Operadic Dold-Kan correspondence and mapping spaces between enriched operads

Hoang Truong

DEPARTMENT OF MATHEMATICS, FPT UNIVERSITY, HA NOI, VIET NAM.

[2020]18G31, 18M60, 18M75, 18N40.

According to the Dold-Kan correspondence, the normalization functor determines a categorical equivalence between simplicial abelian groups and non-negatively graded chain complexes. Nevertheless, the same statement is no longer true when extending to the context of algebras over an operad. In spite of this, the existence of the Dold-Kan correspondence for operadic algebras can still be realized from a homotopy-theoretic perspective. This was indeed observed for $\sA_\infty$-algebras by Schwede-Shipley, and for $\E_\infty$-algebras by Mandell, and more generally, for algebras over any $\Sigma$-cofibrant operad due to White-Yau. In the first part of the paper, we shall prove further that the normalization functor induces an equivalence between the $\infty$-categories of operads with non-fixed sets of colors, in the Dwyer-Kan homotopy theory framework. Subsequently, we delve into investigating the mapping spaces between enriched operads by relating them to mapping spaces between operadic bimodules via nice fiber sequences. The validity of the main statements is examined first for simplicial operads, and then for connective dg operads due to the operadic Dold-Kan correspondence.

§ INTRODUCTION

We will assume that $\textbf{k}$ is a commutative ring. The Dold-Kan correspondence asserts the existence of an adjoint equivalence

$$ \Gamma : \adjunction*{\simeq}{\C_{\geqslant0}(\textbf{k})}{\sMod_{\textbf{k}}}{} : \sN $$

between the category of connective dg $\textbf{k}$-modules (i.e. the differential graded $\textbf{k}$-modules concentrated in non-negative degrees) and the category of simplicial $\textbf{k}$-modules, in which the functor $\sN$ is the well known normalization functor (cf. [1, 2]). It is important to note that the functors $\Gamma$ and $\sN$ are no longer inverses of each other (or even adjunction) when descending to the categories of monoids. Despite this inconvenience, the work of Schwede-Shipley [3] shows that the functor $\sN$ is the right adjoint of a Quillen equivalence when considered as a functor between monoid-categories (cf. Theorem 3.12 of loc.cit). In other words, their result tells us that the homotopy theory of $\sA_\infty$-algebras in $\sMod_{\textbf{k}}$ is equivalent to that of $\sA_\infty$-algebras in $\C_{\geqslant0}(\textbf{k})$, and this can be viewed as the first operadic version of the Dold-Kan correspondence. Moreover, the same statement holds for $\E_\infty$-algebras as well, as shown by Mandell ([[4], Theorem 1.3(i)]).

In fact, the Dold-Kan correspondence of operadic algebras holds true for every $\Sigma$-cofibrant operad. Let $\P$ be a (symmetric) $C$-colored operad in $\sMod_{\textbf{k}}$ with $C$ being some fixed set of colors.  We assume that $\P$ is $\Sigma$-cofibrant, i.e. $\P$ is cofibrant as a $C$-collection. Applying the functor $\sN$ to $\P$ levelwise produces an operad in $\C_{\geqslant0}(\textbf{k})$, denoted  $\sN\P$. The categories $\Alg_\P(\sMod_{\textbf{k}})$ and $\Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$ come equipped with a semi-model structure transferred from the projective model structure on the base category. The following theorem can be derived as a special case of [[5], Theorem 4.3.2].

(<ref>)(White-Yau) The functor
$$\sN_\P : \Alg_\P(\sMod_{\textbf{k}}) \lrar \Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$$
defined by applying the normalization functor levelwise is the right adjoint of a Quillen equivalence between semi-model categories.

Nonetheless, we will provide an alternative proof of this theorem somewhat more concretely. Our approach will make use of a cellular argument, similarly as in the proof of [[3], Proposition 5.1]. In particular, for the inductive step, our proof utilize the description of free extensions of operads presented in, e.g., [6, 7, 8], and free extensions of operadic algebras presented in [7].

(<ref>, <ref>, <ref>, <ref>) The above theorem leads to several interesting examples. First, we obtain that the normalization functor induces a right Quillen equivalence

$$ \sN^{b}_\P : \BMod(\P) \lrarsimeq \BMod(\sN\P) $$

from the category of $\P$-bimodules to that of  $\sN\P$-bimodules. An analogue holds for infinitesimal bimodules, and for left (right) modules as well. Another consequence asserts the existence of a Quillen equivalence

$$ 	\L_C : \x{\simeq}{\adjunction*{}{\Op_C(\C_{\geqslant0}(\textbf{k}))}{\Op_C(\sMod_{\textbf{k}})}{}} : \sN_C $$

between $C$-colored operads in $\sMod_{\textbf{k}}$ and $C$-colored operads in $\C_{\geqslant0}(\textbf{k})$, where the right adjoint is induced by the normalization functor again.

(<ref>, <ref>, <ref>) A version of Theorem <ref> is as follows. Let $\O$ be a cofibrant $C$-colored operad in $\C_{\geqslant0}(\textbf{k})$. Then there exists a Quillen equivalence

such that the right adjoint is given by applying the normalization functor levelwise. Again, an analogue holds for various types of operadic modules.

On a different aspect, the work of Tabuada [9] shows that the functor

\begin{equation}\label{eq:catDK}
\sN : \Cat(\sMod_{\textbf{k}}) \lrar \Cat(\C_{\geqslant0}(\textbf{k}))
\end{equation}

defined by applying the normalization functor levelwise is the right adjoint of a Quillen equivalence between (small) enriched categories, with respect to the Dwyer-Kan model structure on both sides. We generalize this result into the context of enriched operads (i.e. with non-fixed sets of colors). To this end, we first show that the collection of adjunctions
$$\{\L_C : \adjunction*{}{\Op_C(\C_{\geqslant0}(\textbf{k}))}{\Op_C(\sMod_{\textbf{k}})}{} : \sN_C \; | \; C\in\Sets\}$$
can be integrated into a single adjunction

\begin{equation}\label{eq:inte}
\L : \adjunction*{}{\Op(\C_{\geqslant0}(\textbf{k}))}{\Op(\sMod_{\textbf{k}})}{} : \sN
\end{equation}

between the categories of enriched operads. Note that while the Dwyer-Kan model structure on $\Op(\sMod_{\textbf{k}})$ exists for every commutative ring $\textbf{k}$, the same statement does not hold for the category $\Op(\C_{\geqslant0}(\textbf{k}))$. Nevertheless, we do not expect to work within the framework of model categories. Indeed, we will consider these two categories only as relative categories with weak equivalences being precisely the Dwyer-Kan equivalences (see $\S$<ref>). Our result is then stated as follows.
(<ref>) The functor $\sN$ (<ref>) induces an equivalence
$$\Op(\sMod_{\textbf{k}})_\infty \lrarsimeq \Op(\C_{\geqslant0}(\textbf{k}))_\infty$$
between the underlying $\infty$-categories (cf. [10, 11]), with the inverse being induced by the functor $\L$.

The proof of this theorem is straightforward using the fact that the (co)unit of $\L\dashv\sN$ is made up together by the (co)units of the adjunctions $\L_C\dashv\sN_C$ for $C\in\Sets$.

(<ref>) When the Dwyer-Kan model structure on $\Op(\C_{\geqslant0}(\textbf{k}))$ exists, then the adjunction $\L\dashv\sN$ becomes further a Quillen equivalence. For example, this is the case when $\textbf{k}$ is assumed to contain the field $\mathbb{Q}$ of rational numbers.

The last section is dedicated to investigating the mapping spaces of operads, building upon the insights from Dwyer-Hess [12].

We will assume that $\cS$ is a symmetric monoidal model category satisfying some mild conditions. Let $\P\in\Op_C(\mathcal{S})$ be a fixed $C$-colored operad in $\cS$.

Now suppose we are given a map $f : \P\lrar\O$ in $\Op_C(\mathcal{S})$. We also regard $\O$ as a $\P$-bimodule with the structure induced by $f$. Here is the first assertion.

(<ref>) Suppose that $\P$ is a good cofibrant operad. Then there is a fiber sequence of spaces

\begin{equation}\label{eq:goodnessmain2}
\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \O) \lrar \Map^{\h}_{\BMod(\P)}(\P, \O) \lrar \underset{c\in C}{\prod} |\O(c;c)|
\end{equation}

in which the loop space $\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \O)$ is taken at the base point $f$, and while $|\O(c;c)| := \Map^{\h}_{\cS}(1_{\cS}, \O(c;c))$ signifies the underlying space of $\O(c;c) \in \cS$.

This theorem extends a result presented in the loc.cit, which is particularly applicable to the case where $\cS = \Set_\Delta$ and $\P$ is  a non-symmetric operad. Furthermore, for the case $\cS = \Set_\Delta$ and $\O$ being such that $\O(c;c)\simeq *$ for every color $c$, then (<ref>) yields a weak equivalence

$$ \Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \O) \lrarsimeq \Map^{\h}_{\BMod(\P)}(\P, \O),$$

that was observed by Ducoulombier [13].

Moreover, a relation between the mapping spaces of enriched operads (with non-fixed sets of colors) and mapping spaces of operadic bimodules is achieved by combining (<ref>) with the statement below.

Let $\Q\in\Op_{D}(\mathcal{S})$ be another operad colored in some set $D$. Suppose we are given a map $g : C \lrar D$ between the sets of colors. We will write $g^*\Q$ for the $C$-colored operad induced by the restriction of colors, and write $\Q_1$ for the underlying category of $\Q$.
(<ref>)  There is a fiber sequence of spaces

\begin{equation}\label{eq:simplifying1}
\Map^{\h}_{\Op_C(\cS)}(\P,g^*\Q) \lrar \Map^{\h}_{\Op(\cS)}(\P,\Q) \lrar \underset{C}{\prod}\cor(\Q_1)
\end{equation}

in which $\cor(\Q_1)$ refers to the core of $\Q_1\in\Cat(\cS)$ (see Notation <ref>).

For an illustration, when $\C$ is a (fibrant) simplicial category, then we have $\cor(\Q_1) \simeq \sN(\Q_1)^\simeq$, i.e. the maximal $\infty$-groupoid contained in the  nerve of $\C$.

Combining the two theorems above, we obtain a significant result as follows. We will assume further that $\cS$ is subject to an invertibility hypothesis due to Lurie. Now suppose we are given a map $f : \P \lrar \O$ in $\Op(\cS)$.

(<ref>) Suppose that $\P$ is a good cofibrant operad. Then there is a natural weak equivalence

$$  \Omega^2\Map^{\h}_{\Op(\cS)}(\P,\O) \simeq \Omega\Map^{\h}_{\BMod(\P)}(\P,f^*\O) $$

in which the double loop space on the left is taken at the base point $f$, and while the loop space on the right is taken at the induced map $\P\lrar f^*\O$.

When $\cS$ is any of the categories in $\{\Set_\Delta, \sMod_{\textbf{k}}, \C_{\geqslant0}(\textbf{k})\}$, then Theorems <ref> and <ref> remain valid for every $\Sigma$-cofibrant operad $\P$.

The results obtained from section <ref> provide a refined version of the previously presented results in the author's thesis, in which the base ring $\textbf{k}$ is particularly assumed to be a field of characteristic zero. Whilst section <ref> is a preparation for the study of cotangent complex of dg operads. Moreover, Theorem <ref> provides an example for the (nonlinear version of) Quillen principle that we have been developing (cf. [14]).

Acknowledgements. This paper is part of the author's PhD thesis, written under the supervision of Yonatan Harpaz at Université Sorbonne Paris Nord. The author expresses gratitude to his advisor for providing insightful comments and suggestions, and for conscientiously reviewing the paper's earlier drafts.

§ BACKGROUND AND NOTATIONS

§.§ Operads and modules over an operad

Let $(\cS, \otimes, 1_\cS)$ be a symmetric monoidal category such that the monoidal product $\otimes$ distributes over colimits. For a set $C$, regarded as a set of colors, we denote by $\Op_C(\cS)$ the category of symmetric $C$-colored operads in $\cS$. For more information, $\Op_C(\cS)$ agrees with the category of monoids in the monoidal category $(\Coll_C(\cS), \circ, \I_C)$ in which

$\bullet$ an object of $\Coll_C(\cS)$ is a collection
$$M=\{M(c_1,\cdots,c_n;c) \, | \, c_i,c\in C, n\geq 0\}$$
of objects in $\cS$ equipped with a right $\Sigma_n$-action whose data consist of maps of the form

$$\sigma^{*} : M(c_1,\cdots,c_n;c) \lrar M(c_{\sigma(1)},\cdots,c_{\sigma(n)};c)$$

for every sequence $(c_1,\cdots,c_n;c)$ and every permutation $\sigma\in \Sigma_n$,

$\bullet$ $-\circ-$ is the well known composite product of $C$-collections, and

$\bullet$ the monoidal unit $\I_C$ is given by $\emptyset_\cS$ (i.e. the initial object of $\cS$) on all the levels except that $\I_C(c;c)=1_{\cS}$ for every $c\in C$.

Let $\P\in\Op_C(\cS)$ be a $C$-colored operad in $\cS$.  By considering $\P$ as a monoid object, it is natural to define the notion of left (right) $\P$-modules. We will denote by $\LMod(\P)$ the category of left $\P$-modules. One of the main interests in the study of operads is the notion of algebras over them. Recall that a $\P$-algebra is simply a left $\P$-module that is concentrated in level $0$. We let $\Alg_\P(\cS)$ denote the category of $\P$-algebras.

We will denote by $\BMod(\P)$ the  category of $\P$-bimodules and moreover, denote by $\IbMod(\P)$ the  category of infinitesimal $\P$-bimodules (cf. [15], $\S$6.1).

Due to technical requirements, we will need the following construction. Denote by $\Pairs_C(\cS)$ the category whose objects are pairs of the form $(\P , A)$ in which $\P\in \Op_C(\mathcal{S})$ and $A \in \Alg_\P(\SS)$, and such that a morphism $(\P , A) \lrar (\Q , B)$ is a pair $(\varphi , f)$ with $\varphi : \P \rar \Q$ being a map in $\Op_C(\mathcal{S})$ and $f : A \rar B$ a map of $\P$-algebras. Consider the functor
$$ \delta : \Op_C(\mathcal{S}) \lrar \Pairs_C(\cS)$$
defined by sending $\P$ to the pair $(\P , \P_0)$ where $\P_0$ refers to the initial $\P$-algebra, which is given by $\P_0(c)=\P(c)$ for every $c\in C$. According to [16], the functor $\delta$ admits a left adjoint $\Env : \Pairs_C(\cS) \lrar \Op_C(\mathcal{S})$ called the enveloping functor.

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

Notice that we have an identification $\Env(\P , \P_0) \cong \P$, from which we may define a natural map $j_A : \P \lrar \Env(\P , A)$ for any $A \in \Alg_\P(\SS)$ given by applying $\Env(-)$ to the obvious map $(\P , \P_0) \lrar (\P , A)$ between pairs.

One can organize all the operads enriched in $\cS$, with arbitrary sets of colors, into a single category as follows. We associate to each map $\alpha : C \rar D$ of sets a  functor $\alpha^{*} : \Op_D(\SS) \lrar \Op_C(\SS)$ given by taking $\Q\in\Op_D(\SS)$ to $\alpha^{*}\Q$ with

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

The category of $\cS$-enriched operads, denoted $\Op(\cS)$, is defined to be the contravariant Grothendieck construction

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

More precisely, an object of $\Op(\SS)$ is a pair $(C,\P)$ with $C\in \Sets$ and $\P\in \Op_C(\SS)$, and such that a morphism $(C,\P) \lrar (D,\Q)$ consists of a map $\alpha : C \rar D$ of sets and a map $f :  \P \lrar \alpha^{*}\Q$ of $C$-colored operads.

For a $C$-colored operad $\P$ in $\cS$, there exists a significant Quillen adjunction

\begin{equation}\label{eq:signi}
\adjunction*{}{\Op_C(\cS)_{\P/}}{\Op(\cS)_{\P/}}{}
\end{equation}

in which the left adjoint is given by the obvious embedding, and while the right adjoint is induced by the restriction of colors.

