# Relative Calabi–Yau structures and perverse schobers on surfaces

Merlin Christ

###### Abstract

We discuss relative Calabi–Yau structures on functors between $R$-linear
stable $\infty$-categories, with $R$ any $\mathbb{E}_{\infty}$-ring spectrum,
generalizing previous treatments in the setting of dg-categories. Using their
gluing properties, we furthermore construct relative Calabi–Yau structures on
the global sections of perverse schobers, i.e. categorified perverse sheaves,
on surfaces with boundary. We apply this to examples related to Fukaya
categories and representation theory.

###### Contents

  1. 1 Introduction
    1. 1.1 Relative Calabi–Yau structures
    2. 1.2 Perverse schobers and relative Calabi–Yau structures
    3. 1.3 Examples
    4. 1.4 Acknowledgments
    5. 1.5 Notation
  2. 2 Linear $\infty$-categories and Hochschild homology
    1. 2.1 Linear $\infty$-categories
    2. 2.2 Dualizable $\infty$-categories
    3. 2.3 Duals of bimodules
    4. 2.4 Smooth and proper linear $\infty$-categories
    5. 2.5 Traces
    6. 2.6 Hochschild homology
  3. 3 Relative Calabi–Yau structures
    1. 3.1 Left Calabi–Yau structures
    2. 3.2 Right Calabi–Yau structures
    3. 3.3 Behavior under tensor products
    4. 3.4 Gluing Calabi–Yau structures
  4. 4 Perverse schobers on surfaces with boundary
    1. 4.1 Surfaces and ribbon graphs
    2. 4.2 Perverse schobers
    3. 4.3 Monodromy of perverse schobers
  5. 5 Calabi–Yau structures and perverse schobers
    1. 5.1 Calabi–Yau structures locally
    2. 5.2 Weak right Calabi–Yau structures on spherical functors
    3. 5.3 Calabi–Yau structures on global sections
  6. 6 Examples
    1. 6.1 Periodic topological Fukaya categories
    2. 6.2 Spherical objects and Fukaya–Seidel categories
    3. 6.3 Relative Ginzburg algebras of surfaces

##  1 Introduction

Let $k$ be a field. A $k$-linear triangulated category $C$ with finite
dimensional Hom’s is called $n$-Calabi–Yau if there exists an isomorphism of
vector spaces

$\operatorname{Ext}_{C}^{i}(X,Y)\simeq\operatorname{Ext}^{n-i}_{C}(Y,X)^{*}\,,$

bifunctorial in $X,Y\in C$. To obtain a well-behaved notion of $n$-Calabi–Yau
structure on a proper $k$-linear stable $\infty$-category $\mathcal{C}$, one
can ask for a trivialization $S\simeq[n]$ of the Serre functor $S$, i.e. the
functor satisfying the following duality for $R$-linear derived Homs in
$\mathcal{C}$:

$\operatorname{Mor}_{\mathcal{C}}(X,Y)\simeq\operatorname{Mor}_{\mathcal{C}}(Y,S(X))^{*}\in\mathcal{D}(k)\,.$

The trivialization $S\simeq[n]$ is called a weak right $n$-Calabi–Yau
structure on $\mathcal{C}$. Note that the derived natural transformations
$\operatorname{Mor}(\operatorname{id}_{\mathcal{C}},S)$ describe the dual
Hochschild homology $\operatorname{HH}(\mathcal{C})^{*}$ of $\mathcal{C}$. The
identification $S\simeq[n]$ may thus additionally be required to be
$S^{1}$-invariant, i.e. to lie in the image of a dual cyclic homology class
under the morphism
$\operatorname{HH}_{S^{1}}(\mathcal{C})^{*}\to\operatorname{HH}(\mathcal{C})^{*}$.
This leads to the notion of a right $n$-Calabi–Yau structure on $\mathcal{C}$.
The importance of this $S^{1}$-invariance comes from the relation with
topological field theories [Lur09b, Cos07].

There is a similar notion of left $n$-Calabi–Yau structure on a smooth
$k$-linear stable $\infty$-category $\mathcal{C}$, where instead of the Serre
functor, one asks for a trivialization of the bimodule left dual
$\operatorname{id}_{\mathcal{C}}^{!}$ of the evaluation bimodule. The
endofunctor $\operatorname{id}_{\mathcal{C}}^{!}$ is sometimes called the
inverse dualizing bimodule, as it is inverse to the Serre functor if
$\mathcal{C}$ is smooth and proper.

In this paper, we will be concerned with generalizations of left and right
Calabi–Yau structures to $R$-linear functors between $R$-linear stable
$\infty$-categories, where $R$ is the base $\mathbb{E}_{\infty}$-ring
spectrum. Calabi–Yau structures on functors are also referred to as relative
Calabi–Yau structures. This notion was suggested by Toën [Toë14] and fully
worked out in the setting of dg-categories by Brav-Dyckerhoff [BD19]. One of
the main achievements of this paper is a careful lift of this theory to the
setting of $R$-linear stable $\infty$-categories.

There are many natural sources of examples of relative Calabi–Yau structures,
the known ones usually coming from Fukaya categories, representation theory,
topology and algebraic geometry. Further, relative Calabi–Yau structures have
found applications in different areas; applications include the construction
of shifted sympletic structures and Lagrangian structures on moduli spaces of
objects [BD21], of additive categorifications of cluster algebras with
coefficients [Wu23, Chr22a, FKW23], of $2$-Calabi–Yau exact
$\infty$-categories/extriangulated categories from right $2$-Calabi–Yau
functors [Chr22a], as well as the construction of framed $E_{2}$-algebra
structures on Hochschild cohomology [BR23], to name a few.

Relative Calabi–Yau structures posses the remarkable feature that they can be
glued together along suitable pushouts or pullbacks of categories to produce
new relative or absolute Calabi–Yau structures. As emphasized by Brav-
Dyckerhoff [BD19], Calabi–Yau structures should be considered as
noncommutative orientations and their gluing property as a noncommutative
version of the gluing property of oriented manifolds with boundary along
boundary components.

The gluing properties of relative Calabi–Yau structures can be used to
construct relative Calabi–Yau structures on functors with target the
topological Fukaya categories associated with framed marked surfaces, see
[BD19]. These topological Fukaya categories can be seen as the global sections
of perverse schobers on surfaces, i.e. categorified perverse sheaves in the
sense of [KS14]. Using the framework of [Chr22b], we parametrize these
perverse schobers by a ribbon graph homotopic to the surface. Such a
parametrized perverse schobers then amounts by definition to a certain
constructible sheaf of stable $\infty$-categories on the ribbon graph, which
is locally described by spherical adjunctions. In this paper, we discuss ways
to construct relative Calabi–Yau structures on the global sections of more
general perverse schobers on surfaces.

The main results of this paper can be summarized as follows:

  * •

We give an $\infty$-categorical treatment of left and right relative
Calabi–Yau structures. This allows us to define relative Calabi–Yau structures
for stable $\infty$-categories which are linear over an arbitrary base
$\mathbb{E}_{\infty}$-ring spectrum, instead of only a field $k$.

  * •

We introduce a notion of monodromy of parametrized perverse schobers along
loops in the surface and establish the expected properties, such as the
independence on the chosen parametrizing ribbon graph. We prove that
parametrized perverse schobers without singularities are determined, up to
equivalence, by their monodromy.

  * •

In the special case of perverse schobers without singularities, we prove the
existence of a relative Calabi–Yau structure on its $\infty$-category of
global sections given the invariance of a local (negative) cyclic homology
class under the monodromy action. This recovers as a special case Brav-
Dyckerhoff’s result [BD19] on topological Fukaya categories of framed
surfaces.

  * •

We describe further conditions which guarantee the existence of relative
Calabi–Yau structures on the local and on the global sections of singular
parametrized perverse schobers.

  * •

We describe new classes of examples admitting relative Calabi–Yau structures,
including periodic topological Fukaya categories of marked surfaces,
Fukaya–Seidel categories and generalization thereof, the derived categories of
relative Ginzburg algebras associated with $n$-angulated surfaces, as well as
variants of the latter which are linear over an arbitrary
$\mathbb{E}_{\infty}$-ring spectrum.

The remainder of the introduction is structured as follows. We begin in
Section 1.1 by reviewing the notion of a relative Calabi–Yau structure over a
base $\mathbb{E}_{\infty}$-ring spectrum. We proceed in Section 1.2 by stating
our results on relative Calabi–Yau structures on perverse schobers. Finally,
we describe our examples in Section 1.3.

###  1.1 Relative Calabi–Yau structures

Let $R$ be the base $\mathbb{E}_{\infty}$-ring spectrum. The setting for the
definition of relative Calabi–Yau structures is that of stable, presentable,
$R$-linear, dualizable $\infty$-categories and colimit preserving, $R$-linear,
dualizable functors between them. While the definition works for dualizable
$R$-linear $\infty$-categories, in this article, we will often restrict
ourselves to compactly generated $\infty$-categories. In the following, we
sketch the definition of relative Calabi–Yau structure and describe the gluing
properties. The definition makes use of the functoriality of $R$-linear
Hochschild homology, as well as its $S^{1}$-action, which we obtain from the
formalism of traces [HSS17, TV15].

Consider a dualizable $R$-linear functor $F\colon\mathcal{D}\to\mathcal{C}$
between proper, dualizable $R$-linear $\infty$-categories. To define the
notion of a right Calabi–Yau structure on $F$ (also sometimes called a
relative right Calabi–Yau structure on $F$), we assume that
$\mathcal{C},\mathcal{D}$ are proper as $R$-linear $\infty$-categories. The
$R$-linear $\infty$-category $\mathcal{C}$ being proper means that the
evaluation functor
$\operatorname{ev}_{\mathcal{C}}\colon\mathcal{C}^{\vee}\otimes\mathcal{C}\to\operatorname{RMod}_{R}$
admits an $R$-linear right adjoint, which can be identified with an
endofunctor $\operatorname{id}_{\mathcal{C}}^{*}$ of $\mathcal{C}$. If
$\mathcal{C}$ is compactly generated, the functor
$\operatorname{id}_{\mathcal{C}}^{*}$ is a Serre functor on $\mathcal{C}$. The
natural transformations between the Serre functor and the identity functor are
described by the dual Hochschild homology
$\operatorname{HH}(\mathcal{C})^{*}$. In a similar way, a class $\sigma\colon
R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}\coloneqq\operatorname{cof}(\operatorname{HH}(\mathcal{C})^{*}\to\operatorname{HH}(\mathcal{D})^{*})$
in the dual relative Hochschild homology of $F$ defines a map
$\alpha\colon\operatorname{id}_{\mathcal{C}}\to\operatorname{id}_{\mathcal{C}}^{*}[1-n]$
together with a null-homotopy of the composite map
$\operatorname{id}_{\mathcal{D}}\to\operatorname{id}_{\mathcal{D}}^{*}[1-n]$
contained in the following diagram

${{\operatorname{id}_{\mathcal{D}}}}$${{F^{*}(\operatorname{id}_{\mathcal{C}})}}$${\operatorname{cof}}$${\operatorname{fib}}$${F^{*}(\operatorname{id}_{\mathcal{C}}^{*})[1-n]}$${\operatorname{id}_{\mathcal{D}}^{*}[1-n]}$$\scriptstyle{F^{*}(\alpha)}$

with horizontal fiber and cofiber sequences. This null-homotopy allows us to
fill in the dashed arrows. We call $\sigma$ a weak right $n$-Calabi–Yau
structure on $F$ if the vertical maps in the above diagram are equivalences. A
right $n$-Calabi–Yau structure on $F$ then further consists of a lift of
$\sigma$ to a relative dual cyclic homology class. If $F$ admits a right
$n$-Calabi–Yau structure, we also say that $\mathcal{D}$ is relative right
$n$-Calabi–Yau. Non-relative right Calabi–Yau structures correspond to the
case $\mathcal{C}=0$.

We can summarize the definition as follows: If $\mathcal{D}$ is right
$n$-Calabi–Yau, we have a trivialization
$\operatorname{id}_{\mathcal{D}}^{*}[-n]\simeq\operatorname{id}_{\mathcal{D}}$
of the shifted Serre functor. If instead $\mathcal{D}$ is relative right
Calabi–Yau, we have some map
$\operatorname{id}_{\mathcal{D}}^{*}[-n]\to\operatorname{id}_{\mathcal{D}}$,
together with an identification of its cofiber with
$F^{*}(\operatorname{id}_{\mathcal{C}})=GF$, where $G$ is the right adjoint of
$F$. To get a well behaved theory, it is however important that this is not
just any identification of the cofiber of
$\operatorname{id}_{\mathcal{D}}^{*}[-n]\to\operatorname{id}_{\mathcal{D}}$
with $GF$, but rather that there is a specific such cofiber sequence coming
from a relative dual Hochschild class.

Left Calabi–Yau structures for functors between smooth $R$-linear
$\infty$-categories are defined similarly, by replacing the right adjoint of
the evaluation functor by the left adjoint, corresponding to an endofunctor
$\operatorname{id}_{\mathcal{C}}^{!}$, and dual cyclic homology by negative
cyclic homology.

Gluing Calabi–Yau structures

Relative Calabi–Yau structure can be glued together along pushout or pullbacks
of $\infty$-categories. For the gluing of left Calabi–Yau structures, consider
a pushout diagram of smooth, compactly generated $R$-linear
$\infty$-categories and compact objects preserving functors:

${\mathcal{B}_{3}}$${\mathcal{B}_{2}}$${\mathcal{C}_{2}}$${\mathcal{B}_{1}}$${\mathcal{C}_{1}}$${\mathcal{D}}$${\ulcorner}$

######  Theorem 1 (Theorem 3.14,[BD19] for $R=k$ a field).

If the functors $\mathcal{B}_{1}\times\mathcal{B}_{2}\to\mathcal{C}_{1}$ and
$\mathcal{B}_{2}\times\mathcal{B}_{3}\to\mathcal{C}_{2}$ carry $R$-linear left
$n$-Calabi–Yau structures, which are compatible at $\mathcal{B}_{2}$, then the
functor $\mathcal{B}_{1}\times\mathcal{B}_{3}\to\mathcal{D}$ inherits an
$R$-linear left $n$-Calabi–Yau structure.

For the gluing of right Calabi–Yau structures, we consider a pullback diagram
of proper, compactly-generated $R$-linear $\infty$-categories and compact
objects preserving functor functors as follows:

${\mathcal{D}}$${\mathcal{C}_{2}}$${\mathcal{B}_{3}}$${\mathcal{C}_{1}}$${\mathcal{B}_{2}}$${\mathcal{B}_{1}}$${\lrcorner}$

######  Theorem 2 (Theorem 3.15).

If the functors $\mathcal{C}_{1}\to\mathcal{B}_{1}\times\mathcal{B}_{2}$ and
$\mathcal{C}_{2}\to\mathcal{B}_{2}\times\mathcal{B}_{3}$ carry $R$-linear
right $n$-Calabi–Yau structures, which are compatible at $\mathcal{B}_{2}$,
then the functor $\mathcal{D}\to\mathcal{B}_{1}\times\mathcal{B}_{3}$ inherits
an $R$-linear right $n$-Calabi–Yau structure.

###  1.2 Perverse schobers and relative Calabi–Yau structures

Perverse schobers are a, in general conjectural, categorification of perverse
sheaves, proposed by Kapranov-Schechtman [KS14]. In this paper, we use the
framework of parametrized perverse schobers of [Chr21], which describes
perverse schobers on marked surfaces with boundary in terms of constructible
sheaves valued in stable $\infty$-categories, defined on spanning graphs
embedded in the surfaces. Concretely, a perverse schober $\mathcal{F}$
parametrized by a graph ${{\bf G}}$ is encoded as a functor
$\operatorname{Exit}({{\bf G}})\to\operatorname{St}$, where
$\operatorname{St}$ denotes the $\infty$-category of stable
$\infty$-categories, and the domain denotes the exit path category of ${\bf
G}$, whose objects are the vertices and edges of ${\bf G}$ and whose morphisms
describe the incidence between vertices and edges. The limit of this functor,
denoted by $\Gamma({\bf G},\mathcal{F})$, is called the $\infty$-category of
global sections of $\mathcal{F}$.

Perverse sheaves on surfaces are locally constant away from their
$0$-dimensional singularities. If we only consider connected surfaces, all
generic, i.e. non-singular, stalks of a given perverse sheaf are equivalent.
The same is true for a ${\bf G}$-parametrized perverse schober $\mathcal{F}$:
the value of $\mathcal{F}$ at any edge of ${\bf G}$ is independent of the
chosen edge, up to equivalence, and should be considered as the generic stalk.
We usually denote the generic stalk by $\mathcal{N}$. Locally at each vertex
$v$ of $\Gamma$, a perverse schober is described by a spherical adjunction
$\mathcal{V}\leftrightarrow\mathcal{N}$, with $\mathcal{V}$ called the
$\infty$-category of vanishing cycles at $v$. If $\mathcal{V}\neq 0$, we call
the vertex $v$ a singularity of $\mathcal{F}$. Given any based loop $\gamma$
in ${\bf S}$ not intersecting any singularities of $\mathcal{F}$, we associate
in Section 4.3 a monodromy autoequivalence of $\mathcal{N}$ along $\gamma$. We
show that the monodromy does not depend on the homotopy class, relative the
singularities of $\mathcal{F}$, of the based loop. We further prove that two
perverse schobers without singularities are equivalent if and only if their
generic stalk and monodromy are equivalent, see Proposition 4.26. This shows
that the notion of non-singular perverse schober is equivalent to the notion
of a local system of stable $\infty$-categories on the surface. However, we
stress that what we refer to as the global sections of the non-singular
perverse schober does not coincide with the global sections of this local
system.

