Differential Calculus of Hochschild Pairs for Infinity-Categories

Differential Calculus of Hochschild Pairs
for Infinity-Categories††This paper is a contribution to the Special Issue on
Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th
birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Saito.html

Isamu IWANARI

I. Iwanari

Mathematical Institute, Tohoku University, 6-3 Aramakiaza, Sendai, Miyagi,
980-8578, Japan<EMAIL_ADDRESS>
Received February 25, 2020, in final form September 04, 2020; Published online
October 02, 2020

In this paper, we provide a conceptual new construction of the algebraic
structure on the pair of the Hochschild cohomology spectrum (cochain complex)
and Hochschild homology spectrum, which is analogous to the structure of
calculus on a manifold. This algebraic structure is encoded by a two-colored
operad introduced by Kontsevich and Soibelman. We prove that for a stable
idempotent-complete infinity-category, the pair of its Hochschild cohomology
and homology spectra naturally admits the structure of algebra over the
operad. Moreover, we prove a generalization to the equivariant context.

Hochschild cohomology; Hochschild homology; operad; $\infty$-category

16E40; 18N60; 18M60

To Kyoji Saito on his 77th birthday

##  1 Introduction

Let $M$ be a smooth real manifold. Let $T_{M}^{\bullet}=\bigoplus_{p\geq
0}\wedge^{p}T_{M}$ and $\Omega_{M}^{\bullet}=\bigoplus_{q\geq
0}\Omega_{M}^{q}$ denote the graded vector space of multivector fields and
differential forms on $M$, respectively. By convention,
$T_{M}^{p}=\wedge^{p}T_{M}$ has homological degree $-p$ while $\Omega^{q}_{M}$
has homological degree $q$: we adopt the reverse grading. There are several
algebraic structures on the pair $(T_{M}^{\bullet},\Omega_{M}^{\bullet})$. The
graded vector space $T_{M}^{\bullet}$ has a graded commutative (associative)
product given by $\wedge$. Further, the shifted graded vector space
$T_{M}^{\bullet+1}$ inherits the structure of a graded Lie algebra defined by
the Schouten–Nijenhuis bracket $[-,-]$. On the other hand,
$\Omega_{M}^{\bullet}$ has the de Rham differential ${\rm d}_{\rm DR}$ (we do
not consider the obvious graded commutative algebra structure on
$\Omega_{M}^{\bullet}$ because it is irrelevant to the noncommutative
context). Since $\Omega_{M}^{q}$ is the dual vector space of $T^{q}_{M}$, the
contraction morphisms $T^{p}_{M}\otimes\Omega_{M}^{q}\to\Omega_{M}^{q-p}$ give
rise to a $(T_{M}^{\bullet},\wedge)$-module structure on
$\Omega_{M}^{\bullet}$:

$\displaystyle i\colon\
(T_{M}^{\bullet},\wedge)\otimes\Omega^{\bullet}_{M}\to\Omega_{M}^{\bullet},$

where we regard $(T_{M}^{\bullet},\wedge)$ as the graded algebra determined by
$\wedge$. The Lie derivative on $M$ defines a Lie algebra action of
$\big{(}T_{M}^{\bullet+1},[-,-]\big{)}$ on $\Omega_{M}^{\bullet}$:

$\displaystyle l\colon\
\big{(}T_{M}^{\bullet+1},[-,-]\big{)}\otimes\Omega_{M}^{\bullet}\to\Omega_{M}^{\bullet}.$

The tuple $(\wedge,[-,-],{\rm d}_{\rm DR},i,l)$ constitutes fundamental
calculus operations on the manifold $M$. These operations are subject under
certain relations such as ${\rm d}_{\rm DR}^{2}=0$, the Cartan homotopy/magic
formula, the compatibility between the Lie algebra action $l$ and the de Rham
differential ${\rm d}_{\rm DR}$, and so on. If $X$ is a smooth algebraic
variety over a field of characteristic zero, the pair
$(\mathcal{T}_{X}^{\bullet},\Omega_{X}^{\bullet})$ of sheaves of multivector
fields (given by the exterior products of the tangent sheaf) and differential
forms admit such an algebraic structure of calculus.

Let us shift our interest to noncommutative algebraic geometry in which stable
$\infty$-categories, (pretriangulated) differential graded (dg) categories or
the likes play the roles of fundamental geometric objects. From the
Hochschild–Kostant–Rosenberg theorem, an analogue of
$(\mathcal{T}_{X}^{\bullet},\Omega_{X}^{\bullet})$ is the Hochschild pair,
that is, the pair of Hochschild cohomology (cochain complex) and Hochschild
homology (chain complex). In [23, Section 11.2], Kontsevich and Soibelman
introduced a two-colored topological operad which we shall refer to as the
Kontsevich–Soibelman operad and denote here by $\mathbf{KS}$. It can be used
to encode all the structures on the Hochschild pair, which are analogous to
the structure of calculus. The operad $\mathbf{KS}$ generalizes the little
$2$-disks (cubes) operad, i.e., the $\mathbf{E}_{2}$-operad. It contains two
colors, $D$ and $C_{M}$, such that the full suboperad spanned by the color $D$
is the $\mathbf{E}_{2}$-operad. In [11], it was shown that a combinatorial dg
operad $\mathcal{KS}_{{\rm comb}}$, which is a certain dg version of
$\mathbf{KS}$, acts on the pair of the Hochschild cochain complex and the
Hochschild chain complex of an associative algebra ($A_{\infty}$-sense), and
in [16] an algebraic structure on the Hochschild pair over $\mathbf{KS}$ was
constructed for a ring spectrum by means of the Swiss-cheese operad conjecture
[34].

In this paper, we provide a conceptual new construction of the structure of an
algebra over $\mathbf{KS}$ that yields the following result (see Theorem
7.14):

######  Theorem 1.1.

Let $R$ be a commutative ring spectrum. Let $\mathcal{C}$ be a small
$R$-linear stable indempotent-complete $\infty$-category. Let
$\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$ be the Hochschild
cohomology $R$-module spectrum and
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ the Hochschild homology
$R$-module spectrum. Then the pair
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$
is promoted to an algebra over $\mathbf{KS}$, namely, it is a
$\mathbf{KS}$-algebra in the $\infty$-category of $R$-module spectra
$\operatorname{Mod}_{R}$.

The structure of the $\mathbf{KS}$-algebra on
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$
in Theorem 1.1 induces the action morphism
$u\colon\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})\otimes\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})\to\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$,
which is a counterpart to $i$ in the aforementioned tuple. In the classical
differential graded algebraic situation, we further prove that $u$ can be
described by means of well-known algebraic constructions. Thus, $u$ can be
considered a noncommutative contraction morphism, see Section 8, Theorem 8.2
and Proposition 8.8. In other words, the underlying morphism given by actions
is an expected one. By considering Cartan’s homotopy formula built in
$\mathbf{KS}$, an analogue
$L\colon\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})[1]\otimes\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})\to\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$
of the Lie derivative map $l$ is also an expected morphism, cf. Remark 8.10.

In Section 2, we briefly describe the idea and approach of our construction,
which is based on a simple observation. To achieve this, we prove the
following (see Corollary 4.21 for details):

######  Theorem 1.2.

Let $\mathcal{M}^{\otimes}$ be a symmetric monoidal $\infty$-category such
that it admits small colimits and the tensor product functor
$\otimes\colon\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ preserves small
colimits separately in each variable. $($The typical examples of
$\mathcal{M}^{\otimes}$ we should keep in mind are the $\infty$-category of
spectra, and the derived $\infty$-category of vector spaces.$)$

Let $\operatorname{Fun}\big{(}BS^{1}\\!,\mathcal{M}\big{)}\\!$ be the functor
category from the classifying space $BS^{1}$ of the circle $S^{1}$ to
$\mathcal{M}$, which inherits a pointwise symmetric monoidal structure from
the structure on $\mathcal{M}^{\otimes}$. Namely, an object of
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$ can be viewed as an
object $M$ of $\mathcal{M}$ equipped with an $S^{1}$-action. Let
$\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})$ be the $\infty$-category of
$\mathbf{KS}$-algebras in $\mathcal{M}$. Let
$\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})$ be the $\infty$-category of
$\mathbf{E}_{2}$-algebras $($i.e., algebras over the little $2$-disks
operad$)$ in $\mathcal{M}$. Similarly,
$\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
denotes the $\infty$-category of associative algebras
$($$\mathbf{E}_{1}$-algebras$)$ in
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$. We denote by
$\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
the $\infty$-category of pairs $(A,M)$ such that
$A\in\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
and $M$ is a left $A$-module object in
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$.

Then there exists a canonical equivalence of $\infty$-categories

$\displaystyle\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})\simeq\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\times_{\operatorname{Alg}_{\operatorname{As}}(\operatorname{Fun}(BS^{1},\mathcal{M}))}\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}.$

See Section 4 for the definition of the fiber product on the right-hand side.

This result means that $\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})$,
$\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
and
$\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
form building blocks for $\mathbf{KS}$-algebras. This allows us to describe
the structure of a $\mathbf{KS}$-algebra as a collection of more elementary
algebraic data involving associative algebras, left modules, circle actions,
and $\mathbf{E}_{2}$-algebras. As for $\mathbf{E}_{2}$-algebras, thanks to
Dunn additivity theorem for $\infty$-operads proved by Lurie [25], a canonical
equivalence
$\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\simeq\operatorname{Alg}_{\operatorname{As}}(\operatorname{Alg}_{\operatorname{As}}(\mathcal{M}))$
exists. While we make use of Theorem 1.2 in the construction process, it would
be generally useful in the theory of $\mathbf{KS}$-algebras since the notion
of $\mathbf{KS}$-algebras is complicated as is. For example, when
$\mathcal{M}$ is the derived $\infty$-category $\mathcal{D}(k)$ of vector
spaces over a field $k$ of characteristic zero, i.e., in the differential
graded context, there is a quite elementary interpretation. One may take
$\operatorname{Fun}\big{(}BS^{1},\mathcal{D}(k)\big{)}$ to be the
$\infty$-category obtained from the category of mixed complexes in the sense
of Kassel (see, e.g., [24]) by localizing quasi-isomorphisms. Therefore, an
object of
$\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{D}(k)\big{)}\big{)}$
may be regarded as an associative algebra in the monoidal ($\infty$-)category
of mixed complexes. Objects of
$\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{D}(k)\big{)}\big{)}$
can be described in a similar way. Moreover, dg $\mathbf{E}_{2}$-operad is
formal in characteristic zero.

As we will describe in the following section, our method consists of only
natural procedures. In particular, by contrast with previous work, it does not
involve/use complicated resolutions of operads or genuine chain complexes.
Thus, we hope that our proposed approach can be applicable to other settings
and generalizations such as $(\infty,n)$-categories. Indeed, the method allows
us to prove an equivariant generalization of Theorem 1.1 (see Theorem 9.1):

######  Theorem 1.3.

Let $G$ be a group object in the $\infty$-category $\mathcal{S}$ of spaces,
that is, a group-like $\mathbf{E}_{1}$-space. Let $\mathcal{C}$ be a small
$R$-linear stable idempotent-complete $\infty$-category. Suppose that $G$ acts
on $\mathcal{C}$ $($namely, it gives a left action$)$. Then
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$
is promoted to a $\mathbf{KS}$-algebra in
$\operatorname{Fun}(BG,\operatorname{Mod}_{R})$. Namely,
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$
is a $\mathbf{KS}$-algebra in $\operatorname{Mod}_{R}$, which comes equipped
with a left action of $G$.

We would like to invite the reader’s attention to the noteworthy features of
our method:

  * •

Our construction of the structure of an algebra on the Hochschild pair over
$\mathbf{KS}$ starts with an $R$-linear stable $\infty$-category
$\mathcal{C}$. Consequently, if we have an equivalence
$\mathcal{C}\simeq\mathcal{C}^{\prime}$, we have a canonical equivalence
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))\simeq(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}^{\prime}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}^{\prime}))$
as algebras over $\mathbf{KS}$ (see Remark 9.4). Consider the situation that
the associative algebra $A$ and $A^{\prime}$ in the $\infty$-category of
$R$-modules have the equivalent module category $\operatorname{LMod}_{A}$ and
$\operatorname{LMod}_{A^{\prime}}$, that is, $A$ and $A^{\prime}$ are
(derived) Morita equivalent to one another. Here $\operatorname{LMod}_{A}$ and
$\operatorname{LMod}_{A^{\prime}}$ denote the $\infty$-categories of left
$A$-module spectra and left $A^{\prime}$-module spectra, respectively (cf.
Section 3). Then
$\operatorname{LMod}_{A}\simeq\operatorname{LMod}_{A^{\prime}}$ induces the
canonical equivalence of Hochschild pairs as algebras over $\mathbf{KS}$. In
other words, our method provides a natural Morita invariant structure. This
invariance has a fundamental importance in noncommutative geometry and is
reasonable to expect, whereas the Morita invariant property of algebra
structures in [11] and [16] remains an open problem. We would like to mention
a recent work [1] in which the authors prove the Morita invariance of the
calculus structure at the level of graded vector spaces of homology.

  * •

The base ring can be any commutative ring spectrum. Note that it is not
possible to use the dg operad $\mathcal{KS}_{{\rm comb}}$ in the
generalization to the “spectral” setting.

  * •

As stated Theorem 1.3, our functorial method allows us to generalize to
equivariant situations. The $\infty$-categories with group actions naturally
appear in future applications (see below).

  * •

As revealed the outline in Section 2, our construction works well not with
algebras but with $\infty$-categories. Thus, even when one is ultimately
interested in algebras, it is important to consider the $\infty$-category of
their modules.

We would like to view our results from the perspective of noncommutative
algebraic geometry. As mentioned above, the notion of $\mathbf{KS}$-algebra
structures is a counterpart to the calculus on manifolds. Thus,
$\mathbf{KS}$-algebras are central objects in “noncommutative calculus”. We
refer the reader to [12] and references therein for this point of view.

Recall the algebro-geometric interpretations of the Hochschild cohomology
$\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$ and Hochschild homology
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ for stable
$\infty$-categories $\mathcal{C}$ or dg categories (somewhat more precisely,
we assume that they are “linear” over a field of characteristic zero). The
$\mathbf{E}_{2}$-algebra $\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$
governs the deformations theory of the stable $\infty$-category $\mathcal{C}$
in the derived geometric formulation. The Hochschild homology
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ (more precisely, the
Hochschild chain complex) inherits an $S^{1}$-action that corresponds to the
Connes operator. Then $\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$
with $S^{1}$-action gives rise to an analogue of the Hodge filtration: the
pair of the negative cyclic homology and the periodic cyclic homology can be
thought of as such a structure. (These algebraic structures are contained in
the $\mathbf{KS}$-algebra
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$.)
As revealed in [20] in the case of associative (dg) algebras $A$, the action
of $\operatorname{\mathcal{HH}}^{\bullet}(A)$ on
$\operatorname{\mathcal{HH}}_{\bullet}(A)$ encoded by the
$\mathbf{KS}$-algebra structure at the operadic level is a key algebraic datum
that describes variations of the (analogue of) Hodge filtration along
noncommutative (curved) deformations. Namely, the period map for
noncommutative deformations (of an associative algebra) is controlled by the
$\mathbf{KS}$-algebra of the Hochschild cohomology and Hochschild homology.
Therefore, the $\mathbf{KS}$-algebra
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$
will provide a crucial algebraic input for the theory of period maps for
deformations of the stable $\infty$-category $\mathcal{C}$. The significance
of the generalization to the equivariant context (Theorem 1.3) will be seen
when one comes to consider fruitful examples. The motivations partly come from
mirror symmetry. For example, stable $\infty$-categories endowed with
$S^{1}$-actions or some algebraic actions, that are interesting from the
viewpoint of $S^{1}$-equivariant deformation theory, naturally appear from
Landau–Ginzburg models in the context of matrix factorizations. Its
equivariant deformations together with the associated Hodge structure should
provide a categorification of the theory of Landau–Ginzburg models. As a
second example, if $X$ is a sufficiently nice algebraic stack (more generally,
a derived stack), one can consider the derived free loop space
$LX=\operatorname{Map}\big{(}S^{1},X\big{)}$ of $X$ (see, e.g., [5]). The
stable $\infty$-category $\operatorname{Perf}(LX)$ of perfect complexes on
$LX$ comes equipped with the natural $S^{1}$-action. Finally, we would also
like to mention that main results in this paper form the basis for our recent
work [19].

##  2 Strategy and organization

The purpose of this section is to outline the strategy of a construction of a
$\mathbf{KS}$-algebra structure on the pair of Hochschild cohomology and
Hochschild homology and to give the brief organization of this paper. This
section is something like the second part of introduction. We hope that the
following outline will be helpful in understanding the content of the sequel.
However, this section is independent with the rest of this paper so that the
reader can skip it.

