We make use of the cotangent complex formalism developed by Lurie to formulate Quillen cohomology of algebras over an operad in a general base category. Moreover, using the same machinery, we introduce a spectral Hochschild cohomology theory for enriched operads and algebras over them. We prove further that both the Quillen and Hochschild cohomologies of algebras over an operad can be controlled by the corresponding cohomologies of the operad itself. When passing to the category of simplicial sets, we assert that both these cohomology theories for operads, as well as their associated algebras, can be calculated in the same framework of spectrum valued functors on the twisted arrow $\infty$-category of the operad of interest. This assertion allows us to establish a connection between Hochschild and Quillen cohomologies of simplicial operads. Additionally, we formulate the Quillen principle for $\E_n$-algebras in simplicial sets, which provides a convenient fiber sequence relating the Hochschild and cotangent complexes of such objects. This determines an unstable-analogue of a significant result obtained by Francis and Lurie. Our strategy introduces a novel perspective, focusing solely on the intrinsic properties of the twisted arrow $\infty$-categories of operads.

§ INTRODUCTION

In our prior work [1], we have formulated Quillen cohomology of operads enriched over a general base category, thus expanding the work of Harpaz-Nuiten-Prasma [2] on Quillen cohomology of enriched categories. The aim of the present paper is to consider further Quillen cohomology of algebras over an operad, and in addition, to introduce a spectral Hochschild-cohomology theory specifically tailored for such objects.

Another significant goal of the paper is to delve into exploring the cohomologies of simplicial operads and their associated algebras. More specifically, we aim to provide additional clarification while simultaneously broadening the scope of the results from [[1], $\S$6].

§.§ Classical Hochschild and Quillen cohomologies

The cohomology theories mentioned have garnered widespread attention in literature due to their versatility and broad applications. Starting from the deformations of various algebraic structures to the obstruction theory within homotopical algebra, through realms like representation theory  and noncommutative geometry, Hochschild and Quillen cohomologies stand as pivotal tools indispensable in the study of these theories.

Hochschild cohomology (cf. [3]) was initially established as a fundamental cohomology theory for associative algebras.  Let $\textbf{k}$ be a commutative ring and let $A$ be a $\textbf{k}$-algebra. First recall that the structure of a bimodule over $A$ is equivalent to that of a (left) module over $A\otimes A^{\op}$. Conceptually, the $n$'th Hochschild cohomology group of $A$ with coefficients in an $A$-bimodule $N$ is given by

$$  \HHH^n(A;N) := \Ext^n_{A\otimes A^{\op}}(A,N).$$

Furthermore, the Hochschild-cohomology formalism can be applied in a natural fashion to the broader framework of dg categories  (cf., e.g. [4]).

Whilst Quillen cohomology (cf. [5]) is more universal. Let $X$ be an object living in a model category $\M$. Consider the category $\Ab(\M_{/X})$ of abelian group objects over $X$. Let us assume that $\Ab(\M_{/X})$ comes equipped with a model structure transferred along the free-forgetful adjunction $\F :  \adjunction*{}{\M_{/X}}{\Ab(\M_{/X})}{} : \U$. Then one defines the cotangent complex of $X$ to be $\rL_X := \LL\F(X)$ the derived abelianization of $X$. Now the $n$'th Quillen cohomology group of $X$ with coefficients in some $M \in \Ab(\M_{/X})$ is given by the formula

$$ \HHQ^n(X;M) := \pi_0 \Map^{\h}_{\Ab(\M_{/X})}(\rL_X , M[n]) $$

in which $M[n]$ signifies the $n$-suspension of $M$ in $\Ab(\M_{/X})$. We will refer to $\Ab(\M_{/X})$ as the abelianization of $\M$ at $X$.
In practice, the above notion usually leads to canonical representations of the object of interest. For example, the abelianization of the category of groups at some group $G$ agrees with the category of $G$-modules. An analogue holds for commutative algebras, and for Lie algebras as well. On other hand, the abelianization of  $\textbf{k}$-algebras at $A$ is nothing but the category of $A$-bimodules. More generally, if we let $\P$ be an operad in chain complexes and let $B$ be a $\P$-algebra then the abelianization of $\P$-algebras at $B$ is equivalent to the category $B$-modules over $\P$.

Beyond their flexibility and diverse applications, what stands out remarkably is the intimate relationship between the two cohomologies. More precisely, it was observed by Quillen [6] that there is a fiber sequence of $A$-bimodules of the form

\begin{equation}\label{eq:Qprinori}
\rL_A \lrar A \otimes A \lrar A
\end{equation}

in which $A \otimes A$ represents the free $A$-bimodule generated by $\textbf{k}$ and the second map is classified by the unit $\textbf{k} \lrar A$ of $A$. This is an important one because, while Quillen cohomology of $A$ can be described via Hochschild cohomology of $A$,  the latter is really accessible using the well known bar resolution.

To broaden the range of examples for these two cohomology theories, covering not only algebraic objects but also those residing within a topological setting, we will need to use the tangent category formalism, as discussed in the next subsection.

§.§ From tangent categories to spectral cotangent complexes

Despite the successes, there are certain constraints within Quillen's theory (cf. [2, 1]). In response, Lurie contributes to the further advancement of Quillen's abelianization through the stabilization of $\infty$-categories.

Let $X$ be an object living within a presentable $\infty$-category $\C$. The tangent category to $\C$ at $X$ is by definition $\T_X\C := \Stab((\C_{/X})_*)$ the stabilization of $(\C_{/X})_*$, i.e. the pointed $\infty$-category associated to the over category $\C_{/X}$. Due to the presentability of $\C$, the canonical functor $\T_X\C \lrar \C_{/X}$ admits a left adjoint called the suspension spectrum functor, and denoted by
$$\Sigma^{\infty}_+ : \C_{/X} \lrar \T_X\C.$$
In light of this, the (spectral) cotangent complex of $X$ is defined to be $\rL_X := \Sigma^{\infty}_+(X)$. Now, the Quillen cohomology of $X$ with coefficients in some $M \in \T_X\C$ is the space

$$ \HHQ^\star(X;M)  := \Map_{\T_X\C}(\rL_X , M),$$

and the $n$'th Quillen cohomology group is given by the formula $\HHQ^{n}(X;M)  := \pi_0 \Map_{\T_X\C}(\rL_X , M[n])$.

For more details about the above constructions, we refer the readers to [7]. Additionally, according to the recent works of Harpaz-Nuiten-Prasma [8, 2], one may opt to work within the framework of model categories.

§.§ Cohomologies of enriched operads and algebras over them

Suppose we are given a sufficiently nice symmetric monoidal category $\cS$ equipped with a compatible homotopy theory. Let $\P$ be a $C$-colored operad enriched over $\cS$. As per Quillen-Lurie thinking, a cohomology theory for $\P$ depends on only the choice of a homotopy theory on the category containing $\P$. What makes the situation intriguing is that $\mathcal{P}$ resides in various noteworthy categories. Here are the ones of interest:

$\bullet$ $\P \in \Op(\cS)$: the category of $\cS$-enriched operads with non-fixed sets of colors,

$\bullet$ $\P \in \Op_C(\cS)$: the category of $C$-colored operads in $\cS$,

$\bullet$ $\P \in \BMod(\P)$:  the category of $\P$-bimodules, and

$\bullet$ $\P \in \IbMod(\P)$:  the category of infinitesimal $\P$-bimodules.

In each case, an important invariant specific to $\mathcal{P}$ will emerge, as discussed in what below.

We will now endow $\Op(\cS)$ with the Dwyer-Kan homotopy theory, while the others are all equipped with the standard homotopy theory transferred from that on $\cS$. First, we will refer to the Quillen cohomology of $\P\in\Op(\cS)$ as the proper Quillen cohomology of $\P$. Besides that, the corresponding cohomology of $\P\in\Op_C(\cS)$ will be called the reduced Quillen cohomology of $\P$.

The next step is to decide what we mean by the (spectral) Hochschild cohomology of $\P$. Naively thinking, if we consider $\P$ as a monoid  (in the monoidal category $\Coll_C(\cS)$ of $C$-collections) then it is natural to let $\P \in \BMod(\P)$ represent the Hochschild complex of the operad $\P$, just as in the definition of Hochschild cohomology of associative algebras. Nevertheless, this falls short of being a good choice because the structure of operadic bimodules is not linear in general, and hence resulting in possible challenges in calculations. From this point of view, we will let the Hochschild complex of the operad $\P$ be classified by $\P \in \IbMod(\P)$, based on the fact that infinitesimal $\P$-bimodules are exactly the linearization of the usual notion of $\P$-bimodules. Indeed, the structure of infinitesimal $\P$-bimodules is linear in the sense that the forgetful functor

$$ \IbMod(\P) \lrar \Coll_C(\cS) $$

preserves all colimits as long as the monoidal product in $\cS$ distributes over colimits. Nonetheless, note that the category $\IbMod(\P)$ remains unstable unless $\cS$ is stable. Therefore, to assure that the obtained cohomology theory is abelian (i.e. the cohomology groups are all abelian), we will let the cotangent complex of $\P \in \IbMod(\P)$, denoted by  $\rL_{\P^{\IB}} \in \T_\P\IbMod(\P)$, represent the Hochschild complex of the operad $\P$. More precisely, the Hochschild cohomology of $\P$ with coefficients in some $M\in \T_\P\IbMod(\P)$ is the space

$$  \HHH^\star(\P;M)  := \Map^{\h}_{\T_\P\IbMod(\P)}(\rL_{\P^{\IB}} , M).$$

The above definition has full generality because when the base category $\cS$ is stable then we obtain an equivalence $\T_\P\IbMod(\P) \simeq \IbMod(\P)$, which identifies $\rL_{\P^{\IB}}$ with $\P \in \IbMod(\P)$ itself.

Now let $A\in\Alg_\P(\cS)$ be an algebra over $\P$. It is important to note that $A$ may also be regarded as an object of $\Mod_\P^A$ the category of $A$-modules over $\P$. This latter notion provides canonical representations for $A\in\Alg_\P(\cS)$, just like the classical notions of bimodules over an associative algebra, modules over a group or a Lie algebra, etc. First, since we are mainly interested in the $\P$-algebra structure on $A$, it is obvious to exhibit the Quillen cohomology of $A\in\Alg_\P(\cS)$ as the proper Quillen cohomology of $A$. Moreover, according to the logic as in the above paragraph, we will exhibit $\rL_{A^{\sm}} \in \T_{A} \Mod_\P^A$, i.e. the cotangent complex of $A\in\Mod_\P^A$, as the Hochschild complex of the $\P$-algebra $A$. Now the Hochschild cohomology of $A$ with coefficients in some $N\in \T_{A} \Mod_\P^A$ is the space

$$ \HHH^\star(A;N)  := \Map^{\h}_{\T_{A} \Mod_\P^A}(\rL_{A^{\sm}},N).$$

Further discussion on the naturality of the operadic Hochschild cohomology proposed will be given in $\S$<ref>.

§.§ Main statements

We shall now summarize our main results, divided into three parts as follows.

§.§.§ Formulations and comparison theorem

It is extremely noteworthy that there exists a natural passage between infinitesimal $\P$-bimodules (resp. $\P$-bimodules) and $A$-modules over $\P$  (resp. $\P$-algebras) performed by the adjunction $(-) \circ_\P A \dashv \End_{A,-}$ (see $\S$<ref>). Moreover, there is a commutative square of adjunctions between tangent categories:

\begin{equation}\label{eq:introkey}
\begin{tikzcd}[row sep=3.5em, column sep =3.5em]
\mathcal{T}_{\P}\IbMod(\P) \arrow[r, shift left=1.5] \arrow[d, shift right=1.5, "(-)\circ^{\st}_\P A"'] & \mathcal{T}_{\P}\BMod(\P) \arrow[l, shift left=1.5, "\perp"'] \arrow[d, shift left=1.5, "(-)\circ^{\st}_\P A"]  & \\
\T_{A} \Mod_\P^A \arrow[r, shift right=1.5] \arrow[u, shift right=1.5, "\dashv", "\End_{A,-}^{\st}"'] & \T_A\Alg_\P(\cS)  \arrow[l, shift right=1.5, "\downvdash"] \arrow[u, shift left=1.5, "\vdash"', "\End_{A,-}^{\st}"] &
\end{tikzcd}
\end{equation}

in which the vertical pairs are all induced by the adjunction $(-)\circ_\P A \dashv \End_{A,-}$ (see Notation <ref>), and the horizontal pairs are given by the induction-restriction adjunctions.

We will assume that the base category $\cS$ comes equipped with a compatible model structure. Under some mild assumptions on the triple $(\cS,\P,A)$, we obtain the following results.

Let $\sS_C^{n}$ denote the $C$-collection given by $\emptyset_\cS$ on all levels except that $\sS_C^{n}(c;c)=\sS^{n}$, i.e. the pointed $n$-sphere in $\cS$, for every $c\in C$. Moreover, we write $\P\circ\sS_C^{\bullet}\in \T_\P\BMod(\P)$ for the prespectrum with  $(\P\circ\sS_C^{\bullet})_{n,n} := \P\circ\sS_C^{n}$. This is in fact a suspension spectrum model for $\rL_{\P^{\sB}}\in\T_\P\BMod(\P)$, i.e. the cotangent complex of $\P\in\BMod(\P)$ (see Proposition <ref>). The following will play an active role.
(<ref>, <ref>) There are weak equivalences
$$\rL_{\P^{\sB}}\circ^{\st}_\P A \simeq \rL_A \;\; \text{and} \;\; \rL_{\P^{\IB}}\circ^{\st}_\P A \simeq \rL_{A^{\sm}}$$

in $\T_A\Alg_\P(\cS)$ and $\T_A\Mod_\P^A$ respectively. Consequently, the cotangent complex $\rL_A \in \T_A\Alg_\P(\cS)$ admits a suspension spectrum model: $(\P\circ\sS_C^{\bullet})\circ^{\st}_\P A \simeq \rL_A$.

Suppose further that the base category $\cS$ is stable. We have an extended commutative diagram

\begin{equation}\label{eq:introkey1}
\begin{tikzcd}[row sep=3.5em, column sep =3.5em]
\IbMod(\P)	 \arrow[r, shift left=1.5, "\simeq"] \arrow[d, shift right=1.5, "(-)\circ_\P A"'] & \mathcal{T}_{\P}\IbMod(\P) \arrow[l, shift left=1.5, "\perp"'] \arrow[d, shift left=1.5, "(-)\circ^{\st}_\P A"] \arrow[r, shift left=1.5, "\simeq"]  & \mathcal{T}_{\P}\BMod(\P)  \arrow[d, shift left=1.5, "(-)\circ^{\st}_\P A"] \arrow[l, shift left=1.5,  "\perp"']   \\
\Mod_\P^A \arrow[r, shift right=1.5] \arrow[u, shift right=1.5, "\dashv", "\End_{A,-}"'] & \T_{A} \Mod_\P^A  \arrow[l, shift right=1.5, "\downvdash", "\simeq" '] \arrow[u, shift left=1.5, "\vdash"', "\End_{A,-}^{\st}"] \arrow[r, shift right=1.5] & \T_A\Alg_\P(\cS)  \arrow[u, shift left=1.5, "\vdash"', "\End_{A,-}^{\st}"] \arrow[l, shift right=1.5, "\downvdash", "\simeq" ']
\end{tikzcd}
\end{equation}

in which the horizontal pairs are all equivalences. We denote by $\ovl{\rL}_\P \in \IbMod(\P)$ with

$$\ovl{\rL}_\P(c_1,\cdots,c_m;c) = \P(c_1,\cdots,c_m;c)\otimes \hocolim_n\Omega^{n} [\, (\sS^{n})^{\otimes m}  \times_{1_\mathcal{S}}^{\h}  0\,]$$

where the desuspension $\Omega(-)$ is taken in $\cS$. The object $\ovl{\rL}_\P$ is exactly the image of $\rL_{\P^{\sB}}$ through the identification $\mathcal{T}_{\P}\BMod(\P)\simeq\IbMod(\P)$ (see Remark <ref>).
There is a weak equivalence in $\Mod_\P^A$ of the form

$$ \ovl{\rL}_\P\circ_\P A \simeq \rL_A.$$

Here we use the same notation to denote the image of $\rL_A \in \T_A\Alg_\P(\cS)$ under the identification $\T_A\Alg_\P(\cS)\simeq\Mod_\P^A$.

Consider the case $\cS=\C(\textbf{k})$ with $\textbf{k}$ being any commutative ring (see Examples <ref>). We can prove that the object $\ovl{\rL}_\P\in\IbMod(\P)$ is weakly equivalent to the infinitesimal composite product $\P\circ_{(1)}\I_C$ (see $\S$<ref> and $\S$<ref> for notations). Furthermore, it can be shown that the object
$$\ovl{\rL}_\P\circ_\P A \simeq (\P\circ_{(1)}\I_C)\circ_\P A$$
is a model for the module of Kähler differentials $\Omega_A\in\Mod_\P^A$. Due to this, we may recreate a classical result in literature asserting that, in a differential graded setting, the cotangent complex of an operadic algebra is exactly equivalent to its module of Kähler differentials (cf., e.g. [2, 9, 10]). Detailed discussion regarding this topic will be included in a subsequent paper.

The following comparison theorem proves that Hochschild and Quillen cohomologies of $A$ can be encoded by the corresponding cohomologies of the operad $\P$ itself.

(<ref>, <ref>)  Suppose given a fibrant object $N\in \T_{A} \Mod_\P^A$.

(i) There is a natural weak equivalence

$$ \HHQ^\star(A ; N)  \simeq \Omega\HHQ^\star(\P ; \End^{\st}_{A,N})$$

between the Quillen cohomology of $A$ with coefficients in $N$ and the loop space of Quillen cohomology of $\P$ with coefficients in $\End^{\st}_{A,N} \in \mathcal{T}_{\P}\IbMod(\P)$.

(ii) There is a natural weak equivalence

$$ \HHH^\star(A ; N)  \simeq \HHH^\star(\P ; \End^{\st}_{A,N}) $$

between the Hochschild cohomology of $A$ with coefficients in $N$ and Hochschild cohomology of $\P$ with coefficients in $\End^{\st}_{A,N}$.

This theorem is proved using Proposition <ref>, and along with the assertion that under the identification $\T_\P\Op(\mathcal{S}) \simeq \T_{\P}\BMod(\P)$, the cotangent complex $\rL_\P\in\T_\P\Op(\mathcal{S})$ is identified with $\rL_{\P^{\sB}}[-1] \in \T_{\P}\BMod(\P)$ (see Proposition <ref>).

When the base category $\cS$ is stable then we obtain weak equivalences that are similar to those above:

$$ \HHQ^\star(A ; N)  \simeq \Omega\HHQ^\star(\P ; \End_{A,N})  \;\; \text{and} \;\; \HHH^\star(A ; N)  \simeq \HHH^\star(\P ; \End_{A,N}) $$

with the only difference being that $N$ is now an object of $\Mod_\P^A$.

§.§.§ Simplicial operads and algebras over them

In what follows, we will assume that $\cS$ is the category $\Set_\Delta$ of simplicial sets, and assume that $\P$ is a fibrant and $\Sigma$-cofibrant $C$-colored operad in $\Set_\Delta$. Recall from [1] that there is a chain of equivalences of $\infty$-categories

$$ \T_\P\Op(\Set_\Delta)_\infty \simeq \T_\P\Op_C(\Set_\Delta)_\infty \simeq \T_{\P}\BMod(\P)_\infty \simeq \T_{\P}\IbMod(\P)_\infty \simeq \Fun(\Tw(\P) , \Sp)  $$
where $\Tw(\P)$ signifies the twisted arrow $\infty$-category of $\P$ and $\Sp$ denotes the $\infty$-category of spectra.

