# Cluster theory of topological Fukaya categories

Merlin Christ

###### Abstract

We study a class of generalized cluster categories arising from relative
Ginzburg algebras of triangulated marked surfaces without punctures. We show
that these categories describe $1$-periodic versions of the topological Fukaya
categories of these surfaces. These cluster categories hence arise as the
global sections of perverse schobers, meaning certain constructible sheaves of
stable $\infty$-categories.

We exhibit such a cluster category as an additive categorification of a
cluster algebra with coefficients associated to the marked surface. For this,
we make use of an exact $\infty$-structure arising from a relative right
$2$-Calabi-Yau structures in the sense of Brav-Dyckerhoff. We classify the
cluster-tilting objects in this exact cluster category and furthermore
describe a cluster character to the upper cluster algebra by passing through
the commutative Skein algebra of the surface.

###### Contents

  1. 1 Introduction
    1. 1.1 Background on cluster categories
    2. 1.2 1-periodic topological Fukaya categories
    3. 1.3 The generalized cluster categories of marked surfaces and perverse schobers
    4. 1.4 Additive categorification of cluster algebras with coefficients
    5. 1.5 Future work and open problems
    6. 1.6 Acknowledgements
    7. 1.7 Notations and conventions
  2. 2 Preliminaries
    1. 2.1 $\infty$-categories of $\infty$-categories
    2. 2.2 Semiorthogonal decompositions and Verdier quotients
    3. 2.3 Monadicity and derived categories of graded Laurent algebras
    4. 2.4 Relative Calabi-Yau structures
  3. 3 Perverse schobers on surfaces
    1. 3.1 Marked surfaces and ribbon graphs
    2. 3.2 Perverse schobers on marked surfaces
    3. 3.3 Semiorthogonal decompositions of perverse schobers
  4. 4 Generalized cluster categories of surfaces and perverse schobers
    1. 4.1 A semiorthogonal decomposition of perverse schobers
    2. 4.2 The generalized cluster category of a triangulated surface
    3. 4.3 Relation to the 1-periodic topological Fukaya category
  5. 5 Exact structures from relative Calabi-Yau structures
    1. 5.1 Exact $\infty$-categories
    2. 5.2 Extriangulated categories
    3. 5.3 Exact structures via relative Calabi-Yau structures
  6. 6 The geometric model
    1. 6.1 Matching curves
    2. 6.2 Objects and Homs
    3. 6.3 Skein relations from mapping cones
    4. 6.4 Review of the gluing construction of global sections
    5. 6.5 The geometrization Theorem
  7. 7 Categorification of cluster algebras with coefficients
    1. 7.1 Cluster algebras of marked surfaces
    2. 7.2 Commutative Skein algebras
    3. 7.3 Classification of cluster-tilting objects
    4. 7.4 A cluster character on $\mathcal{C}_{\bf S}$
  8. 8 Further directions
    1. 8.1 The stable $\infty$-category of $\mathcal{C}_{\bf S}$ and the ’usual’ $2$-CY cluster category
    2. 8.2 Generalized $n$-cluster categories of marked surfaces
    3. 8.3 Generalized cluster categories of arbitrary ice quivers

##  1 Introduction

Cluster algebras are a class of combinatorially defined commutative algebras
introduced around the year 2000 by Fomin-Zelevinski [FZ02]. The main feature
of cluster algebras is that they come with a notion of mutation, which relates
certain subsets of the cluster algebras called clusters. It was soon observed
that cluster algebras admit a very rich theory of categorification and this
has lead to many fruitful interactions with the representation theory of
algebras. A particular class of (additive) categorifications of cluster
algebras are formed by so called cluster categories, which are triangulated
$2$-Calabi-Yau categories. Cluster categories of acyclic quivers were
introduced in [BMR+06] as certain orbit categories. A generalization to
arbitrary quivers with potential was achieved by Amiot [Ami09]. One starts
with a quiver with potential $(Q,W)$ and considers the perfect derived
category of the $3$-Calabi-Yau Ginzburg dg-algebra $\mathscr{G}(Q,W)$. Amiot’s
generalized cluster category $\mathcal{C}_{Q,W}$ assigned to
$\mathscr{G}(Q,W)$ is then defined as the Verdier quotient

$\mathcal{C}_{Q,W}\coloneqq\mathcal{D}^{\operatorname{perf}}(\mathscr{G}(Q,W))/\mathcal{D}^{\operatorname{fin}}(\mathscr{G}(Q,W))$

of the perfect derived category by the finite derived category. Recently,
generalizations of Ginzburg algebras have emerged [Wu21b, Chr22b], called
relative Ginzburg algebras, which can also be seen as cases of Yeung’s
deformed relative Calabi-Yau completion [Yeu16]. An interesting class of
relative Ginzburg algebras, studied in some detail by the author in [Chr22b,
Chr21], arises from marked surfaces equipped with an ideal triangulation. In
this paper, we suppose that the marked surfaces has no marked points in the
interior, i.e. no punctures. We denote by $\mathscr{G}_{\mathcal{T}}$ the
relative Ginzburg algebra associated with a marked surface ${\bf S}$ with an
ideal triangulation $\mathcal{T}$. We show that Amiot’s formula for the
generalized cluster category

$\mathcal{C}_{\bf
S}\coloneqq\mathcal{D}^{\operatorname{perf}}(\mathscr{G}_{\mathcal{T}})/\mathcal{D}^{\operatorname{fin}}(\mathscr{G}_{\mathcal{T}})$
(1)

applied to the relative Ginzburg algebra $\mathscr{G}_{\mathcal{T}}$ yields an
additive categorification of the cluster algebra of the marked surface with
coefficients in the boundary arcs. This class of cluster algebras was studied
in detail in Fomin-Thurston’s beautiful paper [FT18]. Fomin-Thurston realize
the cluster variables as certain coordinates on the decorated Teichmüller
space of the surface, called lambda lenghts.

To obtain a categorification of a cluster algebra with coefficients, we equip
the generalized cluster category $\mathcal{C}_{\bf S}$, considered as a stable
$\infty$-category, with the structure of a Frobenius exact $\infty$-category.
This structure gives rise to a Frobenius $2$-Calabi-Yau extriangulated
structure on the homotopy $1$-category $\operatorname{ho}\mathcal{C}_{\bf S}$.
This exact $\infty$-structure is obtained from pulling back a split-exact
structure along a functor carrying a right $2$-Calabi-Yau structure in the
sense of Brav-Dyckerhoff [BD19]. The relevance of this exact structure is to
encode a certain maximal subfunctor
$\operatorname{Ext}^{1,\operatorname{CY}}_{\mathcal{C}_{\bf S}}(\mathchar
45\relax,\mathchar 45\relax)\subset\operatorname{Ext}^{1}_{\mathcal{C}_{\bf
S}}(\mathchar 45\relax,\mathchar 45\relax)$ equipped with a $2$-Calabi-Yau
duality

$\operatorname{Ext}^{1,\operatorname{CY}}_{\mathcal{C}_{\bf
S}}(X,Y)\simeq\operatorname{Ext}^{1,\operatorname{CY}}_{\mathcal{C}_{\bf
S}}(Y,X)^{*}\,.$

We give a bijective correspondence between the clusters of the cluster algebra
and the cluster-tilting objects in the exact $\infty$-category $\mathcal{C}$
or equivalently in the extriangulated category
$\operatorname{ho}\mathcal{C}_{\bf S}$. We further describe a cluster
character to the commutative Skein algebra of the surface, which embeds into
the upper cluster algebra of the surface by work of Muller [Mul16].

A further main achievement of this paper is the demonstration that
$\mathcal{C}_{\bf S}$ can be understood both conceptually and computationally
in terms of perverse schobers, meaning categorified perverse sheaves in the
sense of [KS14]. To describe the perverse schobers, we use the framework
described in [Chr22b], which realizes perverse schobers as certain
constructible sheaves of stable $\infty$-categories living on ribbon graphs.
The derived category $\mathcal{D}(\mathscr{G}_{\mathcal{T}})$ of the relative
Ginzburg algebra arises as the global sections of such a perverse schober
$\mathcal{F}_{\mathcal{T}}$, meaning as a certain $\infty$-categorical
(co)limit. We show that the passage to Amiot’s Verdier quotient (1) can be
realized on the level of perverse schobers in terms of the passage to a
quotient perverse schober $\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ of
$\mathcal{F}_{\mathcal{T}}$. By this, we mean that the generalized cluster
category $\mathcal{C}_{\bf S}$ arises as the $\infty$-category of global
sections of $\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$. From this, we
will deduce the remarkable observation that the $\infty$-category
$\mathcal{C}_{\bf S}$ describes the topological Fukaya category of the surface
with coefficients in the derived category of $1$-periodic chain complexes. By
the relation between topological Fukaya categories and the derived categories
of gentle algebras, the generalized cluster category $\mathcal{C}_{\bf S}$ can
hence also be seen as a $1$-periodic version of the derived category of a
gentle algebra. We thereby uncover a precise relation between cluster
categories and topological Fukaya categories/derived categories of gentle
algebras.

The perverse schober description allows us to study $\mathcal{C}_{\bf S}$
using powerful and flexible local-to-global arguments. Results include a
classification of all indecomposable objects in $\mathcal{C}_{\bf S}$ in terms
of suitable curves in ${\bf S}$ and a description of the Hom’s in terms of
intersections of the curves. This classification can be considered as a
$1$-periodic version of the classification of indecomposable objects in the
derived category of a gentle algebra, compare to [HKK17, OPS18].

In Section 1.1, we provide background on the standard approach of additive
categorification of cluster algebras (without coefficients) in terms of
cluster categories. This section may be skipped by experts on cluster
categorification. In Section 1.2, we introduce the $1$-periodic topological
Fukaya categories and their sheaf description. In Section 1.3, we explain how
the passage to the generalized cluster category $\mathcal{C}_{\bf S}$ can be
understood in the framework of perverse schobers. We proceed in Section 1.4 by
describing how the generalized cluster category categorifies a cluster algebra
with coefficients when considered as an exact $\infty$-category. In the final
Section 1.5, we discuss future work and open problems.

###  1.1 Background on cluster categories

In the following, we survey the approach to the additive categorification of
cluster algebras without coefficients in terms of triangulated cluster
categories equipped with cluster-tilting objects and cluster characters.

We begin by recalling Amiot’s construction of the generalized cluster category
[Ami09]. One takes as input a smooth dg-algebra $B$ which is concentrated in
positive degrees (in the homological grading convention) and such that the
Jacobian algebra $\operatorname{H}_{0}(B)$ is finite dimensional. Further, we
ask that $B$ is bimodule $3$-Calabi-Yau, also called weak left $3$-Calabi-Yau,
meaning that the inverse dualizing bimodule $B^{!}$ is equivalent to $B[3]$. A
typical choice for $B$ would be the Ginzburg algebra of a Jacobi-finite quiver
with potential. The smoothness of $B$ implies that the derived category of
finite dimensional modules $\mathcal{D}^{\operatorname{fin}}(B)$ is contained
in the perfect derived category $\mathcal{D}^{\operatorname{perf}}(B)$. We may
thus form the Verdier quotient

$\mathcal{C}_{B}=\mathcal{D}^{\operatorname{perf}}(B)/\mathcal{D}^{\operatorname{fin}}(B)$
(2)

and obtain a triangulated category. This category is called the generalized
cluster category of $B$. The first remarkable property of $\mathcal{C}_{B}$ is
that it is triangulated $2$-Calabi-Yau, meaning that, for all
$X,Y\in\mathcal{C}_{B}$, there exists an isomorphism

$\operatorname{Ext}^{i}_{\mathcal{C}_{B}}(X,Y)\simeq\operatorname{Ext}^{2-i}_{\mathcal{C}_{B}}(Y,X)^{\ast}\,.$

The second remarkable property is that the image $\pi(B)$ of $B$ in
$\mathcal{C}_{B}$ is a cluster-tilting object, meaning that

  * •

$\pi(B)$ is rigid, i.e. $\operatorname{Ext}^{1}(\pi(B),\pi(B))\simeq 0$, and

  * •

$\operatorname{Ext}^{1}(\pi(B),X)\simeq 0$ implies that $X$ is a finite direct
sum of direct summands of $\pi(B)$, for any $X\in\mathcal{C}_{B}$.