2.1. We will give an outline of the construction. Let $\mathcal{C}$ be a small
stable $\infty$-category. If $\mathcal{C}$ is not idempotent-complete, we
replace $\mathcal{C}$ by its idempotent-completion: we assume that
$\mathcal{C}$ is idempotent-complete. While we work with stable
$\infty$-categories over a commutative ring spectrum $R$ in the paper, for
simplicity we here work with plain stable idempotent-complete
$\infty$-categories (equivalently, we assume that $R$ is the sphere spectrum).
We let $\mathcal{D}=\operatorname{Ind}(\mathcal{C})$ denote the Ind-category
that is a compactly generated stable $\infty$-category. The $\infty$-category
$\mathcal{D}$ is also equivalent to the functor category
$\operatorname{Fun}^{\textup{ex}}\big{(}\mathcal{C}^{\rm
op},\operatorname{Sp}\big{)}$ of exact functors, where $\operatorname{Sp}$ is
the stable $\infty$-category of spectra.

Let $\operatorname{Fun}^{\textup{L}}(\mathcal{D},\mathcal{D})$ be the functor
category from $\mathcal{D}$ to itself that consists of those functors which
preserve small colimits. This functor category is a compactly generated stable
$\infty$-category and inherits an associative monoidal structure given by the
composition of functors. We denote by
$\mathcal{E}\textup{nd}(\mathcal{D})^{\otimes}$ the presentable stable
$\infty$-category $\operatorname{Fun}^{\textup{L}}(\mathcal{D},\mathcal{D})$
endowed with the (associative) monoidal structure.

Let $\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})$ denote the
$\infty$-category of associative ring spectra. Given
$A\in\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})$, we define
$\operatorname{RMod}_{A}$ to be the $\infty$-category of right $A$-module
spectra. Let us regard $A$ as a right $A$-module in an obvious way. Then there
is an essentially unique colimit-preserving functor
$p_{A}\colon\operatorname{Sp}\to\operatorname{RMod}_{A}$ which sends the
sphere spectrum $\mathbb{S}$ in $\operatorname{Sp}$ to $A$. Let
$\operatorname{Pr}^{\textup{L}}_{\textup{St}}$ be the $\infty$-category of
presentable stable $\infty$-categories in which morphisms are colimit-
preserving functors. The assignment
$A\mapsto\\{p_{A}\colon\operatorname{Sp}\to\operatorname{RMod}_{A}\\}$ induces
$I\colon\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})\to\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}_{\operatorname{Sp}/}$.
The right adjoint
$E\colon\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}_{\operatorname{Sp}/}\to\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})$
of $I$ carries $p\colon\operatorname{Sp}\to\mathcal{P}$ in
$\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}_{\operatorname{Sp}/}$
to the endomorphism (associative) algebra
$\operatorname{End}_{\mathcal{P}}(P)$, where $P$ is the image $p(\mathbb{S})$.
Note that $\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})$ has a
natural symmetric monoidal structure whose tensor product is induced by the
tensor product $A\otimes B$ in $\operatorname{Sp}$. The $\infty$-category
$\operatorname{Pr}^{\textup{L}}_{\textup{St}}$ also admits an appropriate
symmetric monoidal structure in which $\operatorname{Sp}$ is a unit object,
and
$I\colon\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})\to\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}_{\operatorname{Sp}/}$
can be promoted to a symmetric monoidal functor. Applying
$\operatorname{Alg}_{\operatorname{As}}(-)$ to
$I\colon\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})\leftrightarrows\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}_{\operatorname{Sp}/}\,{\colon}\\!E$,
we obtain

$\displaystyle I\colon\
\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})\simeq\operatorname{Alg}_{\operatorname{As}}(\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp}))\rightleftarrows\operatorname{Alg}_{\operatorname{As}}\big{(}\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}_{\operatorname{Sp}/}\big{)}\simeq\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}\
{\colon}E,$

where $\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})$ is the
$\infty$-category of $\mathbf{E}_{2}$-algebras, and
$\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})\simeq\operatorname{Alg}_{\operatorname{As}}(\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp}))$
and the left equivalence follows from Dunn additivity theorem. Here we abuse
notation by writing $I$ and $E$ for the induced functors. Let us regard
$\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Pr}^{\textup{L}}_{\textup{St}}\big{)}$
as the $\infty$-category of monoidal presentable stable $\infty$-categories.
The left adjoint sends an $\mathbf{E}_{2}$-algebra $A$ to the associative
monoidal $\infty$-category $\operatorname{RMod}_{A}^{\otimes}$. The right
adjoint carries a monoidal presentable stable $\infty$-category
$\mathcal{M}^{\otimes}$ to the endomorphism spectrum
$\operatorname{End}_{\mathcal{M}}(\mathsf{1}_{\mathcal{M}})$ of the unit
object $\mathsf{1}_{\mathcal{M}}$, endowed with an $\mathbf{E}_{2}$-algebra
structure.

We define Hochschild cohomology spectrum
$\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})\\!=\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{D})$
as
$E\big{(}\mathcal{E}\textup{nd}(\mathcal{D})^{\otimes}\big{)}\\!\in\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})$.
The underlying associative algebra
$\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$ is the endomorphism
algebra of the identity functor $\mathcal{D}\to\mathcal{D}$ in
$\operatorname{Fun}^{\textup{L}}(\mathcal{D},\mathcal{D})$.

Consider the counit map of the adjunction:

$\displaystyle\operatorname{RMod}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{D})}^{\otimes}\longrightarrow\mathcal{E}\textup{nd}(\mathcal{D})^{\otimes},$

which is a monoidal functor. Since
$\mathcal{E}\textup{nd}(\mathcal{D})^{\otimes}$ naturally acts on
$\mathcal{D}$, it gives rise to an action of
$\operatorname{RMod}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{D})}^{\otimes}$
on $\mathcal{D}$:

$\displaystyle\mathcal{E}\textup{nd}(\mathcal{D})^{\otimes}\curvearrowright\mathcal{D}\
\ \Rightarrow\ \
\operatorname{RMod}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{D})}^{\otimes}\curvearrowright\mathcal{D}.$

In other words, $\mathcal{D}$ is a left
$\operatorname{RMod}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{D})}^{\otimes}$-module
object in $\operatorname{Pr}^{\textup{L}}_{\textup{St}}$. Let
$\operatorname{RPerf}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})}^{\otimes}\subset\operatorname{RMod}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{D})}^{\otimes}$
be the monoidal full subcategory that consists of compact objects. By the
restrictions, it gives rise to a left
$\operatorname{RPerf}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})}^{\otimes}$-module
object $\mathcal{C}$:

$\displaystyle\operatorname{RPerf}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})}^{\otimes}\curvearrowright\mathcal{C},$

in the $\infty$-category $\operatorname{\mathcal{S}t}$ of small stable
idempotent-complete $\infty$-categories in which morphisms are exact functors
($\operatorname{\mathcal{S}t}$ also admits a suitable symmetric monoidal
structure). Informally, we think of it as a categorical associative action of
$\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$ on $\mathcal{C}$. This is
induced by the adjunction so that it has an evident universal property.

Construct a functor $\operatorname{\mathcal{S}t}\to\operatorname{Sp}$ which
carries $\mathcal{C}$ to the Hochschild homology spectrum
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$. In the classical
differential graded context, Hochschild chain complex comes equipped with the
Connes operator. In our general setting, it is natural to encode such
structures by means of circle actions: Hochschild homology spectrum
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ is promoted to a spectrum
with an $S^{1}$-action, that is, an object of
$\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}$. Thus we configure
the assignment
$\mathcal{C}\mapsto\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ as a
symmetric monoidal functor

$\displaystyle\operatorname{\mathcal{HH}}_{\bullet}(-)\colon\
\operatorname{\mathcal{S}t}^{\otimes}\longrightarrow\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}^{\otimes},$

where $\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}$ inherits a
pointwise symmetric monoidal structure from the structure on
$\operatorname{Sp}$.

Applying the symmetric monoidal functor
$\operatorname{\mathcal{HH}}_{\bullet}(-)$ to the left
$\operatorname{RPerf}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})}^{\otimes}$-module
object $\mathcal{C}$, we obtain a left
$\operatorname{\mathcal{HH}}_{\bullet}\big{(}\\!\operatorname{RPerf}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})}\\!\big{)}$-module
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ in
$\operatorname{Fun}\big{(}\\!BS^{1}\\!,\operatorname{Sp}\\!\big{)}$. Note that
$\operatorname{\mathcal{HH}}_{\bullet}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}))$
$\simeq\operatorname{\mathcal{HH}}_{\bullet}\big{(}\operatorname{RPerf}_{\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})}\big{)}$
is an associative algebra object in
$\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}$. In other words, it
is an associative ring spectrum equipped with an $S^{1}$-action.

There is a topological operad ($\infty$-operad) $\mathbf{Cyl}$ (defined in a
geometric way). We have a canonical equivalence
$\operatorname{Alg}_{\mathbf{Cyl}}(\operatorname{Sp})\simeq\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}\big{)}$
between $\infty$-categories of algebras, where
$\operatorname{Alg}_{\mathbf{Cyl}}(\operatorname{Sp})$ is the
$\infty$-category of algebras over $\mathbf{Cyl}$. There is another two-
colored topological operad ($\infty$-operad) $\mathbf{DCyl}$ such that operads
$\mathbf{E}_{2}^{\otimes}$ and $\mathbf{Cyl}$ are contained in $\mathbf{DCyl}$
as full suboperads:
$\mathbf{E}_{2}^{\otimes}\subset\mathbf{DCyl}\supset\mathbf{Cyl}$. Let
$i\colon\mathbf{E}_{2}^{\otimes}\hookrightarrow\mathbf{DCyl}$ denote the
inclusion, and let
$i_{!}\colon\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})\to\operatorname{Alg}_{\mathbf{DCyl}}(\operatorname{Sp})$
be the left adjoint of the forgetful functor
$i^{*}\colon\operatorname{Alg}_{\mathbf{DCyl}}(\operatorname{Sp})\to\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})$.
Consider the sequence

$\displaystyle\operatorname{Alg}_{\mathbf{E}_{2}}(\operatorname{Sp})\stackrel{{\scriptstyle
i_{!}}}{{\longrightarrow}}\operatorname{Alg}_{\mathbf{DCyl}}(\operatorname{Sp})\stackrel{{\scriptstyle\textup{forget}}}{{\longrightarrow}}\operatorname{Alg}_{\mathbf{Cyl}}(\operatorname{Sp})\simeq\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}\big{)}$

and denote by $i_{!}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}))_{C}$
the image of $\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$ under the
composite. We construct a canonical equivalence
$i_{!}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}))_{C}\simeq\operatorname{\mathcal{HH}}_{\bullet}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}))$
in
$\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}\big{)}$.

Assembling the constructions, we obtain a triple

$\displaystyle(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),i_{!}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}))_{C}\simeq\operatorname{\mathcal{HH}}_{\bullet}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})),$

where $\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C})$ is the
$\mathbf{E}_{2}^{\otimes}$-algebra,
$\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ is the left
$\operatorname{\mathcal{HH}}_{\bullet}(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}))$-module
(in $\operatorname{Fun}\big{(}BS^{1},\operatorname{Sp}\big{)}$). As mentioned
in Theorem 1.2, we prove that the triple exactly amounts to a
$\mathbf{KS}$-algebra
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$,
that is, the structure of an algebra over the Kontsevich–Soibelman operad
$\mathbf{KS}$ on the pair
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$.
This completes the construction.

2.2. This paper is organized as follows: Section 3 collects conventions and
some of the notation that we will use. In Section 4, we discuss algebras over
the Kontsevich–Soibelman operad. The main result of Section 4 is Corollary
4.21 (= Theorem 1.2). Along the way, we introduce several topological colored
operads ($\infty$-operads). In Section 5, we give a brief review of Hochschild
cohomology spectra that we will use. In Section 6, we give a construction of
the assignment
$\mathcal{C}\mapsto\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C})$ which
satisfies the requirements for our goal (partly because we are not able to
find a suitable construction in the literature). The results of this section
will be quite useful for various purposes other than the subject of this
paper. In Section 7, we prove Theorem 7.14 (= Theorem 1.1). Namely, we
construct a $\mathbf{KS}$-algebra
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$.
In Section 8, we study the action morphisms determined by the structure of the
$\mathbf{KS}$-algebra on
$(\operatorname{\mathcal{HH}}^{\bullet}(\mathcal{C}),\operatorname{\mathcal{HH}}_{\bullet}(\mathcal{C}))$.
In Section 9, we give a generalization to an equivariant setting (cf. Theorem
1.3): $\mathcal{C}$ is endowed with the action of a group (a group object in
the $\infty$-category of spaces).

##  3 Notation and convention

Throughout this paper we use the theory of quasi-categories. We assume that
the reader is familiar with this theory and operads. A quasi-category is a
simplicial set which satisfies the weak Kan condition of Boardman–Vogt. The
theory of quasi-categories from the viewpoint of models of
$(\infty,1)$-categories were extensively developed by Joyal and Lurie [21, 25,
27]. Following [27], we shall refer to quasi-categories as
$\infty$-categories. Our main references are [25] and [27]. Given an ordinary
category $\mathcal{C}$, by passing to the nerve
$\operatorname{N}(\mathcal{C})$, we think of $\mathcal{C}$ as the
$\infty$-category $\operatorname{N}(\mathcal{C})$. We usually abuse notation
by writing $\mathcal{C}$ for $\operatorname{N}(\mathcal{C})$ even when
$\mathcal{C}$ should be thought of as a simplicial set or an
$\infty$-category.

We use the theory of $\infty$-operads which is thoroughly developed in [25].
The notion of $\infty$-operads gives one of the models of colored operads.
Thanks to Hinich [15], there is a comparison between algebras over
differential graded operads and algebras over $\infty$-operads in values in
chain complexes. In particular, in characteristic zero, [15] establishes an
equivalence between two notions of algebras, see loc. cit.

Here is a list of some of the conventions and notation that we will use:

  * •

$\mathbf{Z}$: the ring of integers, $\mathbf{R}$ denotes the set of real
numbers which we regard as either a topological space or a ring.

  * •

$\Delta$: the category of linearly ordered non-empty finite sets (consisting
of [0], [1], …, $[n]=\\{0,\dots,n\\}$, …).

  * •

$\Delta^{n}$: the standard $n$-simplex.

  * •

N: the simplicial nerve functor (cf. [27, Section 1.1.5]).

  * •

$\mathcal{S}$: $\infty$-category of small spaces. We denote by
$\widehat{\mathcal{S}}$ the $\infty$-category of large spaces (cf. [27,
Section 1.2.16]).

  * •

$\mathcal{C}^{\simeq}$: the largest Kan subcomplex of an $\infty$-category
$\mathcal{C}$.

  * •

$\mathcal{C}^{\rm op}$: the opposite $\infty$-category of an
$\infty$-category. We also use the superscript “op” to indicate the opposite
category for ordinary categories and enriched categories.

  * •

$\operatorname{Cat}_{\infty}$: the $\infty$-category of small
$\infty$-categories.

  * •

$\operatorname{Sp}$: the stable $\infty$-category of spectra.

  * •

$\operatorname{Fun}(A,B)$: the function complex for simplicial sets $A$ and
$B$.

  * •

$\operatorname{Fun}_{C}(A,B)$: the simplicial subset of
$\operatorname{Fun}(A,B)$ classifying maps which are compatible with given
projections $A\to C$ and $B\to C$.

  * •

$\operatorname{Map}(A,B)$: the largest Kan subcomplex of
$\operatorname{Fun}(A,B)$ when $A$ and $B$ are $\infty$-categories.

  * •

$\operatorname{Map}_{\mathcal{C}}(C,C^{\prime})$: the mapping space from an
object $C\in\mathcal{C}$ to $C^{\prime}\in\mathcal{C}$, where $\mathcal{C}$ is
an $\infty$-category. We usually view it as an object in $\mathcal{S}$ (cf.
[27, Section 1.2.2]).

  * •

$\operatorname{Fin}_{\ast}$: the category of pointed finite sets $\langle
0\rangle,\langle 1\rangle,\dots\langle n\rangle,\dots$, where $\langle
n,\rangle=\\{*,1,\dots,n\\}$ with the base point $*$. We write $\Gamma$ for
$\operatorname{N}(\operatorname{Fin}_{\ast})$. $\langle
n\rangle^{\circ}=\langle n\rangle\backslash*$. Notice that the (nerve of)
Segal’s gamma category is the opposite category of our $\Gamma$.

  * •

$\mathcal{P}^{\textup{act}}$: If $\mathcal{P}$ is an $\infty$-operad, we write
$\mathcal{P}^{\textup{act}}$ for the subcategory of $\mathcal{P}$ spanned by
active morphisms.

  * •

$\operatorname{Triv}^{\otimes}$: the trivial $\infty$-operad [25, Example
2.1.1.20].

  * •

$\operatorname{As}^{\otimes}$: the associative operad [25, Section 4.1.1], we
use the notation slightly different from loc. cit. Informally, an
$\operatorname{As}$-algebra (an algebra over $\operatorname{As}^{\otimes}$) is
an unital associative algebra. For a symmetric monoidal $\infty$-category
$\mathcal{C}^{\otimes}$, we write
$\operatorname{Alg}_{\operatorname{As}}(\mathcal{C})$ for the
$\infty$-category of $\operatorname{As}$-algebra objects. We refer to an
object of $\operatorname{Alg}_{\operatorname{As}}(\mathcal{C})$ as an
associative algebra object in $\mathcal{C}^{\otimes}$. We refer to a monoidal
$\infty$-category over $\operatorname{As}^{\otimes}$ as an associative
monoidal $\infty$-category.