Due to the above, both Quillen and Hochschild cohomologies of $\P$ can be calculated in the framework of functors $\Tw(\P) \lrar \Sp$. In the loc.cit, we have proved that $\rL_\P \in  \T_\P\Op(\Set_\Delta)_\infty$ is identified with $\F_\P[-1] \in \Fun(\Tw(\P) , \Sp)$ where the functor $\F_\P : \Tw(\P) \lrar \Sp$ takes any operation $\mu\in\P$ of arity $m$ to $\F_\P(\mu) = \mathbb{S}^{\oplus m}$. Whilst the Hochschild complex $\rL_{\P^{\IB}} \in \T_{\P}\IbMod(\P)_\infty$ is identified with $\ovl{\mathbb{S}}: \Tw(\P) \lrar \Sp$ the constant functor on the sphere spectrum (see Proposition <ref>). Thus, after sending coefficients into $\Fun(\Tw(\P) , \Sp)$, the two cohomologies of $\P$ are formulated as follows.
Suppose given a functor  $\F : \Tw(\P) \lrar \Sp$.

(i) ([1]) The Quillen cohomology of $\P$ with coefficients in $\F$ is computed by

$$ \HHQ^\star(\P ; \F) \simeq \Map_{\Fun(\Tw(\P) , \Sp)}(\F_\P, \F[1]).$$

(ii) (<ref>) Whilst the Hochschild cohomology of $\P$ with coefficients in $\F$ is given by

$$ \HHH^\star(\P ; \F) \simeq \Map_{\Fun(\Tw(\P) , \Sp)}(\ovl{\mathbb{S}}, \F).$$

In particular, the $n$'th Hochschild cohomology group is computed by

$$ \HHH^n(\P ; \F) \cong  \pi_0\Map_{\Fun(\Tw(\P) , \Sp)}(\ovl{\mathbb{S}}, \F[n]) \cong  \pi_{-n} \, \lim\F. $$

Next let $A$ be a cofibrant $\P$-algebra classified by a map $\ell_A : \P \lrar \End_A$ of operads. The two cohomologies of $A$ will take coefficients in $\T_{A} \Mod_\P^A$. For each levelwise $\Omega$-spectrum $N \in \T_{A} \Mod_\P^A$, we may associate to it a functor

$$\widetilde{\End}^{\st}_{A,N} : \Tw(\P) \lrar \Sp.$$

Explicitly, for each operation $\mu\in \P(c_1,\cdots,c_m;c)$ regarded as an object of $\Tw(\P)$, then $\widetilde{\End}^{\st}_{A,N}(\mu)$ is an $\Omega$-spectrum such that  $\widetilde{\End}^{\st}_{A,N}(\mu)_{k,k}$ agrees with the fiber over $\ell_A(\mu)$ of the simplicial map

$$ \Map_{\Set_\Delta}(A(c_1)\times\cdots\times A(c_m) , N_{k,k}(c)) \lrar \Map_{\Set_\Delta}(A(c_1)\times\cdots\times A(c_m) , A(c))$$

that is induced by the structure map $N_{k,k}(c) \lrar  A(c)$ of $N$ (see Construction <ref>).

(<ref>) (i) The Quillen cohomology of $A$ with coefficients in $N$ is computed by

$$ \HHQ^\star(A ; N) \simeq \Omega\HHQ^\star(\P ; \End^{\st}_{A,N}) \simeq \Map_{\Fun(\Tw(\P) , \Sp)}(\F_\P, \widetilde{\End}^{\st}_{A,N}).$$

(ii) The Hochschild cohomology of $A$ with coefficients in $N$ is computed by

$$ \HHH^\star(A ; N) \simeq \HHH^\star(\P ; \End^{\st}_{A,N}) \simeq \Map_{\Fun(\Tw(\P) , \Sp)}(\ovl{\mathbb{S}}, \widetilde{\End}^{\st}_{A,N}).$$

In particular, the $n$'th Hochschild cohomology group is simply given by

$$   \HHH^n(A ; N) \cong \pi_{-n} \, \lim\widetilde{\End}^{\st}_{A,N}.$$

§.§.§ Quillen principle for topological $\E_n$-algebras
While Quillen cohomology plays a central role in the study of deformation theory and obstruction theory, its actual computation is far from straightforward. In addressing this complexity, one approach is to establish a connection between Quillen cohomology and the notably more accessible Hochschild cohomology.  This approach aligns directly with the Quillen principle that we have been developing. To be more precise, we will say that a given object of interest is subject to the Quillen principle if there exists a convenient fiber sequence relating the cotangent and Hochschild complexes of that object.

The most basic example is the Quillen principle for associative algebras, as expressed by the fiber sequence (<ref>). Moreover, a variant of this principle for simplicial groups exists as well, according to [2]. (See also [[1], $\S$1.2] for more details).

More generally, the Quillen principle for $\E_n$-algebras in stable categories was established by Francis [11] and Lurie [7]. Let us now recall their result. Let $\C$ be a stable presentable symmetric monoidal $\infty$-category whose monoidal product distributes over colimits, and let $A$ be an $\E_n$-algebra in $\C$. Denote by $\Mod_A^{\E_n}(\C)$ the $\infty$-category of $A$-modules over $\E_n$. The stability of the base category $\C$ implies the existence of an equivalence of $\infty$-categories:

\begin{equation}\label{eq:composedeq}
\T_A\Alg_{\E_n}(\C) \lrarsimeq \Mod_A^{\E_n}(\C).
\end{equation}

(Francis [11], Lurie [7]) There is a fiber sequence in $\Mod_A^{\E_n}(\C)$ of the form

\begin{equation}\label{eq:cofibEnalg}
\Free_A(1_\C) \x{\eta_A}{\lrar} A \lrar \rL_A[n]
\end{equation}

in which $\Free_A(1_\C)$ signifies the free $A$-module over $\E_n$ generated by the monoidal unit and the map $\eta_A$ is classified by the unit of $A$.

Note that $A \in \Mod_A^{\E_n}(\C)$ represents the Hochschild complex of the $\E_n$-algebra $A$. Thus, the fiber sequence (<ref>) yields for each $N\in\Mod_A^{\E_n}(\C)$ a fiber sequence

$$ \Omega^n \HHQ^\star(A;N)  \lrar \HHH^\star(A;N) \lrar |N|$$

in which $|N|:= \Map_\C(1_\C,N)$ represents the underlying space of $N$.

We prove the topological-analogue of (<ref>) (i.e. when the base category $\C$ is the $\infty$-category of spaces). The progression of the proof is outlined in the content below.

Now suppose that $\P$ is a fibrant and $\Sigma$-cofibrant single-colored operad in $\Set_\Delta$. Moreover, we will assume that $\P(0) \simeq \P(1) \simeq *$. Let us write $\id\in\P(1)$ for the identity operation of $\P$ regarded as an object of $\Tw(\P)$, and write $\id_*[\mathbb{S}]$ for the right Kan extension of the embedding $\{\mathbb{S}\} \lrar \Sp$ along the inclusion $\{\id\} \lrar \Tw(\P)$. The following establishes a connection between the Quillen and Hochschild cohomologies of $\P$ (see also Remark <ref>).

(<ref>) There is a fiber sequence in $\Fun(\Tw(\P),\Sp)$ of the form

\begin{equation}\label{eq:introfundp}
\F_\P \x{\iota}{\lrar} \id_*[\mathbb{S}] \x{\rho}{\lrar} \ovl{(\Sigma^\infty_+\P(2))^\vee}
\end{equation}

where $\ovl{(\Sigma^\infty_+\P(2))^\vee}$ refers to the constant functor $\Tw(\P) \lrar \Sp$ with value $(\Sigma^\infty_+\P(2))^\vee$, i.e. the dual spectrum of the suspension spectrum $\Sigma^\infty_+\P(2)$.

Clearly the above statement holds for the little $n$-discs operad $\E_n$. But more is true about this operad. We let $\mu_0\in\E_n(0)$ denote the unique nullary operation regarded as an object of $\Tw(\E_n)$, and let $(\mu_0)_![T] : \Tw(\E_n) \lrar \Sp$ signify, for some spectrum $T$, the left Kan extension of the embedding $\{T\} \lrar \Sp$ along the inclusion $\{\mu_0\} \lrar \Tw(\E_n)$.
(<ref>) There is a diagram of Cartesian squares in $\Fun(\Tw(\E_n),\Sp)$:

$$ \xymatrix{
\F_{\E_n} \ar[r]\ar[d] & (\mu_0)_!\left[\mathbb{S}[-n+1]\right] \ar[r]\ar[d] & \id_*[\mathbb{S}] \ar[d] \\
0 \ar[r] & \ovl{\mathbb{S}}[-n+1] \ar[r] & \ovl{\mathbb{S}}\oplus\ovl{\mathbb{S}}[-n+1] \\
} $$

where as usual $\ovl{\mathbb{S}}$ denotes the constant functor $\Tw(\E_n) \lrar \Sp$ with value $\mathbb{S}$.

In the above diagram, the outer Cartesian square is due to Proposition <ref>, and along with the fact that $\E_n(2) \simeq \sS^{n-1}$; while the Cartesian square on the right follows from Proposition <ref>. In particular, the left square is also Cartesian, and thus, it allows us to recover an important result of [1] asserting the existence of a fiber sequence in $\Fun(\Tw(\E_n),\Sp)$:

$$ (\mu_0)_![\mathbb{S}]  \lrar \ovl{\mathbb{S}} \lrar \F_{\E_n}[n].$$

According to Theorem <ref>, the above is exactly the Quillen principle for the topological operad $\E_n$. Moreover, combining this with the key proposition <ref>, we may formulate the Quillen principle for topological $\E_n$-algebras as follows.

Let $A$ be a cofibrant $\E_n$-algebra in $\Set_\Delta$. We let $\Free_A(*)$ denote the free $A$-module over $\E_n$ generated by a singleton, and let $\eta_A : \Free_A(*) \lrar A$ be the map in $\Mod_{\E_n}^A$ classified by the unit of $A$. We have a canonical map $\Sigma^\infty_+(\eta_A) \lrar \rL_{A^{\sm}}$ where $\Sigma^\infty_+(\eta_A)$ refers to the image of $\eta_A$ through the functor
$$\Sigma^\infty_+ : (\Mod_{\E_n}^A)_{/A} \lrar \T_{A}\Mod_{\E_n}^A.$$

(<ref>)  There is a fiber sequence in $\T_{A}\Mod_{\E_n}^A$ of the form

\begin{equation}\label{eq:introQprin3}
\Sigma^\infty_+(\eta_A) \lrar \rL_{A^{\sm}} \lrar \rL_A[n]
\end{equation}

where we use the same notation for the derived image of $\rL_A\in\T_{A}\Alg_{\E_n}(\Set_\Delta)$ under the equivalence $\T_{A}\Alg_{\E_n}(\Set_\Delta) \simeq \T_{A}\Mod_{\E_n}^A$. Consequently, for a given fibrant object $N \in \T_{A}\Mod_{\E_n}^A$, we obtain a fiber sequence connecting the two cohomologies

$$  \Omega^n\HHQ^\star(A;N) \lrar \HHH^\star(A;N) \lrar \Map^{\h}_{(\Mod_{\E_n}^A)_{/A}}(\eta_A, \Omega^\infty_+N).$$

When passing to a stable base category $\cS$ (e.g. chain complexes and symmetric spectra), then we obtain for each $A \in \Alg_{\E_n}(\cS)$ a fiber sequence in $\Mod_{\E_n}^A(\cS)$ that is exactly the same as (<ref>) (see Theorem <ref>). Moreover, when working in the framework of Lurie's  $\infty$-operads, we may also recover that fiber sequence of Francis-Lurie using the fiber sequence (<ref>).

§ BACKGROUND AND CONVENTIONS

This section is devoted to some preliminaries relevant to the theory of enriched  operads and their associated modules, as well as the standard homotopy theories of such objects. We also revisit the tangent category formalism, from which we may obtain the central concepts of the paper including the spectral cotangent complex and Quillen cohomology.  Additionally, we present some main results from our previous work concerning some operadic versions of the well known Dold-Kan correspondence.

§.§ Enriched operads

We will assume that $(\SS, \otimes, 1_\cS)$ is a symmetric monoidal category such that the monoidal product distributes over colimits. By definition, a symmetric $C$-collection in $\SS$ consists of a collection
$$M = \{M(c_1,\cdots,c_n;c) \, | \, c_i,c \in C , n\geqslant0  \}$$
of objects in $\cS$ such that for each sequence $(c_1,\cdots,c_n;c)$ and each permutation $\sigma\in \Sigma_n$, there is a map of the form
$$\sigma^{*} : M(c_1,\cdots,c_n;c) \lrar M(c_{\sigma(1)},\cdots,c_{\sigma(n)};c) .$$
Such maps establish a right action on $M$ by $\Sigma_n$. We let $\Coll_C(\SS)$ denote  the category of symmetric $C$-collections in $\SS$.

When $C$ is a singleton then a symmetric $C$-collection will be called a $\Sigma_*$-object. More concretely, a $\Sigma_*$-object $M$ consists of a collection $\{M(n)\}_{n\geq0}$ such that each $M(n)$ is a right $\Sigma_n$-object in $\cS$. The category of $\Sigma_*$-objects in $\cS$ will be denoted by $\Sigma_*(\cS)$.

The category $\Coll_C(\SS)$ admits a monoidal structure given by the composite product

$$ -\circ- : \Coll_C(\SS) \times \Coll_C(\SS) \lrar \Coll_C(\SS).$$

We will denote by $\I_C$ the monoidal unit which agrees with $\emptyset_\cS$ (i.e. the initial object of $\cS$) on all the levels except that $\I_C(c;c)=1_{\cS}$ for every $c\in C$.

A symmetric $C$-colored operad in $\cS$ is a monoid in the monoidal category $(\Coll_C(\SS),-\circ-,\I_C)$. We will write $\Op_C(\SS)$ to denote the category of symmetric $C$-colored operads in $\cS$. When $C$ is a singleton, then we will write $\Op_*(\SS)$ to denote the corresponding category.

In the above definitions, if we forget the action by symmetric groups, then we obtain the notions of nonsymmetric $C$-collections and nonsymmetric $C$-colored operads. Given our focus solely on the symmetric context, henceforth, we will exclude the term $``$symmetric” when referring to any object within $\Op_C(\SS)$ (or $\Coll_C(\SS)$).

An $\cS$-enriched category $\C$ can be identified with an $\cS$-enriched operad concentrated in arity $1$, i.e. defined by letting $\C(c;d) := \Map_\C(c,d)$. On the other hand, every $\cS$-enriched operad $\P$  has an underlying category $\P_1$ such that $\Map_{\P_1}(c,d) := \P(c;d)$. Moreover, these two constructions determine an adjunction

$$ \adjunction*{}{\Cat_C(\SS)}{\Op_C(\SS)}{} $$

where $\Cat_C(\SS)$ denotes the category of $\cS$-enriched categories with $C$ being the fixed set of objects.

Consider the functor $\iota : \Sets \lrar \cS$ that is characterized by the requirement that $\iota$ preserves coproducts and that $\iota$ sends every singleton to $1_\cS$. Since $\iota$ is a (strongly) symmetric monoidal functor, it sends each operad in $\Sets$ to an operad in $\cS$. This allows us to define two typical operads being the associative operad $\Ass$ and commutative operad $\Com$ as objects in $\Op_*(\cS)$. More concretely, for each $n\in\NN$ we have $\Ass(n) = \underset{\Sigma_n}{\bigsqcup}1_\cS$ with the symmetric-action being induced by the right regular representation of $\Sigma_n$ and with the composition induced by the concatenation of linear orders. Whilst the operad $\Com$ is defined by letting $\Com(n) = 1_\cS$ for every $n\in\NN$, endowed with the trivial action by the symmetric groups; and such that the composition is induced by the isomorphisms $1_\cS\otimes \cdots \otimes 1_\cS \lrar 1_\cS$. In line with the terminologies, the operad $\Ass$ (resp. $\Com$) encodes the category of associative monoids (resp. commutative monoids) in $\cS$.

For each map $\alpha : C \lrar D$ of sets, there is a changing-colors functor $\alpha^{*} : \Op_D(\SS) \lrar \Op_C(\SS)$ defined by sending $\Q\in\Op_D(\SS)$ to $\alpha^{*}\Q$ with

$$ \alpha^{*}\Q(c_1,\cdots,c_n;c) := \Q(\alpha(c_1),\cdots,\alpha(c_n);\alpha(c)) .$$

Due to this, one can organize all the colored operads in $\SS$, for arbitrary sets of colors, into a single category $\Op(\SS)$ by taking the Grothendieck construction

$$ \Op(\SS) := \int_{C\in \Sets} \Op_C(\SS).$$

We refer to $\Op(\SS)$ as the category of $\cS$-enriched operads.

For more details, an object of $\Op(\SS)$ is the choice of a pair $(C,\P)$ with $C\in \Sets$ and $\P\in \Op_C(\SS)$. Moreover, a morphism $(C,\P) \lrar (D,\Q)$ consists of a map $\alpha : C \lrar D$ of sets and a map $f :  \P \lrar \alpha^{*}\Q$ in $\Op_C(\SS)$.

§.§ Modules over an operad

In what follows, we will recall various types of modules over an operad. Let $\P$ be a $C$-colored operad in $\cS$. We will write $\LMod(\P)$, $\RMod(\P)$ and $\BMod(\P)$ respectively standing for the categories of left $\P$-modules, right $\P$-modules and $\P$-bimodules. These notions of modules come naturally when considering $\P$ as a monoid in the monoidal category of $C$-collections.

The structure of right $\P$-modules is linear in the sense that the forgetful functor
$$\RMod(\P) \lrar \Coll_C(\SS)$$
preserves all colimits. However, an analogue in general does not hold for left $\P$-modules, and thus, does not hold for $\P$-bimodules as well. This promotes the development of infinitesimal left modules (bimodules) over an operad (cf. [12]), essentially coming as the linearization of the former. We will denote by $\IbMod(\P)$ (resp. $\ILMod(\P)$) the category of infinitesimal $\P$-bimodules (resp. infinitesimal left $\P$-modules). We will revisit these two notions in $\S$<ref>.

It is noteworthy that each of the categories $\LMod(\P)$, $\BMod(\P)$, $\IbMod(\P)$ and $\RMod(\P)$ can be represented as the category of algebras over an enriched operad (or $\cS$-valued functors on an enriched category). For more details about these, we refer the readers to [1, 13, 14]. Moreover, according to [15], there exists a discrete operad, by which we denote $\textbf{O}_C$, that encodes the category of $C$-colored operads in $\cS$. For the case $C = \{*\}$, we will write $\textbf{O}_*$ standing for that.

The main interest in the study of operads is the notion of algebras over an operad. By definition, a $\P$-algebra is simply a left $\P$-module concentrated in level $0$. More explicitly, a $\P$-algebra $A$ is a collection $A=\{A(c)\}_{c\in C}$ of objects in $\cS$ equipped, for each $(c_1,\cdots,c_n;c)$, with an action map

$$ \P(c_1,\cdots,c_n;c)\otimes A(c_1)\otimes\cdots\otimes A(c_n) \lrar A(c).$$

These maps must satisfy the essential axioms of associativity, unitality and equivariance. We denote by $\Alg_\P(\SS)$ the category of $\P$-algebras.

Just like left modules, the structure of algebras over $\P$ is not linear. In response, the notion of modules over a $\P$-algebra $A$ essentially provides the representations for $A$, as well as the classical notions of bimodules over an associative ring, modules over a group or a Lie algebra, etc.

An $A$-module over $\P$ is a collection  $M=\{M(c)\}_{c\in C}$ of objects in $\cS$ equipped, for each sequence $(c_1,\cdots,c_n;c)$, with an action map of the form

$$ \P(c_1,\cdots,c_n;c) \otimes \;  \bigotimes _{i \in \{1,\cdots,n\} \setminus \{k\}} \; A(c_i) \otimes M(c_k) \lrar M(c) .$$

These maps must satisfy the essential axioms of associativity, unitality and equivariance. The category of $A$-modules over $\P$ will be denoted by $\Mod_\P^{A}$.