We will write $\Cat(\cS)$ for the category of $\cS$-enriched categories. Clearly each $\cS$-enriched category can be identified with an $\cS$-enriched operad concentrated in arity $1$. Conversely, each object $\P \in \Op(\cS)$ admits an underlying category, denoted $\P_1 \in \Cat(\cS)$, and formed by the collection of $1$-ary operations of $\P$. Moreover, these two  constructions determine an adjunction

\begin{equation}\label{eq:signi1}
\adjunction*{}{\Cat(\cS)}{\Op(\cS)}{}.
\end{equation}

§.§ Operadic model structures

We now recall the homotopy theory of algebras over an operad and the homotopy theory of enriched operads themselves.

Let us assume further that $\cS$ is a symmetric monoidal model category. Let $\P$ be a $C$-colored operad in $\cS$. Under some suitable conditions on the base category $\cS$, the category $\Alg_\P(\cS)$ may inherit a model structure transferred from the projective model structure on $\cS^{\times C}$ along the free-forgetful adjunction

$$ \adjunction*{}{\cS^{\times C}}{\Alg_\P(\cS)}{}.$$

Nevertheless, to be able to encompass more examples, we just require the existence of a semi-model structure on $\Alg_\P(\cS)$. By convention, we will say that $\P$ is $\Sigma$-cofibrant if it is cofibrant as a $C$-collection with respect to the projective model structure. We find the following in literature.

(Spitzweck [8], Fresse [17]) Suppose further that the model structure on $\cS$ is cofibrantly generated and such 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 $\cS^{\times C}$.

Note that for every set $C$, there is an $\cS$-enriched operad, denoted $\textbf{O}_C$, with the property that the structure of an $\textbf{O}_C$-algebra is the same as that of a $C$-colored operad in $\cS$, that is to say there is an equivalence of categories $\Alg_{\textbf{O}_C}(\cS) \simeq \Op_C(\cS)$ (cf. [18], $\S$3). Thus the above theorem already includes the transferred model structure on the category $\Op_C(\cS)$.

In a cofibrantly generated semi-model category, not all cofibrations can be expressed as the retract of a cellular map, but only those cofibrations whose domain is cofibrant. In particular, every cofibrant object can be expressed as the retract of a cellular object (cf. [17], $\S$12.1.3).

Suppose given a Quillen adjunction $\rL : \adjunction*{}{\M}{\N}{} : \rR$ between semi-model categories such that the right adjoint $\rR$ preserves and detects weak equivalences. Then $\L\dashv\R$ is a Quillen equivalence if and only if for every cofibrant object $A\in \M$, the unit map $A \lrar \rR\rL(A)$ is a weak equivalence.
This observation is well known within the realm of (full) model categories. The same argument can be applied without any impediment.

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
$$\Ho(\P) := \Ho(\P_1)$$

(see Remark <ref> for notation). 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 $\cS$. In what follows, we recall the two model structures on enriched operads, according to Caviglia's [19].

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.

The canonical model structure on $\Op(\cS)$ is the one whose fibrant objects are the levelwise fibrant operads and whose trivial fibrations are the same as those of the Dwyer-Kan model structure.

The two model structures on $\Op(\cS)$ are a natural generalization of the corresponding model structure on $\Cat(\cS)$. For more details, the Dwyer-Kan model structure on $\Cat(\cS)$ is developed in [20, 21], and can be thought of as a generalization of the Dwyer-Kan homotopy theory of simplicial categories introduced by Bergner ([22]). On the other hand, the canonical model structure on $\Cat(\cS)$ is established by Berger-Moerdijk [23], as another approach to the homotopy theory of enriched categories. Nevertheless, in practice the Dwyer-Kan and canonical model structures usually coincide (cf. [[19], $\S$8]).

A map $f$ in $\Op(\cS)$ is a Dwyer-Kan equivalence if and only if it is a levelwise weak equivalence and such that its underlying map in $\Cat(\cS)$ is a Dwyer-Kan equivalence. On other hand, as originally introduced in [19], $f$ is a canonical fibration (i.e. a fibration with respect to the canonical model structure) if and only if it is a levelwise fibration and such that its underlying map in $\Cat(\cS)$ is a canonical fibration.

We discuss the three base categories that we are concerned with in this paper. Let $\textbf{k}$ be a commutative ring.

(i) The first one is $\Set_\Delta$ the category of simplicial sets, equipped with the Cartesian monoidal product and with the classical (Kan-Quillen) model structure. The transferred model structure on $\Alg_\P(\Set_\Delta)$ exists for every  operad $\P$. Moreover, both the Dwyer-Kan and canonical model structures on $\Op(\Set_\Delta)$ exist and, in fact, coincide with each other.

(ii)  One of the main interests in this paper is $\sMod(\textbf{k})$ the category of simplicial $\textbf{k}$-modules, equipped with the (degreewise) tensor product over $\textbf{k}$ and with the model structure transferred from the classical model structure on $\Set_\Delta$. As well as the case above, all the model category structures of interest are available and, in addition, the two model structures on  $\Op(\sMod(\textbf{k}))$ coincide.

(iii) Another main interest is the category $\C_{\geqslant0}(\textbf{k})$ of connective dg $\textbf{k}$-modules, equipped with the usual tensor product over $\textbf{k}$ and with the projective model structure. In general, for an operad $\O$ in $\C_{\geqslant0}(\textbf{k})$, the transferred homotopy theory on $\Alg_\O(\C_{\geqslant0}(\textbf{k}))$ exists only as a semi-model category, provided $\O$ is $\Sigma$-cofibrant (see Theorem <ref>). We would guess that the Dwyer-Kan homotopy theory on $\Op(\C_{\geqslant0}(\textbf{k}))$ exists as a semi-model category for any commutative ring $\textbf{k}$. We would like to leave it to the readers.

§ OPERADIC DOLD-KAN CORRESPONDENCE

§.§ Dold-Kan correspondence of operadic algebras

In this subsection, we shall present a proof of Theorem <ref>. But first, let us recall some fundamental facts relevant to the classical Dold-Kan correspondence.

As usual, we let $\sN$ denote the normalization (or normalized complex) functor and let $\Gamma$ denote the inverse of $\sN$, as parts of the Dold-Kan correspondence. Let $R$ be a unital ring and let $A$ and $B$ be the simplicial right and left $R$-modules, respectively. Recall that the Alexander-Whitney map

$$ \AW : \sN(A\otimes_R B) \lrar \sN A \otimes_R \sN B $$

and the shuffle map

$$ \nabla : \sN A \otimes_R \sN B \lrar \sN(A\otimes_R B) $$

form the homotopy inverses of each other. For more details, we refer the readers to, e.g., [[3],  $\S$2.3], [[24],  $\S$8.5] and [[25], $\S$4.2.2].

Let $K$ and $L$ be connective dg right and left $R$-modules, respectively. There is an induced natural map

\begin{equation}\label{eq:oplagamma}
\Gamma(K \otimes_R L) \lrar \Gamma(K) \otimes_R \Gamma(L)
\end{equation}

given by the adjoint of the composed map

$$ K \otimes_R L \lrar \sN \Gamma(K) \otimes_R \sN \Gamma(L) \x{\nabla}{\lrar} \sN(\Gamma(K) \otimes_R \Gamma(L)) $$

where the first map is the tensor product over $R$ of the unit maps of $\Gamma \dashv \sN$. This composed map is in particular a weak equivalence.  Thus, the map (<ref>) is as well a weak equivalence, because the functor $\Gamma$ sends every weak equivalence to a weak equivalence.

Now let $\textbf{k}$ be a fixed commutative ring. The tensor product over $\textbf{k}$ will be abbreviated as $``\otimes$".

Let $A$ and $B$ be two simplicial $\textbf{k}$-modules. The maps
$$\nabla : \sN A \otimes \sN B \lrar \sN(A\otimes B) \, \; \text{and} \; \AW : \sN(A\otimes B) \lrar \sN A \otimes \sN B$$
respectively equip the functor $\sN : \sMod_{\textbf{k}} \lrar \C_{\geqslant0}(\textbf{k})$ with a lax and an oplax monoidal structure. Note that while the lax monoidal structure of $\sN$ is symmetric, however, its oplax monoidal structure is not so (cf. [[3], 2.3]). In particular, the canonical lax monoidal structure of $\Gamma : \C_{\geqslant0}(\textbf{k}) \lrar \sMod_{\textbf{k}}$, that is induced by $\AW$, is not symmetric.

Due to the symmetricity of the lax monoidal structure of $\sN : \sMod_{\textbf{k}} \lrar \C_{\geqslant0}(\textbf{k})$, the lifted functor $\sN_C : \Coll_C(\sMod_{\textbf{k}}) \lrar \Coll_C(\C_{\geqslant0}(\textbf{k}))$ admits an extended lax monoidal structure

$$ \sN_C(X) \circ \sN_C(Y) \lrar \sN_C(X \circ Y) $$

for $X, Y \in \Coll_C(\sMod_{\textbf{k}})$. As a consequence, $\sN_C$ descends to a functor

\begin{equation}\label{eq:opnormalized}
\sN_C : \Op_C(\sMod_{\textbf{k}}) \lrar \Op_C(\C_{\geqslant0}(\textbf{k}))
\end{equation}

between $C$-colored operads.

On other hand, the  functor $\Gamma : \C_{\geqslant0}(\textbf{k}) \lrar \sMod_{\textbf{k}}$ does not admit any canonical symmetric lax monoidal structure. Consequently, in general, one can not produce $\sMod_{\textbf{k}}$-enriched operads just by applying $\Gamma$ to the $\C_{\geqslant0}(\textbf{k})$-enriched operads levelwise.

Despite that inconvenience, we can give a description of the left adjoint of (<ref>). In fact, we will proceed in a more general context of operadic algebras, from which the case of (<ref>) can be derived as a particular example (cf. Remark <ref>). To this end, we will need to use the following construction (cf. [[3], $\S$3]).

Let $\C$ and $\D$ be two monoidal categories. We use the same notation $``\odot$" to denote the monoidal product in both these categories. Suppose we are given an adjunction
$$\L : \adjunction*{}{\C}{\D}{} : \R$$
between the underlying categories such that the right adjoint $\R$ is lax monoidal. In particular, the left adjoint $\L$ inherits an induced oplax monoidal structure. Since $\R$ is lax monoidal, it descends to a functor between monoids. Let $A$ be a monoid in $\D$. Again, the functor $\R$ descends to a functor $\R_A : \Mod_A \lrar \Mod_{\R A}$ from left $A$-modules to left $\R A$-modules. Assume that coequalizers of left $A$-modules exist. Let us denote by $\L_{A} : \Mod_{\R A} \lrar \Mod_A$ the functor given by sending $M \in \Mod_{\R A}$ to the coequalizer

\begin{equation}\label{eq:coeq}
A\odot\L(\R A\odot M) \; \doublerightarrow{}{} A\odot \L M \; \lrar \L_{A}M.
\end{equation}

In this diagram, one of the two parallel arrows is given by applying $A\odot\L(-)$ to the structure map $\R A\odot M \lrar M$ of $M$, while the other arrow is the unique map of left $A$-modules extending the composed map

$$ \L(\R A\odot M) \lrar \L\R A \odot \L M \lrar A\odot\L M $$

where the first map is an oplax monoidal map of $\L$ and the second map is the monoidal product of the counit map $\L\R A \lrar A$ with $\Id_{\L M}$.

The main interest in this construction is as follows.
(Schwede-Shipley [3]) The functors $\L_{A}$ and $\R_A$ form an adjunction

$$ \L_{A} : \adjunction*{}{\Mod_{\R A}}{\Mod_A}{} : \R_A .$$

Moreover, the module left adjoint $\L_{A}$ and the original left adjoint $\L$ are related via the identification

\begin{equation}\label{eq:identifi}
\L_{A} \circ (\R A \odot -) \cong (A \odot -) \circ \L \, .
\end{equation}

Suppose given $M \in \Mod_{\R A}$ and $N \in \Mod_A$. For a map $f : A\odot \L M \lrar N$ of left $A$-modules, we denote by $\ovl{f} : \L M \lrar N$ the corresponding map in $\D$ and furthermore, let $\ovl{f}^{\ad} : M \lrar \R N$ denote the adjoint of the latter map. For the first claim, it suffices to argue that the map $f$ is compatible with the coequalizer diagram (<ref>) if and only if $\ovl{f}^{\ad}$ is a map of left $\R A$-modules.

For the second claim, notice first that there is a commutative square of right adjoints

$$ \xymatrix{
\Mod_A \ar^{\R_A}[r]\ar_{\forget}[d] & \Mod_{\R A} \ar^{\forget}[d] \\
\D \ar_{\R}[r] & \C \;  \\
} $$

Therefore, the associated square of  left adjoints is commutative as well. The latter commutativity is equivalent to the identification (<ref>).

Now, let $\P \in \Op_C(\sMod_{\textbf{k}})$ be a fixed $C$-colored operad in $\sMod_{\textbf{k}}$. Consider the functor $\sN_\P : \LMod(\P) \lrar \LMod(\sN\P)$ from left $\P$-modules to left $\sN\P$-modules defined by applying the normalization functor levelwise. According to Construction <ref> and Lemma <ref>, $\sN_\P$ admits a left adjoint
$$\L_\P : \LMod(\sN\P) \lrar \LMod(\P).$$
It is clear that $\sN_\P$ descends to a functor
$$\sN_\P : \Alg_\P(\sMod_{\textbf{k}}) \lrar \Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$$
between the full subcategories of $\P$-algebras and of $\sN\P$-algebras. Moreover, note that the embedding functor $\Alg_\P(\sMod_{\textbf{k}}) \lrar \LMod(\P)$ admits a right adjoint given by removing all the elements of level $\geq 1$ in each left $\P$-module. In particular, that embedding functor preserves colimits. This implies that $\L_\P$ descends to a functor
$$\L_\P : \Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k})) \lrar \Alg_\P(\sMod_{\textbf{k}}).$$
We thus obtain an adjunction $\L_\P \dashv \sN_\P$ between $\sN\P$-algebras and $\P$-algebras.

As discussed earlier, we aim to reproduce a proof of the following theorem, which is a particular case of White-Yau's [[5], Theorem 4.3.2].
The adjunction $\L_\P : \adjunction*{}{\Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))}{\Alg_\P(\sMod_{\textbf{k}})}{} : \sN_\P$ is a Quillen equivalence provided that $\P$ is $\Sigma$-cofibrant.

Let $A$ be an $\sN\P$-algebra. Recall from Remark <ref> that there is a natural map
$$j_{\L_\P A} : \P \lrar \Env(\P , \L_\P A)$$
in $\Op_C(\sMod_{\textbf{k}})$ with $\Env(\P , \L_\P A)$ being the enveloping operad associated to the pair $(\P , \L_\P A)$. On other hand, note that the initial $\sN\Env(\P , \L_\P A)$-algebra is nothing but $\sN\Env(\P , \L_\P A)_0 = \sN_\P\L_\P A$. Now, due to the adjunction $\Env \dashv \delta$, the map $\sN(j_{\L_\P A})$ and the unit map $A \lrar \sN_\P\L_\P A$ together induce a canonical map
$$\varphi_A : \Env(\sN\P , A) \lrar \sN\Env(\P , \L_\P A)$$
in $\Op_{C}(\C_{\geqslant0}(\textbf{k}))$. Taking the adjoint of $\varphi_A$ (considered as a map of $C$-collections), we obtain a map
$$\Psi_A : \Gamma \, \Env(\sN\P , A) \lrar \Env(\P , \L_\P A)$$

in $\Coll_C(\sMod_{\textbf{k}})$, where $\Gamma \, \Env(\sN\P , A)$ is simply given by applying $\Gamma$ to $\Env(\sN\P , A)$ levelwise. The proof of Theorem <ref> will require verifying that $\Psi_A$ is a weak equivalence whenever $A$ is a cofibrant $\sN\P$-algebra.