  * •

$\operatorname{\mathcal{LM}}^{\otimes}$: the $\infty$-operad defined in [25,
Definition 4.2.1.7]. An algebra over $\operatorname{\mathcal{LM}}^{\otimes}$
is a pair $(A,M)$ such that an unital associative algebra $A$ and a left
$A$-module $M$. For a symmetric monoidal $\infty$-category
$\mathcal{C}^{\otimes}\to\Gamma$, we write
$\operatorname{LMod}\big{(}\mathcal{C}^{\otimes}\big{)}$ or
$\operatorname{LMod}(\mathcal{C})$ for
$\operatorname{Alg}_{\operatorname{\mathcal{LM}}^{\otimes}}\big{(}\mathcal{C}^{\otimes}\big{)}$.

  * •

$\mathbf{E}^{\otimes}_{n}$: the $\infty$-operad of little $n$-cubes. For a
symmetric monoidal $\infty$-category $\mathcal{C}^{\otimes}$, we write
$\operatorname{Alg}_{\mathbf{E}_{n}}(\mathcal{C})$ for the $\infty$-category
of $\mathbf{E}_{n}$-algebra objects.

##  4 Operads

4.1. We will define several simplicial colored operads which are relevant to
us. By a simplicial colored operad, we mean a colored operad in the symmetric
monoidal category of simplicial sets. A simplicial colored operad is also
referred to as a symmetric multicategory enriched over the category of
simplicial sets.

######  Definition 4.1.

Let $(0,1)$ denote the open interval $\\{x\in\mathbf{R}\,|\,0<x<1\\}$. For
$n\geq 0$, let $(0,1)^{n}$ be the $n$-fold product, i.e., the $n$-dimensional
cube. An open embedding $f\colon(0,1)^{n}\to(0,1)^{n}$ is said to be
rectilinear if it is given by

$\displaystyle f(x_{1},\dots,x_{n})=(a_{1}x_{1}+b_{1},\dots,a_{n}x_{n}+b_{n})$

for some real constants $0<a_{1},\dots,a_{n}\leq 1,\ 0\leq
b_{1},\dots,b_{n}<1$, provided that the formula defines an embedding. An
embedding $f\colon(0,1)^{n}\to(0,1)^{n}$ is said to be shrinking if it is
given by $f(x_{1},\dots,x_{n})=(a_{1}x_{1},\dots,a_{n}x_{n})$ for some
$0<a_{1},\dots,a_{n}\leq 1$.

Let $S^{1}$ denote the circle $\mathbf{R}/\mathbf{Z}$ which we regard as a
topological space. A continuous map $f\colon(0,1)^{n}\times
S^{1}\to(0,1)^{n}\times S^{1}$ is said to be rectilinear (resp. shrinking) if
$f=(\phi,\psi)$ such that $\phi\colon(0,1)^{n}\to(0,1)^{n}$ is rectilinear
(resp. shrinking) and $\psi\colon S^{1}\to S^{1}$ is given by a rotation

$\displaystyle S^{1}=\mathbf{R}/\mathbf{Z}\ni x\mapsto
x+r\in\mathbf{R}/\mathbf{Z}=S^{1}$

with $r\in\mathbf{R}/\mathbf{Z}$. In particular, when $n=0$, $f\colon S^{1}\to
S^{1}$ is rectilinear if it is given by a rotation.

Let $n\geq 1$. If a continuous map $f\colon(0,1)^{n}\to(0,1)^{n-1}\times
S^{1}$ is said to be rectilinear if it factors as
$(0,1)^{n}\stackrel{{\scriptstyle
g}}{{\to}}(0,1)^{n-1}\times\mathbf{R}\stackrel{{\scriptstyle
h}}{{\to}}(0,1)^{n-1}\times\mathbf{R}/\mathbf{Z}$, where $h$ is the
projection, and $g$ is given by
$f(x_{1},\dots,x_{n})=(a_{1}x_{1}+b_{1},\dots,a_{n}x_{n}+b_{n})$ for some real
constants $0<a_{1},\dots,a_{n}\leq 1$, $0\leq b_{1},\dots,b_{n-1}<1$,
$b_{n}\in\mathbf{R}$, provided that the formula defines an open embedding.

######  Definition 4.2.

Let $\operatorname{Cyl}$ be a simplicial colored operad defined as follows:

  1. (i)

The set of colors of $\operatorname{Cyl}$ has a single element, which we will
denote by $C$.

  2. (ii)

Let $I=\langle r\rangle^{\circ}$ be a finite set and let $\\{C\\}_{I}$ be a
set of colors indexed by $I$. By abuse of notation, we write $C^{\sqcup r}$
for $\\{C\\}_{I}$, where $r$ is the number of elements of $I$. We remark that
$C^{\sqcup r}$ does not mean the coproduct. We define
$\operatorname{Mult}_{\operatorname{Cyl}}(\\{C\\}_{I},C)=\operatorname{Mult}_{\operatorname{Cyl}}\big{(}C^{\sqcup
r},C\big{)}$ to be the singular simplicial complex of the space

$\displaystyle\textup{Emb}^{\rm rec}\big{(}\big{(}(0,1)\times
S^{1}\big{)}^{\sqcup r},(0,1)\times S^{1}\big{)}$

of embeddings $\big{(}(0,1)\times S^{1}\big{)}^{\sqcup r}\to(0,1)\times S^{1}$
such that the restriction to each component $(0,1)\times S^{1}\to(0,1)\times
S^{1}$ is rectilinear. Here $\big{(}(0,1)\times S^{1}\big{)}^{\sqcup r}$ is
the disjoint union of $(0,1)\times S^{1}$, whose set of connected components
is identified with $I$. The space $\textup{Emb}^{\rm
rec}\big{(}\big{(}(0,1)\times S^{1}\big{)}^{\sqcup r},(0,1)\times
S^{1}\big{)}$ is endowed with the standard topology, that is, the subspace of
the mapping space with compact-open topology.

  3. (iii)

The composition law in $\operatorname{Cyl}$ is given by the composition of
rectilinear embeddings, and a unit map is the identity map.

The color $C$ together with
$\operatorname{Mult}_{\operatorname{Cyl}}(\\{C\\}_{I},C)$ constitutes a
fibrant simplicial colored operad. By a fibrant simplicial colored operad we
mean that every simplicial set
$\operatorname{Mult}_{\operatorname{Cyl}}(\\{C\\}_{I},C)$ is a Kan complex.
Note that the singular simplicial complex of a topological space is a Kan
complex.

######  Definition 4.3.

Let $\overline{\operatorname{Cyl}}$ be a simplicial colored operad defined as
follows:

  1. (i)

The set of colors of $\overline{\operatorname{Cyl}}$ has two elements denoted
by $C$ and $C_{M}$.

  2. (ii)

Let $I=\langle r\rangle^{\circ}$ be a finite set and let $\\{C,C_{M}\\}_{I}$
be a set of colors indexed by $I$, which we think of as a map $p\colon
I\to\\{C,C_{M}\\}$. We also write $C^{\sqcup m}\sqcup C_{M}^{\sqcup n}$ for
$\\{C,C_{M}\\}_{I}$ when $p^{-1}(C)$ (resp. $p^{-1}(C_{M})$) has $m$ elements
(resp. $n$ elements). Let

$\displaystyle\textup{Emb}^{\rm rec}\big{(}\big{(}(0,1)\times
S^{1}\big{)}^{\sqcup m}\sqcup\big{(}(0,1)\times S^{1}\big{)}^{\sqcup
n},(0,1)\times S^{1}\big{)}$

denote the space of embeddings $\big{(}(0,1)\times S^{1}\big{)}^{\sqcup
m}\sqcup\big{(}(0,1)\times S^{1}\big{)}^{\sqcup n}\to(0,1)\times S^{1}$ such
that the restriction to each component is rectilinear (the topology is induced
by compact-open topology). We refer to it as the space of rectilinear
embeddings. For $n\geq 1$, let

$\displaystyle\operatorname{Mult}_{\overline{\operatorname{Cyl}}}^{t}\big{(}C^{\sqcup{m}}\sqcup
C_{M}^{\sqcup n},C_{M}\big{)}$ $\displaystyle\qquad{}:={\textup{Emb}^{\rm
rec}\big{(}\big{(}(0,1)\times S^{1}\big{)}\times
p^{-1}(C)\sqcup\big{(}(0,1)\times S^{1}\big{)}\times p^{-1}(C_{M}),(0,1)\times
S^{1}\big{)}}$

be its subspace that consists of those rectilinear embeddings $f$ such that
each restriction to any component in $\big{(}(0,1)\times S^{1}\big{)}\times
p^{-1}(C_{M})$ is (not only rectilinear but also) shrinking. Here
$\big{(}(0,1)\times S^{1}\big{)}\times p^{-1}(C)\sqcup\big{(}(0,1)\times
S^{1}\big{)}\times p^{-1}(C_{M})$ denotes the finite disjoint union of
$(0,1)\times S^{1}$ indexed by $p^{-1}(C)\sqcup p^{-1}(C_{M})\simeq I$, but we
distinguish between components indexed by $p^{-1}(C)$ and those indexed in
$p^{-1}(C_{M})$ since they play different roles. Notice that
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}^{t}\big{(}C^{\sqcup{m}}\sqcup
C_{M}^{\sqcup n},C_{M}\big{)}$ is the empty space for $n\geq 2$. We define
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}C^{\sqcup{m}}\sqcup
C_{M}^{\sqcup n},C_{M}\big{)}$ to be the singular simplicial complex of
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}^{t}\big{(}C^{\sqcup{m}}\sqcup
C_{M}^{\sqcup n},C_{M}\big{)}$. When $n=0$, we define
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}C^{\sqcup{m}},C_{M}\big{)}$
to be the empty simplicial set.

  3. (iii)

We set
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}(C^{\sqcup{m}},C)=\operatorname{Mult}_{\operatorname{Cyl}}\big{(}C^{\sqcup{m}},C\big{)}$.
If $n\neq 0$,
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}C^{\sqcup{m}}\sqcup
C_{M}^{\sqcup n},C\big{)}$ is the empty set.

  4. (iv)

The composition law is given by the composition of rectilinear embeddings, and
a unit map is the identity map.

The colors $C$, $C_{M}$ together with simplicial sets of maps constitute a
fibrant simplicial colored operad.

######  Definition 4.4.

Let $\operatorname{DCyl}$ be a simplicial colored operad defined as follows:

  1. (i)

The set of colors of $\operatorname{DCyl}$ has two elements, which we denoted
by $D$ and $C$.

  2. (ii)

Let $I=\langle r\rangle^{\circ}$ be a finite set and let $\\{D,C\\}_{I}$ be a
set of colors indexed by $I$, that is, a map $p\colon I\to\\{D,C\\}$. By abuse
of notation we write $D^{\sqcup{l}}\sqcup C^{\sqcup m}$ for $\\{D,C\\}_{I}$
when $p^{-1}(D)$ (resp. $p^{-1}(C)$) has $l$ elements (resp. $m$ elements). We
define
$\operatorname{Mult}_{\operatorname{DCyl}}\big{(}D^{\sqcup{l}},D\big{)}$ to be
the singular simplicial complex of the space

$\displaystyle\textup{Emb}^{\rm rec}\big{(}\big{(}(0,1)^{2}\big{)}^{\sqcup
l},(0,1)^{2}\big{)}$

of embeddings from the disjoint union $(0,1)^{2}\times p^{-1}(D)$ to
$(0,1)^{2}$ such that the restriction to each component is rectilinear, where
the space comes equipped with the subspace topology of the mapping space with
compact-open topology.

If $m\geq 1$,
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m},D\big{)}$ is the empty set.

  3. (iii)

We define $\operatorname{Mult}_{\operatorname{DCyl}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m},C\big{)}$ to be the singular complex of the space of embeddings

$\displaystyle\textup{Emb}^{\rm rec}\big{(}(0,1)^{2}\times
p^{-1}(D)\sqcup\big{(}(0,1)\times S^{1}\big{)}\times p^{-1}(C),(0,1)\times
S^{1}\big{)}$

such that the rescriction to a component $(0,1)^{2}$ is rectilinear, and the
restriction to a component $(0,1)\times S^{1}$ is rectilinear.

  4. (iv)

The composition law and the unit are defined in an obvious way.

The colors $D$, $C$ together with simplicial sets of maps constitute a fibrant
simplicial colored operad.

######  Definition 4.5.

Let $\overline{\operatorname{DCyl}}$ be a simplicial colored operad defined as
follows.

  1. (i)

The set of colors of $\overline{\operatorname{DCyl}}$ has three elements,
which we denote by $D$, $C$, and $C_{M}$.

  2. (ii)

Let $I=\langle r\rangle^{\circ}$ be a finite set and let $\\{D,C,C_{M}\\}_{I}$
be a set of colors indexed by $I$, that is, a map $p\colon
I\to\\{D,C,C_{M}\\}$. By abuse of notation, we write $D^{\sqcup{l}}\sqcup
C^{\sqcup m}\sqcup C_{M}^{\sqcup n}$ for $\\{D,C,C_{M}\\}_{I}$ when
$p^{-1}(D)$ (resp. $p^{-1}(C)$, $p^{-1}(C_{M})$) has $l$ elements (resp. $m$
elements, $n$ elements). We set
$\operatorname{Mult}_{\overline{\operatorname{DCyl}}}\big{(}D^{\sqcup{l}},D\big{)}=\operatorname{Mult}_{\operatorname{DCyl}}\big{(}D^{\sqcup{l}},D\big{)}$.
If $m+n\geq 1$,
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m}\sqcup C_{M}^{\sqcup n},D\big{)}$ is the empty set.

  3. (iii)

We set
$\operatorname{Mult}_{\overline{\operatorname{DCyl}}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup
m},C\big{)}=\operatorname{Mult}_{\operatorname{DCyl}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m},C\big{)}$. If $n\geq 1$,

$\displaystyle\operatorname{Mult}_{\overline{\operatorname{DCyl}}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m}\sqcup C_{M}^{\sqcup n},C\big{)}$

is the empty set.

  4. (iv)

For $n\neq 1$,
$\operatorname{Mult}_{\overline{\operatorname{DCyl}}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m}\sqcup C_{M}^{\sqcup n},C_{M}\big{)}$ is the empty set. For $n=1$,

$\displaystyle\operatorname{Mult}_{\overline{\operatorname{DCyl}}}\big{(}D^{\sqcup{l}}\sqcup
C^{\sqcup m}\sqcup C_{M},C_{M}\big{)}$

is the singular simplicial complex of the space of embeddings

$\displaystyle(0,1)^{2}\times p^{-1}(D)\sqcup\big{(}(0,1)\times
S^{1}\big{)}\times p^{-1}(C)\sqcup\big{(}(0,1)\times S^{1}\big{)}\times
p^{-1}(C_{M})\to(0,1)\times S^{1}$

such that the restriction to a component in $(0,1)^{2}\times p^{-1}(D)$ is
rectilinear, the restriction to each component in $\big{(}(0,1)\times
S^{1}\big{)}\times p^{-1}(C)$ is rectilinear, and the restriction to
$\big{(}(0,1)\times S^{1}\big{)}\times p^{-1}(C_{M})\simeq(0,1)\times S^{1}$
is shrinking. By definition, if $l=0$,
$\operatorname{Mult}_{\overline{\operatorname{DCyl}}}(C^{\sqcup m}\sqcup
C_{M},C_{M})=\operatorname{Mult}_{\overline{\operatorname{Cyl}}}(C^{\sqcup
m}\sqcup C_{M},C_{M})$.

  5. (v)

The composition law and the unit map are defined in an obvious way.

The colors $D$, $C$, $C_{M}$ together with simplicial sets of maps constitute
a fibrant simplicial colored operad.

######  Remark 4.6.

There is a commutative diagram of inclusions of simplicial colored operads:

$\textstyle{\operatorname{Cyl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\operatorname{Cyl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{DCyl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\operatorname{DCyl}}.}$

Each inclusion determines a simplicial colored full suboperad.

We obtain an $\infty$-operad from a fibrant simplicial colored operad. We
recall the construction from [25, Notation 2.1.1.22].

######  Definition 4.7.

Let $P$ be a simplicial colored operad. Let $P_{\rm col}$ be the set of colors
of $P$. We let $P_{\Delta}$ be a simplicial category defined as follows:

  1. (i)

The objects of $P_{\Delta}$ are maps $a\colon\langle n\rangle^{\circ}\to
P_{\rm col}$, that is, pairs $(\langle n\rangle,(C_{1},\dots,C_{n}))$, where
$\langle n\rangle\in\operatorname{Fin}_{\ast}$ and $(C_{1},\dots,C_{n})$ is a
finite sequence $(a(1),\dots,a(n))$ of colors.