Given a ${\bf G}$-parametrized perverse schober $\mathcal{F}$, we can evaluate
global sections at the external (i.e. boundary) edges of ${\bf G}$, whose set
is denoted by ${\bf G}_{1}^{\partial}$. This yields a functor

$\prod_{e\in{{\bf G}}_{1}^{\partial}}\operatorname{ev}_{e}\colon\Gamma({{\bf
G}},\mathcal{F})\longrightarrow\prod_{e\in{{\bf
G}}_{1}^{\partial}}\mathcal{F}(e)\simeq\prod_{e\in{{\bf
G}}_{1}^{\partial}}\mathcal{N}\,.$

The right adjoint of this functor is denoted by $\partial\mathcal{F}$.

Our goal is to study when the $\infty$-category of global sections
$\Gamma({{\bf G}},\mathcal{F})$ admits a relative Calabi–Yau structure, i.e.
in the smooth setting when $\partial\mathcal{F}$ admits a left Calabi–Yau
structure and in the proper setting when $\prod_{e\in{{\bf
G}}_{1}^{\partial}}\operatorname{ev}_{e}$ admits a right Calabi–Yau structure.

Note that finite limits of proper $R$-linear $\infty$-categories in
$\operatorname{St}$ (or equivalently in the $\infty$-category
$\operatorname{LinCat}_{R}$ of $R$-linear $\infty$-categories) are not
necessarily again proper. We can fix this issue by forming the limits in the
$\infty$-category $\operatorname{LinCat}_{R}^{\operatorname{cpt-gen}}$ of
compactly generated $R$-linear $\infty$-categories. The arising notion of
$\infty$-category of global section is denoted by $\Gamma^{\operatorname{cpt-
gen}}(\Gamma,\mathcal{F})$, we call these the locally compact global sections.
In the proper setting, we thus ask for the restriction of $\prod_{e\in{{\bf
G}}_{1}^{\partial}}\operatorname{ev}_{e}$ to $\Gamma^{\operatorname{cpt-
gen}}(\Gamma,\mathcal{F})$ to be relative right Calabi–Yau.

Before we state our results, we describe the local relative Calabi–Yau
structures from which we obtain the global ones via gluing.

Local Calabi–Yau structures

Locally at any vertex $v$ of the graph ${{\bf G}}$, with incident edges
$e_{1},\dots,e_{m}$, the perverse schober $\mathcal{F}$ is given by a
collection of functors $\mathcal{F}(v)\to\mathcal{F}(e_{i})\simeq\mathcal{N}$,
$1\leq i\leq m$. One can show that these functors arise, up to suitable
equivalence, from a single spherical adjunction
$F\colon\mathcal{V}\leftrightarrow\mathcal{N}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}G$
via an explicit construction based on the relative Waldhausen
$S_{\bullet}$-construction, see Proposition 4.7. We say that the adjunction
$F\dashv G$ underlies $\mathcal{F}$ at $v$.

######  Proposition 1 (Combine Propositions 4.7 and 5.2).

Let $\mathcal{F}$ be a ${{\bf G}}$-parametrized perverse schober, with
underlying spherical adjunction
$F\colon\mathcal{V}\leftrightarrow\mathcal{N}\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}G$
near a vertex $v$ of ${{\bf G}}$.

  1. (1)

If $F$ admits a right $n$-Calabi–Yau structure, which restricts to a right
$(n-1)$-Calabi–Yau structure on $\mathcal{N}$, then the functor
$\mathcal{F}(v)\to\prod_{i=1}^{n}\mathcal{F}(e_{i})$ also admits a right
$n$-Calabi–Yau structure.

  2. (2)

If $G$ admits a left $n$-Calabi–Yau structure, which restricts to a left
$(n-1)$-Calabi–Yau structure of $\mathcal{N}$, then the right adjoint
$\prod_{i=1}^{n}\mathcal{F}(e_{i})\to\mathcal{F}(v)$ of the above functor also
admits a left $n$-Calabi–Yau structure.

We further prove a novel criterion for a spherical functor
$F\colon\mathcal{V}\to\mathcal{N}$ between compactly generated, proper
$R$-linear $\infty$-categories, where $\mathcal{N}$ is weakly right
$(n-1)$-Calabi–Yau, to admit a weak right $n$-Calabi–Yau structure: this is
the case if and only if its twist functor $T_{V}$ is equivalent to the shifted
Serre functor $\operatorname{id}_{\mathcal{D}}^{*}[1-n]$, see Proposition 5.7.

Global Calabi–Yau structures

For perverse schobers without singularities, we prove the following.

######  Theorem 3 (Theorem 5.11).

Let $\mathcal{F}$ be a ${\bf G}$-parametrized perverse schober without
singularities that is valued in compactly generated $R$-linear
$\infty$-categories. Suppose that the generic stalk $\mathcal{N}$ is smooth
and admits a left $(n-1)$-Calabi–Yau structure

$\eta\colon R[n-1]\to\operatorname{HH}(\mathcal{N})^{S^{1}}\,.$

Suppose further that the action of the monodromy equivalence along each loop
$\gamma$ in ${\bf S}$ on the cyclic homology
$\operatorname{HH}(\mathcal{N})^{S^{1}}$ preserves $\eta$. Then the functor

$\partial\mathcal{F}\colon\prod_{e\in{{\bf
G}}_{1}^{\partial}}\mathcal{F}(e)\longrightarrow\Gamma({{\bf G}},\mathcal{F})$

admits a left $n$-Calabi–Yau structure.

A similar statement holds for relative right Calabi–Yau structures on the
locally compact global sections $\Gamma^{\operatorname{cpt-gen}}({\bf
G},\mathcal{F})$.

We discuss in Section 1.3 below, how 3 generalizes Brav-Dyckerhoff’s result on
relative Calabi–Yau structures on topological Fukaya categories of framed
marked surfaces.

There is no direct analogue of 3 for perverse schobers with singularities, at
least without fixing the equivalence class of the perverse schober by
requiring it to be of a specific form. Essentially, this is because a perverse
schober is not determined up to equivalence by the separate records of
monodromy data and local singularity data, see Example 4.27. There is however
an almost immediate consequence of the gluing property of relative Calabi–Yau
structures for global sections of perverse schobers, see Theorem 5.10, which
is suitable for being applied in practice.

###  1.3 Examples

Fukaya–Seidel-type categories

Let $\mathcal{C}$ be a proper $k$-linear $\infty$-category with a (weak) right
$n$-Calabi–Yau structure. Given a spherical object $X\in\mathcal{C}$, we show
in Example 6.4 that the derived tensor product functor $X\otimes(\mathchar
45\relax)\colon\mathcal{D}(k)\to\mathcal{C}$ admits a (weak) right
$(n+1)$-Calabi–Yau structure. Gluing together these relative Calabi–Yau
structures yields relative Calabi–Yau structures on directed
$\infty$-categories, equipped with an exceptional collection. This type of
category generalizes the Fukaya–Seidel categories associated with Lefschetz
fibrations [Sei08]. We refer to Section 6.2 for more details.

Relative Ginzburg algebras over any base ring spectrum

The derived $\infty$-categories of relative Ginzburg algebras of $n$-angulated
surface arise as the global sections of parametrized perverse schobers, see
[Chr22b, Chr22a]. In Section 6.3, we construct relative left $n$-Calabi–Yau
structures on these derived $\infty$-categories. In the case $n=3$, this
result is a special case of results [Yeu16, Wu23] on relative left
$3$-Calabi–Yau structures on relative Ginzburg algebras of ice quivers with
potentials or equivalently relative Calabi–Yau completions.

The relevant perverse schobers are locally near each vertex described by the
spherical adjunction
$f^{*}\colon\mathcal{D}(k)\leftrightarrow\operatorname{Fun}(S^{n-1},\mathcal{D}(k))\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}f_{*}$,
where $f\colon S^{n-1}\to\ast$ is the map from the singular simplicial set of
the $(n-1)$-sphere to the point. The functor $f_{*}$ is left $n$-Calabi–Yau.
Furthermore, the functor $\bar{f}^{*}$, obtained by restricting $f^{*}$ to a
functor
$\mathcal{D}(k)\to\operatorname{Ind}\operatorname{Fun}(S^{n-1},\mathcal{D}^{\operatorname{perf}}(k))$,
is right $n$-Calabi–Yau.

We can replace $\mathcal{D}(k)$ by $\operatorname{RMod}_{R}$, with $R$ an
$\mathbb{E}_{\infty}$-ring spectrum, to obtain an $R$-linear version of the
adjunction. We expect that the both the Calabi–Yau structures of $f_{*}$ and
$\bar{f}^{*}$ can be lifted to the $R$-linear setting, but only prove that we
have a weak right Calabi–Yau structure on $\bar{f}^{*}$. It’s existence is
proven using the criterion for the existence of weak right Calabi–Yau
structures on spherical functors of Proposition 5.7. Via the gluing of
relative Calabi–Yau structures, the Calabi–Yau structure on $\bar{f}^{*}$
allows us to construct relative weak right $n$-Calabi–Yau structures on the
locally compact global sections of the $R$-linear versions of the perverse
schobers with global sections the ($\operatorname{Ind}$-completed) finite
derived categories of the relative Ginzburg algebras, at least for $n$ odd.

Besides the left Calabi–Yau structures for this class of examples, also many
other classes of examples of $R$-linear relative Calabi–Yau structures remain
to be worked out.

Periodic topological Fukaya categories

The author’s initial motivating example for treating relative Calabi–Yau
structures over an arbitrary base was the construction of relative right
$2$-Calabi–Yau structures on $1$-periodic topological Fukaya categories of
marked surfaces. Their construction is the topic of Section 6.1. These
relative Calabi–Yau structures in turn induce $2$-Calabi–Yau Frobenius exact
$\infty$-structures on these $\infty$-categories, which can be used for the
additive categorification of cluster algebras with coefficients of surfaces,
see [Chr22a].

As $k$-linear $\infty$-categories, with $k$ a field, these periodic categories
are smooth but not proper, since the Ext-groups are non-zero in infinitely
many degrees. This changes when we work with respect to a different base. For
$n$ an integer, the derived category of $n$-periodic chain complexes is
equivalent to the derived $\infty$-category of the dg-algebra $k[t_{n}^{\pm}]$
of graded Laurent algebras, with generator $t_{n}$ in degree $n$. If $n$ is
even, then $k[t_{n}^{\pm}]$ is graded commutative and thus gives rise to an
$\mathbb{E}_{\infty}$-ring spectrum. If $n$ is odd, we can consider
$k[t_{n}^{\pm}]$ as a $k[t_{2n}^{\pm}]$-linear algebra object. Over the base
$k[t_{n}^{\pm}]$, or $k[t_{2n}^{\pm}]$ if $n$ is odd, the derived
$\infty$-category $\mathcal{D}(k[t_{n}^{\pm}])$ is both smooth and proper and
admits left and right $n$-Calabi–Yau structures.

The $n$-periodic topological Fukaya category can be associated with any marked
surface equipped with a $(n-1)$-spin structure, or a $(2n-1)$-spin structure
if $n$ is odd, see [DK15]. This $\infty$-category arises as the global
sections of a perverse schober without singularities and generic stalk
$\mathcal{D}(k[t_{n}^{\pm}])$. Such a perverse schober is fully determined by
its monodromy around the loops of the surface, given by each by a suspension
functor $[i]$, with $0\leq i\leq n-1$ if $n$ is even and $0\leq i\leq 2n-1$ if
$n$ is odd. We suppose that all monodromy is given by an even integer $[2i]$,
since $[2i]$ acts trivially on (negative) cyclic homology. Note that the
remaining choices of monodromy are canonically in bijection with the possible
choices of $(n-1)$\- or $(2n-1)$-spin structure of the surface. Considering
the $n$-periodic topological Fukaya category over the base $k[t_{n}^{\pm}]$,
or $k[t_{2n}^{\pm}]$ if $n$ is odd, and using 3, we obtain on it the desired
relative $(n+1)$-Calabi–Yau structure.

###  1.4 Acknowledgments

This paper is based on, and refines, results from the author’s Ph.D. thesis,
advised by Tobias Dyckerhoff. I thank him for helpful discussions and
feedback. I further thank Chris Brav, Fabian Haiden, Gustavo Jasso and
Bernhard Keller for helpful discussions. The author acknowledges support by
the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC
2121 “Quantum Universe” – 390833306. This project has received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No 101034255.

###  1.5 Notation

We generally follow the notation and conventions of [Lur09a, Lur17]. In
particular, we use the homological grading convention. Given an
$\infty$-category $\mathcal{C}$ and two objects $X,Y\in\mathcal{C}$, we denote
by $\operatorname{Map}_{\mathcal{C}}(X,Y)$ the mapping space. We denote the
$\operatorname{Ind}$-completion of $\mathcal{C}$ by
$\operatorname{Ind}(\mathcal{C})=\operatorname{Ind}_{\omega}(\mathcal{C})$ and
the subcategory of ($\omega$-)compact objects of $\mathcal{C}$ by
$\mathcal{C}^{\operatorname{c}}$. Given a functor
$F\colon\mathcal{C}\to\mathcal{D}$, we denote its left and right adjoints, if
existent, by $\operatorname{ladj}(F)$ and $\operatorname{radj}(F)$,
respectively.

##  2 Linear $\infty$-categories and Hochschild homology

In this section, we review background material on $R$-linear
$\infty$-categories, with $R$ an $\mathbb{E}_{\infty}$-ring spectrum,
different notions of duals of bimodules, smooth and proper $R$-linear
$\infty$-categories and $R$-linear traces and Hochschild homology. Much parts
of this material appears in a similar form in [Lur17, HSS17, Lur18, BD19,
BD21], though partly with less general proofs. The reader may find that most
things work in the $R$-linear setting very similarly to those in the setting
of dg-categories, though also that the material will be rather technical.

###  2.1 Linear $\infty$-categories

Let $\operatorname{Cat}_{\infty}$ be the $\infty$-category of
$\infty$-categories and $\mathcal{S}$ the $\infty$-category of spaces. We
denote by $\mathcal{P}r^{L}\subset\operatorname{Cat}_{\infty}$ the subcategory
of presentable $\infty$-categories and left adjoint functors and by
$\mathcal{P}r^{R}\subset\operatorname{Cat}_{\infty}$ the subcategory of
presentable $\infty$-categories and right adjoint functors. The
$\infty$-category $\mathcal{P}r^{L}$ admits a symmetric monoidal structure,
such that a commutative algebra object in $\mathcal{P}r^{L}$ amounts to a
symmetric monoidal presentable $\infty$-category $\mathcal{C}$, satisfying
that its tensor product $\mathchar 45\relax\otimes\mathchar
45\relax\colon\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ preserves
colimits in both entries, see [Lur17, Section 4.8]. An example of a
commutative algebra object in $\mathcal{P}r^{L}$ is the $\infty$-category
$\operatorname{RMod}_{R}$ of right module spectra over an
$\mathbb{E}_{\infty}$-ring spectrum $R$. Note that if $R=k$ is a commutative
ring, then $\operatorname{RMod}_{k}$ is equivalent as a symmetric monoidal
$\infty$-category to the (unbounded) derived $\infty$-category
$\mathcal{D}(k)$, see [Lur17, 7.1.2.13].

######  Definition 2.1.

Let $R$ be an $\mathbb{E}_{\infty}$-ring spectrum. The $\infty$-category

$\operatorname{LinCat}_{R}\coloneqq\operatorname{Mod}_{\operatorname{RMod}_{R}}(\mathcal{P}r^{L})$

of modules in $\mathcal{P}r^{L}$ over $\operatorname{RMod}_{R}$ is called the
$\infty$-category of $R$-linear $\infty$-categories.

As noted in [Lur18, Section D.1.5], $R$-linear $\infty$-categories in the
above sense are automatically stable. Given
$\mathcal{C}\in\operatorname{LinCat}_{R}$, we denote the result of the action
of an element $C\in\operatorname{RMod}_{R}$ on $X\in\mathcal{C}$ by $C\otimes
X\in\mathcal{C}$.

######  Definition 2.2 ([Lur17, 4.2.1.28]).

Let $R$ be an $\mathbb{E}_{\infty}$-ring spectrum. Let $\mathcal{C}$ be an
$R$-linear $\infty$-category and let $X,Y\in\mathcal{C}$. A morphism object is
an $R$-module
$\operatorname{Mor}_{\mathcal{C}}(X,Y)\in\operatorname{RMod}_{R}$ equipped
with a map $\alpha\colon\operatorname{Mor}_{\mathcal{C}}(X,Y)\otimes
X\rightarrow Y$ in $\mathcal{C}$ such that for every object
$C\in\operatorname{RMod}_{R}$, the following composite morphisms is an
equivalence of spaces

$\operatorname{Map}_{\operatorname{RMod}_{R}}(C,\operatorname{Mor}_{\mathcal{C}}(X,Y))\rightarrow\operatorname{Map}_{\mathcal{C}}(C\otimes
X,\operatorname{Mor}_{\mathcal{C}}(X,Y)\otimes
X)\xrightarrow{\alpha\circ\mathchar
45\relax}\operatorname{Map}_{\mathcal{C}}(C\otimes X,Y)\,.$

We thus have
$\pi_{i}\operatorname{Mor}_{\mathcal{C}}(X,Y)\simeq\pi_{0}\operatorname{Map}_{\mathcal{C}}(X[i],Y)$
for all $i\in\mathbb{Z}$.