For the latter, we prefer to make use of the description of free extensions of operads  (cf. [6, 7, 8]). Suppose we are given a pushout square of $C$-colored operads in some symmetric monoidal category $\cS$ of the form

\begin{equation}\label{eq:freeext3}
\xymatrix{
\Free(M) \ar^{\Free(f)}[r]\ar[d] & \Free(N) \ar[d] \\
\O \ar[r] & \O' \\
\end{equation}

where $\Free(f) : \Free(M) \lrar \Free(N)$ is the image under the free-operad functor of a given map $f : M \rar N$ between $C$-collections. Fix a sequence $\ovl{c} := (c_1,\cdots,c_n;c)$ of colors in $C$. According to [[7], Proposition 5.2], the object $\O'(\ovl{c})\in\cS$ is given by the colimit of a sequence of maps of the form

$$ \O(\ovl{c}) = \O'^{(0)}(\ovl{c}) \lrar \cdots \lrar \O'^{(k)}(\ovl{c}) \lrar \O'^{(k+1)}(\ovl{c}) \lrar \cdots$$

such that each arrow  $\O'^{(k)}(\ovl{c}) \lrar \O'^{(k+1)}(\ovl{c})$ is given by the pushout of a map of the form

\begin{equation}\label{eq:freeext4}
H : \bigsqcup_{[T]}  f^{*}_T \otimes_{\Aut(T)} \Sigma_{\{c_1,\cdots,c_n\}} \lrar \bigsqcup_{[T]} f_T \otimes_{\Aut(T)} \Sigma_{\{c_1,\cdots,c_n\}}.
\end{equation}

Let us explain the map $H$ shortly. Firstly, the coproducts range over all the isomorphism classes of $\textbf{Tree}^{(k)}_{\ovl{c}}$ the collection of planar trees with $k$ marked vertices and with valency $\ovl{c}$ (i.e. with the root edge being colored by $c$ and the leaves being colored from the choices in $\{c_1,\cdots,c_n\}$). Moreover, $\Sigma_{\{c_1,\cdots,c_n\}}$ denotes a certain product of symmetric groups depending on $\{c_1,\cdots,c_n\}$ (cf. the notation $\Sigma_s$ in [7]). Let $T\in\textbf{Tree}^{(k)}_{\ovl{c}}$ be given. Assume that $T$ admits a canonical grafting operation of the form
$$T = r_T \, (T_1,\cdots,T_r)$$
with $r_T$ being the root vertex of $T$ and with $T_i$ being the subtree of $T$ grafted to the $i$'th leaf of $r_T$ for each $i\in\{1,\cdots,r\}$. In order to determine the group $\Aut(T)$, one partitions the set $\{T_1,\cdots,T_r\}$ into subsets of pairwise isomorphic trees

$$ \{T^{1}_1 , \cdots , T^{1}_{r_1}\} \sqcup \cdots \sqcup \{T^{q}_1 , \cdots , T^{q}_{r_q}\} .$$

Then one defines

$$ \Aut(T) := \bigsqcap_{i=1}^{q} \Aut(T^{i})^{r_i} \, \rtimes \,  \bigsqcap_{i=1}^{q} \Sigma_{r_i}$$

in which $T^{i} \in \{T^{i}_1 , \cdots , T^{i}_{r_i}\}$ is any representative. Finally, to understand the map $H$ (<ref>), it remains to define the map
$$\varepsilon_T : f^{*}_T \lrar f_T.$$

This is inductively built as an iterated pushout-product of the form

\begin{equation}\label{eq:epsilonT}
\varepsilon_T := \varepsilon_{r_T} \square \varepsilon_{T_1} \square \cdots \square \varepsilon_{T_r}
\end{equation}

with $\varepsilon_{r_T}$ being given by

$$    \varepsilon_{r_T} :=  \begin{cases}
f(val(r_T)) & \text{ if } \, r_T \, \text{is marked} \\
\eta_\O(val(r_T)) & \text{ if } \, r_T \, \text{is unmarked}
\end{cases} $$

in which $val(r_T)$ refers to the valency of $r_T$ and $\eta_\O : \I_C \lrar \O$ is the unit of $\O$.

For the proof below, it suffices for us to focus only on analyzing the map $\varepsilon_T$. We refer the readers to [6, 7, 8] for more details about the above constructions.

Recall by definition that $\P$ is said to be well-pointed if the unit map $\I_C \lrar \P$ is a cofibration of $C$-collections. Assume that $\P\in \Op_C(\sMod_{\textbf{k}})$ is well-pointed and $A\in\Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$ is a cofibrant object. Then the operad $\sN\P$ is also well-pointed. Moreover, due to [[7], Lemma 6.1(ii)], we have further that the operads $\Env(\sN\P , A)$ and $\Env(\P , \L_\P A)$ are  well-pointed as well.

Suppose that $\P$ is well-pointed and let $A$ be a cofibrant $\sN\P$-algebra. Then the map
$$\Psi_A : \Gamma \, \Env(\sN\P , A) \lrarsimeq \Env(\P , \L_\P A)$$
is a weak equivalence in $\Coll_C(\sMod_{\textbf{k}})$.

It will suffice to treat only the case where $A$ is a cellular $\sN\P$-algebra, that is to say $A$ is given by the colimit of a sequence of maps of the form

$$ \sN\P_0 = A^{(0)} \lrar \cdots \lrar A^{(k)} \lrar A^{(k+1)} \lrar \cdots  $$

in which $\sN\P_0$ represents the initial $\sN\P$-algebra and such that each arrow is the pushout of a free cofibration, i.e. the image of a cofibration in $\C_{\geqslant0}(\textbf{k})^{\times \, C}$ through the free $\sN\P$-algebra functor (see Remark <ref>). By [[7], Proposition 4.4(ii)], along with noting the fact that the forgetful functor $\Op_C(\C_{\geqslant0}(\textbf{k})) \lrar \Coll_C(\C_{\geqslant0}(\textbf{k}))$ preserves filtered colimits, both the functors $\Gamma \, \Env(\sN\P , -)$ and $\Env(\P , \L_\P(-))$ preserve that filtration of $A$. Combined with the fact that weak equivalences in $\Coll_C(\sMod_{\textbf{k}})$ are closed under filtered colimits, that observation tells us that it suffices to prove the map $\Psi_{A^{(k)}}$ is a weak equivalence for every $k\geq0$. Firstly, by the adjunction $\Env \dashv \delta$, the map $\Psi_{\sN\P_0}$ agrees with the counit map $\Gamma\sN\P \lrar \P$, which is an isomorphism. By an inductive argument, the proof will be completed after showing that for any pushout square in $\Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$ of the form

\begin{equation}\label{eq:freeext2'}
\xymatrix{
(\sN\P) \circ X \ar^{(\sN\P) \circ f}[r]\ar[d] & (\sN\P) \circ Y \ar[d] \\
A \ar[r] & B \\
\end{equation}

such that the top horizontal map is given by the image of a given cofibration $f:X \lrar Y$ in $\C_{\geqslant0}(\textbf{k})^{\, \times C}$ through the free $\sN\P$-algebra functor, the map $\Psi_{B}$ is a weak equivalence as long as $\Psi_{A}$ is one. Moreover, we can assume without loss of generality that $X$ and $Y$ are cofibrant objects, because the map $f$ can be chosen from the class of generating cofibrations.

According to [[7], Proposition 4.4(iii)], the (<ref>) gives rise to a pushout square in $\Op_C(\C_{\geqslant0}(\textbf{k}))$ of the form

\begin{equation}\label{eq:freeext3'}
\xymatrix{
\Free(X) \ar^{\Free(f)}[r]\ar[d] & \Free(Y) \ar[d] \\
\Env(\sN\P , A) \ar[r] & \Env(\sN\P , B) \\
\end{equation}

where $\Free$ signifies the free-operad functor $\Coll_C(\C_{\geqslant0}(\textbf{k})) \lrar \Op_C(\C_{\geqslant0}(\textbf{k}))$. As discussed earlier, for each $\ovl{c} := (c_1,\cdots,c_n;c)$, the object $\Env(\sN\P , B) (\ovl{c})$ is given by the sequential colimit of maps

\begin{equation}\label{eq:envk}
\Env(\sN\P , B)^{(k)}(\ovl{c}) \lrar \Env(\sN\P , B)^{(k+1)}(\ovl{c})
\end{equation}

defined as the pushout of a map of the form

\begin{equation}\label{eq:freeext4'}
H : \bigsqcup_{T \in \textbf{Tree}^{(k)}_{\ovl{c}}/\cong}  f^{*}_T \otimes_{\Aut(T)} \Sigma_{\{c_1,\cdots,c_n\}} \lrar \bigsqcup_{T \in \textbf{Tree}^{(k)}_{\ovl{c}}/\cong} f_T \otimes_{\Aut(T)} \Sigma_{\{c_1,\cdots,c_n\}}.
\end{equation}

Due to [[6], Lemma 5.10], we can show that $\varepsilon_T : f^{*}_T \lrar f_T$ is an $\Aut(T)$-cofibration between $\Aut(T)$-cofibrant objects. For this, we will need the assumption that $f$ is a cofibration between cofibrant objects, and that $\Env(\sN\P , A)$ is well-pointed by Remark <ref>. (See also the proof of [[6], Lemma 5.9]). Now, the $\Aut(T)$-cofibrancy of $\varepsilon_T$ implies that $H$ is a cofibration as well. So the map (<ref>) is in fact a homotopy cobase change of $H$.

On other hand, by applying $\L_\P$ to (<ref>) and noting that the map $\L_\P((\sN\P) \circ f)$ is isomorphic to the free cofibration $ \P \circ (\Gamma f) : \P \circ (\Gamma X) \lrar \P \circ (\Gamma Y)$ (cf. Lemma <ref>), we get that the map $\L_\P A \lrar \L_\P B$ is a cobase change of $\P \circ (\Gamma f)$. As in the above paragraph, $\Env(\P , \L_\P B) (\ovl{c}) $ is given by the sequential colimit of maps

$$ \Env(\P , \L_\P B)^{(k)}(\ovl{c}) \lrar \Env(\P , \L_\P B)^{(k+1)}(\ovl{c})$$
which are a homotopy cobase change of maps of the form

\begin{equation}\label{eq:freeext5'}
\widetilde{H} : \bigsqcup_{T \in \textbf{Tree}^{(k)}_{\ovl{c}}/\cong}  (\Gamma f)^{*}_T \otimes_{\Aut(T)} \Sigma_{\{c_1,\cdots,c_n\}} \lrar \bigsqcup_{T \in \textbf{Tree}^{(k)}_{\ovl{c}}/\cong} (\Gamma f)_T \otimes_{\Aut(T)} \Sigma_{\{c_1,\cdots,c_n\}}.
\end{equation}

As argued in the first paragraph, we have to prove that for any sequence $\ovl{c}$ the map
$$\Psi_{B}(\ovl{c}) : \Gamma\Env(\sN\P , B) (\ovl{c}) \lrar \Env(\P , \L_\P B) (\ovl{c})$$
is a weak equivalence, provided that $\Psi_{A}(\ovl{c})$ is one. The map $\Psi_{B}(\ovl{c})$ is compatible with the filtrations of $\Env(\sN\P , B) (\ovl{c})$ and $\Env(\P , \L_\P B) (\ovl{c})$ mentioned in the two paragraphs above. Due to this, we may deduce the lemma after verifying that the map
$$ \Psi_{B}^{(k)}(\ovl{c}) : \Gamma\Env(\sN\P , B)^{(k)} (\ovl{c}) \lrar \Env(\P , \L_\P B)^{(k)} (\ovl{c}) $$

is a weak equivalence for every $k\in\mathbb{N}$. We will proceed by an inductive argument. Firstly, by construction the map $\Psi_{B}^{(0)}(\ovl{c})$ agrees with $\Psi_{A}(\ovl{c})$, which is assumed to be a weak equivalence. For the inductive step, observe first that we have a  commutative cube-diagram

[row sep=1.0em, column sep = 1.0em]
(ΓH) [rr] [dr] [dd,swap]
(ΓH) [dd,dashed, swap] [dr]

Γ(, B)^(k) (c) [rr] [dd]
Γ(, B)^(k+1) [dd]

(H) [rr,dashed, swap] [dr]        (H) [dr,dashed, swap]

(, Ł_B)^(k)(c) [rr]        (, Ł_B)^(k+1)(c)

such that the top and bottom squares are homotopy pushouts, in which $\dom(-)$ (resp. $\codom(-)$) signifies the domain (resp. codomain) of maps, and $\Gamma H$ is the image through $\Gamma$ of the map $H$ (<ref>). Therefore, to complete the inductive step, it will suffice to show that the two maps

\begin{equation}\label{eq:twomaps}
\dom(\Gamma H) \lrar \dom(\widetilde{H})  \; \; \text{and} \; \; \codom(\Gamma H) \lrar \codom(\widetilde{H})
\end{equation}

are weak equivalences. Due to the fact that the objects $f^{*}_T$, $(\Gamma f)^{*}_T$, $f_T$ and $(\Gamma f)_T$ are all $\Aut(T)$-cofibrant (see the second paragraph), we just need to verify the following:

($\star$) For every $k \in \mathbb{N}$, every sequence $\ovl{c} := (c_1,\cdots,c_n;c)$ and for every $T \in \textbf{Tree}^{(k)}_{\ovl{c}}$, the canonical maps
$$\Gamma(f^{*}_T) \lrar (\Gamma f)^{*}_T \; \; \text{and} \; \; \Gamma(f_T) \lrar (\Gamma f)_T$$
are weak equivalences.

Recall by construction that $f^{*}_T$ and $f_T$ are respectively the domain and codomain of the map $\varepsilon_T$, which is inductively given by

$$ \varepsilon_T = \varepsilon_{r_T} \square \varepsilon_{T_1} \square \cdots \square \varepsilon_{T_r}  $$

where $\varepsilon_{r_T}$ is defined as

$$    \varepsilon_{r_T} :=  \begin{cases}
f(val(r_T)) & \text{ if } \, r_T \, \text{is marked} \\
\eta_{\Env(\sN\P , A)}(val(r_T)) & \text{ if } \, r_T \, \text{is unmarked.}
\end{cases} $$

Similarly, $(\Gamma f)^{*}_T$ and $(\Gamma f)_T$ are respectively the domain and codomain of the map

$$ \widetilde{\varepsilon}_T := \widetilde{\varepsilon}_{r_T} \square \widetilde{\varepsilon}_{T_1} \square \cdots \square \widetilde{\varepsilon}_{T_r}  $$

where $\widetilde{\varepsilon}_{r_T}$ is given by

$$    \widetilde{\varepsilon}_{r_T} :=  \begin{cases}
\Gamma f(val(r_T)) & \text{ if } \, r_T \, \text{is marked} \\
\eta_{\Env(\P , \L_\P A)}(val(r_T)) & \text{ if } \, r_T \, \text{is unmarked.}
\end{cases} $$

We shall now prove the ($\star$) in inductive style (again). Firstly, in the case where $T$ is the tree $``d \, |$" (i.e. with no vertex and just one edge colored by some $d\in C$), the claim is verified because the map $\Gamma(f^{*}_T) \lrar (\Gamma f)^{*}_T$ agrees with $\Id_0$ (i.e. the identity on zero) and the map $\Gamma(f_T) \lrar (\Gamma f)_T$ agrees with $\Psi_A(d)$, which is a weak equivalence by assumption. For the remainder, we consider only trees that contain at least one vertex. We shall end the proof after the three steps below.

Step 1. First, assume  that $T$ is an elementary tree, by which we mean a planar tree with the root vertex as its unique vertex.  When $r_T$ is unmarked, then the map $\Gamma(f^{*}_T) \lrar (\Gamma f)^{*}_T$ agrees with $\Id_\textbf{k}$ if $val(r_T) = (c;c)$ and agrees with $\Id_0$ otherwise; while the map $\Gamma(f_T) \lrar (\Gamma f)_T$ agrees with $\Psi_A(val(r_T))$ (and thus, is a weak equivalence). Conversely, when $r_T$ is marked, then the map $\Gamma(f^{*}_T) \lrar (\Gamma f)^{*}_T$ is simply given by $\Id_{\Gamma X(val(r_T))}$ and the map $\Gamma(f_T) \lrar (\Gamma f) _T$ coincides with $\Id_{\Gamma Y(val(r_T))}$. In summary, we just showed that ($\star$) holds for every elementary tree.