  2. (ii)

Let $C=(\langle n\rangle,(C_{1},\dots,C_{n}))$ and $C^{\prime}=(\langle
m\rangle,(C_{1}^{\prime},\dots,C_{m}^{\prime}))$ be two objects. The hom
simplicial set $\operatorname{Map}_{P_{\Delta}}(C,C^{\prime})$ is given by

$\displaystyle\coprod_{\alpha\colon\langle n\rangle\to\langle
m\rangle}\prod_{j\in\langle
m\rangle^{\circ}}\operatorname{Mult}_{P}\left(\\{C_{i}\\}_{i\in\alpha^{-1}(j)},C^{\prime}_{j}\right)\\!.$

  3. (iii)

Composition is determined by the composition laws on
$\operatorname{Fin}_{\ast}$ and on $P$ in an obvious way.

There is a canonical simplicial functor
$P_{\Delta}\to\operatorname{Fin}_{\ast}$ which sends $(\langle
n\rangle,(C_{1},\dots,C_{n}))$ to $\langle n\rangle$. If $P$ is fibrant, the
map of simplicial nerves
$\mathbf{P}:=\operatorname{N}(P_{\Delta})\to\operatorname{N}(\operatorname{Fin}_{\ast})=\Gamma$
constitutes an $\infty$-operad (cf. [25, Proposition 2.1.1.27]). We shall
refer to
$\operatorname{N}(P_{\Delta})\to\operatorname{N}(\operatorname{Fin}_{\ast})=\Gamma$
(or $\operatorname{N}(P_{\Delta})$) as the operadic nerve of $P$. We shall
denote by $\mathsf{P}_{\langle n\rangle}$ the fiber
$\mathbf{P}\times_{\Gamma}\\{\langle n\rangle\\}$ over $\langle n\rangle$. We
usually identify colors with objects in $\mathbf{P}_{\langle 1\rangle}$.

######  Definition 4.8.

We apply the construction in Definiton 4.7 to $\operatorname{Cyl}$,
$\overline{\operatorname{Cyl}}$, $\operatorname{DCyl}$, and
$\overline{\operatorname{DCyl}}$ to obtain $\infty$-operads.

  * •

Let $\mathbf{Cyl}$ be the operadic nerve of $\operatorname{Cyl}$.

  * •

Let $\overline{\mathbf{Cyl}}$ be the operadic nerve of
$\overline{\operatorname{Cyl}}$.

  * •

Let $\mathbf{DCyl}$ be the operadic nerve of $\operatorname{DCyl}$.

  * •

Let $\overline{\mathbf{DCyl}}$ be the operadic nerve of
$\overline{\operatorname{DCyl}}$.

We now recall Kontsevich–Soibelman operad [23].

######  Definition 4.9.

Let KS be the simplicial colored full suboperad of
$\overline{\operatorname{DCyl}}$ which consists of colors $D$, $C_{M}$. We
refer to KS as Kontsevich–Soibelman operad. Let $\mathbf{KS}$ be the operadic
nerve of KS (the notation is slightly diffrent from Introduction). We abuse
terminology by referring to it as Kontsevich–Soibelman operad. In a nutshell,
$\mathbf{KS}\subset\overline{\mathbf{DCyl}}$ is the maximal simplicial
subcomplex spanned by vertices correponding to those tuples which do not
contain the color $C$. It is not difficult to check that KS is equivalent to
that of [16] or [23, Section 11.2]. In [11], a version of KS is called the
cylinder operad.

######  Remark 4.10.

We note that $\operatorname{DCyl}$ has a simplicial full suboperad $\langle
D\rangle\subset\operatorname{DCyl}$ which consists of the single color $D$.
This operad $\langle D\rangle$ is a version of the little $2$-cube operad
(e.g., [25, Definition 5.1.0.1]). Let $\mathbf{E}_{2}^{\otimes}$ be the
operadic nerve of $\langle D\rangle$, which we shall refer to as the
$\infty$-operad of little $2$-cubes.

######  Remark 4.11.

We have the diagram in Remark 4.6 and inlusions
$\textup{KS}\subset\overline{\operatorname{DCyl}}$, $\langle
D\rangle\subset\operatorname{DCyl}$. These inclusions determine the following
diagram of $\infty$-operads:

$\textstyle{\mathbf{E}_{2}^{\otimes}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{Cyl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{DCyl}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbf{Cyl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{KS}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbf{DCyl}}.}$

Let $\textup{E}_{1}$ be the simplicial operad of little $1$-cubes. The
definition is similar to the case of little $2$-cubes (see, e.g., [25,
Proposition 5.1.0.3]). Namely, $\textup{E}_{1}$ has a single color $D_{1}$,
and for a finite sequence $(D_{1},\dots,D_{1})$, the simplicial set
$\operatorname{Mult}_{\textup{E}_{1}}\big{(}D_{1}^{\sqcup n},D_{1}\big{)}$ is
defined to be the singular simplicial complex of the space $\textup{Emb}^{\rm
rec}\big{(}(0,1)^{\sqcup n},(0,1)\big{)}$ of rectilinear embeddings. The
composition law and the unit are defined in the obvious way. Let
$\mathbf{E}_{1}^{\otimes}$ denote the $\infty$-operad of little $1$-cubes,
that is, the operadic nerve of $\textup{E}_{1}$.

######  Definition 4.12.

Let $\overline{\textup{E}}_{1}$ be a simplicial colored operad defined as
follows.

  1. (i)

The set of colors of $\overline{\textup{E}}_{1}$ has two elements which we
denote by $D_{1}$ and $M$.

  2. (ii)

Let $I=\langle r\rangle^{\circ}$ be a finite set and let $\\{D_{1},M\\}_{I}$
be a set of colors indexed by $I$, which is a map $p\colon I\to\\{D_{1},M\\}$.
We write $D_{1}^{\sqcup{m}}\sqcup M^{\sqcup n}$ for $\\{D_{1},M\\}_{J}$ when
$p^{-1}(D_{1})$ (resp. $p^{-1}(M)$) has $m$ elements (resp. $n$ elements). Let
$\textup{Emb}^{\rm rec}\big{(}(0,1)^{\sqcup m}\sqcup(0,1),(0,1)\big{)}$ be the
topological space of embeddings $(0,1)^{\sqcup m}\sqcup(0,1)\to(0,1)$ such
that the restriction to each component is rectilinear. We define
$\operatorname{Mult}_{\overline{\textup{E}}_{1}}\big{(}D_{1}^{\sqcup{m}}\sqcup
M,M\big{)}$ to be the singular simplicial complex of the subspace

$\displaystyle\textup{Emb}^{\rm rec}\big{(}(0,1)\times
p^{-1}(D_{1})\sqcup(0,1)\times p^{-1}(M),(0,1)\big{)}\subset\textup{Emb}^{\rm
rec}\big{(}(0,1)^{\sqcup m}\sqcup(0,1),(0,1)\big{)}.\\!$

The subspace consists of those rectilinear embeddings such that the
restriction to $(0,1)\times p^{-1}(M)\simeq(0,1)$ is shrinking. If $n\neq 1$,
$\operatorname{Mult}_{\overline{\textup{E}}_{1}}\big{(}D_{1}^{\sqcup{m}}\sqcup
M^{\sqcup n},M\big{)}$ is the empty set.

  3. (iii)

We set
$\operatorname{Mult}_{\overline{\textup{E}}_{1}}\big{(}D_{1}^{\sqcup{n}},D_{1}\big{)}=\operatorname{Mult}_{\textup{E}_{1}}\big{(}D_{1}^{\sqcup{n}},D_{1}\big{)}$.
If $m\neq 0$,
$\operatorname{Mult}_{\overline{\textup{E}}_{1}}\big{(}D_{1}^{\sqcup{n}}\sqcup
M^{\sqcup m},D_{1}\big{)}$ is the empty set.

  4. (iv)

The composition law is given by the composition of embeddings, and a unit map
is the identity map.

Let $\overline{\mathbf{E}}_{1}^{\otimes}$ be the operadic nerve of
$\overline{\textup{E}}_{1}$.

######  Remark 4.13.

There is an equivalence from the $\infty$-operad $\mathbf{E}_{1}^{\otimes}$ to
the associative $\infty$-operad $\textup{Assoc}^{\otimes}$, see [25,
Definition 4.1.1.3]. Indeed, if $f\colon(0,1)^{\sqcup n}\to(0,1)$ is a
rectilinear map, then it determines a linear ordering on the set of connected
component $\pi_{0}\big{(}(0,1)^{\sqcup n}\big{)}$ such that $I_{1}>I_{2}$ for
two components $I_{1}$ and $I_{2}$ in $(0,1)^{\sqcup n}$ if $a<b$ for any
$a\in f(I_{1})$ and any $b\in f(I_{2})$. It gives rise to a map from
$\textup{Emb}^{\rm rec}\big{(}(0,1)^{\sqcup n},(0,1)\big{)}$ to the set of
linear ordering on $\pi_{0}\big{(}(0,1)^{\sqcup n}\big{)}$. It is a homotopy
equivalence so that we have an equivalence
$\mathbf{E}_{1}^{\otimes}\to\textup{Assoc}^{\otimes}$ (for details, see [25,
Example 5.1.0.7]).

This equivalence
$\mathbf{E}_{1}^{\otimes}\stackrel{{\scriptstyle\sim}}{{\to}}\textup{Assoc}^{\otimes}$
is extended to an equivalence
$\overline{\mathbf{E}}_{1}^{\otimes}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{\mathcal{LM}}^{\otimes}$
of $\infty$-operads, where $\operatorname{\mathcal{LM}}^{\otimes}$ is the
$\infty$-operad (having two colors) which we use to describe pairs of
associative algebras and left modules [25, Definition 4.2.1.7]. Indeed, as
above, any map $(0,1)^{\sqcup n}\sqcup(0,1)\to(0,1)$ in (ii) Definition 4.12
determines a linear ordering on $\pi_{0}((0,1)^{\sqcup n}\sqcup(0,1))$ such
that the black component is the maximal element. It is easy to see that this
ordering induces an equivalence
$\overline{\mathbf{E}}_{1}^{\otimes}\to\operatorname{\mathcal{LM}}^{\otimes}$
which extends
$\mathbf{E}_{1}^{\otimes}\stackrel{{\scriptstyle\sim}}{{\to}}\textup{Assoc}^{\otimes}$.

4.2. Following [25], we recall the notion of algebras over an $\infty$-operad.
Let $\mathcal{O}\to\Gamma$ be an $\infty$-operads. Let
$\mathcal{M}^{\otimes}\to\Gamma$ be a symmetric monoidal $\infty$-category
whose underlying $\infty$-category is $\mathcal{M}=\mathcal{M}_{\langle
1\rangle}$. An $\mathcal{O}$-algebra in $\mathcal{M}$ is a map
$f\colon{\mathcal{O}}\to\mathcal{M}^{\otimes}$ over $\Gamma$ which preserve
inert morphisms, that is, a map of $\infty$-operads. We define
$\operatorname{Alg}_{\mathcal{O}}\big{(}\mathcal{M}^{\otimes}\big{)}\subset\operatorname{Fun}_{\Gamma}\big{(}\mathcal{O},\mathcal{M}^{\otimes}\big{)}$
to be the full subcategory of
$\operatorname{Fun}_{\Gamma}\big{(}\mathcal{O},\mathcal{M}^{\otimes}\big{)}$
spanned by ${\mathcal{O}}$-algebras. We often write
$\operatorname{Alg}_{\mathcal{O}}(\mathcal{M})$ for
$\operatorname{Alg}_{\mathcal{O}}\big{(}\mathcal{M}^{\otimes}\big{)}$ when the
structure on $\mathcal{M}$ is clear. We refer to
$\operatorname{Alg}_{\mathcal{O}}\big{(}\mathcal{M}^{\otimes}\big{)}$ as the
$\infty$-category of ${\mathcal{O}}$-algebra objects in $\mathcal{M}$, cf.
[25, Definition 2.1.3.1]. When $\mathcal{O}$ is $\Gamma$, we write
$\operatorname{CAlg}\big{(}\mathcal{M}^{\otimes}\big{)}$ for
$\operatorname{Alg}_{\Gamma}\big{(}\mathcal{M}^{\otimes}\big{)}$.

Let $\mathbf{E}_{1}^{\otimes}\to\Gamma$ be the $\infty$-operad of little
$1$-cubes with the natural projection. Let $\big{(}BS^{1}\big{)}_{\Delta}$ be
the simplicial category having a single object $*$ and Hom simplicial set
$\operatorname{Hom}_{(BS^{1})_{\Delta}}(*,*)$. The simplicial set
$\operatorname{Hom}_{(BS^{1})_{\Delta}}(*,*)$ is the simplicial complex of
$S^{1}=\mathbf{R}/\mathbf{Z}$, and the composition is induced by the ordinary
multiplication $S^{1}\times S^{1}\to S^{1}$. We denote by $BS^{1}$ the
simplicial nerve of $\big{(}BS^{1}\big{)}_{\Delta}$. It can also be regarded
as the classifying space of $S^{1}$ in $\mathcal{S}$. Let
$p\colon\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\Gamma$ be the composite
$\mathbf{E}_{1}^{\otimes}\times
BS^{1}\stackrel{{\scriptstyle\textup{pr}_{1}}}{{\to}}\mathbf{E}_{1}^{\otimes}\to\Gamma$.
We note that $\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\Gamma$ is not an
$\infty$-operad. Let $\mathcal{M}^{\otimes}\to\Gamma$ be a symmetric monoidal
$\infty$-category. Though the above definition of algebra objects is not
applicable to $\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\Gamma$, we define
$\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$ as follows (cf. [25, Definition
2.3.3.20]). Let $\rho_{i}\colon\langle n\rangle\to\langle 1\rangle$ be the
unique inert morphism which sends $i\in\langle n\rangle$ to $1\in\langle
1\rangle$. Then $\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$ is the full subcategory of
$\operatorname{Fun}_{\Gamma}\big{(}\mathbf{E}_{1}^{\otimes}\times
BS^{1},\mathcal{M}^{\otimes}\big{)}$ spanned by those maps
$F\colon\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathcal{M}^{\otimes}$
satisfying the condition: If $C$ is an object of
$\mathbf{E}_{1}^{\otimes}\times BS^{1}$ lying over $\langle n\rangle$, and for
$1\leq i\leq n$ $\alpha_{i}\colon C\to C_{i}$ is a locally $p$-coCartesian
morphism covering $\rho_{i}\colon\langle n\rangle\to\langle 1\rangle$, then
$F(\alpha_{i})$ is an inert morphism in $\mathcal{M}^{\otimes}$. Let
$\Delta^{0}\to BS^{1}$ be the natural functor which induces
$\mathbf{E}_{1}^{\otimes}\to\mathbf{E}_{1}^{\otimes}\times BS^{1}$. By the
construction of $\mathbf{E}_{1}^{\otimes}\times BS^{1}$, $F$ belongs to
$\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$ if and only if the composite
$\mathbf{E}_{1}^{\otimes}\to\mathbf{E}_{1}^{\otimes}\times
BS^{1}\stackrel{{\scriptstyle F}}{{\to}}\mathcal{M}^{\otimes}$ carries any
inert morphism in $\mathbf{E}_{1}^{\otimes}$ lying over $\langle
n\rangle\to\langle 1\rangle$ to an inert morphism in $\mathcal{M}^{\otimes}$.
As observed in [25, Remark 2.1.2.9], it is equivalent to the condition that
$\mathbf{E}_{1}^{\otimes}\to\mathbf{E}_{1}^{\otimes}\times
BS^{1}\stackrel{{\scriptstyle F}}{{\to}}\mathcal{M}^{\otimes}$ carries any
inert morphism in $\mathbf{E}_{1}^{\otimes}$ to an inert morphism in
$\mathcal{M}^{\otimes}$.

######  Lemma 4.14.

Let $\mathcal{M}^{\otimes}$ be a symmetric monoidal $\infty$-category whose
underlying category we denote by $\mathcal{M}$. Let
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$ denote the functor
category $($function complex$)$ which is endowed with the pointwise symmetric
monoidal structure induced by that of $\mathcal{M}^{\otimes}$. Namely, the
symmetric monoidal structure on
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$ is given by the
projection
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}:=\operatorname{Fun}\big{(}BS^{1},\mathcal{M}^{\otimes}\big{)}\times_{\operatorname{Fun}(BS^{1},\Gamma)}\Gamma\to\Gamma$.
Then there is a canonical equivalence of $\infty$-categories

$\displaystyle\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\mathbf{E}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}.$

Similarly, there is a canonical equivalence
$\operatorname{Alg}_{\overline{\mathbf{E}}_{1}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\overline{\mathbf{E}}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$.