######  Remark 2.3.

Morphism objects always exist and the formation of morphism objects forms a
functor

$\operatorname{Mor}_{\mathcal{C}}(\mathchar 45\relax,\mathchar
45\relax)\colon\mathcal{C}^{\operatorname{op}}\times\mathcal{C}\longrightarrow\operatorname{RMod}_{R}$

which preserves limits in both entries, see [Lur17, 4.2.1.31].

The $\infty$-category $\operatorname{LinCat}_{R}$ inherits a symmetric
monoidal structure, as the module category over a commutative algebra object.
We will make use of this monoidal product most often, and thus denote it by
$\otimes$. The tensor product of
$\mathcal{C},\mathcal{D}\in\operatorname{LinCat}_{R}$ arises as the geometric
realization of the two-sided bar construction
$\operatorname{Bar}(\mathcal{C},\mathcal{D})_{\ast}\colon\Delta^{\operatorname{op}}\to\mathcal{P}r^{L}$,
given informally by the formula
$\operatorname{Bar}(\mathcal{C},\mathcal{D})_{n}=\mathcal{C}\otimes_{\mathcal{P}r^{L}}\operatorname{RMod}_{R}^{\otimes_{\mathcal{P}r^{L}}n}\otimes_{\mathcal{P}r^{L}}\mathcal{D}$,
where $\otimes_{\mathcal{P}r^{L}}$ denotes the symmetric monoidal product of
$\mathcal{P}r^{L}$. The $\infty$-category $\operatorname{LinCat}_{R}$ admits
an internal morphism object:

######  Lemma 2.4.

There is a functor

$\operatorname{Lin}_{R}(\mathchar 45\relax,\mathchar
45\relax)\colon\operatorname{LinCat}_{R}^{\operatorname{op}}\times\operatorname{LinCat}_{R}\longrightarrow\operatorname{LinCat}_{R}$

satisfying

$\operatorname{Lin}_{R}(\mathcal{B},\operatorname{Lin}_{R}(\mathcal{C},\mathcal{D}))\simeq\operatorname{Lin}_{R}(\mathcal{B}\otimes\mathcal{C},\mathcal{D})\,,$
(1)

functorial in
$\mathcal{B},\mathcal{C}\in\operatorname{LinCat}_{R}^{\operatorname{op}}$ and
$\mathcal{D}\in\operatorname{LinCat}_{R}$. We call
$\operatorname{Lin}_{R}(\mathcal{C},\mathcal{D})$ the $R$-linear
$\infty$-category of $R$-linear functors from $\mathcal{C}$ to $\mathcal{D}$.

###### Proof.

To construct the functor $\operatorname{Lin}_{R}(\mathchar 45\relax,\mathchar
45\relax)$, we follow [Lur17, 4.2.1.31]. Consider the functor

$\operatorname{Map}_{\operatorname{LinCat}_{R}}(\mathchar
45\relax\otimes\mathchar 45\relax,\mathchar
45\relax)\colon\operatorname{LinCat}_{R}^{\operatorname{op}}\times\operatorname{LinCat}_{R}^{\operatorname{op}}\times\operatorname{LinCat}_{R}\longrightarrow\mathcal{S}\,.$

Moving the second copy of $\operatorname{LinCat}_{R}^{\operatorname{op}}$ to
the other side, we obtain a functor

$\operatorname{LinCat}_{R}^{\operatorname{op}}\times\operatorname{LinCat}_{R}\rightarrow\operatorname{Fun}(\operatorname{LinCat}_{R}^{\operatorname{op}},\mathcal{S})\,,$

whose image image lies in the full subcategory of representable presheaves,
since $(\mathchar 45\relax)\otimes(\mathchar 45\relax)$ preserves colimits in
the second entry and thus admits a right adjoint. Composing with the inverse
of the Yoneda embedding
$\operatorname{LinCat}_{R}\to\operatorname{Fun}(\operatorname{LinCat}_{R}^{\operatorname{op}},\mathcal{S})$
yields the functor $\operatorname{Lin}_{R}(\mathchar 45\relax,\mathchar
45\relax)$. By construction, we have

$\operatorname{Map}_{\operatorname{LinCat}_{R}}(\mathcal{B}\otimes\mathcal{C},\mathcal{D})\simeq\operatorname{Map}_{\operatorname{LinCat}_{R}}(\mathcal{B},\operatorname{Lin}_{R}(\mathcal{C},\mathcal{D}))\,,$

functorial in $\mathcal{B},\mathcal{C},\mathcal{D}$. It follows that

$\displaystyle\operatorname{Map}_{\operatorname{LinCat}_{R}}(\mathcal{A},\operatorname{Lin}_{R}(\mathcal{B},\operatorname{Lin}_{R}(\mathcal{C},\mathcal{D})))$
$\displaystyle\simeq\operatorname{Map}_{\operatorname{LinCat}_{R}}(\mathcal{A}\otimes\mathcal{B}\otimes\mathcal{C},\mathcal{D})$
$\displaystyle\simeq\operatorname{Map}_{\operatorname{LinCat}_{R}}(\mathcal{A},\operatorname{LinCat}_{R}(\mathcal{B}\otimes\mathcal{C},\mathcal{D}))\,,$

functorial in
$\mathcal{A},\mathcal{B},\mathcal{C}\in\operatorname{LinCat}_{R}^{\operatorname{op}}$
and $\mathcal{D}\in\operatorname{LinCat}_{R}$. Composing again with the
inverse of the Yoneda embedding shows (1), concluding the proof. ∎

###  2.2 Dualizable $\infty$-categories

We fix an an $\mathbb{E}_{\infty}$-ring spectrum $R$. Recall that an
$R$-linear $\infty$-category $\mathcal{C}\in\operatorname{LinCat}_{R}$ is
called dualizable if it admits a duality datum consisting of evaluation and
coevaluation functors

$\operatorname{ev}_{\mathcal{C}}\colon\mathcal{C}^{\vee}\otimes\mathcal{C}\longrightarrow\operatorname{RMod}_{R}$

and

$\operatorname{coev}_{\mathcal{C}}\colon\operatorname{RMod}_{R}\longrightarrow\mathcal{C}\otimes\mathcal{C}^{\vee}\,,$

satisfying the triangle identities.

Recall further that an $\infty$-category $\mathcal{C}$ is called compactly
generated, if
$\mathcal{C}\simeq\operatorname{Ind}(\mathcal{C}^{\operatorname{c}})$ is
equivalent to the $\operatorname{Ind}$-completion of its subcategory of
compact objects.

######  Definition 2.5.

We denote by
$\operatorname{LinCat}^{\operatorname{dual}}_{R}\subset\operatorname{LinCat}_{R}$
the subcategory consisting of dualizable $R$-linear $\infty$-categories and
dualizable functors, meaning $R$-linear functors whose right adjoint $G$
preserves colimits and is thus again $R$-linear.

We denote by $\operatorname{LinCat}_{R}^{\operatorname{cpt-
gen}}\subset\operatorname{LinCat}_{R}$ the subcategory consisting of compactly
generated $\infty$-categories and compact objects preserving functors.

An $R$-linear $\infty$-category $\mathcal{C}\in\operatorname{LinCat}_{R}$ is
dualizable if and only if it is compactly assembled, see [Lur18, D.7.0.7],
which is equivalent to $\mathcal{C}$ being a retract of a compactly generated,
presentable and stable $\infty$-category in the $\infty$-category
$\mathcal{Pr}^{L}_{\operatorname{St}}\subset\mathcal{P}r^{L}$ of stable,
presentable $\infty$-categories, see [Lur18, Prop. D.7.3.1].

In particular, any compactly generated $R$-linear $\infty$-category
$\mathcal{C}$ is dualizable. In this case, the dual is given by the Ind-
completion
$\mathcal{C}^{\vee}\coloneqq\operatorname{Ind}(\mathcal{C}^{\operatorname{c},\operatorname{op}})$,
where $\mathcal{C}^{\operatorname{c}}$ denotes the subcategory of compact
object and $\mathcal{C}^{\operatorname{c},\operatorname{op}}$ its opposite
category. If $\mathcal{C}$ is compactly generated, the evaluation functor
$\operatorname{ev}_{\mathcal{C}}$ restricts along

$\mathcal{C}^{\operatorname{c},\operatorname{op}}\times\mathcal{C}^{\operatorname{c}}\subset\mathcal{C}^{\vee}\times\mathcal{C}\longrightarrow\mathcal{C}^{\vee}\otimes\mathcal{C}$

to the restriction of the morphism object functor
$\operatorname{Mor}_{\mathcal{C}}(\mathchar 45\relax,\mathchar 45\relax)$, see
[Lur18, D.7.2.3, D.7.7.6]. Under the assumption that $\mathcal{C}$ is
compactly generated, an $R$-linear functor $F\colon\mathcal{C}\to\mathcal{D}$
is dualizable if and only if $F$ preserves compact objects, see [Lur09a,
5.5.7.2]. We thus have a fully faithful inclusion
$\operatorname{LinCat}_{R}^{\operatorname{cpt-
gen}}\subset\operatorname{LinCat}^{\operatorname{dual}}_{R}$.

######  Remark 2.6.

The $\infty$-category $\operatorname{LinCat}^{\operatorname{cpt-gen}}_{R}$
admits (small) colimits and these are preserved by the forgetful functor
$\operatorname{LinCat}_{R}^{\operatorname{cpt-
gen}}\to\operatorname{LinCat}_{R}$. This follows from combining [Lur17,
3.4.4.6], which states that
$\operatorname{LinCat}_{R}\to\mathcal{P}r^{L}\simeq(\mathcal{P}r^{R})^{\operatorname{op}}$
preserves colimits, with the statement that
$\mathcal{P}r^{R}\to\operatorname{Cat}_{\infty}$ preserves limits, see
[Lur09a, 5.5.3.18], and finally the statement that limits in
$\mathcal{P}r^{R}$ of compactly generated $\infty$-categories along right
adjoints of compact objects preserving functors are again compactly generated,
see [Lur09a, 5.5.7.6].

The $\infty$-category $\operatorname{LinCat}^{\operatorname{cpt-gen}}_{R}$
also admits (small) limits. On the level of the underlying
$\infty$-categories, the limit of a diagram $D\colon
Z\to\operatorname{LinCat}^{\operatorname{cpt-gen}}$ is computed by first
restricting the values $D(z)$ to compact objects, for all $z\in Z$, computing
the limit of the resulting diagram in $\operatorname{Cat}_{\infty}$, and then
passing to the $\operatorname{Ind}$-completion. This essentially follows from
the fact that limits in the subcategory $\mathcal{P}r^{L,\operatorname{cpt-
gen}}_{\operatorname{St}}\subset\mathcal{P}r^{L}$ of compactly generated,
stable $\infty$-categories and compact objects preserving functors are
computed this way.

######  Definition 2.7.

Given a dualizable $R$-linear functor $F\colon\mathcal{C}\to\mathcal{D}$
between dualizable $R$-linear $\infty$-categories with right adjoint $G$, we
define the functor $F^{\vee}\colon\mathcal{C}^{\vee}\to\mathcal{D}^{\vee}$ as
the composite

$\mathcal{C}^{\vee}\xlongrightarrow{\operatorname{id}_{\mathcal{C}^{\vee}}\otimes\operatorname{coev}_{\mathcal{D}}}\mathcal{C}^{\vee}\otimes\mathcal{D}\otimes\mathcal{D}^{\vee}\xlongrightarrow{\operatorname{id}_{\mathcal{C}^{\vee}}\otimes
G\otimes\operatorname{id}_{\mathcal{D}^{\vee}}}\mathcal{C}^{\vee}\otimes\mathcal{C}\otimes\mathcal{D}^{\vee}\xrightarrow{\operatorname{ev}_{\mathcal{C}^{\vee}}\otimes\operatorname{id}_{\mathcal{D}^{\vee}}}\mathcal{D}^{\vee}\,.$

Note that if $\mathcal{C},\mathcal{D}$ are compactly generated, then

$F^{\vee}\simeq\operatorname{Ind}(f^{\operatorname{op}})$

is obtained taking the opposite functor of the restriction
$f\colon\mathcal{C}^{\operatorname{c}}\to\mathcal{D}^{\operatorname{c}}$ of
$F$ to compact objects and then $\operatorname{Ind}$-completing.

######  Definition 2.8.

Given a dualizable $R$-linear $\infty$-category $\mathcal{C}$, we denote by

$\mathcal{Y}\colon\mathcal{C}\simeq\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C})\xlongrightarrow{\operatorname{ev}_{\mathcal{C}}\circ(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes(\mathchar
45\relax))}\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})$

the $R$-linear Yoneda embedding. It’s inverse is given by

$\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})\xlongrightarrow{((\mathchar
45\relax)\otimes\operatorname{id}_{\mathcal{C}})\circ\operatorname{coev}_{\mathcal{C}}}\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C})\simeq\mathcal{C}\,.$

######  Lemma 2.9.

Let $\mathcal{C}$ be a dualizable $R$-linear $\infty$-category. There exist an
equivalence of $R$-linear functors
$\operatorname{RMod}_{R}^{\vee}\otimes\mathcal{C}^{\vee}\otimes\mathcal{C}\to\operatorname{RMod}_{R}$:

$\operatorname{ev}_{\mathcal{C}}(C\otimes
Y,Z)\simeq\operatorname{ev}_{\operatorname{RMod}_{R}}(C,\operatorname{ev}_{\mathcal{C}}(Y,Z))\,.$
(2)

###### Proof.

We note that $\operatorname{RMod}_{R}^{\vee}\simeq\operatorname{RMod}_{R}$ and
thus
$\mathcal{C}^{\vee}\otimes\mathcal{C}\simeq\operatorname{RMod}_{R}^{\vee}\otimes\mathcal{C}^{\vee}\otimes\mathcal{C}$.
Composing with this equivalence, both functors in (2) yield
$\operatorname{ev}_{\mathcal{C}}(\mathchar 45\relax,\mathchar 45\relax)$,
showing their equivalence. ∎

###  2.3 Duals of bimodules

We again fix a base $\mathbb{E}_{\infty}$-ring spectrum $R$. Suppose we are
given two $R$-linear ring spectra $A,A^{\prime}$. The $\infty$-category of
$A$-$A^{\prime}$-bimodules
${}_{A}\\!\operatorname{BMod}_{A^{\prime}}(\operatorname{RMod}_{R})$ is
equivalent to the $\infty$-category
$\operatorname{Lin}_{R}(\operatorname{RMod}_{A},\operatorname{RMod}_{A^{\prime}})$
of $R$-linear functors between the respective right module
$\infty$-categories, see [Lur17, 4.8.4.1, 4.3.2.7]. In terms of functors, left
and right duals of bimodules, if they exist, correspond to left and right
adjoints of the corresponding functors. In the following, we will work with
functors instead of bimodules, as these have the advantage that we do no need
to assume that our given $R$-linear $\infty$-categories admit a compact
generator, and thus be equivalent to $\operatorname{RMod}_{A}$ for some ring
spectrum $A$.

Let $\mathcal{C}$ be a dualizable $R$-linear $\infty$-category. We are
especially interested in the adjoints of functors
$\mathcal{C}\otimes\mathcal{C}^{\vee}\to\operatorname{RMod}_{R}$ or
$\operatorname{RMod}_{R}\to\mathcal{C}\otimes\mathcal{C}^{\vee}$. This
corresponds as a special case to studying modules over the enveloping algebra
$A^{e}=A\otimes_{R}A^{\operatorname{rev}}$ of some $R$-linear ring spectrum
$A$. We have the following equivalences.

######  Lemma 2.10.

  1. (1)

The $R$-linear functor $\Theta_{\mathcal{C}}$, defined as the composite

$\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})\xlongrightarrow{\operatorname{id}_{\mathcal{C}}\otimes(\mathchar
45\relax)}\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee}\otimes\mathcal{C},\mathcal{C})\xlongrightarrow{(\mathchar
45\relax)\circ(\operatorname{coev}_{\mathcal{C}}\otimes\operatorname{id}_{\mathcal{C}})}\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\,,$

is an equivalence with inverse $\Theta_{\mathcal{C}}^{-1}$ given by

$\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\xlongrightarrow{\operatorname{id}_{\mathcal{C}^{\vee}}\otimes(\mathchar
45\relax)}\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\mathcal{C}^{\vee}\otimes\mathcal{C})\xlongrightarrow{\operatorname{ev}_{\mathcal{C}}\circ(\mathchar
45\relax)}\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})\,.$

  2. (2)

The $R$-linear functor $\Xi_{\mathcal{C}}$, defined as the composite

$\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}\otimes\mathcal{C}^{\vee})\xlongrightarrow{(\mathchar
45\relax)\otimes\operatorname{id}_{\mathcal{C}}}\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C}\otimes\mathcal{C}^{\vee}\otimes\mathcal{C})\xlongrightarrow{\operatorname{id}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{C}}\circ(\mathchar
45\relax)}\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\,,$

is an equivalence with inverse $\Xi_{\mathcal{C}}^{-1}$ given by

$\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\xlongrightarrow{(\mathchar
45\relax)\otimes\operatorname{id}_{\mathcal{C}^{\vee}}}\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\mathcal{C}\otimes\mathcal{C}^{\vee})\xrightarrow{(\mathchar
45\relax)\circ\operatorname{coev}_{\mathcal{C}}}\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}\otimes\mathcal{C}^{\vee})\,.$

###### Proof.

We begin by proving part (1). The equivalence of $\infty$-categories

$\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\xrightarrow{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{Y})}\operatorname{Lin}_{R}(\mathcal{C},\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R}))\simeq\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})\,,$

with $\mathcal{Y}$ the $R$-linear Yoneda embedding, maps an endofunctor
$X\colon\mathcal{C}\to\mathcal{C}$ to
$\operatorname{ev}_{\mathcal{C}}\circ(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes
X)$. This shows that $\Theta_{\mathcal{C}}^{-1}$ is essentially surjective.
The triangle identity for $\operatorname{ev}_{\mathcal{C}}$ and
$\operatorname{coev}_{\mathcal{C}}$ implies that
$\Theta_{\mathcal{C}}\circ\Theta_{\mathcal{C}}^{-1}\simeq\operatorname{id}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}$.
It follows that $\Theta_{\mathcal{C}}^{-1}$ is faithful, and in fact a split
inclusion on Hom spaces. Using that all objects
$Y\in\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})$
are of the form
$Y\simeq\operatorname{ev}_{\mathcal{C}}\circ(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes
X)$, we find $\Theta_{\mathcal{C}}^{-1}\circ\Theta_{\mathcal{C}}(Y)\simeq Y$.
Using that $\Theta_{\mathcal{C}}$ and $\Theta_{\mathcal{C}}^{-1}$ are exact,
it follows that $\Theta_{\mathcal{C}}^{-1}$ is also full, showing that
$\Theta_{\mathcal{C}}^{-1}$ is an equivalence. Since
$\Theta_{\mathcal{C}}\circ\Theta_{\mathcal{C}}^{-1}\simeq\operatorname{id}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}$,
the inverse of $\Theta_{\mathcal{C}}^{-1}$ is given by $\Theta_{\mathcal{C}}$.