Cluster-tilting objects in triangulated categories admit a well behaved theory
of mutations, see [IY08], which categorifies the mutations of clusters in the
cluster algebra. For an additive categorification of a cluster algebra, one
typically asks that the cluster-tilting objects in $\mathcal{C}_{B}$ are in
bijection with the clusters of the cluster algebra.

For a detailed description of $\mathcal{C}_{B}$ when $B$ arises from an
acyclic quiver, see for example [CK08], or when $B$ arises as the (non-
relative) Ginzburg algebra of a triangulated marked surfaces, see [QZ17] and
references therein.

The degree $0$ homology of the endomorphism dg-algebra of $\pi(B)$ is
equivalent to the Jacobian algebra $\operatorname{H}_{0}(B)$. In the cases
where $B$ is the Ginzburg algebra of a triangulated marked surface, the
corresponding Jacobian algebra is a gentle algebra.

Given a $2$-Calabi-Yau triangulated category $\mathcal{C}$ with a cluster-
tilting object, there is an associated cluster character, see [Pal08]. A
cluster character on $\mathcal{C}$ with values in a commutative ring $A$ is a
function $\chi:\operatorname{ob}(\mathcal{C})\rightarrow A$, on the set of
isomorphism classes of objects in $\mathcal{C}$. A typical choice of $A$ would
be a cluster algebra. The cluster character is a kind of exponential map, in
the sense that $\chi(X\oplus Y)\simeq X\cdot Y$ for any two
$X,Y\in\mathcal{C}$. We note that this is the main difference between an
additive categorification and another approach called monoidal
categorification of cluster algebras; in the latter the multiplication of the
cluster algebra is realized in terms of a symmetric monoidal product (instead
of the direct sum). To connect to the cluster algebra, the cluster character
has to satisfy a further relation, called the cluster multiplication formula.
It states that if
$\operatorname{dim}_{k}\operatorname{Ext}^{1}(Y,X)=\operatorname{dim}_{k}\operatorname{Ext}^{1}(X,Y)=1$,
and $X\rightarrow B\rightarrow Y$ and $Y\rightarrow B^{\prime}\rightarrow X$
are the corresponding non-split extensions, then

$\chi(X\oplus Y)\simeq\chi(B)+\chi(B^{\prime})\,.$

This allows to recover the cluster exchange relations in the cluster algebra
in terms of the mutation of cluster-tilting objects in $\mathcal{C}$ and the
cluster character.

To summarize, an additive categorification of a cluster algebra $A$ consists
of a $2$-Calabi-Yau triangulated category, whose cluster-tilting objects are
in bijection with the clusters of $A$ and which is further equipped with a
cluster character to $A$.

###  1.2 1-periodic topological Fukaya categories

We fix an oriented compact surface ${\bf S}$ with non-empty boundary and a set
$M\subset\partial{\bf S}$ of marked points on the boundary, such that each
boundary component contains at least one marked point. The simplest examples
of such surfaces are the $n$-gons. We call ${\bf S}$ a marked surface. We
further choose an ideal triangulation of ${\bf S}$. As will be explained in
Section 1.3, the generalized cluster category associated to the relative
Ginzburg algebra arising from the ideal triangulation turns out to be
equivalent to the $1$-periodic topological Fukaya category of ${\bf S}$. In
this section, we describe the construction and properties of this topological
Fukaya category.

Let $\mathcal{T}$ be the trivalent ribbon graph dual to the triangulation. A
constructible sheaf of stable $\infty$-categories on the ribbon graph
$\mathcal{T}$ is a gadget which associates to each edge or vertex of
$\mathcal{T}$ a stable $\infty$-category and to each incidence of an edge and
a vertex a functor. This defines a diagram in the $\infty$-category
$\operatorname{St}$ of stable $\infty$-categories and its $\infty$-categorical
limit is called the global sections of the constructible sheaf. One may
equivalently consider the dual notion of a constructible cosheaf, where all
functors go in the opposite direction and the global sections arise as a
colimit. We call an $\infty$-category a topological Fukaya category of ${\bf
S}$ valued in a stable $\infty$-category $\mathcal{D}$ if it is equivalent to
the global sections of a suitable constructible (co)sheaf on $\mathcal{T}$,
which associates

  * •

to each edge of $\mathcal{T}$ the stable $\infty$-category $\mathcal{D}$ and

  * •

to each (by assumption trivalent) vertex of $\mathcal{T}$ the stable
$\infty$-category of $\mathcal{D}$-valued representations of the
$A_{2}$-quiver.

The precise conditions on the constructible (co)sheaf involve local
descriptions in terms of Waldhausen’s $S_{\bullet}$-construction applied to
$\mathcal{D}$, see [DK18]. To summarize, topological Fukaya categories arises
via gluing copies of the $\mathcal{D}$-valued representations of the
$A_{2}$-quiver. Choosing $\mathcal{D}=\mathcal{D}(k)$ the derived
$\infty$-category of a field, one recovers the
$\operatorname{Ind}$-completions of the usual $\mathbb{Z}$-graded topological
Fukaya categories, considered for example in [DK15, HKK17], which are also
equivalent to both the partially wrapped Fukaya categories of the surfaces
with stops at the marked points and the derived categories of graded gentle
algebras.

In this paper, we mainly consider the choice
$\mathcal{D}=\mathcal{D}(k[t_{1}^{\pm}])$, the derived $\infty$-category of
the graded Laurent algebra with generator in degree $|t_{1}|=1$ over an
algebraically closed field $k$. This $\infty$-category can also be described
as the derived $\infty$-category of $1$-periodic chain complexes, meaning
chain complexes which are isomorphic to their shift. Such $1$-periodic
complexes are also known under the name of modules with differentials and go
back at least to [CE56]. We call the topological Fukaya categories arsing from
$\mathcal{D}(k[t_{1}^{\pm}])$ the $1$-periodic topological Fukaya categories.
We warn the reader that contrary to the name, neither the $1$-periodic
topological Fukaya categories, nor $\mathcal{D}(k[t_{1}^{\pm}])$, are
$1$-periodic $\infty$-categories in the sense of admitting a trivialization of
the suspension functor $[1]\simeq\operatorname{id}$. Nevertheless, the
suspension functor acts on objects as the identity. Furthermore, these
$\infty$-categories are $2$-periodic, meaning that
$[2]\simeq\operatorname{id}$. In Section 8.2, we also consider $n$-periodic
topological Fukaya categories, which describe the $(n+1)$-cluster categories
of marked surfaces.

As with the $2$-periodic topological Fukaya category, the $1$-periodic
topological Fukaya category can be uniquely associated to an oriented marked
surface, up to equivalence of $\infty$-categories, see [DK18]. The
$\mathbb{Z}$-graded topological Fukaya category instead requires the input of
a grading (also called framing) of the surface, see [DK15].

The representation type of a $\mathcal{D}(k)$-valued topological Fukaya
category is famously tame, its indecomposable compact object have been
classified in [HKK17] in terms of suitable curves in the underlying surface
(these curves are Lagrangians with respect to the volume $2$-form). The
dimensions of the Hom’s between two objects corresponding to two curves can be
described in terms of counts of intersections of the curves. In representation
theory, these and further results relating the category to the underlying
surface are commonly referred to as a geometric model for the topological
Fukaya category, see [OPS18]. In Section 6, we give a similar geometric model
for the $1$-periodic topological Fukaya category. The construction of objects
from suitable curves, called matching curves, and the description of the Hom’s
was given in the prequel paper [Chr21] for the derived $\infty$-category of a
relative Ginzburg algebra $\mathcal{D}(\mathscr{G}_{\mathcal{T}})$, into which
the $1$-periodic topological Fukaya category embeds fully faithfully (the
embedding does not preserve compact objects). We prove the following
geometrization Theorem.

######  Theorem 1 (Theorem 6.12).

Let ${\bf S}$ be a marked surface and $\mathcal{C}_{\bf S}$ the associated
$1$-periodic topological Fukaya category. Every compact object
$X\in\mathcal{C}_{\bf S}$ splits uniquely into the direct sum of
indecomposable objects associated to matching curves.

Note that the restriction to compact objects in 1 arises from the fact that
$\mathcal{C}_{\bf S}$ is an $\operatorname{Ind}$-complete category, the
’perfect’ version is the full subcategory of compact objects.

A similar classification result of the objects in the generalized cluster
categories of non-relative Ginzburg algebras arising from triangulated
surfaces without punctures appears in [BZ11], the case with punctures was
treated in [AP21, QZ17]. We take an original approach to the proof of Theorem
6.12 in that we employ general $\infty$-categorical techniques on the
description of objects in limits of diagrams of $\infty$-categories in terms
of sections of the Grothendieck construction and use the gluing properties of
the objects arising from matching curves.

In their paper [BD19], Brav and Dyckerhoff propose a notion of a Calabi-Yau
structure associated with a functor between dg-categories, called a relative
Calabi-Yau structure. They further show that these relative Calabi-Yau
structures have nice gluing properties and use this to construct $1$-Calabi-
Yau structures on certain functors with target a $\mathcal{D}(k)$-valued
topological Fukaya category. In the work in progress [Chr22c], the author
extends this construction to topological Fukaya categories valued in an
arbitrary Calabi-Yau $\infty$-category which is linear over a commutative base
dg-algebra (instead of linear over $k$). The derived $\infty$-category
$\mathcal{D}(k[t_{1}^{\pm}])$ of $1$-periodic chain complexes is $2$-periodic
and thus linear over the commutative dg-algebra $k[t_{2}^{\pm}]$ of graded
Laurent polynomials with generator in degree $|t_{2}|=2$. As a
$k[t_{2}^{\pm}]$-linear $\infty$-category, $\mathcal{D}(k[t_{1}^{\pm}])$ is
smooth and proper and admits a (right) $1$-Calabi-Yau structure. Applying the
gluing results yields a relative (right) $2$-Calabi-Yau structure on the
$\mathcal{D}(k[t_{1}^{\pm}])$-valued topological Fukaya category, considered
as a $k[t_{2}^{\pm}]$-linear smooth and proper $\infty$-category. More
precisely, the (right) $2$-Calabi-Yau structure is defined on the functor

$G=\prod_{e\in\mathcal{T}_{1}^{\partial}}\operatorname{ev}_{e}:\mathcal{C}_{\bf
S}\longrightarrow\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{D}(k[t_{1}^{\pm}])\,,$
(3)

which takes a global section and evaluates it at all external edges of the
ribbon graph $\mathcal{T}$ (the set of whish is denoted by
$\mathcal{T}_{1}^{\partial}$). In symplectic geometry, this functor is also
known as the cap functor and is adjoint to the Orlov functor or cup functor,
see [Syl19] for background.

Finally, we note that the results of [DK15] show that the $1$-periodic
topological Fukaya category $\mathcal{C}_{\bf S}$ admits an action of the
mapping class group of the surface, see Theorem 4.17. Mapping class group
actions on cluster categories of marked surfaces have not previously been
constructed.