###### Proof.

We prove that there is an isomorphism of simplicial sets

$\displaystyle\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\mathbf{E}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}.$

Observe that the symmetric monoidal $\infty$-category
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}$ is defined by
the following universal property: for a simplicial set $K$, there is a natural
bijection of

$\operatorname{Hom}_{\operatorname{Set}_{\Delta}}\big{(}K,\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$

with the set of pairs $(\alpha,\beta)$ which makes the diagram commute

$\textstyle{\mathcal{M}^{\otimes}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{BS^{1}\times
K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textup{pr}_{2}}$$\scriptstyle{\beta}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\Gamma.}$

The assignment $(\alpha,\beta)\mapsto\alpha$ induces
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\to\Gamma$.
Therefore, for a simplicial set $L$, a map
$L\to\operatorname{Alg}_{\mathbf{E}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$
amounts to a map $f\colon BS^{1}\times
L\times\mathbf{E}_{1}^{\otimes}\to\mathcal{M}^{\otimes}$ over $\Gamma$ such
that for any vertex $(a,l)$ in $BS^{1}\times L$ and for any inert morphism $i$
in $\mathbf{E}_{1}^{\otimes}$, the image $f((a,l,i))$ is an inert morphism in
$\mathcal{M}^{\otimes}$ (note also that by construction $BS^{1}$ has a single
vertex). Next, we consider the universal property of
$\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$. By the observation before this
lemma, for a simplicial set $L$, a map
$L\to\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$ amounts to a map $g$ such that the
diagram

$\textstyle{\mathcal{M}^{\otimes}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\times\mathbf{E}_{1}^{\otimes}\times
BS^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathbf{E}_{1}^{\otimes}\times
BS^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma}$

commutes and for any vertex $(l,a)$ in $L\times BS^{1}$ and for any inert
morphism $i$ in $\mathbf{E}_{1}^{\otimes}$, the image $g((l,i,a))$ is an inert
morphism in $\mathcal{M}^{\otimes}$. Comparing universal properties of
$\operatorname{Alg}_{\mathbf{E}_{1}}\\!\big{(}\operatorname{Fun}\big{(}BS^{1}\\!,\mathcal{M}\big{)}\big{)}$
and $\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$, we have a canonical isomorphism
of simplicial sets
$\operatorname{Alg}_{\mathbf{E}_{1}}\\!\big{(}\operatorname{Fun}\big{(}BS^{1}\\!,\mathcal{M}\big{)}\big{)}$
$\simeq\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$. The final assertion also follows
from an argument similar to this proof. ∎

######  Construction 4.15.

We will define a functor $\mathbf{E}_{1}^{\otimes}\times
BS^{1}\to\mathbf{Cyl}$ over $\Gamma$. To this end, we consider the following
simplicial categories $\big{(}\textup{E}_{1}\times BS^{1}\big{)}_{\Delta}$ and
$\operatorname{Cyl}_{\Delta}$. Let $\big{(}\textup{E}_{1}\times
BS^{1}\big{)}_{\Delta}$ be a simplicial category defined as follows.

  * •

The objects of $\big{(}\textup{E}_{1}\times BS^{1}\big{)}_{\Delta}$ are
objects $\langle n\rangle$ in $\operatorname{Fin}_{\ast}$ (which we regard as
formal symbols).

  * •

For $\langle n\rangle,\langle m\rangle\in\big{(}\textup{E}_{1}\times
BS^{1}\big{)}_{\Delta}$, the Hom simplicial set
$\operatorname{Hom}_{(\textup{E}_{1}\times BS^{1})_{\Delta}}(\langle
n\rangle,\langle m\rangle)$ is the singular complex of the space

$\displaystyle\coprod_{\alpha\colon\langle n\rangle\to\langle
m\rangle}\bigg{(}\prod_{1\leq j\leq m}\textup{Emb}^{\rm
rec}\big{(}(0,1)\times\alpha^{-1}(j),(0,1)\big{)}\bigg{)}\times S^{1}.$

  * •

The composition is determined by the composition of embeddings and the
multiplication $S^{1}\times S^{1}\to S^{1}$.

There is a canoncial projection $\big{(}\textup{E}_{1}\times
BS^{1}\big{)}_{\Delta}\to\operatorname{Fin}_{\ast}$. The simplicial nerve
$\operatorname{N}\big{(}\big{(}\textup{E}_{1}\times
BS^{1}\big{)}_{\Delta}\big{)}\to\operatorname{N}(\operatorname{Fin}_{\ast})=\Gamma$
is $p\colon\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\Gamma$.

  * •

The objects of $\operatorname{Cyl}_{\Delta}$ are objects $\langle n\rangle$ in
$\operatorname{Fin}_{\ast}$ (which we regard as formal symbols).

  * •

For $\langle n\rangle,\langle m\rangle\in\operatorname{Cyl}_{\Delta}$, the Hom
simplicial set $\operatorname{Hom}_{\operatorname{Cyl}_{\Delta}}(\langle
n\rangle,\langle m\rangle)$ is the singular simplicial complex of the space

$\displaystyle\coprod_{\alpha\colon\langle n\rangle\to\langle
m\rangle}\prod_{1\leq j\leq m}\textup{Emb}^{\rm rec}\big{(}(0,1)\times
S^{1}\times\alpha^{-1}(j),(0,1)\times S^{1}\big{)}.$

The composition is determined by the composition of embeddings.

Passing to simplicial nerves, we get
$\operatorname{N}(\operatorname{Cyl}_{\Delta})\simeq\mathbf{Cyl}\to\Gamma$.
Note that there is a canonical homeomorphism of spaces

$\displaystyle\textup{Emb}^{\rm rec}\big{(}(0,1)\times
S^{1}\times\alpha^{-1}(j),(0,1)\times S^{1}\big{)}\simeq\textup{Emb}^{\rm
rec}\big{(}(0,1)\times\alpha^{-1}(j),(0,1)\big{)}\times\big{(}S^{1}\big{)}^{\times\sharp\alpha^{-1}(j)}$

(each factor $a\in(S^{1})^{\times\sharp\alpha^{-1}(j)}$ determines the
rotation of the restriction $(0,1)\times S^{1}\to(0,1)\times S^{1}$). The
diagonal map $S^{1}\to\big{(}S^{1}\big{)}^{\times\sharp\alpha^{-1}(j)}$
induces a map

$\displaystyle\operatorname{Hom}_{(\textup{E}_{1}\times
BS^{1})_{\Delta}}(\langle n\rangle,\langle
m\rangle)\to\operatorname{Hom}_{\operatorname{Cyl}_{\Delta}}(\langle
n\rangle,\langle m\rangle)$

in the natural way. It gives rise to a functor
$Z\colon\big{(}\textup{E}_{1}\times
BS^{1}\big{)}_{\Delta}\to\operatorname{Cyl}_{\Delta}$ of simplicial
categories. Then it induces a map

$\displaystyle z\colon\ \mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathbf{Cyl}$

over $\Gamma$. Let $\mathcal{M}^{\otimes}$ be a symmetric monoidal
$\infty$-category whose underlying category is $\mathcal{M}$. The restriction
along $\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathbf{Cyl}$ induces a functor
$z^{*}\colon\operatorname{Alg}_{\mathbf{Cyl}}\big{(}\mathcal{M}^{\otimes}\big{)}\to\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$.

######  Proposition 4.16.

The functor

$\displaystyle z^{*}\colon\
\operatorname{Alg}_{\mathbf{Cyl}}\big{(}\mathcal{M}^{\otimes}\big{)}\to\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$

is an equivalence of $\infty$-categories.

Observe that
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}C^{\sqcup{n}}\sqcup
C_{M},C_{M}\big{)}$ is the singular complex of the space which is homeomorphic
to $\operatorname{Mult}_{\overline{E}_{1}}\big{(}D_{1}^{\sqcup{n}}\sqcup
M,M\big{)}\times\big{(}S^{1}\big{)}^{\times n+1}$. As in the case of
$\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathbf{Cyl}$, using
$\operatorname{Mult}_{\overline{E}_{1}}\big{(}D_{1}^{\sqcup{n}}\sqcup
M,M\big{)}\times
S^{1}\to\operatorname{Mult}_{\overline{E}_{1}}\big{(}D_{1}^{\sqcup{n}}\sqcup
M,M\big{)}\times\big{(}S^{1}\big{)}^{\times n+1}$ induced by the diagonal map
$S^{1}\to\big{(}S^{1}\big{)}^{\times n+1}$ we obtain a morphism
$\bar{z}\colon\overline{\mathbf{E}}_{1}^{\otimes}\times
BS^{1}\to\overline{\mathbf{Cyl}}$ over $\Gamma$. The restriction along
$\bar{z}$ gives rise to
$\bar{z}^{*}\colon\operatorname{Alg}_{\overline{\mathbf{Cyl}}}\big{(}\mathcal{M}^{\otimes}\big{)}\to\operatorname{Alg}_{\overline{\mathbf{E}}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$.

######  Proposition 4.17.

The functor

$\displaystyle\bar{z}^{*}\colon\
\operatorname{Alg}_{\overline{\mathbf{Cyl}}}\big{(}\mathcal{M}^{\otimes}\big{)}\to\operatorname{Alg}_{\overline{\mathbf{E}}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$

is an equivalence of $\infty$-categories.

######  Corollary 4.18.

There are canonical equivalences of $\infty$-categories

$\displaystyle\operatorname{Alg}_{\mathbf{Cyl}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\mathbf{E}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$
$\displaystyle\hphantom{\operatorname{Alg}_{\mathbf{Cyl}}\big{(}\mathcal{M}^{\otimes}\big{)}}{}\simeq\operatorname{Alg}_{\textup{Assoc}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$

and

$\displaystyle\operatorname{Alg}_{\overline{\mathbf{Cyl}}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\overline{\mathbf{E}}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}\simeq\operatorname{Alg}_{\overline{\mathbf{E}}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$
$\displaystyle\hphantom{\operatorname{Alg}_{\overline{\mathbf{Cyl}}}\big{(}\mathcal{M}^{\otimes}\big{)}}{}\simeq\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}.$

Here
$\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}=\operatorname{Alg}_{\operatorname{\mathcal{LM}}^{\otimes}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}^{\otimes}\big{)}$.

###### Proof.

It follows from Lemma 4.14, Propositions 4.16 and 4.17, and Remark 4.13. ∎

The proof of Proposition 4.17 is similar to Proposition 4.16. We prove
Proposition 4.16.

###### Proof of Proposition 4.16.

We use the notion of a weak approximation in the sense of [25, Definition
2.3.3.6] (since $z\colon\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathbf{Cyl}$
is not an equivalence of $\infty$-operads, it is necessary to use more
delicate notion). According to [25, Theorem 2.3.3.23], if two conditions

  1. (i)

$\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathbf{Cyl}$ is a weak
approximation,

  2. (ii)

$z\colon\mathbf{E}_{1}^{\otimes}\times BS^{1}\to\mathbf{Cyl}$ induces an
equivalence between the fiber over $\langle 1\rangle$

hold, then
$z^{*}\colon\operatorname{Alg}_{\mathbf{Cyl}}\big{(}\mathcal{M}^{\otimes}\big{)}\to\operatorname{Alg}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}\mathcal{M}^{\otimes}\big{)}$ is an equivalence. We first prove
(ii). Let $z_{\langle 1\rangle}\colon\big{(}\mathbf{E}_{1}^{\otimes}\times
BS^{1}\big{)}_{\langle 1\rangle}\to\mathbf{Cyl}_{\langle 1\rangle}$ be the map
of fibers over $\langle 1\rangle$. Both fibers consist of a unique object
(here we denote it by $*$ whose mapping space $\operatorname{Map}(*,*)$ is
(homotopy) equivalent to $S^{1}$). Taking into account our construction of
$Z\colon\big{(}\textup{E}_{1}\times
BS^{1}\big{)}_{\Delta}\to\operatorname{Cyl}_{\Delta}$, we see that $z_{\langle
1\rangle}$ is a homotopy equivalence $BS^{1}\to BS^{1}$. This proves (ii).
Next we will prove (i). Let $p\colon\mathbf{E}_{1}^{\otimes}\times
BS^{1}\to\Gamma$ be the projection. Let $\textup{Tup}_{n}$ be the subcategory
of $\Gamma_{/\langle n\rangle}$ whose objects are active morphisms $\langle
m\rangle\to\langle n\rangle$ and whose morphisms are equivalences. According
to a criterion [25, Proposition 2.3.3.14], to prove (i), it is enough to prove
that for any $X\in\mathbf{E}_{1}^{\otimes}\times BS^{1}$ with $\langle
n\rangle=p(X)$, $z$ induces a weak homotopy equivalence

$\displaystyle u\colon\ A(X):=\big{(}\mathbf{E}_{1}^{\otimes}\times
BS^{1}\big{)}_{/X}\times_{\Gamma_{/\langle
n\rangle}}\textup{Tup}_{n}\to\mathbf{Cyl}_{/z(X)}\times_{\Gamma_{/\langle
n\rangle}}\textup{Tup}_{n}=:B(z(X)).$

Note that both domain and target are $\infty$-categories. Consequently, it
will suffice to show that $u$ is a categorical equivalence. Clearly, $u$ is
essentially surjective. We prove that $u$ is fully faithful. The general case
is essentially the same as the case $n=1$ except for a more complicated
notation, so that we treat the case of $n=1$. We think of $D_{1}$ as the
unique object of $\big{(}\mathbf{E}_{1}^{\otimes}\times BS^{1}\big{)}_{\langle
1\rangle}$. Also, we write $D_{1}^{m}$ for the unique object of the fiber
$\big{(}\mathbf{E}_{1}^{\otimes}\times BS^{1}\big{)}_{\langle m\rangle}$ over
$\langle m\rangle$ (namely, $D_{1}=D_{1}^{1}$). Let $f\colon D_{1}^{m}\to
D_{1}$ be a map in $\mathbf{E}_{1}^{\otimes}\times BS^{1}$ lying over an
active morphism $\alpha\colon\langle m\rangle\to\langle 1\rangle$ of $\Gamma$.
We regard $f$ as the product $g\times h\colon(0,1)^{\sqcup m}\times
S^{1}\to(0,1)\times S^{1}$ of a rectilinear map $\phi\colon(0,1)^{\sqcup
m}\to(0,1)$ and a rectilinear map $h\colon S^{1}\to S^{1}$. Let
$f^{\prime}\colon D_{1}^{m}\to D_{1}$ be another map in
$\mathbf{E}_{1}^{\otimes}\times BS^{1}$ lying over $\alpha$. We have

$\displaystyle\begin{split}&\operatorname{Map}_{A(D_{1})}(f,f^{\prime})\simeq\operatorname{Map}^{\rm
eq}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}D_{1}^{m},D_{1}^{m}\big{)}\times_{\operatorname{Map}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}(D_{1}^{m},D_{1})}\\{f\\}\\\
&\hphantom{\operatorname{Map}_{A(D_{1})}(f,f^{\prime})}{}\simeq\big{(}S^{1}\times\Sigma_{m}\big{)}\times_{(S^{1}\times\Sigma_{m})}\\{f\\},\end{split}$

where $\operatorname{Map}^{\rm eq}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}D_{1}^{m},D_{1}^{m}\big{)}$ is the full subcategory of
$\operatorname{Map}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}D_{1}^{m},D_{1}^{m}\big{)}$ spanned by equivalences, and
$\operatorname{Map}^{\rm eq}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}D_{1}^{m},D_{1}^{m}\big{)}\to\operatorname{Map}_{\mathbf{E}_{1}^{\otimes}\times
BS^{1}}\big{(}D_{1}^{m},D_{1}\big{)}$ is induced by the composition with
$f^{\prime}$. Here $\Sigma_{m}$ denotes the symmetric group (which comes from
permutations of components). Thus, the mapping space is contractible because
$(S^{1}\times\Sigma_{m})\to\big{(}S^{1}\times\Sigma_{m}\big{)}$ is an
equivalence. Next, we regard the color $C$ in $\operatorname{Cyl}$ as an
object in $\mathbf{Cyl}$ that lies over $\langle 1\rangle$. We denote by
$C^{m}$ the unique object of $\mathbf{Cyl}$ that lies over $\langle m\rangle$.
Let $z(f),z(f^{\prime})\colon C^{m}\to C$ be the images of $f$ and
$f^{\prime}$, respectively. Then we have equivalences in $\mathcal{S}$

$\displaystyle\operatorname{Map}_{B(C)}(z(f),z(f^{\prime}))\simeq\operatorname{Map}^{\rm
eq}_{\mathbf{Cyl}}\big{(}C^{m},C^{m}\big{)}\times_{\operatorname{Map}_{\mathbf{Cyl}}(C^{m},C)}\\{z(f)\\}$
$\displaystyle\hphantom{\operatorname{Map}_{B(C)}(z(f),z(f^{\prime}))}{}\simeq\big{(}\big{(}S^{1}\big{)}^{\times
m}\times\Sigma_{m}\big{)}\times_{((S^{1})^{\times
m}\times\Sigma_{m})}\\{z(f)\\},$

where $\operatorname{Map}^{\rm eq}_{\mathbf{Cyl}}\big{(}C^{m},C^{m}\big{)}$ is
the full subcategory of $\operatorname{Map}_{\mathbf{Cyl}}(C^{m},C^{m})$
spanned by equivalences. It follows from the canonical equivalence
$\big{(}S^{1}\big{)}^{\times m}\times\Sigma_{m}\to\big{(}S^{1}\big{)}^{\times
m}\times\Sigma_{m}$ that $\operatorname{Map}_{B(C)}(z(f),z(f^{\prime}))$ is
contractible. Thus,
$\operatorname{Map}_{A(D_{1})}(f,f^{\prime})\to\operatorname{Map}_{B(C)}(z(f),z(f^{\prime}))$
is an equivalence. We conclude that $u$ is a categorical equivalence. ∎

4.3. Let $\mathcal{M}^{\otimes}$ be a symmetric monoidal $\infty$-category. If
$\mathbf{P}^{\otimes}$ and $\mathbf{Q}^{\otimes}$ are small $\infty$-operads
and $c\colon\mathbf{P}^{\otimes}\to\mathbf{Q}^{\otimes}$ is a morphism of
$\infty$-operads (over $\Gamma$), then we denote by
$c^{*}\colon\operatorname{Alg}_{\mathbf{Q}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{P}}(\mathcal{M})$
the restriction/forgetful functor along $c$. If there exists a left adjoint of
$c^{*}$, we denote it by
$c_{!}\colon\operatorname{Alg}_{\mathbf{P}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{Q}}(\mathcal{M})$.
If $\mathcal{M}^{\otimes}$ admits small colimits and its monoidal
multiplication functor $\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ preserves
small colimits separately in each variable, then there exists a left adjoint
$c_{!}$, cf. [25, Corollary 3.1.3.5].

