathcal{P}(m))\otimes\mathbb{S}_{m}$ is
the free right $\mathbb{S}_{m}$-module on
$\mathcal{E}_{\mathrm{pl}}(m)\otimes\mathcal{P}(m)$. This isomorphism is
described in Proposition 1.

Notice that this isomorphism commutes with $\mathrm{Id}_{\mathcal{E}}\otimes
d_{\mathcal{P}}$. This allows one to filter the graded $\Bbbk$-module
$s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})(m)$, for all $m\geq 0$, by the
degree of the elements of $\mathcal{E}_{\mathrm{pl}}(m)$:

$(\tilde{F}_{n}s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P}))(m)\coloneqq
s(\mathcal{E}_{\mathrm{pl},\leq
n-1}(m)\otimes\mathcal{P}(m))\leavevmode\nobreak\ ,\quad
n\in\mathbb{N}\leavevmode\nobreak\ ,$

where $\mathcal{E}_{\mathrm{pl},\leq n}(m)$ is the graded $\mathbb{S}$-module
that consists in elements of $\mathcal{E}_{\mathrm{pl}}(m)$ whose degree is at
most $n$

$\mathcal{E}_{\mathrm{pl},\leq
n}(m)_{k}=\begin{cases}\mathcal{E}_{\mathrm{pl}}(m)_{k}\text{ if }k\leq n;\\\
0\text{ otherwise.}\end{cases}$

One can notice that

$(\tilde{F}_{0}s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P}))(m)=0\leavevmode\nobreak\
,$

Then, one filters $s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I}$ as

$\displaystyle\tilde{F}_{0}(s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I})=0$
$\displaystyle\tilde{F}_{n}(s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I})=\tilde{F}_{n}(s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I},\quad n\geq 1.$

Thus, one can also filter
$\mathbb{T}_{\mathrm{pl}}(s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I})$ by

$\tilde{F}_{n}\mathbb{T}_{\mathrm{pl}}(s\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})(m)=\bigoplus_{t}\sum_{i_{1}+\ldots+i_{k}=n}\bigotimes_{j=1}^{k}\leavevmode\nobreak\
\tilde{F}_{i_{j}}(s(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\oplus
s^{2}\mathcal{I})(l_{j})\leavevmode\nobreak\ ,$

where the first sum is taken over the planar trees $t$ with $m$-leaves ; it
has $k$ nodes whose arities are $l_{1},\ldots,l_{k}$.

Finally, this induces a filtration
$\tilde{F}_{n}B(\mathcal{E}\otimes\mathcal{P})$ on the underlying graded
conilpotent cooperad of $B(\mathcal{E}\otimes\mathcal{P})$ that is canonically
isomorphic to
$\mathbb{T}_{\mathrm{pl}}(s\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\otimes\mathbb{S}$,
where the isomorphism is induced the above isomorphism
$\mathcal{E}\otimes\mathcal{P}\cong(\mathcal{E}_{\mathrm{pl}}\otimes\mathcal{P})\otimes\mathbb{S}$.

######  Proposition 15.

For every natural integer $n$, $\tilde{F}_{n}B(\mathcal{E}\otimes\mathcal{P})$
is a conilpotent curved subcooperad of $B(\mathcal{E}\otimes\mathcal{P})$.
Moreover, the diagram

$\tilde{F}_{0}B(\mathcal{E}\otimes\mathcal{P})\longrightarrow\tilde{F}_{1}B(\mathcal{E}\otimes\mathcal{P})\longrightarrow\cdots\longrightarrow\tilde{F}_{n}B(\mathcal{E}\otimes\mathcal{P})\longrightarrow\cdots$

is a quasi-planar $\omega$-ladder whose colimit is
$B(\mathcal{E}\otimes\mathcal{P})$.

###### Proof.

The only point that is not clear is the fact that the non-planar component of
the pre-differential of $\tilde{F}_{n+1}B(\mathcal{E}\otimes\mathcal{P})$
actually targets $\tilde{F}_{n}B(\mathcal{E}\otimes\mathcal{P})$. This follows
from the fact that this non-planar component is given by applying the non-
planar part of the differential of $\mathcal{E}$. ∎

Let $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad that is the
colimit of a quasi-planar cooperad ladder $(\mathcal{C}^{(i)})_{i\in\alpha}$.
We built in the previous subsection 2.8 a morphism of dg operads

$\Omega\mathcal{C}\longrightarrow\mathcal{E}\otimes\Omega\mathcal{C}$

whose restriction to $s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}$ factors
through

$s^{-1}\overline{\mathcal{C}}_{\mathrm{pl}}\longrightarrow\mathcal{E}_{\mathrm{pl}}\otimes
s^{-1}\overline{\mathcal{C}}.$

Subsequently, the restriction of the adjoint map $\mathcal{C}\longrightarrow
B(\mathcal{E}\otimes\Omega\mathcal{C})$ to $\mathcal{C}_{\mathrm{pl}}$ factor
through

$\mathcal{C}_{\mathrm{pl}}\longrightarrow\mathbb{T}_{\mathrm{pl}}(\mathcal{E}_{\mathrm{pl}}\otimes\Omega\mathcal{C})\leavevmode\nobreak\
.$

######  Definition 38.

Let $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad. For every
natural integer $n$, we define the quasi-planar ladder
$F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}$ of $\mathcal{C}_{\mathrm{pl}}$
as the following pullback in the category of graded conilpotent cooperads

${F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}}$${\mathcal{C}_{\mathrm{pl}}}$${\tilde{F}_{n}B(\mathcal{E}\otimes\Omega\mathcal{C})_{\mathrm{pl}}}$${B(\mathcal{E}\otimes\Omega\mathcal{C})_{\mathrm{pl}}.}$${\lrcorner}$

It is the largest sub-planar conilpotent cooperad of
$\mathcal{C}_{\mathrm{pl}}$ whose image in
$B(\mathcal{E}\otimes\Omega\mathcal{C})_{\mathrm{pl}}$ lies inside
$\tilde{F}_{n}B(\mathcal{E}\otimes\Omega\mathcal{C})_{\mathrm{pl}}$.

######  Lemma 19.

The above square that defines $F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}$
is a pullback in the category of graded $\Bbbk$-modules. In particular, the
map
$F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}\rightarrowtail\mathcal{C}_{\mathrm{pl}}$
is degree-wise injective.

###### Proof.

This is a direct consequence of the fact that the tensor product of graded
$\Bbbk$-modules preserves intersections. ∎

Notice that the diagram of graded planar conilpotent cooperads

$F_{0}^{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}\rightarrowtail
F_{1}^{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}\rightarrowtail\cdots\rightarrowtail
F_{n}^{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}\rightarrowtail\cdots$

is an $\omega$-ladder whose colimit is $\mathcal{C}_{\mathrm{pl}}$.

######  Definition 39 (Canonical quasi-planar ladder).

Let $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad. For every
natural integer $n$, we define the canonical quasi-planar ladder
$F_{n}^{\mathrm{qp}}\mathcal{C}$ of $\mathcal{C}$ as the following pullback in
the category of conilpotent curved cooperads

${F_{n}^{\mathrm{qp}}\mathcal{C}}$${\mathcal{C}}$${\tilde{F}_{n}B(\mathcal{E}\otimes\Omega\mathcal{C})}$${B(\Omega(\mathcal{C})\otimes\mathcal{E})\leavevmode\nobreak\
.}$${\lrcorner}$

######  Proposition 16.

Let $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad.

  1. (1)

There is a canonical isomorphism of graded conilpotent cooperads

$(F_{n}^{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}})\otimes\mathbb{S}\cong
F_{n}^{\mathrm{qp}}\mathcal{C}\leavevmode\nobreak\ .$

Furthermore, it is natural with respect to the inclusions
$F_{n}^{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}\rightarrowtail
F_{n+1}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}$ and
$F_{n}^{\mathrm{qp}}\mathcal{C}\rightarrowtail
F_{n+1}^{\mathrm{qp}}\mathcal{C}$.

  2. (2)

The diagram of conilpotent curved cooperads

$F_{0}^{\mathrm{qp}}\mathcal{C}\rightarrowtail
F_{1}^{\mathrm{qp}}\mathcal{C}\rightarrowtail\cdots\rightarrowtail
F_{n}^{\mathrm{qp}}\mathcal{C}\rightarrowtail\cdots$

is a quasi-planar $\omega$-ladder whose colimit is $\mathcal{C}$.

###### Proof.

Let us prove the first point: let $\mathcal{D}$ be the following pullback in
the category of conilpotent curved cooperads

${\mathcal{D}}$${\mathcal{C}}$${\tilde{F}_{n}B(\mathcal{E}\otimes\Omega\mathcal{C})}$${B(\Omega(\mathcal{C})\otimes\mathcal{E}).}$${\lrcorner}$

Both the functor $-\otimes\mathbb{S}$ from graded planar conilpotent cooperads
to graded conilpotent cooperads and the forgetful functor from pdg conilpotent
cooperads to graded conilpotent cooperads preserve limits. Thus there is a
canonical isomorphism of graded conilpotent cooperads

$(F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}})\otimes\mathbb{S}\cong\mathcal{D}.$

The map $\mathcal{D}\rightarrowtail\mathcal{C}$ is degree-wise injective.
Subsequently $\mathcal{D}$ is curved and the above square is also a pullback
in the category of conilpotent curved cooperads. Hence $\mathcal{D}\cong
F_{n}^{\mathrm{qp}}\mathcal{C}$.

The fact that the diagram forms quasi-planar ladder follows from the fact that

$\tilde{F}_{0}B(\mathcal{E}\otimes\mathcal{P})\longrightarrow\tilde{F}_{1}B(\mathcal{E}\otimes\mathcal{P})\longrightarrow\cdots\longrightarrow\tilde{F}_{n}B(\mathcal{E}\otimes\mathcal{P})\longrightarrow\cdots$

is a quasi-planar $\omega$-ladder. The fact that its colimit is $\mathcal{C}$
is a direct consequence of Lemma 19. ∎

######  Remark 28.

Notice that for a dg operad $\mathcal{P}$, we have defined two canonical
quasi-planar cooperad $\omega$-ladder on $B(\mathcal{E}\otimes\mathcal{P})$
that are $\tilde{F}_{n}B(\mathcal{E}\otimes\mathcal{P})$ and
$F_{n}^{\mathrm{qp}}B(\mathcal{E}\otimes\mathcal{P})$. With some more work,
one can check that these two quasi-planar cooperad ladders are equal. This
will be done in a future work.

######  Proposition 17.

Let $g:\mathcal{C}\longrightarrow\mathcal{D}$ be a quasi-planar morphism of
quasi-planar conilpotent curved cooperads. Then the restriction of $g$ to
$F_{n}^{\mathrm{qp}}\mathcal{C}$ factors through
$F_{n}^{\mathrm{qp}}\mathcal{D}$, and the following diagram commutes

${F_{n}^{\mathrm{qp}}\mathcal{C}}$${F_{n}^{\mathrm{qp}}\mathcal{D}}$${\mathcal{C}}$${\mathcal{D}\leavevmode\nobreak\
.}$$\scriptstyle{F_{n}^{\mathrm{qp}}g}$$\scriptstyle{g}$

###### Proof.

It is straightforward to check that any quasi-planar morphism
$g:\mathcal{C}\longrightarrow\mathcal{D}$ induces a morphism of left
$\mathcal{E}$-comodules
$\Omega(g):\Omega\mathcal{C}\longrightarrow\Omega\mathcal{D}$. Therefore it
induces a morphism of diagrams between the pullbacks that define
$F_{n}^{\mathrm{qp}}\mathcal{C}$ and $F_{n}^{\mathrm{qp}}\mathcal{D}$, which
allows us to conclude. ∎

##  3\. Algebras, coalgebras, and Bar-Cobar adjunctions

In this section, we recall the various notions of (co)algebras over a
(co)operads, as well as the bar-cobar adjunctions that interrelate them. Along
the way, we develop some of their main categorical and algebraic properties.
Finally, we review the different methods that can give a homotopical meaning
to the categories of dg (co)algebras over an dg operad. Finally, we review the
different methods that can give a homotopical meaning to the categories of dg
(co)algebras over an dg operad.

###  3.1. From $\mathbb{S}$-modules to functors

The category of dg modules is enriched and tensored in dg $\mathbb{S}$-modules
as follows.

  1. (1)

For every dg modules $X,Y$, the mapping dg $\mathbb{S}$-module is
$\mathrm{Mult}(X,Y)$ given by

$\mathrm{Mult}(X,Y)(n)\coloneqq[X^{\otimes n},Y].$

In the case where $Y=X$, we set $\mathrm{End}(X)\coloneqq\mathrm{Mult}(X,X)$.

  2. (2)

For every dg module $X$ and every dg $\mathbb{S}$-module $M$, the
tensorisation of $X$ by $M$ is just given by $M\circ X$ as dg module, where
$X$ is considered as dg $\mathbb{S}$-modules concentrated in arity zero. Thus
for every additional dg $\mathbb{S}$-module $N$, the structural map

$(N\circ M)\circ X\longrightarrow N\circ(M\circ X)$

is an isomorphism.

It is also enriched and cotensored in dg $\mathbb{S}$-modules as follows.

  1. (1)

For every dg modules $X,Y$, the mapping dg $\mathbb{S}$-module is
$\mathrm{coMult}(X,Y)$ given by

$\mathrm{coMult}(X,Y)(n)\coloneqq[X,Y^{\otimes n}].$

In the case where $Y=X$, we set
$\mathrm{coEnd}(X)\coloneqq\mathrm{coMult}(X,X)$.

  2. (2)

For every dg module $X$ and every dg $\mathbb{S}$-module $M$, the
cotensorisation of $X$ by $M$ is $X^{M}$ given by

$X^{M}=\prod_{n\geq 0}\left[M(n),X^{\otimes
n}\right]^{\mathbb{S}_{n}}\leavevmode\nobreak\ .$

For every additional dg $\mathbb{S}$-module $N$, the structural map

$\varphi_{M,N}:\left(X^{M}\right)^{N}\longrightarrow X^{N\circ M}$

is a degree-wise injection.

Similarly, dg modules are on the one hand enriched and tensored over dg
$\mathbb{N}$-module through the bifunctors

$X,Y\mapsto\mathrm{Mult}_{\mathrm{pl}}(X,Y)\coloneqq[X^{\otimes-},Y];\quad
M,X\mapsto M\circ_{\mathrm{pl}}X$

and they are on the other hand enriched and cotensored over dg
$\mathbb{N}$-module through the bifunctors

$X,Y\mapsto\mathrm{coMult}_{\mathrm{pl}}(X,Y)\coloneqq[X,Y^{\otimes-}];\quad
X,M\mapsto X^{M}\prod_{n}[M(n),X^{\otimes n}]\leavevmode\nobreak\ .$

There are canonical isomorphisms of dg $\mathbb{N}$-modules
$\mathrm{Mult}_{\mathrm{pl}}(-,-)\cong U_{\mathbb{S}}\mathrm{Mult}(-,-)$ and
$\mathrm{coMult}_{\mathrm{pl}}(-,-)\cong U_{\mathbb{S}}\mathrm{coMult}(-,-)$.
We obtain by transposing along the adjunction two canonical natural
isomorphisms

$(M\otimes\mathbb{S})\circ X\simeq M\circ_{\mathrm{pl}}X\leavevmode\nobreak\
,\quad X^{M\otimes\mathbb{S}}\simeq X^{M}\leavevmode\nobreak\ ,$

for every dg module $X$ and every dg $\mathbb{N}$-module $M$.