###  1.3 The generalized cluster categories of marked surfaces and perverse
schobers

We again fix a marked surface ${\bf S}$ and choose an ideal triangulation with
dual trivalent ribbon graph $\mathcal{T}$. Associated to the ideal
triangulation is a smooth relative $3$-Calabi-Yau Ginzburg algebra
$\mathscr{G}_{\mathcal{T}}$, see [Chr22b]. It’s derived $\infty$-category
$\mathcal{D}(\mathscr{G}_{\mathcal{T}})$ arises as the global sections of
another constructible sheaf of stable $\infty$-categories on $\mathcal{T}$,
denoted $\mathcal{F}_{\mathcal{T}}$. This sheaf is an example of a perverse
schober, meaning a categorified perverse sheaf in the sense of [KS14]. At each
vertex of the ribbon graph, a perverse schober is described by a spherical
adjunction $\mathcal{V}\leftrightarrow\mathcal{N}$, meaning an adjunction
where the left adjoint (or equivalently the right adjoint) is a spherical
functor in the sense of [AL17]. The constructible sheaves giving rise to
$\mathcal{D}$-valued topological Fukaya categories are also examples of
perverse schobers, where the spherical adjunctions at all vertices are
trivial, i.e. given by $0\leftrightarrow\mathcal{D}$. We refer to [Chr22b] for
more background on perverse schobers and how to describe them as constructible
sheaves on ribbon graphs. The spherical adjunction describing the perverse
schober $\mathcal{F}_{\mathcal{T}}$ at any vertex of $\mathcal{T}$ is given by

$\phi^{*}:\mathcal{D}(k)\leftrightarrow\mathcal{D}(k[t_{1}]):\phi_{*}\,,$

where $k[t_{1}]$ denotes the graded polynomial algebra with generator in
degree $1$ and $\phi$ is the morphism of dg-algebras
$k[t_{1}]\xrightarrow{t_{1}\mapsto 0}k$.

Since $\mathscr{G}_{\mathcal{T}}$ is smooth, we may formally apply Amiot’s
quotient construction and define the $\operatorname{Ind}$-complete version of
the generalized cluster category as

$\mathcal{C}_{\bf
S}\coloneqq\mathcal{D}(\mathscr{G}_{\mathcal{T}})/\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(\mathscr{G}_{\mathcal{T}})\simeq\operatorname{Ind}\left(\mathcal{D}^{\operatorname{perf}}(\mathscr{G}_{\mathcal{T}})/\mathcal{D}^{\operatorname{fin}}(\mathscr{G}_{\mathcal{T}})\right)\,.$

We consider the $\operatorname{Ind}$-complete version of the generalized
cluster category because of the superior formal properties of presentable
$\infty$-categories.

The surprising observation which sparked this project is that
$\mathcal{C}_{\bf S}$ and
$\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(\mathscr{G}_{\mathcal{T}})$
arise as the global sections of two perverse subschobers
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ and
$\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}$ of
$\mathcal{F}_{\mathcal{T}}$ and which fit into a cofiber sequence of perverse
schobers:

${\mathcal{F}_{\mathcal{T}}^{\operatorname{Ind-
fin}}}$${\mathcal{F}_{\mathcal{T}}}$${0}$${\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}}$${\ulcorner}$
(4)

At the vertices of $\mathcal{T}$, the perverse schober
$\mathcal{F}_{\mathcal{T}}^{\operatorname{Ind-fin}}$ is described by the
spherical adjunction

$\mathcal{D}(k)\longleftrightarrow\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{1}])$
(5)

arising from restricting $\phi^{*}\dashv\phi_{*}$ and
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ is described by the
’quotient’ spherical adjunction

$0\longleftrightarrow\mathcal{D}(k[t_{1}])/\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{1}])\simeq\mathcal{D}(k[t_{1}^{\pm}])\,,$

which is the trivial spherical adjunction from above with
$\mathcal{D}=\mathcal{D}(k[t_{1}^{\pm}])$. The derived $\infty$-category of
$1$-periodic chain complexes
$\mathcal{D}(k[t_{1}^{\pm}])\simeq\mathcal{D}(k[t_{1}])/\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{1}])$
arises here naturally as the quotient of the derived category of the graded
polynomial algebra $k[t_{1}]$ by its $\operatorname{Ind}$-complete finite
derived category and may also be seen as the ’$1$-Calabi-Yau cluster category’
of the $A_{1}$-quiver. In this way, we arrive at the observation that the
generalized cluster category $\mathcal{C}_{\bf S}$ associated to the relative
Ginzburg algebra $\mathscr{G}_{\mathcal{T}}$ is the $1$-periodic topological
Fukaya category.

We formulate this relation in terms of a notion of semiorthogonal
decomposition of the perverse schober $\mathcal{F}_{\mathcal{T}}$ into
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ and
$\mathcal{F}_{\mathcal{T}}^{\operatorname{Ind-fin}}$. We will see that the
spherical adjunction (5) is the monadic adjunction associated to the
adjunction monad $\phi_{*}\phi^{*}$ of the adjunction
$\phi^{*}\dashv\phi^{*}$. We expect the semiorthogonal decomposition of
$\mathcal{F}_{\mathcal{T}}$ to be a special case of a general principle, by
which, under some assumptions such as commutating monodromy equivalences, a
perverse schober decomposes into subschobers with monadic or trivial spherical
adjunctions. The passage from $\mathcal{D}(\mathscr{G}_{\mathcal{T}})$ to the
generalized cluster category $\mathcal{C}_{\bf S}$ can hence be motivated
intrinsically in the framework of perverse schobers.

###  1.4 Additive categorification of cluster algebras with coefficients

Let again ${\bf S}$ be a marked surface and choose an auxiliary ideal
triangulation with dual trivalent ribbon graph $\mathcal{T}$. As explained in
Section 1.3, the generalized cluster category $\mathcal{C}_{\bf S}$ arising
from the relative Ginzburg algebra $\mathscr{G}_{\mathcal{T}}$ is the
$1$-periodic topological Fukaya category of the marked surface ${\bf S}$. In
this section, we explain how we equip the stable $\infty$-category
$\mathcal{C}_{\bf S}$ with the structure of an exact $\infty$-category and how
this exact $\infty$-category is an additive categorification of a cluster
algebra with coefficients.

Recall from Section 1.1, that we ask any cluster category to be triangulated
$2$-Calabi-Yau, meaning that is satisfies the $2$-Calabi-Yau duality
$\operatorname{Ext}^{i}(X,Y)\simeq\operatorname{Ext}^{2-i}(Y,X)^{\ast}$. The
$\infty$-category $\mathcal{C}_{\bf S}$ does not directly satisfy this,
instead the functor $G:\mathcal{C}_{\bf
S}\rightarrow\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{D}(k[t_{1}^{\pm}])$
from (3) has a (right) $2$-Calabi-Yau structure. As it turns out, one can use
this relative Calabi-Yau structure to identify a set of morphisms in
$\mathcal{C}_{\bf S}$ which satisfy the $2$-Calabi-Yau duality.

######  Definition 1.1 (Definition 5.18 and Lemma 5.19).

Let

$\operatorname{Mor}_{\mathcal{C}_{\bf S}}^{\operatorname{CY}}(\mathchar
45\relax,\mathchar 45\relax)\subset\operatorname{Mor}_{\mathcal{C}_{\bf
S}}(\mathchar 45\relax,\mathchar 45\relax)$

be the maximal subfunctor of the derived Hom-functor satisfying that

$\operatorname{Mor}_{\mathcal{C}_{\bf S}}^{\operatorname{CY}}(G(\mathchar
45\relax),G(\mathchar 45\relax))\simeq 0\,.$

For $X,Y\in\mathcal{C}_{\bf S}$, we call the $k$-vector space
$\operatorname{Ext}^{i,\operatorname{CY}}(X,Y)\coloneqq\operatorname{H}_{0}\operatorname{Mor}_{\mathcal{C}_{\bf
S}}^{\operatorname{CY}}(X,Y[i])$ the Calabi-Yau extensions. Calabi-Yau
extensions form a subfunctor of the $\operatorname{Ext}$-functor.

######  Proposition 1.2 (Proposition 5.20).

The Calabi-Yau extensions satisfy the $2$-Calabi-Yau duality

$\operatorname{Ext}^{i,\operatorname{CY}}_{\mathcal{C}_{\bf
S}}(X,Y)\simeq\operatorname{Ext}^{2-i,\operatorname{CY}}_{\mathcal{C}_{\bf
S}}(Y,X)^{*}\,,$

bifunctorially in $X$ and $Y$.

In terms of the geometric model for $\mathcal{C}_{\bf S}$ based on curves in
${\bf S}$, Calabi-Yau extensions have a very simple interpretation. Given two
objects arising from curves in ${\bf S}$, a basis of the extensions arises
from crossings and directed boundary intersections. The extensions arising
from crossings are Calabi-Yau extensions, whereas the extensions arising from
directed boundary intersections are not Calabi-Yau.

For the cluster categorification, we would like $\mathcal{C}_{\bf S}$ to have
cluster-tilting objects, which are in particular rigid, i.e. satisfy
$\operatorname{Ext}^{1}_{\mathcal{C}_{\bf S}}(X,X)\simeq 0$. Because of the
$1$-periodicity, we however have $\operatorname{Ext}^{1}_{\mathcal{C}_{\bf
S}}(X,X)\simeq\operatorname{Ext}^{0}_{\mathcal{C}_{\bf S}}(X,X)$ for any
$X\in\mathcal{C}_{\bf S}$. Thus, any rigid object in $\mathcal{C}_{\bf S}$ is
automatically zero. Instead, we should consider rigid objects with respect to
the Calabi-Yau extensions subfunctor, meaning that
$\operatorname{Ext}^{1,\operatorname{CY}}_{\mathcal{C}_{\bf S}}(X,X)\simeq 0$.
Similarly, we consider cluster-tilting objects, where in the definition the
$\operatorname{Ext}^{1}$’s are replaced by
$\operatorname{Ext}^{1,\operatorname{CY}}$’s.

This version of rigid or cluster-tilting can be naturally interpreted as being
rigid or cluster-tilting with respect to an exact $\infty$-structure on
$\mathcal{C}_{\bf S}$, or equivalently as being rigid or cluster-tilting with
respect to an induced extriangulated structure on the homotopy $1$-category.
The exact $\infty$-structure arises from the Calabi-Yau functor
$G:\mathcal{C}_{\bf
S}\rightarrow\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{D}(k[t_{1}^{\pm}])$
as follows. We consider
$\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{D}(k[t_{1}^{\pm}])$ as an
exact $\infty$-category with the split-exact structure, meaning that the exact
sequences are the split sequences. We can now pull back the exact structure
along $G$, to an exact structure on $\mathcal{C}_{\bf S}$, see Example 5.9.
The exact sequences in $\mathcal{C}_{\bf S}$ are by definition those which are
mapped under $G$ to split-exact sequences in
$\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{D}(k[t_{1}^{\pm}])$. One find
the isomorphism classes of such exact sequences $X\rightarrow Z\rightarrow Y$
to be described by the Calabi-Yau extensions
$\operatorname{Ext}^{1,\operatorname{CY}}_{\mathcal{C}_{\bf S}}(X,Y)$. Pulling
exact structures is a general source of exact structures. In the setting of
extriangulated categories, this is known as ’relative theory’, see [HLN21,
Section 3.2]. Proposition 1.2 implies a $2$-Calabi-Yau property of the exact
$\infty$-category $\mathcal{C}_{\bf S}$.

If the functor along which the split-exact structure is pulled back is a
spherical functor, the exact $\infty$-category is furthermore Frobenius,
meaning that it has enough injective and enough projective objects and that
the two classes of these objects coincide, see Proposition 5.10. In the
setting of triangulated categories, this was also observed in [BS21]. The
functor $G$ is indeed spherical, see Corollary 4.16. The injective and
projective indecomposable objects in a Frobenius exact category appear as a
direct summand of each cluster-tilting object, these summands thus serve to
model the frozen cluster variables. Because of this, Frobenius categories are
considered the natural framework for the categorification of cluster algebra
with coefficient, see also [FK10] for general background on this.

The following Theorem is the first part of the categorification of a cluster
algebra in terms of $\mathcal{C}_{\bf S}$.

######  Theorem 1.3 (Corollary 7.15).

There are canonical bijections between the sets of the following objects.

  * •

Clusters of the cluster algebra with coefficients in the boundary arcs
associated to ${\bf S}$.

  * •

Ideal triangulations of ${\bf S}$.

  * •

Cluster-tilting objects in the exact $\infty$-category $\mathcal{C}_{\bf S}$.

In Section 7.3, we also describe the endomorphism algebras of the cluster-
tilting objects. They are given by finite-dimensional non-smooth gentle
algebras.