The diagram $\mathbf{DCyl}\stackrel{{\scriptstyle
a}}{{\leftarrow}}\mathbf{Cyl}\stackrel{{\scriptstyle
b}}{{\to}}\overline{\mathbf{Cyl}}$ induces
$a^{*}\colon\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$
and $b^{*}\colon$
$\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$.
It gives rise to the fiber product
$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})$.
We also have the inclusions $\mathbf{DCyl}\to\overline{\mathbf{DCyl}}$ and
$\overline{\mathbf{Cyl}}\to\overline{\mathbf{DCyl}}$. Then the
restriction/forgetful functors induce

$\displaystyle\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M}).$

Now suppose that $\mathcal{M}^{\otimes}$ admits small colimits and the tensor
product functor $\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ preserves small
colimits separately in each variable. Let
$i\colon\mathbf{E}_{2}^{\otimes}\to\mathbf{DCyl}$ be the canonical inclusion.
We then have an adjoint pair

$\displaystyle i_{!}\colon\
\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\rightleftarrows\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\
\,{\colon}i^{*},$

where $i_{!}$ is fully faithful (indeed, the left adjoint is given by the
operadic left Kan extension so that the unit map of the adjunction is the
identity). We write
$\operatorname{Alg}^{\mathbf{D}}_{\mathbf{DCyl}}(\mathcal{M})$ for the
essential image of $i_{!}$. Note that
$\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\simeq\operatorname{Alg}^{\mathbf{D}}_{\mathbf{DCyl}}(\mathcal{M})$.
We set
$\operatorname{Alg}^{\mathbf{D}}_{\overline{\mathbf{DCyl}}}(\mathcal{M}):=\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})}\operatorname{Alg}^{\mathbf{D}}_{\mathbf{DCyl}}(\mathcal{M})$.

######  Proposition 4.19.

Suppose that $q\colon\mathcal{M}^{\otimes}\to\Gamma$ is a symmetric monoidal
$\infty$-category such that the underlying $\infty$-category $\mathcal{M}$ has
small colimits and the tensor product functor
$\otimes\colon\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ preserves small
colimits separately in each variable. Then the functor

$\displaystyle\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})$

is an equivalence of $\infty$-categories. Moreover, it induces an equivalence
of $\infty$-categories

$\displaystyle\operatorname{Alg}^{\mathbf{D}}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Alg}^{\mathbf{D}}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M}).$

######  Proposition 4.20.

The restriction along the inclusion $\mathbf{KS}\to\overline{\mathbf{DCyl}}$
induces an equivalence of $\infty$-categories

$\displaystyle\operatorname{Alg}^{\mathbf{D}}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M}).$

Now we consider the diagram

$\textstyle{\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})}$

and its restriction to
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}^{\mathbf{D}}(\mathcal{M})$.

######  Corollary 4.21.

We have canonical categorical equivalences

$\displaystyle\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\simeq\operatorname{Alg}^{\mathbf{D}}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\simeq\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M}).$

Moreover, by Corollary 4.18, the $\infty$-category on the left-hand side is
equivalent to

$\displaystyle\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{E}_{1}}(\operatorname{Fun}(BS^{1},\mathcal{M}))}\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)},$

where
$\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{E}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
is the composite

$\displaystyle\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\stackrel{{\scriptstyle
i_{!}}}{{\simeq}}\operatorname{Alg}^{\mathbf{D}}_{\mathbf{DCyl}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\simeq\operatorname{Alg}_{\mathbf{E}_{1}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}.$

In particular, we have an equivalence of $\infty$-categories

$\displaystyle\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})\simeq\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{E}_{1}}(\operatorname{Fun}(BS^{1},\mathcal{M}))}\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}.$

This equivalence commutes with projections to
$\operatorname{Alg}_{\mathbf{E}_{2}}(\mathcal{M})$ in the natural way.

The proof of Proposition 4.19 requires Lurie–Barr–Beck theorem [25, Corollary
4.7.3.16]. Let us consider the comutative diagram

$\textstyle{\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\scriptstyle{G}$$\textstyle{\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G^{\prime}}$$\textstyle{\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times\mathcal{M}.}$

The functor $U$ is the functor in the statement in Proposition 4.19, and $G$
is determined by the forgetful/restriction functor
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})$
and the functor
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\to\mathcal{M}$
given by the evaluation at $C_{M}$. The functor $G^{\prime}$ is determined by
the first projection and the functor
$\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\to\mathcal{M}$
given by the evaluation at $C_{M}$. According to Lurie–Barr–Beck theorem [25,
Corollary 4.7.3.16], $U$ is a categorical equivalence if the following
conditions are satisfied:

  1. (i)

The functors $G$ and $G^{\prime}$ admit left adjoints $F$ and $F^{\prime}$,
respectively.

  2. (ii)

$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})$ admits geometric
realizations of simplicial objects, which are preserved by $G$.

  3. (iii)

$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})$
admits geometric realizations of simplicial objects, which are preserved by
$G^{\prime}$.

  4. (iv)

The functors $G$ and $G^{\prime}$ are conservative.

  5. (v)

For any object $X$ in
$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times\mathcal{M}$, the unit
map $X\to GF(X)\simeq G^{\prime}UF(X)$ induces an equivalence
$G^{\prime}F^{\prime}(X)\to GF(X)$ in
$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times\mathcal{M}$.

###### Proof of Proposition 4.19.

We will prove the conditions (i), (ii), (iii), (iv), (v).

A morphism $f$ in $\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})$
is an equivalence if and only if the evaluations at $D$, $C$, $C_{M}$ are
equivalences. Similarly, a morphism $f$ in
$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})$ is an equivalence if and
only if the evaluations at objects $D,C$ are equivalences. It follows that $G$
is conservative. Similarly, we see that $G^{\prime}$ is conservative. Hence
(iv) is proved. The conditions (ii) and (iii) follow from the existence and
the compatibility of sifted colimits [25, Proposition 3.2.3.1] and the
conservativity in (iv). (If $\mathcal{M}^{\otimes}$ is a presentably symmetric
monoidal $\infty$-category, the condition (i) follows from adjoint functor
theorem since $G$ and $G^{\prime}$ preserve small limit and filtered colimit.
We prove (i) in full generality below.)

We prove the conditions (v) and (i). For this purpose, we first consider the
left adjoint $F^{\prime}$ of $G^{\prime}$. The values under $F^{\prime}$ can
be described in terms of operadic colimits if we assume the existence of a
left adjoint $F^{\prime}$. Let $j\colon\mathbf{Cyl}\to\overline{\mathbf{Cyl}}$
be the canonical inclusion. Let
$j^{\prime}\colon\operatorname{Triv}^{\otimes}\to\overline{\mathbf{Cyl}}$ be a
morphism from the trivial $\infty$-operad to $\overline{\mathbf{Cyl}}$ that is
determined by $C_{M}$. Let $\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}$
denote the coproduct of $\mathbf{Cyl}$ and $\operatorname{Triv}^{\otimes}$.
Namely, it is a coproduct of $\mathbf{Cyl}$ and
$\operatorname{Triv}^{\otimes}$ in the $\infty$-category of $\infty$-operads,
but we use its explicit construction in [25, Construction 2.2.3.3]. By the
universal property, the morphism $j$ and $j^{\prime}$ induces
$k\colon\mathbf{Cyl}\boxplus\textup{Triv}^{\otimes}\to\overline{\mathbf{Cyl}}$.
By [25, Corollary 3.1.3.5], we have an adjoint pair

$\displaystyle k_{!}\colon\
\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\times\mathcal{M}\simeq\operatorname{Alg}_{\mathbf{Cyl}\boxplus\textup{Triv}^{\otimes}}(\mathcal{M})\rightleftarrows\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\
{\colon}\\!k^{*}.$

Here we use the canonical categorical equivalence
$\operatorname{Alg}_{\operatorname{Triv}}(\mathcal{M})\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{M}$.
There are categorical fibrations
$\textup{pr}_{1}\colon\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\times\mathcal{M}\to\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$
and
$\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$
induced by the inclusion $\mathbf{Cyl}\hookrightarrow\overline{\mathbf{Cyl}}$.
The right adjoint $k^{*}$ commutes with these projections to
$\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$. Let
$Y\colon\mathbf{Cyl}\boxplus\textup{Triv}^{\otimes}\to\mathcal{M}^{\otimes}$
be a map of $\infty$-operads, which we regard as an object $Y$ of
$\operatorname{Alg}_{\mathbf{Cyl}\boxplus\textup{Triv}^{\otimes}}(\mathcal{M})$.
For $A\in\overline{\mathbf{Cyl}}$, the evaluation of $k_{!}(Y)$ at $A$ is an
operadic $q$-colimit of the map

$\displaystyle\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}_{/A}^{\textup{act}}:=\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}\times_{\overline{\mathbf{Cyl}}}\overline{\mathbf{Cyl}}_{/A}^{\textup{act}}\to\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\stackrel{{\scriptstyle
Y}}{{\to}}\mathcal{M}^{\otimes},$

lying over
$\big{(}(\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes})_{/A}^{\textup{act}}\big{)}^{\triangleright}\to\Gamma$.
See [25, Sections 3.1.1–3.1.3] for operadic left Kan extensions and operadic
colimits. Note that
$\operatorname{Mult}_{\overline{\operatorname{Cyl}}}\big{(}C^{\sqcup{n}}\sqcup
C_{M}^{\sqcup m},C\big{)}$ is the empty set for $m\neq 0$. Hence
$\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}\times_{\overline{\mathbf{Cyl}}}\overline{\mathbf{Cyl}}_{/C}^{\textup{act}}\simeq\mathbf{Cyl}\times_{\overline{\mathbf{Cyl}}}\overline{\mathbf{Cyl}}_{/C}^{\textup{act}}$
so that there is a final object determined by the identity $C\to C$. It
follows that $\textup{pr}_{1}(Y)\to\textup{pr}_{1}k^{*}k_{!}(Y)$ is an
equivalence in $\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$ for each
$Y\in\operatorname{Alg}_{\mathbf{Cyl}\boxplus\textup{Triv}^{\otimes}}(\mathcal{M})\simeq\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\times\mathcal{M}$.
Thus $(k_{!},k^{*})$ is an adjunction relative to
$\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$. See [25, Definition 7.3.2.2]
for the notion of relative adjunctions. The base change of $(k_{!},k^{*})$
along
$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\to\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})$
gives rise to an adjunction

$\displaystyle
l_{!}\colon\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\\!\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}(\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\\!\times\mathcal{M})\rightleftarrows\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\\!\times_{\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})}\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\
{\colon}\\!l^{*}$

relative to $\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})$, where the right
adjoint is $G^{\prime}$. This shows that there exists a left adjoint
$F^{\prime}$ of $G^{\prime}$. In addition, $F^{\prime}\simeq l_{!}$. We denote
informally by $X=(A,B,M)$ an object of
$\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times\mathcal{M}\simeq\operatorname{Alg}_{\mathbf{Cyl}\boxplus\textup{Triv}^{\otimes}}(\mathcal{M})$,
where $(A,B)\in\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})$,
$M\in\mathcal{M}$, $A$ is the restriction to
$\mathbf{E}_{2}^{\otimes}\subset\mathbf{DCyl}$, and $B$ is the restriction to
$\mathbf{Cyl}\subset\mathbf{DCyl}$. We compute the image of $(A,B,M)$ under
the left adjoint $l_{!}$. The left adjoint $F^{\prime}=l_{!}$ is induced by
$k_{!}$ so that $l_{!}(A,B,M)=((A,B),k_{!}(B,M))$, where $((A,B),k_{!}(B,M))$
indicates the object of the fiber product on the right-hand side. The pair
$(B,M)$ is an object of
$\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\times\mathcal{M}$. Thus we
will compute $k_{!}(B,M)$ in terms of the operadic left Kan extension; we
describe it as a colimit of a certain diagram. Let
$(B,M)\colon\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\to\mathcal{M}^{\otimes}$
be a morphism of $\infty$-operads which corresponds to
$(B,M)\in\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\times\mathcal{M}$. Let
$\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\to\overline{\mathbf{Cyl}}$
be a morphism determined by $\mathbf{Cyl}\to\overline{\mathbf{Cyl}}$ and the
morphism $\operatorname{Triv}^{\otimes}\to\overline{\mathbf{Cyl}}$ classified
by the object $C_{M}$ in the fiber $\overline{\mathbf{Cyl}}_{\langle
1\rangle}$ (by using the same symbol $C_{M}$ we abuse notation). Let
$\overline{\mathbf{Cyl}}^{\textup{act}}$ be the subcategory spanned by those
morphisms whose images in $\Gamma$ are active (cf. [25, Definition 2.1.2.1]).
Let us consider
$\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}^{\textup{act}}_{/C_{M}}=\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}\times_{\overline{\mathbf{Cyl}}}\overline{\mathbf{Cyl}}^{\textup{act}}_{/C_{M}}$.
We have a morphism

$\displaystyle p\colon\
K:=\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}^{\textup{act}}_{/C_{M}}\to\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\stackrel{{\scriptstyle(B,M)}}{{\longrightarrow}}\mathcal{M}^{\otimes}$

that extends
$(\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes})^{\textup{act}}_{/C_{M}}\to\Gamma$.
According to [25, Proposition 3.1.3.2], the operadic left Kan extension
$k_{!}(B,M)\colon\overline{\mathbf{Cyl}}\to\mathcal{M}^{\otimes}$ of $k$
carries $C_{M}$ to a colimit of $p$, that is, the image of the cone point
under an operadic $q$-colimit diagram $p^{\prime}\colon
K^{\triangleright}\to\mathcal{M}^{\otimes}$ that extends $p$. Let $\star$
denote the unique object of $\operatorname{Triv}^{\otimes}$ lying over
$\langle 1\rangle$. Let Sub be a category defined as follows [25, Definition
2.2.3.2]: The objects of Sub are triples $(\langle n\rangle,S,T)$ such that
$\langle n\rangle\in\operatorname{Fin}_{\ast}$, and $S,T\subset\langle
n\rangle$ are subsets such that $S\cap T=\ast$ and $S\cup T=\langle n\rangle$.
A morphism $(\langle n\rangle,S,T)\to(\langle
n^{\prime}\rangle,S^{\prime},T^{\prime})$ is a morphism $\langle
n\rangle\to\langle n\rangle$ in $\operatorname{Fin}_{\ast}$ such that
$f(S)\subset S^{\prime}$ and $f(T)\subset T^{\prime}$. Let

$\displaystyle(\langle n\rangle,S,T,C^{n-1}=(C,\dots,C),\star)$

be an object of $\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}$ lying over
$\langle n\rangle$ such that $(\langle n\rangle,S,T)$ is an object of Sub,
$C^{n-1}$ is the unique object of $\mathbf{Cyl}_{\langle n-1\rangle}$ (lying
over $\langle n-1\rangle$), and
$\star\in\operatorname{Triv}=\operatorname{Triv}^{\otimes}_{\langle
1\rangle}$. This presentation is based on the explicit construction of the
coproducts in [25, Construction 2.2.3.3]. For our purpose below, we may assume
that $T\subset\langle n\rangle$ is of the form $T=\\{*,i\\}$ so that by
default $T$ in the above object is of the form $T=\\{*,i\\}$. The mapping
space from $\big{(}\langle n\rangle,S,T,C^{n-1},\star\big{)}$ to
$\big{(}\langle m\rangle,S^{\prime},T^{\prime},C^{m-1},\star\big{)}$ is given
by