According to [16], the notion above can be reformulated as follows. We write $\Pairs_C(\cS)$ for the category whose objects are the pairs $(\P , A)$ with $\P\in \Op_C(\mathcal{S})$ and $A \in \Alg_\P(\SS)$, and such that a morphism $(\P , A) \lrar (\Q , B)$ consists of a map $\P \rar \Q$ in $\Op_C(\mathcal{S})$, and along with a map $A \rar B$ of $\P$-algebras. There exists an evident embedding $ \delta : \Op_C(\mathcal{S}) \lrar \Pairs_C(\cS)$ sending $\P$ to the pair $(\P , \P_0)$ where $\P_0$ is the collection of nullary ($=0$-ary) operations of $\P$ regarded as an initial $\P$-algebra. The functor $\delta$ admits a left adjoint denoted $\Env : \Pairs_C(\cS) \lrar \Op_C(\mathcal{S})$, and called the enveloping functor.

The enveloping operad associated to a  pair $(\P , A) \in \Pairs_C(\cS)$ is $\Env(\P , A)$ the image of $(\P , A)$ through the enveloping functor.

Due to [Theorem 1.10, [16]], we have a canonical isomorphism

\begin{equation}\label{eq:AmodP}
\Mod_\P^{A} \cong \Fun(\Env(\P,A)_1 , \cS)
\end{equation}

between the category of $A$-modules over $\P$ and the category of $\cS$-valued enriched functors on $\Env(\P,A)_1$ the underlying category of $\Env(\P,A)$.

As a standard notation, we will write $\sU_\P(A) := \Env(\P,A)_1$ and refer to it as the enveloping category of $A$. The set of objects of $\sU_\P(A)$ is $C$ (since $\Env(\P,A)$ is a $C$-colored operad). In particular, when $\P$ is a single-colored operad then $\sU_\P(A)$ forms an associative monoid in $\cS$. In this situation, that object is usually called the universal enveloping algebra of $A$ (cf. e.g., [9, 10]).

The following is a fundamental result.
Suppose that $\P$ is a single-colored operad. Then $\sU_\P(A)$ can be modeled by $\Free_A(1_{\cS})$ the free $A$-module over $\P$ generated by $1_{\cS}$.
As discussed above, the category $\Mod_\P^{A}$ is identified with the category of left modules over $\sU_\P(A)$. Note that $\sU_\P(A)$ is the free left module over itself that is generated by $1_{\cS}$. Thus we obtain an identification $\sU_\P(A)\cong\Free_A(1_{\cS})$ of $A$-modules over $\P$.

Note that when $C$ contains more than one color, then $\sU_\P(A)$ cannot even be modeled as an object of $\Fun(\sU_\P(A), \cS)$. Thus the same statement as above no longer holds in the multi-colored setting.

§.§ Operadic model structures

We assume further that $\SS$ is a symmetric monoidal model category (cf. [17]). Let $\P$ be a $C$-colored operad in $\cS$. We will usually require the existence of a transferred model (or semi-model) structure on the category $\Alg_\P(\cS)$. This is based on the following result.

(Spitzweck [18], Fresse [10]) Suppose further that the model structure on $\cS$ is cofibrantly generated, that the monoidal unit $1_\cS$ is cofibrant, and that $\P$ is $\Sigma$-cofibrant. Then the category $\Alg_\P(\cS)$ carries a cofibrantly generated semi-model structure transferred from the projective model structure on $\cS^{\times C}$.

In particular, with the same assumptions on $\cS$, the transferred semi-model structure on the category $\Op_C(\SS)$ does exist. Additionally, with the same assumptions on $\cS$ and $\P$, then the categories $\LMod(\P)$ and $\BMod(\P)$ admit a transferred semi-model structure as well, while $\RMod(\P)$ and $\IbMod(\P)$ inherit a transferred (full) model structure. These observations follow from the above theorem and Remark <ref>.

We now recall the homotopy theory of $\cS$-enriched operads, according to the work of Caviglia [19]. For $\C \in \Cat(\cS)$ an $\cS$-enriched category, the homotopy category of $\C$, denoted $\Ho(\C)$, is the ordinary category whose objects are the same as those of $\C$ and such that, for $x,y\in \Ob(\C)$, we have
$$\Hom_{\Ho(\C)}(x,y) := \Hom_{\Ho(\cS)}(1_\cS,\Map_\C(x,y)) .$$

By convention, the homotopy category of an operad $\P$ is
$$\Ho(\P) := \Ho(\P_1).$$

Moreover, a map $f : \P \rar \Q$ in $\Op(\cS)$ is called a levelwise weak equivalence (resp. fibration, trivial fibration, etc) if for every sequence $(c_1,\cdots,c_n;c)$ of colors of $\P$, the induced map
$$\P(c_1,\cdots,c_n;c) \lrar \Q(f(c_1),\cdots,f(c_n) ; f(c))$$
is a weak equivalence (resp. fibration, trivial fibration, etc) in $\cS$.

A map $f : \P \rar \Q$ in $\Op(\cS)$ is called a Dwyer-Kan equivalence if it is a levelwise weak equivalence and such that the induced functor
between homotopy categories is essentially surjective.

The Dwyer-Kan model structure on $\Op(\cS)$ is the one whose weak equivalences are the Dwyer-Kan equivalences and whose trivial fibrations are the levelwise trivial fibrations surjective on colors.

In practice, the Dwyer-Kan model structure usually coincides with the canonical model structure, as also presented in [19]. In particular, if this is the case then Dwyer-Kan fibrant objects in $\Op(\cS)$ are precisely the levelwise fibrant operads.

Here are some base categories that we are concerned with in this paper. Let $\textbf{k}$ be a (unital) commutative ring.

(i) Firstly, we let $\Set_\Delta$ denote the category of simplicial sets. This category comes equipped with the Cartesian monoidal structure and with the standard (Kan-Quillen) model structure.

(ii) Another instance is the category of simplicial $\textbf{k}$-modules, denoted $\sMod(\textbf{k})$, and equipped with the (degreewise) tensor product over $\textbf{k}$, and with the standard model structure transferred from that on $\Set_\Delta$.

(iii) We will denote by $\C(\textbf{k})$ (resp. $\C_{\geqslant0}(\textbf{k})$) the category of dg $\textbf{k}$-modules (resp. connective dg $\textbf{k}$-modules), endowed with the usual tensor product, and with the projective model structure.

When $\cS = \Set_\Delta$ or $\sMod(\textbf{k})$, then for every operad $\P$ the transferred homotopy theory on $\Alg_\P(\cS)$ exists as a (full) model category. Moreover, the two model structures on $\Op(\cS)$ exist, and in fact coincide with each other.

For the case $\cS = \C(\textbf{k})$ or $\C_{\geqslant0}(\textbf{k})$, due to Theorem <ref> the transferred homotopy theory on $\Alg_\P(\cS)$ exists as a semi-model category when provided $\P$ is $\Sigma$-cofibrant. Additionally, it can be shown that the Dwyer-Kan homotopy theory on $\Op(\cS)$ exists as a semi-model category as well. (This is in fact made up together by the Dwyer-Kan model structure on $\Cat(\cS)$ and by the transferred semi-model structure on the categories $\Op_C(\cS)$ for $C\in\Sets$). In particular, in this situation $\cS$ is not sufficient in the sense of [[1], $\S$3]. In spite of this, the results needed from the loc.cit remain valid if we bring them in the framework of semi-model categories.

§.§ Spectral cotangent complex and  Quillen cohomology

We recall briefly some fundamental notions relevant to the spectral cotangent complex and Quillen cohomology, according to the works of Harpaz-Nuiten-Prasma [8, 2]. Nevertheless, our setting is a bit more flexible, so that we may define tangent categories of (semi) model categories of interest without requiring the left properness. This enables us to cover a wider range of examples. We also refer the readers to [[1], $\S$2.5] for more details.

Recall by definition that a (semi) model category $\textbf{M}$ is said to be weakly pointed if it contains a weak zero object $0$, i.e. $0$ is both homotopy initial and terminal. When $\textbf{M}$ is weakly pointed, we may define the suspension and desuspension of an object $X\in\Ob(\textbf{M})$ respectively as

$$ \Sigma(X) := 0 \, \x{\h}{\underset{X}{\bigsqcup}} \, 0  \; \; \; \text{and} \; \; \; \Omega(X) := 0 \, \x{\h}{\underset{X}{\times}} \, 0. $$

Moreover, when passing to the homotopy category, these constructions determine an adjunction
$$\Sigma : \adjunction*{}{\Ho(\textbf{M})}{\Ho(\textbf{M})}{}  : \Omega.$$

A (semi) model category $\textbf{M}$ is said to be stable if the following equivalent conditions hold:

(1) The underlying $\infty$-category $\textbf{M}_\infty$ is stable in the sense of [7].

(2) $\textbf{M}$ is weakly pointed and such that a square in $\textbf{M}$ is homotopy coCartesian if and only if it is homotopy Cartesian.

(3) $\textbf{M}$ is weakly pointed and such that the self-adjunction $\Sigma \dashv \Omega$ on $\Ho(\textbf{M})$ is an adjoint equivalence.

We now recall the procedure of taking stabilizations. Let $\textbf{M}$ be a weakly pointed model category and let $X$ be an ($\mathbb{N}\times\mathbb{N} $)-diagram in $\textbf{M}$. A square of $X$ that takes the form

$$ \xymatrix{
X_{n,n} \ar[r]\ar[d] & X_{n,n+1} \ar[d] \\
X_{n+1,n} \ar[r] & X_{n+1,n+1} \\

will be called a diagonal square.
An  ($\mathbb{N}\times\mathbb{N} $)-diagram in $\textbf{M}$ is called

(1) a prespectrum if all its off-diagonal entries are weak zero objects in $\textbf{M}$,

(2) an $\Omega$-spectrum if it is a prespectrum and all its diagonal squares are homotopy Cartesian,

(3) a suspension spectrum if it is a prespectrum and all its diagonal squares are homotopy coCartesian.

A map $f : X\rar Y$ in $\textbf{M}^{\mathbb{N}\times\mathbb{N}}$ is called a stable equivalence if for every $\Omega$-spectrum $Z$ the induced map

$$ \Map^{\der}_{\textbf{M}^{\mathbb{N}\times\mathbb{N}}_{\proj}}(Y,Z) \lrar  \Map^{\der}_{\textbf{M}^{\mathbb{N}\times\mathbb{N}}_{\proj}}(X,Z) $$

is a weak equivalence, where $\textbf{M}^{\mathbb{N}\times\mathbb{N}}_{\proj}$ signifies the projective model category of ($\mathbb{N}\times\mathbb{N} $)-diagrams in $\textbf{M}$.

We will assume further that $\textbf{M}$ is a weakly pointed combinatorial model category such that the domains of generating cofibrations are cofibrant.

The stabilization of $\textbf{M}$, denoted $\Sp(\textbf{M})$, is the left Bousfield localization of $\textbf{M}^{\mathbb{N}\times\mathbb{N}}_{\proj}$ with $\Omega$-spectra as the local objects. More precisely, $\Sp(\textbf{M})$ is a cofibrantly generated semi-model category such that

$\bullet$ weak equivalences are the stable equivalences, and

$\bullet$ (generating) cofibrations are the same as those of $\textbf{M}^{\mathbb{N}\times\mathbb{N}}_{\proj}$.

In particular, fibrant objects of $\Sp(\textbf{M})$ are precisely the levelwise fibrant $\Omega$-spectra.

The above definition is based on the main result of [20], and together with the fact that the $\Omega$-spectra in $\textbf{M}$ can be characterized as local objects against a certain set of maps (cf. [8], Lemma 2.1.6). Moreover, it suffices to require $\textbf{M}$ to be a semi-model category with the same properties as above.

By construction, there is a Quillen adjunction $\Sigma^{\infty} : \adjunction*{}{\textbf{M}}{\Sp(\textbf{M})}{} : \Omega^{\infty}$ where $\Omega^{\infty}(X)=X_{0,0}$ and $\Sigma^{\infty}(X)$ is the constant diagram with value $X$.

For an object $A\in\Ob(\textbf{M})$, the tangent category to $\textbf{M}$ at $A$ is

$$ \mathcal{T}_A\textbf{M}:=\Sp(\textbf{M}_{A//A})  $$

the stabilization of $\textbf{M}_{A//A}$, i.e. the model category of objects over and under $A$.

There is a Quillen adjunction  $ \adjunction{\Sigma^{\infty}_+}{\textbf{M}_{/A}}{\mathcal{T}_A\textbf{M}}{\Omega^{\infty}_+} $ defined as the composed adjunction

$$ \adjunction*{A\sqcup (-)}{\textbf{M}_{/A}}{\textbf{M}_{A//A}}{\forget} \adjunction*{\Sigma^{\infty}}{}{\mathcal{T}_A\textbf{M}}{\Omega^{\infty}} .$$

For more details, the left adjoint $\Sigma^{\infty}_+$ sends each $B \in \textbf{M}_{/A}$ to $ \Sigma^{\infty}_+(B) = \Sigma^{\infty}(A\longrightarrow A\sqcup B \longrightarrow A)$, while the right adjoint $\Omega^{\infty}_+$ sends $X \in \mathcal{T}_A\textbf{M}$ to $\Omega^{\infty}_+(X) = (X_{0,0} \lrar A)$.

Here is the central concept of the paper.
For an object $A\in\Ob(\textbf{M})$, the cotangent complex of $A$ is

$$\rL_A:=\mathbb{L}\Sigma^{\infty}_+(A) \in  \mathcal{T}_A\textbf{M},$$

i.e. the derived suspension spectrum of $A$.

Accordingly, the cotangent complex of $A$ can be modeled by the constant spectrum $\Sigma^{\infty}(A\sqcup A^{\cof})$ where $A^{\cof}$ signifies a cofibrant resolution of $A\in\textbf{M}$. Furthermore, $\rL_A = \Sigma^{\infty}(A\sqcup A^{\cof})$ admits a suspension spectrum replacement determined by fixing $A\sqcup A^{\cof}$ as its value at the bidegree $(0,0)$. In particular, the value at the bidegree $(n,n)$ is given by $\Sigma^n(A\sqcup A^{\cof})$ the $n$-suspension of $A\sqcup A^{\cof}\in \textbf{M}_{A//A}$ (see [[8], Corollary 2.3.3]). In practice, it is usually convenient to exhibit that suspension spectrum as a model for $\rL_A$.

For a map $f : A \rar B$ in $\textbf{M}$, the relative cotangent complex of $f$ is

$$\rL_{B/A}:=\hocofib \, [\,\mathbb{L}\Sigma^{\infty}_+(f) \lrar \rL_B\,]$$

the homotopy cofiber of the map $\mathbb{L}\Sigma^{\infty}_+(f) \lrar \rL_B$ in $\mathcal{T}_B\textbf{M}$.

When $A$ is the initial object then $\rL_{B/A}$ is weakly equivalent to $\rL_B$. Thus, on the one hand, the relative cotangent complex can be viewed as a natural extension of the standard notion of cotangent complex. On other hand, for any map $f : A \rar B$, we can identify $\rL_{B/A}$ with the (absolute) cotangent complex of $f$ considered as an object of $\textbf{M}_{A/}$ (see [[2], Proposition 2.2.10]).

Let $X$ be a fibrant object of $\textbf{M}$. The (spectral) Quillen cohomology of $X$ with coefficients in a given object $M\in\mathcal{T}_X\textbf{M}$ is the space

$$ \HHQ^\star(X ; M) :=  \Map^{\der}_{\mathcal{T}_X\textbf{M}} (\rL_X , M).$$

Moreover, for each $n\in\mathbb{Z}$, the $n$'th Quillen cohomology group of $X$ with coefficients in $M$ is given by

$$ \HHQ^n(X  ; M) := \pi_0 \Map^{\der}_{\mathcal{T}_X\textbf{M}} (\rL_X , M[n])$$

where $M[n] : = \Sigma^{n}M$ the $n$-suspension of $M$ in $\mathcal{T}_X\textbf{M}$.

We will regard $\HHQ^\star(X ; M)$ as a pointed space with the base point being the zero map $\rL_X \x{0}{\lrar} M$. Note also that this base point represents the zero element of the group $\HHQ^0(X  ; M)$.

By adjunction, we have a weak equivalence

$$ \Map^{\der}_{\mathcal{T}_X\textbf{M}} (\rL_X , M) \simeq \Map^{\der}_{\textbf{M}_{/X}} (X, \RR\Omega^{\infty}_+(M)).$$

This reveals how the spectral Quillen cohomology can be put into practical use. Moreover, it implies that Quillen cohomology is invariant under weak equivalences between fibrant objects. Nevertheless, note also that when $\textbf{M}$ is in addition right proper then the model category $\textbf{M}_{/X}$ has already the right homotopy type, and therefore, in this situation the requirement for fibrancy on $X$ is no longer needed.

§.§ Operadic Dold-Kan correspondence

We recall from [21] some results concerning a generalization of the Dold-Kan correspondence into the operadic context. As the standard notations, we write
$$\Gamma : \adjunction*{\simeq}{\C_{\geqslant0}(\textbf{k})}{\sMod_{\textbf{k}}}{} : \sN $$

for the  Dold-Kan correspondence, where $\textbf{k}$ is a commutative ring and $\sN$ refers to the normalization functor. An important fact is that the functors $\Gamma$ and $\sN$ are no longer inverses of each other (or even adjunction) when descending to the categories of monoids. Thus, the Dold-Kan correspondence certainly fails when extending to the framework of algebras over an operad. In spite of this, the existence of a Dold-Kan correspondence for operadic algebras can still be realized from a homotopy-theoretic perspective. Concretely, the result is stated as follows.

Let $\P$ be a $\Sigma$-cofibrant operad in $\sMod_{\textbf{k}}$. Applying the normalization functor to $\P$ levelwise produces an operad in $\C_{\geqslant0}(\textbf{k})$, denoted $\sN\P$.
([21], [22]) The functor $\sN_\P : \Alg_\P(\sMod_{\textbf{k}}) \lrar \Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$ given by applying the normalization functor levelwise is the right adjoint of a Quillen equivalence between semi-model categories.

Several interesting corollaries can be derived from the above theorem, as discussed in what follows.

Firstly, for each set $C$ we may show that the functor $\sN_C : \Op_C(\sMod_{\textbf{k}}) \lrar \Op_C(\C_{\geqslant0}(\textbf{k}))$ (again induced by the normalization functor) is a right Quillen equivalence. Let us denote by $\L_C$ the left adjoint to $\sN_C$. The collection of adjunctions $\{\L_C \dashv\sN_C  \, | \, C \in\Sets\}$ can then be organized into a single adjunction

$$ \L : \adjunction*{}{\Op(\C_{\geqslant0}(\textbf{k}))}{\Op(\sMod_{\textbf{k}})}{} : \sN.$$

We consider these two categories as a relative category with weak equivalences being precisely the Dwyer-Kan equivalences. Another important consequence is as follows.
[[21], $\S$3.2] The functor $\sN$ above induces an equivalence
$$\Op(\sMod_{\textbf{k}})_\infty \lrarsimeq \Op(\C_{\geqslant0}(\textbf{k}))_\infty$$
between the underlying $\infty$-categories with the inverse being induced by the functor $\L$.

Nevertheless, the corollary of interest in this paper is as follows.
[[21], $\S$3.1] Let $\P$ be a levelwise cofibrant operad in $\sMod_{\textbf{k}}$. Then there is a Quillen equivalence
$$\L^{\IB}_\P : \adjunction*{\simeq}{\IbMod(\sN\P)}{\IbMod(\P)}{} :  \sN^{\IB}_\P$$
in which the right adjoint is as usual defined by applying the normalization functor levelwise.

§.§ More conventions and notations

Let $\cS$ be a suitable symmetric monoidal model category, $\P$ a $C$-colored operad in $\cS$, and let $A$ be a $\P$-algebra. To proceed, we will use the following notations and conventions.

(i) We will say that the pair $(\cS, \P)$ is sufficient (resp. stably sufficient) if the base category $\cS$ is sufficient (resp. stably sufficient) in the sense of [[1], $\S$3], and in addition, $\P$ is $\Sigma$-cofibrant.