For part (2), a similar argument as above applies, using that the equivalence
of $\infty$-categories

$\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\simeq\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C}))\simeq\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}\otimes\mathcal{C}^{\vee})$

maps a functor $X\colon\mathcal{C}\to\mathcal{C}$ to
$(X\otimes\operatorname{id}_{\mathcal{C}^{\vee}})\circ\operatorname{coev}_{\mathcal{C}}$.
∎

######  Notation 2.11.

We denote by $\tau$ the $R$-linear equivalence
$\mathcal{C}\otimes\mathcal{C}^{\vee}\simeq\mathcal{C}^{\vee}\otimes\mathcal{C}$
which permutes the factors.

We can use the equivalences $\Theta_{\mathcal{C}}$ and $\Xi_{\mathcal{C}}$ to
define the dual of an $R$-linear endofunctor $\mathcal{C}\to\mathcal{C}$,
considered as a functor
$\mathcal{C}^{\vee}\otimes\mathcal{C}\to\operatorname{RMod}_{R}$ or
$\operatorname{RMod}_{R}\to\mathcal{C}\otimes\mathcal{C}^{\vee}$.

######  Definition 2.12.

Let $X\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$ be an $R$-linear
endofunctor.

  1. 1.

We call $X$ left dualizable if $\Theta_{\mathcal{C}}^{-1}(X)$ admits an
$R$-linear left adjoint. In this case, we call

$X^{!}\coloneqq\Xi_{\mathcal{C}}(\tau\circ\operatorname{ladj}(\Theta_{\mathcal{C}}^{-1}(X)))\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$

the left dual of $X$.

  2. 2.

We call $X$ right dualizable if $\Theta_{\mathcal{C}}^{-1}(X)$ admits an
$R$-linear right adjoint. In this case, we call

$X^{\ast}\coloneqq\Xi_{\mathcal{C}}(\tau\circ\operatorname{radj}(\Theta_{\mathcal{C}}^{-1}(X)))\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$

the right dual of $X$.

######  Lemma 2.13.

Let $X\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$.

  1. (1)

$X$ is left dualizable if and only if $\Xi_{\mathcal{C}}^{-1}(X)$ admits a
right adjoint, and in this case

$X^{!}\simeq\Theta_{\mathcal{C}}(\tau\circ\operatorname{radj}(\Xi_{\mathcal{C}}^{-1}(X)))\,.$

  2. (2)

$X$ is right dualizable if and only if $\Xi_{\mathcal{C}}^{-1}(X)$ admits a
left adjoint, and in this case

$X^{*}\simeq\Theta_{\mathcal{C}}(\tau\circ\operatorname{ladj}(\Xi_{\mathcal{C}}^{-1}(X)))\,.$

###### Proof.

The lemma follows from Lemma 2.14 below and the observations that for any
dualizable $R$-linear functor $F\colon\mathcal{A}\to\mathcal{B}$, we have

$\operatorname{Lin}_{R}(\operatorname{radj}(F),\operatorname{RMod}_{R})\dashv\operatorname{Lin}_{R}(F,\operatorname{RMod}_{R})\,.$

∎

######  Lemma 2.14.

Let $X\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$. There are
commutative diagrams,

${\operatorname{RMod}_{R}}$${\mathcal{C}\otimes\mathcal{C}^{\vee}}$${{\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\operatorname{RMod}_{R})}}$${{\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})}}$$\scriptstyle{\Xi_{\mathcal{C}}^{-1}(X)}$$\scriptstyle{\mathcal{Y}}$$\scriptstyle{\mathcal{Y}}$$\scriptstyle{\operatorname{Lin}_{R}(\Theta_{\mathcal{C}}^{-1}(X),\operatorname{RMod}_{R})}$

and

${\mathcal{C}^{\vee}\otimes\mathcal{C}}$${\operatorname{RMod}_{R}}$${{\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})}}$${{\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\operatorname{RMod}_{R})}}$$\scriptstyle{\mathcal{Y}}$$\scriptstyle{\Theta_{\mathcal{C}}^{-1}(X)}$$\scriptstyle{\mathcal{Y}}$$\scriptstyle{\operatorname{Lin}_{R}(\Xi_{\mathcal{C}}^{-1}(X),\operatorname{RMod}_{R})}$

with $\mathcal{Y}$ the $R$-linear Yoneda embedding, see Definition 2.8.

###### Proof.

The evaluation functor

$\operatorname{ev}_{\mathcal{C}\otimes\mathcal{C}^{\vee}}\colon\mathcal{C}^{\vee}\otimes\mathcal{C}\otimes\mathcal{C}\otimes\mathcal{C}^{\vee}\simeq(\mathcal{C}\otimes\mathcal{C}^{\vee})^{\vee}\otimes\mathcal{C}\otimes\mathcal{C}^{\vee}\longrightarrow\operatorname{RMod}_{R}$

is, after reordering the factors of the tensor product, given by the tensor
product of the evaluation functors of $\mathcal{C}$ and $\mathcal{C}^{\vee}$.
Note that these two evaluation functors are themselves equivalent, up to
composition with
$\tau\colon\mathcal{C}\otimes\mathcal{C}^{\vee}\simeq\mathcal{C}^{\vee}\otimes\mathcal{C}$.
The Yoneda embedding

$\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}\otimes\mathcal{C}^{\vee})\simeq\mathcal{C}\otimes\mathcal{C}^{\vee}\xlongrightarrow{\mathcal{Y}}\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})$

is thus given by the functor

$\left(\operatorname{ev}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{C}}\right)\circ\left(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes(\mathchar
45\relax)\otimes\operatorname{id}_{\mathcal{C}}\right)\,.$ (3)

Using this, the commutativity directly follows from the triangle identities
for the evaluation and coevaluation functors when inserting the descriptions
of $\Theta^{-1}_{\mathcal{C}}(X),\Xi^{-1}_{\mathcal{C}}(X)$ in Lemma 2.10. ∎

######  Remark 2.15.

We denote by
$\operatorname{Lin}_{R}^{\operatorname{ld}}(\mathcal{C},\mathcal{C})\subset\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$
the stable subcategories of left dualizable functors. We similarly denote by
$\operatorname{Lin}_{R}^{\operatorname{rd}}(\mathcal{C},\mathcal{C})\subset\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$
the stable subcategory of right dualizable functors. Since passing to adjoints
is functorial, see [Lur09a, 5.2.6.2], there are exact functors

$(\mathchar
45\relax)^{!}\colon\operatorname{Lin}_{R}^{\operatorname{ld}}(\mathcal{C},\mathcal{C})\rightarrow\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})^{\operatorname{op}}$

and

$(\mathchar
45\relax)^{*}\colon\operatorname{Lin}_{R}^{\operatorname{rd}}(\mathcal{C},\mathcal{C})\rightarrow\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})^{\operatorname{op}}\,.$

More concretely, we find that
$\operatorname{Lin}_{R}^{\operatorname{ld}}(\mathcal{C},\mathcal{C})=\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})^{\operatorname{c}}$
is given by the subcategory of compact objects. This follows from the
observation that a compact object in

$\mathcal{C}\otimes\mathcal{C}^{\vee}\overset{\mathcal{Y}}{\simeq}\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})\simeq\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$

gives via the Yoneda embedding rise to an exact functor
$\mathcal{C}^{\vee}\otimes\mathcal{C}\to\operatorname{RMod}_{R}$ which also
preserves filtered limits, and hence all limits, and thus admits a left
adjoint by the adjoint functor theorem. An endofunctor is right dualizable if
and only if its image under $\Theta_{\mathcal{C}}^{-1}$ in
$\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})$
preserves compact objects, since in this case the right adjoint preserves
colimits and is thus $R$-linear.

######  Lemma 2.16.

Let $F\colon\mathcal{C}\to\mathcal{D}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ with $R$-linear right
adjoint $G$.

  1. (1)

There exists a commutative diagram

${{\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}\otimes\mathcal{C}^{\vee})}}$${{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}}$${{\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{D}\otimes\mathcal{D}^{\vee})}}$${{\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})}}$$\scriptstyle{(F\otimes
F^{\vee})\circ(\mathchar
45\relax)}$$\scriptstyle{\Xi_{\mathcal{C}}}$$\scriptstyle{F\circ(\mathchar
45\relax)\circ G}$$\scriptstyle{\Xi_{\mathcal{D}}}$

In particular, it follows that the $R$-linear functor

$F_{!}\coloneqq F\circ(\mathchar 45\relax)\circ
G\colon\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\rightarrow\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})$

preserves compact objects.

  2. (2)

There exists a commutative diagram

${{\operatorname{Lin}_{R}(\mathcal{D}^{\vee}\otimes\mathcal{D},\operatorname{RMod}_{R})}}$${{\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})}}$${{\operatorname{Lin}_{R}(\mathcal{C}^{\vee}\otimes\mathcal{C},\operatorname{RMod}_{R})}}$${{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}}$$\scriptstyle{(\mathchar
45\relax)\circ(F^{\vee}\otimes
F)}$$\scriptstyle{\Theta_{\mathcal{D}}}$$\scriptstyle{G\circ(\mathchar
45\relax)\circ F}$$\scriptstyle{\Theta_{\mathcal{C}}}$

In particular, it follows that the $R$-linear functor

$F^{*}\coloneqq G\circ(\mathchar 45\relax)\circ
F\colon\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})\rightarrow\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$

preserves compact objects.

###### Proof.

We only prove part (1), part (2) is analogous. It follows from

$\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes\operatorname{id}_{\mathcal{D}})\simeq\operatorname{ev}_{\mathcal{C}}\circ(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes\
G)$

that

$\displaystyle\Xi_{\mathcal{D}}((F\otimes F^{\vee})\circ\alpha)$
$\displaystyle\simeq(\operatorname{id}_{\mathcal{D}}\otimes\operatorname{ev}_{\mathcal{D}})\circ(F\otimes
F^{\vee}\otimes\operatorname{id}_{\mathcal{D}})\circ(\alpha\otimes\operatorname{id}_{\mathcal{D}})$
$\displaystyle\simeq(\operatorname{id}_{\mathcal{D}}\otimes\operatorname{ev}_{\mathcal{C}})\circ(F\otimes\operatorname{id}_{\mathcal{D}^{\vee}}\otimes\
G)\circ(\alpha\otimes\operatorname{id}_{\mathcal{D}})$ $\displaystyle\simeq
F\circ\Xi_{\mathcal{C}}(\alpha)\circ G\,,$

functorial in
$\alpha\colon\operatorname{RMod}_{R}\to\mathcal{C}\otimes\mathcal{C}^{\vee}$.
∎

###  2.4 Smooth and proper linear $\infty$-categories

We fix an $\mathbb{E}_{\infty}$-ring spectrum $R$ and a dualizable $R$-linear
$\infty$-category $\mathcal{C}$.

######  Definition 2.17.

  1. (1)

The $\infty$-category $\mathcal{C}$ is called smooth if
$\operatorname{id}_{\mathcal{C}}\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$
is left dualizable. In this case, the left dual
$\operatorname{id}_{\mathcal{C}}^{!}$ is also called the inverse dualizing
bimodule.

  2. (2)

The $\infty$-category $\mathcal{C}$ is called proper if the functor
$\operatorname{id}_{\mathcal{C}}$ is right dualizable.

Note that we do not require smooth or proper $\infty$-categories to be
compactly generated. However, if $\mathcal{C}$ is compactly generated, then
$\mathcal{C}$ being proper is equivalent to the assertion, that for any two
compact objects $X,Y\in\mathcal{C}^{c}$ the $R$-linear morphism object
$\operatorname{Mor}_{\mathcal{C}}(X,Y)\in\operatorname{RMod}_{R}$ is compact.

We denote $(\mathchar
45\relax)^{*}=\operatorname{Mor}_{\operatorname{RMod}_{R}}(\mathchar
45\relax,R)\colon(\operatorname{RMod}_{R}^{\operatorname{c}})^{\operatorname{op}}\to\operatorname{RMod}_{R}^{\operatorname{c}}$

######  Definition 2.18.

Suppose that $\mathcal{C}$ is compactly generated and proper. We call an
$R$-linear endofunctor $U\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$ a
Serre functor of $\mathcal{C}$ if there exists a natural equivalence

$\operatorname{Mor}_{\mathcal{C}}(\mathchar 45\relax_{1},\mathchar
45\relax_{2})\simeq\operatorname{Mor}_{\mathcal{C}}(\mathchar
45\relax_{2},U(\mathchar
45\relax_{1}))^{\ast}\colon\mathcal{C}^{\operatorname{c},\operatorname{op}}\times\mathcal{C}^{\operatorname{c}}\rightarrow\operatorname{RMod}_{R}^{\operatorname{c}}\,.$

######  Lemma 2.19.

Suppose that $\mathcal{C}$ is compactly generated and proper and let
$U,U^{\prime}$ be two Serre functors of $\mathcal{C}$. Then $U\simeq
U^{\prime}\in\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$.

###### Proof.

Since $U$ and $U^{\prime}$ are both Serre functors, there exist natural
equivalences

$\operatorname{Mor}_{\mathcal{C}}(\mathchar 45\relax_{1},U(\mathchar
45\relax_{2}))\simeq\operatorname{Mor}_{\mathcal{C}}(\mathchar
45\relax_{2},\mathchar
45\relax_{1})^{\ast}\simeq\operatorname{Mor}_{\mathcal{C}}(\mathchar
45\relax_{1},U^{\prime}(\mathchar 45\relax_{2}))\,.$

Applying $\operatorname{Map}_{\operatorname{RMod}_{R}}(R,\mathchar 45\relax)$
to this equivalence yields

$\operatorname{Map}_{\mathcal{C}^{\operatorname{c}}}(\mathchar
45\relax_{1},U(\mathchar
45\relax_{2}))\simeq\operatorname{Map}_{\mathcal{C}^{\operatorname{c}}}(\mathchar
45\relax_{1},U^{\prime}(\mathchar 45\relax_{2}))\,.$

It follows that $U\simeq U^{\prime}$ on $\mathcal{C}^{\operatorname{c}}$ by (a
corollary of) the Yoneda lemma, see for instance [Cis19, Cor. 5.8.14]. Passing
to $\operatorname{Ind}$-completions shows $U\simeq U^{\prime}$. ∎

######  Lemma 2.20.

  1. (1)

If $\mathcal{C}$ is proper and compactly generated, the right dual
$\operatorname{id}_{\mathcal{C}}^{\ast}$ is a Serre functor of $\mathcal{C}$.

  2. (2)

If $\mathcal{C}$ is smooth and proper, the functors
$\operatorname{id}_{\mathcal{C}}^{\ast}$ and
$\operatorname{id}_{\mathcal{C}}^{!}$ are inverse equivalences.

######  Remark 2.21.

Part (1) of Lemma 2.20 is stated without proof in [Lur18, 11.1.5.2].

###### Proof of Lemma 2.20.

We begin by proving part (1). We denote by

$\widehat{\operatorname{Mor}}_{\mathcal{C}}(\mathchar 45\relax,\mathchar
45\relax)\colon\mathcal{C}^{\vee}\times\mathcal{C}\to\operatorname{RMod}_{R}$

the functor obtained by passing to $\operatorname{Ind}$-completions from the
restriction of $\operatorname{Mor}_{\mathcal{C}}(\mathchar 45\relax,\mathchar
45\relax)$ to
$\mathcal{C}^{\operatorname{c},\operatorname{op}}\times\mathcal{C}^{\operatorname{c}}$.
Let $X\in\mathcal{C}^{\operatorname{c}}$ and consider the adjunction

$(\mathchar
45\relax)\otimes\widehat{\operatorname{Mor}}_{\mathcal{C}}(X,\mathchar
45\relax)\colon\operatorname{RMod}_{R}\longleftrightarrow\operatorname{Lin}_{R}(\mathcal{C},\operatorname{RMod}_{R})\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\operatorname{RMod}_{R})}(\widehat{\operatorname{Mor}}_{\mathcal{C}}(X,\mathchar
45\relax),\mathchar 45\relax)\,.$

Using that $\widehat{\operatorname{Mor}}_{\mathcal{C}}(X,\mathchar
45\relax)\simeq\widehat{\operatorname{Mor}}_{\mathcal{C}^{\vee}}(\mathchar
45\relax,X)$ and the fully faithfulness of the $R$-linear Yoneda embedding of
$\mathcal{C}^{\vee}$, we find that the right adjoint is given by evaluation at
$X$, i.e.