Let us state some categorical properties that the constructions perfomed so
far satisfy.

######  Lemma 20.

Let $M$ be a dg $\mathbb{S}$-module. The endofunctor $M\circ-$ commutes with
sifted colimits and the endofunctor $(-)^{M}$ commutes with $\beta$-filtered
colimits for a small regular cardinal $\beta$.

###### Proof.

The assertion about $M\circ-$ can be deduced from the following facts

  1. (1)

the tensor product $X\mapsto X^{\otimes n}$ commutes with sifted colimits;

  2. (2)

the functor $Y\mapsto M(n)\otimes_{\mathbb{S}}Y$ commutes with colimits for
every natural integer $n$;

  3. (3)

coproducts commute with colimits.

Now, let $\beta$ be a regular cardinal such that $\beta>\omega$ and such that
every $M(n)$, for every $n$ in $\mathbb{N}$, is $\beta$-small. Then the
endofunctor $(-)^{M}$ commutes with $\beta$-filtered colimits since

  1. (1)

the tensor product $X\mapsto X^{\otimes n}$ commutes with sifted colimits, in
particular $\beta$-filtered colimits;

  2. (2)

the construction $Y\mapsto[M(n),Y]$ commutes with $\beta$-filtered colimits;

  3. (3)

the construction $Z\mapsto Z^{\mathbb{S}_{n}}$ commutes with filtered
colimits;

  4. (4)

countable products commute with $\beta$-filtered colimits since
$\beta>\omega$.

∎

######  Lemma 21.

Let $M$ be a dg $\mathbb{S}$-module. The endofunctor $(-)^{M}$ commutes with
finite cosifted limits.

###### Proof.

This can be deduced from the following facts:

  1. (1)

the tensor product $X\mapsto X^{\otimes n}$ commutes with finite cosifted
limits,

  2. (2)

the functor $Y\mapsto[M(n),Y]^{\mathbb{S}_{n}}$ commutes, limits for every
natural integer $n$;

  3. (3)

products commute with limits.

∎

One can perform the same constructions with graded modules or pdg modules
instead of dg modules. Moreover, these constructions commute strictly with the
forgetful functors $\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\longrightarrow\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}$.

Finally, let us notice that when a dg $\mathbb{S}$-module $M$ is planar as a
graded $\mathbb{S}$-module, one has extra properties on these endofunctors.

######  Lemma 22.

Let $M$ be a dg $\mathbb{S}$-module. Let us suppose that the underlying graded
$\mathbb{S}$-module of $M$ has the form $M_{\mathrm{pl}}\otimes\mathbb{S}$.
Equivalently, $M(n)$ is a quasi-free dg $\Bbbk[\mathbb{S}_{n}]$-module for all
$n\geq 0$. Then

  1. (1)

the endofunctor $(-)^{M}$ commutes with finite sifted colimits;

  2. (2)

the endofunctor $M\circ(-)$ commutes with finite cosifted limits.

###### Proof.

  1. (1)

For the endofunctor $(-)^{M}$, it is a consequence of the fact that the tensor
product, the functors $[M_{\mathrm{pl}}(n),-]$ and products commute with
finite sifted colimits.

  2. (2)

For the endofunctir $M\circ(-)$, it is a consequence of the fact that the
tensor product, the functors $M_{\mathrm{pl}}(n)\otimes-$ and coproducts
commute with finite cosifted limits.

∎

######  Notation.

Let $f:X\longrightarrow Y$ be a map of degree $0$ and $g:X\longrightarrow Y$
be a map of degree $p$ between graded modules $X,Y$. Let us consider

$\shuffle_{n}(f,g)\coloneqq\sum_{i=0}^{n}f^{\otimes i-1}\otimes g\otimes
f^{\otimes n-i}:X^{\otimes n}\longrightarrow Y^{\otimes n}\leavevmode\nobreak\
,$

which is an $\mathbb{S}_{n}$-equivariant map of degree $p$. Let $M$ be an
graded $\mathbb{S}$-module.

  1. (1)

It induces a map of degree $p$

$\bigoplus_{n\geq
0}\mathrm{Id}_{M(n)}\otimes_{\mathbb{S}_{n}}\shuffle_{n}(f,g):\bigoplus_{n\geq
0}M(n)\otimes_{\mathbb{S}_{n}}X^{\otimes n}\longrightarrow\bigoplus_{n\geq
0}M(n)\otimes_{\mathbb{S}_{n}}Y^{\otimes n}\leavevmode\nobreak\ .$

By a slight abuse of notation, we denote this morphism by
$M\circ\shuffle(f,g)$.

  2. (2)

It induces a map of degree $p$

$\prod_{n\geq
0}[\mathrm{Id}_{M(n)},\shuffle_{n}(f,g)]^{\mathbb{S}_{n}}:\prod_{n\geq
0}[M(n),X^{\otimes n}]^{\mathbb{S}_{n}}\longrightarrow\prod_{n\geq
0}[M(n),Y^{\otimes n}]^{\mathbb{S}_{n}}$

By a slight abuse of notation, this map will be denoted by
$\shuffle(f,g)^{M}$.

###  3.2. Algebras and coalgebras over (co)operads

Let $\mathcal{P}$ be a dg operad and let $\mathcal{C}$ be a dg cooperad. We
recall the definitions of a dg $\mathcal{P}$-algebra and dg
$\mathcal{P}$-coalgebra, as well as the definitions of a dg
$\mathcal{C}$-coalgebra and of a dg $\mathcal{C}$-algebra. See [GL22, Section
3].

######  Definition 40 (dg $\mathcal{P}$-algebra).

A dg $\mathcal{P}$-algebra $A$ amounts to the data of a dg module $(A,d_{A})$
equipped with a morphism of dg operads
$\Gamma_{A}:\mathcal{P}\longrightarrow\mathrm{End}(A)$.

This data is equivalent to a structural map $\gamma_{A}:\mathcal{P}\circ
A\longrightarrow A$ that satisfies unitality and associativity conditions. In
other words, it is equivalent to the data of an algebra over the monad
$\mathcal{P}\circ-$.

######  Remark 29.

More precisely, the data of a dg $\mathcal{P}$-algebra structure on
$(A,d_{A})$ amounts to a collection of maps

$\gamma_{A}^{n}:\mathcal{P}\otimes_{\mathbb{S}_{n}}A^{\otimes
n}\longrightarrow A\leavevmode\nobreak\ ,$

for all $n\geq 0$, which satisfy compatibility conditions. Since we consider
on the left-hand side coinvariants with respect to the action of
$\mathbb{S}_{n}$, no divided power operations appear. The definition of a dg
$\mathcal{P}$-algebra encodes classical types of algebraic structures over a
positive characteristic field. For instance, dg Lie-algebras, dg commutative
algebras, dg Poisson algebras, etc.

######  Definition 41 (dg $\mathcal{P}$-coalgebra).

A dg $\mathcal{P}$-coalgebra $V$ amounts to the data of a dg module
$(V,d_{V})$ equipped with a morphism of dg operads
$\delta_{V}:\mathcal{P}\longrightarrow\mathrm{coEnd}(V)$.

This data is equivalent to a structural map
$\Delta_{V}:V\longrightarrow\displaystyle V^{\mathcal{P}}$ such that the
following diagrams commute

${V}$${V^{\mathcal{P}}}$${(V^{\mathcal{P}})^{\mathcal{P}}}$${V^{\mathcal{P}}}$${V^{\mathcal{P}\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \mathcal{P}}\leavevmode\nobreak\
,}$$\scriptstyle{\Delta_{V}}$$\scriptstyle{\Delta_{V}}$$\scriptstyle{(\Delta_{V})^{\mathrm{Id}}}$$\scriptstyle{\varphi_{\mathcal{P},\mathcal{P}}}$$\scriptstyle{(\mathrm{id})^{\gamma}}$
${V}$${V^{\mathcal{P}}}$${V\leavevmode\nobreak\
,}$$\scriptstyle{\Delta_{V}}$$\scriptstyle{\mathrm{Id}}$$\scriptstyle{(\mathrm{id})^{\eta}}$

where $\gamma:\mathcal{P}\circ\mathcal{P}\longrightarrow\mathcal{P}$ and
$\eta:\mathcal{I}\longrightarrow\mathcal{P}$ are, respectively, the
composition morphism and the unit of the dg operad $\mathcal{P}$.

######  Remark 30.

The data of a dg $\mathcal{P}$-coalgebra structure on $(V,d_{V})$ also amounts
to a collection of maps

$\Delta_{V}^{n}:V\longrightarrow[\mathcal{P}(n),V^{\otimes
n}]^{\mathbb{S}_{n}}\leavevmode\nobreak\ ,$

for all $n\geq 0$, which satisfy compatibility conditions. Since we consider
on the right-hand side invariants with respect to the action of
$\mathbb{S}_{n}$, no divided power operations appear. The definition of a dg
$\mathcal{P}$-coalgebra encodes classical types of coalgebraic structures,
without any conilpotency condition, over a positive characteristic field. For
instance, dg Lie-coalgebras, dg cocommutative coalgebras, dg Poisson
coalgebras, etc.

Even though $(-)^{\mathcal{P}}$ fails to be a comonad, the forgetful functor
from dg $\mathcal{P}$-coalgebras to dg modules is comonadic.

######  Theorem 3 ([Ane14, Theorem 2.7.11]).

The forgetful functor from dg $\mathcal{P}$-coalgebras to dg modules is
comonadic. The related comonad $L^{\mathcal{P}}(-)$ is given, for a dg module
$X$, by the following pullback square

${L^{\mathcal{P}}X}$${(X^{\mathcal{P}})^{\mathcal{P}}}$${X^{\mathcal{P}}}$${X^{\mathcal{P}\leavevmode\nobreak\
\circ\leavevmode\nobreak\
\mathcal{P}}.}$${\lrcorner}$$\scriptstyle{\varphi_{\mathcal{P},\mathcal{P}}}$$\scriptstyle{(\mathrm{id})^{\mathcal{P}}}$

in the category of dg modules, where
$\gamma:\mathcal{P}\circ\mathcal{P}\longrightarrow\mathcal{P}$ is the
composition map of $\mathcal{P}$.

The structure of a comonad on $L^{\mathcal{P}}$ is given by the decomposition
map $L^{\mathcal{P}}X\longrightarrow(X^{\mathcal{P}})^{\mathcal{P}}$, which
factors through $L^{\mathcal{P}}L^{\mathcal{P}}X$, and by the following counit
map $L^{\mathcal{P}}X\longrightarrow X^{\mathcal{P}}\longrightarrow X$.

######  Definition 42 (dg $\mathcal{C}$-coalgebra).

A dg $\mathcal{C}$-coalgebra $W$ amounts to the data of a dg module
$(W,d_{W})$ equipped with structural map
$\Delta_{W}:W\longrightarrow\mathcal{C}\circ W$ such that the following
diagrams commute

${W}$${\mathcal{C}\circ W}$${\mathcal{C}\circ
W}$${\mathcal{C}\circ\mathcal{C}\circ
W}$$\scriptstyle{\Delta_{W}}$$\scriptstyle{\Delta_{W}}$$\scriptstyle{\Delta\leavevmode\nobreak\
\circ\leavevmode\nobreak\
\mathrm{Id}}$$\scriptstyle{\mathrm{Id}\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \Delta_{W}}$ ${W}$${\mathcal{C}\circ
W}$${W.}$$\scriptstyle{\Delta_{W}}$$\scriptstyle{\mathrm{Id}}$$\scriptstyle{\epsilon\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \mathrm{Id}}$

where $\Delta:\mathcal{C}\longrightarrow\mathcal{C}\circ\mathcal{C}$ and
$\epsilon:\mathcal{C}\longrightarrow\mathcal{I}$ are, respectively, the
decompositon map and the counit map of the cooperad $\mathcal{C}$. In other
words, it is a coalgebra over the comonad $\mathcal{C}\circ-$.

######  Remark 31.

The structural map of a dg $\mathcal{C}$-coalgebra $W$

$\Delta_{W}:W\longrightarrow\bigoplus_{n\geq
0}\mathcal{C}(n)\otimes_{\mathbb{S}_{n}}W^{\otimes n}\leavevmode\nobreak\ ,$

lands on the direct sum. Therefore any element $w$ in $W$ can only be
decomposed into a finite sum. This implies that a dg $\mathcal{C}$-coalgebras
always satisfies some type conilpotency condition. Furthermore, the fact that
$\Delta_{W}$ lands on the coinvariants on the right-hand side implies that
divided power operations will appear for this type of structures. Thus the
definition of a dg $\mathcal{C}$-coalgebra encodes divided powers conilpotent
types of coalgebraic structures.

######  Definition 43 (dg $\mathcal{C}$-algebra).

A dg $\mathcal{C}$-algebra $\Lambda$ amounts to the data of a dg module
$(\Lambda,d_{\Lambda})$ equipped with a structural morphism
$\gamma_{\Lambda}:\Lambda^{\mathcal{C}}\longrightarrow\Lambda$ such that the
following diagrams commute

${(\Lambda^{\mathcal{C}})^{\mathcal{C}}}$${\Lambda^{\mathcal{C}\leavevmode\nobreak\
\circ\leavevmode\nobreak\
\mathcal{C}}}$${\Lambda^{{\mathcal{C}}}}$${\Lambda^{\mathcal{C}}}$${\Lambda}$$\scriptstyle{\varphi_{\mathcal{C},\mathcal{C}}}$$\scriptstyle{(\mathrm{Id})^{\gamma_{\Lambda}}}$$\scriptstyle{(\mathrm{Id})^{\Delta}}$$\scriptstyle{\gamma_{\Lambda}}$$\scriptstyle{\gamma_{B}}$
${\Lambda}$${\Lambda^{\mathcal{C}}}$${\Lambda}$$\scriptstyle{(\mathrm{Id})^{\epsilon}}$$\scriptstyle{\mathrm{Id}}$$\scriptstyle{\gamma_{\Lambda}}$

where $\Delta:\mathcal{C}\longrightarrow\mathcal{C}\circ\mathcal{C}$ and
$\epsilon:\mathcal{C}\longrightarrow\mathcal{I}$ are, respectively, the
decompositon morphism and the counit morphism of the cooperad $\mathcal{C}$.
In other words, it is an algebra over the monad $(-)^{\mathcal{C}}$.

######  Remark 32.

The structural map of a dg $\mathcal{C}$-algebra $\Lambda$

$\gamma_{\Lambda}:\prod_{n\geq 0}[\mathcal{C}(n),\Lambda^{\otimes
n}]^{\mathbb{S}_{n}}\longrightarrow\Lambda\leavevmode\nobreak\ ,$

comes from an infinite product. Therefore any formal infinite sum of
operations admits a well-defined image in $\Lambda$ by definition, without
presupposing any topology. We refer to these phenomena as absolute algebraic
structures. They generalize the notion of a contramodule, see [Pos21] for a
precise definition of them. Furthermore, the presence of invariants on the
left-hand side implies that these algebraic structures are also endowed with
divided powers operations. Therefore the definition of a dg
$\mathcal{C}$-dalgebra encodes divided powers absolute types of algebraic
structures.

######  Remark 33 (Planar definitions).

One can also define the dg (co)algebras over a planar dg operad $\mathcal{P}$
as the (co)algebras over the dg operad $\mathcal{P}\otimes\mathbb{S}$. And the
dg (co)algebras over a dg planar cooperad $\mathcal{C}$ as the (co)algebras
over the dg cooperad $\mathcal{C}\otimes\mathbb{S}$. In this case, no divided
powers operations appear in dg $\mathcal{C}$-coalgebras nor in dg
$\mathcal{C}$-algebras.