(ii) Moreover, the pair $(\cS, \P)$ is said to be abundant (resp. stably abundant) if the base category $\cS$ is abundant (resp. stably abundant) in the sense of [[1], $\S$3] and such that $\P$ is cofibrant, and is good in the sense of [[1], $\S$5.1].

(iii) We will say that the triple $(\cS,\P, A)$ is sufficient (resp. stably sufficient) if the pair $(\cS, \P)$ is sufficient (resp. stably sufficient) in the above sense, and in addition, $A\in \Alg_\P(\cS)$ is cofibrant.

(iv) Furthermore, $(\cS,\P, A)$ is abundant (resp. stably abundant) if the pair $(\cS, \P)$ is abudant  (resp. stably abundant) and $A\in \Alg_\P(\cS)$ is cofibrant.

The hypotheses presented above allow us to inherit some main results of [23, 1] (see $\S$<ref> for more details). Therefore, readers may safely disregard the details of these hypotheses.

(v)  To avoid confusion, we will write $\P^{\sB}$ (resp. $\P^{\IB}, \P^{\sr}$ and $\P^{i\ell}$) to denote $\P$ in its role as an object of $\BMod(\P)$ (resp. $\IbMod(\P), \RMod(\P)$ and $\ILMod(\P)$). We also write $A^{\sm}$ standing for $A$ in its role as an object of $\Mod_\P^A$.

(vi) We will refer to different cotangent complexes of $\P$ using the following notations:

$\bullet$ $\rL_\P \in \T_\P\Op(\cS)$: the cotangent complex of $\P$ considered as an object of $\Op(\cS)$.

$\bullet$ $\rL^{\red}_\P \in \T_\P\Op_C(\cS)$: the cotangent complex of $\P$ considered as an object of $\Op_C(\cS)$.

$\bullet$ $\rL_{\P^{\sB}} \in \T_{\P^{\sB}}\BMod(\P)$: the cotangent complex of $\P$ when regarded as a bimodule over itself.

$\bullet$ $\rL_{\P^{\IB}} \in \T_{\P^{\IB}}\IbMod(\P)$: the cotangent complex of $\P$ as an infinitesimal bimodule over itself.

(vii) To denote different cotangent complexes of $A$, we will use the following:

$\bullet$ $\rL_A \in \T_A\Alg_\P(\cS)$: the cotangent complex of $A$ considered as an algebra over $\P$, and

$\bullet$ $\rL_{A^{\sm}} \in \T_{A^{\sm}} \Mod_\P^A$: the cotangent complex of $A$ considered as an object of $\Mod_\P^A$.

(viii) We will write

$$ \HHQ^\star(\P; -) \;\; \text{and} \;\; \HHQ^\star_{\red}(\P; -)$$

respectively standing for Quillen cohomology of $\P$ when considered as an object of $\Op(\cS)$ and $\Op_C(\cS)$. Besides that, we will write $\HHQ^\star(A; -)$ to denote Quillen cohomology of $A$ as a $\P$-algebra.

(ix) Moreover, we will refer to $\HHQ^\star(\P; -)$ as the (proper) Quillen cohomology of $\P$, and refer to $\HHQ^\star_{\red}(\P; -)$ as the reduced Quillen cohomology of $\P$. By notation the former is classified by  $\rL_\P \in \T_\P\Op(\cS)$, and while the latter is classified by $\rL^{\red}_\P \in \T_\P\Op_C(\cS)$.

(x) We will write
$$\Free_\P^{\IB} : \Coll_C(\cS) \lrar \IbMod(\P)$$

$$(\text{resp.} \Free_\P^{i\ell} : \Coll_C(\cS) \lrar \ILMod(\P))$$

standing for the functor that assigns to each $C$-collection $M$ the free infinitesimal $\P$-bimodule (resp. free infinitesimal left $\P$-module) generated by $M$.

(xi) We will write
$$\Free_A : \cS^{\times C} \lrar \Mod_A^\P$$
to denote the functor that assigns to each object $X = \{X(c)\}_{c\in C} \in \cS^{\times C}$ the free $A$-module over $\P$ generated by $X$.

(xii) We will denote by $\mathcal{E}_C$ the $C$-collection that is $\emptyset_\cS$ on all levels except that $\mathcal{E}_C(c) = 1_\cS$ for every $c\in C$. When $C$ is a singleton, then we will signify that using the notation $\mathcal{E}_*$.

(xiii) Finally, for a (semi) model category $\textbf{M}$ containing two objects $X$ and $Y$, we will sometimes abbreviate the derived mapping space $\Map^{\h}_{\textbf{M}}(X,Y)$ simply by $\Map^{\h}(X,Y)$, as long as the category $\textbf{M}$ is understood.

§ OPERADIC TANGENT CATEGORIES AND COTANGENT COMPLEX OF ENRICHED OPERADS

We recall from  [23, 1] some fundamental results concerning various operadic tangent categories and cotangent complex of operads. Let $\cS$ be a symmetric monoidal model category, $\P$ a $C$-colored operad in $\cS$, and let $A$ be a $\P$-algebra.

(i) First, assume that the triple $(\cS,\P,A)$ is sufficient. The following theorem plays a key role in the investigation of operadic tangent categories.

[Harpaz-Nuiten-Prasma [23]] There is a Quillen equivalence

\begin{equation}\label{eq:Quillenalg}
\adjunction*{\simeq}{\T_{A^{\sm}} \Mod_\P^A}{\T_A\Alg_\P(\cS)}{}
\end{equation}

induced by the adjunction $\adjunction*{}{(\Mod_\P^A)_{A^{\sm}/}}{(\Alg_\P(\cS))_{A/}}{}$ of induction-restriction functors. Moreover, when $(\cS,\P,A)$ is stably sufficient then there is a chain of Quillen equivalences

\begin{equation}\label{eq:Quillenalg1}
\adjunction*{\simeq}{\Mod_\P^A}{}{}\adjunction*{\simeq}{\T_{A^{\sm}} \Mod_\P^A}{\T_A\Alg_\P(\cS)}{}
\end{equation}

where the first adjunction is given by the composed Quillen equivalence

\begin{equation}\label{eq:seqQuillenstab1}
\adjunction*{(-) \sqcup A^{\sm}}{\Mod_\P^A}{(\Mod_\P^A)_{A^{\sm}//A^{\sm}}}{\ker} \adjunction*{\Sigma^{\infty}}{}{\T_{A^{\sm}} \Mod_\P^A}{\Omega^{\infty}}
\end{equation}

(see $\S$<ref> for notations).

(ii) A version of the above theorem for various tangent categories at $\P$ is as follows.

[1] Assume that the pair $(\cS, \P)$ is sufficient. There is a chain of Quillen equivalences

\begin{equation}\label{eq:seqQuillen}
\adjunction*{\simeq}{\T_{\P^{\IB}}\IbMod(\P)}{\T_{\P^{\sB}}\BMod(\P)}{} \adjunction*{\simeq}{}{\T_\P\Op_C(\mathcal{S})}{} \adjunction*{\simeq}{}{\T_\P\Op(\mathcal{S})}{}
\end{equation}

that is induced by the adjunctions of induction-restriction functors

$$ \adjunction*{}{\IbMod(\P)_{\P^{\IB}/}}{\BMod(\P)_{\P^{\sB}/}}{} \adjunction*{}{}{\Op_C(\mathcal{S})_{\P/}}{} \adjunction*{}{}{\Op(\mathcal{S})_{\P/}}{}.$$
Furthermore, when $(\cS, \P)$ is stably sufficient, we have a prolonged chain of Quillen equivalences:

\begin{equation}\label{eq:seqQuillenstab}
\adjunction*{\simeq}{\IbMod(\P)}{}{}\adjunction*{\simeq}{\T_{\P^{\IB}}\IbMod(\P)}{\T_{\P^{\sB}}\BMod(\P)}{} \adjunction*{\simeq}{}{\T_\P\Op_C(\mathcal{S})}{} \adjunction*{\simeq}{}{\T_\P\Op(\mathcal{S})}{}
\end{equation}

where the first adjunction is given by the composed Quillen equivalence

\begin{equation}\label{eq:seqQuillenstab2}
\adjunction*{(-) \sqcup \P^{\IB}}{\IbMod(\P)}{\IbMod(\P)_{\P^{\IB}//\P^{\IB}}}{\ker} \adjunction*{\Sigma^{\infty}}{}{\T_{\P^{\IB}}\IbMod(\P)}{\Omega^{\infty}}
\end{equation}

(see $\S$<ref> for notations).

(iii) In what follows, we shall recall from [1] a description of the cotangent complex of $\P$. The key proposition is as follows.

[[1], $\S$5.2] Suppose that the pair $(\cS, \P)$ is abundant. Under the right Quillen equivalence
$$\T_\P\Op(\mathcal{S}) \lrarsimeq \T_{\P^{\sB}}\BMod(\P),$$

the cotangent complex $\rL_\P \in \T_\P\Op(\cS)$ is identified with $\rL_{\P^{\sB}}[-1] \in \T_{\P^{\sB}}\BMod(\P)$.

Next we recall how the object $\rL_{\P^{\sB}}$ can be well calculated. Assume that $\P$ is $\Sigma$-cofibrant and that $\cS$ is combinatorial. In this situation, for any cofibrant resolution $(\P^{\sB})^{\cof} \lrarsimeq \P^{\sB}$ of $\P^{\sB}\in\BMod(\P)$, the induced map
$$ (\P^{\sB})^{\cof}  \sqcup  \P^{\sB} \lrarsimeq \P^{\sB} \sqcup \P^{\sB}$$

is a weak equivalence. Thus we may exhibit the constant spectrum $\Sigma^\infty(\P^{\sB} \sqcup \P^{\sB}) \in \T_{\P^{\sB}}\BMod(\P)$ as a model for $\rL_{\P^{\sB}}$. For more convenience, we find a suspension spectrum replacement for $\rL_{\P^{\sB}}$. To this end, we just need to compute $\Sigma^n(\P^{\sB} \sqcup \P^{\sB})$ for each $n\geq0$, i.e. the $n$-suspension  of $\P^{\sB} \sqcup \P^{\sB} \in \BMod(\P)_{\P^{\sB}//\P^{\sB}}$ (see Remark <ref>).

For each $n \geqslant 0$, we denote by $\sS^{n}:=\Sigma^{n}(1_\mathcal{S}\sqcup 1_\mathcal{S})\in\cS$ with the suspension $\Sigma(-)$ taken in $\mathcal{S}_{1_\mathcal{S}//1_\mathcal{S}}$, and refer to $\sS^{n}$ as the pointed $n$-sphere in $\cS$. Moreover, we will write $\sS_C^{n}$ to denote the $C$-collection that is $\emptyset_\cS$ on all levels except that $\sS_C^{n}(c;c)=\sS^{n}$ for every $c\in C$.

We have that $\sS_C^{0} \cong \I_C \sqcup \I_C$, and for every $n\geq0$, we have $\sS_C^{n} \simeq \Sigma^{n}(\I_C \sqcup \I_C)$ where the suspension  $\Sigma(-)$ is taken in $\Coll_C(\cS)_{\I_C//\I_C}$.

In [[1], $\S$5.3], we showed that $\P\circ\sS_C^{0}$ is a model for the coproduct $\P^{\sB} \sqcup \P^{\sB}\in\BMod(\P)$, and furthermore, for every $n\geq0$ we have

$$ \Sigma^n(\P^{\sB} \sqcup \P^{\sB}) \simeq \P\circ\sS_C^{n}.$$

We have on each level that

$$ (\P\circ\sS_C^{n})(c_1,\cdots,c_m;c) = \P(c_1,\cdots,c_m;c)\otimes (\sS^{n})^{\otimes m}.$$

Let us denote by $\P\circ\sS_C^{\bullet}\in\T_\P\BMod(\P)$ the suspension spectrum with $(\P\circ\sS_C^{\bullet})_{n,n} = \P\circ\sS_C^{n}$. (This agrees with the object $\widetilde{\rL}_\P$ in the loc.cit). Thus we obtain the following.
There is a weak equivalence $\Sigma^\infty(\P^{\sB} \sqcup \P^{\sB}) \simeq \P\circ\sS_C^{\bullet}$ in $\T_\P\BMod(\P)$, which exhibits $\P\circ\sS_C^{\bullet}$ as a suspension spectrum model for $\rL_{\P^{\sB}}$.

Furthermore, we obtain the following.

Assume that $(\cS, \P)$ is sufficient. Then under the right Quillen equivalence
$$\T_{\P^{\sB}}\BMod(\P) \lrarsimeq \T_{\P^{\IB}}\IbMod(\P),$$

the object $\P\circ\sS_C^{\bullet}$ is identified to itself $\P\circ\sS_C^{\bullet}$ considered as a prespectrum of infinitesimal $\P$-bimodules. Hence the latter is a model for the derived image of $\rL_{\P^{\sB}}$ in $\T_{\P^{\IB}}\IbMod(\P)$.

Assume further that $(\cS, \P)$ is stably sufficient, so that we obtain the chain of Quillen equivalences (<ref>). Then under the composed right Quillen equivalence
$$ \ker\circ \, \Omega^\infty : \T_{\P^{\IB}}\IbMod(\P) \lrarsimeq \IbMod(\P),$$

the prespectrum $\P\circ\sS_C^{\bullet}$ is identified with $\ovl{\rL}_\P \in \IbMod(\P)$ which is given on each level by

\begin{equation}\label{eq:barL}
\ovl{\rL}_\P(c_1,\cdots,c_m;c) \simeq \P(c_1,\cdots,c_m;c)\otimes \hocolim_n\Omega^{n} [\, (\sS^{n})^{\otimes m}  \times_{1_\mathcal{S}}^{\h}  0\,]
\end{equation}

where the desuspension $\Omega(-)$ is taken in $\cS$. Thus in this situation, $\ovl{\rL}_\P$ is a model for the derived image of $\rL_{\P^{\sB}}$ in $\IbMod(\P)$.

In summary, the cotangent complex $\rL_\P \in \T_\P\Op(\cS)$ is described as follows.

Suppose that the pair $(\cS, \P)$ is abundant. Then under the right Quillen equivalence

$$ \T_\P\Op(\cS) \lrarsimeq \T_{\P^{\IB}}\IbMod(\P),$$

the cotangent complex $\rL_\P$ is identified with $(\P\circ\sS_C^{\bullet})[-1]\in\T_{\P^{\IB}}\IbMod(\P)$. Moreover, when $(\cS, \P)$ is stably abundant, we obtain that under the right Quillen equivalence

$$ \T_\P\Op(\cS) \lrarsimeq \IbMod(\P),$$

$\rL_\P \in \T_\P\Op(\cS)$ is identified with $\ovl{\rL}_\P[-1]\in\IbMod(\P)$.

An immediate consequence of this theorem is as follows. After sending coefficients into $\T_{\P^{\IB}}\IbMod(\P)$, we have a weak equivalence

$$ \HHQ^\star(\P;\M) \simeq \Map^{\h}(\P\circ\sS_C^{\bullet},\M[1])$$

for every $\M\in\T_{\P^{\IB}}\IbMod(\P)$. When $(\cS, \P)$ is in addition stably abundant then we have

$$ \HHQ^\star(\P;M) \simeq \Map^{\h}(\ovl{\rL}_\P,M[1])$$

for every $M\in\IbMod(\P)$.

(iv) Various cotangent complexes of $\P$ are related via a cofiber sequence as follows. We will  write $\theta_\P: \Free^{\IB}_\P(\I_C) \lrar \P$ for the map in $\IbMod(\P)$ classified by the unit map $\I_C \lrar \P$.

[[1], $\S$5.3] Suppose that the pair $(\cS, \P)$ is abundant. There is a cofiber sequence in $\T_{\P^{\IB}}\IbMod(\P)$ of the form

$$ \rL^{\red}_\P \lrar \Sigma^\infty_+(\theta_\P) \lrar \P\circ\sS_C^{\bullet} $$

where we use the same notation to denote the derived image of $\rL^{\red}_\P$ under the Quillen equivalence $\T_\P\Op_C(\mathcal{S})\simeq\T_{\P^{\IB}}\IbMod(\P)$, and the middle term is the image of $\theta_\P$ through the functor
$$\Sigma^\infty_+ : \IbMod(\P)_{/\P^{\IB}} \lrar \T_{\P^{\IB}}\IbMod(\P).$$

§ MORE ABOUT INFINITESIMAL BIMODULES OVER AN OPERAD

As before, we let $\P$ be a fixed $C$-colored operad in $\cS$. Recall by definition that an infinitesimal left $\P$-module is a $C$-collection $M$ endowed with a map $\P\circ_{(1)}M \lrar M$ of $C$-collections where $``\circ_{(1)}"$ signifies the infinitesimal composite product
$$ - \circ_{(1)} - : \Coll_C(\cS) \times \Coll_C(\cS) \lrar \Coll_C(\cS)$$
(see [9], $\S$6.1). The action map must satisfy the usual axioms of the associativity and unitality. We let $\ILMod(\P)$ denote the category of infinitesimal left $\P$-modules.

An infinitesimal $\P$-bimodule is a $C$-collection $M$ equipped with an infinitesimal left $\P$-module structure, exhibited by a map $\mathcal{P}\circ_{(1)}M \lrar M$; and with a right $\P$-module structure, exhibited by a map $M \circ \P \lrar M$. Moreover, these two are subject to the usual axiom of the compatibility. (See [[1], $\S$2.2] for more details).

The data of an infinitesimal left $\P$-module structure $\P\circ_{(1)}M \lrar M$ consist of $\Sigma_*$-equivariant maps of the form

$$ \P(c_1,\cdots,c_n;c) \otimes M(d_1,\cdots,d_m;c_i) \lrar M(c_1,\cdots,c_{i-1},d_1,\cdots,d_m,c_{i+1},\cdots,c_n;c).$$

Whilst the data of a right $\P$-module structure $M \circ \P \lrar M$ consist of $\Sigma_*$-equivariant maps of the form

$$  M(c_1,\cdots,c_n;c) \otimes \P(d_{1,1},\cdots,d_{1,k_1};c_1) \otimes \cdots \otimes \P(d_{n,1},\cdots,d_{n,k_n};c_n) \lrar M(d_{1,1},\cdots,d_{1,k_1},\cdots,d_{n,1},\cdots,d_{n,k_n};c).$$

As discussed in [[1], $\S$2.2], we have for each $C$-collection $M$ that
$$ \Free_\P^{\IB}(M) \cong \mathcal{P}\circ_{(1)}( M \circ \mathcal{P}) \;\;\; \text{and} \;\;\; \Free_\P^{i\ell}(M) \cong \mathcal{P}\circ_{(1)}M.$$
Taking $M = \mathcal{E}_C$, we have

$$ \Free_\P^{i\ell}(\mathcal{E}_C) \cong \mathcal{P}\circ_{(1)}\mathcal{E}_C \cong \mathcal{P}\circ_{(1)}( \mathcal{E}_C \circ \mathcal{P}) \cong \Free_\P^{\IB}(\mathcal{E}_C)$$

(see $\S$<ref> for notations). For the case $C =\{*\}$, note that $\mathcal{P}\circ_{(1)}\mathcal{E}_*$ agrees with the $``$shifted object” $\P[1]$ of [[10], $\S$17.3], which is given on each level by $\P[1](n)= \mathcal{P}(n+1)$. Thus we may write
$$ \Free_\P^{i\ell}(\mathcal{E}_*) \cong  \P[1] \cong \Free_\P^{\IB}(\mathcal{E}_*).$$

A basic fact is that $\P$ is free (generated by $\I_C$) when regarded as a right module over itself. Nonetheless, $\P$ is in general not a free object when living in $\ILMod(\P)$ (or $\IbMod(\P)$). A noteworthy exception is the case of the commutative operad $\Com$. Indeed, according to the above remark, we obtain that the operad $\Com$ is free (generated by $\mathcal{E}_*$) when considered as an object of $\IbMod(\Com)$ (or $\ILMod(\Com)$).