Step 2. Suppose now that $T$ is not elementary. In particular, $T$ has at least one internal edge $e$. Decomposing $T$ at this edge generates two subtrees $T_1$ and $T_2$ such that the color assigned to $e$ is saved in both of them, and the root vertex of $T_1$ coincides with that of $T$. We also write $T=T_1\circ T_2$ standing for such a decomposition. The goal of this step is to show that if the ($\star$) holds for both $T_1$ and $T_2$, then it holds for $T$ as well. To this end, observe first that $\varepsilon_T = \varepsilon_{T_1} \square \varepsilon_{T_2}$ and $\widetilde{\varepsilon}_T = \widetilde{\varepsilon}_{T_1} \square \widetilde{\varepsilon}_{T_2}$. So the map $\Gamma(f_T) \lrar (\Gamma f) _T$ is isomorphic to the map

$$ \Gamma(f_{T_1} \otimes f_{T_2}) \lrar (\Gamma f)_{T_1} \otimes (\Gamma f)_{T_2}, $$

which agrees with the composed map

$$ \Gamma(f_{T_1} \otimes f_{T_2}) \lrar \Gamma(f_{T_1}) \otimes \Gamma(f_{T_2}) \lrar (\Gamma f)_{T_1} \otimes (\Gamma f)_{T_2} .$$

The first map is an oplax monoidal map of $\Gamma$ and hence, is a weak equivalence. Thus, $\Gamma(f_T) \lrar (\Gamma f) _T$ is a weak equivalence as soon as  $\Gamma(f_{T_i})  \lrar (\Gamma f)_{T_i}$ is one for $i=1,2$. On the other hand, the map $\Gamma(f^{*}_T) \lrar (\Gamma f)^{*}_T$ is isomorphic to the map between pushouts:

$$ \Gamma(  f^{*}_{T_1} \otimes f_{T_2} ) \bigsqcup_{ \Gamma( f^{*}_{T_1}  \otimes f^{*}_{T_2} )} \Gamma( f_{T_1} \otimes f^{*}_{T_2}  ) \; \lrar \;   (\Gamma f)^{*}_{T_1} \otimes (\Gamma f)_{T_2} \bigsqcup_{(\Gamma f)^{*}_{T_1} \otimes (\Gamma f)^{*}_{T_2}} (\Gamma f)_{T_1} \otimes (\Gamma f)^{*}_{T_2}.$$

Since both $\varepsilon_{T_i}$ and $\widetilde{\varepsilon}_{T_i}$ are a cofibration between cofibrant objects (see the second paragraph), it implies that each of the two pushouts represents the corresponding homotopy pushout. Therefore, by the same argument as above, $\Gamma(f^{*}_T) \lrar (\Gamma f)^{*}_T$ is a weak equivalence as soon as the maps $\Gamma(f_{T_i}) \lrar (\Gamma f)_{T_i}$ and $\Gamma(f^{*}_{T_i}) \lrar (\Gamma f)^{*}_{T_i}$  are all weak equivalences for $i=1,2$.

By convention, we will say that a decomposition $T = T_1 \circ T_2$ is elementary if both $T_1$ and $T_2$ are so.

Step 3. Finally, let $T$ be any tree. If $T$ is elementary then we are done, as argued the first step. Otherwise, we take a decomposition $T = T_1 \circ T_2$ of $T$ as in the second step. If this decomposition is elementary then we get the claim by combining the first and second steps. Otherwise, we decompose $T_i$ at any internal edge (if this subtree is not elementary). But this machinery must be stable at elementary decompositions. Hence, we may deduce the ($\star$) again combining the two steps above.

Let $\rho : \Q \lrarsimeq \P$ be a cofibrant resolution of $\P\in\Op_C(\sMod_{\textbf{k}})$. Note that since $\Q$ is cofibrant, it is automatically well-pointed; and hence, so is the operad $\sN\Q$. In particular, the map $\sN\rho : \sN\Q \lrarsimeq \sN\P$ is a weak equivalence between $\Sigma$-cofibrant operads. According to [[17], Theorem 12.5.A], the map $\sN\rho$ gives rise to a Quillen equivalence between $\sN\Q$-algebras and $\sN\P$-algebras, and the same statement holds for the map $\rho$. These observations prove that if the theorem holds for $\Q$ then it holds for $\P$ as well. Thus we may assume without of loss generality that $\P$ is well-pointed.

Since the functor $\sN_\P$ creates fibrations and weak equivalences, it implies that the adjunction $\L_\P \dashv \sN_\P$ indeed forms a Quillen adjunction. In order to prove further $\L_\P \dashv \sN_\P$ is a Quillen equivalence, we just need to verify that for every cofibrant $\sN\P$-algebra $A$ the unit map $\eta_A : A \lrar \sN_\P \L_\P (A)$ is a weak equivalence (see Observation <ref>). Considering $\eta_A$ as a map in $\C_{\geqslant0}(\textbf{k})^{\times \, C}$, it suffices to verify that the adjoint of $\eta_A$,

$$ \eta^{*}_A : \Gamma A   \lrar  \L_\P A, $$

is a weak equivalence in $\sMod_{\textbf{k}}^{\times \, C}$. As in the above proof, we can assume that $A$ is a cellular $\sN\P$-algebra and then, argue by induction on the filtration of $A$. Observe first that the map $\eta^{*}_{\sN\P_0}$ coincides with the counit map $\Gamma\sN\P_0 \lrar \P_0$, which is an isomorphism. Hence, it remains to show that for any pushout square in $\Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$ of the form

\begin{equation}\label{eq:freeext1}
\xymatrix{
(\sN\P) \circ X \ar^{(\sN\P) \circ f}[r]\ar[d] & (\sN\P) \circ Y \ar[d] \\
A \ar[r] & B \\
\end{equation}

such that the top horizontal map is given by the image of a given cofibration $f:X \lrar Y$ in $\C_{\geqslant0}(\textbf{k})^{\, \times C}$ through the free $\sN\P$-algebra functor, the map $\eta^{*}_{B}$ is a weak equivalence when provided that $\eta^{*}_{A}$ is one. As in the previous proof, we may assume that $X$ and $Y$ are cofibrant objects in $\C_{\geqslant0}(\textbf{k})^{\, \times C}$.

For simplicity of equations, in the remainder, we will assume that $\P$ is a single-colored operad. According to [[7], Proposition 5.7], the underlying dg $\textbf{k}$-module of $B$ is given by the sequential colimit of maps $B^{(n-1)} \lrar B^{(n)}$ with $B_0 = A$ and such that each map $B^{(n-1)} \lrar B^{(n)}$ is a cobase change of the map
$$\varphi_n := \Env(\sN\P , A) (n) \otimes_{\Sigma_n} f^{\square \, n}$$
with $f^{\square \, n}$ being the $n$-fold pushout-product of $f$. Note first that the map $f^{\square \, n}$ is a cofibration due to that property of $f$. Furthermore, by the assumption that $\P$ is $\Sigma$-cofibrant and that $A\in\Alg_{\sN\P}(\C_{\geqslant0}(\textbf{k}))$ is cofibrant, it implies that $\Env(\sN\P , A)$ is $\Sigma$-cofibrant as well (cf. [[7], Lemma 6.1(ii)]). So we get that $\varphi_n$ is a cofibration. In particular, the map $B^{(n-1)} \lrar B^{(n)}$ is in fact a homotopy cobase change of  $\varphi_n$. Moreover, by extracting the iterated pushout-product $f^{\square \, n}$, the map $\varphi_n$ is of the form

$$ \varphi_n : \Env(\sN\P , A) (n) \otimes_{\Sigma_n} Q_n(X,Y) \lrar \Env(\sN\P , A) (n) \otimes_{\Sigma_n} Y^{\otimes \, n} $$

in which $Q_n(X,Y)$ is the domain of $f^{\square \, n}$. We may regard $Q_n(X,Y)$ as the colimit of a punctured $n$-cube (i.e. the $n$-cube with the terminal vertex removed) $W_n(X,Y)$ whose vertices are of the form $C_1 \otimes C_2 \otimes \cdots \otimes C_n$ such that each $C_i$ is either $X$ or $Y$, and such that each edge is given by an iterated tensor product of maps of the types: $f$, $\Id_X$ and $\Id_Y$. In fact, $Q_n(X,Y)$ is a model for the homotopy colimit of $W_n(X,Y)$, due to the pushout-product axiom again.

On other hand, by applying the left adjoint $\L_\P$ to (<ref>), we obtain a pushout square in $\Alg_\P(\sMod_{\textbf{k}})$:

\begin{equation}\label{eq:freeext2}
\xymatrix{
\L_\P((\sN\P) \circ X) \ar[r]\ar[d] & \L_\P((\sN\P) \circ Y) \ar[d] \\
\L_\P A \ar[r] & \L_\P B \\
\end{equation}

By Lemma <ref>, the top horizontal map is canonically isomorphic to the free cofibration $ \P \circ (\Gamma f) : \P \circ (\Gamma X) \lrar \P \circ (\Gamma Y)$. As in the above paragraph, we get that the underlying simplicial $\textbf{k}$-module of $\L_\P B$ is given by the sequential colimit of maps $(\L_\P B)^{(n-1)} \lrar (\L_\P B)^{(n)}$ with $(\L_\P B)^{(0)} = \L_\P A$ and such that each map $(\L_\P B)^{(n-1)} \lrar (\L_\P B)^{(n)}$ is a homotopy cobase change of a map of the form

$$ \widetilde{\varphi}_n : \Env(\P , \L_\P A) (n) \otimes_{\Sigma_n} Q_n(\Gamma X,\Gamma Y) \lrar \Env(\P , \L_\P A) (n) \otimes_{\Sigma_n} (\Gamma Y)^{\otimes \, n} $$

such that $Q_n(\Gamma X,\Gamma Y) \in \sMod_{\textbf{k}}$ is a homotopy colimit of the punctured $n$-cube $W_n(\Gamma X,\Gamma Y)$.

The map $\eta^{*}_B : \Gamma B   \lrar  \L_\P B$ is compatible with the filtrations of $B$ and $\L_\P B$ constructed in the two above paragraphs. Thus, to show that  $\eta^{*}_{B}$ is a weak equivalence, it suffices to verify that the component map
$$(\eta^{*}_{B})_{n} : \Gamma \, B^{(n)}  \lrar  (\L_\P B)^{(n)}$$
is one for every $n$. We will prove the latter in inductive style, again. Notice first that when $n=0$, the map $(\eta^{*}_{B})_0$ coincides with  $\eta^{*}_A$, which was assumed to be a weak equivalence. Now assume by induction that $(\eta^{*}_{B})_{n-1}$ is a weak equivalence. According to the two paragraphs above, the object $\Gamma \, B^{(n)}$ (resp. $(\L_\P B)^{(n)}$) is a homotopy pushout of  $\Gamma \, B^{(n-1)}$ (resp. $(\L_\P B)^{(n-1)}$) along $\Gamma(\varphi_n)$ (resp. $\widetilde{\varphi}_n$). Therefore, the proof will be completed after showing that the two canonical maps:

$$ \Gamma(\Env(\sN\P , A) (n) \otimes_{\Sigma_n} Q_n(X,Y)) \x{\Theta_n}{\xrightarrow{\hspace*{0.8cm}}} \Env(\P , \L_\P A) (n) \otimes_{\Sigma_n} Q_n(\Gamma X,\Gamma Y) $$

$$ \text{and} \; \; \Gamma(\Env(\sN\P , A) (n) \otimes_{\Sigma_n} Y^{\otimes \, n}) \x{\Phi_n}{\xrightarrow{\hspace*{0.8cm}}} \Env(\P , \L_\P A) (n) \otimes_{\Sigma_n} (\Gamma Y)^{\otimes \, n} $$

are a weak equivalence.

$\bullet$  Firstly, $\Theta_n$ is given by taking the composition of two maps:

$$ \Gamma \, \left(\Env(\sN\P , A)(n) \otimes_{\Sigma_n} Q_n(X,Y)\right) \lrar \Gamma\Env(\sN\P , A)(n) \otimes_{\Sigma_n} \Gamma \, Q_n(X,Y)$$

$$ \text{and} \; \; \Gamma\Env(\sN\P , A)(n) \otimes_{\Sigma_n} \Gamma \, Q_n(X,Y) \lrar \Env(\P , \rL_\P A) (n) \otimes_{\Sigma_n} Q_n(\Gamma X,\Gamma Y).$$

The first one is a canonical map of the type (<ref>), and hence is a weak equivalence. Whilst the second map is given by the tensor product $\Psi_A(n)\otimes_{\Sigma_n}h_{X,Y}$ in which

$$\Psi_A(n) : \Gamma\Env(\sN\P , A) (n) \lrar \Env(\P , \rL_\P A) (n)$$

is the map of Lemma <ref>, which is a weak equivalence as indicated there, and

$$ h_{X,Y} : \Gamma \, Q_n(X,Y) \lrar Q_n(\Gamma X,\Gamma Y)$$

is a canonical map induced by $f$ and the oplax monoidal structure of $\Gamma$. Note that both the domain and codomain of $h_{X,Y}$ (resp. $\Psi_A(n)$) are cofibrant (resp. $\Sigma_n$-cofibrant). Thus, $\Psi_A(n)\otimes_{\Sigma_n}h_{X,Y}$ will be a weak equivalence as soon as $h_{X,Y}$ is one. We shall now verify the latter. As discussed in the second paragraph, $Q_n(-,-)$ is a homotopy colimit of $W_n(-,-)$ and moreover, the restriction of $h_{X,Y}$ to each vertex of the punctured $n$-cube diagram  is an (iterated) oplax monoidal map of $\Gamma$ of the form
$$ \Gamma(C_1 \otimes C_2 \otimes \cdots \otimes C_n) \lrar \Gamma C_1 \otimes \Gamma C_2 \otimes \cdots \otimes \Gamma C_n $$

such that each $C_i$  is either $X$ or $Y$. The latter is certainly a weak equivalence and hence, $h_{X,Y}$ is indeed a weak equivalence.

$\bullet$ The map $\Phi_n$ can also be shown to be a weak equivalence using a similar argument as in the case above.

In light of Theorem <ref>, we obtain some significant corollaries as follows.

Let $\P$ be a $\Sigma$-cofibrant operad in $\sMod_{\textbf{k}}$. Then the functor

$$ \sN^{b}_\P : \BMod(\P) \lrar \BMod(\sN\P) $$

defined by applying the normalization functor levelwise is the right adjoint of a Quillen equivalence.
Recall from [[14], $\S$2.2] that we can associate to $\P$ an operad, denoted $\textbf{B}^\P$, that encodes the category of $\P$-bimodules, i.e. there is an isomorphism:
$$\Alg_{\textbf{B}^\P}(\sMod_{\textbf{k}}) \cong \BMod(\P).$$
Additionally, $\textbf{B}^\P$ is $\Sigma$-cofibrant due to that property of $\P$. Applying Theorem <ref> to $\textbf{B}^\P$ gives us a right Quillen equivalence:

$$ \Alg_{\textbf{B}^\P}(\sMod_{\textbf{k}}) \lrarsimeq \Alg_{\sN\textbf{B}^\P}(\C_{\geqslant0}(\textbf{k})).$$

On other hand, the shuffle map $\nabla$ induces a map $\nabla^{*} : \textbf{B}^{\sN\P} \lrar \sN\textbf{B}^\P$ of operads in $\C_{\geqslant0}(\textbf{k})$. It is clear that $\nabla^{*}$ is a weak equivalence, and besides that, $\nabla^{*}$ goes between $\Sigma$-cofibrant operads. By [[17], Theorem 12.5.A], we obtain another right  Quillen equivalence:

$$ \Alg_{\sN\textbf{B}^\P}(\C_{\geqslant0}(\textbf{k})) \lrarsimeq \Alg_{\textbf{B}^{\sN\P}}(\C_{\geqslant0}(\textbf{k})).$$

We complete the proof by observing that the composition of the two right Quillen equivalences obtained above is exactly the functor $\sN^{b}_\P$.