$\displaystyle\Bigg{(}\coprod_{\begin{subarray}{c}\alpha\colon\langle
n\rangle\to\langle m\rangle\\\ \alpha(S)\subset S^{\prime},\ \alpha(T)\subset
T^{\prime}\end{subarray}}\prod_{j\in\langle m\rangle\backslash
T^{\prime}}\textup{Emb}^{\rm rec}\big{(}\big{(}(0,1)\times
S^{1}\big{)}\times\alpha^{-1}(j),(0,1)\times S^{1}\big{)}\Bigg{)}\times*,$

which we regard as an object in $\mathcal{S}$, and $*$ indicates the
contractible space which we regard as the mapping space from $\star$ to
$\star$. Using this description we consider mapping spaces in
$(\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes})^{\textup{act}}_{/C_{M}}$.
We abuse notation by writing $\big{(}\langle
n\rangle,S,T,C^{n-1},\star,\big{(}C^{n-1},C_{M}\big{)}\to C_{M}\big{)}$ for an
object of $K$, where $\big{(}\langle
n\rangle,S,T,C^{n-1},\star\big{)}\in\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}$
and $(C,\dots,C,C_{M})=\big{(}C^{n-1},C_{M}\big{)}\to C_{M}$ is a morphism in
$\overline{\mathbf{Cyl}}^{\textup{act}}$ lying over the active morphism
$\langle n\rangle\to\langle 1\rangle$, where $\big{(}C^{n-1},C_{M}\big{)}$ is
a sequence of $n-1$ $C$’s and a single $C_{M}$ which we regard as an object in
$\overline{\mathbf{Cyl}}_{\langle n\rangle}$. Now it is easy to compute the
mapping space from $H=\big{(}\langle
n\rangle,S,T,C^{n-1},\star,\big{(}C^{n-1},C_{M}\big{)}\stackrel{{\scriptstyle
f}}{{\to}}C_{M}\big{)}$ to $H^{\prime}=\big{(}\langle
m\rangle,S^{\prime},T^{\prime},C^{m-1},\star,\big{(}C^{m-1},C_{M}\big{)}\stackrel{{\scriptstyle
g}}{{\to}}C_{M}\big{)}$ in
$\big{(}\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}\big{)}\times_{\overline{\mathbf{Cyl}}}\overline{\mathbf{Cyl}}^{\textup{act}}_{/C_{M}}$.
We think of $f$ as an embedding $\big{(}(0,1)\times S^{1}\big{)}^{\sqcup
n-1}\sqcup(0,1)\times S^{1}\to(0,1)\times S^{1}$ that belongs to
$\operatorname{Mult}_{\overline{\mathbf{Cyl}}}\big{(}C^{\sqcup n-1}\sqcup
C_{M},C_{M}\big{)}$ (namely, the restricition to the “right component”
$(0,1)\times S^{1}$ is shrinking). Consider the restriction
$\big{(}(0,1)\times S^{1}\big{)}^{\sqcup n-1}\to(0,1)\times S^{1}$ and its
projection $\overline{f}\colon(0,1)^{\sqcup n-1}\to(0,1)$ obtained by
forgetting the $S^{1}$-factor, which is a rectilinear embedding. If we denote
by $D_{1}^{n}$ the unique object in the fiber
$\big{(}\mathbf{E}_{1}^{\otimes}\big{)}_{\langle n\rangle}$ over $\langle
n\rangle\in\Gamma$, we can regard $\overline{f}$ as a map $D_{1}^{n-1}\to
D_{1}:=D_{1}^{1}$ in $\mathbf{E}_{1}^{\otimes}$. Let
$\operatorname{Map}_{(\mathbf{E}_{1}^{\otimes})_{/D_{1}}}(\overline{f},\overline{g})$
be the discrete mapping space from $\overline{f}\colon D_{1}^{n-1}\to D_{1}$
to $\overline{g}\colon D_{1}^{m-1}\to D_{1}$. Given a morphism $H\to
H^{\prime}$, by applying the same procedure to the induced morphism
$C^{n-1}\to C^{m-1}$, we obtain a map
$\operatorname{Map}_{K}(H,H^{\prime})\to\operatorname{Map}_{(\mathbf{E}_{1}^{\otimes})_{/D_{1}}}(\overline{f},\overline{g})$.
Note that
$\operatorname{Map}_{\overline{\mathbf{Cyl}}^{\textup{act}}_{/C_{M}}}(C_{M},C_{M})$
is contractible. Consequently, the restriction to the component $C_{M}$ gives
rise to

$\displaystyle\operatorname{Map}_{K}(H,H^{\prime})\to\operatorname{Map}_{\operatorname{Triv}}(\star,\star)\times_{\operatorname{Map}_{\overline{\mathbf{Cyl}}}(C_{M},C_{M})}\operatorname{Map}_{\overline{\mathbf{Cyl}}^{\textup{act}}_{/C_{M}}}(C_{M},C_{M})\simeq*\times_{S^{1}}*=\mathbf{Z}.$

Taking account of definitions of $\infty$-operads
$\mathbf{Cyl}\boxplus\operatorname{Triv}^{\otimes}$ and
$\overline{\mathbf{Cyl}}$, we see that

$\displaystyle\operatorname{Map}_{K}(H,H^{\prime})\to\operatorname{Map}_{(\mathbf{E}_{1}^{\otimes})_{/D_{1}}}(\overline{f},\overline{g})\times\mathbf{Z}$

is an equivalence in $\mathcal{S}$. Let $L$ be the full subcategory of $K$
spanned by the single object

$\displaystyle\mathsf{Z}=\big{(}\langle 2\rangle,S\simeq\langle
1\rangle,T\simeq\langle 1\rangle,C,\star,(C,C_{M})\stackrel{{\scriptstyle
j}}{{\to}}C_{M}\big{)}$

(a morphism $j$ is uniquely determined up to homotopy). We now claim that
$L\subset K$ is cofinal. It will suffice to prove that for each $\mathsf{V}\in
K$, the $\infty$-category $L\times_{K}K_{\mathsf{V}/}$ is weakly contractible,
see [25, Definition 4.1.3.1]. Let $\mathsf{V}=\big{(}\langle
n\rangle,S,T,C^{n-1},\star,\big{(}C^{n-1},C_{M}\big{)}\stackrel{{\scriptstyle
f}}{{\to}}C_{M}\big{)}$. By the above discussion about mapping spaces, a
morphism $u\colon\mathsf{V}\to\mathsf{Z}$ is uniquely determined by
$a\in\mathbf{Z}$ since
$\operatorname{Map}_{(\mathbf{E}_{1}^{\otimes})_{/D_{1}}}(\overline{f},\overline{j})$
is contractible. Let $u^{\prime}\colon\mathsf{V}\to\mathsf{Z}$ be another
object of $L\times_{K}K_{\mathsf{V}/}$ that is determined by
$a^{\prime}\in\mathbf{Z}$. Note that
$\operatorname{Map}_{K}(\mathsf{Z},\mathsf{Z})\simeq\mathbf{Z}$ and the
composition
$\operatorname{Map}_{K}(\mathsf{Z},\mathsf{Z})\times\operatorname{Map}_{K}(\mathsf{V},\mathsf{Z})\simeq\mathbf{Z}\times\mathbf{Z}\to\operatorname{Map}_{K}(\mathsf{V},\mathsf{Z})\simeq\mathbf{Z}$
can be identified with the additive operation
$+\colon\mathbf{Z}\times\mathbf{Z}\to\mathbf{Z}$ (up to automorphisms of
$\mathbf{Z}$). It follows that
$\operatorname{Map}_{L\times_{K}K_{\mathsf{V}/}}(u,u^{\prime})$ is a
contractible space so that the $\infty$-category $L\times_{K}K_{\mathsf{V}/}$
is contractible. Let $p^{\prime}\colon
K^{\triangleright}\to\mathcal{M}^{\otimes}$ be the operadic $q$-colimit
diagram. Let $p^{\prime\prime}\colon K^{\triangleright}\to\mathcal{M}$ be the
diagram obtained by a $q$-coCartesian natural transformation from
$p^{\prime}$. Since we assume that
$\otimes\colon\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ preserves small
colimits separately in each variable, then by [25, Propositions 3.1.1.15 and
3.1.1.16], $p^{\prime\prime}$ is a colimit diagram of
$p^{\prime\prime}|_{K}\colon$ $K\to\mathcal{M}$, and the image of the cone
point under $p^{\prime\prime}$ is naturally equivalent to the image of the
cone point under $p^{\prime}$. Since $L\subset K$ is cofinal, we have a
canonical equivalence
$\operatorname{colim}p^{\prime\prime}|_{K}\simeq\operatorname{colim}p^{\prime\prime}|_{L}$.

Indeed, $\operatorname{colim}p^{\prime\prime}|_{K}$ is equivalent to $B\otimes
M\otimes S^{1}$ (this computation is not necessary to the proof so that the
reader may skip this paragraph, but it may be helpful to get feeling for the
operadic left Kan extension $F^{\prime}$). By construction, the composite
$L\hookrightarrow K\stackrel{{\scriptstyle p}}{{\to}}\mathcal{M}^{\otimes}$ is
equivalent to a contant diagram so that $p^{\prime\prime}|_{L}\colon
L\to\mathcal{M}$ is a contant diagram taking the value $B\otimes M$. Note that
$\operatorname{Map}_{K}(\mathsf{Z},\mathsf{Z})=\mathbf{Z}$ and there is a
categorical equivalence $L\simeq B\mathbf{Z}\simeq S^{1}$. We deduce that
$\operatorname{colim}p^{\prime\prime}|_{L}\simeq(B\otimes M)\otimes S^{1}$.
Namely, the evaluation $F^{\prime}(A,B,M)(C_{M})$ of $F^{\prime}(A,B,M)$ at
$C_{M}$ is
$\operatorname{colim}p^{\prime\prime}|_{K}\simeq\operatorname{colim}p^{\prime\prime}|_{L}\simeq(B\otimes
M)\otimes S^{1}$. Another way to compute it is as follows. By Corollary 4.18,
we have
$\operatorname{Alg}_{\mathbf{Cyl}}(\mathcal{M})\simeq\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
and
$\operatorname{Alg}_{\overline{\mathbf{Cyl}}}(\mathcal{M})\simeq\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$.
These equivalences commute with forgetful functors arising from the inclusions
$\operatorname{As}\to\operatorname{\mathcal{LM}}$ and
$\mathbf{Cyl}\to\overline{\mathbf{Cyl}}$. The adjunction $(k_{!},k^{*})$ can
be identified with the composite of adjunctions

$\displaystyle\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}\times\mathcal{M}\rightleftarrows\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}\times\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$
$\displaystyle\hphantom{\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}\times\mathcal{M}}{}\rightleftarrows\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)},$

where the left adjunction is induced by the adjunction
$\mathcal{M}\rightleftarrows\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$
which consists of the forgetful functor
$\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\to\mathcal{M}$ and the
left adjoint free functor which sends $M$ to $S^{1}\otimes M$. The right
adjoint in the right adjunction is given by the evaluation of the module
objects
$\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}\\!\to\\!\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}$
and
$\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}\\!\to\\!\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}$
induced by $\operatorname{As}\hookrightarrow\operatorname{\mathcal{LM}}$. The
left adjoint functors carry $(B,M)$ to

$\displaystyle\big{(}B,B\otimes S^{1}\otimes
M\big{)}\in\operatorname{LMod}\big{(}\operatorname{Fun}\big{(}BS^{1},\mathcal{M}\big{)}\big{)}.$

Next we will consider $F(A,B,M)$. Let
$r\colon\mathbf{DCyl}\boxplus\textup{Triv}^{\otimes}\to\overline{\mathbf{DCyl}}$
be a morphism of $\infty$-operads induced by
$\mathbf{DCyl}\hookrightarrow\overline{\mathbf{DCyl}}$ and
$\operatorname{Triv}^{\otimes}\to\overline{\mathbf{DCyl}}$ determined by
$C_{M}\in\overline{\mathbf{DCyl}}_{\langle 1\rangle}$ correpondings to the
color $C_{M}$ (we slightly abuse notation again). By [25, Corollary 3.1.3.5],
we have an adjunction
$r_{!}\colon\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times\mathcal{M}\rightleftarrows\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M}){\colon}\\!r^{*}$.
This shows that there exists a left adjoint $F=r_{!}$ of $G$. Consider

$\displaystyle e\colon\
P:=\mathbf{DCyl}\boxplus\textup{Triv}^{\otimes}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/C_{M}}\stackrel{{\scriptstyle\textup{pr}_{1}}}{{\longrightarrow}}\mathbf{DCyl}\boxplus\textup{Triv}^{\otimes}\stackrel{{\scriptstyle(A,B,M)}}{{\longrightarrow}}\mathcal{M}^{\otimes},$

where the right arrow classifies the algebra object
$(A,B,M)\in\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})\times\mathcal{M}$.
The evaluation of $r_{!}(A,B,M)$ at $C_{M}$ is a colimit of $e$, that is, the
image of the cone point under an operadic $q$-colimit diagram
$e^{\prime}\colon P^{\triangleright}\to\mathcal{M}^{\otimes}$ that extends
$e$. The computation of the colimit of $e$ is similar to that of $p$. We
infomally denote by $(\langle n\rangle,S,T,D^{d},C^{c},\star)$ an object of
$\mathbf{DCyl}\boxplus\textup{Triv}^{\otimes}$, where
$\star\in\operatorname{Triv}$, and $(D^{d},C^{c})$ indicates the sequence of
colors which consists of $d$ $D$’s and $c$ $C$’s which we regard as an object
in $\mathbf{DCyl}_{\langle n-1\rangle}$ ($d+c=n-1$). By abuse of notation, we
write

$\displaystyle R=\big{(}\langle
n\rangle,S,T,D^{d},C^{c},\star,(D^{d},C^{c},C_{M})\stackrel{{\scriptstyle
f}}{{\to}}C_{M}\big{)}$

for an object of $P$, where $f\colon\big{(}D^{d},C^{c},C_{M}\big{)}\to C_{M}$
is a morphism in $\overline{\mathbf{DCyl}}^{\textup{act}}$ that lies over the
active morphism $\langle n\rangle\to\langle 1\rangle$. We compute the mapping
space from $R$ to another object

$\displaystyle R^{\prime}=\big{(}\langle
m\rangle,S^{\prime},T^{\prime},D^{d^{\prime}},C^{c^{\prime}},\star,(D^{d^{\prime}},C^{c^{\prime}},C_{M})\stackrel{{\scriptstyle
g}}{{\to}}C_{M}\big{)}.$

Given a morphism $\phi\colon R\to R^{\prime}$ we have the induced morphism
$\big{(}D^{d},C^{c},C_{M}\big{)}\to\big{(}D^{d^{\prime}},C^{c^{\prime}},C_{M}\big{)}$
in $\overline{\mathbf{DCyl}}^{\textup{act}}_{/C_{M}}$. Notice that it is given
by the union of
$\big{(}D^{d},C^{c}\big{)}\to\big{(}D^{d^{\prime}},C^{c^{\prime}}\big{)}$ and
$C_{M}\to C_{M}$ over $C_{M}$. Moreover, we can think of
$\big{(}D^{d},C^{c}\big{)}\to\big{(}D^{d^{\prime}},C^{c^{\prime}}\big{)}$ as a
rectilinear embedding $\big{(}(0,1)^{2}\big{)}^{\sqcup
d}\sqcup\big{(}(0,1)\times S^{1}\big{)}^{\sqcup
c}\to\big{(}(0,1)^{2}\big{)}^{\sqcup d^{\prime}}\sqcup\big{(}(0,1)\times
S^{1}\big{)}^{\sqcup c^{\prime}}$ over $(0,1)\times S^{1}$. In this way, we
obtain the induced morphism

$\displaystyle\operatorname{Map}_{P}(R,R^{\prime})\to\operatorname{Map}_{\mathbf{DCyl}^{\textup{act}}_{/C}}\big{(}\big{(}D^{d},C^{c}\big{)},\big{(}D^{d^{\prime}},C^{c^{\prime}}\big{)}\big{)}.$

As in the case of $K$, the restriction to $C_{M}$ gives rise to a morphism

$\displaystyle\operatorname{Map}_{P}(R,R^{\prime})\to\operatorname{Map}_{\operatorname{Triv}}(\star,\star)\times_{\operatorname{Map}_{\overline{\mathbf{DCyl}}}(C_{M},C_{M})}\operatorname{Map}_{\overline{\mathbf{DCyl}}^{\textup{act}}_{/C_{M}}}(C_{M},C_{M})\simeq*\times_{S^{1}}*=\mathbf{Z}.$

It gives rise to an equivalence in $\mathcal{S}$:

$\displaystyle\operatorname{Map}_{P}(R,R^{\prime})\to\operatorname{Map}_{\mathbf{DCyl}^{\textup{act}}_{/C}}\big{(}\big{(}D^{d},C^{c}\big{)},\big{(}D^{d^{\prime}},C^{c^{\prime}}\big{)}\big{)}\times\mathbf{Z}.$

Let $Q\subset P$ be the full subcategory spanned by $\mathsf{Z}$ which we
think of as an object of $P$ in the obvious way. As in the case of $K$, using
the above description of $\operatorname{Map}_{P}(R,R^{\prime})$ we see that
for any $\mathsf{V}\in P$, $Q\times_{P}P_{\mathsf{V}/}$ is weakly contractible
so that $Q\subset P$ is cofinal. Let $e^{\prime}\colon
P^{\triangleright}\to\mathcal{M}^{\otimes}$ be an operadic $q$-colimit diagram
that extends $e$. Let $e^{\prime\prime}\colon
P^{\triangleright}\to\mathcal{M}=\mathcal{M}_{\langle 1\rangle}$ be the
diagram obtained by a $q$-coCartesian natural transformation from
$e^{\prime}$. Then the image of the cone point under $e^{\prime}$ is
$\operatorname{colim}e^{\prime\prime}|_{P}\simeq\operatorname{colim}e^{\prime\prime}|_{Q}$.
(We can also deduce that $r_{!}(A,B,M)(C_{M})=F(A,B,M)(C_{M})$ is $B\otimes
M\otimes S^{1}\in\mathcal{M}$ in the same way as described above.)