######  Proposition 18.

The category of dg $\mathcal{P}$-algebras is $\omega$-presentable. The
categories of dg $\mathcal{P}$-coalgebras, dg $\mathcal{C}$-coalgebras and dg
$\mathcal{C}$-algebras are presentable.

###### Proof.

The monad $\mathcal{P}\circ-$ and the comonad $\mathcal{C}\circ-$ are
$\omega$-accessible. The monad $(-)^{\mathcal{C}}$ and the comonad
$L^{\mathcal{P}}$ are also accessible but for a larger small cardinal
($\aleph_{1}$). We conclude by the fact that for a regular small cardinal
$\alpha$, the category of algebras over an $\alpha$-accessible monad in an
$\alpha$-presentable category is $\alpha$-presentable. See [AR94] for more
details. Moreover, the category of coalgebras over an accessible comonad in a
presentable category is presentable, see [CR14]. ∎

One can consider the same definitions and perform the same constructions with
graded modules or pdg modules instead of dg module. These constructions
commute strictly with the forgetful functors $\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\longrightarrow\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}$.

Curved algebras and coalgebras. Let $\mathcal{C}$ be a curved cooperad. It is
in particular a pdg cooperad, and one can define the categories of pdg
$\mathcal{C}$-coalgebras and of pdg $\mathcal{C}$-algebras. Among these, the
full subcategories of curved objects will be of particular interest.

######  Definition 44 (Curved $\mathcal{C}$-coalgebra).

Let $\mathcal{C}$ be a curved cooperad. A pdg coalgebra $(W,\Delta_{W},d_{W})$
is curved if the following diagram commutes

${W}$${\mathcal{C}\circ W}$${W\leavevmode\nobreak\
,}$$\scriptstyle{\Delta_{W}}$$\scriptstyle{-d_{W}^{2}}$$\scriptstyle{\theta\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \mathrm{Id}}$

where $\theta$ denotes the curvature of the cooperad $\mathcal{C}$.

The category $\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}$ of curved $\mathcal{C}$-coalgebras is the
full subcategory of pdg $\mathcal{C}$-coalgebras spanned by those which are
curved.

######  Remark 34.

If the curvature of $\mathcal{C}$ is zero, then the category of curved
$\mathcal{C}$-coalgebras in pdg modules is isomorphic to the category of dg
$\mathcal{C}$-coalgebras.

######  Definition 45 (Curved $\mathcal{C}$-algebra).

Let $\mathcal{C}$ be a curved cooperad. A pdg $\mathcal{C}$-algebra
$(\Lambda,\gamma_{\Lambda},d_{\Lambda})$ is curved if the following diagram
commutes:

${\Lambda}$${\Lambda^{\mathcal{C}}}$${\Lambda\leavevmode\nobreak\
.}$$\scriptstyle{\Lambda^{\theta}}$$\scriptstyle{d_{\Lambda}^{2}}$$\scriptstyle{\gamma_{\Lambda}}$

The category $\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}$ of curved $\mathcal{C}$-algebras is the full
subcategory of pdg $\mathcal{C}$-algebras spanned by those which are curved.

######  Remark 35.

If the curvature of $\mathcal{C}$ is zero, then the category of curved
$\mathcal{C}$-algebras in pdg modules is isomorphic to the category of dg
$\mathcal{C}$-algebras.

###  3.3. Categorical properties of the category of (co)algebras over a operad

Let $\mathcal{P}$ be a dg operad. Let us consider the following commutative
diagram of forgetful functors

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{gr}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}.}$

All these categories are presentable. Moreover, all the functors are right
adjoints because they preserve limits and filtered colimits. They are
conservative and they also preserve sifted colimits. Therefore, all these
forgetful functors are monadic.

Let us consider the following commutative diagram of forgetful functors

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{gr}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$

All these categories are presentable. Moreover, all the functors are left
adjoints since they preserve colimits. They are also conservative and preserve
finite cosifted limits. Therefore, all the functors are comonadic.

###  3.4. Categorical properties of coalgebras over a conilpotent curved
cooperad

Let $\mathcal{C}$ be a conilpotent curved cooperad whose underlying graded
conilpotent cooperad is the image through $-\otimes\mathbb{S}$ of a graded
planar conilpotent cooperad $\mathcal{C}_{\mathrm{pl}}$.

######  Proposition 19.

The forgetful functor from curved $\mathcal{C}$-coalgebras to pdg
$\mathcal{C}$-coalgebras admits a right adjoint denoted by $\mathrm{Curv}$.
This gives an adjunction

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}\leavevmode\nobreak\
.}$$\scriptstyle{\mathrm{U}}$$\scriptstyle{\mathrm{Curv}}$${\dashv}$

Hence the category of curved $\mathcal{C}$-coalgebras forms a coreflexive full
subcategory of the category of pdg $\mathcal{C}$-coalgebras. In other words,
they are coalgebras over an idempotent comonad in the category of pdg
$\mathcal{C}$-coalgebras.

###### Proof.

Let $(W,\Delta_{W},d_{W})$ be a pdg $\mathcal{C}$-coalgebra, we consider the
following two maps of graded $\Bbbk$-modules

${W}$${s^{2}W\leavevmode\nobreak\ ,}$$\scriptstyle{(\theta\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \mathrm{Id})\Delta_{W}}$$\scriptstyle{-d_{W}^{2}}$

where $\theta:\mathcal{C}\longrightarrow\mathcal{\mathcal{I}}$ is the
curvature of the cooperad $\mathcal{C}$. Let $Z_{W}$ be the cofree pdg
$\Bbbk$-module obtained from the graded $\Bbbk$ module $s^{2}W$. Since a pdg
$\Bbbk$-module is a graded $\Bbbk$-module together with a degree $-1$
endomorphism, $Z_{W}$ is given by $s^{2}W\otimes\Bbbk[\mathrm{u}]$, where
$\mathrm{u}$ is a generator in degree $-1$. From the above two maps, one gets
a natural coreflexive pair of maps of pdg $\mathcal{C}$-coalgebras

${W}$${(\mathcal{C}\circ Z_{W})\times W}$

where the rightward maps are obtained from the previous diagram of graded
$\Bbbk$-modules, using the forgetful-cofree adjunction between graded
$\Bbbk$-modules and pdg $\Bbbk$-modules, and where the leftward map is just
the projection onto $W$.

This pair has an equaliser $(H,\Delta_{H},d_{H})$ in pdg
$\mathcal{C}$-coalgebras, which can be computed in graded $\Bbbk$-modules. In
particular, the morphism of pdg $\mathcal{C}$-coalgebras $H\longrightarrow W$
is a degree-wise monomorphism. Furthermore, the following diagram commutes

${H}$${W}$${s^{2}H}$${s^{2}W\leavevmode\nobreak\
.}$$\scriptstyle{(\theta\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\mathrm{Id})\Delta_{H}+d_{H}^{2}}$$\scriptstyle{(\theta\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \mathrm{Id})\Delta_{W}+d_{W}^{2}}$

The map from $H$ to $s^{2}W$ is zero by the universal property of $H$. Since
the bottom map is a monomorphism, the left map is also zero. So $H$ is curved.
We set $\mathrm{Curv}(W)\coloneqq H$, and it is straightforward to show that
it defines a right adjoint to the forgetful functor. ∎

######  Corollary 2.

The category of curved $\mathcal{C}$-coalgebras is the category of coalgebra
over an idempotent comonad over pdg $\mathcal{C}$-coalgebras which is
accessible and preserves coreflexive equalisers. Hence, the category of curved
$\mathcal{C}$-coalgebras is presentable.

###### Proof.

Let $W$ be a pdg $\mathcal{C}$-coalgebra. Its image by the functor
$\mathrm{Curv}$ is also given by the coreflexive equaliser of a the pair of
maps of the form

${\mathcal{C}\circ W}$${\mathcal{C}\circ((\mathcal{C}\circ W)\oplus
Z_{W})\leavevmode\nobreak\ .}$

This coreflexive equaliser is constructed from the one that defines the
functor $\mathrm{Curv}$ and from the one that gives $W$ as a pdg
$\mathcal{C}$-coalgebra. The comonad that defines curved
$\mathcal{C}$-coalgebras preserves finite cosifted limits since its
composition with the the forgetful functor to graded $\Bbbk$-modules does.
Indeed, we have that:

  1. (1)

the forgetful functor preserves and creates finite cosifted limits;

  2. (2)

the constructions $X\mapsto\mathcal{C}\circ X$ and
$W\mapsto\mathcal{C}\circ((\mathcal{C}\circ W)\oplus Z_{W})$ preserves finite
cosifted limits;

  3. (3)

then the construction $W\mapsto\lim(W\rightrightarrows\mathcal{C}\circ W\oplus
Z_{W})$ preserves finite cosifted limits.

To prove that this comonad preserves filtered colimits, it suffices to prove
that its composition with the forgetful functor towards pdg $\Bbbk$-modules
preserves filtered colimits. This follows from:

  1. (1)

the forgetful functor preserves and creates filtered colimits and coreflexive
equalisers;

  2. (2)

the constructions $W\mapsto\mathcal{C}\circ W$ and
$W\mapsto\mathcal{C}\circ((\mathcal{C}\circ W)\oplus Z_{W})$ preserve filtered
colimits;

  3. (3)

filtered colimits and coreflexive equalisers commute in graded
$\Bbbk$-modules.

∎

Let us consider the following commutative diagram of forgetful functors

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{gr}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\leavevmode\nobreak\ .}$

All the categories are presentable and all the functors are left adjoint
(since they preserve colimits), conservative and they also preserve
coreflexive equalisers. Therefore all the functors are comonadic.

###  3.5. Curved algebras over a conilpotent curved cooperad

Again, let $\mathcal{C}$ be a conilpotent curved cooperad whose underlying
graded conilpotent cooperad is the image through $-\otimes\mathbb{S}$ of a
graded planar conilpotent cooperad $\mathcal{C}_{\mathrm{pl}}$.

######  Proposition 20.

The forgetful functor from curved $\mathcal{C}$-algebras to pdg
$\mathcal{C}$-algebras admits a left adjoint denoted by $\mathrm{Curv}$. This
gives an adjunction

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\leavevmode\nobreak\
.}$$\scriptstyle{\mathrm{U}}$$\scriptstyle{\mathrm{Curv}}$${\dashv}$

Hence the category of curved $\mathcal{C}$-algebras forms a reflexive full
subcategory of the category of pdg $\mathcal{C}$-algebras. In other words,
they are algebras over an idempotent monad in the category of pdg
$\mathcal{C}$-algebras.

###### Proof.

Let $(\Lambda,\gamma_{\Lambda},d_{\Lambda})$ be a pdg $\mathcal{C}$-algebra.
We consider the two maps of graded $\Bbbk$-modules

${s^{-2}\Lambda}$${\Lambda\leavevmode\nobreak\
,}$$\scriptstyle{\gamma_{\Lambda}\leavevmode\nobreak\
\Lambda^{\theta}}$$\scriptstyle{d^{2}_{\Lambda}}$

where $\theta:\mathcal{C}\longrightarrow\mathcal{\mathcal{I}}$ is the
curvature of the cooperad $\mathcal{C}$. Let $Z_{\Lambda}$ be the free pdg
$\Bbbk$-module obtained from the graded $\Bbbk$ module $s^{-2}\Lambda$. Since
a pdg $\Bbbk$-module is a graded $\Bbbk$-module together with a degree $-1$
endomorphism, $Z_{\Lambda}$ is given by
$s^{-2}\Lambda\otimes\Bbbk[\mathrm{u}]$, where $\mathrm{u}$ is a generator in
degree $-1$. From the above two maps, one gets a natural reflexive pair of
maps of pdg $\mathcal{C}$-algebras

${\Lambda\sqcup(Z_{\Lambda})^{\mathcal{C}}}$${\Lambda}$

where the leftward maps are obtained from the previous diagram of graded
$\Bbbk$-modules, using the free-forgetful adjunction between graded
$\Bbbk$-modules and pdg $\Bbbk$-modules, and where the rightward map is just
the inclusion of $\Lambda$.

This pair has a coequaliser $(\Xi,\gamma_{\Xi},d_{\Xi})$ in the category of
pdg $\mathcal{C}$-algebras, which can be computed in graded $\Bbbk$-modules.
In particular, the morphism of pdg $\mathcal{C}$-algebras
$\Lambda\twoheadrightarrow\Xi$ is a degree-wise epimorphism. Furthermore, the
following square commutes:

${s^{-2}\Lambda}$${s^{-1}\Xi}$${\Lambda}$${\Xi.}$$\scriptstyle{\gamma_{\Lambda}\leavevmode\nobreak\
\Lambda^{\theta}-d^{2}_{\Lambda}}$$\scriptstyle{\gamma_{\Xi}\leavevmode\nobreak\
\Xi^{\theta}-d^{2}_{\Xi}}$

The map from $s^{-2}\Lambda$ to $\Xi$ is zero by the universal property of
$\Xi$. Since the bottom map is an epimorphism, the left map is also zero. So
$\Xi$ is curved. We set $\mathrm{Curv}(W)\coloneqq\Xi$, and it is
straightforward to show that it defines a left adjoint to the forgetful
functor. ∎

######  Corollary 3.

The category of curved $\mathcal{C}$-algebras is the category of algebras over
idempotent monad over pdg $\mathcal{C}$-algebras which is accessible and
preserves reflexive coequalisers. Hence, the category of curved
$\mathcal{C}$-algebras is presentable.

###### Proof.

Let $\Lambda$ be a pdg $\mathcal{C}$-algebra. Its image by the functor
$\mathrm{Curv}$ is also given by the reflexive coequaliser of a the pair of
maps of the form

${(Z_{\Lambda}\oplus\Lambda^{\mathcal{C}})^{\mathcal{C}}}$${\Lambda^{\mathcal{C}}.}$

This reflexive coequaliser is constructed from the one that defines the
functor $\mathrm{Curv}$ and from the one that gives $\Lambda$ as a pdg
$\mathcal{C}$-algebra. This monad preserves finite sifted colimits since its
composition with the forgetful functor towards graded $\Bbbk$-modules does.
Indeed, we have that:

  1. (1)

this forgetful functor preserves and creates finite sifted colimits;

  2. (2)

the constuction $X\mapsto X^{\mathcal{C}}$ preserves finite sifted colimits;

  3. (3)

the construction $\Lambda\mapsto Z_{\Lambda}$ preserves finite sifted
colimits;

  4. (4)

then the construction
$\Lambda\mapsto\varinjlim((Z_{\Lambda}\oplus\Lambda^{\mathcal{C}})^{\mathcal{C}}\rightrightarrows\Lambda^{\mathcal{C}})$
preserves finite sifted colimits.

Let us prove that this monad is accessible. Let $\beta$ be a small regular
cardinal so that the monad $(-)^{\mathcal{C}}$ is $\beta$-accessible. To prove
that the idempotent "curvature" monad preserves $\beta$-filtered colimits, it
suffices to prove that its composition with the forgetful functor towards pdg
$\Bbbk$-modules perserves these colimits. This follows from the following
facts:

  1. (1)

the forgetful functor preserves and creates reflexive coequalisers and
$\beta$-filtered colimits;

  2. (2)

the constructions $\Lambda\mapsto\Lambda^{\mathcal{C}}$ and
$\Lambda\mapsto\mathcal{C}\circ((\mathcal{C}\circ\Lambda)\oplus Z_{\Lambda})$
preserve filtered colimits;

  3. (3)

reflexive coequalisers and $\beta$-filtered colimits commute in pdg
$\Bbbk$-modules.