In [[14], $\S$2.2], we also associate to each enriched operad an enriched category which encodes the category of infinitesimal bimodules over that operad. Hence, the same machinery as in the corollary above can be applied to obtain the following.

Let $\P$ be a levelwise cofibrant operad in $\sMod_{\textbf{k}}$. Then the functor $\sN^{ib}_\P : \IbMod(\P) \lrar \IbMod(\sN\P)$ defined by applying the normalization functor levelwise is a right Quillen equivalence.

According to the loc.cit again,  the category of left (resp. right) $\P$-modules can be as well represented as the category of algebras (resp. enriched functors) over a certain operad (resp. category). Due to this, we can prove an analogue of the two corollaries above for left (right) $\P$-modules.

Here is the corollary that is of most interest.

The normalization functor induces a right Quillen equivalence

\begin{equation}\label{eq:opnormalized1}
\sN_C : \Op_C(\sMod_{\textbf{k}}) \lrarsimeq \Op_C(\C_{\geqslant0}(\textbf{k}))
\end{equation}

between $C$-colored operads.
Consider the operad $\textbf{O}_C$ that encodes the category of $C$-colored operads in some symmetric monoidal category (see Remark <ref>). Note that $\textbf{O}_C$ comes from an operad in $\Sets$, and equipped with a free-action by the symmetric groups. In particular, $\textbf{O}_C$ is $\Sigma$-cofibrant when considered in any base category as long as the monoidal unit is cofibrant. Applying Theorem <ref> to this operad, we obtain a right Quillen equivalence:

$$ \sN_{\textbf{O}_C} : \Alg_{\textbf{O}_C}(\sMod_{\textbf{k}}) \lrarsimeq \Alg_{\sN\textbf{O}_C}(\C_{\geqslant0}(\textbf{k})). $$

Now, since $\textbf{O}_C$ is discrete, it implies that $\sN\textbf{O}_C$ coincides with $\textbf{O}_C$ itself considered as an operad in $\C_{\geqslant0}(\textbf{k})$. So the two functors $\sN_C$ and $\sN_{\textbf{O}_C}$ are isomorphic.

We let $\L_C := \L_{\textbf{O}_C}$ denote the left adjoint to $\sN_C$. An explicit description of $\L_C$ is based on Construction <ref>.

The above result leads to another consequence as follows.

Let $\O\in\Op_C(\C_{\geqslant0}(\textbf{k})$ be a cofibrant $C$-colored operad in $\C_{\geqslant0}(\textbf{k})$. Then there exists a Quillen equivalence

\begin{equation}\label{eq:opnormalized2}
\adjunction*{\simeq}{\Alg_{\O}(\C_{\geqslant0}(\textbf{k}))}{\Alg_{\L_C\O}(\sMod_{\textbf{k}})}{}
\end{equation}

between $\O$-algebras and $\L_C\O$-algebras.
First, argue that the unit map $u_{\O} : \O\lrar\sN_C\L_C(\O)$ is a weak equivalence. Indeed, this follows from the fact that $\L_C\dashv\sN_C$ is a Quillen equivalence, that $\sN_C$ creates weak equivalences, and that $\O$ is cofibrant by assumption. We thus obtain a Quillen equivalence induced by $u_{\O}$:

$$ (u_{\O})_! : \adjunction*{\simeq}{\Alg_{\O}(\C_{\geqslant0}(\textbf{k}))}{\Alg_{\sN_C\L_C(\O)}(\C_{\geqslant0}(\textbf{k}))}{} : u_{\O}^*. $$

Moreover, by Theorem <ref> we get another Quillen equivalence

$$ \adjunction*{\simeq}{\Alg_{\sN_C\L_C(\O)}(\C_{\geqslant0}(\textbf{k}))}{\Alg_{\L_C\O}(\sMod_{\textbf{k}})}{}.$$

Finally, the expected Quillen equivalence (<ref>) is given by the composition of the two above.

Using a similar argument, and along with Corollary <ref>, we obtain the following.
Let $\O\in\Op_C(\C_{\geqslant0}(\textbf{k})$ be a cofibrant $C$-colored operad in $\C_{\geqslant0}(\textbf{k})$. Then there is a Quillen equivalence

\begin{equation}\label{eq:opnormalized3}
\adjunction*{\simeq}{\BMod(\O)}{\BMod(\L_C\O)}{}
\end{equation}

between $\O$-bimodules and $\L_C\O$-bimodules.

An analogue of this corollary holds for infinitesimal bimodules and for lelft (right) modules as well.

§.§ Dold-Kan correspondence of enriched operads

In this subsection, we shall give a proof of the statement <ref>. To this end, we first need to show that applying the normalization functor levelwise determines a right adjoint functor $\Op(\sMod_{\textbf{k}}) \lrar \Op(\C_{\geqslant0}(\textbf{k}))$.

For the latter, it is needed to understand how the Grothendieck construction is compatible with adjunctions. Let $\F : \C^{\op} \lrar \Cat$ be a contravariant functor on a given category $\C$, and valued in the category of small categories $\Cat$. For each map $\alpha : c \rar d$ in $\C$, we will write $\alpha^{*} : \F(d) \lrar \F(c)$ standing for the functor corresponding to $\alpha$ through $\F$. As usual, we denote by $\int_{\C}\F$ the (contravariant) Grothendieck construction of $\F$.

Suppose we are given two functors $\F,\G : \C^{\op} \lrar \Cat$ and suppose that for every object $c\in\Ob(\C)$, there is an adjunction

$$ \rL_c : \adjunction*{}{\F(c)}{\G(c)}{} : \rR_c $$

such that the collection of right adjoints $\{\rR_c\}_{c\in\C}$ forms a natural transformation from $\G$ to $\F$. We establish two  integrated functors

$$ \rL : \int_{\C}\F \lrar \int_{\C}\G \; \; \text{and} \; \; \rR : \int_{\C}\G \lrar \int_{\C}\F $$

as follows.

$\bullet$ Recall that an object of $\int_{\C}\F$ is the choice of a pair $(c, X)$ with $c\in\Ob(\C)$ and with $X$ being an object of $\F(c)$. The functor $\rL$ is given on  objects by sending $(c, X)$ to the pair $(c, \rL_c X)$. The functor $\rR$ is defined on objects in a similar way.

$\bullet$ A map $(c,A) \lrar (d,B)$ in $\int_{\C}\G$ is by definition the choice of a pair $(\alpha,f)$ with $\alpha : c \rar d$ being a map in $\C$ and  $f : A \lrar \alpha^{*}B$ a map in $\G(c)$. We define $\rR (\alpha,f)$ to be the pair
$$\left( \alpha , \; \rR_c (A) \x{\rR_c (f)}{\xrightarrow{\hspace*{1cm}}} \rR_c \alpha^{*}(B) \x{\cong}{\lrar} \alpha^{*}\rR_d (B) \right).$$

(The latter isomorphism is due to the assumption that $\{\rR_c\}_{c\in\C}$ forms a natural transformation from $\G$ to $\F$). Now let $(\beta, g) : (c,X) \lrar (d,Y)$ be a map in $\int_{\C}\F$. We define $\rL (\beta, g)$ to be the pair $(\beta, \widetilde{g})$ in which $\widetilde{g}$ is given by the adjoint of the composed map

$$ X \x{g}{\lrar} \beta^{*}Y \x{\beta^{*}(u_Y)}{\xrightarrow{\hspace*{1cm}}} \beta^{*} \rR_d \rL_d (Y) \x{\cong}{\lrar} \rR_c \beta^{*} \rL_d (Y) $$

where $u_Y$ signifies a unit map of the adjunction $\rL_d\dashv \rR_d$.

The main interest in this construction is as follows.

The obtained integrated functors form an adjunction between the Grothendieck constructions:

$$ \rL : \adjunction*{}{\int_{\C}\F \; }{ \; \int_{\C}\G}{} : \rR \, .$$

The proof is straightforward using the definition and basic properties of adjoint functors.

Let us now apply Construction <ref> to the collection of adjunctions

$$ \{\L_C : \adjunction*{}{\Op_C(\C_{\geqslant0}(\textbf{k}))}{\Op_C(\sMod_{\textbf{k}})}{} : \sN_C \}_{C\in \Sets}.$$

Here we note that since the functors $\sN_C$'s are given by applying the normalization functor levelwise, they indeed commute with the changing-colors functors. Thus by Proposition <ref>, we obtain an integrated adjunction between the categories of enriched operads:

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

The functor $\sN : \Op(\sMod_{\textbf{k}}) \lrar \Op(\C_{\geqslant0}(\textbf{k}))$ preserves and detects the Dwyer-Kan equivalences.
According to [[9], $\S$5], the Dwyer-Kan model structure on $\Cat(\sMod_{\textbf{k}})$ can be obtained by transferring that model structure on $\Cat(\C_{\geqslant0}(\textbf{k}))$ along the normalization functor. In particular, a map $f$ in $\Cat(\sMod_{\textbf{k}})$ is a Dwyer-Kan equivalence if and only if $\sN(f)$ is one. We complete the proof by combining the latter fact with Remark <ref>, and along with the fact that the functor $\sN$ preserves and detects the levelwise weak equivalences.

In order to state the main theorem, we will regard $\Op(\sMod_{\textbf{k}})$ and $\Op(\C_{\geqslant0}(\textbf{k}))$ as relative categories with weak equivalences being precisely the Dwyer-Kan equivalences. To avoid confusion, we will write $\Op'(\sMod_{\textbf{k}})$ and $\Op'(\C_{\geqslant0}(\textbf{k}))$ standing for the corresponding relative categories. Since $\sN$ preserves Dwyer-Kan equivalences, it determines a relative functor from $\Op'(\sMod_{\textbf{k}})$ to $\Op'(\C_{\geqslant0}(\textbf{k}))$.

The functor $\sN : \Op'(\sMod_{\textbf{k}}) \lrar  \Op'(\C_{\geqslant0}(\textbf{k}))$ is an equivalence of relative categories, meaning that it induces an equivalence between the underlying $\infty$-categories (or simplicial localizations).

The proof of this theorem will require a more technical lemma. Let $\O$ be a $C$-colored operad in $\C_{\geqslant0}(\textbf{k})$. We will say that $\O$ is relative cofibrant  if it is cofibrant as an object of $\Op_C(\C_{\geqslant0}(\textbf{k}))$ with respect to the transferred semi-model structure.

The functor $\L$ preserves Dwyer-Kan equivalences between relative cofibrant operads.
Let $f : \O \lrar \O'$ be any Dwyer-Kan equivalence between relative cofibrant objects. Since $\sN$ detects Dwyer-Kan equivalences (see Lemma <ref>), the proof will be completed after showing that the map $\sN\L(f) : \sN\L(\O) \lrar \sN\L(\O')$ is one. By construction, the unit map $\O  \lrar \sN\L(\O)$ agrees with the unit $\O  \lrar \sN_C\L_C(\O)$  of the adjunction $\L_C\dashv \sN_C$. By Corollary <ref>, the latter is a weak equivalence in $\Op_C(\C_{\geqslant0}(\textbf{k}))$ and in particular, a Dwyer-Kan equivalence. Thus by naturality, the two maps $\sN\L(f)$ and $f$ are related via a pair of Dwyer-Kan equivalences and hence, $\sN\L(f)$ is indeed a Dwyer-Kan equivalence.

When the Dwyer-Kan model structure on $\Op(\C_{\geqslant0}(\textbf{k}))$ is available, the class of relative cofibrant operads is precisely that of cofibrant objects in $\Op(\C_{\geqslant0}(\textbf{k}))$ (cf. [14], Observation 4.1.1(ii)). In this situation, the statement of Lemma <ref> becomes evident because $\L$ is then a left Quillen functor.

For each set $C$, there exists a canonical functorial cofibrant replacement $Q_C$ on the category $\Op_C(\C_{\geqslant0}(\textbf{k}))$ which is natural in $``C$". We can then integrate the collection $\{Q_C \, | \, C\in\Sets \}$ into a single functorial relative cofibrant replacement:
$$Q : \Op(\C_{\geqslant0}(\textbf{k})) \lrar \Op(\C_{\geqslant0}(\textbf{k})).$$

We denote by $\mathbb{L}\L := \L\circ Q$ and refer to it the relative left derived functor of $\L$.

Due to Lemma <ref>, the functor $\mathbb{L}\L$  determines a functor between relative categories:

$$ \mathbb{L}\L : \Op'(\C_{\geqslant0}(\textbf{k})) \lrar \Op'(\sMod_{\textbf{k}}).$$

It now suffices to argue that the composed functor $\mathbb{L}\L\circ\sN$ is weakly equivalent to the identity functor $\Id_{\Op'(\sMod_{\textbf{k}})}$, and that $\sN\circ \, \mathbb{L}\L$ is related to the corresponding identity via a chain of natural Dwyer-Kan equivalences. These follow from the fact that the (co)unit of $\L\dashv \sN$ is simply given by the (co)units of the adjunctions $\L_C \dashv \sN_C$ for  $C\in\Sets$, that the functor $\sN_C$ preserves and detects weak equivalences, and that $\L_C \dashv \sN_C$ is a Quillen equivalence for every $C$.

Assume that the Dwyer-Kan model structure on $\Op(\C_{\geqslant0}(\textbf{k}))$ exists. Then it automatically coincides with the canonical model structure, according to [[19], Proposition 4.25] and the fact that these two model structures have the same trivial fibrations. Therefore, a map in $\Op(\C_{\geqslant0}(\textbf{k}))$ is a Dwyer-Kan fibration if and only if it is a levelwise fibration such that its underlying map in $\Cat(\C_{\geqslant0}(\textbf{k}))$ is a Dwyer-Kan fibration (see also Remark <ref>). The same statement holds for Dwyer-Kan fibrations in $\Op(\sMod_{\textbf{k}})$ as well. We hence deduce that the functor $\sN$ preserves and detects the Dwyer-Kan fibrations, similarly as in Lemma <ref>.

Suppose that the Dwyer-Kan model structure on the category $\Op(\C_{\geqslant0}(\textbf{k}))$ exists. Then $\L\dashv\sN$ forms a Quillen equivalence.
First, the adjunction $\L\dashv\sN$ forms a Quillen adjunction by Lemma <ref> and by Remark <ref>. Theorem <ref> proves further that $\L\dashv\sN$ is a Quillen equivalence as expected.

§ MAPPING SPACES OF OPERADS IN RELATION TO OPERADIC BIMODULES

In this section, we will let $\cS$ be a fixed symmetric monoidal category and let $\P$ be a $C$-colored operad in $\cS$. To avoid confusion, we will write

$$ \sqcup \; , \; \bigsqcup \; \text{and} \; \coprod $$

standing for the coproduct operations (and also, pushouts) in the categories $\BMod(\mathcal{P})$, $\Op_C(\mathcal{S})$ and $\Op(\mathcal{S})$, respectively.

§.§ The first fiber sequence

First, let us consider the case where the set of colors is fixed.

We are recalling a hypothesis introduced by Dwyer-Hess, that has played a very active role in their work [12]. Recall by definition that a pointed $\P$-bimodule is a $\P$-bimodule $M$ endowed with a map $\I_C \lrar M$ of $C$-collections. We will let $\BMod(\mathcal{P})^{*}$ denote the category of such objects with morphisms being the basepoint-preserving maps.

Note that $\mathcal{P}$ itself is an object of $\BMod(\mathcal{P})^{*}$ with the base point given by the unit of $\P$. Notice also that $\BMod(\mathcal{P})^{*}$ agrees with the under-category $\BMod(\mathcal{P})_{\mathcal{P}\circ \mathcal{P}/}$ where $\mathcal{P}\circ \mathcal{P}$ represents the free $\P$-bimodule generated by $\mathcal{I}_C$.