The projection of $G^{\prime}F^{\prime}(A,B,M)\to GF(A,B,M)$ to $\mathcal{M}$
is the canonical map
$\operatorname{colim}p^{\prime\prime}|_{K}\to\operatorname{colim}e^{\prime\prime}|_{P}$.
We note that the canonical functor $K\to P$ induces an equivalence
$L\stackrel{{\scriptstyle\sim}}{{\to}}Q$. Consequently,
$\operatorname{colim}p^{\prime\prime}|_{K}\to\operatorname{colim}e^{\prime\prime}|_{P}$
can be identified with the equivalence
$\operatorname{colim}p^{\prime\prime}|_{L}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{colim}e^{\prime\prime}|_{Q}$.
Next we consider the projection of $G^{\prime}F^{\prime}(A,B,M)\to GF(A,B,M)$
to $\operatorname{Alg}_{\mathbf{DCyl}}(\mathcal{M})$. Taking into account the
equivalences
$\mathbf{DCyl}\boxplus\textup{Triv}^{\otimes}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/D}\simeq\mathbf{DCyl}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/D}$
and
$\mathbf{DCyl}\boxplus\textup{Triv}^{\otimes}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/C}\simeq\mathbf{DCyl}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/C}$
and the presentation of $F$ in terms of operadic colimits, we see that the
evaluations of unit maps $A\to GF(A,B,M)(D)$ and $B\to GF(A,B,M)(C)$ are
equivalence so that the evaluations $A\simeq G^{\prime}F^{\prime}(A,B,M)(D)\to
GF(A,B,M)(D)$ and $B\simeq G^{\prime}F^{\prime}(A,B,M)(C)\to GF(A,B,M)(C)$ of
$G^{\prime}F^{\prime}(A,B,M)\to GF(A,B,M)$ are also equivalences.
Consequently, $G^{\prime}F^{\prime}(A,B,M)\to GF(A,B,M)$ is an equivalence
since evaluations at $D$, $C$, and the projection to $\mathcal{M}$ are
equivalences. This proves (v). We also have proved the existence of $F$ and
$F^{\prime}$, that is, (i). The final assertion is clear. ∎

###### Proof of Proposition 4.20.

The inclusion $j\colon\mathbf{KS}\to\overline{\mathbf{DCyl}}$ induces an
adjunction

$\displaystyle j_{!}\colon\
\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})\rightleftarrows\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\
{\colon}\\!j^{*}.$

Since $j_{!}$ is fully faithful, it is enough to prove that the essential
image of $j_{!}$ is
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}^{\mathbf{D}}(\mathcal{M})$. Let
$X=(A,B,M)\colon\overline{\mathbf{DCyl}}\to\mathcal{M}^{\otimes}$ be an object
of $\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})$ whose
evaluations at $D$, $C$ and $C_{M}$ are $A$, $B$ and $M$, respectively. Note
that the forgetful functor
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}(\mathcal{M})\to\mathcal{M}\times\mathcal{M}\times\mathcal{M}$
induced by evaluations at $D$, $C$, and $C_{M}$ is conservative. Thus if we
write $q\colon\mathcal{M}^{\otimes}\to\Gamma$ for the structure map, $X$
belongs to
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}^{\mathbf{D}}(\mathcal{M})$ if
and only if

$\displaystyle p^{\prime}\colon\
K^{\triangleright}:=\big{(}\mathbf{E}_{2}^{\otimes}\times_{\mathbf{DCyl}}\mathbf{DCyl}^{\textup{act}}_{/C}\big{)}^{\triangleright}\to\overline{\mathbf{DCyl}}\to\mathcal{M}^{\otimes}$

is an operadic $q$-colimit diagram that extends $p=p^{\prime}|_{K}$, see [25,
Definition 3.1.2.1]. Let $Y\colon\mathbf{KS}\to\mathcal{M}^{\otimes}$ be an
object of $\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})$. Let us consider the
image of $C$ under
$j_{!}(Y)\colon\overline{\mathbf{DCyl}}\to\mathcal{M}^{\otimes}$. It is an
operadic $q$-colimit of

$\displaystyle e\colon\
L:=\mathbf{KS}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/C}\to\mathbf{KS}\to\mathcal{M}^{\otimes}.$

In view of [25, Propositions 3.1.1.15 and 3.1.1.16], to prove the image of
$j_{!}$ is contained in
$\operatorname{Alg}_{\overline{\mathbf{DCyl}}}^{\mathbf{D}}(\mathcal{M})$, it
is enough to observe that the natural functor
$\mathbf{E}_{2}^{\otimes}\times_{\mathbf{DCyl}}\mathbf{DCyl}^{\textup{act}}_{/C}\to\mathbf{KS}\times_{\overline{\mathbf{DCyl}}}\overline{\mathbf{DCyl}}^{\textup{act}}_{/C}$
is a categorical equivalence. It follows from the fact that if a sequence
$E=(D,\dots,D$, $C,\dots,C,C_{M},\dots C_{M})$ (regarded as an object of
$\overline{\mathbf{DCyl}}$) contains $C_{M}$, then there is no morphism from
$E$ to $C$ in $\overline{\mathbf{DCyl}}^{\textup{act}}$. Consequently, we have
a new adjunction
$j_{!}\colon\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})\rightleftarrows\operatorname{Alg}_{\overline{\mathbf{DCyl}}}^{\mathbf{D}}(\mathcal{M})\,{\colon}\\!j^{*}$.
By the comparison of operadic $q$-colimit diagrams from $K^{\triangleright}$
and $L^{\triangleright}$, the counit map of this adjunction is an equivalence.
Therefore, we obtain a categorical equivalence
$\operatorname{Alg}_{\mathbf{KS}}(\mathcal{M})\simeq\operatorname{Alg}_{\overline{\mathbf{DCyl}}}^{\mathbf{D}}(\mathcal{M})$
induced by $j_{!}$ (or $j^{*}$). ∎

##  5 Hochschild cohomology

In this section, we recall Hochschild cohomology spectra of stable
$\infty$-categories $\mathcal{C}$. The definition is based on the principle
that, under a suitable condition on $\mathcal{C}$, Hochschild cohomology of
$\mathcal{C}$ is the endomorphism algebra of the identity functor
$\mathcal{C}\to\mathcal{C}$. Moreover, since
$\operatorname{Fun}(\mathcal{C},\mathcal{C})$ has the monoidal structure given
by the composition, Hochschild cohomology is the endomorphism algebra of the
unit object of $\operatorname{Fun}(\mathcal{C},\mathcal{C})$ so that it comes
equipped with the structure of an $\mathbf{E}_{2}$-algebra, cf. [4, 22] (see
also references cited in loc. cit. for Deligne conjecture concerning
Hochschild cochains). We establish some notation. Let $R$ be a commutative
ring spectrum. Let $\operatorname{Mod}_{R}^{\otimes}$ be the symmetric
monoidal $\infty$-category of $R$-module spectra whose underlying category we
denote by $\operatorname{Mod}_{R}$. Let
$\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})$ be the
$\infty$-category of the associative algebra objects in
$\operatorname{Mod}_{R}$. Let $\operatorname{Pr}^{\textup{L}}$ be the
$\infty$-category of presentable $\infty$-categories whose morphisms are those
functors that preserve small colimits. This category
$\operatorname{Pr}^{\textup{L}}$ admits a symmetric monoidal structure, see
[25, Notation 4.8.1.7 and Proposition 4.8.1.15]. The $\infty$-category of
small spaces $\mathcal{S}$ is a unit object in
$\operatorname{Pr}^{\textup{L}}$. For
$\mathcal{D},\mathcal{D}^{\prime}\in\operatorname{Pr}^{\textup{L}}$, the
tensor product $\mathcal{D}\otimes\mathcal{D}^{\prime}$ comes equipped with a
functor
$\mathcal{D}\times\mathcal{D}^{\prime}\to\mathcal{D}\otimes\mathcal{D}^{\prime}$
which preserves small colimits separately in each variable and satisfies the
following universal property: for any
$\mathcal{F}\in\operatorname{Pr}^{\textup{L}}$, the composition induces a
fully faithful functor

$\displaystyle\operatorname{Fun}^{\textup{L}}(\mathcal{D}\otimes\mathcal{D}^{\prime},\mathcal{F})\to\operatorname{Fun}(\mathcal{D}\times\mathcal{D}^{\prime},\mathcal{F})$

whose essential image is spanned by those functors
$\mathcal{D}\times\mathcal{D}^{\prime}\to\mathcal{F}$ which preserves small
colimits separately in each variable, where
$\operatorname{Fun}^{\textup{L}}(-,-)$ indicates the full subcategory of
$\operatorname{Fun}(-,-)$ spanned by those functors which preserves small
colimits. The underlying associative monoidal $\infty$-category
$\operatorname{Mod}_{R}^{\otimes}$ can be regarded as an associative algebra
object in $\operatorname{Pr}^{\textup{L}}$ since $\operatorname{Mod}_{R}$ is
presentable and the tensor product functor
$\operatorname{Mod}_{R}\times\operatorname{Mod}_{R}\to\operatorname{Mod}_{R}$
preserves small colimits separately in each variable. We denote by
$\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$
the $\infty$-category of left $\operatorname{Mod}_{R}^{\otimes}$-module
objects in $\operatorname{Pr}^{\textup{L}}$. Given
$A\in\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})$, we write
$\operatorname{RMod}_{A}:=\operatorname{RMod}_{A}(\operatorname{Mod}_{R})$ for
the $\infty$-category of right module objects of $A$ in
$\operatorname{Mod}_{R}$. We remark that the forgetful lax symmetric monoidal
functor $\operatorname{Mod}_{R}\to\operatorname{Sp}$ induces
$\operatorname{Mod}_{A}(\operatorname{Mod}_{R})\to\operatorname{Mod}_{A}(\operatorname{Sp})$,
where we use the same notation $A$ to indicate the image in
$\operatorname{Alg}_{\operatorname{As}}(\operatorname{Sp})$. The functor
$\operatorname{Mod}_{A}(\operatorname{Mod}_{R})\to\operatorname{Mod}_{A}(\operatorname{Sp})$
is an equivalence of $\infty$-categories (for example, apply Lurie–Barr–Beck
theorem to this functor endowed with projections to $\operatorname{Sp}$) so
that the notation $\operatorname{RMod}_{A}$ is consistent with that of [25].
The category $\operatorname{RMod}_{A}$ has a natural left module structure
$\operatorname{Mod}_{R}\times\operatorname{RMod}_{A}\to\operatorname{RMod}_{A}$
informally given by $(M,N)\mapsto M\otimes_{R}N$. In what follows, when we
treat the tensor product of objects in $\operatorname{Mod}_{R}$ (over $R$), we
write $\otimes$ for $\otimes_{R}$. The assignment
$A\mapsto\operatorname{RMod}_{A}$ gives rise to a functor

$\displaystyle\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})\to\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$

that sends $A$ to $\operatorname{RMod}_{A}$ and carries a morphism $f\colon
A\to B$ to the base change functor
$\operatorname{RMod}_{A}\to\operatorname{RMod}_{B};\ N\mapsto N\otimes_{A}B$,
that is, a left adjoint of the forgetful functor
$\operatorname{RMod}_{B}\to\operatorname{RMod}_{A}$, see [25, Section 4.8.3,
Notation 4.8.5.10 and Theorem 4.8.5.11]. We have the induced functor

$\displaystyle I\colon\
\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})\simeq\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})_{R/}\to\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)_{\operatorname{Mod}_{R}/},$

which sends $A$ to the base change functor
$\operatorname{Mod}_{R}=\operatorname{RMod}_{R}\to\operatorname{RMod}_{A}$.
The functor $I$ is fully faithful and admits a right adjoint $E$. A morphism
$f\colon\operatorname{Mod}_{R}\to\mathcal{D}$ in
$\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$
is determined by the image $f(R)$ of $R\in\operatorname{Mod}_{R}$ in an
essentially unique way (up to a contractible space of choices). Therefore, an
object of
$\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)_{\operatorname{Mod}_{R}/}$
is regarded as a pair $(\mathcal{D},D)$ such that $\mathcal{D}$ belongs to
$\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$
and $D$ is an object of $\mathcal{D}$. The essential image of $I$ can
naturally be identified with
$\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})$. Namely, it
consists of pairs of the form $(\operatorname{RMod}_{A},A)$: $I$ carries $A$
to $(\operatorname{RMod}_{A},A)$. Put another way, the essential image is
spanned by pairs $(\mathcal{D},D)$ such that $\mathcal{D}$ is a compactly
generated stable $\infty$-category equipped with a single compact generator
$D$. The right adjoint $E$ sends $(\mathcal{D},D)$ to an endormorphism algebra
object
$\textup{End}(D)\in\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})$
[25, Theorem 4.8.5.11]. Since the left adjoint $I$ is fully faithful, the unit
map $\textup{id}\to E\circ I$ is a natural equivalence. Namely, the adjunction
$(I,E)$ is a colocalization. If we denote
$\mathcal{A}\subset\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)_{\operatorname{Mod}_{R}/}$
by the essential image of $I$, then $E$ induces a categorical equivalence
$\mathcal{A}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})$.

The functor $I$ is extended to a symmetric monoidal functor. To explain this,
note that $\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})$
comes equipped with a symmetric monoidal structure induced by that of
$\operatorname{Mod}_{R}^{\otimes}$, see [25, Section 3.2.4] or Construction
7.9. Since $\operatorname{Mod}_{R}^{\otimes}$ is a symmetric monoidal
$\infty$-category such that $\operatorname{Mod}_{R}$ has small colimits and
the tensor product functor
$\operatorname{Mod}_{R}\times\operatorname{Mod}_{R}\to\operatorname{Mod}_{R}$
preserves small colimits separately in each variable, we define
$\operatorname{Mod}^{\otimes}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$
to be the symmetric monoidal $\infty$-category of
$\operatorname{Mod}_{R}^{\otimes}$-module objects in
$\operatorname{Pr}^{\textup{L}}$, cf. [25, Section 3.3.3]. If we denote the
underlying $\infty$-category by
$\operatorname{Mod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$,
then
$\operatorname{Mod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)\simeq\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)$.
Hence
$\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)_{\operatorname{Mod}_{R}/}\simeq\operatorname{Mod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)_{\operatorname{Mod}_{R}/}\simeq\operatorname{Alg}_{\mathbf{E}_{0}^{\otimes}}(\operatorname{Mod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right))$
inherits a symmetric monoidal structure. In summary, we have the adjunction

$\displaystyle I\colon\
\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R})\rightleftarrows\operatorname{LMod}_{\operatorname{Mod}_{R}^{\otimes}}\left(\operatorname{Pr}^{\textup{L}}\right)_{\operatorname{Mod}_{R}/}\
{\colon}\\!E$

whose left adjoint is symmetric monoidal and fully faithful, and whose right
adjoint is lax symmetric monoidal. It gives rise to an adjunction

$\displaystyle\begin{array}[]{ll}I\colon\
\operatorname{Alg}_{\operatorname{As}}(\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R}))\\!\\!&\rightleftarrows\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Mod}_{\operatorname{Mod}_{R}^{\otimes}}\big{(}\operatorname{Pr}^{\textup{L}}\big{)}_{\operatorname{Mod}_{R}/}\big{)}\vspace{.5ex}\\\
\hphantom{I\colon\operatorname{Alg}_{\operatorname{As}}(\operatorname{Alg}_{\operatorname{As}}(\operatorname{Mod}_{R}))}&\simeq\operatorname{Alg}_{\operatorname{As}}\big{(}\operatorname{Mod}_{\operatorname{Mod}_{R}^{\otimes}}\big{(}\operatorname{Pr}^{\textup{L}}\big{)}\big{)}\
{\colon}\\!E,\end{array}$

where we abuse notation by writing $(I,E)$ for the induced adjunction. By
virtue of the canonical equivalence
$\mathbf{E}_{1}^{\otimes}\simeq\operatorname{As}^{\otimes}$ and the
$\infty$-operad version of Dunn additivity theorem [25, Theorem 5.1.2.2], we
have a canonical equivalence