The notion of cluster character naturally extends to exact $\infty$-categories
or extriangulated categories. The second part of the additive categorification
of a cluster algebra in terms of $\mathcal{C}_{\bf S}$ is the description of a
cluster character on $\mathcal{C}_{\bf S}$ with values in a the commutative
Kauffman Skein algebra $\operatorname{Sk}^{1}_{\bf S}$ of the surface ${\bf
S}$ (with parameter $q=1$). The elements of this algebra are links in ${\bf
S}$, meaning superpositions of curves, modulo relations such as the Kauffman
Skein relation, see Definition 7.10. This algebra embeds into the upper
cluster algebra of the ${\bf S}$ with coefficients in the boundary arcs, see
[Mul16].

######  Theorem 1.4 (Theorem 7.22).

There is a cluster character $\chi:\operatorname{obj}(\mathcal{C}_{\bf
S}^{\operatorname{c}})\rightarrow\operatorname{Sk}^{1}_{\bf S}$ with values in
the commutative Skein algebra of links in ${\bf S}$. The character maps an
object corresponding to a matching curve in ${\bf S}$ to the matching curve
considered as a link with a single component. Furthermore, composing $\chi$
with the inclusion of $\operatorname{Sk}^{1}_{\bf S}$ into the upper cluster
algebra yields a cluster character to the upper cluster algebra.

As shown in [JKPW22], the stable category of the Frobenius exact
$\infty$-category $\mathcal{C}_{\bf S}$, obtained by a localization removing
the injective projective objects, is a stable $\infty$-category, see also
Proposition 5.5. We conjecture that this stable $\infty$-category is
equivalent to an enhancement of the standard $2$-Calabi-Yau triangulated
cluster category arising from ${\bf S}$, see 8.2. In the case that the (non-
frozen) quiver arising from the triangulation is acyclic, we apply a result of
Keller-Reiten [KR08] to show that the two triangulated homotopy categories are
equivalent, see Theorem 8.3.

The results of this paper might suggest that Amiot’s construction may be
applied to classes of smooth relative $3$-Calabi-Yau dg-categories to obtain
categorifications of cluster algebras with coefficients. We discuss the
limitations of this approach in Section 8.3.

Finally, we wish to mention Wu’s recent approach to the categorification of
cluster algebras with coefficients called relative cluster categories and
Higgs categories, see [Wu21b]. In loc. cit., Amiot’s generalized cluster
category is generalized in a different way than in this paper, however the
outcome is also a Frobenius $2$-Calabi-Yau extriangulated category, called the
Higgs category. The extriangulated structure of the Higgs category arises from
the ambient triangulated category, called the generalized cluster category, in
which the Higgs category lies as an extension closed subcategory.

###  1.5 Future work and open problems

In a sequel to this paper [Chr22a], we extend the results of this paper to
marked surfaces with boundary and punctures. The main difference there is that
the generalized cluster category is not constructed starting with a relative
Ginzburg algebra. This is due to the fact that the relative Ginzburg algebra
of a marked surface with punctures (with the canonical non-degenerate
potential) does not seem to arise in the desired way from a perverse schober.
Instead, the generalized cluster category arises from a perverse schober which
is a modification of the schober $\mathcal{F}_{\mathcal{T}}$ at the punctures.
This modification is already discussed in [Chr21].

This paper opens possibilities for constructing and studying new classes of
cluster categories. We expect that suitable ’higher rank’ topological Fukaya
categories categorify higher rank cluster algebras of surfaces, in the sense
of Fock-Goncharov [FG06]. It is currently an open problem to treat perverse
schobers on closed surfaces. Supposing this is solved, one should also be able
to obtain cluster categories of closed surfaces from perverse schobers.

Finally, we remark that the Fock-Goncharov higher Teichmüller spaces arise via
a monoidal version of ’gluing’ along a triangulation, see for instance
[JLSS21] for a formulation in terms of stratified factorization homology. It
is an intriguing question, whether this gluing can be related to the gluing of
the cluster category $\mathcal{C}_{\bf S}$.

###  1.6 Acknowledgements

I wish to thank my advisor Tobias Dyckerhoff for his support. It is my
pleasure to further thank Gustavo Jasso for suggesting to me the use of exact
$\infty$-categories. I also thank Martin Herschend, Bernhard Keller and Qiu Yu
for stimulating discussions. The author acknowledges support by the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306.

###  1.7 Notations and conventions

In this paper, we freely employ the langue of $\infty$-categories, as
developed in [Lur09, Lur17]. We generally follow the notation and conventions
of [Lur09, Lur17]. In particular, we use the homological grading. 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 further use
the following conventions.

  * •

We denote the homotopy $1$-category of an $\infty$-category $\mathcal{C}$ by
$\operatorname{ho}\mathcal{C}$.

  * •

We denote the right adjoint of a functor $F$ by $\operatorname{radj}(F)$ and
the left adjoint by $\operatorname{ladj}(F)$ (if existent).

  * •

Given a functor $F:\mathcal{A}\rightarrow\mathcal{B}$, we denote by
$F(\mathcal{A})$ the essential image of $F$, meaning the smallest full
subcategory of $\mathcal{B}$ which is closed under equivalences in
$\mathcal{B}$ and contains the objects in the image of $F$.

  * •

Matrices can be used to describe morphisms between direct sums of objects in
additive $\infty$-categories, such as stable $\infty$-categories. For this we
use the convention of multiplication with a matrix from the left.

##  2 Preliminaries

###  2.1 $\infty$-categories of $\infty$-categories

We use the following notation for different $\infty$-categories of
$\infty$-categories. We denote by

  * •

$\mathcal{S}$ the $\infty$-category of spaces.

  * •

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

  * •

$\operatorname{St}$ the $\infty$-category of stable $\infty$-categories and
exact functors.

  * •

$\mathcal{P}r^{L}$ the $\infty$-category of presentable $\infty$-categories
and left adjoint functors.

  * •

$\mathcal{P}r^{R}$ the $\infty$-category of presentable $\infty$-categories
and right adjoint functors.

  * •

by $\mathcal{P}r^{L}_{\operatorname{St}}\subset\mathcal{P}r^{L}$ and
$\mathcal{P}r^{R}_{\operatorname{St}}\subset\mathcal{P}r^{R}$ the full
subcategories consisting of stable $\infty$-categories.

We remark that passing to adjoint functors defines an equivalence of
$\infty$-categories
$\mathcal{P}r^{L}\simeq(\mathcal{P}r^{R})^{\operatorname{op}}$, see [Lur09,
5.5.3.4].

Let $\mathcal{C}$ be an $\infty$-category. An object $x$ in an
$\infty$-category $\mathcal{C}$ is called compact if
$\operatorname{Map}_{\mathcal{C}}(x,\mathchar
45\relax):\mathcal{C}\rightarrow\mathcal{S}$ preserves filtered colimits. We
denote by $\mathcal{C}^{\operatorname{c}}\subset\mathcal{C}$ the full
subcategory of compact objects. We denote by $\operatorname{Ind}(\mathcal{C})$
the $\operatorname{Ind}$-completion of $\mathcal{C}$. If $\mathcal{C}$ is
stable and idempotent complete, then
$\mathcal{C}\simeq\operatorname{Ind}(\mathcal{C})^{\operatorname{c}}$.

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 $\mathchar 45\relax\otimes\mathchar
45\relax:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ preserves
colimits in both entries, see [Lur17, Section 4.7]. An example of such a
commutative algebra object in $\mathcal{P}r^{L}$ is the derived
$\infty$-category $\mathcal{D}(R)$ of a commutative dg-algebra $R$.

######  Definition 2.1.

Let $R$ be a commutative dg-algebra. We call the $\infty$-category
$\operatorname{Mod}_{\mathcal{D}(R)}(\mathcal{P}r^{L})$ of modules, see
[Lur17, Def. 4.5.1.1], the $\infty$-category of $R$-linear presentable
$\infty$-categories and denote it by $\operatorname{Lin}_{R}$.

As noted in [Lur18, Section D.1.5], $R$-linear $\infty$-categories in the
above sense are automatically stable.

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

Let $R$ be a commutative dg-algebra. Let $\mathcal{A}$ be an $R$-linear
$\infty$-category and let $X,Y\in\mathcal{A}$. A morphism object for $X,Y$ is
an $R$-module $\operatorname{Mor}_{\mathcal{C}}(X,Y)\in\mathcal{D}(R)$
equipped with a map
$\alpha:\operatorname{Mor}_{\mathcal{C}}(X,Y)\otimes_{R}X\rightarrow Y$ in
$\mathcal{C}$ such that for every object $C\in\mathcal{D}(R)$ composition with
$\alpha$ induces an equivalence of spaces

$\operatorname{Map}_{\mathcal{D}(R)}(C,\operatorname{Mor}_{\mathcal{C}}(X,Y))\rightarrow\operatorname{Map}_{\mathcal{C}}(C\otimes_{R}X,Y)\,.$

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

######  Remark 2.3.

Morphism objects always exist and the formation of morphism objects forms an
exact functor

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

Given an $R$-linear stable $\infty$-category $\mathcal{C}$ and two objects
$A,B\in\mathcal{C}$, the $n$-th $\operatorname{Ext}$-group is defined as

$\operatorname{Ext}^{n}_{\mathcal{C}}(A,B)\coloneqq\operatorname{H}_{-n}\operatorname{Mor}_{\mathcal{C}}(A,B)\simeq\operatorname{H}_{0}\operatorname{Mor}_{\mathcal{C}}(A,B[n])\,.$

###  2.2 Semiorthogonal decompositions and Verdier quotients

Given a stable $\infty$-category $\mathcal{C}$ and a full subcategory
$\mathcal{A}\subset\mathcal{C}$, we call $\mathcal{A}$ a stable subcategory of
$\mathcal{C}$ if $\mathcal{A}$ is stable, the inclusion is an exact functor
and $\mathcal{A}$ is closed under equivalences in $\mathcal{C}$.

######  Definition 2.4.

Let $\mathcal{C}$ be a stable $\infty$-category and let
$\mathcal{A},\mathcal{B}\subset\mathcal{B}$ be stable subcategories. We say
that the pair $(\mathcal{A},\mathcal{B})$ forms a semiorthogonal decomposition
of $\mathcal{C}$ if

  * •

for all $a\in\mathcal{A}$ and $b\in\mathcal{B}$, the mapping space
$\operatorname{Map}_{\mathcal{C}}(b,a)$ is contractible and

  * •

for all $x\in\mathcal{C}$, there exists a fiber and cofiber sequence
$b\rightarrow x\rightarrow a$ in $\mathcal{C}$ with $a\in\mathcal{A}$ and
$b\in\mathcal{B}$.

######  Definition 2.5.

An exact sequence of stable, presentable $\infty$-categories consists of a
cofiber sequence in $\mathcal{P}r^{L}_{\operatorname{St}}$

${\mathcal{A}}$${\mathcal{C}}$${0}$${\mathcal{B}}$$\scriptstyle{\iota}$${\ulcorner}$

such that $i$ is a fully faithful functor.

######  Remark 2.6.

In the setting of Definition 2.5, the triangulated homotopy category of
$\mathcal{B}$ is equivalent to the triangulated Verdier quotient of
$\mathcal{C}$ by $\mathcal{A}$, see [BGT13, Prop. 5.9]. We thus call
$\mathcal{B}$ the Verdier quotient of $\mathcal{C}$ by $\mathcal{A}$.

The following Lemma shows that the datum of a semiorthogonal decomposition of
a stable, presentable $\infty$-category is equivalent to the datum of an exact
sequence in $\mathcal{P}r^{L}_{\operatorname{St}}$.

######  Lemma 2.7.

Consider a diagram in $\mathcal{P}r^{L}_{\operatorname{St}}$

$\mathcal{A}\xrightarrow{i}\mathcal{C}\xrightarrow{\pi}\mathcal{B}\,.$ (6)

Suppose that $i$ and $\operatorname{radj}(\pi)$ are fully faithful. Then the
diagram can be extended to an exact sequence if and only if
$\left(\operatorname{radj}(\pi)(\mathcal{B}),i(\mathcal{A})\right)$ forms a
semiorthogonal decomposition of $\mathcal{C}$.