In what follows, we shall recall from [[1], $\S$2.2] how infinitesimal $\P$-bimodules can be represented as $\cS$-valued enriched functors. We will need the following notations and conventions.

$\bullet$ We will write $\Fin$ to denote the smallest skeleton of the category of finite sets whose objects are $\underline{0}:=\emptyset$ and $\underline{m}:=\{1,\cdots,m\}$ for $m\geqslant1$.

$\bullet$ We write $\Fin_*$ to denote the smallest skeleton of the category of finite pointed sets. The objects of $\Fin_*$ will be written as $\left \langle m \right \rangle := \{0,1,\cdots,m\}$  with $0$ as the basepoint, for every $m\geqslant0$.

$\bullet$ A morphism $f: \l m \r \lrar \l n \r$ in $_*$ is called an \textbf{inert map} if $f^-1(i)$ is a singleton for every $i∈{1,⋯,n}$, and an \textbf{active map} if $f^-1(0) = {0}$.

$∙$ We will write $_*^$ (resp. $_*^$)  for the subcategory of $_*$ consisting of all the inert maps (resp. active maps).

\begin{rem}\label{r:finact} There is an obvious embedding $\Fin \lrar \Fin_*$ that sends each $\underline{m}$ to $\l m \r$. Moreover, we may show that the essential image of this functor coincides with $\Fin_*^{\act}$.
\end{rem}

\begin{rem}\label{r:facfin} It is known that the pair $(\Fin_*^{\ine}, \Fin_*^{\act})$ form a \textbf{factorization system} on $\Fin_*$. That is to say, both $\Fin_*^{\ine}$ and $\Fin_*^{\act}$ are a \textit{wide subcategory} and such that every morphism $\l m \r \lrar \l n \r$  in $\Fin_*$ can be factored as a composed map
$$ \l m \r \x{f}{\lrar} \l p \r \x{g}{\lrar} \l n \r $$
with $f$ (resp. $g$) being inert (resp. active); and moreover, this factorization is unique up to unique isomorphism. In an opposite convention, we have that $\left((\Fin_*^{\act})^{\op}, (\Fin_*^{\ine})^{\op}\right)$ form a factorization system on $(\Fin_*)^{\op}$.
\end{rem}

\begin{conss}\label{infbimod} We construct three categories in $\cS$, denoted $\textbf{Ib}^{\mathcal{P}}$, $\textbf{R}^{\mathcal{P}}$ and  ${\textbf{I}\ell}^{\mathcal{P}}$, as follows. Firstly, the three categories have the same objects given by $C$-sequences $\{(c_1,\cdots,c_n;c) \, | \, c_i,c \in C , n\geqslant0  \}$.

$\bullet$ The mapping spaces of $\textbf{Ib}^{\mathcal{P}}$ are defined as follows. For each map $f : \l m \r \lrar \l n \r$ in $\Fin_*$, we define
$$  \Map^f_{\textbf{Ib}^{\mathcal{P}}}((c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;d)) := \mathcal{P}\left (c,\{d_j\}_{j\in f^{-1}(0)};d \right ) \otimes \bigotimes_{i=1,\cdots,n} \mathcal{P} \left (\{d_j\}_{j\in f^{-1}(i)};c_i \right ).$$
Then we define
$$ \Map_{\textbf{Ib}^{\mathcal{P}}}((c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;d)) := \bigsqcup_{\left \langle m \right \rangle  \overset{f }{\rar} \left \langle n \right \rangle} \Map^f_{\textbf{Ib}^{\mathcal{P}}}((c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;d) ) $$
where the coproduct ranges over $\Hom_{\Fin_*}(\l m \r, \l n \r)$. The unit morphisms of $\textbf{Ib}^{\mathcal{P}}$ are defined via the unit operations of $\mathcal{P}$, and moreover, the categorical structure maps are induced by the composition in $\mathcal{P}$.

$\bullet$ The mapping spaces of $\textbf{R}^{\mathcal{P}}$ are concentrated in objects of the form
$$ \Map_{\textbf{R}^{\mathcal{P}}}(\, (c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;c) \,) := \bigsqcup_{\left \langle m \right \rangle  \underset{\text{active}}{\overset{f}{\lrar}} \left \langle n \right \rangle} \left [  \bigotimes_{i=1,\cdots,n} \mathcal{P} \left (\{d_j\}_{j\in f^{-1}(i)};c_i \right )  \right ] $$
where the coproduct ranges over those maps $f\in\Hom_{\Fin_*^{\act}}(\l m \r,\l n \r)$. Observe that there is a map
$$ \Map_{\textbf{R}^{\mathcal{P}}}( \, (c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;c) \, ) \lrar \Map_{\textbf{Ib}^{\mathcal{P}}}(\, (c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;c) \,) $$
induced by the embedding $\Fin_*^{\act} \lrar \Fin_*$ and by inserting the unit operation $\id_c$ into the factor $\P(c;c)$ of the right hand side. The categorical structure of $\textbf{R}^{\mathcal{P}}$ is then defined via the operad structure of $\P$, so that $\textbf{R}^{\mathcal{P}}$ forms a subcategory of $\textbf{Ib}^{\mathcal{P}}$.

$\bullet$ The mapping spaces of ${\textbf{I}\ell}^{\mathcal{P}}$ are given by
$$  \Map_{\textbf{I}\ell^{\mathcal{P}}}((c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;d)) := \bigsqcup_{f} \mathcal{P}\left (c,\{d_j\}_{j\in f^{-1}(0)};d \right )$$
where the coproduct ranges over the subset of $\Hom_{\Fin_*^{\ine}}(\l m \r,\l n \r)$ consisting of those maps $f$ with the property that the two colors $c_i$ and $d_{f^{-1}(i)}$ coincide for every $i\in\{1,\cdots,n\}$. As above, there is an embedding $\textbf{I}\ell^{\mathcal{P}} \lrar \textbf{Ib}^{\mathcal{P}}$ induced by the canonical embedding $\Fin_*^{\ine} \lrar \Fin_*$ and by inserting the identity operation $\id_{c_i}$ into the factor $\P(c_i;c_i)$ for $i=1,\cdots,n$.
\end{conss}

The following proposition expresses our main interest in the above constructions.
\begin{prop}\label{p:ibfunc} There are natural isomorphisms of categories

(i)  $ \Fun(\textbf{Ib}^{\mathcal{P}}, \cS) \cong \IbMod(\P)$,

(ii)  $ \Fun(\textbf{R}^{\mathcal{P}}, \cS) \cong \RMod(\P)$, and

(iii)  $ \Fun(\textbf{I}\ell^{\mathcal{P}}, \cS) \cong \ILMod(\P)$.
\begin{proof} The first two isomorphisms are included in [[1], $\S$2.2]. The other can be proved in the same manner.
\end{proof}
\end{prop}

We provide additional examples and remarks regarding Constructions \ref{infbimod}.

\begin{example}\label{ex:ibcom} It is clear that $\textbf{Ib}^{\Com}$ is isomorphic to $\Fin_*^{\op}$, and while $\textbf{I}\ell^{\Com}$ (resp. $\textbf{R}^{\Com}$) is isomorphic to $(\Fin_*^{\ine})^{\op}$ (resp. $(\Fin_*^{\act})^{\op}$). Here we made use of the functor $\iota : \Sets \lrar \cS$ (see Remark \ref{r:iota}) to associate to each ordinary category the $\cS$-enriched version of that category.
\end{example}

\begin{rem}\label{ex:ibnucom} Let $\P$ be such that $\P(c) = \emptyset_\cS$ for every color $c$. Then by construction, we have that $$\Map^f_{\textbf{Ib}^{\mathcal{P}}}((c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;d)) = \emptyset_\cS$$ whenever $f$ is not a surjection. Thus, the mapping space $\Map_{\textbf{Ib}^{\mathcal{P}}}((c_1,\cdots,c_n;c) , (d_1,\cdots,d_m;d))$ consists of only the summands that correspond to a surjective map $\l m \r \lrar \l n \r$. The same observation holds for $\textbf{I}\ell^{\P}$ and  $\textbf{R}^{\P}$. Let us now denote by $\nucom$ the \textbf{nonunital commutative operad} which coincides with $\Com$ except that $\nucom(0)=\emptyset_\cS$. Accordingly, we can see that $\textbf{Ib}^{\nucom} \cong (\Fin_*^{\sur})^{\op}$ where $\Fin_*^{\sur}$ denotes the subcategory of $\Fin_*$ consisting of surjective maps. We have also that
$$ \textbf{I}\ell^{\nucom} \cong (\Fin_*^{\ine, \sur})^{\op} \;\; \text{and} \;\; \textbf{R}^{\nucom} \cong (\Fin_*^{\act, \sur})^{\op}$$
in which $\Fin_*^{\ine, \sur} := \Fin_*^{\ine} \cap \Fin_*^{\sur}$ and $\Fin_*^{\act, \sur} := \Fin_*^{\act} \cap \Fin_*^{\sur}$.
\end{rem}

\begin{example}\label{ex:ibass} Consider the case where $\P=\Ass$. By construction, the category $\textbf{Ib}^{\Ass}$ has the same objects as $\Fin_*$. Moreover, the data of a morphism in $\Map_{\textbf{Ib}^{\Ass}}(\l n \r,\l m \r)$ consist of a map $f : \l m \r \lrar \l n \r$ in $\Fin_*$, and in addition, a tuple of permutations $$(\sigma_0,\sigma_1,\cdots,\sigma_n) \in \Sigma_{k_0}\times\Sigma_{k_1}\times\cdots\times\Sigma_{k_n}$$
with $k_i$ being the cardinality of $f^{-1}(i)$. We can see that $\textbf{Ib}^{\Ass}$ is isomorphic to $\Gamma(\as)^{\op}$ the opposite category of the category $\Gamma(\as)$ of [24]. Moreover, the significant functor $\bar{\sB} : \Gamma(\as)^{\op} \lrar \cS$ considered in the loc.cit is nothing but a model for $\Ass^{\IB} \in \IbMod(\Ass)$.
\end{example}

\begin{rem}\label{r:esimage} From a set-theoretic perspective, a morphism in $\textbf{Ib}^{\mathcal{P}}$ can be represented as a tuple of operations of $\mathcal{P}$ of the form $(\mu ; \nu_1,\cdots,\nu_n)$. In this way, the image of the embedding $\textbf{R}^{\mathcal{P}}\lrar\textbf{Ib}^{\mathcal{P}}$ consists of those morphisms of the form $(\id_c ; \nu_1,\cdots,\nu_n)$; whilst the image of the embedding $\textbf{I}\ell^{\mathcal{P}}\lrar\textbf{Ib}^{\mathcal{P}}$ consists of those morphisms of the form $(\mu ; \id_{c_1},\cdots,\id_{c_n})$.
\end{rem}

\begin{rem} We have seen from Example \ref{ex:ibcom} that the triple $\left(\textbf{Ib}^{\Com}, \textbf{R}^{\Com}, \textbf{I}\ell^{\Com}\right)$ agrees with the triple $$\left(\Fin_*^{\op}, (\Fin_*^{\act})^{\op}, (\Fin_*^{\ine})^{\op}\right).$$ An immediate consequence is that the pair $\left(\textbf{R}^{\Com}, \textbf{I}\ell^{\Com}\right)$ form a \textit{factorization system} on $\textbf{Ib}^{\Com}$ (see Remark \ref{r:facfin}).
\end{rem}

In fact, this phenomenon can be generalized to any operad that comes from an operad in $\Sets$, as demonstrated in the statement below.

\begin{prop} Suppose that $\P$ comes from an operad in $\Sets$. Then the pair $\left(\textbf{R}^{\P}, \textbf{I}\ell^{\P}\right)$ form a factorization system on $\textbf{Ib}^{\P}$.
\begin{proof} Clearly $\textbf{R}^{\P}$ and $\textbf{I}\ell^{\P}$ are a wide subcategory of $\textbf{Ib}^{\P}$. Let $\widetilde{f}$ be a morphism in $\textbf{Ib}^{\P}$ that lies above a map $f$ in $\Fin_*$. As discussed in Remark \ref{r:facfin}, $f$ admits a factorization $f = g\circ h$ with $h\in \Fin_*^{\ine}$ and $g \in \Fin_*^{\act}$, and this factorization is unique (up to unique isomorphism). For each such a factorization, there exists a pair $(\widetilde{g}, \widetilde{h}) \in \textbf{R}^{\P}\times\textbf{I}\ell^{\P}$ lying above the pair $(g, h)$ and such that $\widetilde{h} \circ \widetilde{g} = \widetilde{f}$. Moreover, the choice of $(\widetilde{g}, \widetilde{h})$ is (absolutely) unique, as described in Remark \ref{r:esimage}.
\end{proof}
\end{prop}

\begin{rem} The category $\textbf{Ib}^{\P}$ is the same as the category $``\mathcal{U}(\P)"$ of [25], and moreover, the above factorization system coincides with that of Proposition 2.42 of the loc.cit. As in the latter, we may obtain further a factorization system on the twisted arrow category $\Tw(\P)$ (see $\S$\ref{s:twis}) by taking the (covariant) unstraightening of the functors $\P^{\sr} : \textbf{R}^{\P} \lrar \Sets$ and $\P^{i\ell} : \textbf{I}\ell^{\P} \lrar \Sets$. The situation would become more interesting when passing to the framework of simplicial (or topological) operads, at which $\Tw(\P)$ is considered as an $\infty$-category.
\end{rem}

\section{Hochschild and  Quillen cohomologies of operadic algebras}\label{s:op}

In this section, we will assume that the base category $\cS$ is \textbf{sufficient} in the sense of [[1], $\S$3], and we let $\P$ be a fixed $C$-colored operad in $\cS$.

We shall first introduce \textbf{spectral Hochschild cohomology} of enriched operads and operadic algebras, with the aim of generalizing the classical Hochschild cohomology to a broader framework.

Then we will give a discussion on various \textbf{endomorphism constructions}, which describe the natural passage from operadic bimodules to operadic algebras, as well as the natural passage from  operadic infinitesimal bimodules to modules over operadic algebras.

The endomorphism constructions provide a machinery for how various cohomologies of $\P$-algebras can be governed by the corresponding cohomologies of the operad $\P$ itself, as will be discussed in the last subsection.

\subsection{Spectral Hochschild cohomology}\label{s:spechoch}

Let us start with the notion of spectral Hochschild cohomology of operads.

\begin{dfn} The \textbf{spectral Hochschild cohomology} of $\P$ with coefficients in a given object $M\in\T_{\P^{\IB}}\IbMod(\P)$ is the space
$$ \HHH^\star(\P, M) := \Map^{\h}(\rL_{\P^{\IB}}, M).$$
For each $n\in\ZZ$, the \textit{$n$'th spectral Hochschild cohomology group of $\P$ with coefficients in $M$} is
$$ \HHH^n(\P  , M) := \pi_0 \Map^{\der} (\rL_{\P^{\IB}} , M[n]).$$
\end{dfn}

\begin{rem} By definition, the spectral Hochschild cohomology of $\P$ coincides with Quillen cohomology of $\P$ considered as an infinitesimal bimodule over itself. To avoid confusion, we will refer to $\rL_{\P^{\IB}}$ as the \textbf{Hochschild complex of} $\P$.
\end{rem}

Now let $A\in \Alg_\P(\cS)$ be an algebra over $\P$. The corresponding cohomology of $A$ is defined as follows.
\begin{dfn} The \textbf{spectral Hochschild cohomology} of $A$ with coefficients in a given object $N\in\T_{A^{\sm}} \Mod_\P^A$ is the space
$$ \HHH^\star(A, N) := \Map^{\h}(\rL_{A^{\sm}}, N).$$
For each $n\in\ZZ$, the \textit{$n$'th spectral Hochschild cohomology group of $A$ with coefficients in $N$} is given by
$$ \HHH^n(A, N) := \pi_0 \Map^{\der}(\rL_{A^{\sm}}, N[n]).$$
\end{dfn}

\begin{rem} Accordingly, the spectral Hochschild cohomology of $A$ agrees with Quillen cohomology of $A$ regarded as an object of $\Mod_\P^A$. As above, we will also refer to $\rL_{A^{\sm}}$ as the \textbf{Hochschild complex of} $A$.
\end{rem}

\begin{conv} From now on, we will usually refer to \textit{spectral Hochschild cohomology} simply as \textit{Hochschild cohomology}, except for cases of inconsistency.
\end{conv}

\begin{example} Recall from Remark \ref{r:encode} that there exists an operad $\textbf{O}_C$ such that the category of $\textbf{O}_C$-algebras is isomorphic to the category of $C$-colored operads in $\cS$. Thus we may consider $\P$ as an object of $\Alg_{\textbf{O}_C}(\cS)$. It can be proved further that there is a categorical isomorphism
$$  \Mod_{\textbf{O}_C}^\P  \cong \IbMod(\P) $$
between $\P$-modules over $\textbf{O}_C$ and infinitesimal $\P$-bimodules, which identifies $\P^{\IB}\in\IbMod(\P)$ with $\P$ itself as an object of $\Mod_{\textbf{O}_C}^\P$. Accordingly, we obtain that Hochschild cohomology of the operad $\P$ agrees with Hochschild cohomology of $\P\in\Alg_{\textbf{O}_C}(\cS)$. (On other hand, we can see that Quillen cohomology of $\P\in\Alg_{\textbf{O}_C}(\cS)$ coincides with the reduced Quillen cohomology of the operad $\P$).
\end{example}

\begin{notns} By abuse of notation, we will write $\P\otimes\mathbb{S}\in\T_{\P^{\IB}}\IbMod(\P)$ for the suspension spectrum with value on each level $(c_1,\cdots,c_m;c)$ and at the bidegree $(n,n)$ being given by
$$ (\P\otimes\mathbb{S})_{n,n}(c_1,\cdots,c_m;c) := \P(c_1,\cdots,c_m;c)\otimes \sS^{n}.$$
In the same fashion, we let $A\otimes\mathbb{S}\in\T_{A^{\sm}} \Mod_\P^A$ denote the suspension spectrum with
$$ (A\otimes\mathbb{S})_{n,n}(c) := A(c)\otimes \sS^{n}$$
for every $n\in\mathbb{N}$ and $c\in C$.
\end{notns}

\begin{prop}\label{p:hhop} Suppose that $\P$ is levelwise cofibrant. Then $\P\otimes\,\mathbb{S}$ is a suspension spectrum model for the Hochschild complex $\rL_{\P^{\IB}}$. Likewise, when $A$ is levelwise cofibrant then $A\otimes\mathbb{S}$ is a suspension spectrum model for $\rL_{A^{\sm}}$.
\begin{proof} When $\P$ is levelwise cofibrant then the coproduct $\P^{\IB} \sqcup \P^{\IB}\in\IbMod(\P)$ has already the right homotopy type. Thus we may exhibit the constant spectrum $\Sigma^\infty(\P^{\IB} \sqcup \P^{\IB})$ as a model for $\rL_{\P^{\IB}}\in\T_{\P^{\IB}}\IbMod(\P)$. Moreover, the $n$-suspension $\Sigma^n(\P^{\IB} \sqcup \P^{\IB})\in\IbMod(\P)_{\P^{\IB}//\P^{\IB}}$ is given on each level by
$$ \Sigma^n(\P^{\IB} \sqcup \P^{\IB})(c_1,\cdots,c_m;c) \simeq \P(c_1,\cdots,c_m;c)\otimes \sS^{n}$$
(see Notation \ref{no:sphere}). This follows from the fact that colimits of infinitesimal $\P$-bimodules can be taken levelwise. We just showed that $\P\otimes\mathbb{S}$ is a suspension spectrum model for $\rL_{\P^{\IB}}$ (see also Remark \ref{r:sus}). The second claim is very similar.
\end{proof}
\end{prop}