$\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\operatorname{RMod}_{R})}(\widehat{\operatorname{Mor}}_{\mathcal{C}}(X,\mathchar
45\relax),\mathchar 45\relax)\simeq\operatorname{ev}_{X}\,.$

Using the identification

$\operatorname{Lin}_{R}(\mathcal{C},\operatorname{RMod}_{R})\otimes\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})\simeq\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})$

we define the functor

$\operatorname{ev}_{X}^{\prime}\colon\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})\xlongrightarrow{\operatorname{ev}_{X}\otimes\operatorname{id}_{\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})}}\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})$

with left adjoint

$\operatorname{ladj}(\operatorname{ev}_{X}^{\prime})\colon\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})\xlongrightarrow{\left((\mathchar
45\relax)\otimes\widehat{\operatorname{Mor}}_{\mathcal{C}}(X,\mathchar
45\relax)\right)\otimes\operatorname{id}}\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})\,.$

Informally, the left adjoint
$\operatorname{ladj}(\operatorname{ev}_{X}^{\prime})$ is given by

$(c^{\prime}\mapsto F(c^{\prime}))\mapsto\left((c\otimes
c^{\prime})\mapsto\widehat{\operatorname{Mor}}_{\mathcal{C}}(X,c)\otimes
F(c^{\prime})\right)\,.$

The right adjoint of the evaluation functor
$\operatorname{ev}_{\mathcal{C}}\colon\mathcal{C}^{\vee}\otimes\mathcal{C}\to\operatorname{RMod}_{R}$
is equivalent to
$(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes\operatorname{id}_{\mathcal{C}}^{*})\circ\operatorname{coev}_{\mathcal{C}^{\vee}}$.
Using the description the the Yoneda embedding in (3), it follows that the
right adjoint of the functor

$\widetilde{\operatorname{ev}}\colon\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})\simeq\mathcal{C}^{\vee}\otimes\mathcal{C}\xlongrightarrow{\operatorname{ev}_{\mathcal{C}}}\operatorname{RMod}_{R}$

is equivalent to $(\mathchar
45\relax)\otimes\operatorname{ev}_{\mathcal{C}^{\vee}}\circ(\operatorname{id}^{*}_{\mathcal{C}}\otimes\operatorname{id}_{\mathcal{C}^{\vee}})$.

In total, we obtain that the right adjoint of

$\operatorname{Lin}_{R}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})\xlongrightarrow{\operatorname{ladj}(\operatorname{ev}_{X}^{\prime})}\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})\xlongrightarrow{\widetilde{\operatorname{ev}}}\operatorname{RMod}_{R}$

is given by $(\mathchar
45\relax)\otimes\widehat{\operatorname{Mor}}_{\mathcal{C}}(\mathchar
45\relax,\operatorname{id}_{\mathcal{C}}^{*}(X))$. To make the notation more
transparent, we denote in the following by
$\operatorname{Mor}^{\prime}_{\mathcal{C}}(\mathchar 45\relax,\mathchar
45\relax)\colon(\mathcal{C}^{\vee}\otimes\mathcal{C})^{\operatorname{c}}\to\operatorname{RMod}_{R}$
the restriction of $\operatorname{ev}_{\mathcal{C}}$. Using the above
adjunctions, the fully faithfulness of the $R$-linear Yoenda embedding and
Lemma 2.9, we find the following equivalences in $\operatorname{RMod}_{R}$,
functorial in $X\otimes
Y\in(\mathcal{C}\otimes\mathcal{C}^{\vee})^{\operatorname{c}}$,

$\displaystyle\operatorname{Mor}^{\prime}_{\mathcal{C}}(X,Y)^{*}=$
$\displaystyle\operatorname{Mor}_{\operatorname{RMod}_{R}}(\operatorname{Mor}^{\prime}_{\mathcal{C}}(X,Y),R)$
$\displaystyle\simeq$
$\displaystyle\operatorname{Mor}_{\operatorname{RMod}_{R}}(\widetilde{\operatorname{ev}}\circ\operatorname{ladj}(\operatorname{ev}_{X}^{\prime})(\widehat{\operatorname{Mor}}_{\mathcal{C}}(\mathchar
45\relax,Y)),R)$ $\displaystyle\simeq$
$\displaystyle\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C}\otimes\mathcal{C}^{\vee},\operatorname{RMod}_{R})}(\operatorname{ladj}(\operatorname{ev}_{X}^{\prime})(\widehat{\operatorname{Mor}}_{\mathcal{C}}(\mathchar
45\relax,Y)),\operatorname{ev}_{\mathcal{C}}\circ(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes\operatorname{id}^{*}_{\mathcal{C}}(\mathchar
45\relax)))$ $\displaystyle\simeq$
$\displaystyle\operatorname{Mor}_{\operatorname{Lin}(\mathcal{C}^{\vee},\operatorname{RMod}_{R})}(\widehat{\operatorname{Mor}}_{\mathcal{C}}(\mathchar
45\relax,Y),\widehat{\operatorname{Mor}}_{\mathcal{C}}(\mathchar
45\relax,\operatorname{id}_{\mathcal{C}}^{*}(X)))$ $\displaystyle\simeq$
$\displaystyle\operatorname{Mor}_{\mathcal{C}}^{\prime}(Y,\operatorname{id}_{\mathcal{C}}^{*}(X))\,.$

Restricting the above equivalence of functors along
$\mathcal{C}^{\operatorname{c}}\times\mathcal{C}^{\operatorname{c},\operatorname{op}}\to(\mathcal{C}\otimes\mathcal{C}^{\vee})^{\operatorname{c}}$
shows that that $\operatorname{id}_{\mathcal{C}}^{*}$ is indeed a Serre
functor.

Finally, we note that the above computation actually gives a slightly stronger
statement, namely an equivalence of functors
$\mathcal{C}^{\operatorname{c}}\times\mathcal{C}^{\vee}\to\operatorname{RMod}_{R}$.

We proceed with proving part (2). We have

$\operatorname{id}_{\mathcal{C}}\simeq(\operatorname{id}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{C}})\circ(\operatorname{coev}_{\mathcal{C}}\otimes\operatorname{id}_{\mathcal{C}})$
(4)

and passing to the right adjoint yields

$\operatorname{id}_{\mathcal{C}}\simeq(\operatorname{radj}(\operatorname{coev}_{\mathcal{C}})\otimes\operatorname{id}_{\mathcal{C}})\circ(\operatorname{id}_{\mathcal{C}}\otimes\operatorname{radj}(\operatorname{ev}_{\mathcal{C}}))\,.$

We have

$\operatorname{radj}(\operatorname{ev}_{\mathcal{C}})\simeq(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes\operatorname{id}_{\mathcal{C}}^{*})\circ\operatorname{coev}_{\mathcal{C}^{\vee}}$

and by Lemma 2.13 further

$\operatorname{radj}(\operatorname{coev}_{\mathcal{C}})\simeq\operatorname{ev}_{\mathcal{C}^{\vee}}\circ(\operatorname{id}_{\mathcal{C}}^{!}\otimes\operatorname{id}_{\mathcal{C}^{\vee}})\,.$

Combining the above equivalences yields
$\operatorname{id}_{\mathcal{C}}^{*}\circ\operatorname{id}_{\mathcal{C}}^{!}\simeq\operatorname{id}_{\mathcal{C}}$.
The identity
$\operatorname{id}_{\mathcal{C}}^{!}\circ\operatorname{id}_{\mathcal{C}}^{*}\simeq\operatorname{id}_{\mathcal{C}}$
arises from a similar argument by passing to the left adjoint of (4). ∎

######  Definition 2.22.

Given a compactly generated $R$-linear $\infty$-category $\mathcal{C}$, we
denote by $\mathcal{C}^{\operatorname{fin}}\subset\mathcal{C}$ the full
subcategory of objects $Y$, satisfying that
$\operatorname{Mor}_{\mathcal{C}}(X,Y)\in\operatorname{RMod}_{R}$ is compact
for all $X\in\mathcal{C}^{\operatorname{c}}$. We also refer to the objects of
$\mathcal{C}^{\operatorname{fin}}$ as finite.

The following lemma provides the analog of part (1) of Lemma 2.20 for smooth,
but not necessarily proper $R$-linear $\infty$-categories.

######  Lemma 2.23.

Let $\mathcal{C}$ be a compactly generated and smooth $R$-linear
$\infty$-category. Then

$\operatorname{Mor}_{\mathcal{C}}(X,Y)^{*}\simeq\operatorname{Mor}_{\mathcal{C}}(\operatorname{id}^{!}(Y),X)\,,$

functorial in $X\in\mathcal{C}^{\operatorname{c}}$ and
$Y\in(\mathcal{C}^{\operatorname{fin}})^{\operatorname{op}}$.

In particular, this shows that if $\operatorname{id}_{\mathcal{C}}^{!}$ is an
equivalence, then
$\mathcal{C}^{\operatorname{fin}}\subset\mathcal{C}^{\operatorname{c}}$.

###### Proof of Lemma 2.23..

The exact inclusion $\mathcal{C}^{\operatorname{fin}}\subset\mathcal{C}$ gives
rise to an $R$-linear functor
$\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\to\mathcal{C}$. The
$R$-linear functor

$\operatorname{ev}_{\mathcal{C}}^{\operatorname{fin}}\colon\mathcal{C}^{\vee}\otimes\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\longrightarrow\mathcal{C}^{\vee}\otimes\mathcal{C}\xlongrightarrow{\operatorname{ev}_{\mathcal{C}}}\operatorname{RMod}_{R}$

preserves compact objects by the definition of
$\mathcal{C}^{\operatorname{fin}}$ and thus admits an $R$-linear right adjoint
$\operatorname{radj}(\operatorname{ev}_{\mathcal{C}}^{\operatorname{fin}})$.
We define the $R$-linear functor
$U\colon\mathcal{C}\to\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}$ as
the composite

$\mathcal{C}\xlongrightarrow{\operatorname{id}_{\mathcal{C}}\otimes\operatorname{radj}(\operatorname{ev}_{\mathcal{C}}^{\operatorname{fin}})}\mathcal{C}\otimes\mathcal{C}^{\vee}\otimes\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\xlongrightarrow{\operatorname{ev}_{\mathcal{C}}\otimes\operatorname{id}_{\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}}}\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\,.$

The functor $U$ admits a left adjoint, given by the composite

$\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\xlongrightarrow{\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\otimes\operatorname{id}_{\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}}}\mathcal{C}\otimes\mathcal{C}^{\vee}\otimes\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\xlongrightarrow{\operatorname{id}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{C}}^{\operatorname{fin}}}\mathcal{C}\,,$

which describes the composite of $\operatorname{id}_{\mathcal{C}}^{!}$ with
the $R$-linear functor
$\operatorname{Ind}\mathcal{C}^{\operatorname{fin}}\to\mathcal{C}$.

The proof of part (1) of Lemma 2.20 adapts with minimal changes to this
setting and shows that

$\operatorname{Mor}_{\mathcal{C}}(X,Y)^{*}\simeq\operatorname{Mor}_{\mathcal{C}}(Y,U(X))\,,$
(5)

functorial in $Y\in\mathcal{C}^{\operatorname{fin},\operatorname{op}}$ and
$X\in\mathcal{C}^{\operatorname{c}}$. By the above adjunction, we have

$\operatorname{Mor}_{\mathcal{C}}(Y,U(X))\simeq\operatorname{Mor}_{\mathcal{C}}(\operatorname{id}_{\mathcal{C}}^{!}(Y),X),$

which combined with (5) yields the desired equivalence. ∎

###  2.5 Traces

We fix an $\mathbb{E}_{\infty}$-ring spectrum $R$.

######  Definition 2.24.

Let $\mathcal{C}$ be a dualizable $R$-linear $\infty$-category. Let
$E\colon\mathcal{C}\to\mathcal{C}$ be an $R$-linear endomorphism. The trace
$\operatorname{Tr}(E)$ of $E$ is defined as the $R$-linear endomorphism

$\operatorname{RMod}_{R}\xrightarrow{\operatorname{coev}_{\mathcal{C}}}\mathcal{C}\otimes\mathcal{C}^{\vee}\xrightarrow{E\otimes\operatorname{id}_{\mathcal{C}^{\vee}}}\mathcal{C}\otimes\mathcal{C}^{\vee}\xrightarrow{\tau}\mathcal{C}^{\vee}\otimes\mathcal{C}\xrightarrow{\operatorname{ev}_{\mathcal{C}}}\operatorname{RMod}_{R}\,.$

In the following, we recall the construction of Hoyois-Scherotzke-Sibilla
[HSS17] of the $S^{1}$-equivariant $(\infty,1)$-categorical trace functor.
This functor will also give rise to the Hochschild homology functor.

As a model for $(\infty,2)$-categories, we use Barwick’s complete $2$-fold
Segal spaces. We let $\mathcal{C}$ be a symmetric monoidal
$(\infty,2)$-category. We will primarily be interested in the case where
$\bm{C}={\bf LinCat}_{R}$ is the $(\infty,2)$-category of $R$-linear stable
and presentable $\infty$-categories, see [HSS17, Section 4.4], where the
notation ${\bf Pr}^{L}(\operatorname{RMod}_{R})$ is used. We have an
associated complete ($1$-fold) Segal space (which models an
$(\infty,1)$-category) of symmetric monoidal oplax transfors [JFS17]

$\operatorname{End}(\bm{C})=\operatorname{Fun}^{\operatorname{oplax}}_{\otimes}(\operatorname{Fr}^{\operatorname{rig}}(B\mathbb{N}),\bm{C})\,,$

where $\operatorname{Fr}^{\operatorname{rig}}(B\mathbb{N})$ denotes the free
rigid111By rigid, we mean that all objects are dualizable.
$(\infty,2)$-category generated by $B\mathbb{N}$, see also [HSS17, Def. 2.2].
The objects and morphisms in $\operatorname{End}(\bm{C})$ can be concretely
described as follows:

  * •

Objects are given by pairs $(\mathcal{C},E)$ with $\mathcal{C}\in\bm{C}$
dualizable and $E\colon\mathcal{C}\to\mathcal{C}$ an endomorphism in $\bm{C}$.

  * •

A morphism $(F,\alpha)\colon(\mathcal{C},E)\to(\mathcal{D},E^{\prime})$
corresponds to a right dualizable morphism $F\colon\mathcal{C}\to\mathcal{D}$
in $\bm{C}$ together with an oplax-commutative square

${\mathcal{C}}$${\mathcal{C}}$${\mathcal{D}}$${\mathcal{D}}$$\scriptstyle{E}$$\scriptstyle{F}$$\scriptstyle{F}$$\scriptstyle{\alpha}$$\scriptstyle{E^{\prime}}$

meaning a $2$-morphism $\alpha\colon FE\Rightarrow E^{\prime}F$ in $\bm{C}$.

We also consider the complete Segal space $\Omega\bm{C}$ of endomorphisms of
the monoidal unit of $\bm{C}$. The trace defines by [HSS17, Def. 2.9,2.11] a
symmetric monoidal functor between complete Segal spaces

$\operatorname{Tr}\colon\operatorname{End}(\bm{C})\to\Omega\bm{C}\,,$

which is natural in $\bm{C}$. Given a morphism
$(F,\alpha)\colon(\mathcal{C},E)\to(\mathcal{D},E^{\prime})$ in
$\operatorname{End}({\bf LinCat}_{R})$, the morphism

$\operatorname{Tr}(F,\alpha)\colon\operatorname{Tr}(E)\to\operatorname{Tr}(E^{\prime})$

in
$\operatorname{Map}_{\operatorname{LinCat}_{R}}(\operatorname{RMod}_{R},\operatorname{RMod}_{R})$
can be identified with the composition of the following natural
transformations.

${\operatorname{RMod}_{R}}$${\mathcal{C}\otimes\mathcal{C}^{\vee}}$${\mathcal{C}\otimes\mathcal{C}^{\vee}}$${\mathcal{C}^{\vee}\otimes\mathcal{C}}$${\operatorname{RMod}_{R}}$${\operatorname{RMod}_{R}}$${\mathcal{D}\otimes\mathcal{D}^{\vee}}$${\mathcal{D}\otimes\mathcal{D}^{\vee}}$${\mathcal{D}^{\vee}\otimes\mathcal{D}}$${\operatorname{RMod}_{R}}$$\scriptstyle{\operatorname{coev}_{\mathcal{C}}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{F\otimes
F^{\vee}}$$\scriptstyle{E\otimes\operatorname{id}_{\mathcal{C}^{\vee}}}$$\scriptstyle{\nu}$$\scriptstyle{F\otimes
F^{\vee}}$$\scriptstyle{\alpha\otimes\operatorname{id}}$$\scriptstyle{\tau}$$\scriptstyle{F^{\vee}\otimes
F}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{ev}_{\mathcal{C}}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\epsilon}$$\scriptstyle{\operatorname{coev}_{\mathcal{D}}}$$\scriptstyle{E^{\prime}\otimes\operatorname{id}_{\mathcal{D}^{\vee}}}$$\scriptstyle{\tau}$$\scriptstyle{\operatorname{ev}_{\mathcal{D}}}$

Using the triangle identities, the natural transformation $\nu$ above is
defined as the composite of

$\displaystyle(F\otimes F^{\vee})\circ\operatorname{coev}_{\mathcal{C}}$
$\displaystyle\simeq(F\otimes\operatorname{id}_{\mathcal{D}^{\vee}})\circ(\operatorname{id}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{C}^{\vee}}\otimes\operatorname{id}_{\mathcal{D}^{\vee}})\circ(\operatorname{id}_{\mathcal{C}}\otimes\operatorname{id}_{\mathcal{C}^{\vee}}\otimes
G\otimes\operatorname{id}_{\mathcal{D}^{\vee}})\circ(\operatorname{id}_{\mathcal{C}}\otimes\operatorname{id}_{\mathcal{C}^{\vee}}\otimes\operatorname{coev}_{\mathcal{D}})\circ\operatorname{coev}_{\mathcal{C}}$
$\displaystyle\simeq(FG\otimes\operatorname{id}_{\mathcal{D}^{\vee}})\circ\operatorname{coev}_{\mathcal{D}}$

and

$(FG\otimes\operatorname{id}_{\mathcal{D}^{\vee}})\circ\operatorname{coev}_{\mathcal{D}}\xlongrightarrow{(\operatorname{cu}\otimes\operatorname{id}_{\mathcal{D}^{\vee}})\circ\operatorname{coev}_{\mathcal{D}}}\operatorname{coev}_{\mathcal{D}}\,,$

where $G$ denotes the right adjoint of $F$ and $\operatorname{cu}$ the counit.
The natural transformation $\epsilon$ is defined similarly. Further,
$\operatorname{Tr}(F,\alpha)$ is also equivalent to the composite

$\operatorname{Tr}(E)\xrightarrow{\operatorname{Tr}(E\operatorname{u})}\operatorname{Tr}(EGF)\xrightarrow{\beta}\operatorname{Tr}(FEG)\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},\alpha
G)}\operatorname{Tr}(E^{\prime}FG)\xrightarrow{\operatorname{Tr}(E^{\prime}\operatorname{cu})}\operatorname{Tr}(E^{\prime})\,,$
(6)

where $\operatorname{u}$ denotes the unit and $\beta$ is the apparent
morphism, see [BD21, Lemma 4.1].