###### Proof.

Suppose that (6) is part of an exact sequence. We find that for all
$a\in\mathcal{A}$ and $b\in\mathcal{B}$, the mapping space
$\operatorname{Map}_{\mathcal{C}}(\iota(a),\operatorname{radj}(\pi)(b))\simeq\operatorname{Map}_{\mathcal{B}}(\pi\iota(a),b)\simeq\operatorname{Map}_{\mathcal{B}}(0,b)$
is contractible. Let $c\in\mathcal{C}$ and
$\operatorname{cu}_{c}:c\rightarrow\operatorname{radj}(\pi)(\pi(c))$ be the
counit. The map $\pi(\operatorname{cu}_{c})$ is an equivalence by the
2/3-property and the fact that $\operatorname{radj}(\pi)$ is fully faithful.
We hence have $\pi(\operatorname{fib}(\operatorname{cu}_{c}))\simeq 0$. It
follows that
$\operatorname{fib}(\operatorname{cu}_{c})\in\operatorname{Im}(i)$. There thus
exists an object $a\in\mathcal{A}$ and an exact sequence $a\rightarrow
c\rightarrow\operatorname{radj}(\pi)\circ\pi(c)$ in $\mathcal{C}$. We have
shown that $\left(i(\mathcal{A}),\operatorname{radj}(\pi)(\mathcal{B})\right)$
forms a semiorthogonal decomposition of $\mathcal{C}$.

For the converse implication, assume that
$(\operatorname{radj}(\pi)(\mathcal{B}),i(\mathcal{A}))$ forms a
semiorthogonal decomposition of $\mathcal{C}$. The assertion that (6) can be
extended to an exact sequence is via the equivalence
$\operatorname{radj}:\mathcal{P}r^{L}_{\operatorname{St}}\simeq(\mathcal{P}r^{R}_{\operatorname{St}})^{\operatorname{op}}$
equivalent to the assertion that there exists a pullback diagram in
$\operatorname{Cat}_{\infty}$ as follows:

${\mathcal{B}}$${\mathcal{C}}$${0}$${\mathcal{A}}$$\scriptstyle{\operatorname{radj}(\pi)}$${\lrcorner}$$\scriptstyle{\operatorname{radj}(i)}$

This is in turn equivalent to the assertion that $\operatorname{radj}(\pi)$
defines an equivalence between $\mathcal{B}$ and the full subcategory of
$\mathcal{C}$ spanned by objects $c\in\mathcal{C}$, satisfying that
$\operatorname{Map}_{\mathcal{C}}(i(a),c)$ is contractible for all
$a\in\mathcal{A}$. Proposition 2.2.4 from [DKSS21] shows that this is a
property of the semiorthogonal decomposition, concluding the proof of the
converse implication. ∎

###  2.3 Monadicity and derived categories of graded Laurent algebras

Let $\mathcal{C}$ be an $\infty$-category. A monad $M$ on $\mathcal{C}$ is an
associative algebra object in the monoidal $\infty$-category
$\operatorname{End}(\mathcal{C})=\operatorname{Fun}(\mathcal{C},\mathcal{C})$
of endofunctors. In other words, a monad is a functor
$M:\mathcal{C}\rightarrow\mathcal{C}$, together with a unit
$u:\operatorname{id}_{\mathcal{C}}\rightarrow M$ and a multiplication map
$M\circ M\rightarrow M$ equipped with data exhibiting coherent associativity
and unitality. A main source of monads are adjunctions: given an adjunction
$F:\mathcal{C}\leftrightarrow\mathcal{D}:G$ of $\infty$-categories, there is
an associated monad $M=GF$, see [Lur17, Section 4.7.3], which we call the
adjunction monad. Associated to a monad $M$ on $\mathcal{C}$ is its
$\infty$-category $\operatorname{LMod}_{M}(\mathcal{C})$ of left modules in
$\mathcal{C}$ and the free-forget adjunction

$\operatorname{Free}:\mathcal{C}\longleftrightarrow\operatorname{LMod}_{M}(\mathcal{C}):\operatorname{Forget}\,,$

whose adjunction monad is equivalent to $M$. The $\infty$-category
$\operatorname{LMod}_{M}(\mathcal{C})$ is also called the Eilenberg-Moore
$\infty$-category of $M$. The definition of
$\operatorname{LMod}_{M}(\mathcal{C})$ and this adjunction is rather involved,
based on a very general machinery of algebra objects and module categories
developed in [Lur17]. Instead of recalling its exact definition, it may be
more instructive to present a typical example. If $\mathcal{C}=\mathcal{D}(k)$
is the derived $\infty$-category of a field and $M=A\otimes_{k}\mathchar
45\relax$ is the monad arising from tensoring with a dg-algebra $A$, then
$\operatorname{LMod}_{M}(\mathcal{D}(k))\simeq\mathcal{D}(A)$. The associated
adjunction is equivalent to the usual free-forget adjunction
$\mathcal{D}(k)\leftrightarrow\mathcal{D}(A)$.

Given an adjunction $F:\mathcal{C}\leftrightarrow\mathcal{D}:G$ with monad
$M=GF$, there is an associated functor
$\mathcal{D}\rightarrow\operatorname{LMod}_{M}(\mathcal{C})$, making the
following diagram commute:

${\mathcal{D}}$${\operatorname{LMod}_{M}(\mathcal{C})}$${\mathcal{C}}$$\scriptstyle{G}$$\scriptstyle{\operatorname{Forget}}$

######  Definition 2.8.

Let $G:\mathcal{D}\rightarrow\mathcal{C}$ be a functor between
$\infty$-categories.

  1. 1.

The functor $G$ is called monadic if it admits a left adjoint $F$ with
adjunction monad $M=GF$ and the associated functor
$\mathcal{D}\rightarrow\operatorname{LMod}_{M}(\mathcal{C})$ is an equivalence
of $\infty$-categories.

  2. 2.

The functor $G$ is called comonadic if the opposite functor
$G^{\operatorname{op}}:\mathcal{D}^{\operatorname{op}}\rightarrow\mathcal{C}^{\operatorname{op}}$
is monadic.

If $\mathcal{C},\mathcal{D}$ are presentable $\infty$-categories and $G$
preserves (sufficient) colimits, then
$\mathcal{D}\rightarrow\operatorname{LMod}_{M}(\mathcal{C})$ admits a fully
faithful left adjoint, see [Lur17, Lemma 4.7.3.13]. In this case,
$\operatorname{LMod}_{M}(\mathcal{C})$ is presentable and if $\mathcal{C}$ and
$\mathcal{D}$ are furthermore stable, then
$\operatorname{LMod}_{M}(\mathcal{C})$ is also stable, see [Lur17, Prop.
4.2.3.4].

To determine whether a functor is monadic, one can use the
$\infty$-categorical Barr-Beck monadicity theorem, see [Lur17, Thm. 4.7.3.5].
Assuming that all involved $\infty$-categories are presentable and that the
functor preserves all colimits, the theorem reduces to the statement that a
right adjoint $G:\mathcal{D}\rightarrow\mathcal{C}$ is monadic if and only if
it is conservative, i.e. reflects isomorphisms. Similarly, in this setting a
left adjoint $F$ which preserves sufficient limits is comonadic if and only if
it is conservative.

We now turn to a specific example of a monadic adjunction. Let $k$ be a field
and $n\geq 2$. We are interested in the monadic adjunction arising from the
spherical adjunction

$f^{*}:\mathcal{D}(k)\longleftrightarrow\operatorname{Fun}(S^{n},\mathcal{D}(k)):f_{*}\,,$

where $\operatorname{Fun}(S^{n},\mathcal{D}(k))$ is the $\infty$-category of
$\mathcal{D}(k)$-valued local systems on the $n$-sphere $S^{n}$ and $f^{*}$ is
the pullback along $f:S^{n}\rightarrow\ast$, see [Chr22d] for a detailed study
of this adjunction. In the case $n=2$, this adjunction describes the
singularities of the perverse schober $\mathcal{F}_{\mathcal{T}}$ considered
in Section 4.1. Let $M=f_{*}f^{*}$ be the adjunction monad and denote by
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\coloneqq\operatorname{LMod}_{M}(\mathcal{D}(k))$
the stable, presentable $\infty$-category of left modules over $M$. We denote
the associated fully faithful functor
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\rightarrow\operatorname{Fun}(S^{n},\mathcal{D}(k))$
by $i_{\operatorname{mnd}}$ and we will identify
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}$ with its
essential image under $i_{\operatorname{mnd}}$.

######  Lemma 2.9.

The object $f^{*}(k)$ is a compact generator of the $\infty$-category
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}$.

###### Proof.

The restriction of $f_{*}$ to
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}$ is by
definition monadic. Its left adjoint $(f^{*})^{\operatorname{mnd}}$ is
obtained from $f^{*}$ by restricting the target. The $k$-linear functor
$(f^{*})^{\operatorname{mnd}}$ is fully determined by the image of $k$, which
is $f^{*}(k)$. It is thus equivalent to the functor $(\mathchar
45\relax)\otimes f^{*}(k)$ (defined using the $k$-linear structure). The
adjunction

$(\mathchar 45\relax)\otimes
f^{*}(k)\dashv\operatorname{Mor}_{\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}}(f^{*}(k),\mathchar
45\relax)\,,$

the facts that the right adjoint
$\operatorname{Mor}_{\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}}(f^{*}(k),\mathchar
45\relax)\simeq(f_{*})^{\operatorname{mnd}}$ is conservative, because monadic,
and that $k\in\mathcal{D}(k)$ is compact generator, now imply that $f^{*}(k)$
compactly generates
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}$. ∎

As shown in [Chr22b, Prop. 5.5], there exists an equivalence of $k$-linear
$\infty$-categories

$\operatorname{Fun}(S^{n},\mathcal{D}(k))\simeq\mathcal{D}(k[t_{n-1}])\,,$

where $k[t_{n-1}]$ denotes the graded polynomial algebra with generator in
degree $|t_{n-1}|=n-1$.

######  Lemma 2.10.

There exist equivalences of $k$-linear $\infty$-categories

$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\simeq\operatorname{Ind}\operatorname{Fun}(S^{n},\mathcal{D}^{\operatorname{perf}}(k))\simeq\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])\,,$

compatible with their inclusions into
$\operatorname{Fun}(S^{n},\mathcal{D}(k))$, where
$\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$ denotes the derived
$\infty$-category of $k[t_{n-1}]$-modules with finite dimensional homology.

###### Proof.

Under the equivalence
$\operatorname{Fun}(S^{n},\mathcal{D}(k))\simeq\mathcal{D}(k[t_{n-1}])$, the
forgetful functor $\mathcal{D}(k[t_{n-1}])\rightarrow\mathcal{D}(k)$
corresponds to the evaluation functor
$\operatorname{Fun}(S^{n},\mathcal{D}(k))\rightarrow\mathcal{D}(k)$ at any
point $x\in S^{n}$. This implies
$\operatorname{Fun}(S^{n},\mathcal{D}^{\operatorname{perf}}(k))\simeq\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$,
which yields
$\operatorname{Ind}\operatorname{Fun}(S^{n},\mathcal{D}^{\operatorname{perf}}(k))\simeq\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$
by passing to $\operatorname{Ind}$-completions.