\begin{rem}\label{r:same1} Suppose that the pair $(\cS, \P)$ is sufficient, so that we obtain Quillen equivalences
\begin{equation}\label{eq:seqQuillen1}
\adjunction*{\simeq}{\T_{\P^{\IB}}\IbMod(\P)}{\T_{\P^{\sB}}\BMod(\P)}{} \adjunction*{\simeq}{}{\T_\P\Op_C(\mathcal{S})}{} \adjunction*{\simeq}{}{\T_\P\Op(\mathcal{S})}{}
\end{equation}
(see $\S$\ref{s:rv}). In particular, this allows us to calculate both Quillen and Hochschild cohomologies of $\P$ within the same category $\T_{\P^{\IB}}\IbMod(\P)$, and furthermore, the same category $\IbMod(\P)$ when $(\cS, \P)$ is stably sufficient.
\end{rem}

\begin{rem}\label{r:same2} Similarly, when the triple $(\cS,\P,A)$ is sufficient, then due to the Quillen equivalence
$$ \adjunction*{\simeq}{\T_{A^{\sm}} \Mod_\P^A}{\T_A\Alg_\P(\cS)}{},$$
we may also consider both Quillen and Hochschild cohomologies of $A$ in the same framework of $\T_{A^{\sm}} \Mod_\P^A$. Moreover, when $(\cS,\P,A)$ is stably sufficient then the two cohomologies can be brought into the same category $\Mod_\P^A$.
\end{rem}

\begin{rem} In any base category $\cS$, Hochschild cohomology of the operad $\Com$ is not interesting at all. This is because of the fact that $\Com^{\IB} \in \IbMod(\Com)$ is a free object generated by $\mathcal{E}_*$ (see Example \ref{r:comfree}). An analogous observation holds for every commutative algebra $A$, due to the fact that $A^{\sm} \in \Mod_{\Com}^A$ coincides with the free object generated by $1_\cS$.
\end{rem}

\begin{example}\label{ex:hhpstab} Assume that the pair $(\cS,\P)$ is stably sufficient. Then under the right Quillen equivalence
$$ \ker\circ \, \Omega^\infty : \T_{\P^{\IB}}\IbMod(\P) \lrarsimeq \IbMod(\P),$$
the object $\rL_{\P^{\IB}}$ is simply identified with $\P^{\IB}\in\IbMod(\P)$ (cf. [[23], Corollary 2.2.4]). Therefore, after sending coefficients into $\IbMod(\P)$, the Hochschild cohomology of $\P$ with coefficients in some $M\in\IbMod(\P)$ is given by
$$ \HHH^\star(\P ; M) \simeq \Map^{\h}_{\IbMod(\P)}(\P^{\IB}, M).$$
In particular, for the case in which $\cS = \C(\textbf{k})$ and $\P$ is a dg operad concentrated in arity $1$ (i.e. $\P$ is identified with a dg category), then we recover the usual \textbf{Hochschild cohomology of a dg category}, that has been widely considered in literature.
\end{example}

\begin{example}\label{ex:hhastab} In the same fashion, when the triple $(\cS,\P, A)$ is stably sufficient, then we obtain that the right Quillen equivalence
$$ \ker\circ \, \Omega^\infty : \T_{A^{\sm}}\Mod_\P^A \lrarsimeq \Mod_\P^A$$
identifies $\rL_{A^{\sm}} \in \T_{A^{\sm}}\Mod_\P^A$ to $A\in\Mod_\P^A$. Thus in this situation, the Hochschild cohomology of $A$ with coefficients in a given object $N\in\Mod_\P^A$ is computed by
$$ \HHH^\star(A ; N) = \Map^{\h}_{\Mod_\P^A}(A^{\sm}, N).$$
\end{example}

\begin{example} Consider the case where $\cS = \C(\textbf{k})$ (see Examples \ref{ex:base})  and $\P=\Ass$. Then we have that $\Mod_{\Ass}^A$ agrees with the category of $A$-\textbf{bimodules}, and thus, in this situation we recover the classical Hochschild cohomology of dg associative algebras.
\end{example}

The following example will be clarified in $\S$\ref{s:cotan}.

\begin{example} In the case where $\cS = \Set_\Delta$ and $\P$ is a simplicial operad which is fibrant, we have an equivalence of $\infty$-categories
$$\T_{\P^{\IB}}\IbMod(\P)_\infty \simeq \Fun(\Tw(\P), \Sp)$$
in which $\Tw(\P)$ is the twisted arrow $\infty$-category of $\P$ and $\Sp$ denotes the $\infty$-category of spectra. Moreover, it can be shown that under this equivalence, $\rL_{\P^{\IB}} \in \T_{\P^{\IB}}\IbMod(\P)$ is identified with the constant functor on the sphere spectrum $\mathbb{S}\in\Sp$. Thus after sending coefficients into $\Fun(\Tw(\P), \Sp)$, the Hochschild cohomology of $\P$ with coefficients in a given functor $\F:\Tw(\P) \lrar \Sp$ is simply classified by
$\lim\F$. In particular, when $\P$ is concentrated in arity $1$ (i.e. $\P$ is identified with a simplicial category or an $\infty$-category), then we obtain the usual cohomology of $\infty$-categories which generalizes the \textbf{Baues-Wirsching cohomology} of small categories (cf. [26]).
\end{example}

\smallskip

\subsection{Endomorphism constructions}\label{s:endo}

\smallskip

It is known that the composite product $ - \circ - : \Coll_C(\cS) \times \Coll_C(\cS) \lrar \Coll_C(\cS)$ is \textit{left linear}, i.e. for every $A \in \Coll_C(\cS)$ the functor $ (-) \circ A : \Coll_C(\cS) \lrar \Coll_C(\cS)$ preserves all colimits. Consider the case where $A \in \cS^{\times C}$ regarded as a $C$-collection concentrated in level $0$. Taking composite product with $A$ determines a functor $$ (-) \circ A : \Coll_C(\cS) \lrar  \cS^{\times C}.$$
Of course, this functor admits a right adjoint, as described in the following construction.

\begin{cons} Let $A=\{A(c)\}_{c\in C}$ and $B=\{B(c)\}_{c\in C}$ be two objects of $\cS^{\times C}$. The \textbf{endomorphism object} associated to the  pair $(A,B)$ is a $C$-collection denoted by $\End_{A,B}$, and defined by letting
$$ \End_{A,B}(c_1,\cdots,c_n;c) := \Map_{\cS}(A(c_1)\otimes\cdots\otimes A(c_n),B(c)).$$
The right $\Sigma_n$-action is naturally given by the permutation on the factors $A(c_i)$'s. In particular, the \textbf{endomorphism operad associated to $A$} is given by $\End_A:=\End_{A,A}$. (See also [10, 27]).
\end{cons}

We have indeed the following result, which can be readily verified using definition.

\begin{prop} The functors $ (-) \circ A$ and $\End_{A,-}$ determine an adjunction
\begin{equation}\label{eq:endbasic}
(-) \circ A : \adjunction*{}{\Coll_C(\cS)}{\cS^{\times C}}{} : \End_{A,-}.
\end{equation}
\end{prop}

One may obtain relative versions of \eqref{eq:endbasic} when endowing the involving objects with additional structures, as discussed in what follows.

Let us assume further that $A$ comes equipped with a $\P$-algebra structure classified by a map $\ell_A : \P \lrar \End_A$ of operads. The \textbf{relative composite product} determines a functor $$(-)\circ_\P A : \RMod(\P)\lrar \cS^{\times C}.$$
On other hand, for any $B \in \cS^{\times C}$, one observes that $\End_{A,B}$ carries a canonical right module structure over $\End_A$.  Thus, $\End_{A,B}$ carries a right $\P$-module structure given by the restriction along $\ell_A$. Furthermore, when $B$ comes equipped with the structure of a $\P$-algebra then $\End_{A,B}$ carries a canonical $\P$-bimodule structure. The relative versions of \eqref{eq:endbasic} are as follows.

\begin{prop}[10, 27] \label{p:end} The constructions $(-)\circ_\P A$ and  $\End_{A,-}$ determine the adjunctions:
\begin{equation}\label{eq:end1}
(-)\circ_\P A : \adjunction*{}{\RMod(\P)}{\cS^{\times C}}{} : \End_{A,-}, \;\; \text{and}
\end{equation}
\begin{equation}\label{eq:end2}
(-)\circ_\P A : \adjunction*{}{\BMod(\P)}{\Alg_\P(\cS)}{} : \End_{A,-}
\end{equation}
\end{prop}

Another important adjunction, which interposes the two above, is given as follows.

\begin{prop}\label{p:end3} The constructions $(-)\circ_\P A$ and  $\End_{A,-}$ determine an adjunction:
\begin{equation}\label{eq:end3}
(-)\circ_\P A : \adjunction*{}{\IbMod(\P)}{\Mod_\P^A}{} : \End_{A,-}
\end{equation}
\begin{proof}[Sketch of proof] Let $N\in\Mod_\P^A$ and $M\in \IbMod(\P)$ be given. In order to see how $\End_{A,N}$ comes equipped with a canonical infinitesimal $\P$-bimodule structure, we may use Proposition \ref{p:ibfunc}, so that it suffices to represent $\End_{A,N}$ as a functor $\textbf{Ib}^{\mathcal{P}} \lrar \cS$. On other hand, note that the infinitesimal left $\P$-module structure on $M$ can be equivalently encoded by a map of the form
$$ \P\circ (\P;M) \lrar M$$
where the left hand side is the sub-collection of $\P\circ (\P\sqcup M)$ that is linear in $``M$'' (see [9], $\S$6.1 and 12.6.14). The above map induces in an obvious way a map of the form
$$ \P\circ (A;M\circ_\P A) \cong \P\circ (\P\circ_\P A;M\circ_\P A) \lrar M\circ_\P A.$$
This latter map encodes the induced $A$-module structure on $M\circ_\P A$ (see also $\S$12.3.1 of the loc.cit). Finally, to verify that the obtained functors indeed form an adjoint pair, we may make use of the adjunction \eqref{eq:end1}; so that it will suffice to argue that the unit (resp. counit) of \eqref{eq:end1} forms a map of infinitesimal $\P$-bimodules (resp. $A$-modules) when descending to $\IbMod(\P)$ (resp. $\Mod_\P^A$).
\end{proof}
\end{prop}

\begin{rem} We describe an alternative approach for how the adjunctions \eqref{eq:end2} and \eqref{eq:end3} can be established. For simplicity, assume that $\P$ is a single-colored operad. Recall from [[10], $\S$6.2] that the category $\RMod(\P)$ comes equipped with a symmetric monoidal structure whose monoidal product is given by the usual tensor product of underlying $\Sigma_*$-objects. The monoidal unit is nothing but $\mathcal{E}_*$. Observe then that the embedding $ \delta :  \cS \lrar \RMod(\P),$ which takes each $X\in\cS$ to  $X$ itself concentrated in level $0$, is symmetric monoidal. Let us now denote by $\widetilde{\P} \in \Op_*(\RMod(\P))$ the image of $\P$ through the induced functor $\delta : \Op_*(\cS) \lrar \Op_*(\RMod(\P))$. According to the loc.cit, there is an identification $\BMod(\P) \cong \Alg_{\widetilde{\P}}(\RMod(\P))$. In particular,  $\P^{\sB}\in\BMod(\P)$ itself is a $\widetilde{\P}$-algebra. Pick up now the adjunction \eqref{eq:end1}:
$$ (-)\circ_\P A : \adjunction*{}{\RMod(\P)}{\cS}{} : \End_{A,-}.$$
According to Lemma 6.2.1 of the loc.cit, the left adjoint above is symmetric monoidal. The induced functor $(-)\circ_\P A : \Op_*(\RMod(\P)) \lrar \Op_*(\cS)$ sends $\widetilde{\P}$ to nothing but $\P$. So we obtain an induced adjunction
$$ (-)\circ_\P A : \adjunction*{}{\Alg_{\widetilde{\P}}(\RMod(\P))}{\Alg_\P(\cS)}{} : \End_{A,-} \, ,$$
which coincides with \eqref{eq:end2}. Furthermore, since the left adjoint above sends $\P^{\sB}\in\Alg_{\widetilde{\P}}(\RMod(\P))$ to $A\in\Alg_\P(\cS)$, we obtain another adjunction
$$ (-)\circ_\P A : \adjunction*{}{\Mod_{\widetilde{\P}}^{\P^{\sB}}}{\Mod_\P^A}{} : \End_{A,-}.$$
This adjunction agrees with \eqref{eq:end3}, after applying the obvious identification $\Mod_{\widetilde{\P}}^{\P^{\sB}} \cong \IbMod(\P)$ between $\P^{\sB}$-modules over $\widetilde{\P}$ and infinitesimal $\P$-bimodules.
\end{rem}

A noteworthy consequence is as follows.

\begin{cor}\label{r:freeA} For an object $X\in\cS^{\times C} \subseteq \Coll_C(\cS)$, there is an isomorphism
$$ \Free_A(X) \cong (\mathcal{P}\circ_{(1)}X) \circ_\P A$$
of $A$-modules over $\P$. In particular, when $C = \{*\}$ we obtain that $\Free_A(1_\cS) \cong  \mathcal{P}[1]\circ_\P A$ (see $\S$\ref{s:infbi} for notations).
\begin{proof} Consider the following square of left adjoints
$$ \xymatrix{
\Coll_C(\cS) \ar[r]^{(-) \circ A}\ar[d]_{\Free_\P^{\IB}} & \cS^{\times C} \ar[d]^{\Free_A} \\
\IbMod(\P) \ar[r]_{(-) \circ_\P A} & \Mod_\P^A \\
which is commutative due to that property of the associated square of right adjoints. Combined with Remark \ref{r:freeinf}, this yields for each $X\in\cS^{\times C} \subseteq \Coll_C(\cS)$ a chain of isomorphisms in $\Mod_\P^A$:
$$ \Free_A(X) \cong \Free_A(X \circ A) \cong \Free_\P^{\IB}(X)\circ_\P A \cong (\mathcal{P}\circ_{(1)}X) \circ_\P A.$$
For the case $C=\{*\}$, we obtain that $\Free_A(1_\cS) \cong (\mathcal{P}\circ_{(1)}\mathcal{E}_*)\circ_\P A \cong \mathcal{P}[1]\circ_\P A$.
\end{proof}
\end{cor}

\begin{rem} The latter identification can also be found in [[10], $\S$4.3]. Due to this, one may obtain a description of $\Free_A(1_\cS)$, which models the universal enveloping algebra of $A$ (see Proposition \ref{p:uenv}), as in the loc.cit.
\end{rem}

\begin{example} An interesting example concerns the two operads $\Lie$ and $\Ass$ enriched over $\cS = \C(\textbf{k})$ (see Examples \ref{ex:base}). The canonical embedding $\Lie \lrar \Ass$ endows $\Ass$ with an infinitesimal $\Lie$-bimodule structure. Moreover, it can be shown that the map $$\Free_{\Lie}^{\IB}(\mathcal{E}_*) = \Lie[1]  \overset{\cong}{\lrar} \Ass$$ classified by the unique nullary operation of $\Ass$ is an isomorphism in $\IbMod(\Lie)$. Combined with Corollary \ref{r:freeA}, this yields for each $\Lie$-algebra $\g$ a chain of natural isomorphisms in $\Mod^\g_{\Lie}$:
$$ \Ass\circ_{\Lie}\g \cong \Free_{\Lie}^{\IB}(\mathcal{E}_*)\circ_{\Lie}\g \cong \Free_\g(\textbf{k}) \cong \sU_{\Lie}(\g).$$
This allows us to recover a fundamental result asserting that the universal enveloping algebra of $\g$ can be naturally modeled by $\Ass\circ_{\Lie}\g$, i.e. the induction of $\g$ along the map $\Lie \lrar \Ass$.
\end{example}

To be able to do homotopy theory, we will need the following assertion.

\begin{prop} The adjunctions \eqref{eq:end1}, \eqref{eq:end2} and \eqref{eq:end3} are all Quillen adjunctions when provided that $A$ is levelwise cofibrant.
\begin{proof} Clearly with the assumption that $A$ is levelwise cofibrant, the construction $\End_{A,-}$ preserves fibrations and trivial fibrations, due to basic properties of a symmetric monoidal model category.
\end{proof}
\end{prop}

\begin{rem}\label{r:keydiagram} By the identification $\P\circ_\P A \cong A$, we obtain further a commutative diagram of adjunctions:
\begin{equation}\label{eq:prekey}
\begin{tikzcd}[row sep=3.5em, column sep =3.5em]
\RMod(\P)_{\P^{\sr}/}	 \arrow[r, shift left=1.5] \arrow[d, shift right=1.5] & \IbMod(\P)_{\P^{\IB}/} \arrow[l, shift left=1.5, "\perp"'] \arrow[d, shift left=1.5] \arrow[r, shift left=1.5]  & \BMod(\P)_{\P^{\sB}/}  \arrow[d, shift left=1.5] \arrow[l, shift left=1.5, "\perp"']   \\
(\cS^{\times C})_{A/} \arrow[r, shift right=1.5] \arrow[u, shift right=1.5, "\dashv"] & (\Mod_\P^A)_{A^{\sm}/}  \arrow[l, shift right=1.5, "\downvdash"] \arrow[u, shift left=1.5, "\vdash"'] \arrow[r, shift right=1.5] & (\Alg_\P(\cS))_{A/}  \arrow[u, shift left=1.5, "\vdash"'] \arrow[l, shift right=1.5, "\downvdash"]
\end{tikzcd}
\end{equation}
in which the vertical adjunctions are all induced by the adjunction $(-)\circ_\P A \dashv \End_{A,-}$, and the horizontal pairs are given by the induction-restriction adjunctions.
\end{rem}

\smallskip

\subsection{Formulations and comparison theorems}\label{s:op2}

\smallskip

We are interested in the right square of \eqref{eq:prekey}, from which we obtain a commutative square of Quillen adjunctions between tangent categories:
\begin{equation}\label{eq:key}
\begin{tikzcd}[row sep=3.5em, column sep =3.5em]
\mathcal{T}_{\P^{\IB}}\IbMod(\P) \arrow[r, shift left=1.5] \arrow[d, shift right=1.5] & \mathcal{T}_{\P^{\sB}}\BMod(\P) \arrow[l, shift left=1.5, "\perp"'] \arrow[d, shift left=1.5]  & \\
\T_{A^{\sm}} \Mod_\P^A \arrow[r, shift right=1.5] \arrow[u, shift right=1.5, "\dashv"] & \T_A\Alg_\P(\cS)  \arrow[l, shift right=1.5, "\downvdash"] \arrow[u, shift left=1.5, "\vdash"'] &
\end{tikzcd}
\end{equation}

In light of this, we will show that the cotangent complex $\rL_A$ (resp. Hochschild complex $\rL_{A^{\sm}}$) can be represented via the cotangent complex $\rL_{\P^{\sB}}$ (resp. Hochschild complex $\rL_{\P^{\IB}}$), in a very natural fashion.

\begin{notn}\label{no:circend} We will use the same notation $(-)\circ^{\st}_\P A \dashv \End^{\st}_{A,-}$ to signify the two vertical adjunctions of \eqref{eq:key}:
$$ (-)\circ^{\st}_\P A : \adjunction*{}{\mathcal{T}_{\P^{\IB}}\IbMod(\P)}{\T_{A^{\sm}} \Mod_\P^A}{} : \End^{\st}_{A,-}, \;\; \text{and} \;\; $$
$$ (-)\circ^{\st}_\P A : \adjunction*{}{\mathcal{T}_{\P^{\sB}}\BMod(\P)}{\T_A\Alg_\P(\cS)}{} : \End^{\st}_{A,-}.$$
\end{notn}

\begin{rem}\label{r:endcons} By construction, for each $M\in\mathcal{T}_{\P^{\sB}}\BMod(\P)$ (or $\mathcal{T}_{\P^{\IB}}\IbMod(\P)$), the spectrum object $M\circ^{\st}_\P A$ is simply taken degreewise, i.e.  $$(M\circ^{\st}_\P A)_{m,n}=M_{m,n}\circ_\P A.$$
The case of $\End^{\st}_{A,-}$ is rather different. Namely, for each $N\in\T_A\Alg_\P(\cS)$ (or $\T_{A^{\sm}}\Mod_\P^A$), the spectrum object $\End^{\st}_{A,N}$ is given at each bidegree $(m,n)$ as the pullback
$$ \xymatrix{
(\End^{\st}_{A,N})_{m,n} \ar[r]\ar[d] & \End_{A,N_{m,n}} \ar[d] \\
\P \ar[r]^{\ell_A} & \End_A \\
where the map $\End_{A,N_{m,n}} \lrar \End_A$ ($\x{\defi}{=} \End_{A,A}$) is induced by the structure map $N_{m,n} \lrar A$.
\end{rem}