∎

Let us consider the following commutative diagram of forgetful functors:

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{gr}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{pdg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{gr}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}\leavevmode\nobreak\ .}$

All the categories are presentable. Moreover all the functors are right
adjoint (they are accessible and preserve limits), conservative and they also
preserve finite sifted colimits. Therefore they are monadic.

###  3.6. Complete algebras over the canonical ladder of a quasi-planar
cooperad

This section is the mirror of [GL22, Section 4.4] in any characteristic for a
quasi-planar cooperad ladder equipped with its canonical quasi-planar
filtration. For the rest of this section, we fix a quasi-planar curved
conilpotent cooperad $\mathcal{C}$.

######  Lemma 23.

Let $n$ be a natural integer and let $X$ be a pdg $\Bbbk$ module. Let us
consider the inclusion
$i_{n}:F_{n}^{\mathrm{qp}}\mathcal{C}\rightarrowtail\mathcal{C}$. The natural
map

$(\mathrm{Id})^{i_{n}}:X^{\mathcal{C}}\twoheadrightarrow
X^{F_{n}^{\mathrm{qp}}\mathcal{C}}$

is a degree-wise epimorphism. Subsequently, the forgetful functor from pdg
$F_{n}^{\mathrm{qp}}\mathcal{C}$-algebras to pdg $\mathcal{C}$-algebras is
fully faithful.

###### Proof.

In the category of graded $\Bbbk$-modules, this map may be rewritten as

$X^{\mathcal{C}_{\mathrm{pl}}}\twoheadrightarrow
X^{F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}}$

Moreover, the map of graded $\mathbb{N}$-module
$F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}\rightarrowtail\mathcal{C}_{\mathrm{pl}}$
has a left inverse. Therefore the above map has a section. It implies that it
is a degree-wise epimorphism. We thus get a morphism of monads that is object-
wise an epimorphism. Therefore the related right adjoint forgetful functor is
fully faithful. ∎

For every natural integer $n$, we denote by

$F^{n}_{\mathrm{qp}}:\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
F_{n}^{\mathrm{qp}}\mathcal{C}\text{-}\mathsf{alg}$

be the functor that is left adjoint to the forgetful functor. It sends every
pdg $\mathcal{C}$-algebra $(B,\gamma_{B},d_{B})$ to the coequaliser of the
following pair

${(\Lambda^{\mathcal{C}})^{F_{n}^{\mathrm{qp}}\mathcal{C}}}$${\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}}\leavevmode\nobreak\
.}$$\scriptstyle{(\gamma_{\Lambda})^{\mathrm{Id}}}$$\scriptstyle{(\mathrm{Id})^{\Delta}\leavevmode\nobreak\
\circ\leavevmode\nobreak\ (\mathrm{Id})^{i_{n}}}$

This coequalizer is also the pushout of the span

${\Lambda^{\mathcal{C}}}$${\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}}}$${\Lambda\leavevmode\nobreak\
.}$$\scriptstyle{(\mathrm{Id})^{i_{n}}}$$\scriptstyle{\gamma_{\Lambda}}$

Both of these colimits can be computed in the category of pdg
$F_{n}^{\mathrm{qp}}\mathcal{C}$-algebras or in the underlying category of
graded $\Bbbk$-modules.

This gives a natural diagram

$\Lambda\longrightarrow\cdots\longrightarrow
F^{n}_{\mathrm{qp}}\Lambda\longrightarrow
F^{n-1}_{\mathrm{qp}}\Lambda\longrightarrow\cdots
F^{1}_{\mathrm{qp}}\Lambda\longrightarrow
F^{0}_{\mathrm{qp}}\Lambda\leavevmode\nobreak\ ,$

where we set $F^{-1}_{\mathrm{qp}}(\Lambda)=0$. For every natural integer $n$
in $\mathbb{N}$, the associated graded is given by

$\mathrm{gr}^{n}_{\mathrm{qp}}\Lambda\coloneqq\mathrm{Ker}\left(F^{n}_{\mathrm{qp}}(\Lambda)\longrightarrow
F^{n-1}_{\mathrm{qp}}(\Lambda)\right)\leavevmode\nobreak\ .$

######  Definition 46 (qp-completion functor).

The qp-completion functor, denoted by $\widehat{(-)}$, is the endofunctor of
pdg $\mathcal{C}$-algebras that sends an algebra $\Lambda$ to the limit

$\widehat{\Lambda}\coloneqq\lim_{n\in\omega^{\mathrm{op}}}F^{n}_{\mathrm{qp}}(\Lambda)\leavevmode\nobreak\
.$

######  Definition 47 (qp-complete pdg $\mathcal{C}$-algebra).

A pdg $\mathcal{C}$-algebra $\Lambda$ is called qp-complete if the canonical
morphism $\Lambda\longrightarrow\widehat{\Lambda}$ is an isomorphism.

We denote $\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}$ the full
subcategory of that of pdg $\mathcal{C}$-algebras spanned by the qp-complete
ones.

######  Lemma 24 (After [GL22, Lemma 4.22]).

Let $f:\Lambda\longrightarrow\Gamma$ be a morphism of pdg
$\mathcal{C}$-algebras. Let us suppose that for every natural integer $n$, the
induced map

$\mathrm{gr}^{n}(f):\mathrm{gr}^{n}_{\mathrm{qp}}\Lambda\twoheadrightarrow\mathrm{gr}^{n}_{\mathrm{qp}}\Gamma$

is a degree-wise epimorphism.

  1. (1)

For every $n$, the map

$F^{n+1}_{\mathrm{qp}}\Lambda\twoheadrightarrow
F^{n}_{\mathrm{qp}}\Lambda\times_{F^{n}_{\mathrm{qp}}\Gamma}F^{n+1}_{\mathrm{qp}}\Gamma$

is a degree-wise epimorphism.

  2. (2)

The map $\widehat{f}:\widehat{\Lambda}\twoheadrightarrow\widehat{\Gamma}$ is a
degree-wise epimorphism.

###### Proof.

Let us prove the first assertion. Let us consider the square diagram

${F^{n+1}_{\mathrm{qp}}\Lambda}$${F^{n}_{\mathrm{qp}}\Lambda}$${F^{n+1}_{\mathrm{qp}}\Gamma}$${F^{n}_{\mathrm{qp}}\Gamma}$$\scriptstyle{p_{n}}$$\scriptstyle{F^{n+1}_{\mathrm{qp}}(f)}$$\scriptstyle{F^{n}_{\mathrm{qp}}(f)}$$\scriptstyle{q_{n}}$

Given an element $(y,z)\in
F^{n}_{\mathrm{qp}}\Lambda\times_{F^{n}_{\mathrm{qp}}\Gamma}F^{n+1}_{\mathrm{qp}}\Gamma$
one can find $x^{\prime}$ so that $p_{n}(x^{\prime})=y$. Then
$q_{n}(z-F^{n+1}_{rad}(f)(x^{\prime}))=0$. In other words

$z-F^{n+1}_{\mathrm{qp}}(f)(x^{\prime})\in\mathrm{gr}^{n+1}_{\mathrm{qp}}\Gamma\leavevmode\nobreak\
.$

Let $x^{\prime\prime}$ be one of its antecedent in
$\mathrm{gr}^{n+1}_{\mathrm{qp}}\Lambda$. Then $x=x^{\prime}+x^{\prime\prime}$
is an antecedent of $(y,z)$. This proves the first point.

Let us prove the second point. Let $z$ be a degree $k$ element of
$\widehat{\Lambda}$. It is a sequence
$(z_{n})_{n\in\mathbb{N}}\in\prod_{n}F^{n}_{\mathrm{qp}}\widehat{\Lambda}_{k}$,
where each $z_{n}$ is the image of $z_{n+1}$ via the projection
$F^{n+1}_{\mathrm{qp}}(\widehat{\Lambda})\twoheadrightarrow
F^{n}_{\mathrm{qp}}(\widehat{\Lambda})$. Using

  1. (1)

the fact that the map $\mathrm{gr}^{0}(f)$ is a degree-wise epimorphism;

  2. (2)

the first assertion of the lemma;

  3. (3)

the axiom of choice;

one can build a sequence
$x=(x_{n})_{n\in\mathbb{N}}\in\prod_{n}F_{\mathrm{qp}}^{n}\Lambda_{k}$, where
each $x_{n}$ is the image of $x_{n+1}$ via the projection
$F^{n+1}_{\mathrm{qp}}(\Lambda)\twoheadrightarrow
F^{n}_{\mathrm{qp}}(\Lambda)$ and such that
$F^{n}_{\mathrm{qp}}(f)(x_{n})=z_{n}$. Thus $x$ is an antecedent of $z$ in
$\widehat{\Lambda}$. ∎

######  Lemma 25 (After [GL22, Proposition 4.23]).

The completion endofunctor preserves degree-wise epimorphisms.

###### Proof.

Let $f:\Lambda\longrightarrow\Gamma$ be a degree-wise epimorphism. In order to
show that $\widehat{f}$ is a degree-wise epimorphism, it suffices using Lemma
24 to prove that $\mathrm{gr}^{n}_{\mathrm{qp}}(f)$ is a degree-wise
epimorphism for every natural integer $n$.

Infinite products in graded $\Bbbk$-modules are exact, the contravariant
functor $[-,X]$ for every $X$ is also exact, therefore the sequence

$0\longrightarrow\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}/F_{n-1}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}}\longrightarrow\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}}\longrightarrow\Lambda^{F_{n-1}^{\mathrm{qp}}\mathcal{C}_{\mathrm{pl}}}\longrightarrow
0$

is exact. Thus, the following commutative diagram of graded $\Bbbk$-modules is
a pushout

${\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}/F_{n-1}^{\mathrm{qp}}\mathcal{C}}}$${0}$${\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}}}$${\Lambda^{F_{n-1}^{\mathrm{qp}}\mathcal{C}}}$${F^{n}_{\mathrm{qp}}\Lambda}$${F^{n-1}_{\mathrm{qp}}\Lambda.}$

Therefore the kernel $\mathrm{gr}^{n}_{\mathrm{qp}}\Lambda$ is the image in
$F^{n}_{\mathrm{qp}}\Lambda$ of the map
$\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}/F_{n-1}^{\mathrm{qp}}\mathcal{C}}\longrightarrow
F^{n}_{\mathrm{qp}}\Lambda$. Let us consider the following commutative diagram
of graded $\Bbbk$-modules

${\Lambda^{F_{n}^{\mathrm{qp}}\mathcal{C}/F_{n-1}^{\mathrm{qp}}\mathcal{C}}}$${\Gamma^{F_{n}^{\mathrm{qp}}\mathcal{C}/F_{n-1}^{\mathrm{qp}}\mathcal{C}}}$${\mathrm{gr}^{n}_{\mathrm{qp}}\Lambda}$${\mathrm{gr}^{n}_{\mathrm{qp}}\Gamma.}$$\scriptstyle{f^{F_{n}^{\mathrm{qp}}\mathcal{C}/F_{n-1}^{\mathrm{qp}}\mathcal{C}}}$$\scriptstyle{\mathrm{gr}^{n}_{\mathrm{qp}}(f)}$

Since the top map and the right map are degree-wise epimorphisms, so is bottom
map, which proves the result. ∎

######  Lemma 26 (After [GL22, Proposition 4.21 and 4.24]).

  1. (1)

The $X$ be a pdg module $X$, the free pdg $\mathcal{C}$-algebra
$X^{\mathcal{C}}$ is qp-complete.

  2. (2)

The natural map

$\Lambda\twoheadrightarrow\widehat{\Lambda}$

is an degree-wise epimorphism for every dg $\mathcal{C}$-algebra $\Lambda$.

###### Proof.

For every free pdg $\mathcal{C}$-algebra $X^{\mathcal{C}}$ on a pdg module
$X$, one has
$F^{n}_{\mathrm{qp}}X^{\mathcal{C}}=X^{F_{n}^{\mathrm{qp}}\mathcal{C}}$ and
that it is complete:

$\widehat{X^{\mathcal{C}}}=\lim_{n\in\omega^{\mathrm{op}}}X^{F_{n}^{\mathrm{qp}}\mathcal{C}}\cong
X^{\mathcal{C}}.$

Now, let $\Lambda$ be any pdg $\mathcal{C}$-algebra. We consider the following
commutative square

${\Lambda^{\mathcal{C}}}$${\widehat{\Lambda^{\mathcal{C}}}}$${\Lambda}$${\widehat{\Lambda}.}$

The top horizontal map is an isomorphism and the right vertical map is a
degree-wise epimorphism by Lemma 25. The map from $\Lambda^{\mathcal{C}}$ to
$\Lambda$ is also epimorphism. Therefore so is the map
$\Lambda\longrightarrow\widehat{\Lambda}$. ∎

######  Lemma 27.

The canonical map

$\widehat{\Lambda}\longrightarrow\widehat{\widehat{(\Lambda)}}$

is an isomorphism. Therefore the completion functor equipped with the natural
map $\Lambda\longrightarrow\widehat{\Lambda}$ is an idempotent monad in the
category of pdg $\mathcal{C}$-algebras.

###### Proof.

For every natural integer $n$, both maps
$\Lambda\longrightarrow\widehat{\Lambda}\longrightarrow
F_{n}^{\mathrm{qp}}\Lambda$ are degree-wise epimorphisms. Applying the left
adjoint functor $F^{n}_{\mathrm{qp}}$ which preserves degree-wise
epimorphisms, one gets a diagram

${F_{n}^{\mathrm{qp}}\Lambda}$${F_{n}^{\mathrm{qp}}\widehat{\Lambda}}$${F_{n}^{\mathrm{qp}}\Lambda.}$$\scriptstyle{\mathrm{Id}}$

The first map is both a degree-wise epimorphism and a degree-wise
monomorphism. Hence it is an isomorphism and so is the second map. Taking the
limit over $n\in\omega^{\mathrm{op}}$, one gets the required isomorphism. ∎

######  Proposition 21.

The category of qp-complete pdg $\mathcal{C}$-algebra is presentable.

###### Proof.

It is a reflexive full subcategory of the category of pdg
$\mathcal{C}$-algebras which is presentable. The reflector is the completion
$\Lambda\mapsto\widehat{\Lambda}$. One can find a small regular cardinal
$\beta$ so that every functor $F^{n}_{\mathrm{qp}}$ is $\beta$-accessible and
so that $\beta$-filtered colimits commute with limits of diagrams indexed by
$\omega^{op}$. Then, the completion functor is $\beta$-accessible. Hence the
category of qp-complete pdg $\mathcal{C}$-algebras is presentable. ∎

######  Remark 36.

The category of qp-complete pdg $\mathcal{C}$-algebra is the homotopy limit of
the (pseudo) diagram of categories

$\cdots\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
F_{n}^{\mathrm{qp}}\mathcal{C}\text{-}\mathsf{alg}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
F_{n-1}^{\mathrm{qp}}\mathcal{C}\text{-}\mathsf{alg}\longrightarrow\cdots\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
F_{1}^{\mathrm{qp}}\mathcal{C}\text{-}\mathsf{alg}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
F_{0}^{\mathrm{qp}}\mathcal{C}\text{-}\mathsf{alg}.$

It is presentable as a small limit of presentable categories and left adjoint
functors.

######  Remark 37.

The analogue definitions and results are valid for dg $\mathcal{C}$-algebras
or graded $\mathcal{C}$-algebras as well.