We proceed by showing the equivalence
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\simeq\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$.
Consider the trivial $k[t_{n-1}]$-module with homology $k$ (corresponding to
$f^{*}(k)$ under the equivalence
$\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])\simeq\operatorname{Fun}(S^{n},\mathcal{D}(k)^{\operatorname{perf}})$.
Let $X\in\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$. We show by induction
on the dimension of the homology of $X$, that $X$ lies in the stable closure
$\langle k\rangle\subset\mathcal{D}(k[t_{n-1}])$ of $k$, i.e. the smallest
stable subcategory containing $k$. The induction beginning is the case
$X\simeq 0$ and thus clear. For the induction step, suppose that
$m\in\mathbb{Z}$ is the maximal degree in which the homology
$\operatorname{H}_{*}\operatorname{Mor}_{\mathcal{D}(k[t_{n-1}])}(k[t_{n-1}],X)$
of $X$ is nontrivial. We find a corresponding non-zero morphism
$\alpha:k[t_{n-1}][m]\rightarrow X$, such that the composite with
$k[t_{n-1}][m+n-1]\rightarrow k[t_{n-1}][m]$ is zero. The morphism $\alpha$
thus induces a non-zero morphism $\alpha^{\prime}:k\rightarrow X$, which is
injective on homology. Taking the fiber of $\alpha^{\prime}$ we obtain a
module $\operatorname{fib}(\alpha^{\prime})$, whose homology has one dimension
less. We thus have by the induction assumption, that
$\operatorname{fib}(\alpha^{\prime})\in\langle k\rangle$. From
$X\simeq\operatorname{cof}(\operatorname{fib}(\alpha^{\prime})\rightarrow k)$,
it follows that $X$ also lies in $\langle k\rangle$. Since any object in
$\langle k\rangle$ also lies in
$\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$, we obtain $\langle
k\rangle=\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$. This $\infty$-category
is stable and also idempotent-complete, as having finite dimensional homology
is a condition preserved under retracts. Consider the $k$-linear endomorphism
dg-algebras
$\operatorname{End}_{\mathcal{D}(k[t_{n-1}])}(k)\simeq\operatorname{End}_{\operatorname{Fun}(S^{n},\mathcal{D}(k))}(f^{*}(k))\simeq
k\oplus k[-n]$. Since $f^{*}(k)$ is by Lemma 2.9 a compact generator of
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}$, we have
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\simeq\mathcal{D}(\operatorname{End}_{\operatorname{Fun}(S^{n},\mathcal{D}(k))}(f^{*}(k)))\simeq\mathcal{D}(\operatorname{End}_{\mathcal{D}(k[t_{n-1}])}(k))$,
see [Lur17, 7.1.2.1]. The perfect derived $\infty$-category
$\mathcal{D}^{\operatorname{perf}}(\operatorname{End}_{\mathcal{D}(k[t_{n-1}])}(k))$
(consisting of compact objects) is by [Lur17, 7.2.4.1, 7.2.4.4] the smallest
stable subcategory of
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\simeq\mathcal{D}(\operatorname{End}_{\mathcal{D}(k[t_{n-1}])}(k))$
containing $k$ which is closed under retracts and thus equivalent to $\langle
k\rangle=\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$. It follows that

$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}\simeq\mathcal{D}(\operatorname{End}_{\mathcal{D}(k[t_{n-1}])}(k))\simeq\operatorname{Ind}\mathcal{D}^{\operatorname{perf}}(\operatorname{End}_{\mathcal{D}(k[t_{n-1}])}(k))\simeq\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])\,,$

concluding the proof. ∎

We denote by $k[t_{n-1}^{\pm}]$ the dg-algebra of graded Laurent polynomials
with generator in degree $|t_{n-1}|=n-1$. Note that if $i$ is even, then
$k[t_{n-1}]$ and $k[t_{n-1}^{\pm}]$ are graded commutative dg-algebras,
whereas if $i$ is odd, then $k[t_{n-1}]$ and $k[t_{n-1}^{\pm}]$ are not graded
commutative, because $t_{n-1}^{2}\neq 0$. By Lemma 2.7, the following Lemma
shows that $\mathcal{D}(k[t_{n-1}^{\pm}])$ arises as the Verdier quotient of
$\mathcal{D}(k[t_{n-1}])$ by
$\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$, or by Lemma
2.10 equivalently as the Verdier quotient of
$\operatorname{Fun}(S^{n},\mathcal{D}(k))$ by
$\operatorname{Fun}(S^{n},\mathcal{D}(k))^{\operatorname{mnd}}$.

######  Lemma 2.11.

Let $n\geq 1$. The $\infty$-category $\mathcal{D}(k[t_{n-1}])$ admits a
semiorthogonal decomposition
$(\mathcal{D}(k[t_{n-1}^{\pm}]),\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}]))$.

###### Proof.

For the proof, we show that
$\mathcal{D}(k[t_{n-1}^{\pm}])=\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])^{\perp}$
is the stable subcategory of $\mathcal{D}(k[t_{n-1}])$ consisting of objects
$b$, such that $\operatorname{Mor}_{\mathcal{D}(k[t_{n-1}])}(a,b)\simeq 0$ for
all $a\in\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])$.
Using that the inclusion
$\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])\subset\mathcal{D}(k[t_{n-1}])$
admits a right adjoint, the Lemma then follows from [DKSS21, Prop. 2.2.4,
Prop. 2.3.2].

The objects of $\mathcal{D}(k[t_{n-1}])$ can be identified with dg-modules
over the dg-algebra $k[t_{n-1}]$. Under this identification, we observe that
$\mathcal{D}(k[t_{n-1}^{\pm}])\subset\mathcal{D}(k[t_{n-1}])$ is the full
subcategory consisting of $k[t_{n-1}]$-modules $M_{\bullet}$, such that
$t_{n-1}:M_{i}\rightarrow M_{i+n-1}$ is an isomorphism of $k$-vector spaces
for all $i\in\mathbb{Z}$. We have

$\operatorname{Mor}_{\mathcal{D}(k[t_{n-1}])}(k[t_{n-1}][i],M_{\bullet})\simeq
M_{\bullet+i}\,.$

Using that $k\simeq\operatorname{cof}(k[t_{n-1}][n-1]\rightarrow k[t_{n-1}])$,
we thus have

$\operatorname{Mor}_{\mathcal{D}(k[t_{n-1}])}(k[i],M_{\bullet})\simeq\operatorname{cof}(t_{n-1}:M_{\bullet+i}\rightarrow
M_{\bullet+i+n-1})\in\mathcal{D}(k)\,.$

We thus find that $t_{n-1}:M_{i}\rightarrow M_{i+n-1}$ is an isomorphism for
all $i\in\mathbb{Z}$ if and only if
$M_{\bullet}\in\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])^{\perp}$.
This shows the desired equality

$\mathcal{D}(k[t_{n-1}^{\pm}])=\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}(k[t_{n-1}])^{\perp}\,,$

concluding the proof. ∎

######  Lemma 2.12.

There exists an equivalence of $1$-categories
$\operatorname{Vect}_{k}\simeq\operatorname{ho}\mathcal{D}(k[t_{1}^{\pm}])$,
where $\operatorname{Vect}_{k}$ denotes the $1$-category of $k$-vector spaces
and $\operatorname{ho}\mathcal{D}(k[t_{1}^{\pm}])$ denotes the homotopy
category of $\mathcal{D}(k[t_{1}^{\pm}])$.

###### Proof.

Since any complex of $k$-vector spaces is quasi-isomorphic to its homology,
any $1$-periodic complex with values in $k$ is equivalent to an object in the
image of
$N(\operatorname{Vect}_{k})\hookrightarrow\mathcal{D}(k)\xrightarrow{\mathchar
45\relax\otimes k[t_{1}^{\pm}]}\mathcal{D}(k[t_{1}^{\pm}])$. This functor is
thus essentially surjective, and using that
$\pi_{0}\operatorname{Map}_{\mathcal{D}(k[t_{1}^{\pm}])}(k[t_{1}^{\pm}],k[t_{1}^{\pm}])\simeq\operatorname{H}_{0}(k[t_{1}^{\pm}])\simeq
k$, one sees that this functor is also fully faithful on the level of homotopy
categories. ∎

In the triangulated homotopy category
$\operatorname{ho}\mathcal{D}(k[t_{1}^{\pm}])$, distinguished triangles take
the form

$\operatorname{ker}(\alpha)\oplus\operatorname{coker}(\alpha)\xlongrightarrow{(\iota,0)}k[t_{1}^{\pm}]^{\oplus
I}\xlongrightarrow{\alpha}k[t_{1}^{\pm}]^{\oplus
J}\xrightarrow{(0,\pi)}\operatorname{ker}(\alpha)\oplus\operatorname{coker}(\alpha)$

with $\iota$ being the kernel map and $\pi$ being the cokernel map and $I,J$
two sets.

###  2.4 Relative Calabi-Yau structures

Let $k$ be a field and $\kappa=k$ or $\kappa=k[t_{2m}^{\pm}]$ the dg-algebra
of graded Laurent polynomials, see Section 2.3, for some $m\in\mathbb{N}$. In
this section, we sketch the definition of a right $\kappa$-linear Calabi-Yau
structure. The case $\kappa=k$ is described in detail in [BD19]. The case
$\kappa=k[t_{2m}^{\pm}]$ is analogous and will also be treated in more detail
in [Chr22c]. For simplicity, we assume that all $\kappa$-linear
$\infty$-categories in this section are $\kappa$-linear smooth and proper.

######  Definition 2.13.

Let $\mathcal{A}$ be a $\kappa$-linear $\infty$-category. A Serre functor on
$\mathcal{A}$ is a $\kappa$-linear endofunctor
$U:\mathcal{A}\rightarrow\mathcal{A}$, satisfying that there exists a natural
equivalence of functors

$\operatorname{Mor}_{\mathcal{A}}(\mathchar 45\relax_{1},\mathchar
45\relax_{2})\simeq\operatorname{Mor}_{\mathcal{A}}(\mathchar
45\relax_{2},U(\mathchar
45\relax_{1}))^{\ast}:\mathcal{A}^{\operatorname{c},\operatorname{op}}\times\mathcal{A}^{\operatorname{c}}\rightarrow\mathcal{D}(\kappa)\,.$

Here $\mathcal{A}^{\operatorname{c}}$ denotes the full subcategory of
$\mathcal{A}$ of compact objects and $(\mathchar
45\relax)^{*}=\operatorname{Mor}_{\mathcal{D}(\kappa)}(\mathchar
45\relax,\kappa):\mathcal{D}(\kappa)^{\operatorname{op}}\rightarrow\mathcal{D}(\kappa)$.

Every $\kappa$-linear $\infty$-category $\mathcal{A}$ (by assumption smooth
and proper) admits a Serre functor $U$, given by the right bimodule dual
$\operatorname{id}_{\mathcal{A}}^{\ast}$ of $\operatorname{id}_{\mathcal{A}}$.
The Serre functor $U$ is an equivalence, its inverse is the left bimodule dual
$\operatorname{id}_{\mathcal{A}}^{!}$. A weak right $n$-Calabi-Yau structure
on such an $\infty$-category $\mathcal{A}$ consists of a natural equivalence
$U\simeq\operatorname{id}_{\mathcal{A}}[n]$. Such an equivalence arises from a
$\kappa$-linear dual Hochschild homology class

$\sigma\in\operatorname{H}_{n}\left(\operatorname{HH}(\mathcal{A})^{*}\right)\simeq\operatorname{H}_{n}\operatorname{Mor}_{\operatorname{Fun}(\mathcal{A},\mathcal{A})}(\operatorname{id}_{\mathcal{A}},\operatorname{id}_{\mathcal{A}}^{\ast})\,.$

For simplicity, we do not consider cyclic enhancements of weak Calabi-Yau
structures in this paper.

Consider a $\kappa$-linear $\infty$-category $\mathcal{B}$ equipped with a
$\kappa$-linear functor $G:\mathcal{B}\rightarrow\mathcal{A}$ with right
adjoint $H$. The functor $G$ induces a functor
$\operatorname{HH}(G)^{*}:\operatorname{HH}(\mathcal{A})^{*}\rightarrow\operatorname{HH}(B)^{*}$
on the dual of Hochschild homology and we define
$\operatorname{HH}(\mathcal{B},\mathcal{A})^{*}=\operatorname{fib}(\operatorname{HH}(G)^{*})$.
A relative Hochschild class
$\sigma\in\operatorname{H}_{n-1}(\operatorname{HH}(\mathcal{B},\mathcal{A})^{*})$
induces a diagram

${{\operatorname{id}_{\mathcal{B}}[n-1]}}$${{HG[n-1]}}$${H\operatorname{id}_{\mathcal{A}}^{*}G}$${\operatorname{id}_{\mathcal{B}}^{*}}$$\scriptstyle{u[n-1]}$

where $u$ is the unit of $G\dashv H$, together with a choice of null-homotopy
of the composite morphism.