Finally, we turn to the $S^{1}$-functoriality of the trace functor. For this,
consider the complete Segal space
$\operatorname{Aut}(\bm{C})=\operatorname{Fun}^{\operatorname{oplax}}_{\otimes}(\operatorname{Fr}^{\operatorname{rig}}(B\mathbb{Z}),\bm{C})$,
which comes with a symmetric monoidal inclusion
$\operatorname{Aut}(\bm{C})\subset\operatorname{End}(\bm{C})$. The objects of
$\operatorname{Aut}(\bm{C})$ are given by pairs $(\mathcal{C},E)$ with
$\mathcal{C}\in\bm{C}$ dualizable and $E\colon\mathcal{C}\to\mathcal{C}$ an
equivalence. The self-action of the circle group $S^{1}=B\mathbb{Z}$ induces
an action on $\operatorname{Aut}(\bm{C})$, which in turn induces an action on
the space $\operatorname{Map}(\operatorname{Aut}(\bm{C}),\Omega(\bm{C}))$ of
functors, natural in $\bm{C}$. The trace can be exhibited as a homotopy fixed
point of the $S^{1}$-action on
$\operatorname{Map}(\operatorname{Aut}(\bm{C}),\Omega(\bm{C}))$, see [HSS17,
Thm. 2.14]. This $S^{1}$-invariance datum results in an $S^{1}$-action on the
trace of any pair
$(\mathcal{C},\operatorname{id}_{\mathcal{C}})\in\operatorname{End}(\bm{C})$
as well as an $S^{1}$-equivariant map
$\operatorname{Tr}(F,\operatorname{id}_{F})$ for any morphism
$(F,\operatorname{id}_{F})\colon(\mathcal{C},\operatorname{id}_{\mathcal{C}})\to(\mathcal{D},\operatorname{id}_{\mathcal{D}})$
in $\operatorname{End}(\bm{C})$.

######  Remark 2.25.

The datum of the $S^{1}$-invariance of the trace functor is natural in the
symmetric monoidal $(\infty,2)$-category $\bm{C}$. Thus, given a symmetric
monoidal functor $F\colon\bm{C}\to\bm{D}$ and a dualizable object
$\mathcal{C}\in\bm{C}$, we have an $S^{1}$-equivariant equivalence
$F(\operatorname{Tr}(\mathcal{C},\operatorname{id}_{\mathcal{C}}))\simeq\operatorname{Tr}(F(\mathcal{C}),\operatorname{id}_{F(\mathcal{C})})$.

We can specialize this to the symmetric monoidal functor $\ast\to\bm{C}$ from
the trivial single object symmetric monoidal $(\infty,2)$-category, mapping
$\ast$ to the unit object $1_{C}$ of $\bm{C}$. Since the $S^{1}$-action on the
trace $\operatorname{Tr}(\ast,\operatorname{id}_{\ast})$ is necessarily
trivial, this shows that the $S^{1}$-action on
$\operatorname{Tr}(1_{C},\operatorname{id}_{1_{C}})$ is also trivial. In
particular, this holds for $\bm{C}={\bf LinCat}_{R}$ with
$1_{C}=\operatorname{RMod}_{R}$ and
$\operatorname{Tr}(\operatorname{RMod}_{R},\operatorname{id}_{\operatorname{RMod}_{R}})\simeq
R$.

###  2.6 Hochschild homology

We fix an $\mathbb{E}_{\infty}$-ring spectrum $R$. Given a dualizable
$R$-linear $\infty$-category $\mathcal{C}$, its $R$-linear Hochschild homology
is defined as the value of the trace at $R$:

$\operatorname{HH}(\mathcal{C})\coloneqq\operatorname{ev}_{\mathcal{C}}\circ\
\tau\circ\operatorname{coev}_{\mathcal{C}}(R)=\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})(R)\in\operatorname{RMod}_{R}\,.$

Note that $\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})$ is an
$R$-linear functor and thus fully determined by its value
$\operatorname{HH}(\mathcal{C})$. When $R$ is the sphere spectrum,
$\operatorname{HH}(\mathcal{C})$ is also called topological Hochschild
homology. When $R=k$ is a commutative ring, $\operatorname{HH}(\mathcal{C})$
describes the usual $k$-linear Hochschild homology.

The dual Hochschild homology is defined as

$\operatorname{HH}(\mathcal{C})^{\ast}\coloneqq\operatorname{Mor}_{\operatorname{RMod}_{R}}(\operatorname{HH}(\mathcal{C}),R)\,.$

The fixed points (limit over $BS^{1}$) of the $S^{1}$-action on
$\operatorname{HH}(\mathcal{C})$ are by denoted
$\operatorname{HH}(\mathcal{C})^{S^{1}}$, we will refer to
$\operatorname{HH}(\mathcal{C})^{S^{1}}$ as the negative cyclic homology of
$\mathcal{C}$, as $\operatorname{HH}(\mathcal{C})^{S^{1}}$ recovers the usual
negative cyclic homology when $R=k$ is a commutative ring. The orbits (colimit
over $BS^{1}$) of the $S^{1}$-action on $\operatorname{HH}(\mathcal{C})$ are
denoted $\operatorname{HH}(\mathcal{C})_{S^{1}}$, and we similarly call
$\operatorname{HH}(\mathcal{C})_{S^{1}}$ the cyclic homology. The dual cyclic
homology is given by
$\operatorname{HH}(\mathcal{C})_{S^{1}}^{*}=\operatorname{Mor}_{\operatorname{RMod}_{R}}(\operatorname{HH}(\mathcal{C})_{S^{1}},R)$.
There are natural maps
$\operatorname{HH}(\mathcal{C})^{S^{1}}\to\operatorname{HH}(\mathcal{C})$ and
$\operatorname{HH}(\mathcal{C})_{S^{1}}^{*}\to\operatorname{HH}(\mathcal{C})^{*}$.

######  Notation 2.26.

Let $F\colon\mathcal{C}\to\mathcal{D}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$. We denote by

  1. (1)

$\operatorname{HH}(F)\colon\operatorname{HH}(\mathcal{C})\to\operatorname{HH}(\mathcal{D})$
the evaluation at $R$ of $\operatorname{Tr}(F,\operatorname{id}_{F})$. We
further define
$\operatorname{HH}(\mathcal{D},\mathcal{C})=\operatorname{cof}\operatorname{HH}(F)$.

  2. (2)

$\operatorname{HH}(F)^{S^{1}}\colon\operatorname{HH}(\mathcal{C})^{S^{1}}\to\operatorname{HH}(\mathcal{D})^{S^{1}}$
the induced map. We similarly define
$\operatorname{HH}(\mathcal{D},\mathcal{C})^{S^{1}}=\operatorname{cof}(\operatorname{HH}(F)^{S^{1}})$.

Let $F\colon\mathcal{D}\to\mathcal{C}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$. We denote by

  1. (3)

$\operatorname{HH}(F)^{\ast}\colon\operatorname{HH}(\mathcal{C})^{\ast}\to\operatorname{HH}(\mathcal{D})^{\ast}$
the dual map obtained by precomposition with $\operatorname{HH}(F)$. We
further define
$\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}=\operatorname{cof}\operatorname{HH}(F)^{*}$.

  2. (4)

$\operatorname{HH}(F)_{S^{1}}^{*}\colon\operatorname{HH}(\mathcal{C})_{S^{1}}^{*}\to\operatorname{HH}(\mathcal{D})_{S^{1}}^{*}$
the induced map. We similarly define
$\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}_{S^{1}}=\operatorname{cof}(\operatorname{HH}(F)_{S^{1}}^{*})$.

The notation of (1) and (2) will usually be used when discussing left
Calabi–Yau structures for $F$, whereas the notation of (3) and (4) will
usually be used when discussing right Calabi–Yau structures for $F$. This is
why we swap the roles of $\mathcal{C}$ (the ’Calabi–Yau-boundary’) and
$\mathcal{D}$ (the relative Calabi–Yau category) as domain and target.

######  Lemma 2.27.

Let $\mathcal{C}$ be a dualizable $R$-linear $\infty$-category and let
$E\colon\mathcal{C}\to\mathcal{C}$ be $R$-linear.

  1. (1)

If $\mathcal{C}$ is smooth, then $\operatorname{Tr}(E)(R)$ is canonically
equivalent to

$\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(\operatorname{id}_{\mathcal{C}}^{!},E)\,.$

In particular, we have
$\operatorname{HH}(\mathcal{C})\simeq\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(\operatorname{id}_{\mathcal{C}}^{!},\operatorname{id}_{\mathcal{C}})$.

  2. (2)

If $\mathcal{C}$ is proper, then

$\operatorname{Tr}(E)(R)^{*}\coloneqq\operatorname{Mor}_{\operatorname{RMod}_{R}}(\operatorname{Tr}(E)(R),R)$

is canonically equivalent to

$\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(E,\operatorname{id}_{\mathcal{C}}^{*})\,.$

In particular, we have
$\operatorname{HH}(\mathcal{C})^{*}\simeq\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(\operatorname{id}_{\mathcal{C}},\operatorname{id}_{\mathcal{C}}^{*})$.

###### Proof.

Suppose that $\mathcal{C}$ is smooth. Then we have an adjunction

$\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\circ(\mathchar
45\relax)\colon\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}^{\vee}\otimes\mathcal{C})\longleftrightarrow\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\operatorname{RMod}_{R})\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}\operatorname{ev}_{\mathcal{C}}\circ(\mathchar
45\relax)\,,$

whose unit is given by precomposition with the unit of
$\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\dashv\operatorname{ev}_{\mathcal{C}}$.
It follows that

$\displaystyle\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(\operatorname{id}_{\mathcal{C}}^{!},E)\simeq$
$\displaystyle\operatorname{Mor}_{\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{C}\otimes\mathcal{C}^{\vee})}(\tau\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}}),(E\otimes\operatorname{id}_{\mathcal{C}^{\vee}})\circ\operatorname{coev}_{\mathcal{C}})$
$\displaystyle\simeq$
$\displaystyle\operatorname{Mor}_{\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\operatorname{RMod}_{R})}(\operatorname{id}_{\operatorname{RMod}_{R}},\operatorname{ev}_{\mathcal{C}}\circ\tau\circ(E\otimes\operatorname{id}_{\mathcal{C}^{\vee}})\circ\operatorname{coev}_{\mathcal{C}})$
$\displaystyle\simeq$ $\displaystyle\operatorname{Tr}(E)(R)\,.$

If $\mathcal{C}$ is proper, a similar argument applies. ∎

######  Remark 2.28.

Suppose the dualizable $R$-linear $\infty$-category $\mathcal{C}$ is smooth.
If we make two different choices of left duals/adjoints

$\operatorname{id}_{\mathcal{C}}^{!}=\Xi_{\mathcal{C}}(\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}}))\,,\quad(\operatorname{id}_{\mathcal{C}}^{!})^{\prime}=\Xi_{\mathcal{C}}(\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})^{\prime})$

and two choices of units, there is a contractible space of equivalences
$\alpha\colon(\operatorname{id}_{\mathcal{C}}^{!})^{\prime}\simeq\operatorname{id}_{\mathcal{C}}^{!}$,
compatible with the unit, see [Cis19, Prop. 6.1.9]. Any such equivalence
$\alpha$ assembles with the equivalences from Lemma 2.27 into to a commutative
diagram as follows:

${{\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(\operatorname{id}_{\mathcal{C}}^{!},E)}}$${{\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}((\operatorname{id}_{\mathcal{C}}^{!})^{\prime},E)}}$${\operatorname{Tr}(E)(R)}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{Mor}_{\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})}(\alpha,E)}$$\scriptstyle{\simeq}$

Stated differently, this means that the equivalence in part (1) of Lemma 2.27
is independent of the choice of left dual. A similar statement holds for the
equivalence in part (2).

######  Construction 2.29.

Let $\mathcal{C},\mathcal{D}$ be dualizable $R$-linear $\infty$-categories.

Case 1: Suppose that $\mathcal{C},\mathcal{D}$ are smooth.

Let $F\colon\mathcal{C}\to\mathcal{D}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ and $G$ the $R$-linear right
adjoint of $F$. We denote by

$F_{!}(\mathchar 45\relax)=F\circ(\mathchar 45\relax)\circ
G\colon\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\longrightarrow\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})$

the functor from Lemma 2.16, and by $\operatorname{cu}\colon
F_{!}(\operatorname{id}_{\mathcal{C}})\to\operatorname{id}_{\mathcal{D}}$ the
counit transformation of $F\dashv G$. We define the unit
$\tilde{\operatorname{u}}\colon\operatorname{id}_{\mathcal{D}}^{!}\to
F_{!}(\operatorname{id}_{\mathcal{C}}^{!})$ as the image under
$\Xi_{\mathcal{D}}$ of the natural transformation

$\displaystyle\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})$
$\displaystyle\rightarrow\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{C}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\rightarrow\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\rightarrow(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$

composed with the equivalence

$\Xi_{\mathcal{D}}((F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}}))\simeq
F_{!}(\operatorname{id}_{\mathcal{C}}^{!})$

from Lemma 2.16. The transformation $\tilde{\operatorname{u}}$ is indeed a
unit if $F$ admits a left adjoint, see Lemma 2.32.

Case 2: Suppose that $\mathcal{C},\mathcal{D}$ are proper.

Let $F\colon\mathcal{D}\to\mathcal{C}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ and $G$ the $R$-linear right
adjoint of $F$. Consider the functor

$F^{*}(\mathchar 45\relax)=G\circ(\mathchar 45\relax)\circ
F\colon\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\longrightarrow\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})$

from Lemma 2.16 and denote by
$\operatorname{u}\colon\operatorname{id}_{\mathcal{D}}\to
F^{*}(\operatorname{id}_{\mathcal{C}})$ the unit of $F\dashv G$. Let
$E^{\vee}$ denote the right adjoint of $F^{\vee}$. Applying
$\Xi_{\mathcal{C}^{\vee}}^{-1}$ to the counit
$F^{\vee}E^{\vee}\to\operatorname{id}_{\mathcal{C}}$ defines a natural
transformation

$(F^{\vee}\otimes
F)\circ\operatorname{coev}_{\mathcal{D}^{\vee}}\longrightarrow\operatorname{coev}_{\mathcal{C}^{\vee}}\,.$

We use this to define the counit $\tilde{\operatorname{cu}}\colon
F^{*}(\operatorname{id}_{\mathcal{C}}^{*})\to\operatorname{id}_{\mathcal{D}}^{*}$
as the image under $\Theta_{\mathcal{D}}$ of the natural transformation

$\displaystyle\operatorname{ladj}(\operatorname{coev}_{\mathcal{C}^{\vee}})\circ(F^{\vee}\otimes
F)$
$\displaystyle\to\operatorname{ladj}(\operatorname{coev}_{\mathcal{C}^{\vee}})\circ(F^{\vee}\otimes
F)\circ\operatorname{coev}_{\mathcal{D}^{\vee}}\circ\operatorname{ladj}(\operatorname{coev}_{\mathcal{D}^{\vee}})$
$\displaystyle\to\operatorname{ladj}(\operatorname{coev}_{\mathcal{C}^{\vee}})\circ\operatorname{coev}_{\mathcal{C}^{\vee}}\circ\operatorname{ladj}(\operatorname{coev}_{\mathcal{D}^{\vee}})$
$\displaystyle\to\operatorname{ladj}(\operatorname{coev}_{\mathcal{D}^{\vee}})$

composed with the identification
$\Theta_{\mathcal{D}}(\operatorname{ladj}(\operatorname{coev}_{\mathcal{C}^{\vee}})\circ(F^{\vee}\otimes
F))\simeq F^{*}(\operatorname{id}_{\mathcal{C}}^{*})$ from Lemma 2.16.

The following proposition describes the Hochschild homology functor in terms
of the corresponding morphisms between bimodules.