###  3.7. Categorical properties of qp-complete curved algebras over a quasi-
planar cooperad

######  Proposition 22.

The qp-completion idempotent monad on pdg $\mathcal{C}$-algebras preserves
curved algebras. Moreover, the restricted monad (to curved algebras) is
accessible. Hence, the category of qp-complete curved $\mathcal{C}$-algebras
is presentable and the forgetful functor towards curved $\mathcal{C}$-algebras
is accessible.

###### Proof.

For every curved $\mathcal{C}$-algebra $\Lambda$, $\widehat{\Lambda}$ is also
curved since the map $\Lambda\longrightarrow\widehat{\Lambda}$ is a degree-
wise epimorphism. Therefore, the qp-completion monad restricts to curved
$\mathcal{C}$-algebras. Let $\beta$ be regular small cardinal such that

  1. (1)

the forgetful functor $\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}$ is a $\beta$-accessible functor between
$\beta$-accessible categories;

  2. (2)

the completion monad on $\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}$ is $\beta$-accessible.

Then, the restriction of this monad to curved algebras is also
$\beta$-accessible. ∎

######  Remark 38.

Let us consider the following commutative diagram of functors.

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}}$${\mathsf{gr}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{gr}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$

The vertical arrows are monadic and accessible but they do not preserve
reflexive coequalisers in general; otherwise all $\mathcal{C}$-algebras would
be qp-complete.

###  3.8. Naturality of (co)algebras of a dg operad.

The definitions of algebras and coalgebras are compatible with morphisms of
operads. This will mean that any such morphism induces an adjunction between
the appropriate categories. Let $f:\mathcal{P}\longrightarrow\mathcal{Q}$ be a
morphism of dg operads.

Naturality of algebras. The morphism $f$ induces a morphism of monads
$\mathcal{P}\circ-\longrightarrow\mathcal{Q}\circ-$, which yields an
adjunction

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

since the category of dg $\mathcal{Q}$-algebras is cocomplete, by the adjoint
lifting theorem. See Appendix A.1 for more details. Since the monad
$\mathcal{Q}\circ-$ commutes with sifted colimits, the left adjoint functor
$f_{!}$ sends a dg $\mathcal{P}$-algebra $A$ to the reflexive coequaliser of
the pair

${\mathcal{Q}\circ(\mathcal{P}\circ A)}$${\mathcal{Q}\circ
A\leavevmode\nobreak\ ,}$$\scriptstyle{\mathrm{Id}\leavevmode\nobreak\
\circ\leavevmode\nobreak\
\gamma_{A}}$$\scriptstyle{\gamma_{\mathcal{Q}}\leavevmode\nobreak\
(f\leavevmode\nobreak\ \circ\leavevmode\nobreak\ \mathrm{Id})}$

between the structural map $\gamma_{A}$ of $A$ and the map induced by the
morphism $f$ composed with the monad structure $\gamma_{\mathcal{Q}}\circ-$ of
$\mathcal{Q}\circ-$. This is a coequalizer both in dg $\mathcal{Q}$-algebras
and in graded $\Bbbk$-modules.

Naturality of coalgebras. The morphism $f$ also induces a morphism of comonads
$L^{\mathcal{P}}\longrightarrow L^{\mathcal{Q}}$, which yields an adjunction

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

since the category of dg $\mathcal{Q}$-coalgebras is complete, again using the
adjoint lifting theorem. Since the comonad $L^{\mathcal{Q}}$ commutes with
finite cosifted limits, the functor $f_{!}$ sends a dg $\mathcal{P}$-coalgebra
$V$ to the coreflexive equaliser of the pair

${L^{\mathcal{Q}}V}$${L^{\mathcal{Q}}L^{\mathcal{P}}V\leavevmode\nobreak\
,}$$\scriptstyle{L^{f}(\mathrm{Id})\leavevmode\nobreak\
\omega_{\mathcal{Q}}(V)}$$\scriptstyle{\gamma_{\mathcal{Q}}\leavevmode\nobreak\
(f\leavevmode\nobreak\ \circ\leavevmode\nobreak\ \mathrm{Id})}$

between the structure map $\Delta_{V}$ of $V$ and the map given by the comonad
structure $\omega_{\mathcal{Q}}$ of $L^{\mathcal{Q}}$ composed with the
morphism induced by $f$. This is an equalizer both in dg
$\mathcal{Q}$-coalgebra and in graded $\Bbbk$-modules.

###  3.9. Naturality of (co)algebras over a cooperad.

The definitions of algebras and coalgebras are compatible with morphisms of of
cooperads. This will mean that any such morphism induces an adjunction between
the appropriate categories. Let $\mathcal{C},\mathcal{D}$ be two conilpotent
curved cooperads whose underlying graded conilpotent cooperads are the image
through $-\otimes\mathbb{S}$ of a graded planar conilpotent cooperads
respectively $\mathcal{C}_{\mathrm{pl}},\mathcal{D}_{\mathrm{pl}}$. Let
$g:\mathcal{C}\longrightarrow\mathcal{D}$ be a morphisms of curved cooperads.

Naturality of coalgebras. The map $g$ induces a morphism of comonads on the
category of pdg $\Bbbk$-modules
$\mathcal{C}\circ-\longrightarrow\mathcal{D}\circ-$, which yields an
adjunction

${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{cog}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}\leavevmode\nobreak\
,}$$\scriptstyle{g^{!}}$$\scriptstyle{g_{*}}$${\dashv}$

by the adjoint lifting theorem. Since the comonad $\mathcal{D}\circ-$
preserves finite sifted limits, the functor $g^{!}$ sends a pdg
$\mathcal{D}$-coalgebra $W$ to the coreflexive equaliser of the pair

${\mathcal{C}\circ W}$${\mathcal{C}\circ\mathcal{D}\circ W\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{Id}\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\Delta_{W}}$$\scriptstyle{(g\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\mathrm{Id})\leavevmode\nobreak\ \Delta_{\mathcal{C}}(W)}$

between the structure map $\Delta_{W}$ of $W$ and the map given by the comonad
structure $\Delta_{\mathcal{C}}(-)$ of $\mathcal{C}\circ-$ composed with the
morphism induced by $g$. The existence of this adjunction and the fact that
this equaliser can be computed in graded $\Bbbk$-modules follows from Lemma
22.

######  Proposition 23.

The adjunction $g_{*}\dashv g^{!}$ restrict to an adjunction

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{cog}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}\leavevmode\nobreak\
,}$$\scriptstyle{g^{!}}$$\scriptstyle{g_{*}}$${\dashv}$

relating curved $\mathcal{D}$-coalgebras and curved $\mathcal{D}$-coalgebras.

###### Proof.

The fact that $g_{*}$ sends curved objects to curved objects is clear. Now,
given a curved $\mathcal{D}$-coalgebra $W$, the squared coderivation $d^{2}$
on $\mathcal{C}\circ W$ is

$\displaystyle d^{2}$
$\displaystyle=d_{\mathcal{C}}^{2}\circ\mathrm{Id}_{W}+\mathrm{Id}_{\mathcal{C}}\circ\shuffle(\mathrm{Id}_{W},d_{W}^{2})$
$\displaystyle=-((\theta_{\mathcal{C}}\circ\mathrm{Id}_{\mathcal{C}})\Delta_{\mathcal{C}})\circ\mathrm{Id}_{W}+((\mathrm{Id}_{\mathcal{C}}\circ\shuffle(\epsilon_{\mathcal{C}},\theta_{\mathcal{C}}))\Delta_{\mathcal{C}})\circ\mathrm{Id}_{W}-\mathrm{Id}_{\mathcal{C}}\circ\shuffle(\mathrm{Id}_{W},(\theta_{\mathcal{D}}\circ\mathrm{Id}_{W})\Delta_{W})\leavevmode\nobreak\
,$

where $\upsilon_{\mathcal{C}}$ denotes the counit of the cooperad
$\mathcal{C}$. If we denote $j$ the inclusion
$g^{!}W\hookrightarrow\mathcal{C}\circ W$, then the pre-composition of $j$
with the sum

$((\mathrm{Id}_{\mathcal{C}}\circ\shuffle(\epsilon_{\mathcal{C}},\theta_{\mathcal{C}}))\Delta_{\mathcal{C}})\circ\mathrm{Id}_{W}-\mathrm{Id}_{\mathcal{C}}\circ\shuffle(\mathrm{Id}_{W},(\theta_{\mathcal{D}}\circ\mathrm{Id}_{W})\Delta_{W})$

is zero. Thus,

$jd^{2}=d^{2}j=-((\theta_{\mathcal{C}}\circ\mathrm{Id}_{\mathcal{C}})\Delta_{\mathcal{C}}\circ\mathrm{Id}_{W})j=-j((\theta_{\mathcal{C}}\circ\mathrm{Id}_{g^{!}_{W}})\Delta_{g^{!}_{W}})\leavevmode\nobreak\
.$

Since $j$ is injective, $g^{!}_{W}$ is curved. ∎

######  Remark 39.

Actually, when dealing with coalgebras, we do not need $\mathcal{C}$ to
proceed from a planar cooperad. We will need it in the context of algebras, in
order to be able to talk about complete $\mathcal{C}$-algebras.

Naturality of algebras. The morphism $g$ induces a morphism of monads on pdg
$\Bbbk$-modules $(-)^{\mathcal{D}}\longrightarrow(-)^{\mathcal{C}}$, which
yields an adjunction

${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{pdg}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{g_{*}}$$\scriptstyle{g^{!}}$${\dashv}$

by the adjoint lifting theorem. The functor $g^{!}$ sends a curved
$\mathcal{D}$-algebra $\Lambda$ to the reflexive coequaliser of the pair

${(\Lambda^{\mathcal{D}})^{\mathcal{C}}}$${\Lambda^{\mathcal{C}}\leavevmode\nobreak\
,}$$\scriptstyle{(\gamma_{\Lambda})^{(\mathrm{Id})}}$$\scriptstyle{(\Lambda)^{\Delta_{\mathcal{C}}}\leavevmode\nobreak\
(\Lambda)^{g}}$

between the structure map $\gamma_{\Lambda}$ of $\Lambda$ and the map induced
by the morphism $g$ composed with the monad structure
$(-)^{\Delta_{\mathcal{C}}}$ of $(-)^{\mathcal{C}}$. The existence of this
adjunction and the fact that this coequaliser can be computed in graded
$\Bbbk$-modules follows from Lemma 22.

######  Proposition 24.

The adjunction $g^{!}\dashv g^{*}$ restrict to an adjunction

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{g^{*}}$$\scriptstyle{g^{!}}$${\dashv}$

relating curved $\mathcal{D}$-algebras and curved $\mathcal{D}$-algebras.

###### Proof.

This follows from dual arguments as those used to prove Proposition 24. The
fact that $g_{*}$ sends curved algebras to curved algebras is clear. Now,
given a curved $\mathcal{D}$-algebra $\Lambda$, the squared coderivation
$d^{2}$ on $\Lambda^{\mathcal{C}}\simeq\Lambda^{\mathcal{C}_{\mathrm{pl}}}$ is

$\displaystyle d^{2}$
$\displaystyle=-\Lambda^{d_{\mathcal{C}}^{2}}+\shuffle(\mathrm{Id}_{\Lambda},d_{\Lambda}^{2})^{\mathcal{C}}\leavevmode\nobreak\
.$

It is thus the sum of the three maps

$\displaystyle\delta^{(1)}:\Lambda^{\mathcal{C}_{\mathrm{pl}}}$
$\displaystyle\xrightarrow{\Lambda^{\theta_{\mathcal{C}}\circ_{\mathrm{pl}}\mathrm{Id}}}\Lambda^{\mathcal{C}_{\mathrm{pl}}\circ_{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}}\longrightarrow\Lambda^{\mathcal{C}_{\mathrm{pl}}};$
$\displaystyle\delta^{(2)}:\Lambda^{\mathcal{C}_{\mathrm{pl}}}$
$\displaystyle\xrightarrow{-\Lambda^{\mathrm{Id}_{\mathcal{C}}\circ_{\mathrm{pl}}\shuffle(\upsilon_{\mathcal{C}},\theta_{\mathcal{C}})}}\Lambda^{\mathcal{C}_{\mathrm{pl}}\circ_{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}}\longrightarrow\Lambda^{\mathcal{C}_{\mathrm{pl}}};$
$\displaystyle\delta^{(3)}:\Lambda^{\mathcal{C}_{\mathrm{pl}}}$
$\displaystyle\xrightarrow{\shuffle(\Lambda^{\upsilon_{\mathcal{D}}},\Lambda^{\theta_{\mathcal{C}}})^{\mathcal{C}}}(\Lambda^{\mathcal{D}_{\mathrm{pl}}})^{\mathcal{C}_{\mathrm{pl}}}\xrightarrow{\gamma_{\Lambda}^{\mathcal{C}}}\Lambda^{\mathcal{C}_{\mathrm{pl}}},$

where $\upsilon_{\mathcal{C}},\upsilon_{\mathcal{D}}$ are the counits of the
cooperads $\mathcal{C},\mathcal{D}$. Since the curvature on cooperads in zero
on arities different from $1$ and since the morphism of cooperads $g$ commutes
with the curvatures and the counits, the map

$\Lambda^{\mathrm{Id}_{\mathcal{C}}\circ_{\mathrm{pl}}\shuffle(\upsilon_{\mathcal{C}},\theta_{\mathcal{C}})}:\Lambda^{\mathcal{C}_{\mathrm{pl}}}\longrightarrow\Lambda^{\mathcal{C}_{\mathrm{pl}}\circ_{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}}$

is equal to the composition

$\Lambda^{\mathcal{C}_{\mathrm{pl}}}\xrightarrow{\shuffle(\Lambda^{\upsilon},\Lambda^{\theta})^{\mathcal{C}}}(\Lambda^{\mathcal{D}_{\mathrm{pl}}})^{\mathcal{C}_{\mathrm{pl}}}\xrightarrow{(\Lambda^{g})^{\mathcal{C}_{\mathrm{pl}}}}(\Lambda^{\mathcal{C}_{\mathrm{pl}}})^{\mathcal{C}_{\mathrm{pl}}}\hookrightarrow\Lambda^{\mathcal{C}_{\mathrm{pl}}\circ_{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}}.$

As a consequence, if we denote $p$ the projection
$p:\Lambda^{\mathcal{C}_{\mathrm{pl}}}\twoheadrightarrow g^{!}(\Lambda)$, then

$p(\delta^{(2)}+\delta^{(3)})=0\leavevmode\nobreak\ .$

The map

$\Lambda^{\theta_{\mathcal{C}}\circ_{\mathrm{pl}}\mathrm{Id}}:\Lambda^{\mathcal{C}_{\mathrm{pl}}}\longrightarrow\Lambda^{\mathcal{C}_{\mathrm{pl}}\circ_{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}}$

is equal to the composition

$\Lambda^{\mathcal{C}_{\mathrm{pl}}}\xrightarrow{(\Lambda^{\mathcal{C}_{\mathrm{pl}}})^{\theta_{\mathcal{C}}}}(\Lambda^{\mathcal{C}_{\mathrm{pl}}})^{\mathcal{C}_{\mathrm{pl}}}\hookrightarrow\Lambda^{\mathcal{C}_{\mathrm{pl}}\circ_{\mathrm{pl}}\mathcal{C}_{\mathrm{pl}}}\leavevmode\nobreak\
.$

Thus
$\delta^{(1)}=\gamma_{\Lambda^{\mathcal{C}_{\mathrm{pl}}}}(\Lambda^{\mathcal{C}_{\mathrm{pl}}})^{\theta_{\mathcal{C}}}$
and therefore

$d^{2}p=pd^{2}=p\delta^{(1)}=p\gamma_{\Lambda^{\mathcal{C}_{\mathrm{pl}}}}(\Lambda^{\mathcal{C}_{\mathrm{pl}}})^{\theta_{\mathcal{C}}}=\gamma_{g^{!}(\Lambda)}(g^{!}(\Lambda))^{\theta_{\mathcal{C}}}p\leavevmode\nobreak\
.$

Since $p$ is surjective, this entails that

$d^{2}=\gamma_{g^{!}(\Lambda)}(g^{!}(\Lambda))^{\theta_{\mathcal{C}}}\leavevmode\nobreak\
,$

which proves the result. ∎

######  Proposition 25.

Let $g:\mathcal{C}\longrightarrow\mathcal{D}$ be a quasi-planar morphism of
curved cooperads. Then the functor $g_{*}$ sends qp-complete
$\mathcal{C}$-algebras to qp-complete $\mathcal{D}$-algebras.