######  Definition 2.14.

A weak right $n$-Calabi-Yau structure on the functor
$G:\mathcal{B}\rightarrow\mathcal{A}$ consists of a class
$\sigma\in\operatorname{H}_{n-1}(\operatorname{HH}(\mathcal{B},\mathcal{A})^{*})$,
such that in the induced diagram with horizontal fiber and cofiber sequences

${{\operatorname{id}_{\mathcal{B}}[n-1]}}$${{HG[n-1]}}$${\operatorname{cof}}$${\operatorname{fib}}$${H\operatorname{id}_{\mathcal{A}}^{*}G}$${\operatorname{id}_{\mathcal{B}}^{*}}$

the vertical morphisms are equivalences.

We will omit the adjectives weak and right and simply refer to a structure as
in Definition 2.14 as an $n$-Calabi-Yau structure on $G$ or as a relative
$n$-Calabi-Yau structure on $\mathcal{B}$.

Informally, relative Calabi-Yau structures can be understood as follows. There
is a morphism
$U\simeq\operatorname{id}_{\mathcal{B}}^{*}\rightarrow\operatorname{id}_{\mathcal{B}}[n]$
of endofunctors, with $U$ the Serre functor, which is not an equivalence but
whose fiber is equivalent to $HG[n-1]$. Thus, a relative Calabi-Yau structure
is a description of how much the Serre functor differs from a suspension of
the identity, in terms of the adjunction $G\dashv H$.

##  3 Perverse schobers on surfaces

###  3.1 Marked surfaces and ribbon graphs

######  Definition 3.1.

A surface is a smooth compact oriented surface ${\bf S}$ with non-empty
boundary $\partial{\bf S}$ and interior ${\bf S}^{\circ}$. Note that this
implies that the boundary of ${\bf S}$ is a disjoint union of circles.

A marked surface $({\bf S},M)$ consists of a surface with nonempty boundary
and a finite set $M\subset\partial{\bf S}$ of marked points on the boundary
such that each connected component of $\partial{\bf S}$ contains at least one
marked point. We always assume that the ${\bf S}$ is not the monogon or digon.

######  Definition 3.2.

  * •

A graph $\Gamma$ consists of two finite sets $\Gamma_{0}$ of vertices and
$\operatorname{H}$ of halfedges together with an involution
$\tau:\operatorname{H}\rightarrow\operatorname{H}$ and a map
$\sigma:\operatorname{H}\rightarrow\Gamma_{0}$.

  * •

Let $\Gamma$ be a graph. We denote the set of orbits of $\tau$ by
$\Gamma_{1}$. The elements of $\Gamma_{1}$ are called the edges of $\Gamma$.
An edge is called internal if the orbit contains two elements and called
external if the orbit contains a single element. An internal edge is called a
loop at $v\in\Gamma_{0}$ if it consists of two halfedges both being mapped
under $\sigma$ to $v$. The set of internal edges of $\Gamma$ is denoted by
$\Gamma_{1}^{\circ}$ and the set of external edges is denoted by
$\Gamma_{1}^{\partial}$.

  * •

A ribbon graph consists of a graph $\Gamma$ together with a choice of a cyclic
order on the set $\operatorname{H}(v)$ of halfedges incident to $v$ for each
$v\in\Gamma_{0}$.

######  Definition 3.3.

Let $\Gamma$ be a graph. We denote by $\operatorname{Exit}(\Gamma)$ the
simplicial set given by the nerve of the $1$-category with

  * •

objects given by the set $\Gamma_{0}\amalg\Gamma_{1}$ and

  * •

non-identity morphisms of the form $v\rightarrow e$ with $v\in\Gamma_{0}$ a
vertex and $e\in\Gamma_{1}$ an edge incident to $v$. If $e$ is a loop at $v$,
then there are two morphisms $v\rightarrow e$.

We call $\operatorname{Exit}(\Gamma)$ the exit path category of $\Gamma$.

The geometric realization $|\Gamma|$ of the graph $\Gamma$ is defined as the
geometric realization $|\operatorname{Exit}(\Gamma)|$ of the simplicial set
$\operatorname{Exit}(\Gamma)$.

######  Remark 3.4.

Let $\Gamma$ be a graph and ${\bf S}$ an oriented surface. An embedding of
$|\Gamma|$ into ${\bf S}$ determines a canonical ribbon graph structure on
$\Gamma$ by requiring the halfedges at any vertex to be ordered in the
counterclockwise direction.

######  Definition 3.5.

Let ${\bf S}$ be a marked surface. We call a graph $\Gamma$ together with an
embedding $i:|\Gamma|\subset{\bf S}\backslash M$ a spanning graph for ${\bf
S}$, if $i$ is a homotopy equivalence and the restriction $i^{-1}(\partial{\bf
S}\backslash M)\rightarrow\partial{\bf S}\backslash M$ of $i$ is also a
homotopy equivalence. We consider spanning graphs as endowed with the
canonical ribbon graph structure arising from the embedding into the oriented
surface ${\bf S}$.

######  Remark 3.6.

Any trivalent spanning graph of a marked surface ${\bf S}$ arises as the dual
graph of an ideal triangulation of ${\bf S}$ in the sense of Definition 6.5
and this defines a bijection between the sets of suitable homotopy classes of
trivalent spanning graph and ideal triangulations.

###  3.2 Perverse schobers on marked surfaces

In this section, we recall the definition of perverse schober parametrized by
a ribbon graph. For more background on perverse schobers on marked surfaces,
we refer to [Chr22b, Sections 3 and 4].

We first briefly recall the definition of spherical functor or a spherical
adjunction. Let $F:\mathcal{A}\leftrightarrow\mathcal{B}:G$ be an adjunction
between stable $\infty$-categories with unit
$\operatorname{u}:\operatorname{id}_{\mathcal{A}}\rightarrow GF$ and counit
$\operatorname{cu}:FG\rightarrow\operatorname{id}_{\mathcal{B}}$. Taking the
cofiber of the unit $\operatorname{u}$ in the stable $\infty$-category
$\operatorname{Fun}(\mathcal{A},\mathcal{A})$ of endofunctors, we obtain a
functor $T_{\mathcal{A}}:\mathcal{A}\rightarrow\mathcal{A}$ called the twist
functor. Similarly, the cotwist functor
$T_{\mathcal{B}}:\mathcal{B}\rightarrow\mathcal{B}$ is defines as the fiber of
the counit. The adjunction $F\dashv G$ is called spherical if both
$T_{\mathcal{A}}$ and $T_{\mathcal{B}}$ are autoequivalences. In this case, we
also call the functors $F$ and $G$ spherical. For treatments of spherical
adjunctions in the setting of stable $\infty$-categories, we refer to [DKSS21,
Chr22d].

We turn to describing the local model of a perverse schober arising from a
spherical functor. Let $\Gamma$ be a ribbon graph and $v$ a vertex of $\Gamma$
of valency $m$. The objects of the under category
$\operatorname{Exit}(\Gamma)_{v/}$ are the vertex $v$ and its incident
halfedges. We choose a labeling of these halfedges by $1,\dots,m$, compatibly
with their given cyclic order. Given a spherical functor
$F_{v}:\mathcal{V}_{v}\rightarrow\mathcal{N}$, we define a functor
$\mathcal{G}_{v}(F_{v}):\operatorname{Exit}(\Gamma)_{/v}\rightarrow\operatorname{St}$
as follows. Consider the diagram
$D:[m-1]\rightarrow\operatorname{Set}_{\Delta}$ given by

$\mathcal{V}_{v}\xrightarrow{F_{v}}\underbrace{\mathcal{N}\xrightarrow{\operatorname{id}}\dots\xrightarrow{\operatorname{id}}\mathcal{N}}_{(m-1)\text{-many}}\,.$

Let $p:\Gamma(D)\rightarrow\Delta^{m-1}\simeq N([m-1])$ be the (covariant)
Grothendieck construction of $D$, see [Lur17, 3.2.5.2], where it is called the
relative nerve. We refer also to [Chr21, Section 3.1] for a brief introduction
of the Grothendieck construction.

  * •

We set
$\mathcal{G}_{v}(F_{v})(v)=\operatorname{Fun}_{\Delta^{m-1}}(\Delta^{m-1},\Gamma(D))$
to be the stable $\infty$-category of sections of $p$. We also denote
$\mathcal{V}_{F_{v}}^{m}=\mathcal{G}_{v}(F_{v})(v)$. We note that if
$\mathcal{V}_{v}\simeq 0$, then
$\mathcal{V}^{m}_{F_{v}}\simeq\operatorname{Fun}(\Delta^{m-2},\mathcal{N})$
should be considered as the $\infty$-category of representations of the
$A_{n-1}$-quiver in $\mathcal{N}$.

  * •

We set $\mathcal{G}_{v}(F_{v})(i)=\mathcal{N}$ for all $1,\dots,m$.

  * •

We denote $\varrho_{i}\coloneqq\mathcal{G}_{v}(F_{v})(v\rightarrow
i):\mathcal{V}^{m}_{F_{v}}\rightarrow\mathcal{N}$. We set

$\varrho_{1}=\operatorname{ev}_{m-1}$

to be the evaluation functor of the sections of $p$ at $m-1\in\Delta^{m-1}$.
For $1\leq i<m$, we set
$\varrho_{i+1}=\operatorname{ladj}(\operatorname{ladj}(\varrho_{i}))$. For a
detailed description (and thus also a proof of existence) of these functors,
we refer to [Chr22b, Section 3].

In the case $m=1$, we have $\operatorname{Exit}(\Gamma)_{/v}\simeq\Delta^{1}$
and the functor $\mathscr{G}_{v}(F_{v})$ is simply the functor $F_{v}$. One
can further show that making a different choice of total order on the
halfedges of $v$ changes the functor $\mathcal{G}_{v}(F_{v})$ only by
composition with an autoequivalence, see [Chr22b, Prop. 3.11].

######  Definition 3.7.

Let $\Gamma$ be a ribbon graph. A $\Gamma$-parametrized perverse schober is a
functor
$\mathcal{F}:\operatorname{Exit}(\Gamma)\rightarrow\mathcal{P}r^{L}_{\operatorname{St}}$
such that for each vertex $v$ of $\mathcal{T}$ there exists a spherical
functor $F_{v}:\mathcal{V}_{v}\rightarrow\mathcal{N}$, satisfying that the
composite of
$\operatorname{Exit}(\Gamma)_{v/}\rightarrow\operatorname{Exit}(\Gamma)$ with
$\mathcal{F}$ is equivalent to $\mathcal{G}_{v}(F_{v})$ in
$\operatorname{Fun}(\operatorname{Exit}(\Gamma)_{v/}\,\operatorname{St})$.

A vertex $v$ of $\Gamma$ satisfying that $\mathcal{V}_{v}\not\simeq 0$ is
called a singularity of $\mathcal{F}$. We call $F_{v}$ the spherical functor
describing $\mathcal{F}$ at $v$ (this functor is only determined up to
equivalence and composition with autoequivalences).

Assuming that the graph $\Gamma$ is connected, a $\Gamma$-parametrized
perverse schober assigns to each edge of $\Gamma$ an equivalent
$\infty$-category $\mathcal{N}$, called the generic stalk of the perverse
schober.

######  Remark 3.8.

We restrict in Definition 3.7 to perverse schobers taking values in stable and
presentable $\infty$-categories and colimit preserving functor, but one could
also consider perverse schobers valued in stable $\infty$-categories. Any
perverse schober taking values in presentable $\infty$-categories also defines
a functor to $\mathcal{P}r^{R}_{\operatorname{St}}$, because the functor
$\varrho_{i}$ defined above admits repeated left and right adjoints, see
[Chr21, Rem. 4.11].