Let $f + g : \mathcal{P}\bigsqcup\mathcal{P} \lrar \Q$ be a map in $\Op_C(\mathcal{S})$. Then $\Q$ inherits a $\P$-bimodule structure with the left (resp. right) $\P$-action induced by $f$ (resp. $g$).  From this, we obtain a restriction functor
$$\Op_C(\mathcal{S})_{\mathcal{P}\bigsqcup\mathcal{P}/} \lrar \BMod(\mathcal{P})^{*} ,$$
which admits a left adjoint, denoted by $\E$.  Observe that the functor $\E$ sends $\P$ to itself $\P\in \Op_C(\mathcal{S})_{\mathcal{P}\bigsqcup\mathcal{P}/}$ equipped with the fold map $\Id+\Id : \mathcal{P}\bigsqcup\mathcal{P} \lrar \P$.

We suppose further that $\cS$ is a  symmetric monoidal model category satisfying the assumptions as in Theorem <ref>.

(Goodness hypothesis)
The operad $\P\in \Op_C(\mathcal{S})$ is said to be good if the left derived  functor  $\LL\E:\BMod(\mathcal{P})^{*}\lrar \Op_C(\mathcal{S})_{\mathcal{P}\bigsqcup\mathcal{P}/}$ sends $\P$ to itself $\mathcal{P}$. Moreover, the base category $\cS$ is good if every cofibrant operad in $\cS$ is good.

Considering operads as monoid objects in the monoidal category of $C$-collections, we can see that the goodness hypothesis is a particular case of what Dwyer and Hess refer to as the stability of distinguished objects with respect to the functor $\E$ (cf. [[12], $\S$3.4]).

We give the following statement as an illustration of how that hypothesis can be applied in practice.

Suppose that $\P\in \Op_C(\mathcal{S})$ is a good cofibrant operad and suppose given a map $f : \P \lrar \O$ in $\Op_C(\mathcal{S})$. We also regard $\O$ as a $\P$-bimodule with the structure induced by $f$. Then there is a fiber sequence of spaces

\begin{equation}\label{eq:goodnessmain}
\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \O) \lrar \Map^{\h}_{\BMod(\P)}(\P, \O) \lrar \underset{c\in C}{\prod}\Map^{\h}_{\cS}(1_{\cS}, \O(c;c))
\end{equation}

in which the loop space $\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \O)$ is taken at the base point $f$.

This theorem is in fact a version of Theorem 3.11 of the loc.cit. Nonetheless, we shall provide an alternative proof that is somewhat more direct.

First, observe that there is a commutative square of left Quillen functors

\begin{equation}\label{eq:goodness}
\xymatrix{
\BMod(\mathcal{P})_{\mathcal{P}\circ \mathcal{P}/} \ar^{\mu_!}[r]\ar_{\E}[d] & \BMod(\mathcal{P})_{\P/} \ar^{\ovl{\E}}[d] \\
\Op_C(\mathcal{S})_{\mathcal{P}\bigsqcup\mathcal{P}/} \ar_{\; \; (\Id+\Id)_!}[r] & \Op_C(\mathcal{S})_{\P/} \\
\end{equation}

where $\ovl{\E}$ denotes the left adjoint to the corresponding restriction functor, the functor $\mu_!$ is induced by the composition $\mu : \mathcal{P}\circ \mathcal{P} \lrar \P$ regarded as a map in  $\BMod(\mathcal{P})$, and $(\Id+\Id)_!$ refers to the functor induced by the fold map $\mathcal{P}\bigsqcup\mathcal{P} \lrar \P$. To verify the commutativity, it suffices to observe that the associated square of right adjoints is commutative.

Let us take a factorization $\mathcal{P}\circ \mathcal{P} \x{\varphi}{\lrar} \P^{b} \lrarsimeq \mathcal{P}$ in $\BMod(\mathcal{P})$ of $\mu$ into a cofibration followed by a weak equivalence. In particular, $\varphi$ exhibits $\P^{b}$ as a cofibrant model for $\P\in\BMod(\mathcal{P})^{*}$. Since $\P$ is good and cofibrant, it implies that the derived image of $\P^{b}\in\BMod(\mathcal{P})^{*}$ through the composed functor $(\Id+\Id)_!\circ\E$ is a model for  $\Sigma(\mathcal{P}\bigsqcup\mathcal{P})$, i.e. the homotopy pushout of the fold map $\mathcal{P}\bigsqcup\mathcal{P} \lrar \P$ with itself. In other direction, the derived image of $\P^{b}$ under $\mu_!$ is the pushout $\P  \underset{\P\circ\P}{\sqcup}  \P^b =: B_\P$. Combined with the commutativity of (<ref>), those observations prove that the derived image of $B_\P \in \BMod(\mathcal{P})_{\P/}$ through $\ovl{\E}$ is given by $\Sigma(\mathcal{P}\bigsqcup\mathcal{P}) \in \Op_C(\mathcal{S})_{\P/}$.

Let us now consider the following coCartesian square in $\BMod(\mathcal{P})$:

\begin{equation}\label{eq:goodness1}
\xymatrix{
\P \sqcup (\P\circ\P)  \ar^{ \; \; \Id\sqcup\varphi}[r]\ar_{\Id+\mu}[d] & \P \sqcup \P^{b} \ar[d] \\
\P \ar[r] & B_\P \\

Note that, when considered as a coCartesian square in $\BMod(\mathcal{P})_{\P/}$, this square is in fact homotopy coCartesian due to the fact that $\varphi : \P\circ\P \lrar \P^{b}$ is a cofibration between cofibrant $\P$-bimodules. Consider $\O$ as an object of $\BMod(\mathcal{P})_{\P/}$. The square (<ref>) yields a fiber sequence:

$$  \Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(B_\P, \O) \lrar \Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(\P \sqcup \P^{b}, \O) \lrar \Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(\P \sqcup (\P\circ\P), \O).$$

We shall complete the proof by showing each of these three spaces, in the order given, is naturally weakly equivalent to the respective space in the sequence (<ref>). First, argue that there exists a chain of weak equivalences:

$$  \Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(B_\P, \O) \simeq \Map^{\h}_{\Op_C(\mathcal{S})_{\P/}}(\Sigma(\mathcal{P}\bigsqcup\mathcal{P}), \O) \simeq \Omega\Map^{\h}_{\Op_C(\mathcal{S})_{\P/}}(\mathcal{P}\bigsqcup\mathcal{P}, \O) .$$

Indeed, the first weak equivalence follows from the second paragraph, while the other is straightforward. Moreover, by adjunction we have a weak equivalence
$$\Map^{\h}_{\Op_C(\mathcal{S})_{\P/}}(\mathcal{P}\bigsqcup\mathcal{P}, \O)\simeq \Map^{\h}_{\Op_C(\mathcal{S})}(\mathcal{P}, \O).$$

We thus obtain a weak equivalence $\Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(B_\P, \O) \simeq \Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\mathcal{P}, \O)$. By adjunction again, we obtain the following chains of weak equivalences:

$$ \Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(\P \sqcup \P^{b}, \O) \simeq \Map^{\h}_{\BMod(\mathcal{P})}(\P^{b}, \O) \simeq \Map^{\h}_{\BMod(\mathcal{P})}(\P, \O), \; \text{and}$$

$$ \Map^{\h}_{\BMod(\mathcal{P})_{\P/}}(\P \sqcup (\P\circ\P), \O) \simeq \Map^{\h}_{\BMod(\mathcal{P})}(\P\circ\P, \O) \simeq \Map^{\h}_{\Coll_C(\cS)}(\I_C, \O).$$

Finally, notice that the latter space is weakly equivalent to $\underset{c\in C}{\prod}\Map^{\h}_{\cS}(1_{\cS}, \O(c;c))$.

According to the proof above, the map

$$ \Map^{\h}_{\BMod(\P)}(\P, \O) \lrar \underset{c\in C}{\prod}\Map^{\h}_{\cS}(1_{\cS}, \O(c;c)) $$

is weakly equivalent to the map $\Map^{\h}_{\BMod(\P)}(\P, \O) \lrar \Map^{\h}_{\BMod(\P)}(\P\circ\P, \O)$ that is induced by the composition $\mu : \P\circ\P \lrar \P$.

The following explains the significance of good base categories.
If the base category $\cS$ is good then the statement of Theorem <ref> holds true for every $\Sigma$-cofibrant operad in $\cS$.
Suppose that $\cS$ is good and let $\P \in \Op_C(\mathcal{S})$ be a $\Sigma$-cofibrant operad. Let us take a cofibrant resolution $f:\Q \lrarsimeq \P$ for $\P$. In particular, the operad $\Q$ is good by assumption. We deduce the lemma by observing that the adjunction $\adjunction*{}{\BMod(\Q)}{\BMod(\P)}{}$ induced by $f$ is a Quillen equivalence.

We will now discuss the verification of the goodness-hypothesis in practice. It was proved by Dwyer-Hess that every non-symmetric simplicial operad is good in our sense (cf. [[12], Proposition 5.4]). Expanding upon their result, we have obtained the following in [[14], $\S$5.1].
Suppose that $\cS$ is either $\Set_\Delta$ or $\sMod_{\textbf{k}}$ with $\textbf{k}$ being any commutative ring. Then we have that $\cS$ is good, i.e. every  cofibrant (symmetric) operad in $\cS$ is good.

Here is an application of the operadic Dold-Kan correspondence.
For any commutative ring $\textbf{k}$, the base category $\C_{\geqslant0}(\textbf{k})$ is good.
Let $\P\in \Op_{C}(\C_{\geqslant0}(\textbf{k}))$ be a cofibrant $C$-colored operad in $\C_{\geqslant0}(\textbf{k})$. Consider the following square of left Quillen functors

$$  \xymatrix{
\BMod(\P)^* \ar[r]\ar_{\E}[d] & \BMod(\L_C\P)^* \ar^{\E}[d] \\
\Op_C(\C_{\geqslant0}(\textbf{k}))_{\P\bigsqcup\P/} \ar[r] & \Op_C(\sMod_{\textbf{k}})_{\L_C(\P\bigsqcup\P)/} \\
}	 $$

where the top horizontal functor is the pointed-analogue of the left Quillen functor $\BMod(\P) \lrarsimeq \BMod(\L_C\P)$ of Corollary <ref>, while  the bottom horizontal functor is induced by $\L_C : \Op_C(\C_{\geqslant0}(\textbf{k})) \lrarsimeq \Op_C(\sMod_{\textbf{k}})$. It can be shown that this square is commutative (again by verifying the commutativity of the associated square of right adjoints). We now claim that $\P$ is good due to the fact that $\L_C\P$ is good by Proposition <ref>, and that both the horizontal functors are a left Quillen equivalence identifying $\P$ to $\L_C\P$.

Combining the above statements, we obtain the following.
Suppose that $\cS$ is any of the categories in $\{\Set_\Delta, \sMod_{\textbf{k}}, \C_{\geqslant0}(\textbf{k})\}$. Let $\P\in\Op_C(\mathcal{S})$ be a $\Sigma$-cofibrant operad and suppose given a map $\P\lrar\O$ between $C$-colored operads. Then there is a fiber sequence of spaces

\begin{equation}\label{eq:goodnessmain1}
\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \O) \lrar \Map^{\h}_{\BMod(\P)}(\P, \O) \lrar \underset{c\in C}{\prod}\Map^{\h}_{\cS}(1_{\cS}, \O(c;c)).
\end{equation}

In the case where $\cS=\Set_\Delta$, the above theorem extends a result of Ducoulombier [13], in which the operads are assumed to have a single color and such that $\O(1) \simeq *$. More precisely, in this situation we obtain a weak equivalence

\begin{equation}\label{eq:sum1}
\Omega\Map^{\h}_{\Op_*(\Set_\Delta)}(\P, \O) \lrarsimeq \Map^{\h}_{\BMod(\P)}(\P, \O)
\end{equation}

where $\Op_*(\Set_\Delta)$ refers to the category of  single-colored simplicial operads.

For the case $\cS=\C_{\geqslant0}(\textbf{k})$ (or $\sMod_{\textbf{k}}$), an analogue of (<ref>) does not hold for nontrivial choices of the operad $\O$. Instead, in several cases of interest where $\O(1) \simeq \textbf{k}$, we have a fiber sequence of the form

$$ \Omega\Map^{\h}_{\Op_*(\C_{\geqslant0}(\textbf{k}))}(\P, \O) \lrar \Map^{\h}_{\BMod(\P)}(\P, \O) \lrar \textbf{k}.$$

Again, let $\cS$ and $\P$ be as in the statement of Theorem <ref>.

For an ordered pair $(A,B)$ of objects in $\cS^{\times C}$, one can associate to it an endomorphism $C$-collection, denoted $\End_{A,B}$, and defined on each level by letting
$$\End_{A,B}(c_1,\cdots,c_n;c) := \Map_{\cS}(A(c_1)\otimes\cdots\otimes A(c_n),B(c)).$$

In particular, the endomorphism operad associated to $A$ is given by $\End_{A}:=\End_{A,A}$. Let us assume that $A$ comes equipped with a $\P$-algebra structure classified by a map $\mu_A : \P\lrar\End_A$ of  operads. Then we have an adjunction

$$ (-)\circ_{\P}A : \adjunction*{}{\BMod(\P)}{\Alg_{\P}(\cS)}{} : \End_{A,-} $$

where the left adjoint is given by the relative composite product over $\P$ (cf., e.g. [17, 26] for more details about these constructions).   Clearly the above adjunction becomes a Quillen adjunction as long as $A$ is levelwise cofibrant. Now assume further that $A$ is levelwise bifibrant, so that
we have a weak equivalence
$$\Map^{\h}_{\BMod(\P)}(\P, \End_A) \simeq \Map^{\h}_{\Alg_{\P}(\cS)}(A,A),$$

and so that the underlying space of $\Map_{\cS}(A(c),A(c))$ is a right model for the corresponding derived mapping space. Applying Theorem <ref> to the map $\mu_A$ hence gives us a fiber sequence:

\begin{equation}\label{eq:sum2}
\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \End_A) \lrar \Map^{\h}_{\Alg_{\P}(\cS)}(A,A) \lrar \underset{c\in C}{\prod} \Map^{\h}_{\cS}(A(c),A(c)).
\end{equation}

We will illustrate how the above example harmonizes naturally with a result obtained by Rezk [26]. For each model category $\textbf{M}$, we denote by $\textbf{M}_\infty^{\simeq}$ the maximal $\infty$-groupoid contained in the underlying $\infty$-category of $\textbf{M}$ (cf. [20]). Moreover, for an object $X\in\textbf{M}$, we will write
$$\Aut^{\h}_{\textbf{M}}(X) \subseteq\Map^{\h}_{\textbf{M}}(X,X)$$
for the subspace consisting of those that are  self-equivalences on $X$. It is known that $\Aut^{\h}_{\textbf{M}}(X)$ is a natural model for the loop space of $\textbf{M}_\infty^{\simeq}$ at $X$ (cf., e.g. [28]). As in [26], we refer to the space $\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \End_A)$ as the moduli space of $\P$-algebra structures on  $A\in\cS$. Now, according to Theorem 1.2.10 (and 1.2.15) of the loc.cit, there is a fiber sequence

\begin{equation}\label{eq:sum3}
\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \End_A) \lrar \Alg_{\P}(\cS)_\infty^{\simeq} \lrar \underset{C}{\prod}\cS_\infty^{\simeq}.
\end{equation}

Suppose further that $A$ is endowed with a $\P$-algebra structure. Then looping down (<ref>) at $A$ produces a fiber sequence

\begin{equation}\label{eq:sum3loop}
\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \End_A) \lrar \Aut^{\h}_{\Alg_{\P}(\cS)}(A) \lrar \underset{c\in C}{\prod}\Aut^{\h}_{\cS}(A(c)).
\end{equation}

Our observation is that the two sequences (<ref>) and (<ref>) fit together into a diagram of homotopy Cartesian squares of the form

\begin{equation}\label{eq:Rezkfiber}
\xymatrix{
\Omega\Map^{\h}_{\Op_C(\mathcal{S})}(\P, \End_A) \ar[r]\ar[d] & \Aut^{\h}_{\Alg_{\P}(\cS)}(A) \ar[r]\ar[d] & \Map^{\h}_{\Alg_{\P}(\cS)}(A,A) \ar[d] \\
\{\Id_A\} \ar[r] & \underset{c\in C}{\prod}\Aut^{\h}_{\cS}(A(c)) \ar[r] & \underset{c\in C}{\prod} \Map^{\h}_{\cS}(A(c),A(c)) \\
\end{equation}

where the right homotopy Cartesian square is due to the fact that the forgetful functor $\Alg_{\P}(\cS) \lrar \underset{C}{\prod} \, \cS$ preserves and detects weak equivalences.