###### Proof.

Since $g$ is quasi-planar, by Proposition 17, the following diagram of
conilpotent curved cooperads

${F_{n}^{\mathrm{qp}}\mathcal{C}}$${F_{n}^{\mathrm{qp}}\mathcal{D}}$${\mathcal{C}}$${\mathcal{D}.}$$\scriptstyle{F_{n}^{\mathrm{qp}}g}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{g}$

commutes for every $n\geq 0$. Let $\Lambda$ be a qp-complete pdg
$\mathcal{C}$-algebra. The $\mathcal{D}$-algebra
$g_{*}(F^{n}_{\mathrm{qp}}\Lambda)$ rewrites as

$g_{*}(F^{n}_{\mathrm{qp}}\Lambda)=g_{*}i^{*}i_{!}(\Lambda)\cong
j^{*}(F_{n}^{\mathrm{qp}}g)^{*}i_{!}(\Lambda)\leavevmode\nobreak\ .$

Thus $g_{*}(F^{n}_{\mathrm{qp}}\Lambda)$ is qp-complete as it is the image of
a pdg $F_{n}^{\mathrm{qp}}\mathcal{D}$-algebra. Since $g_{*}$ preserves limits
and since $\Lambda$ is complete, the map

$g_{*}(\Lambda)\longrightarrow\lim_{n\in\omega^{\mathrm{op}}}g_{*}(F^{n}_{\mathrm{qp}}\Lambda)$

is an isomorphism. So $g_{*}(\Lambda)$ is complete as it is the limit of a
diagram of complete algebras. ∎

######  Proposition 26.

Let $g:\mathcal{C}\longrightarrow\mathcal{D}$ be a quasi-planar morphism of
curved cooperads. There is an adjunction

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{alg}^{\mathsf{qp}\text{-}\mathsf{comp}}\leavevmode\nobreak\
.}$$\scriptstyle{g^{*}}$$\scriptstyle{\widehat{g}^{!}}$${\dashv}$

between qp-complete curved $\mathcal{C}$-algebras and qp-complete curved
$\mathcal{D}$-algebras, where the left adjoint $\widehat{g}^{!}$ is given by
$g^{!}$ post-composed with the completion functor of $\mathcal{C}$-algebras.

###### Proof.

Follows from Proposition 24, using the fact that the completion functor of
$\mathcal{C}$-algebras preserves curved objects. ∎

Embedding of cooperads. Let us assume that
$g:\mathcal{C}\longrightarrow\mathcal{D}$ is a arity-wise degree-wise
injection. Since the underlying graded $\mathbb{S}$-module of $\mathcal{C}$ is
cofree and thus injective, we get a section of graded $\mathbb{S}$-modules
$p:\mathcal{D}\longrightarrow\mathcal{C}$ that is left inverse to $g$.

As a consequence, the morphism of comonads on pdg $\Bbbk$-modules
$\mathcal{C}\circ-\longrightarrow\mathcal{D}\circ-$ is object-wise a
monomorphism and the morphism of monads on pdg $\Bbbk$-modules
$(-)^{\mathcal{D}}\longrightarrow(-)^{\mathcal{C}}$ is object-wise an
epimorphism. This implies that the functors

$\displaystyle g_{*}:$ $\displaystyle\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{cog}$ $\displaystyle g_{*}:$
$\displaystyle\mathsf{pdg}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}\longrightarrow\mathsf{pdg}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{alg}$

are fully faithful. Therefore the first $g_{*}$ is the forgetful functor
related to an idempotent comonad and the second $g_{*}$ is the forgetful
functor related to an idempotent monad. In this case, their respective adjoint
functors admit an easy description in terms of pullbacks and pushouts.

######  Lemma 28.

For every pdg $\mathcal{D}$-coalgebra $(W,\Delta_{W},d_{W})$, $g^{!}(W)$ is
given by the following pullback

${g^{!}(W)}$${\mathcal{C}\circ W}$${W}$${\mathcal{D}\circ
W.}$${\lrcorner}$$\scriptstyle{g\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\mathrm{Id}_{W}}$$\scriptstyle{\Delta_{W}}$

in the category of pdg $\Bbbk$-modules.

###### Proof.

Follows directly from the dual of Proposition 71. ∎

######  Lemma 29.

For every pdg $\mathcal{D}$-algebra $(\Lambda,\gamma_{\Lambda},d_{\Lambda})$,
$g^{!}(\Lambda)$ is given by the following pushout

${\Lambda^{\mathcal{D}}}$${\Lambda^{\mathcal{C}}}$${\Lambda}$${g^{!}(\Lambda).}$$\scriptstyle{\gamma_{\Lambda}}$$\scriptstyle{\Lambda^{g}}$${\ulcorner}$

in the category of pdg $\Bbbk$-modules.

###### Proof.

Follows directly from Proposition 71. ∎

###  3.10. The Bar-Cobar adjunction

Koszul duality between dg operads and conilpotent curved cooperads is encoded
by the notion of a curved twisting morphism, see [Gri21] or [RiL22b]. Any
curved twisting morphism $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$
between a conilpotent curved cooperad and a dg operad induces a bar-cobar
adjunction between the categories of dg $\mathcal{P}$-algebras and of curved
$\mathcal{C}$-coalgebras.

The bar construction relative to $\alpha$. Given a dg $\mathcal{P}$-algebra
$(A,\gamma_{A},d_{A})$, one can construct a curved $\mathcal{C}$-coalgebra
$\mathrm{B}_{\alpha}A$ called the bar construction. The underlying graded
$\mathcal{C}$-coalgebra of $\mathrm{B}_{\mathcal{C}}A$ is given by
$\mathcal{C}\circ A$. It is equipped with the unique coderivation whose
projection onto the generators is the sum of the following maps

${d_{1}:\mathcal{C}\circ A}$${A}$${A\leavevmode\nobreak\
,}$$\scriptstyle{d_{A}}$

${d_{2}:\mathcal{C}\circ A}$${\mathcal{P}\circ A}$${A\leavevmode\nobreak\
.}$$\scriptstyle{\alpha\leavevmode\nobreak\ \circ\leavevmode\nobreak\
\mathrm{Id}}$$\scriptstyle{\gamma_{A}}$

One can check that the pdg $\mathcal{C}$-coalgebra $\mathrm{B}_{\alpha}A$ is
in fact a curved $\mathcal{C}$-coalgebra.

The cobar construction relative to $\alpha$. Given a curved
$\mathcal{C}$-coalgebra $(W,\Delta_{W},d_{W})$, one can construct a dg
$\mathcal{P}$-algebra $\Omega_{\alpha}W$ called de cobar construction. The
underlying graded $\mathcal{P}$-algebra is given by $\mathcal{P}\circ W$. It
is equipped with the unique derivation whose restriction to the generators is
the sum of the maps

${d_{1}:W}$${W}$${\mathcal{P}\circ W\leavevmode\nobreak\
,}$$\scriptstyle{d_{W}}$

${d_{2}:W}$${W}$${\mathcal{C}\circ W}$${\mathcal{P}\circ W\leavevmode\nobreak\
.}$$\scriptstyle{(-1).(-)}$$\scriptstyle{\Delta_{W}}$$\scriptstyle{\alpha\leavevmode\nobreak\
\circ\leavevmode\nobreak\ \mathrm{Id}}$

One can check that the pdg $\mathcal{P}$-algebra $\Omega_{\alpha}W$ is in fact
a dg $\mathcal{P}$-algebra.

######  Definition 48 (Twisting morphism).

Let $(W,\Delta_{W},d_{W})$ be a curved $\mathcal{C}$-coalgebra and let
$(A,\gamma_{A},d_{A})$ be a dg $\mathcal{P}$-algebra. A twisting morphism
relative to $\alpha$ is the data of a graded morphism
$\lambda:W\longrightarrow A$ which satisfies the following equation

$\gamma_{A}\leavevmode\nobreak\ (\alpha\circ\lambda)\leavevmode\nobreak\
\Delta_{W}+d_{A}\leavevmode\nobreak\ \lambda-\lambda\leavevmode\nobreak\
d_{W}=0\leavevmode\nobreak\ .$

We denote the set of twisting morphisms between $W$ and $A$ by
$\mathrm{Tw}^{\alpha}(W,A)$.

######  Proposition 27.

Let $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$ be a curved twisting
morphism between a conilpotent curved cooperad $\mathcal{C}$ and a dg operad
$\mathcal{P}$. There are natural isomorphisms

$\mathrm{Hom}_{\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}(\Omega_{\alpha}W,A)\cong\mathrm{Tw}^{\alpha}(W,A)\cong\mathrm{Hom}_{\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}(W,\mathrm{B}_{\alpha}A)\leavevmode\nobreak\
,$

for any curved $\mathcal{C}$-coalgebra $W$ and any dg $\mathcal{P}$-algebra
$A$. This gives the bar-cobar adjunction relative to $\alpha$

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

between the categories of dg $\mathcal{P}$-algebras and the category of curved
$\mathcal{C}$-coalgebras. Furthermore,

###### Proof.

These constructions can be found in [LG19, Section 4.3]. The case where
$\mathcal{C}$ is a dg cooperad can be found in [LV12, Section 11]. ∎

###  3.11. The complete Bar-Cobar adjunction

Any curved twisting morphism $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$
between a conilpotent curved cooperad and a dg operad also induces a complete
bar-cobar adjunction between the categories of dg $\mathcal{P}$-coalgebras and
of curved $\mathcal{C}$-algebras.

The complete cobar construction. Given a dg $\mathcal{P}$-coalgebra
$(V,\Delta_{V},d_{V})$, one can construct a curved $\mathcal{C}$-algebra
$\widehat{\Omega}_{\alpha}V$ called the complete cobar construction. The
underlying graded $\mathcal{C}$-algebra is given by $V^{\mathcal{C}}$. It is
equipped with the unique derivation whose restriction to the generators is the
sum of the maps

${d_{1}:V}$${V}$${V^{\mathcal{C}}\leavevmode\nobreak\ ,}$$\scriptstyle{d_{V}}$

${d_{2}:V}$${V}$${V^{\mathcal{P}}}$${V^{\mathcal{C}}\leavevmode\nobreak\
.}$$\scriptstyle{(-1).(-)}$$\scriptstyle{\Delta_{V}}$$\scriptstyle{(\mathrm{Id})^{\alpha}}$

One can check that the pdg $\mathcal{C}$-algebra $\widehat{\Omega}_{\alpha}V$
is in fact a qp-complete curved $\mathcal{C}$-algebra.

The complete bar construction. Given a curved $\mathcal{C}$-algebra
$(\Lambda,\gamma_{\Lambda},d_{\Lambda})$, one can construct a dg
$\mathcal{P}$-coalgebra $\widehat{\mathrm{B}}_{\alpha}\Lambda$ called the
complete bar construction. The underlying graded $\mathcal{P}$-coalgebra is
given by $L^{\mathcal{P}}\Lambda$. It is equipped with the unique coderivation
whose projection onto the generators is the sum of the maps

${d_{1}:L^{\mathcal{P}}\Lambda}$${\Lambda}$${\Lambda\leavevmode\nobreak\
,}$$\scriptstyle{d_{\Lambda}}$

${d_{2}:L^{\mathcal{P}}\Lambda}$${\Lambda^{\mathcal{P}}}$${\Lambda^{\mathcal{C}}}$${\Lambda\leavevmode\nobreak\
.}$$\scriptstyle{(\mathrm{Id})^{\alpha}}$$\scriptstyle{\gamma_{B}}$

######  Remark 40.

The cofree $\mathcal{P}$-coalgebra functor $L^{\mathcal{P}}$ behaves well with
respect to coderivations, which can be induced simply by a graded map onto the
cogenerators. We refer to [GL22, Section 6.5] for these type of results.

######  Definition 49 (Twisting morphism).

Let $(\Lambda,\gamma_{\Lambda},d_{\Lambda})$ be a curved $\mathcal{C}$-algebra
and let $(C,\Delta_{C},d_{C})$ be a dg $\mathcal{P}$-coalgebra. A twisting
morphism relative to $\alpha$ is the data of a graded morphism
$\lambda:C\longrightarrow\Lambda$ which satisfies the following equation

$\gamma_{\Lambda}\leavevmode\nobreak\ (\lambda)^{\alpha}\leavevmode\nobreak\
\Delta_{C}+d_{\Lambda}\leavevmode\nobreak\ \lambda-\lambda\leavevmode\nobreak\
d_{C}=0\leavevmode\nobreak\ .$

We denote the set of twisting morphisms between $C$ and $\Lambda$ by
$\mathrm{Tw}^{\alpha}(C,\Lambda)$.

######  Proposition 28.

Let $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$ be a curved twisting
morphism between a conilpotent curved cooperad $\mathcal{C}$ and a dg operad
$\mathcal{P}$. There are natural isomorphisms

$\mathrm{Hom}_{\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{coalg}}\left(C,\widehat{\mathrm{B}}_{\alpha}\Lambda\right)\cong\mathrm{Tw}^{\alpha}(C,\Lambda)\cong\mathrm{Hom}_{\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}\left(\widehat{\Omega}_{\alpha}C,\Lambda\right)\leavevmode\nobreak\
,$

for any curved $\mathcal{C}$-algebra $\Lambda$ and any dg
$\mathcal{P}$-coalgebra $C$. This gives the complete bar-cobar adjunction
relative to $\alpha$

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

between the categories of dg $\mathcal{P}$-coalgebras and the category of
curved $\mathcal{C}$-algebras.

###### Proof.

These constructions can be found in [GL22, Section 8]. ∎

######  Remark 41.

It is also an adjunction

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

since $\widehat{\Omega}_{\alpha}$ naturally lands in qp-complete curved
$\mathcal{C}$-algebras.