The following plays a pivotal role in the study of various cohomologies of operadic algebras.
\begin{prop}\label{p:bialg}  Suppose that $A$ is levelwise cofibrant. Then there are weak equivalences $$\rL_{\P^{\sB}}\circ^{\st}_\P A \simeq \rL_A \;\; \text{and} \;\; \rL_{\P^{\IB}}\circ^{\st}_\P A \simeq \rL_{A^{\sm}}$$
in $\T_A\Alg_\P(\cS)$ and $\T_{A^{\sm}} \Mod_\P^A$ respectively.
\begin{proof} We will prove the first weak equivalence. The other can then be verified in the same fashion. By definition, $\rL_{\P^{\sB}}$ is represented by a constant spectrum of the form $\Sigma^\infty\left(\P^{\sB}\sqcup(\P^{\sB})^{\cof}\right) \in \mathcal{T}_{\P^{\sB}}\BMod(\P)$ in which $(\P^{\sB})^{\cof}$ refers to any cofibrant model for $\P^{\sB} \in \BMod(\P)$, and the coproduct $\P^{\sB}\sqcup(\P^{\sB})^{\cof}$ is considered as an object over and under $\P^{\sB}$. We have a chain of weak equivalences
$$ \rL_{\P^{\sB}}\circ^{\st}_\P A \simeq \Sigma^\infty\left(\P^{\sB}\sqcup(\P^{\sB})^{\cof}\right)\circ^{\st}_\P A \; \x{\defi}{=} \; \Sigma^\infty\left((\P^{\sB}\sqcup(\P^{\sB})^{\cof})\circ_\P A\right) \simeq \Sigma^\infty(A\sqcup A^{\cof})$$
where $A^{\cof} := (\P^{\sB})^{\cof}\circ_\P A$ represents a cofibrant model for $A\in\Alg_\P(\cS)$. As above, $\Sigma^\infty(A\sqcup A^{\cof})$ is a model for $\rL_A$. So we obtain a weak equivalence $\rL_{\P^{\sB}}\circ^{\st}_\P A \simeq \rL_A$ as expected.
\end{proof}
\end{prop}

According to $\S$\ref{s:rv}, $\rL_{\P^{\sB}}\in\mathcal{T}_{\P^{\sB}}\BMod(\P)$ admits a suspension spectrum replacement given by $\P\circ\sS_C^{\bullet}$ with $(\P\circ\sS_C^{\bullet})_{n,n}=\P\circ\sS_C^{n}$, when provided that $\P$ is $\Sigma$-cofibrant. Due to this, we can describe the cotangent complex $\rL_A$ as follows.

\begin{prop}\label{p:cotalg}  Suppose that the triple $(\cS,\P, A)$ is sufficient. Then for every $n\in\mathbb{N}$ we have a weak equivalence in $\Alg_\P(\cS)_{A//A}$:
$$ (\P\circ\sS_C^{n})\circ_\P A \simeq \Sigma^n(A \sqcup A) $$
in which the right hand side is the $n$-suspension of $A\sqcup A \in \Alg_\P(\cS)_{A//A}$. Consequently, the cotangent complex $\rL_A \in \T_A\Alg_\P(\cS)$ admits a suspension spectrum replacement:
$$ (\P\circ\sS_C^{\bullet})\circ^{\st}_\P A \simeq \rL_A.$$
\begin{proof} As discussed in $\S$\ref{s:rv}, we have a weak equivalence $\P\circ\sS_C^{n} \simeq \Sigma^n(\P^{\sB} \sqcup \P^{\sB})$ in $\BMod(\P)_{\P^{\sB}//\P^{\sB}}$. By the assumption that $\P$ is $\Sigma$-cofibrant and that $A$ is cofibrant, we obtain that the relative composite product $(\P\circ\sS_C^{n})\circ_\P A$ has the right homotopy type (cf. [10], $\S$15.2). Thus we obtain a chain of natural weak equivalences
$$ (\P\circ\sS_C^{n})\circ_\P A \simeq \left(\Sigma^n(\P^{\sB} \sqcup \P^{\sB})\right) \circ_\P A \simeq \Sigma^n(A\sqcup A)$$
where the latter weak equivalence is due to the left Quillen functor $$(-)\circ_\P A : \BMod(\P)_{\P//\P} \lrar \Alg_\P(\cS)_{A//A}.$$ The second statement follows from the above, because the functor $(-)\circ^{\st}_\P A$ is computed degreewise. (Of course, it can also be immediately derived from Proposition \ref{p:bialg}).
\end{proof}
\end{prop}

\begin{rem} From a set-theoretic perspective, each element of $\Sigma^n(A \sqcup A)$ can be represented by an operation $\mu \in \P$, and together with a tuple $(a_1,\cdots,a_k) \in A^{\times k}$ (with $k$ being the arity of $\mu$) such that each of these is attached to a specific point on a distinguished $n$-sphere. Formally, such an element can be written as
$$[ \mu ; (\tau_1,a_1),\cdots, (\tau_k,a_k) ] $$
in which each $\tau_i \in \sS^n$ represents the location of $a_i$ on the $n$-sphere it belongs to. Moreover, such elements are subject to an equivalence relation as follows
$$  [\mu ; (\tau_1,a_1),\cdots,(\tau_i, \nu\cdot(b_1,\cdots,b_l)) ,\cdots, (\tau_k,a_k)]  \sim [ \mu\circ_i \nu ; (\tau_1,a_1),\cdots,(\tau_i, b_1),\cdots,(\tau_i, b_l) ,\cdots, (\tau_k,a_k) ].$$
\end{rem}

\begin{rem}\label{r:cotalg} Under the same assumption as in Proposition \ref{p:cotalg}, we have further that under the right Quillen equivalence
$$ \T_A\Alg_\P(\cS) \lrarsimeq \T_A \Mod_\P^A,$$
the object $(\P\circ\sS_C^{\bullet})\circ^{\st}_\P A$ is identified with itself $(\P\circ\sS_C^{\bullet})\circ^{\st}_\P A$ considered as a prespectrum of the underlying $A$-modules. (This follows from [[8], Corollary 2.4.8]). Thus, the latter is a model for the derived image of $\rL_A$ in $\T_A \Mod_\P^A$.
\end{rem}

\begin{rem}\label{r:cotalgstab}
Assume that the triple $(\cS,\P, A)$ is stably sufficient and consider the extended commutative diagram of Quillen adjunctions
\begin{equation}\label{eq:key1}
\begin{tikzcd}[row sep=3.5em, column sep =3.5em]
\IbMod(\P)	 \arrow[r, shift left=1.5, "\simeq"] \arrow[d, shift right=1.5, "(-)\circ_\P A"'] & \mathcal{T}_{\P}\IbMod(\P) \arrow[l, shift left=1.5, "\perp"'] \arrow[d, shift left=1.5, "(-)\circ^{\st}_\P A"] \arrow[r, shift left=1.5, "\simeq"]  & \mathcal{T}_{\P}\BMod(\P)  \arrow[d, shift left=1.5, "(-)\circ^{\st}_\P A"] \arrow[l, shift left=1.5,  "\perp"']   \\
\Mod_\P^A \arrow[r, shift right=1.5] \arrow[u, shift right=1.5, "\dashv", "\End_{A,-}"'] & \T_{A} \Mod_\P^A  \arrow[l, shift right=1.5, "\downvdash", "\simeq" '] \arrow[u, shift left=1.5, "\vdash"', "\End_{A,-}^{\st}"] \arrow[r, shift right=1.5] & \T_A\Alg_\P(\cS)  \arrow[u, shift left=1.5, "\vdash"', "\End_{A,-}^{\st}"] \arrow[l, shift right=1.5, "\downvdash", "\simeq" ']
\end{tikzcd}
\end{equation}
in which the horizontal pairs are all Quillen equivalences. Recall from $\S$\ref{s:rv} that under the right Quillen equivalence $\mathcal{T}_{\P^{\IB}}\IbMod(\P)  \lrarsimeq  \IbMod(\P)$, the object $\P\circ\sS_C^{\bullet}\in\mathcal{T}_{\P^{\IB}}\IbMod(\P)$ is identified with $\ovl{\rL}_\P \in \IbMod(\P)$. Combined with the above remark and with the fact that $\ovl{\rL}_\P$ is $\Sigma$-cofibrant, it yields a weak equivalence in $\Mod_\P^A$:
$$ \ovl{\rL}_\P\circ_\P A \simeq \rL_A.$$
Here we use the same notation to denote the derived image of $\rL_A \in \T_A\Alg_\P(\cS)$ in $\Mod_\P^A$.
\end{rem}

In summary, we obtain the following conclusion.

\begin{thm}\label{t:maincotalg} Suppose that the triple $(\cS,\P, A)$ is sufficient. The Quillen cohomology of $A$ with coefficients in a given object $N\in\T_{A^{\sm}} \Mod_\P^A$ is computed by the formula
$$ \HHQ^\star(A ; N) \simeq \Map^{\h}\left((\P\circ\sS_C^{\bullet})\circ^{\st}_\P A, N \right).$$
When $(\cS,\P, A)$ is in addition stably sufficient, the Quillen cohomology of $A$ with coefficients in some $N\in\Mod_\P^A$ is computed by the formula
$$ \HHQ^\star(A ; N) \simeq \Map^{\h}_{\Mod_\P^A}(\ovl{\rL}_\P\circ_\P A, N).$$
\end{thm}

In what follows, we shall provide some comparison theorems asserting that Hochschild and Quillen cohomologies of $A$ can be encoded by the corresponding cohomologies of $\P$ itself. We will assume that the triple $(\cS,\P,A)$ is (at least) sufficient, so that the horizontal adjunctions of \eqref{eq:key} are all Quillen equivalences.

\begin{thm} \label{co:bialg} Suppose we are given a fibrant object $N\in \T_{A^{\sm}} \Mod_\P^A$.

(i) There is a natural weak equivalence
$$ \HHQ^\star(A ; N)  \simeq  \HHQ^\star(\P^{\sB} ; \End^{\st}_{A,N})$$
between Quillen cohomology of $A$ with coefficients in $N$ and Quillen cohomology of $\P^{\sB}\in\BMod(\P)$ with coefficients in $\End^{\st}_{A,N} \in \mathcal{T}_{\P^{\IB}}\IbMod(\P)$.

(ii) There is a natural weak equivalence
$$ \HHH^\star(A ; N)  \simeq \HHH^\star(\P ; \End^{\st}_{A,N}) $$
between the Hochschild cohomology of $A$ with coefficients in $N$ and Hochschild cohomology of $\P$ with coefficients in $\End^{\st}_{A,N}$.

\begin{proof} After sending coefficients into $\mathcal{T}_{\P^{\IB}}\IbMod(\P)$, we have that
$$ \HHQ^\star(\P^{\sB} ; \End^{\st}_{A,N}) \simeq \Map^{\h}(\P\circ\sS_C^{\bullet},\End^{\st}_{A,N}),$$
and while by Theorem \ref{t:maincotalg} we have $\HHQ^\star(A;N) \simeq \Map^{\h}\left((\P\circ\sS_C^{\bullet})\circ^{\st}_\P A,N \right)$. Combined with the adjunction $(-)\circ^{\st}_\P A\dashv\End^{\st}_{A,-}$, these indeed verify (i). The statement (ii) is verified by combining the adjunction $(-)\circ^{\st}_\P A\dashv\End^{\st}_{A,-}$ with Proposition \ref{p:bialg}.
\end{proof}
\end{thm}

Now assume further that the triple $(\cS,\P,A)$ is abundant.  We arrive at the main result as follows.
\begin{thm}\label{t:opmain} For a fibrant object $N\in \T_{A^{\sm}} \Mod_\P^A$, there is a natural weak equivalence
$$\HHQ^\star(A;N)  \simeq \Omega\HHQ^\star(\P ; \End^{\st}_{A,N})$$
where the loop space on the right is taken at the zero element (see Remark \ref{r:trivclass}). In particular, for every integer $n$ we have an isomorphism
$$ \HHQ^n(A;N)  \cong \HHQ^{n+1}(\P;\End^{\st}_{A,N}).$$
\begin{proof} We have in fact a chain of weak equivalences
$$ \HHQ^\star(A ; N)  \simeq \HHQ^\star(\P^{\sB} ; \End^{\st}_{A,N}) \simeq \Omega\HHQ^\star(\P ; \End^{\st}_{A,N})$$
where the first weak equivalence is due to Theorem \ref{co:bialg}(i), and the second weak equivalence follows from Proposition \ref{p:key}.
\end{proof}
\end{thm}

\begin{rem} The above theorem admits a stable-analogue as follows. Assume that the triple $(\cS,\P,A)$ is stably abundant, so that the horizontal adjunctions of \eqref{eq:key1} are all Quillen equivalences. Similarly as above, we may obtain a chain of weak equivalences
$$\HHQ^\star(A ; N)  \simeq  \HHQ^\star(\P^{\sB} ; \End_{A,N}) \simeq \Omega\HHQ^\star(\P ; \End_{A,N})$$
in which $N$ is now a fibrant object in $\Mod_\P^A$, and $\End_{A,N}$ is simply an infinitesimal $\P$-bimodule.
\end{rem}

\begin{rem} When the triple $(\cS,\P,A)$ is stably sufficient, an analogue for Hochschild cohomology is now evident, as described via the following chain of weak equivalences:
$$\HHH^\star(A;N) \simeq \Map^{\h}_{\Mod_\P^A}(A,N) \simeq \Map^{\h}_{\IbMod(\P)}(\P,\End_{A,N}) \simeq \HHH^\star(\P ; \End_{A,N})$$
(see Examples \ref{ex:hhpstab} and \ref{ex:hhastab}).
\end{rem}

\section{Simplicial operads and algebras over them}\label{s:Qprin}

\smallskip

We assume that the category $\Set_\Delta$ of simplicial sets comes equipped with the Kan-Quillen model structure. We let $\P$ be a fixed $C$-colored simplicial operad, living in the Dwyer-Kan homotopy theory on $\Op(\Set_\Delta)$ (see $\S$\ref{s:optransfermod}).

We shall first recall from [1] the construction $\Tw(\P)$, i.e. the twisted arrow $\infty$-category of $\P$, and recall how the tangent categories at $\P$ can be represented via that construction. Due to this, we may then formulate Hochschild and Quillen cohomologies of $\P$, as well as cohomologies of $\P$-algebras. In the last subsection, we shall formulate the Quillen principle for $\E_n$-operads, and from which we obtain the Quillen principle for algebras over $\E_n$.

\subsection{Operadic twisted arrow $\infty$-categories and tangent categories}\label{s:twis}

We will assume that $\P$ is fibrant (i.e. every space of operations of $\P$ is a Kan complex) and $\Sigma$-cofibrant. Recall from [1] that the twisted arrow $\infty$-category $\Tw(\P)$ is given by the (covariant) unstraightening of the simplicial copresheaf $\P^{\IB} : \textbf{Ib}^{\mathcal{P}} \lrar \Set_\Delta$ which encodes the datum of $\P$ as an infinitesimal bimodule over itself (see $\S$\ref{s:infbi}). In particular, $\Tw(\P)$ is endowed with a left fibration $$\Tw(\P) \lrar \sN(\textbf{Ib}^{\mathcal{P}})$$
where $\sN(-)$ denotes the \textit{simplicial nerve functor}.

\begin{rem} Unwinding the definitions, objects of $\Tw(\P)$ are precisely the \textbf{operations} of $\P$ (i.e. the vertices of the spaces of operations of $\P$). Let $\ovl{c} := (c_1,\cdots,c_n;c)$ and $\ovl{d} := (d_1,\cdots,d_m;d)$ be two $C$-sequences, and let $\mu \in \P(\ovl{c})$ and $\nu\in\P(\ovl{d})$ be two operations. The data of a morphism $\mu \lrar \nu$ in $\Tw(\P)$ consist of

$\bullet$ a map $f:\langle n \rangle \lrar \langle m \rangle$ in $\Fin_*$,

$\bullet$ a tuple of operations  $\alpha = (\alpha_0, \alpha_1,\cdots,\alpha_m) \in \Map^{f}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d})$, and

$\bullet$ an edge $p : \nu \lrar \alpha^{*}(\mu)$ in $\P(\ovl{d})$ where $\alpha^{*} : \P(\ovl{c}) \lrar \P(\ovl{d})$ is the map corresponding to $\alpha$ via the simplicial functor structure of $\P^{\IB} : \textbf{Ib}^{\P} \lrar \Set_\Delta$.
\end{rem}

\begin{rem}\label{r:pcontw} Since the map $\Tw(\P) \lrar \sN(\textbf{Ib}^{\mathcal{P}})$ is a left fibration between $\infty$-categories, it implies that a typical morphism $(f,\alpha,p) : \mu \lrar \nu$ (as in the above remark) is an equivalence in $\Tw(\P)$ if and only if $\alpha \in \Map^{f}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d})$ is an equivalence in $\sN(\textbf{Ib}^{\mathcal{P}})$. This leads to a consequence as follows. Let $\gamma, \gamma' \in \P(\ovl{c})$ be two operations that are in the same path component of $\P(\ovl{c})$. We take an arbitrary edge $q : \gamma \lrar \gamma'$ in $\P(\ovl{c})$. This is part of a morphism $(\Id_{\l n \r}, \Id_{\ovl{c}}, q) : \gamma' \lrar \gamma$ in $\Tw(\P)$ where, by construction, $\Id_{\ovl{c}} = (\id_{c},\id_{c_1},\cdots, \id_{c_n}) \in \Map^{\Id_{\l n \r}}_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{c})$ represents the identity morphism on $\ovl{c}$. Of course, $\Id_{\ovl{c}}$ is an equivalence in the $\infty$-category  $\sN(\textbf{Ib}^{\mathcal{P}})$, and thus, $(\Id_{\l n \r}, \Id_{\ovl{c}}, q)$ is an equivalence in $\Tw(\P)$. Consequently, if the space $\P(\ovl{c})$ is path connected then any two operations in $\P(\ovl{c})$ are equivalent as objects of $\Tw(\P)$.
\end{rem}

\begin{rem}\label{r:pcontw1} On other hand, two operations of $\P$ that have different arities cannot be equivalent as objects of $\Tw(\P)$, because of the fact that two $C$-sequences that do not share the same length cannot be equivalent as objects of $\sN(\textbf{Ib}^{\mathcal{P}})$. (The latter can be readily verified using definition).
\end{rem}

The mapping spaces of $\Tw(\P)$ are given as follows. As above, we regard $\mu \in \P(\ovl{c})$ and $\nu\in\P(\ovl{d})$ as objects of $\Tw(\P)$.

\begin{prop}[1]\label{p:mappingtwp} There is a canonical homotopy equivalence
$$ \{\nu\} \times^{\h}_{\P(\ovl{d})}\Map_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \lrarsimeq \Map_{\Tw(\P)}(\mu , \nu) $$
in which the map $\Map_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \lrar \P(\ovl{d})$ is given by the composed map
$$ \Map_{\textbf{Ib}^{\P}}(\ovl{c} , \ovl{d}) \x{\P^{\IB}}{\lrar} \Map_{\Set_\Delta}(\mathcal{P}(\ovl{c}) , \mathcal{P}(\ovl{d})) \x{ev_\mu}{\lrar} \mathcal{P}(\ovl{d}) $$
with $ev_\mu$ being the evaluation at $\mu$.
\end{prop}

The two most basic examples are: $\Tw(\Ass) \simeq \Delta$ and $\Tw(\Com) \simeq \Fin_*^{\op}$.