######  Proposition 2.30 ([BD21, Prop. 4.4]).

Let $F\colon\mathcal{C}\to\mathcal{D}$ be a map in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ with $\mathcal{C}$ smooth.
Consider a morphism $\xi\colon
R[i]\to\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})$, corresponding via
Lemma 2.27 to a natural transformation
$\alpha\colon\operatorname{id}_{\mathcal{C}}^{!}[i]\to\operatorname{id}_{\mathcal{C}}$.

  1. (1)

The morphism

$R[i]\xlongrightarrow{\xi}\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})\xlongrightarrow{\operatorname{Tr}(F,\operatorname{id}_{F})}\operatorname{Tr}(\operatorname{id}_{\mathcal{D}})$
(7)

is equivalent to the composite

$\displaystyle
R[i]\longrightarrow\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{!})[i]\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\operatorname{id}_{\mathcal{C}}^{!}\operatorname{u})[i]}$
$\displaystyle\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{!}GF)[i]\longrightarrow\operatorname{Tr}(F\operatorname{id}_{\mathcal{C}}^{!}G)[i]$
$\displaystyle\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},F\alpha
G)}$
$\displaystyle\operatorname{Tr}(FG)\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},\operatorname{cu})}\operatorname{Tr}(\operatorname{id}_{\mathcal{D}})\,.$

  2. (2)

Suppose that $\mathcal{D}$ is smooth. Then the morphism (7) can be identified
with

$\operatorname{cu}\circ
F_{!}(\alpha)\circ\tilde{\operatorname{u}}\in\operatorname{Ext}^{-i}_{\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})}(\operatorname{id}_{\mathcal{D}}^{!},\operatorname{id}_{\mathcal{D}})\,.$

###### Proof.

The proof of part (1) can be found in [BD21, Prop. 4.4], we also spell out its
dual version in the proof of Proposition 2.31 below. Part (2) is stated in
loc. cit., we spell the proof out for convenience of the reader. Using part
(1), it suffices to show that the composite

$R[i]\longrightarrow\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{!})[i]\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\operatorname{id}_{\mathcal{C}}^{!}\operatorname{u})[i]}\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{!}GF)[i]\longrightarrow\operatorname{Tr}(F\operatorname{id}_{\mathcal{C}}^{!}G)[i]$

maps $1\in R[i]$ to $\tilde{\operatorname{u}}$ after evaluating the trace at
$R$ and using the identification of Lemma 2.27.

The image under $\Xi_{\mathcal{C}}^{-1}$ of the unit
$\tilde{\operatorname{u}}\colon\operatorname{id}_{\mathcal{D}}^{!}\to
F_{!}(\operatorname{id}_{\mathcal{D}}^{!})$ corresponds via the adjunction

$\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\circ(\mathchar
45\relax)\colon\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\operatorname{RMod}_{R})\longleftrightarrow\operatorname{Lin}_{R}(\operatorname{RMod}_{R},\mathcal{D}^{\vee}\otimes\mathcal{D})\nobreak\mspace{6.0mu}{:}\nonscript\mkern-3.0mu\mathpunct{}\mspace{2.0mu}\operatorname{ev}_{\mathcal{D}}\circ(\mathchar
45\relax)$

to the morphism

$\displaystyle\operatorname{id}_{\operatorname{RMod}_{R}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{C}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\,,$

which is by the triangle identity the same as

$\operatorname{id}_{\operatorname{RMod}_{R}}\to\operatorname{ev}_{\mathcal{C}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\to\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\,.$

The map coming from the trace is given by

$\displaystyle\operatorname{id}_{\operatorname{RMod}_{R}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{C}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\to\operatorname{ev}_{\mathcal{C}}\circ(\operatorname{id}_{\mathcal{C}^{\vee}}\otimes
GF)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
FGF)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\,.$

Applying the triangle identity of the adjunction $F\dashv G$, we find that
this agrees with the above. ∎

Given $(\mathcal{C},E)\in{\bf End}({\bf LinCat}_{R})$, we denote by
$\operatorname{Tr}(\mathcal{C},E)^{*}$ the composite of
$\operatorname{Tr}(\mathcal{C},E)$ and
$\operatorname{Mor}_{\operatorname{RMod}_{R}}(\mathchar
45\relax,R)\colon\operatorname{RMod}_{R}\to\operatorname{RMod}_{R}$. The
analogue of Proposition 2.30 for the dual Hochschild homology is as follows.

######  Proposition 2.31.

Let $F\colon\mathcal{D}\to\mathcal{C}$ be a map in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ with $\mathcal{D}$ proper.
Consider a morphism $\xi\colon
R[i]\to\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})^{*}$, corresponding
via Lemma 2.27 to a natural transformation
$\alpha\colon\operatorname{id}_{\mathcal{C}}\to\operatorname{id}_{\mathcal{C}}^{*}[-i]$.

  1. (1)

The morphism

$R[i]\xlongrightarrow{\xi}\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})^{*}\xlongrightarrow{\operatorname{Tr}(F,\operatorname{id}_{F})^{*}}\operatorname{Tr}(\operatorname{id}_{\mathcal{D}})^{*}$
(8)

is equivalent to the composite

$\displaystyle
R[i]\longrightarrow\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{*})^{*}[i]\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\operatorname{id}_{\mathcal{C}}^{*}\operatorname{cu})^{*}[i]}$
$\displaystyle\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{*}FG)^{*}[i]\longrightarrow\operatorname{Tr}(G\operatorname{id}_{\mathcal{C}}^{*}F)^{*}[i]$
$\displaystyle\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},G\alpha
F)^{*}}$
$\displaystyle\operatorname{Tr}(GF)^{*}\xrightarrow{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},\operatorname{u})^{*}}\operatorname{Tr}(\operatorname{id}_{\mathcal{D}})^{*}\,.$

  2. (2)

Suppose that $\mathcal{C}$ is proper. Then the morphism (8) can be identified
with

$\tilde{\operatorname{cu}}\circ
F^{*}(\alpha)\circ\operatorname{u}\in\operatorname{Ext}^{-i}_{\operatorname{Lin}_{R}(\mathcal{D},\mathcal{D})}(\operatorname{id}_{\mathcal{D}},\operatorname{id}_{\mathcal{D}}^{*})\,.$

###### Proof.

The proof of part (1) is dual to the proof of part (1) of Proposition 2.30.
Using Lemma 2.27, the identity on $\operatorname{id}_{\mathcal{C}}^{*}$ gives
rise to a morphism $\mu\colon
R[i]\to\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{*})[i]$. The
statement now follows from the following commutative diagram.

${{R[i]}}$${{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{*})^{*}[i]}}$${\operatorname{Tr}(\operatorname{id}_{\mathcal{C}})^{*}}$${{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}}^{*}FG)^{*}[i]}}$${\operatorname{Tr}(FG)^{*}}$${{\operatorname{Tr}(G\operatorname{id}_{\mathcal{C}}^{*}F)^{*}[i]}}$${\operatorname{Tr}(GF)^{*}}$${\operatorname{Tr}(\operatorname{id}_{\mathcal{D}})^{*}}$$\scriptstyle{\mu}$$\scriptstyle{\xi}$$\scriptstyle{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\alpha)^{*}}$$\scriptstyle{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\operatorname{id}_{\mathcal{C}}^{*}\operatorname{cu})^{*}[i]}$$\scriptstyle{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\operatorname{cu})^{*}}$$\scriptstyle{\operatorname{Tr}(F,\operatorname{id}_{F})^{*}}$$\scriptstyle{\operatorname{Tr}(\operatorname{id}_{\mathcal{C}},\alpha
FG)^{*}}$$\scriptstyle{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},G\alpha
F)^{*}}$$\scriptstyle{\operatorname{Tr}(\operatorname{id}_{\mathcal{D}},\operatorname{u})^{*}}$

Note that the commutativity of the rightmost triangle follows from dualizing
Equation 6.

Part (2) can be shown as in the proof of Proposition 2.30. ∎

######  Lemma 2.32.

Let
$\mathcal{C},\mathcal{D}\in\operatorname{LinCat}_{R}^{\operatorname{dual}}$.

  1. (1)

Suppose that $\mathcal{C},\mathcal{D}$ are smooth. Let
$F\colon\mathcal{C}\to\mathcal{D}$ be a map in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ which admits a left adjoint
$E$. Then there exists a natural equivalence
$F_{!}(\operatorname{id}_{\mathcal{C}}^{!})\simeq
FE\operatorname{id}_{\mathcal{D}}^{!}$, such that the composite of

$\tilde{\operatorname{u}}\colon\operatorname{id}_{\mathcal{D}}^{!}\to
F_{!}(\operatorname{id}_{\mathcal{C}}^{!})$

with this equivalence describes a unit of $E\dashv F$ composed with
$\operatorname{id}_{\mathcal{D}}^{!}$.

  2. (2)

Suppose that $\mathcal{C},\mathcal{D}$ are proper. Let
$F\colon\mathcal{D}\to\mathcal{C}$ be a map in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ which admits a left adjoint
$E$. Then there exists a natural equivalence
$F^{*}(\operatorname{id}_{\mathcal{C}}^{*})\simeq\operatorname{id}_{\mathcal{D}}^{*}EF$,
such that the composite of

$\tilde{\operatorname{cu}}\colon
F^{*}(\operatorname{id}_{\mathcal{C}}^{*})\to\operatorname{id}_{\mathcal{D}}^{*}$

with this equivalence describes a counit of $E\dashv F$ composed with
$\operatorname{id}_{\mathcal{D}}^{*}$.

###### Proof.

We only prove part (1), part (2) is similar. We have

$\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes\operatorname{id}_{\mathcal{C}})\simeq\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
F)\,.$

The units of the adjunctions

$(F^{\vee}\otimes\operatorname{id}_{\mathcal{C}})\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\dashv\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes\operatorname{id}_{\mathcal{C}})$

and

$(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
E)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\dashv\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
F)$

are therefore equivalent. This gives rise to the following commutative
diagram:

${\operatorname{id}_{\operatorname{RMod}_{R}}}$${\operatorname{ev}_{\mathcal{C}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})}$${\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}F^{\vee}\otimes\operatorname{id}_{\mathcal{C}})\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})}$${\operatorname{ev}_{\mathcal{D}}\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})}$${\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FE)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})}$${\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}\otimes
F)\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

The naturality of the unit
$\operatorname{id}_{\mathcal{D}^{\vee}}\otimes\operatorname{id}_{\mathcal{D}}\to\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FE$ and the counit
$\operatorname{id}_{\mathcal{D}^{\vee}}\otimes\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\operatorname{ev}_{\mathcal{D}}\to\operatorname{id}_{\mathcal{D}^{\vee}}\otimes\operatorname{id}_{\mathcal{D}}$
gives rise to the following commutative diagram:

${\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\operatorname{ev}_{\mathcal{D}}\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})}$${\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\operatorname{ev}_{\mathcal{D}}(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FE)\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})}$${\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})}$${(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FE)\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})}$

Postcomposing the upper diagram with
$\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})$ and combining it with
the lower diagram, we see that the definition of the natural transformation
$\tilde{\operatorname{u}}\colon\operatorname{id}_{\mathcal{D}}^{!}\to
F_{!}(\operatorname{id}_{\mathcal{C}}^{!})$ in 2.29 is equivalent to the image
under $\Xi_{\mathcal{D}}$ of the natural transformation

$\displaystyle\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})$
$\displaystyle\to\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\operatorname{ev}_{\mathcal{D}}\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})$
$\displaystyle\to\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})$
$\displaystyle\to(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FE)\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})$
$\displaystyle\simeq(F^{\vee}\otimes
F)\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})\,.$

The desired description of $\tilde{\operatorname{u}}$ now follows via the
triangle identity for the adjunction
$\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\dashv\operatorname{ev}_{\mathcal{D}}$.
∎

The following lemma shows that the natural transformations
$\tilde{\operatorname{u}}$ and $\tilde{\operatorname{cu}}$ can also be seen as
the adjoints of the counit and unit of $F\dashv G$.

######  Lemma 2.33.

  1. (1)

Let $F\colon\mathcal{C}\to\mathcal{D}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ with
$\mathcal{C},\mathcal{D}$ smooth. Denote the right adjoint of $F$ by $G$. The
image under $\Xi_{\mathcal{C}}^{-1}$ of the unit
$\tilde{\operatorname{u}}\colon\operatorname{id}_{\mathcal{D}}^{!}\to
F\operatorname{id}_{\mathcal{C}}^{!}G$ is left adjoint to the counit
$\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FG)\to\operatorname{ev}_{\mathcal{D}}$.

  2. (2)

Let $F\colon\mathcal{D}\to\mathcal{C}$ be a morphism in
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ with
$\mathcal{C},\mathcal{D}$ proper. Denote the right adjoint of $F$ by $G$. The
image under $\Theta_{\mathcal{C}}^{-1}$ of the counit
$\tilde{\operatorname{cu}}\colon
G\operatorname{id}_{\mathcal{C}}^{*}F\to\operatorname{id}_{\mathcal{C}}^{*}$
is left adjoint to the unit
$\operatorname{ev}_{\mathcal{D}}\to\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
GF)$.

###### Proof.

We only prove part (1), part (2) can be proven similarly. Denote the right
adjoint of $F^{\vee}$ by $E^{\vee}$. Upon passing to right adjoints, the
natural transformation
$\Xi_{\mathcal{C}}^{-1}(\tilde{\operatorname{u}})\colon\operatorname{ladj}(\operatorname{ev}_{\mathcal{D}})\to(F\otimes
F^{\vee})\circ\operatorname{ladj}(\operatorname{ev}_{\mathcal{C}})$ induces
the following natural transformation:

$\displaystyle\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes G)$
$\displaystyle\to\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes
G)\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{C}}\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{C}})\circ\operatorname{ev}_{\mathcal{D}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}$

Unraveling the definition of the above natural transformation222Hint: given a
natural transformation $F\to F^{\prime}$, the induced natural transformations
on the right adjoints $G^{\prime}\to G$ is obtained as the composite
$G^{\prime}\to GFG^{\prime}\to GF^{\prime}G^{\prime}\to G$., one sees that it
is equivalent to the composite

$\displaystyle\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes G)$
$\displaystyle\to\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes
G)\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}E^{\vee}\otimes
FG)\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}$
$\displaystyle\to\operatorname{ev}_{\mathcal{D}}\,,$

where the third natural transformation uses the counits of $F\dashv G$ and
$F^{\vee}\dashv E^{\vee}$. This natural transformation fits into the following
commutative diagram,

${\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FG)}$${\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes
G)}$${\operatorname{ev}_{\mathcal{C}}\circ(E^{\vee}\otimes
G)\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}}$${\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FGFG)}$${\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}E^{\vee}\otimes
FG)}$${\operatorname{ev}_{\mathcal{D}}\circ(F^{\vee}E^{\vee}\otimes
FG)\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}}$${\operatorname{ev}_{\mathcal{D}}}$${\operatorname{ev}_{\mathcal{D}}\circ\operatorname{radj}(\operatorname{ev}_{\mathcal{D}})\circ\operatorname{ev}_{\mathcal{D}}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

exhibiting it as the counit
$\operatorname{ev}_{\mathcal{D}}\circ(\operatorname{id}_{\mathcal{D}^{\vee}}\otimes
FG)\to\operatorname{ev}_{\mathcal{D}}$ by further triangle identities. ∎

##  3 Relative Calabi–Yau structures

The goal of this section is to introduce $R$-linear relative Calabi–Yau
structures and describe essential features of their theory. We begin in
Sections 3.1 and 3.2 with their definitions. After the short Section 3.3 on
the behavior of Calabi–Yau structures under tensor products, we generalize the
gluing properties of relative Calabi–Yau structures of [BD19] to the
$R$-linear setting in Section 3.4. For the entire section, we fix an
$\mathbb{E}_{\infty}$-ring spectrum $R$.

###  3.1 Left Calabi–Yau structures

Let $F\colon\mathcal{C}\rightarrow\mathcal{D}$ be a dualizable $R$-linear
functor between dualizable and smooth $R$-linear $\infty$-categories. Part (2)
of Proposition 2.30 shows that an $R$-linear relative Hochschild homology
class $\sigma\colon R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})$ amounts
to a diagram

${{\operatorname{id}_{\mathcal{D}}^{!}}}$${{F_{!}(\operatorname{id}_{\mathcal{C}}^{!})}}$${{F_{!}(\operatorname{id}_{\mathcal{C}})[1-n]}}$${{\operatorname{id}_{\mathcal{D}}[1-n]}}$$\scriptstyle{\tilde{\operatorname{u}}}$$\scriptstyle{\operatorname{cu}[1-n]}$

together with a choice of null-homotopy of the composite
$\operatorname{id}_{\mathcal{D}}^{!}\to\operatorname{id}_{\mathcal{D}}[1-n]$.
The null-homotopy allows us to extend the diagram to a diagram with horizontal
fiber and cofiber sequences as follows:

${{\operatorname{id}_{\mathcal{D}}^{!}}}$${{F_{!}(\operatorname{id}_{\mathcal{C}}^{!})}}$${\operatorname{cof}}$${\operatorname{fib}}$${{F_{!}(\operatorname{id}_{\mathcal{C}})[1-n]}}$${{\operatorname{id}_{\mathcal{D}}[1-n]}}$$\scriptstyle{\tilde{\operatorname{u}}}$$\scriptstyle{\operatorname{cu}[1-n]}$
(9)

We call the Hochschild homology class $\sigma$ non-degenerate if all vertical
maps in the diagram (9) are equivalences.

######  Definition 3.1.

  1. (1)

A weak left $n$-Calabi–Yau structure on the functor $F$ consists of a non-
degenerate Hochschild homology class $\sigma\colon
R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})$. If $F=0$, we also say that
$\mathcal{D}$ carries a weak left $n$-Calabi–Yau structure.