###  3.12. Compatibility with the morphisms of (co)operads.

Let $f:\mathcal{P}\longrightarrow\mathcal{Q}$ be a morphims of dg operads, let
$g:\mathcal{C}\longrightarrow\mathcal{D}$ be a morphism of conilpotent curved
cooperads, and let $\alpha:\mathcal{C}\longrightarrow\mathcal{P}$ and
$\beta:\mathcal{D}\longrightarrow\mathcal{Q}$ be two curved twisting
morphisms, such that the following diagram commutes:

${\mathcal{C}}$${\mathcal{P}}$${\mathcal{D}}$${\mathcal{Q}\leavevmode\nobreak\
.}$$\scriptstyle{\alpha}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{\beta}$

Then the bar-cobar adjunctions are compatible with the natural adjunctions
induced by the morphisms, meaning that the following square of adjunctions

${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{alg}}$$\scriptstyle{\Omega_{\alpha}}$$\scriptstyle{g^{!}}$$\scriptstyle{f^{*}}$$\scriptstyle{\mathrm{B}_{\alpha}}$$\scriptstyle{\Omega_{\beta}}$$\scriptstyle{g_{*}}$$\scriptstyle{\mathrm{B}_{\beta}}$$\scriptstyle{f_{!}}$${\dashv}$${\dashv}$${\dashv}$${\dashv}$

is commutative. The complete bar-cobar adjunctions are also compatible with
the natural adjunctions induced by the morphisms, meaning that the following
square of adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{cog}}$${\mathsf{curv}\leavevmode\nobreak\
\mathcal{D}\text{-}\mathsf{alg}}$$\scriptstyle{\widehat{\Omega}_{\alpha}}$$\scriptstyle{f_{!}}$$\scriptstyle{g_{*}}$$\scriptstyle{\widehat{\mathrm{B}}_{\alpha}}$$\scriptstyle{\widehat{\Omega}_{\beta}}$$\scriptstyle{f^{*}}$$\scriptstyle{\widehat{\mathrm{B}}_{\beta}}$$\scriptstyle{g^{!}}$${\dashv}$${\dashv}$${\dashv}$${\dashv}$

is commutative.

######  Remark 42.

Under the extra assumption that the morphism
$g:\mathcal{C}\longrightarrow\mathcal{D}$ is quasi-planar, one can consider
qp-complete algebras in the right hand side of the last commutative square of
adjunctions, where $g^{!}$ is replaced by $\widehat{g}^{!}$.

###  3.13. (Co)admissible operads

In this subsection, we review various constructions that allow one the give a
meaning to the homotopy theory of dg $\mathcal{P}$-(co)algebras, where
$\mathcal{P}$ is a dg operad.

######  Definition 50 (Admissible dg operad).

A dg operad $\mathcal{P}$ is called admissible if the category of dg
$\mathcal{P}$-algebras admits a combinatorial model structure on right-
transferred along the free-forgetful adjunction

${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{\mathcal{P}\leavevmode\nobreak\ \circ\leavevmode\nobreak\
-}$$\scriptstyle{\mathrm{U}}$${\dashv}$

determined by the following sets of maps

  1. (1)

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

  2. (2)

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

  3. (3)

the generating cofibrations are the maps $\mathcal{P}\circ
S^{n}\longrightarrow\mathcal{P}\circ D^{n+1}$ for $n\in\mathbb{Z}$;

  4. (4)

the generating acyclic cofibrations are the maps $\mathcal{P}\circ
0\longrightarrow\mathcal{P}\circ D^{n+1}$ for $n\in\mathbb{Z}$.

######  Definition 51 (Coadmissible dg operad).

A dg operad $\mathcal{P}$ is called coadmissible if the category of dg
$\mathcal{P}$-coalgebras admits a combinatorial model structure left-
transferred along the forgetful-cofree adjunction

${\mathsf{dg}\leavevmode\nobreak\
\Bbbk\text{-}\mathsf{mod}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}\leavevmode\nobreak\
,}$$\scriptstyle{L^{\mathcal{P}}}$$\scriptstyle{\mathrm{U}}$${\dashv}$

determined by the following sets of maps

  1. (1)

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

  2. (2)

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

  3. (3)

the generating (acyclic) cofibrations are the maps between small objects so
whose image through the forgetful functor is an (acyclic) cofibration.

######  Remark 43.

Let $\Bbbk$ be a characteristic zero field. Then every dg operad is
admissible. Even in characteristic zero, not every dg operad is coadmissible.
For instance, the operad $\mathrm{u}\mathcal{C}\mathrm{om}$ is not
coadmissible, see [GL22, Section 8.3].

For a dg operad $\mathcal{P}$ to be admissible (resp. coadmissible) it
suffices to provide a natural path object (resp. a natural cylinder object).

######  Proposition 29 ([BM03]).

A dg operad $\mathcal{P}$ that is $\mathcal{E}$-split is admissible and
coadmissible. In particular, a cofibrant operad is admissible and
coadmissible.

###### Proof.

As described in [BF04], the cellular model of the interval $I$ has the
canonical structure of a $\mathcal{E}$-coalgebra.

For every dg $\mathcal{P}$-algebra $A$, the dg module $[I,A]$ inherits the
structure of a dg $\mathcal{E}\otimes\mathcal{P}$-algebra and therefore the
structure of a dg $\mathcal{P}$-algebra. This provides a natural path object,
and proves the existence of the transferred model structure on dg
$\mathcal{P}$-algebras.

For every dg $\mathcal{P}$-coalgebra $V$, the dg module $V\otimes I$ inherits
the structure of a dg $\mathcal{E}\otimes\mathcal{P}$-coalgebra and therefore
the structure of a dg $\mathcal{P}$-coalgebra. This provides a natural
cylinder object, and proves the existence of the transferred model structure
on dg $\mathcal{P}$-coalgebras. ∎

######  Example 4.

For any dg operad $\mathcal{P}$, the dg operad $\mathcal{E}\otimes\mathcal{P}$
is $\mathcal{E}$-split. Indeed, the Barratt-Eccles operad is a dg Hopf operad:
in particular, there exists a morphism of dg operad
$\mathcal{E}\longrightarrow\mathcal{E}\otimes\mathcal{E}$, which induces an
$\mathcal{E}$-splitting on $\mathcal{E}\otimes\mathcal{P}$. Thus,
$\mathcal{E}\otimes\mathcal{P}$ is always admissible and coadmissible.

######  Proposition 30.

Let $f:\mathcal{P}\longrightarrow\mathcal{Q}$ be a morphism of dg operads.

  1. (1)

If $\mathcal{P}$ and $\mathcal{Q}$ are both admissible, the adjunction

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

is a Quillen adjunction.

  2. (2)

If $\mathcal{P}$ and $\mathcal{Q}$ are both coadmissible, the adjunction

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

is a Quillen adjunction.

###### Proof.

This is a consequence of the fact that these model structures are transferred
from dg modules. It is immediate to see that $f^{*}$ preserves quasi-
isomorphisms and fibrations of dg $\mathcal{Q}$-algebras, and that $f^{*}$
preserves quasi-isomorphisms and cofibrations of dg $\mathcal{Q}$-coalgebras.
∎

######  Lemma 30 (After [GL22]).

Let $\mathcal{P}$ be a dg operad and let $K$ be an arity-wise acylic dg
$\mathbb{N}$-module. We denote
$\mathcal{Q}=\mathcal{P}\sqcup\mathbb{T}_{\mathrm{pl}}\left(K\otimes\mathbb{S}\right)$,
$i$ the canonical inclusion of $\mathcal{P}$ into $\mathcal{Q}$ and $p$ the
canonical projection from $\mathcal{Q}$ onto $\mathcal{P}$. This gives the
following diagram

${\mathcal{P}}$${\mathcal{Q}}$${\mathcal{P}\leavevmode\nobreak\
,}$$\scriptstyle{i}$$\scriptstyle{p}$

where the inclusion $i$ is an acyclic cofibration and $p$ is an acyclic
fibration. If $\mathcal{P}$ is $\mathcal{E}$-split, so is $\mathcal{Q}$.
Furthemore, the induced Quillen adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$$\scriptstyle{i_{!}}$$\scriptstyle{i^{*}}$$\scriptstyle{p_{!}}$$\scriptstyle{p^{*}}$${\dashv}$${\dashv}$

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$$\scriptstyle{i_{!}}$$\scriptstyle{i^{*}}$$\scriptstyle{p_{!}}$$\scriptstyle{p^{*}}$${\dashv}$${\dashv}$

are Quillen equivalences.

###### Proof.

The fact that the inclusion $\mathcal{P}\longrightarrow\mathcal{Q}$ is an
acyclic cofibration and that its left inverse
$\mathcal{Q}\longrightarrow\mathcal{P}$ is a acyclic fibration is clear.

Let $e:\mathcal{P}\longrightarrow\mathcal{E}\otimes\mathcal{P}$ be the
$\mathcal{E}$-splitting of the dg operad $\mathcal{P}$. An
$\mathcal{E}$-splitting of the dg operad $\mathcal{Q}$ is given by the
morphism

${\mathcal{P}}$${\mathcal{E}\otimes\mathcal{P}}$${\mathcal{E}\otimes\mathcal{Q}}$$\scriptstyle{e}$$\scriptstyle{\mathrm{Id}\leavevmode\nobreak\
\otimes\leavevmode\nobreak\ i}$

together with the choice of a section $K\longrightarrow\mathcal{E}\otimes K$
of the projection $\mathcal{E}\otimes K\longrightarrow K$ in the category of
dg $\mathbb{N}$-module.

Let us prove that the Quillen adjunctions relating dg $\mathcal{Q}$-algebras
to dg $\mathcal{P}$-algebras are Quillen equivalences. This amounts to prove
that the two right Quillen functors

$\displaystyle\mathrm{Ho}(p^{\ast}):\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg})\longrightarrow\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{alg})$
$\displaystyle\mathrm{Ho}(i^{\ast}):\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{alg})\longrightarrow\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg})$

are equivalences. Let us show that these are inverse equivalences. On the one
hand, we have a canonical isomorphism $i^{\ast}p^{\ast}\simeq\mathrm{Id}$,
thus we get an isomorphism
$\mathrm{Ho}(i^{\ast})\mathrm{Ho}(p^{\ast})\simeq\mathrm{Id}$.

Notice that the data of a dg $\mathcal{Q}$-algebra structure amounts to the
data of a dg $\mathcal{P}$-algebra structure and of a morphism of dg module
$K\circ_{\mathrm{pl}}A\longrightarrow A$. We define an endofunctor
$\tilde{(-)}$ of the category of dg $\mathcal{Q}$-algebras as follows. Let $A$
be a dg $\mathcal{Q}$-algebra. Its image $\tilde{A}$ is the dg
$\mathcal{Q}$-algebra whose underlying dg module is the path object
$[I,A]=I^{\ast}\otimes A$ (where $I$ is the cellular model of the interval).
Its dg $\mathcal{P}$-algebra structure is given by the following maps

${\mathcal{P}(n)\otimes(I^{\ast}\otimes A)^{\otimes
n}}$${\mathcal{E}(n)\otimes\mathcal{P}(n)\otimes(I^{\ast}\otimes A)^{\otimes
n}}$${\mathcal{E}(n)\otimes(I^{\ast})^{\otimes n}\otimes\mathcal{P}(n)\otimes
A^{\otimes n}}$${I^{\ast}\otimes A,}$$\scriptstyle{e(n)\leavevmode\nobreak\
\otimes\leavevmode\nobreak\
\mathrm{Id}}$$\scriptstyle{\simeq}$$\scriptstyle{\gamma_{I}^{n}\leavevmode\nobreak\
\otimes\leavevmode\nobreak\ \gamma_{A}^{n}|_{\mathcal{P}}}$

where $\gamma_{A}^{n}|_{\mathcal{P}}$ is given by the dg $\mathcal{P}$-algebra
structure on $A$ and where $\gamma_{I}^{n}$ is given by the dg
$\mathcal{E}$-algebra structure on $I$ constructed in [BF04]. The action of
$K$ is given as follows

${K(n)\otimes(I^{\ast}\otimes A)^{\otimes n}}$${K(n)\otimes A^{\otimes
n}}$${A}$${I^{\ast}\otimes A}$$\scriptstyle{\gamma_{A}^{n}|_{K}}$

where the first map results from the inclusion of the first point into the
interval $\Bbbk\longrightarrow I$ which gives a map
$I^{\ast}\longrightarrow\Bbbk$, and where $\gamma_{A}^{n}|_{K}$ is the action
of $K$ on $A$ given by its dg $\mathcal{Q}$-algebra structure.

We have canonical natural transformations

$p^{\ast}i^{\ast}\xleftarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\tilde{(-)}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{Id}$

that are object-wise quasi-isomorphisms. Hence we get an isomorphism
$\mathrm{Ho}(p^{\ast})\mathrm{Ho}(i^{\ast})\simeq\mathrm{Id}$. Therefore,
$\mathrm{Ho}(i^{\ast})$ and $\mathrm{Ho}(p^{\ast})$ are inverse to each other
and their related Quillen adjunctions are Quillen equivalences.

Finally, let us prove that the Quillen adjunctions relating dg
$\mathcal{Q}$-coalgebras to dg $\mathcal{P}$-coalgebras are also Quillen
equivalences. Again, this amounts to prove that the two right adjoint functors

$\displaystyle\mathrm{Ho}(p^{\ast}):\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog})\longrightarrow\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{cog})$
$\displaystyle\mathrm{Ho}(i^{\ast}):\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{Q}\text{-}\mathsf{cog})\longrightarrow\mathrm{Ho}(\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog})$

are equivalences. This follows from similar arguments as those used in the
algebra case. There is a canonical isomorphism
$i^{\ast}p^{\ast}\simeq\mathrm{Id}$.

Again, a dg $\mathcal{Q}$-coalgebra structure is completely determined by a dg
$\mathcal{P}$-coalgebra and a morphism of dg modules $V\longrightarrow
V^{K\otimes\mathbb{S}}$. We define an endofunctor $\tilde{(-)}$ category of dg
$\mathcal{Q}$-coalgebras as follows. Let $V$ be a dg $\mathcal{Q}$-coalgebra.
The underlying dg module of $\tilde{V}$ is the cylinder object $I\otimes V$.
Its dg $\mathcal{P}$-coalgebra is given by

${I\otimes V\otimes\mathcal{P}(n)}$${I\otimes
V\otimes\mathcal{E}(n)\otimes\mathcal{P}(n)}$${\mathcal{E}(n)\otimes I\otimes
V\otimes\mathcal{P}(n)}$${I^{\otimes n}\otimes V^{\otimes n}}$${(I\otimes
V)^{\otimes n}\leavevmode\nobreak\
,}$$\scriptstyle{\mathrm{Id}\leavevmode\nobreak\ \otimes\leavevmode\nobreak\
e}$$\scriptstyle{\simeq}$$\scriptstyle{\Delta_{I}^{n}\leavevmode\nobreak\
\otimes\leavevmode\nobreak\
\Delta_{V}^{n}|_{\mathcal{P}}}$$\scriptstyle{\simeq}$

where $\Delta_{V}^{n}|_{\mathcal{P}}$ is the given by the dg
$\mathcal{P}$-coalgebra structure on $A$ and where $\Delta_{I}^{n}$ is the dg
$\mathcal{E}$-algebra on $I$ constructed in [BF04]. The coaction of $K$ is
given by

${I\otimes V\otimes K(n)}$${V\otimes K(n)}$${V^{\otimes n}}$${(I\otimes
V)^{\otimes n}\leavevmode\nobreak\ ,}$$\scriptstyle{\Delta_{V}^{n}|_{K}}$

where $\Delta_{V}^{n}|_{K}$ is the coaction of $K$ on $V$ and where the last
map proceed from the inclusion of the first point into the interval
$\Bbbk\longrightarrow I$. We have canonical natural transformations

$p^{\ast}i^{\ast}\xleftarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\tilde{(-)}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathrm{Id}$

that are object-wise quasi-isomorphisms and yield an isomorphism
$\mathrm{Ho}(p^{\ast})\mathrm{Ho}(i^{\ast})\simeq\mathrm{Id}$. Therefore,
$\mathrm{Ho}(i^{\ast})$ and $\mathrm{Ho}(p^{\ast})$ are inverse equivalences
and their related Quillen adjunctions are Quillen equivalences. ∎

######  Proposition 31 ([BM03, GL22]).

Let
$f:\mathcal{P}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{Q}$
be a quasi-isomorphism of cofibrant dg operads. The induced Quillen
adjunctions

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

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

are both Quillen equivalences.