§.§ The second fiber sequence

As the main interest, we now consider the mapping spaces between operads with non-fixed sets of colors.

As before, we let $\cS$ be a base category satisfying the assumptions as in Theorem <ref>, and let $\P\in\Op_{C}(\cS)$ be a $C$-colored operad in $\cS$. For more convenience, we will assume that the Dwyer-Kan homotopy theory on $\Op(\cS)$ exists (at least) as a semi-model category.

By abuse of notation, for an $\cS$-enriched category $\C$, we will denote by $\cor(\C) := \Map^{\h}_{\Cat(\cS)}([0]_\cS, \C)$ where $[0]_\cS$ denotes the category with a single object and with the mapping object being given by $1_\cS$, and refer to $\cor(\C)$ as the core of $\C$.

We have that  $\pi_0(\cor(\C))$ represents the equivalence classes of objects in $\C$. Here by convention two objects in $\C$ are equivalent if they are isomorphic as objects in the homotopy category $\Ho(\C)$ (see $\S$<ref>).

When $\C$ is a fibrant simplicial category then we have a weak equivalence $\cor(\C) \simeq \sN(\C)^{\simeq}$, i.e. the maximal $\infty$-groupoid contained in the homotopy coherent nerve of $\C$. To see this, it suffices to observe that

$$ \Map^{\h}_{\Cat(\Set_\Delta)}([0]_{\Set_\Delta}, \C) \simeq \Map_{\Cat_\infty}(*, \sN(\C)) $$

where the space on the right is the mapping space in the $\infty$-category of $\infty$-categories from the terminal $\infty$-category $*$ to $\sN(\C)$, which is exactly given by the maximal $\infty$-groupoid contained in $\Fun(*,\sN(\C)) \cong \sN(\C)$.

More generally, suppose that $\cS$ comes equipped with a weak monoidal Quillen adjunction $\adjunction*{}{\Set_\Delta}{\cS}{}$ (cf. [3]), so that we may associate to any category $\C\in\Cat(\cS)$ a simplicial category, denoted by $\C_\Delta$, and defined by the derived image of $\C$ through the induced functor $\Cat(\cS) \lrar \Cat(\Set_\Delta)$. Moreover, by adjunction we have a weak equivalence

$$ \Map^{\h}_{\Cat(\Set_\Delta)}([0]_{\Set_\Delta}, \C_\Delta) \simeq \Map^{\h}_{\Cat(\cS)}([0]_\cS, \C) $$

(here we note that $[0]_\cS \in \Cat(\cS)$ is always cofibrant, for any base category $\cS$). Combining this with the above example, we obtain a chain of weak equivalences
$$ \cor(\C) \simeq \cor(\C_\Delta) \simeq \sN(\C_\Delta)^{\simeq}.$$

Clearly this machinery can be applied to the categories in spectra, categories in $\sMod_{\textbf{k}}$, and moreover, to all categories in $\C_{\geqslant0}(\textbf{k})$ after applying the categorical Dold-Kan correspondence (<ref>).

Now suppose we are given an operad $\Q\in \Op_D(\cS)$ colored in some set $D$, and along with a map $f :  C \lrar D$ between the sets of colors. Recall by notations that $\Q_1 \in \Cat(\cS)$ signifies the underlying category of $\Q$ formed by the collection of unary operations, and $f^*\Q$ denotes the $C$-colored operad determined by the restriction of colors.

It follows immediately from the definition that
$$\Hom_{\Op(\cS)}(\P,\Q) \cong \underset{\alpha}{\coprod} \; \Hom_{\Op_C(\cS)}(\P,\alpha^*\Q)$$
where the coproduct ranges over the set of maps $ C \lrar D$. Nonetheless, when passing to the realm of homotopy theory, the circumstance might vary significantly, as we will see below.

The following is the key step for simplifying the mapping spaces in $\Op(\cS)$.

There is a homotopy Cartesian square of spaces of the form

\begin{equation}\label{eq:simplifying}
\xymatrix{
\Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q)  \ar[r]\ar[d] & \Map^{\h}_{\Op(\cS)}(\P,\Q) \ar[d] \\
\{*\} \ar[r]^{\underset{c\in C}{\prod}\{f(c)\}} &  \underset{C}{\prod}\cor(\Q_1) \;  \\
\end{equation}

in which the bottom horizontal map is given by inserting, for each $c\in C$, the object $f(c) \in \Ob(\Q_1)$ into $\cor(\Q_1)$.
Without loss of generality, we may assume that $\P\in\Op_C(\cS)$ is cofibrant. Since the initial $C$-colored operad $\I_C$ is cofibrant as an object of $\Op(\cS)$, it implies that $\P$ is as well cofibrant within $\Op(\cS)$ (cf. [14], Observation 4.1.1(ii)). Now consider the following coCartesian square in $\Op(\cS)$:

$$ \xymatrix{
\I_C \coprod \I_C  \ar[r]\ar[d] & \I_C \coprod \P \ar[d] \\
\I_C \ar[r] &  \P \;  \\
} $$

Due to the discussion above, this square is automatically homotopy coCartesian when considered as a square in $\Op(\cS)_{\I_C/}$. We also regard $\Q$ as an object of $\Op(\cS)_{\I_C/}$ via the map $\eta_f :\I_C \lrar \Q$ that is determined uniquely by $f:C \lrar D$. Now, the above square yields a homotopy Cartesian square of spaces

$$ \xymatrix{
\Map^{\h}_{\Op(\cS)_{\I_C/}}(\P,\Q) \ar[r]\ar[d] & \Map^{\h}_{\Op(\cS)}(\P,\Q) \ar[d] \\
\{\eta_f\} \ar[r] &  \Map^{\h}_{\Op(\cS)}(\I_C,\Q) \;  \\
} $$

(In fact, a generalization of this holds true in any under $\infty$-category, according to [[20], 5.5.5.12] with the opposite convention). We shall complete the proof by showing the existence of natural weak equivalences:

$$  \Map^{\h}_{\Op(\cS)_{\I_C/}}(\P,\Q) \simeq  \Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q)  \;\; \text{and} \;\; \Map^{\h}_{\Op(\cS)}(\I_C,\Q) \simeq \underset{C}{\prod}\cor(\Q_1).$$

Let us start with the first weak equivalence. Combining the Quillen adjunction (<ref>) with the fact that $\I_C$ represents an initial object in $\Op_C(\cS)$, we obtain natural weak equivalences

$$ \Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q) \simeq \Map^{\h}_{\Op_C(\cS)_{\I_C/}}(\P,f^*\Q) \simeq \Map^{\h}_{\Op(\cS)_{\I_C/}}(\P,\Q),$$

as expected.

For the second one, we will make use of the Quillen adjunction $\adjunction*{}{\Cat(\cS)}{\Op(\cS)}{}$ (see Remark <ref>). Combined with the observation that $\I_C$ can be identified with a category in $\cS$ exhibited as the $C$-fold coproduct of $[0]_\cS\in\Cat(\cS)$, that adjunction yields natural weak equivalences

$$ \Map^{\h}_{\Op(\cS)}(\I_C,\Q) \simeq \Map^{\h}_{\Cat(\cS)}(\I_C,\Q_1) \simeq \underset{C}{\prod}\Map^{\h}_{\Cat(\cS)}([0]_\cS,\Q_1) \, \x{\defi}{=} \, \underset{C}{\prod}\cor(\Q_1).$$

Moreover, under this identification, the map $\eta_f :\I_C \lrar \Q$ corresponds to exactly $\underset{c\in C}{\prod}\{f(c)\} \subseteq \underset{C}{\prod}\cor(\Q_1)$.

Consequently, a connection between the mapping spaces in $\Op(\cS)$ and mapping spaces of operadic bimodules is achieved by combining the above theorem with Theorem <ref>.

To further elaborate, we shall now provide some additional examples derived from our current findings.

First, consider the case where $\cS$ is the category of simplicial sets, $\P$ is $\Sigma$-cofibrant and the operad $\Q$ is such that $\Q(d;d') \simeq *$  for every $d,d'\in D$. Moreover, suppose we are given a map $f : \P \lrar \Q$ in $\Op(\Set_\Delta)$.  Clearly in this case we have $\cor(\Q_1) \simeq *$. Thus due to Theorem <ref>, we get that

$$ \Map^{\h}_{\Op(\cS)}(\P,\Q) \simeq \Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q).$$

Furthermore, combining this with Theorem <ref>, we obtain that

$$ \Omega\Map^{\h}_{\Op(\cS)}(\P,\Q) \simeq \Omega\Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q) \simeq \Map^{\h}_{\BMod(\P)}(\P, f^*\Q)$$

where the first loop space is taken at the base point $f : \P \lrar \Q$, while the second is taken at the map $\P\lrar f^*\Q$ induced by $f$.

Now consider the case where $\cS = \C_{\geqslant0}(\textbf{k})$, while $\P$ is $\Sigma$-cofibrant, and $\Q$ satisfies that $\Q(d;d') \simeq \textbf{k}$ for every $d,d'\in D$. As above, suppose we are given a map $f : \P \lrar \Q$ in $\Op(\C_{\geqslant0}(\textbf{k}))$. According to Example <ref>, we may show that $\cor(\Q_1)$ is weakly equivalent to the Eilenberg–MacLane space $K(\textbf{k}^\times, 1)$ in which as usual $\textbf{k}^\times$ signifies the group of units. Thus, in this case we obtain two fiber sequences

$$ \Omega\Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q) \lrar \Map^{\h}_{\BMod(\P)}(\P, f^*\Q) \lrar \underset{C}{\prod} \, \textbf{k} \,, \; \text{and}$$

$$ \Map^{\h}_{\Op_C(\cS)}(\P,f^*\Q) \lrar \Map^{\h}_{\Op(\cS)}(\P,\Q) \lrar \underset{C}{\prod} \, K(\textbf{k}^\times, 1) $$

respectively according to Theorems <ref> and <ref>.

We will now revisit our examination of the general base category $\cS$. Let us consider the case where $\Q = \End_\cS$, i.e. the operad in $\cS$ whose colors are chosen from the collection of objects in $\cS$, and such that the spaces of operations are given by

$$  \End_\cS(X_1,\cdots,X_n;X)  := \Map_{\cS}(X_1\otimes\cdots\otimes X_n,X).$$

Suppose we are given a collection $A = \{A(c)\}_{c\in C}$ of objects in $\cS$, regarded as a map from $C$ to $\Ob(\cS)$. Then the restriction along $A$ yields a $C$-colored operad $A^*(\End_\cS)$ that exactly coincides with the endomorphism operad $\End_{A}$ (see Example <ref>). On other hand, notice that the underlying category $(\End_\cS)_1$ is nothing but $\cS \in \Cat_\cS$. Accordingly, Theorem <ref> implies the existence of a fiber sequence of the form

\begin{equation}\label{eq:ends}
\Map^{\h}_{\Op_C(\cS)}(\P,\End_{A}) \lrar \Map^{\h}_{\Op(\cS)}(\P,\End_\cS) \lrar \underset{C}{\prod}\cor(\cS).
\end{equation}

One of the main interests in the operad $\End_\cS$ is that each map $\varphi : \P \lrar \End_\cS$ in $\Op(\cS)$ can be identified with a $\P$-algebra in $\cS$, whose underlying objects are given by the collection $\{\varphi(c)\}_{c\in C} \in \cS^{\times C}$. One may expect further that an analogue remains hold true when moving into the realm of homotopy theory, i.e. there might be a weak equivalence of the form

\begin{equation}\label{eq:ends1}
\Map^{\h}_{\Op(\cS)}(\P,\End_\cS) \simeq \Alg_{\P}(\cS)_\infty^\simeq.
\end{equation}

For example, this is the case for every simplicial operad (or $\infty$-operad), according to Lurie's [[27], $\S$2.1.4]. Furthermore, an advantage is that when (<ref>) holds true in $\cS$, then combined with (<ref>), it yields a homotopy Cartesian square

$$ \xymatrix{
\Map^{\h}_{\Op_C(\cS)}(\P,\End_{A})  \ar[r]\ar[d] & \Alg_{\P}(\cS)_\infty^\simeq \ar[d] \\
\{A\} \ar[r] &  \underset{C}{\prod}\cor(\cS) \;  \\
} $$

which will provide a generalization for [[26], Theorem 1.2.10] of Rezk (see also Example <ref>).

§.§ Double loop spaces of mapping spaces between enriched operads

In this last subsection, we shall provide a proof of the statement <ref>. As in the previous subsection, we will assume that the base category $\cS$ satisfies the assumptions as in Theorem <ref> and such that the Dwyer-Kan homotopy theory on $\Op(\cS)$ exists (at least) as a semi-model category. Moreover, we will let $\P\in \Op_C(\cS)$ be a cofibrant operad satisfying the goodness hypothesis (see Definition <ref>).

Additionally, we will assume that $\cS$ satisfies Lurie's invertibility hypothesis [[20], Definition A.3.2.12]. Roughly speaking, this hypothesis requires that, for any category $\C\in\Cat(\cS)$ containing a morphism $f$, localizing $\C$ at $f$ does not change the homotopy type of $\C$ as long as $f$ is an isomorphism in $\Ho(\C)$.

The invertibility hypothesis is in fact popular in practice, as discussed in [[14], $\S$3]. In particular, all the base categories of Examples <ref> satisfy that hypothesis.

Now suppose we are given a map $f : \P \lrar \O$ in $\Op(\cS)$. As usual, $f$ induces a map $\P\lrar f^*\O$ in $\Op_C(\cS)$. We also regard this latter map as a map in $\BMod(\P)$.

There is a natural weak equivalence

$$  \Omega^2\Map^{\h}_{\Op(\cS)}(\P,\O) \simeq \Omega\Map^{\h}_{\BMod(\P)}(\P,f^*\O) $$

in which the double loop space on the left is taken at the base point $f$, and while the loop space on the right is taken at the map $\P\lrar f^*\O$.

Our proof will leverage an approach analogous to the one we used in proving [[14], Proposition 5.2.5]. Let us first pick up several notations from the loc.cit.

We let $[1]_\cS$ denote the category with objects $0, 1$ and with the mapping spaces $\Map_{[1]_\cS}(0,1) = \Map_{[1]_\cS}(0,0) = \Map_{[1]_\cS}(1,1) = 1_\cS$ and  $\Map_{[1]_\cS}(1,0) = \emptyset$. Next, denote by $[1]_\cS^{\sim}$ the category which is the same as the previous one except that $\Map_{[1]^{\sim}_\cS}(1,0) = 1_\cS$. Take a factorization
$$[1]_\cS \lrar \mathcal{E} \lrarsimeq [1]_\cS^{\sim}$$
of the canonical map $[1]_\cS \lrar [1]_\cS^{\sim}$ into a cofibration followed by a trivial fibration. We now obtain a sequence of maps
$$[0]_\cS\coprod[0]_\cS \lrar [1]_\cS \lrar \mathcal{E} \lrarsimeq [1]_\cS^{\sim} \lrarsimeq [0]_\cS$$

such that the first two maps are cofibrations, while the others are weak equivalences.

We will write $-\otimes-$ for the levelwise tensor product of categories in $\cS$. For more details, the tensor product $\C \otimes \D$ of two categories $\C$ and $\D$ has as set of objects $\Ob(\C \otimes \D) := \Ob(\C) \times \Ob(\D)$ and such that for every $c,c'\in \Ob(\C)$ and $d,d'\in\Ob(\D)$ we have

$$ \Map_{\C \otimes \D}((c,d),(c',d')) := \Map_\C(c,c') \otimes \Map_\D(d,d').$$

Now applying $\I_C\otimes(-)$ (where $\I_C$ is viewed as an $\cS$-enriched category) to the above sequence, we obtain another sequence of maps in $\Cat(\mathcal{S})$:

\begin{equation}\label{eq:icotimes}
\I_C \coprod \I_C \lrar \I_C \otimes [1]_\cS \lrar \I_C \otimes \mathcal{E} \lrarsimeq  \I_C \otimes [1]_\cS^{\sim} \lrarsimeq \I_C.
\end{equation}

Observe that the last two maps are weak equivalences, while the others are again cofibrations because $\I_C$ is a discrete category.