For each $1\leq n < \infty$, we will denote by $\E_n$ the simplicial version of the \textbf{little $n$-discs operad}, and denote by $\E_\infty$ a fibrant and  $\Sigma$-cofibrant model for the commutative operad $\Com$. As in the loc.cit, we have an $\infty$-categorical filtration of the nerve of $\Fin_*^{\op}$:
\begin{equation}\label{eq:nervefin}
\sN(\Delta) \simeq \Tw(\E_1) \lrar \Tw(\E_2) \lrar \cdots \lrar \Tw(\E_\infty) \simeq \sN(\Fin_*^{\op})
\end{equation}

\begin{rem}  This sequence may start with $\Tw(\E_0)$ which can be identified with the subcategory $\{[1] \lrar [0]\}$ of the simplex category $\Delta$. Here we recall by definition that $\E_0$ is the single-colored operad such that $\E_0(n)$ is a singleton when $n\leq 1$ and is empty otherwise, and by construction $\E_0$ encodes \textit{pointed simplicial sets}.
\end{rem}

\begin{rem}  Recall that the space $\E_n(k)$ is in particular path connected for every $k\geq0$ and $n\geq2$. According to remarks \ref{r:pcontw} and \ref{r:pcontw1}, we obtain that  $\Tw(\E_n)$ (with $n\geq2$) admits a minimal skeleton given by the full subcategory generated by a collection $\{\mu_k\}_{k\geq0}$ of operations in which each $\mu_k \in \E_n(k)$ is selected arbitrarily. Moreover, in spite of the fact that $\E_1$'s spaces of operations are not path connected except for $k\leq1$, we still have that any two operations in $\E_1(k)$ are equivalent as objects of $\Tw(\E_1)$ for every $k\geq0$ (and as we have seen, $\Tw(\E_1)$ is simply modeled by $\sN(\Delta)$).
\end{rem}

We will write $\Sp$ to denote the \textbf{$\infty$-category of spectra}, defined as the stabilization of the $\infty$-category of pointed simplicial sets. One of the main interests in the construction $\Tw(-)$ is as follows.

\begin{thm}[1]\label{t:maintan}  There is an equivalence of $\infty$-categories
\begin{equation}\label{eq:maintain}
\T_{\P^{\IB}}\IbMod(\P)_\infty \simeq \Fun(\Tw(\P) , \Sp)
\end{equation}
Consequently, there is a chain of equivalences of $\infty$-categories:
$$ \T_\P\Op(\Set_\Delta)_\infty \simeq \T_\P\Op_C(\Set_\Delta)_\infty \simeq \T_{\P^{\sB}}\BMod(\P)_\infty \simeq \T_{\P^{\IB}}\IbMod(\P)_\infty \simeq \Fun(\Tw(\P) , \Sp) . $$
\end{thm}

\begin{rem}\label{r:acompu} Let $M\in \T_{\P^{\IB}}\IbMod(\P)$ be a prespectrum whose datum consists of for each $m\geq0$ a sequence of maps $\P^{\IB} \lrar M_{m,m} \lrar \P^{\IB}$. We will write $\widetilde{M}:\Tw(\P) \lrar \Sp$ for the functor corresponding to $M$. Unwinding the definitions, $\widetilde{M}$ is given at each operation $\mu\in \P(\ovl{c})$ by taking $\widetilde{M}(\mu)\in\Sp$ to be a prespectrum with
$$ \widetilde{M}(\mu)_{m,m} \simeq \{\mu\} \times^{\h}_{\P(\ovl{c})} M_{m,m}(\ovl{c}).$$
Moreover, for a morphism $\mu \lrar \nu$ in $\Tw(\P)$ (with $\nu\in\P(\ovl{d})$) that lies above a morphism $\ovl{c} \lrar \ovl{d}$ in $\Map_{\textbf{Ib}^{\mathcal{P}}}(\ovl{c} , \ovl{d})$, we get a natural map $\widetilde{M}(\mu)_{m,m} \lrar \widetilde{M}(\nu)_{m,m}$ induced by the commutative square
$$ \xymatrix{
M_{m,m}(\ovl{c}) \ar[r]\ar[d] & \P(\ovl{c}) \ar[d] \\
M_{m,m}(\ovl{d}) \ar[r] & \P(\ovl{d}) \\
where the vertical maps are due to the infinitesimal $\P$-bimodule structures on $M_{m,m}$ and $\P^{\IB}$.
\end{rem}

\begin{example}  When $\P=\E_1$ (resp. $\E_\infty$), various tangent categories at this operad are equivalent to $\Fun(\sN(\Delta) , \Sp)$ (resp. $\Fun(\sN(\Fin_*^{\op}) , \Sp)$). In particular, the identification
$$ \T_{\E_\infty}\Op(\Set_\Delta)_\infty \simeq \Fun(\sN(\Fin_*^{\op}) , \Sp)$$
proves that the \textbf{stabilization of $\infty$-operads} is equivalent to $\Fun(\sN(\Fin_*^{\op}) , \Sp)$, because $\E_\infty$ is a terminal object in the $\infty$-category $\Op(\Set_\Delta)_\infty$.
\end{example}

\subsection{Hochschild and Quillen cohomologies of simplicial operads and algebras over them}\label{s:cotan}

As above, we will assume that $\P$ is fibrant and $\Sigma$-cofibrant. According to Theorem \ref{t:maintan}, various cotangent complexes and Hochschild complex of $\P$ can be represented as a functor $\Tw(\P) \lrar \Sp$. As in [[1], $\S$6.3], we let $\F_\P : \Tw(\P) \lrar \Sp$ denote the image of $\rL_{\P^{\sB}} \in \T_{\P^{\sB}}\BMod(\P)_\infty$ through the identification
$$ \T_{\P^{\sB}}\BMod(\P)_\infty  \simeq \Fun(\Tw(\P) , \Sp) .$$
According to $\S$\ref{s:rv}, $\F_\P$ coincides with the image of $\P\circ\sS_C^{\bullet}\in\T_{\P^{\IB}}\IbMod(\P)$ through the equivalence \eqref{eq:maintain}, and alternatively, $\F_\P$ agrees with the image of $\rL_{\P}[1] \in \T_\P\Op(\Set_\Delta)_\infty$ under the identification $$\T_\P\Op(\Set_\Delta)_\infty \simeq \Fun(\Tw(\P) , \Sp).$$
In order to give an explicit description of $\F_\P$, we will make use of the following.

\begin{prop}\label{p:destw} Let $f:\P \lrar \Q$ be a map between fibrant and $\Sigma$-cofibrant simplicial operads. Then the derived functor of the right adjoint $f^* : \T_\Q\Op(\Set_\Delta) \lrar \T_\P\Op(\Set_\Delta)$ sends $\rL_{\Q}$ to $\rL_{\P}$.
\begin{proof} It will suffice to show that the derived functor of the right adjoint $$f^* : \T_{\Q^{\IB}}\IbMod(\Q) \lrar \T_{\P^{\IB}}\IbMod(\P)$$ sends $\Q\circ\sS_D^{\bullet}$ to $\P\circ\sS_C^{\bullet}$, where $D$ refers to the set of colors of $\Q$. Here by construction the above functor is induced by the right adjoint $$ f^* : \IbMod(\Q)_{\Q^{\IB}//\Q^{\IB}} \lrar \IbMod(\P)_{\P^{\IB}//\P^{\IB}}$$
that takes each $M \in \IbMod(\Q)_{\Q^{\IB}//\Q^{\IB}}$ to $f^*(M) := \P^{\IB} \times_{(\Q^{\IB})^*} M^*$, where the pullback is taken in $\IbMod(\P)$ and $M^*$ denotes the infinitesimal $\P$-bimodule with $$M^*(c_1,\cdots,c_n;c) := M(f(c_1),\cdots,f(c_n);f(c)),$$
(while the case of $(\Q^{\IB})^*$ is defined similarly). Unwinding the definitions, we just need to verify for each $m\geq0$ the existence of a natural weak equivalence in $\IbMod(\P)$ of the form
$$ \P\circ\sS_C^{m} \lrarsimeq \P^{\IB} \times^{\h}_{(\Q^{\IB})^*} (\Q\circ\sS_D^{m})^*$$
(see also [[8], Corollary 2.4.8]). Moreover, after having fixed a Kan-model for the simplicial $m$-sphere  $\sS^{m}$, it suffices to show that the canonical map $\P\circ\sS_C^{m} \lrar \P^{\IB} \times_{(\Q^{\IB})^*} (\Q\circ\sS_D^{m})^*$ is an isomorphism. This is verified because we have on each level a Cartesian square of the form
$$ \xymatrix{
\P(c_1,\cdots,c_n;c)\times (\sS^{m})^{\times n} \ar[r]\ar[d] & \Q(f(c_1),\cdots,f(c_n);f(c))\times (\sS^{m})^{\times n} \ar[d] \\
\P(c_1,\cdots,c_n;c) \ar[r] & \Q(f(c_1),\cdots,f(c_n);f(c)) \\
\end{proof}
\end{prop}

\begin{rem} The assertion above reflects one of the hallmarks of the cotangent complex of simplicial operads, or more generally, operads enriched over a Cartesian monoidal category. We also warn the readers that, when working on a base category that is not Cartesian (e.g. dg modules over a commutative ring), the circumstance might vary significantly.
\end{rem}

Note that under the identification of Theorem \ref{t:maintan} the functor $f^* : \T_\Q\Op(\Set_\Delta) \lrar \T_\P\Op(\Set_\Delta)$ corresponds to the functor
$$ \Tw(f)^* : \Fun(\Tw(\Q) , \Sp) \lrar \Fun(\Tw(\P) , \Sp) $$
given by the restriction along $\Tw(f) : \Tw(\P) \lrar \Tw(\Q)$. Combined with the above proposition, it proves that the functor $\F_\P$ is equivalent to the composed functor $\Tw(\P) \lrar \Tw(\Q) \x{\F_{\Q}}{\lrar} \Sp$.

Now, without loss of generality we can assume that $\P$ comes equipped with a map of operads $\P \lrar \E_\infty$. The following is an immediate consequence.
\begin{cor}\label{co:destw} The functor $\F_\P$ is equivalent to the composed functor
$$ \Tw(\P) \lrar \Tw(\E_\infty) \x{\F_{\E_\infty}}{\lrar} \Sp.$$
\end{cor}

\begin{rem}\label{r:equidef0} As presented in [[1], $\S$6.3], the functor $ \F_{\E_\infty} : \sN(\Fin_*^{\op}) \lrar \Sp $
is given on objects by $\F_{\E_\infty}(\l m \r) =\mathbb{S}^{\oplus m}$ (i.e. the $m$-fold coproduct of the sphere spectrum); and moreover, for each map $f : \l n \r \lrar \l m \r$ in $\Fin_*$, the structure map
$$ \F_{\E_\infty}(f) : \mathbb{S}^{\oplus \{1,\cdots,m\}} \lrar \mathbb{S}^{\oplus \{1,\cdots,n\}}  $$
is defined by, for each $i\in \{1,\cdots,m\}$, copying the $i$'th summand to the summands of position $j\in f^{-1}(i)$ when this fiber is nonempty, or collapsing that summand to the zero spectrum otherwise.
\end{rem}

\begin{rem}\label{r:equidef} The functor $\F_{\E_\infty}$ can be equivalently defined by letting $\F_{\E_\infty}(\l m \r) = [\l m \r, \mathbb{S}]_*$ where  $[- , -]_*$ refers to the powering of pointed simplicial sets over spectra, and $\l m \r$ is considered as a (discrete) pointed simplicial set with base point $0$. From this point of view, we may think of $\F_{\E_\infty}$ as the spectral version of Pirashvili's functor $$t : \Fin_*^{\op} \lrar \Mod(\textbf{k}),$$
with $\textbf{k}$ being some commutative ring, defined by taking $t(\l m \r) := [\l m \r, \textbf{k}]_*$, i.e. the $\textbf{k}$-module of based maps $\l m \r \lrar \textbf{k}$ (where $\textbf{k}$ has base point $0_\textbf{k}$). This functor $t$ played a very active role in the author's works, e.g. [28] and [[29], chapter 13]. We shall revisit this remark later.
\end{rem}

Here is the fundamental theorem concerning the cotangent complex of simplicial operads.
\begin{thm}[1]\label{t:cotan} Under the equivalence $\T_{\P^{\sB}}\BMod(\P)_\infty \simeq \Fun(\Tw(\P) , \Sp)$, the cotangent complex $\rL_{\P^{\sB}} \in \T_{\P^{\sB}}\BMod(\P)_\infty$ is identified with $\F_\P : \Tw(\P) \lrar \Sp$ which is given on objects by sending each operation $\mu\in\P$ of arity $m$  to $\F_\P(\mu) = \mathbb{S}^{\oplus m}$. Consequently, under the equivalence $$\T_\P\Op(\Set_\Delta)_\infty \simeq \Fun(\Tw(\P) , \Sp),$$
the cotangent complex $\rL_{\P} \in \T_\P\Op(\Set_\Delta)_\infty$ is identified with $\F_\P[-1]$ the desuspension of $\F_\P$.
\end{thm}

\begin{rem}\label{r:immcons} An immediate consequence is as follows. After sending coefficients into $\Fun(\Tw(\P) , \Sp)$, the Quillen cohomology of $\P$ with coefficients in a given functor $\F : \Tw(\P) \lrar \Sp$ is computed by
$$ \HHQ^\star(\P ; \F) \simeq \Map_{\Fun(\Tw(\P) , \Sp)}(\F_\P[-1], \F).$$
Moreover, the $n$'th Quillen cohomology group of $\P$ with coefficients in $\F$ is computed by
$$ \HHQ^n(\P ; \F) \cong \pi_0\Map_{\Fun(\Tw(\P) , \Sp)}(\F_\P[-1], \F[n]) \cong \pi_0\Map_{\Fun(\Tw(\P) , \Sp)}(\F_\P, \F[n+1]).$$
\end{rem}

\begin{rem} Using Corollary \ref{co:destw}, we obtain a chain of weak equivalences
$$ \HHQ^\star(\P ; \F) \simeq \Map_{\Fun(\Tw(\P) , \Sp)}(\F_\P[-1], \F) \simeq \Map_{\Fun(\Tw(\E_\infty) , \Sp)}(\F_{\E_\infty}[-1], \F_*) \simeq \HHQ^\star(\E_\infty ; \F_*) $$
where $\F_* \in \Fun(\Tw(\E_\infty) , \Sp)$ refers to the right Kan extension of $\F: \Tw(\P) \lrar \Sp$ along the functor $\Tw(\P) \lrar \Tw(\E_\infty)$. This tells us that Quillen cohomology of every operad can be seen as a particular case of the cohomology of $\E_\infty$.
\end{rem}

The statement below proves that Quillen cohomology of $\E_\infty$ can be built up from Quillen cohomology of the operads $\E_n$'s. Let $\F : \Tw(\E_\infty) \lrar \Sp$ be given and let $\F_n$ denote the composed functor
$$ \Tw(\E_n) \x{\varphi_n}{\lrar} \Tw(\E_\infty) \x{\F}{\lrar} \Sp.$$
\begin{prop} For every integer $k$ there is an exact sequence of abelian groups
\begin{equation}\label{eq:milnor}
0 \lrar \underset{n}{\lim}^1 \HHQ^{k-1}(\E_n ; \F_n) \lrar  \HHQ^k(\E_\infty ; \F)  \lrar \underset{n}{\lim} \HHQ^{k}(\E_n ; \F_n) \lrar 0
\end{equation}
where $\lim^1$ denotes the first right derived functor of the limit-functor on towers of abelian groups.
\begin{proof} The filtration \eqref{eq:nervefin} gives rise to a sequential limit of $\infty$-categories:
$$ \Fun(\Tw(\E_\infty),\Sp) \lrar \cdots \lrar \Fun(\Tw(\E_2),\Sp) \lrar \Fun(\Tw(\E_1),\Sp).$$
Combined with Corollary \ref{co:destw}, the sequence above yields a sequential limit of $\infty$-groupoids:
$$ \left[\F_{\E_\infty}, \F[k+1]\right]_{\Sp^{\Tw(\E_\infty)}} \lrar \cdots \lrar [\F_{\E_2}, \F_2[k+1]]_{\Sp^{\Tw(\E_2)}}\lrar [\F_{\E_1}, \F_1[k+1]]_{\Sp^{\Tw(\E_1)}}.$$
(Here for brevity we write $[-,-]_\C$ to denote the mapping space in $\C$). We consider the above spaces as a pointed space with the zero map as its base point. Now, the first \textit{Milnor exact sequence} (see e.g. [[30], $\S$9.3]) determines an exact sequence of abelian groups
$$ 0 \rar \underset{n}{\lim}^1 \pi_1\left[\F_{\E_n}, \F_n[k+1]\right]_{\Sp^{\Tw(\E_n)}} \rar  \pi_0\left[\F_{\E_\infty}, \F[k+1]\right]_{\Sp^{\Tw(\E_\infty)}}  \rar \underset{n}{\lim} \pi_0\left[\F_{\E_n}, \F_n[k+1]\right]_{\Sp^{\Tw(\E_n)}} \rar 0.$$
Applying the formulae given in Remark \ref{r:immcons}, we obtain the expected exact sequence \eqref{eq:milnor}.
\end{proof}
\end{prop}

\begin{rem} Of course, an analogue of \eqref{eq:milnor} holds for $\P$ as long as we obtain a filtration of $\P$. Then we may refer to the obtained exact sequence as the \textbf{Milnor exact sequence} for Quillen cohomology of simplicial operads.
\end{rem}

In what follows, we discuss on Hochschild cohomology of $\P$. To proceed with the two statements below, it is sufficient for $\P$ to be fibrant only.
\begin{prop}\label{p:hoch} Under the equivalence $\T_{\P^{\IB}}\IbMod(\P)_\infty \simeq \Fun(\Tw(\P) , \Sp)$, the Hochschild complex $\rL_{\P^{\IB}}\in\T_{\P^{\IB}}\IbMod(\P)_\infty$ is identified with $\ovl{\mathbb{S}} : \Tw(\P) \lrar \Sp$, i.e. the constant functor with value $\mathbb{S}\in\Sp$.
\begin{proof} According to Proposition \ref{p:hhop}, the Hochschild complex $\rL_{\P^{\IB}}$ can be modeled by the suspension spectrum $\P\times\mathbb{S}$ with
$$ (\P\times\mathbb{S})_{m,m}(c_1,\cdots,c_n;c) = \P(c_1,\cdots,c_n;c) \times \sS^{m}.$$
We write $\mathcal{H}_\P : \Tw(\P) \lrar \Sp$ for the functor corresponding to $\rL_{\P^{\IB}}$. As in Proposition \ref{p:destw}, we may show that the derived functor of the right adjoint $\T_{\E_\infty^{\IB}}\IbMod(\E_\infty) \lrar \T_{\P^{\IB}}\IbMod(\P)$ sends $\rL_{\E_\infty^{\IB}}$ to $\rL_{\P^{\IB}}$. Thus, as in Corollary \ref{co:destw}, we obtain that $\mathcal{H}_\P$ is equivalent to the composed functor $$ \Tw(\P) \lrar \Tw(\E_\infty) \x{\mathcal{H}_{\E_\infty}}{\lrar} \Sp.$$ Therefore, the proof will be completed after showing that $\mathcal{H}_{\E_\infty} : \sN(\Fin_*^{\op}) \lrar \Sp$ is equivalent to the constant functor with value $\mathbb{S}$. For any object $\l n \r$ of $\Fin_*^{\op}$, by construction $\mathcal{H}_{\E_\infty}(\l n \r)\in \Sp$ is the prespectrum with $\mathcal{H}_{\E_\infty}(\l n \r)_{m,m}$ being given by