###### Proof.

Let us denote $QE$ the (large) set of morphisms of cofibrant operads
$f:\mathcal{P}\longrightarrow\mathcal{Q}$ whose induced Quillen adjunctions
relating their algebras and relating their coalgebras are Quillen equivalence.
One can notice that $QE$ is stable through composition and retracts and
satisfies the 2-out-of-3 law. By Lemma 30 is contains morphisms of the form
$\mathcal{P}\longrightarrow\mathcal{P}\sqcup\mathbb{T}_{\mathrm{pl}}(K\otimes\mathbb{S})$,
where $K$ is an acyclic dg $\mathbb{N}$-module. Since every acyclic
cofibration of cofibrant dg operads is a retract of a map having this from,
$QE$ contains all the acyclic cofibration of cofibrant dg operads. Now, let
$f:\mathcal{P}\longrightarrow\mathcal{Q}$ be an weak-equivalence of cofibrant
dg operad. We can factorise the map
$\mathcal{P}\sqcup\mathcal{Q}\longrightarrow\mathcal{Q}$ by a cofibration
followed by an acyclic cofibration

${\mathcal{P}\sqcup\mathcal{Q}}$${\mathcal{R}}$${\mathcal{Q}.}$$\scriptstyle{\sim}$

The maps $\mathcal{P}\longrightarrow\mathcal{R}$ and
$\mathcal{Q}\longrightarrow\mathcal{R}$ belong to $QE$ since they are acyclic
cofibrations. Then, the map $\mathcal{R}\longrightarrow\mathcal{P}$ belongs to
$QE$ by the 2-out-of-3 law. So $f$ belongs to $QE$ as composition of two maps
in $QE$. ∎

Semi-model category structures and reduction the admissible case. Under weaker
assumptions, M. Spitzweck constructed in [Spi01] a semi-model category
structure on the category of dg $\mathcal{P}$-algebras. Nevertheless, up to a
convenient replacement of the initial dg operad $\mathcal{P}$, one can always
assume the existence of a model category structure without losing any
homotopical information.

######  Theorem 4 ([Spi01]).

Let $\mathcal{P}$ be a $\mathbb{S}$-projective dg operad. The category of dg
$\mathcal{P}$-algebras admits a semi-model category structure, determined by
the following sets of maps

  1. (1)

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

  2. (2)

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

  3. (3)

the set of cofibrations is given by morphisms with a left-lifting property
against acyclic fibrations.

These semi-model category structures are stable under quasi-isomorphisms of dg
operads.

######  Theorem 5 ([Fre09]).

Let
$f:\mathcal{P}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{Q}$
be a quasi-isomorphism of $\mathbb{S}$-projective dg operads. The induced
adjunction

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

is an equivalence of semi-model categories.

In particular, for any $\mathbb{S}$-projective dg operad $\mathcal{P}$, the
canonical quasi-isomorphism
$\varphi_{\mathcal{P}}:\mathcal{E}\otimes\mathcal{P}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$
always induces an equivalence of semi-model categories

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{E}\otimes\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}\leavevmode\nobreak\
,}$$\scriptstyle{(\varphi_{\mathcal{P}})_{!}}$$\scriptstyle{\varphi_{\mathcal{P}}^{*}}$${\dashv}$

where on the left-hand side there is a model category structure on the
category of dg $\mathcal{E}\otimes\mathcal{P}$-algebras. Recall that
$\mathcal{E}\otimes\mathcal{P}$ is admissible since it is $\mathcal{E}$-split.
Therefore, by replacing $\mathcal{P}$ with $\mathcal{E}\otimes\mathcal{P}$ we
can always "rectify" the semi-model structure into a model structure, and both
present the same underlying homotopy category.

###  3.14. Quasi-planar Bar-Cobar adjunctions

In the subsequent sections, we will manly work with dg operads of the form
$\Omega\mathcal{C}$, where $\mathcal{C}$ is a quasi-planar conilpotent curved
cooperad. Nevertheless, if one starts with a dg operad $\mathcal{P}$, there is
a canonical quasi-planar conilpotent curved cooperad given by
$\mathrm{B}(\mathcal{E}\otimes\mathcal{P})$. One can then construct quasi-
planar bar-cobar adjunctions between their respective (co)algebras categories.
It will follow from the results in the subsequent sections that these
adjunctions are Quillen adjunctions when $\mathcal{P}$ is (co)admissible and
induce Quillen equivalences when $\mathcal{P}$ is cofibrant.

Let $\mathcal{P}$ be a dg operad. Let
$\varphi_{\mathcal{P}}:\mathcal{E}\otimes\mathcal{P}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\mathcal{P}$
be the canonical quasi-isomorphism of dg operads, where $\mathcal{E}$ is the
Barratt-Eccles operad.

Quasi-planar bar-cobar adjunction. The morphism $\varphi_{\mathcal{P}}$
induces an adjunction
$(\varphi_{\mathcal{P}})_{!}\dashv\varphi_{\mathcal{P}}^{*}$ between dg
$\mathcal{P}$-algebras and dg $\mathcal{E}\otimes\mathcal{P}$-algebras. We
define the quasi-planar bar-cobar adjunction

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

as the following composition of adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{E}\otimes\mathcal{P}\text{-}\mathsf{alg}}$${\mathsf{curv}\leavevmode\nobreak\
\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\text{-}\mathsf{cog}\leavevmode\nobreak\
.}$$\scriptstyle{\varphi_{\mathcal{P}}^{*}}$$\scriptstyle{(\varphi_{\mathcal{P}})_{!}}$$\scriptstyle{\mathrm{B}_{\pi}}$$\scriptstyle{\Omega_{\pi}}$${\dashv}$${\dashv}$

Quasi-planar complete bar-cobar adjunction. The morphism
$\varphi_{\mathcal{P}}$ induces an adjunction
$(\varphi_{\mathcal{P}})_{!}\dashv\varphi_{\mathcal{P}}^{*}$ between dg
$\mathcal{P}$-coalgebras and dg $\mathcal{E}\otimes\mathcal{P}$-coalgebras. We
define the quasi-planar complete bar-cobar adjunction

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

as the following composition of adjunctions

${\mathsf{dg}\leavevmode\nobreak\
\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{dg}\leavevmode\nobreak\
\mathcal{E}\otimes\mathcal{P}\text{-}\mathsf{cog}}$${\mathsf{curv}\leavevmode\nobreak\
\mathrm{B}(\mathcal{E}\otimes\mathcal{P})\text{-}\mathsf{alg}\leavevmode\nobreak\
.}$$\scriptstyle{\varphi_{\mathcal{P}}^{*}}$$\scriptstyle{(\varphi_{\mathcal{P}})_{!}}$$\scriptstyle{\widehat{\Omega}_{\pi}}$$\scriptstyle{\widehat{\mathrm{B}}_{\pi}}$${\dashv}$${\dashv}$

##  4\. Model structure on coalgebras over a cooperad

The goal of this section is to study the homotopical properties of the bar-
cobar adjunction between dg $\Omega\mathcal{C}$-algebras and curved
$\mathcal{C}$-coalgebras, in the case where $\mathcal{C}$ is a quasi-planar
conilpotent curved cooperad.

The dg operad $\Omega\mathcal{C}$ is cofibrant, and therefore admissible by
Proposition 29. This means that the category of dg
$\Omega\mathcal{C}$-algebras admits a model category structure where weak-
equivalences are given by qausi-isomorphisms and fibrations by degree-wise
epimorphisms. Let us consider the bar-cobar adjunction

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

relative to $\iota:\mathcal{C}\longrightarrow\Omega\mathcal{C}$, which will be
denoted by $\Omega_{\mathcal{C}},\mathrm{B}_{\mathcal{C}}$ from now on, since
we will not consider any other curved twisting morphism. Our first goal is
going to be to transfer this model category structure along the bar-cobar
adjunction to the category of dg $\mathcal{C}$-coalgebras.

######  Theorem 6.

Let $\mathcal{C}$ be a quasi-planar conilpotent curved cooperad. The exists a
combinatorial model category structure on the category of curved
$\mathcal{C}$-coalgebras given by the following sets of maps:

  1. (1)

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

  2. (2)

the set of cofibrations is given by morphisms $f$ such that
$\Omega_{\mathcal{C}}(f)$ is a cofibration; they correspond to degree-wise
injections,

  3. (3)

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

######  Remark 44.

Using the standard transfer theorem for model category structures only gives
that cofibrations are morphism which are sent by $\Omega_{\mathcal{C}}$ to
cofibrations. The theorem contains an additional characterization of
cofibrations of curved $\mathcal{C}$-coalgebras as degree-wise injective maps.

###  4.1. Outline of the transfer of model structures

Let us first start by defining the sets of morphisms of curved
$\mathcal{C}$-coalgebras that we intend to study.

######  Definition 52 (Cofibrations).

A morphism $f$ of curved $\mathcal{C}$-coalgebras is a cofibration if
$\Omega_{\mathcal{C}}(f)$ is a cofibration of dg $\Omega\mathcal{C}$-algebras.

######  Definition 53 (Weak-equivalences).

A morphism $f$ of curved $\mathcal{C}$-coalgebras is a weak-equivalence if
$\Omega_{\mathcal{C}}(f)$ is a quasi-isomorphism of dg
$\Omega\mathcal{C}$-algebras.

######  Definition 54 (Fibrations).

A morphism of curved $\mathcal{C}$-coalgebras is a fibration if it has the
right-lifting property against all acyclic cofibrations.

Both categories $\mathsf{curv}\leavevmode\nobreak\
\mathcal{C}\text{-}\mathsf{cog}$ and $\mathsf{dg}\leavevmode\nobreak\
\Omega\mathcal{C}\text{-}\mathsf{alg}$ are presentable, therefore by Appendix
A.2 it suffices to exhibit a natural cofibrant resolution and a natural
cylinder for coalgebras to prove the existence of the transferred model
structure. We will show that cofibrations of Definition 52 are given by
degree-wise injective maps in Proposition 36 and we will construct a natural
cylinder in Proposition 39.

For the rest of this section, let us fix a quasi-planar conilpotent curved
cooperad $\mathcal{C}$ whose quasi-planar ladder is indexed by some small
ordinal $\alpha$.

###  4.2. Elementary cofibrations

Elementary cofibrations are a particularly well-behaved set of cofibrations of
curved $\mathcal{C}$-coalgebras. For any such elementary cofibration, its
cokernel is a dg $\Bbbk$-module.

######  Definition 55 (Elementary cofibrations).

A morphism $f:W^{\prime}\rightarrowtail W$ of curved $\mathcal{C}$-coalgebras
is an elementary cofibration if it is degree-wise injective and if the map
$\overline{\Delta}_{W}:W\longrightarrow\overline{\mathcal{C}}\circ W$ factors
through the sub-object $\overline{\mathcal{C}}\circ
W^{\prime}\subseteq\overline{\mathcal{C}}\circ W$, that is, if there exists a
dashed map

${W}$${\mathcal{C}\circ W}$${\overline{\mathcal{C}}\circ
W^{\prime}}$${\overline{\mathcal{C}}\circ
W}$$\scriptstyle{\Delta_{W}}$$\scriptstyle{\overline{\mathcal{C}}\leavevmode\nobreak\
\circ\leavevmode\nobreak\ f}$

that makes the diagram commute. Here $\Delta_{W}$ denotes the structural map
of $W$.

######  Remark 45.

The induced map $\overline{\mathcal{C}}\circ
W^{\prime}\longrightarrow\overline{\mathcal{C}}\circ W$ is also monomorphism
since it is has a left inverse in the category of graded $\Bbbk$-modules.

######  Lemma 31.

Let $f:W^{\prime}\rightarrowtail W$ be an elementary cofibration curved
$\mathcal{C}$-coalgebras. Then $\mathrm{Coker}(f)$ is a dg module.

###### Proof.

Let us denote $p_{f}:W\twoheadrightarrow\mathrm{Coker}(f)$ the canonical
projection. We have

$d^{2}p_{f}=p_{f}d^{2}=-p_{f}(\theta\circ\mathrm{Id}_{W})\overline{\Delta}_{W}=-(\theta\circ
p_{f})\overline{\Delta}_{W}=-(\theta\circ\mathrm{Id}_{\mathrm{Coker}(f)})(\mathcal{C}\circ
p_{f})\overline{\Delta}_{W}.$

The definition of an elementary cofibration tells us that $(\mathcal{C}\circ
p_{f})\overline{\Delta}_{W}=0$. We conclude by noticing that $p_{f}$ is an
epimorphism. ∎

######  Proposition 32.

Let $f:W^{\prime}\rightarrowtail W$ be an elementary cofibration. Then $f$ is
a cofibration.

###### Proof.

Identifying $W^{\prime}$ with its image in $W$, we can decompose the
underlying graded $\Bbbk$-module of $W$ as the a direct $W\cong
W^{\prime}\oplus U$. The pre-differential of $W$ decomposes as the pre-
differential $d_{W^{\prime}}$ on $V$, the differential $d_{U}$ on $U$ and a
degree $-1$ map $\zeta:U\longrightarrow W^{\prime}$. The map of graded
$\Bbbk$-modules

$g:U\hookrightarrow W\hookrightarrow\mathcal{P}\circ W$

induces a morphism of dg modules

$D^{0}\otimes U\longrightarrow\Omega_{\mathcal{C}}W\leavevmode\nobreak\ ,$

whose restriction to $S^{0}\otimes U$ is $g$ and whose restriction to
$S^{-1}\otimes U$ is $-\mathrm{Id}\otimes\partial(g)$. One can notice that
this restriction to $S^{-1}\otimes U$ factors through the sub-dg module
$\Omega_{\mathcal{C}}W^{\prime}\subset\Omega_{\mathcal{C}}W$. We thus get a
commutative diagram of dg modules

${S^{-1}\otimes U}$${\Omega_{\mathcal{C}}W^{\prime}}$${D^{0}\otimes
U}$${\Omega_{\mathcal{C}}W\leavevmode\nobreak\ ,}$$\scriptstyle{f}$

which gives a commutative diagram of dg $\Omega\mathcal{C}$-algebras

${\Omega\mathcal{C}\circ(S^{-1}\otimes
U)}$${\Omega_{\mathcal{C}}W^{\prime}}$${\Omega\mathcal{C}\circ(D^{0}\otimes
U)}$${\Omega_{\mathcal{C}}W\leavevmode\nobreak\
.}$${\ulcorner}$$\scriptstyle{f}$

This diagram is a pushout square since the underlying diagram of graded
$\Omega\mathcal{C}$-algebras is a pushout square. Since the left vertical map
is a cofibration, so is the right vertical map $f$. ∎

######  Lemma 32.

Let us consider a commutative diagram of curved $\mathcal{C}$-coalgebras

${U}$${U}$${W^{\prime}}$${W}$$\scriptstyle{\mathrm{Id}}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{g}$

where

  1. (1)

$i$ and $j$ are elementary cofibrations,

  2. (2)

the map of dg modules induced by $g$

$\bar{g}:W^{\prime}/U\longrightarrow W/U$

is a quasi-isomorphism.

Then $g$ is a weak-equivalence.

###### Proof.

Let us decompose the underlying graded $\Bbbk$-module of $W$ as a direct sum
$W\cong U\oplus Y$. We decompose the underlying graded $\Bbbk$-module of