We will be particularly concerned with the following objects in $\Op(\cS)_{\P//\P}$:

$$\A := (\P\coprod\P) \underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes [1]_\cS \;\;\; \text{and} \;\;\; \B := (\P\coprod\P) \underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes \mathcal{E} \;.$$

Note first that each of these two pushouts represents the corresponding homotopy pushout, due to the two cofibrations of (<ref>) and along with the cofibrancy of $\P$. In other words, the two objects $\A$ and $\B$ have the right homotopy type.

We regard $\Op(\cS)_{\P//\P}$ as a pointed model category with $\P$ as the base point, so that we may define the suspension functor $\Sigma(-)$ on this category.
Let $\varphi : \A \lrar \B$ be the map induced by the map $[1]_\cS \lrar \mathcal{E}$. Then the induced map $\Sigma(\varphi) : \Sigma(\A) \lrar \Sigma(\B)$ is a weak equivalence in $\Op(\cS)_{\P//\P}$.

In fact, this claim can be derived from the proof of [[14], Lemma 5.2.10]. For readers' convenience, we still present here a detailed proof. Observe first that the map $\varphi$ can be equivalently rewritten as the map

$$ \ovl{\A} \; \underset{\I_C}{\coprod} \P \lrar \ovl{\B} \; \underset{\I_C}{\coprod} \P $$

in which $\ovl{\A} := \P\underset{\I_C}{\coprod} (\I_C\otimes [1]_\cS)$ and  $\ovl{\B} := \P\underset{\I_C}{\coprod} (\I_C\otimes \mathcal{E})$. Accordingly,  $\varphi$ is a homotopy cobase change in $\Op(\cS)_{\P//\P}$ of the map  $\ovl{\varphi} : \ovl{\A} \lrar \ovl{\B}$. Thus we just need to prove that the map $\Sigma(\ovl{\varphi})$ is a weak equivalence. Moreover, note that $\ovl{\varphi}$ coincides with the (derived) image of the map $\psi : \I_C\otimes [1]_\cS   \lrar \I_C\otimes \mathcal{E}$ through the left Quillen functor $\eta_! : \Op(\cS)_{\I_C//\I_C} \lrar \Op(\cS)_{\P//\P}$ that is induced by the unit map $\I_C \lrar \P$. Therefore, it will suffice to verify that $\Sigma(\psi)$ is a weak equivalence (in $\Cat(\cS)_{\I_C//\I_C}$). By the weak equivalence $\I_C\otimes \mathcal{E} \lrarsimeq \I_C$, we just need to show that the fold map

$$\I_C\otimes \mathcal{E} \underset{\I_C\otimes [1]_\cS}{\coprod}\I_C\otimes \mathcal{E} \lrar \I_C\otimes \mathcal{E}$$

is a weak equivalence. But this is equivalent to saying that the fold map $\mathcal{E} \underset{[1]_\cS}{\coprod}\mathcal{E} \lrar \mathcal{E}$ is a weak equivalence, which immediately follows from the invertibility hypothesis on $\cS$.

Notice that there is a weak equivalence $\ovl{\B} \lrarsimeq \P$ that follows from the weak equivalence $\I_C\otimes \mathcal{E} \lrarsimeq \I_C$. This implies the existence of a weak equivalence $\B \lrarsimeq \P\underset{\I_C}{\coprod}\P$ in $\Op(\cS)_{\P//\P}$. We now obtain a sequence of maps $\A \lrar \B \lrarsimeq \P\underset{\I_C}{\coprod}\P$, each of which becomes a weak equivalence after taking suspension:

$$  \Sigma(\A) \lrarsimeq \Sigma(\B) \lrarsimeq \Sigma(\P\underset{\I_C}{\coprod}\P).$$

In light of this, we will regard $\Sigma(\A)$ and $\Sigma(\B)$ as two convenient models for the suspension $\Sigma(\P\underset{\I_C}{\coprod}\P)$.

We will be also interested in the following objects in $\Op(\cS)_{\P//\P}$:

$$ \U := \P \underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes [1]_\cS \; \;\; \text{and} \;\;\; \V := \P \underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes \mathcal{E} \;.$$

As above, these two objects have already the right homotopy type. Additionally, there is a canonical map $\rho : \U \lrar \V$ that is induced by the map $[1]_\cS \lrar \mathcal{E}$. Moreover, a similar argument to that in the above proof shows that the map
$$\Sigma(\rho) : \Sigma(\U) \lrarsimeq \Sigma(\V)$$
is a weak equivalence as well.

We will be in position to prove the main theorem of this subsection after a more comprehensive discussion below.

By adjunctions, we have a chain of weak equivalences
$$\Map^{\h}_{\Op_C(\mathcal{S})}(\P, f^*\O) \, \simeq \, \Map^{\h}_{\Op_C(\mathcal{S})_{\P/}}(\P\bigsqcup\P, f^*\O) \, \simeq \, \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\P, \O).$$

Here we made use of the fact that the coproduct $\P\bigsqcup\P \in \Op_C(\mathcal{S})$, when considered as an object of $\Op(\cS)$, agrees with the pushout $\P\underset{\I_C}{\coprod}\P \in \Op(\mathcal{S})$. In the same fashion, we may obtain that

$$ \underset{c\in C}{\prod}\Map^{\h}_{\cS}(1_{\cS}, f^*\O(c;c))  \cong  \Map^{\h}_{\Op_C(\mathcal{S})}(\Free(\I_C), f^*\O)  \simeq \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\Free(\I_C), \O)$$

in which $\Free(\I_C)$ signifies the free $C$-colored operad generated by the $C$-collection $\I_C$. Thus we may rewrite the fiber sequence (<ref>) for the map $\P\lrar f^*\O$ as

$$  \Omega\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\P, \O) \lrar \Map^{\h}_{\BMod(\P)}(\P,f^*\O) \lrar \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\Free(\I_C), \O).$$

On other hand, the fiber sequence (<ref>) can be equivalently rewritten as

$$ 	\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\P, \O) \lrar \Map^{\h}_{\Op(\cS)}(\P,\O) \lrar \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\coprod\I_C, \O).$$

Consequently, we obtain two fiber sequences as follows

\Omega\Map^{\h}_{\BMod(\P)}(\P,f^*\O) \lrar	 \Omega\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\Free(\I_C), \O) \x{\alpha}{\lrar} \Omega\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\P, \O)$$

$$  \Omega^2\Map^{\h}_{\Op(\cS)}(\P,\O) \lrar \Omega^2\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\coprod\I_C, \O) \x{\beta}{\lrar} \Omega\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\P, \O).$$

For the proof below, the key step is to find convenient models for the two maps $\alpha$ and $\beta$, and additionally, to show how they can be related via a pair of natural weak equivalences.

Notice first that, when considered as an object of $\Op(\mathcal{S})$, the operad $\Free(\I_C)$ can be modeled by

$$\Free(\I_C) \, \cong \, \I_C \, \underset{\I_C \coprod \I_C}{\coprod} \, \I_C\otimes [1]_\cS \in \Op(\mathcal{S}).$$

Due to this, we obtain an identification

\begin{equation}\label{eq:U}
\P\underset{\I_C}{\coprod}\Free(\I_C) \cong \P \underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes [1]_\cS \; \x{\defi}{=} \; \U.
\end{equation}

Next, because of the fact that $\P\coprod\I_C$ agrees with the (derived) image of $\I_C \coprod \I_C$ through the functor $\eta_! : \Op(\cS)_{\I_C//\I_C} \lrar \Op(\cS)_{\P//\P}$, we obtain that

\begin{equation}\label{eq:V}
\Sigma(\P\coprod\I_C) \simeq \eta_!\Sigma(\I_C \coprod \I_C) \simeq \eta_!(\I_C\underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes \mathcal{E}) \cong \P \underset{\I_C\coprod\I_C}{\coprod}\I_C \otimes \mathcal{E} \; \x{\defi}{=} \; \V.
\end{equation}

Let us now consider the following commutative square in $\Op(\mathcal{S})_{\P/}$:

\begin{equation}\label{eq:doudle}
\xymatrix{
\Sigma(\A)  \ar[r]^{\Sigma(\ovl{\alpha})}\ar[d]_{\Sigma(\varphi)}^\simeq & \Sigma(\U) \ar[d]_\simeq^{\Sigma(\rho)} \\
\Sigma(\B) \ar[r]_{\Sigma(\ovl{\beta})} &  \Sigma(\V) \;  \\
\end{equation}

such that the maps $\ovl{\alpha} : \A \lrar \U$ and $\ovl{\beta} : \B \lrar \V$ are both induced by the fold map $\P\coprod\P \lrar \P$, and the vertical maps are all weak equivalences as indicated earlier. The map $\Sigma(\ovl{\alpha})$ induces a map between derived mapping spaces

$$ \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\U), \O)  \lrar \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\A), \O)  $$

whose domain is weakly equivalent to $\Omega\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\Free(\I_C), \O)$ due to (<ref>), and whose codomain is weakly equivalent to  $\Omega\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\P\underset{\I_C}{\coprod}\P, \O)$ by Remark <ref>. Moreover, the above map stands as a natural model for the map $\alpha$. In addition, we may show that the induced map

$$  \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\V), \O)  \lrar \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\B), \O) $$

is weakly equivalent to the map $\beta$, according to (<ref>) and Remark <ref> again. In summary, we now obtain a commutative square

$$ 	\xymatrix{
\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\V), \O)  \ar[r]\ar[d]_\simeq & \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\B), \O) \ar[d]^\simeq \\
\Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\U), \O) \ar[r] &  \Map^{\h}_{\Op(\mathcal{S})_{\P/}}(\Sigma(\A), \O) \;  \\
} $$

such that the top and bottom horizontal maps are respectively weakly equivalent to $\beta$ and $\alpha$, and while the two vertical maps are all weak equivalences. Thus we get a natural weak equivalence between homotopy fibers
$$   \Omega^2\Map^{\h}_{\Op(\cS)}(\P,\O) \lrarsimeq \Omega\Map^{\h}_{\BMod(\P)}(\P,f^*\O)$$

as expected.

As in Lemma <ref>, if $\cS$ is already a good base category then the statement of Theorem <ref> holds for every $\Sigma$-cofibrant operad in $\cS$. In particular, this is the case when $\cS$ is any of the categories in $\{\Set_\Delta, \sMod_{\textbf{k}}, \C_{\geqslant0}(\textbf{k})\}$.

Let us apply Theorem <ref> to the case where $\P = \I$, i.e. the initial single-colored operad in $\cS$, and where $\O = \End_\cS$ (see Example <ref>). In this situation, we  have that

$$ \Map^{\h}_{\Op(\cS)}(\I,\End_\cS) \simeq \Map^{\h}_{\Cat(\cS)}([0]_\cS,(\End_\cS)_1) \, = \, \cor(\cS).$$

Now let $A \in \cS$ be a bifibrant object regarded as a map $\I \lrar \End_\cS$ of operads. Then Theorem <ref> proves further that

$$ \Omega^2\cor(\cS) \simeq  \Omega^2\Map^{\h}_{\Op(\cS)}(\I,\End_\cS) \simeq \Omega\Map^{\h}_{\BMod(\I)}(\I, \End_{A}) \simeq \Omega\Map^{\h}_\cS(A,A).$$

(The third weak equivalence follows from the fact that $\BMod(\I)$ is simply the category of $\Sigma_*$-objects in $\cS$). Here the loop spaces are taken at $A$ or $\Id_A$. Assume further that $\cS$ is nice enough so that its core can be modeled simply by $\cor(\cS) \simeq \cS_\infty^\simeq$. For example, this is the case as soon as $\cS\in\{\Set_\Delta, \sMod_{\textbf{k}}, \C_{\geqslant0}(\textbf{k})\}$ (see Example <ref>). Then we may rewrite

$$\Omega^2\cor(\cS) \; \simeq  \; \Omega(\Omega \, \cS_\infty^\simeq) \; \simeq \; \Omega\Aut_\cS^{\h}(A)$$

(see Example <ref> for notations). These observations tell us that the embedding $\Aut_\cS^{\h}(A) \lrar \Map^{\h}_\cS(A,A)$ becomes a weak equivalence after a single looping at $\Id_A$:

\begin{equation}\label{eq:autend}
\Omega\Aut_\cS^{\h}(A) \lrarsimeq \Omega\Map^{\h}_\cS(A,A).
\end{equation}

This assertion is in general not a trivial one, even though it could be verified somehow more directly.

Continuing from the above example, let us proceed by considering a model category $\textbf{M}$ that is endowed with a right Quillen functor $\sU : \textbf{M}\lrar \cS$. Moreover, we assume that $\sU$ preserves and detects weak equivalences. This implies the existence of a homotopy Cartesian square

$$ \xymatrix{
\Aut^{\h}_{\textbf{M}}(X) \ar[r]\ar[d] & \Map^{\h}_{\textbf{M}}(X,X) \ar[d] \\
\Aut^{\h}_{\cS}(\sU(X)) \ar[r] &  \Map^{\h}_{\cS}(\sU(X),\sU(X)) \;  \\
} $$

for every object $X\in \textbf{M}$. Combining this with (<ref>), we obtain a weak equivalence

$$ \Omega\Aut^{\h}_{\textbf{M}}(X) \lrarsimeq \Omega\Map^{\h}_{\textbf{M}}(X,X)$$

where the loop spaces are taken at $\Id_X$. Numerous categories can be found that align with $\textbf{M}$ such as the category of $C$-colored operads in $\cS$, the category of operadic algebras, and various types of operadic modules in $\cS$, etc.

We may use Theorem <ref> to investigate mapping spaces between $\infty$-categories (or simplicial categories, equivalently). Suppose we are given a functor $ f : \C \lrar \D$ between $\infty$-categories. We will write $\textbf{S} := (\Set_\Delta)_\infty$ to denote the $\infty$-category of spaces. We may think of the category $\Fun(\C^{\op} \times \C , \textbf{S})$ as a model for the category of $\C$-bimodules. Then we may regard $\C$ as a bimodule over itself via the functor

$$ \Map_\C : \C^{\op} \times \C \lrar \textbf{S} \; , \; (x,y) \mapsto \Map_\C(x,y).$$

Also, the restriction along $f$ exhibits $\D$ as a $\C$-bimodule via the functor

$$ \Map_{f^*\D} : \C^{\op} \times \C \lrar \textbf{S} \; , \; (x,y) \mapsto \Map_\D(f(x),f(y)).$$

Accordingly, we may rewrite

$$ \Map^{\h}_{\BMod(\C)}(\C,f^*\D) \simeq \Map_{\Fun(\C^{\op} \times \C , \textbf{S})}(\Map_\C, \Map_{f^*\D}).$$

Next, after passing to the framework of simplicial categories, the unstraightening of $\Map_\C$ is exactly a model for the twisted arrow $\infty$-category $\Tw(\C)$, which comes together with a left fibration $\Tw(\C) \lrar \C^{\op} \times \C$ (cf. [[27], 5.2.1.11]). Let us now write $\D^\star$ for the composed functor

$$ \Tw(\C) \lrar \C^{\op} \times \C \x{\Map_{f^*\D}}{\lrar} \textbf{S},$$

which sends an edge $x \lrar y$ in $\C$ (viewed as an object of $\Tw(\C)$) to $\Map_\D(f(x),f(y))$. Due to the straightening-unstraightening adjunction, we may show that there is a weak equivalence of spaces

$$\lim\D^\star  \simeq  \Map_{\Fun(\C^{\op} \times \C , \textbf{S})}(\Map_\C, \Map_{f^*\D}).$$

Combining this with Theorem <ref>, we now obtain a weak equivalence

$$  \Omega^2\Map_{\Cat_\infty}(\C,\D)  \simeq \Omega\lim\D^\star.$$

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