  2. (2)

A left $n$-Calabi–Yau structure on the functor $F$ consists of a negative
cyclic homology class $\eta\colon
R[n]\to\operatorname{HH}^{S^{1}}(\mathcal{D},\mathcal{C})$, whose composition
with
$\operatorname{HH}^{S^{1}}(\mathcal{D},\mathcal{C})\to\operatorname{HH}(\mathcal{D},\mathcal{C})$
defines a non-degenerate Hochschild homology class. If $F=0$, we also say that
$\mathcal{D}$ carries a left $n$-Calabi–Yau structure.

Weak left $n$-Calabi–Yau structures are also sometimes called bimodule
$n$-Calabi–Yau structures.

######  Remark 3.2.

The notion of weak left Calabi–Yau structure on a functor $F$ only depends on
the functor and the relative Hochschild class and not on any further choices
made in its definition. This includes choices of adjoints and (co)units, for
instance the choice of right adjoint of $F$ together with the counit, of which
there each exists a contractible space of choices. Inspecting the definition
one finds that making a different choice yields an equivalent diagram in (9)
and thus the same condition of the Hochschild class being non-degenerate.

###  3.2 Right Calabi–Yau structures

Let $F\colon\mathcal{D}\rightarrow\mathcal{C}$ be a dualizable $R$-linear
functor between dualizable and proper $R$-linear $\infty$-categories. Part (2)
of Proposition 2.31 shows that an $R$-linear dual relative Hochschild homology
class $\sigma\colon R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}$
amounts to a diagram

${{\operatorname{id}_{\mathcal{D}}}}$${{F^{*}(\operatorname{id}_{\mathcal{C}})}}$${F^{*}(\operatorname{id}_{\mathcal{C}}^{*})[1-n]}$${\operatorname{id}_{\mathcal{D}}^{*}[1-n]}$$\scriptstyle{\operatorname{u}}$$\scriptstyle{\tilde{\operatorname{cu}}[1-n]}$

together with a choice of null-homotopy of the composite
$\operatorname{id}_{\mathcal{D}}\to\operatorname{id}_{\mathcal{D}}^{*}[1-n]$.
We extend the diagram to a diagram with horizontal fiber and cofiber sequences
as follows:

${{\operatorname{id}_{\mathcal{D}}}}$${{F^{*}(\operatorname{id}_{\mathcal{C}})}}$${\operatorname{cof}}$${\operatorname{fib}}$${F^{*}(\operatorname{id}_{\mathcal{C}}^{*})[1-n]}$${\operatorname{id}_{\mathcal{D}}^{*}[1-n]}$$\scriptstyle{\operatorname{u}}$$\scriptstyle{\tilde{\operatorname{cu}}[1-n]}$
(10)

As in the smooth case, we call the dual Hochschild homology class $\sigma$
non-degenerate if all vertical maps in the above diagram are equivalences.

######  Definition 3.3.

  1. (1)

A weak right $n$-Calabi–Yau structure on the functor $F$ consists of a non-
degenerate dual Hochschild homology class $\sigma\colon
R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}$. If $\mathcal{C}=0$, we
also say that $\mathcal{D}$ carries a weak right $n$-Calabi–Yau structure.

  2. (2)

A right $n$-Calabi–Yau structure on the functor $F$ consists of a dual cyclic
homology class $\eta\colon
R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}_{S^{1}}$, whose
composition with
$\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}_{S^{1}}\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}$
defines a non-degenerate dual Hochschild homology class. If $\mathcal{C}=0$,
we also say that $\mathcal{D}$ carries a right $n$-Calabi–Yau structure.

######  Remark 3.4.

Assume that $\mathcal{D}$ is compactly generated. A weak right $n$-Calabi–Yau
structure on $\mathcal{D}$ equivalently consists of an equivalence in
$\operatorname{RMod}_{R}$

$\operatorname{Mor}_{\mathcal{D}}(X,Y)\simeq\operatorname{Mor}_{\mathcal{D}}(Y,X[n])^{*}\,,$
(11)

bifunctorial in $X,Y\in\mathcal{D}^{\operatorname{c}}$.

In good situations, a relative right $n$-Calabi–Yau structure on $\mathcal{D}$
gives rise to a duality of a subfunctor of
$\operatorname{Mor}_{\mathcal{D}}(\mathchar 45\relax,\mathchar 45\relax)$,
generalizing the equivalence (11), see [Chr22a].

######  Remark 3.5.

It is also possible to make sense of relative right Calabi–Yau structures on
some non-proper $k$-linear $\infty$-categories, namely those arising as the
derived $\infty$-category of a dg-algebras whose homology is finite
dimensional in each degree. We refer to the recent work [KL23] for an account
and the construction of many examples related to cluster categories.

###  3.3 Behavior under tensor products

######  Lemma 3.6.

Let $\mathcal{C},\mathcal{D}$ be dualizable $R$-linear $\infty$-categories.
There is a canonical equivalence

$\operatorname{HH}(\mathcal{C})\otimes\operatorname{HH}(\mathcal{D})\simeq\operatorname{HH}(\mathcal{C}\otimes\mathcal{D})\,.$

###### Proof.

Using that
$\operatorname{ev}_{\mathcal{C}\otimes\mathcal{D}}\simeq\operatorname{ev}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{D}}$
and
$\operatorname{coev}_{\mathcal{C}\otimes\mathcal{D}}\simeq\operatorname{coev}_{\mathcal{C}}\otimes\operatorname{coev}_{\mathcal{D}}$,
we find

$\displaystyle\operatorname{HH}(\mathcal{C}\otimes\mathcal{D})$
$\displaystyle=\operatorname{ev}_{\mathcal{C}\otimes\mathcal{D}}\circ\tau\circ\operatorname{coev}_{\mathcal{C}\otimes\mathcal{D}}(R)$
$\displaystyle\simeq(\operatorname{ev}_{\mathcal{C}}\otimes\operatorname{ev}_{\mathcal{D}})\circ\tau\circ(\operatorname{coev}_{\mathcal{C}}\otimes\operatorname{coev}_{\mathcal{D}})(R)$
$\displaystyle\simeq(\operatorname{ev}_{\mathcal{C}}\circ\tau\circ(\operatorname{coev}_{\mathcal{C}})(R)\otimes(\operatorname{ev}_{\mathcal{D}}\circ\tau\circ\operatorname{coev}_{\mathcal{D}})(R)$
$\displaystyle=\operatorname{HH}(\mathcal{C})\otimes\operatorname{HH}(D)\,.$

∎

We note that the equivalence in Lemma 3.6 is $S^{1}$-equivariant.

######  Remark 3.7.

If $\mathcal{C},\mathcal{D}$ are smooth, then $\mathcal{C}\otimes\mathcal{D}$
is also smooth and we have
$\operatorname{id}_{\mathcal{C}\otimes\mathcal{D}}^{!}\simeq\operatorname{id}_{\mathcal{C}}^{!}\otimes\operatorname{id}_{\mathcal{D}}^{!}$.
Similarly, if $\mathcal{C},\mathcal{D}$ are proper then
$\mathcal{C}\otimes\mathcal{D}$ is proper and
$\operatorname{id}_{\mathcal{C}\otimes\mathcal{D}}^{*}\simeq\operatorname{id}_{\mathcal{C}}^{*}\otimes\operatorname{id}_{\mathcal{D}}^{*}$.

In the smooth case, a pair of morphisms
$\alpha\colon\operatorname{id}_{\mathcal{C}}^{!}\to\operatorname{id}_{\mathcal{C}}[-n]$,
$\beta\colon\operatorname{id}_{\mathcal{D}}^{!}\to\operatorname{id}_{\mathcal{D}}[-m]$
gives under the identifications from Lemmas 2.27 and 3.6 rise to the morphism

$\operatorname{id}_{\mathcal{C}\otimes\mathcal{D}}^{!}\simeq\operatorname{id}_{\mathcal{C}}^{!}\otimes\operatorname{id}_{\mathcal{D}}^{!}\xrightarrow{\alpha\otimes\beta}\operatorname{id}_{\mathcal{C}}[-n]\otimes\operatorname{id}_{\mathcal{D}}[-m]\simeq\operatorname{id}_{\mathcal{C}\otimes\mathcal{D}}[-n-m]\,.$

A similar assertion holds in the proper case.

The following proposition shows that the tensor product of a Calabi–Yau
functor with a Calabi–Yau category is again Calabi–Yau. A similar statement is
proven in [BD21, Prop. 6.4].

######  Proposition 3.8.

Let $\mathcal{C},\mathcal{D},\mathcal{E}$ be dualizable $R$-linear
$\infty$-categories.

  1. (1)

Assume that $\mathcal{C},\mathcal{D},\mathcal{E}$ are smooth. Let
$F\colon\mathcal{C}\rightarrow\mathcal{D}$ be a dualizable $R$-linear functor
and let $\eta\colon R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{S^{1}}$
and $\eta^{\prime}\colon R[m]\to\operatorname{HH}(\mathcal{E})^{S^{1}}$ be
left Calabi–Yau-structures on $F$ and $\mathcal{E}$, respectively. The class

$R[n+m]\simeq R[n]\otimes
R[m]\xlongrightarrow{\eta\otimes\eta^{\prime}}\operatorname{HH}(\mathcal{D},\mathcal{C})^{S^{1}}\otimes\operatorname{HH}(\mathcal{E})^{S^{1}}\simeq\operatorname{HH}(\mathcal{D}\otimes\mathcal{E},\mathcal{C}\otimes\mathcal{E})^{S^{1}}$

defines a left $(n+m)$-Calabi–Yau structure on

$F\otimes\mathcal{E}\colon\mathcal{C}\otimes\mathcal{E}\rightarrow\mathcal{D}\otimes\mathcal{E}\,.$

  2. (2)

Assume that $\mathcal{C},\mathcal{D},\mathcal{E}$ are proper. Let
$F\colon\mathcal{D}\rightarrow\mathcal{C}$ be a dualizable $R$-linear functor
and let $\eta\colon
R[n]\to\operatorname{HH}(\mathcal{D},\mathcal{C})^{*}_{S^{1}}$ and
$\eta^{\prime}\colon R[m]\to\operatorname{HH}(\mathcal{E})^{*}_{S^{1}}$ be
right Calabi–Yau-structures on $F$ and $\mathcal{E}$, respectively. The class

$R[n+m]\simeq R[n]\otimes
R[m]\xlongrightarrow{\eta\otimes\eta^{\prime}}\operatorname{HH}(\mathcal{D},\mathcal{C})_{S^{1}}^{*}\otimes\operatorname{HH}(\mathcal{E})^{*}_{S^{1}}\simeq\operatorname{HH}(\mathcal{D}\otimes\mathcal{E},\mathcal{C}\otimes\mathcal{E})^{*}_{S^{1}}$

defines a right $(n+m)$-Calabi–Yau structure on

$F\otimes\mathcal{E}\colon\mathcal{D}\otimes\mathcal{E}\rightarrow\mathcal{C}\otimes\mathcal{E}\,.$

###### Proof.

We only prove part (1), part (2) is analogous. The Hochschild homology class
$R[n+m]\to\operatorname{HH}(\mathcal{D}\otimes\mathcal{E},\mathcal{C}\otimes\mathcal{E})^{S^{1}}\to\operatorname{HH}(\mathcal{D}\otimes\mathcal{E},\mathcal{C}\otimes\mathcal{E})$
gives rise to the following diagram in
$\operatorname{Lin}_{R}(\mathcal{D}\otimes\mathcal{E},\mathcal{C}\otimes\mathcal{E})$,
up to equivalence.

${\operatorname{id}_{\mathcal{D}}^{!}\otimes\operatorname{id}_{\mathcal{E}}^{!}}$${F_{!}(\operatorname{id}_{\mathcal{C}}^{!})\otimes\operatorname{id}_{\mathcal{E}}^{!}}$${\operatorname{cof}\otimes\operatorname{id}_{\mathcal{E}}^{!}}$${{\operatorname{fib}\otimes\operatorname{id}_{\mathcal{E}}[-m]}}$${{F_{!}(\operatorname{id}_{\mathcal{C}})[1-n]\otimes\operatorname{id}_{\mathcal{E}}[-m]}}$${{\operatorname{id}_{\mathcal{D}}[1-n]\otimes\operatorname{id}_{\mathcal{E}}[-m]}}$

The horizontal sequences in the above diagram are fiber and cofiber sequences
as tensor products of such with $\operatorname{id}_{\mathcal{E}}^{!}$ or
$\operatorname{id}_{\mathcal{E}}[m]$. The vertical maps are equivalences as
tensor products of equivalences, showing the non-degeneracy of the Hochschild
homology class. ∎

###  3.4 Gluing Calabi–Yau structures

In this section, we discuss a generalization of the gluing theorem for left
Calabi–Yau structures on $k$-linear dg-categories, see [BD19, Theorem 6.1], to
left and right Calabi–Yau structures on $R$-linear $\infty$-categories, see
Theorems 3.14 and 3.15. The gluing theorems boil down to a simple description
of objects in pullbacks/pushouts of stable, presentable $\infty$-categories in
terms of their restrictions given in Lemma 3.10. We restrict the discussion to
compactly generated $\infty$-categories, as the literature seems to be lacking
results on the computation of limits and colimits in the $\infty$-category
$\operatorname{LinCat}_{R}^{\operatorname{dual}}$ of dualizable $R$-linear
$\infty$-categories. After resolving this, we expect all arguments to apply in
the dualizable case as well.

Consider the simplicial set
$Z=\Delta^{\\{0,1\\}}\amalg_{\Delta^{\\{0\\}}}\Delta^{\\{0,1^{\prime}\\}}$
describing a cospan with objects $0,1,1^{\prime}$ and two non-degenerate
$1$-simplicies $0\to 1,1^{\prime}$. We fix a diagram $D\colon
Z\rightarrow\operatorname{LinCat}_{R}$ with colimit $\mathcal{C}$, satisfying
that $D$ maps each morphism to a functor admitting an $R$-linear right
adjoint. For $z\in Z$, we denote $\mathcal{C}_{z}=D(z)$ and can depict the
colimit diagram of $D$ as follows:

${\mathcal{C}_{0}}$${\mathcal{C}_{1}}$${\mathcal{C}_{1^{\prime}}}$${\mathcal{C}}$$\scriptstyle{i_{0}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{1^{\prime}}}$

Above $i_{z}\colon\mathcal{C}_{z}\rightarrow\mathcal{C}$ denotes the functor
from the colimit cone. We further denote by $j_{z}=\operatorname{radj}(i_{z})$
the right adjoint, and $\operatorname{cu}_{z}$ the counit of $i_{z}\dashv
j_{z}$, for $z\in Z$.

The fact that counits compose to counits provides us with a commutative square

$\phi_{D}\colon
Z^{\triangleright}\rightarrow\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$

which can be depicted as follows:

${i_{0}j_{0}}$${i_{1}j_{1}}$${i_{1^{\prime}}j_{1^{\prime}}}$${\operatorname{id}_{\mathcal{C}}}$$\scriptstyle{\operatorname{cu}_{0}}$$\scriptstyle{\operatorname{cu}_{1}}$$\scriptstyle{\operatorname{cu}_{1^{\prime}}}$

######  Proposition 3.9.

The square $\phi_{D}\colon
Z^{\triangleright}\rightarrow\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})$
is biCartesian.

###### Proof.

Using that the forgetful functor
$\operatorname{Lin}_{R}(\mathcal{C},\mathcal{C})\rightarrow\operatorname{Fun}(\mathcal{C},\mathcal{C})$
reflects finite colimits and that colimits in functor categories are computed
pointwise, see [Lur09a, 5.1.2.3], the statement reduces to Lemma 3.10. ∎

######  Lemma 3.10.

Let $c\in\mathcal{C}$ and consider the diagram

$\phi_{c}\colon Z^{\triangleright}\rightarrow\mathcal{C}\,,$

given by evaluating $\phi_{D}$ at $c$. Then $\phi_{c}$ describes a biCartesian
square in $\mathcal{C}$.

###### Proof.

We can identify the colimit $\mathcal{C}$ of $D$ with the $\infty$-category of
coCartesian sections of the Grothendieck construction
$\pi\colon\Gamma(\operatorname{radj}(D))\rightarrow Z^{\operatorname{op}}$ of
the diagram obtained from $D$ by passing to right adjoint functors, i.e. the
coCartesian fibration classified by $\operatorname{radj}(D)$, see [Lur,
7.4.1.9]. Denote by $\mathcal{L}$ the $\infty$-category of all sections of
$\pi$ and by $\kappa\colon\mathcal{C}\hookrightarrow\mathcal{L}$ the fully
faithful inclusion with left adjoint $\zeta$. For $z\in Z$, we denote by
$\tilde{j}_{z}\colon\mathcal{L}\to\mathcal{C}_{z}$ the evaluation functor at
$z$, satisfying that $\tilde{j}_{z}\circ\kappa\simeq j_{z}$. The left adjoint
of $\tilde{j}_{z}$ is denoted $\tilde{i}_{z}$, it satisfies
$\zeta\circ\tilde{i}_{z}\simeq i_{z}$. An object $c\in\mathcal{C}$
corresponding to a coCartesian section of $\pi$ is of the form

${c_{1}}$${c_{1^{\prime}}}$${c_{0}}$

with $c_{i}\in\mathcal{C}_{i}$. By [Lur09a, 4.3.2.17], $\tilde{i}_{z}$ is
given by the $\pi$-relative left Kan extension functor and the objects
$\tilde{i}_{z}j_{z}(c)$, $z\in Z$ are hence given as follows.

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