######  Definition 3.9.

The $\infty$-categories of left/right $\Gamma$-parametrized perverse schobers
$\mathfrak{P}^{L}(\Gamma)$ and $\mathfrak{P}^{R}(\Gamma)$ are defined as the
full subcategories of
$\operatorname{Fun}(\operatorname{Exit}(\Gamma),\mathcal{P}r^{L}_{\operatorname{St}})$
and
$\operatorname{Fun}(\operatorname{Exit}(\Gamma),\mathcal{P}r^{R}_{\operatorname{St}})$,
respectively, spanned by $\Gamma$-parametrized perverse schobers.

######  Definition 3.10.

Let $\Gamma$ be a ribbon graph and $\mathcal{F}$ a $\Gamma$-parametrized
perverse schober.

  * •

The $\infty$-category of global sections
$\mathcal{H}(\Gamma,\mathcal{F})\in\mathcal{P}r^{L}_{\operatorname{St}}$ of
$\mathcal{F}$ is defined as the limit of $\mathcal{F}$.

  * •

The $\infty$-category of local sections
$\mathcal{L}(\Gamma,\mathcal{F})=\operatorname{Fun}_{\operatorname{Exit}(\Gamma)}(\operatorname{Exit}(\Gamma),\Gamma(\mathcal{F}))\in\mathcal{P}r^{L}_{\operatorname{St}}$
of $\mathcal{F}$ is defined as the $\infty$-category of sections of the
(covariant) Grothendieck construction
$p:\Gamma(\mathcal{F})\rightarrow\operatorname{Exit}(\Gamma)$.

######  Remark 3.11.

We usually identify the limit $\mathcal{H}(\Gamma,\mathcal{F})$ of a
$\Gamma$-parametrized perverse schober $\mathcal{F}$ with the full subcategory
of the $\infty$-category of sections $\mathcal{L}(\Gamma,\mathcal{F})$ of the
Grothendieck construction
$p:\Gamma(\mathcal{F})\rightarrow\operatorname{Exit}(\Gamma)$ of
$\mathcal{F}$, spanned by coCartesian sections. This is possible by [Lur09,
Cor. 3.3.3.2]. A section
$s:\operatorname{Exit}(\Gamma)\rightarrow\Gamma(\mathcal{F})$ of $p$ is called
coCartesian if $s(\alpha)$ is a coCartesian edge in the dual sense of [Lur09,
2.4.1.1] for each $1$-simplex $\alpha$ in $\operatorname{Exit}(\Gamma)$.

Let $\mathcal{F}$ be a $\Gamma$-parametrized perverse schober. Given an edge
$e$ of $\Gamma$, we denote by
$\operatorname{ev}_{e}:\mathcal{H}(\Gamma,\mathcal{F})\rightarrow\mathcal{F}(e)$
the evaluation functor, which maps a coCartesian section $s$ of $p$ to its
value $s(e)$ at $e$. Consider the product of the evaluation functors at the
external edges

$\prod_{e\in\Gamma_{1}^{\partial}}\operatorname{ev}_{e}:\mathcal{H}(\Gamma,\mathcal{F})\rightarrow\prod_{e\in\Gamma_{1}^{\partial}}\mathcal{F}(e)\,.$
(7)

The functor $\prod_{e\in\Gamma_{1}^{\partial}}\operatorname{ev}_{e}$ preserves
all limits and colimits by [Lur09, 5.1.2.3, 4.3.1.10, 4.3.1.16]. By the
$\infty$-categorical adjoint functor theorem,
$\prod_{e\in\Gamma_{1}^{\partial}}\operatorname{ev}_{e}$ thus admits a left
adjoint.

######  Definition 3.12.

The functor

$\partial\mathcal{F}:\prod_{e\in\Gamma_{1}^{\partial}}\mathcal{F}(e)\longrightarrow\mathcal{H}(\Gamma,\mathcal{F})$

is defined as the left adjoint of
$\prod_{e\in\Gamma_{1}^{\partial}}\operatorname{ev}_{e}$.

######  Remark 3.13.

In the study of partially wrapped Fukaya categories, particularly Fukaya-
Seidel categories, the functor $\partial\mathcal{F}$ is often called the Orlov
functor or cup functor and
$\prod_{e\in\Gamma_{1}^{\partial}}\operatorname{ev}_{e}$ is called the cap
functor, see for instance [Syl19] for background.

###  3.3 Semiorthogonal decompositions of perverse schobers

Let $\Gamma$ be a ribbon graph. We note that a morphism
$\alpha:\mathcal{F}\rightarrow\mathcal{G}$ of $\Gamma$-parametrized perverse
schobers in one of the $\infty$-categories $\mathfrak{P}^{L}(\Gamma)$ or
$\mathfrak{P}^{R}(\Gamma)$, see Definition 3.9, is simply a natural
transformation
$\Delta^{1}\times\operatorname{Exit}(\Gamma)\rightarrow\mathcal{P}r^{L}_{\operatorname{St}}$
or
$\Delta^{1}\times\operatorname{Exit}(\Gamma)\rightarrow\mathcal{P}r^{R}_{\operatorname{St}}$
between the diagrams defining $\mathcal{F}$ and $\mathcal{G}$.

######  Definition 3.14.

Let $\Gamma$ be a ribbon graph and let
$\alpha:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism of
$\Gamma$-parametrized perverse schobers in $\mathfrak{P}^{L}(\Gamma)$ or
$\mathfrak{P}^{R}(\Gamma)$.

  1. 1.

We call $\alpha$ an inclusion of perverse schobers in
$\mathfrak{P}^{L}(\Gamma)$ or $\mathfrak{P}^{R}(\Gamma)$ if
$\alpha(x):\mathcal{F}(x)\rightarrow\mathcal{G}(x)$ is the inclusion of a
stable subcategory (in particular fully faithful) for all
$x\in\operatorname{Exit}(\Gamma)$.

  2. 2.

Suppose that $\alpha$ is an inclusion in $\mathfrak{P}^{L}(\Gamma)$. We call
$\alpha$ right admissible if there exists a morphism
$\beta:\mathcal{G}\rightarrow\mathcal{F}$ in $\mathfrak{P}^{R}(\Gamma)$ such
that $\beta(x)$ is right adjoint to $\alpha(x)$ for all
$x\in\operatorname{Exit}(\Gamma)$.

  3. 3.

Suppose that $\alpha$ is an inclusion in $\mathfrak{P}^{R}(\Gamma)$. We call
$\alpha$ left admissible if there exists a morphism
$\beta:\mathcal{G}\rightarrow\mathcal{F}$ in $\mathfrak{P}^{L}(\Gamma)$ such
that $\beta(x)$ is left adjoint to $\alpha(x)$ for all
$x\in\operatorname{Exit}(\Gamma)$.

######  Remark 3.15.

Let $\alpha:\mathcal{F}\rightarrow\mathcal{G}$ be an inclusion of
$\Gamma$-parametrized perverse schobers. Spelling out the condition that
$\alpha$ is left admissible yields the requirement that for each morphism
$v\rightarrow e$ in $\operatorname{Exit}(\Gamma)$ the diagram

${\mathcal{F}(v)}$${\mathcal{G}(v)}$${\mathcal{F}(e)}$${\mathcal{G}(e)}$$\scriptstyle{\mathcal{F}(v\rightarrow
e)}$$\scriptstyle{\alpha(v)}$$\scriptstyle{\mathcal{G}(v\rightarrow
e)}$$\scriptstyle{\alpha(e)}$ (8)

is left adjointable, meaning that the diagram

${\mathcal{F}(v)}$${\mathcal{G}(v)}$${\mathcal{F}(e)}$${\mathcal{G}(e)}$$\scriptstyle{\mathcal{F}(v\rightarrow
e)}$$\scriptstyle{\mathcal{G}(v\rightarrow
e)}$$\scriptstyle{\beta(v)}$$\scriptstyle{\beta(e)}$

commutes, with $\beta(x)$ left adjoint to $\alpha(x)$ for $x=v,e$.
Analogously, the condition that $\alpha$ is right admissible is equivalent to
the right adjointability of the diagram (8). Adjointability of a commutative
square is also called the Beck-Chevalley condition.

######  Definition 3.16.

Let $\Gamma$ be a ribbon graph. A semiorthogonal decomposition
$\\{\mathcal{F}_{1},\mathcal{F}_{2}\\}$ of a $\Gamma$-parametrized perverse
schober $\mathcal{G}$ consists of

  * •

an inclusion $\alpha_{1}:\mathcal{F}_{1}\rightarrow\mathcal{G}$ in
$\mathfrak{P}^{R}(\Gamma)$ and

  * •

an inclusion $\alpha_{2}:\mathcal{F}_{2}\rightarrow\mathcal{G}$ in
$\mathfrak{P}^{L}(\Gamma)$, satisfying that

  * •

$\alpha_{1}$ is left admissible and that $\alpha_{2}$ is right admissible and
that

  * •

the cofiber of $\alpha_{2}$ in
$\operatorname{Fun}(\operatorname{Exit}(\Gamma),\mathcal{P}r^{L}_{\operatorname{St}})$
is given by $\mathcal{F}_{1}$.

######  Remark 3.17.

Consider a semiorthogonal decomposition
$\\{\mathcal{F}_{1},\mathcal{F}_{2}\\}$ of the $\Gamma$-parametrized perverse
schober $\mathcal{G}$.

  1. 1.

Then for each $x\in\operatorname{Exit}(\Gamma)$, the $\infty$-category
$\mathcal{G}(x)$ admits a semiorthogonal decomposition
$(\mathcal{F}_{1}(x),\mathcal{F}_{2}(x))$.

  2. 2.

The condition that $\mathcal{F}_{1}$ is the cofiber of $\alpha_{2}$ in
$\operatorname{Fun}(\operatorname{Exit}(\Gamma),\mathcal{P}r^{L}_{\operatorname{St}})$
is equivalent to the condition that $\mathcal{F}_{2}$ is the fiber of the
pointwise right adjoint $\beta_{1}:\mathcal{G}\rightarrow\mathcal{F}_{1}$ of
$\alpha_{1}$ in
$\operatorname{Fun}(\operatorname{Exit}(\Gamma),\mathcal{P}r^{R}_{\operatorname{St}})$.
This follows from the fact that limits and colimits in functor categories are
determined pointwise, see [Lur09, 5.1.2.3]. Since limits commute with limits,
we find that there exists a fiber sequence in
$\mathcal{P}r^{L}_{\operatorname{St}}$

$\mathcal{H}(\Gamma,\mathcal{F}_{2})\xrightarrow{i}\mathcal{H}(\Gamma,\mathcal{G})\xrightarrow{\pi}\mathcal{H}(\Gamma,\mathcal{F}_{1})\,.$

Passing to right adjoints yields a cofiber sequence in
$\mathcal{P}^{R}_{\operatorname{St}}$, showing by Lemma 2.7 that
$\mathcal{H}(\Gamma,\mathcal{G})$ admits a semiorthogonal decomposition

$\big{(}\operatorname{radj}(\pi)(\mathcal{H}(\Gamma,\mathcal{F}_{1})),i(\mathcal{H}(\Gamma,\mathcal{F}_{2}))\big{)}\,.$

######  Definition 3.18.

Let $\mathcal{F}$ be a $\Gamma$-parametrized perverse schober. We call

  * •

$\mathcal{F}$ comonadic if at each vertex $v$ of $\Gamma$, the spherical
functor $F_{v}:\mathcal{V}_{v}\rightarrow\mathcal{N}$ describing $\mathcal{F}$
at $v$ is a comonadic functor.

  * •

$\mathcal{F}$ monadic if at each vertex $v$ of $\Gamma$, the right adjoint of
the spherical functor $F_{v}$ describing $\mathcal{F}$ at $v$ is a monadic
functor.

  * •

$\mathcal{F}$ locally constant if $\mathcal{F}$ has no singularities.

######  Remark 3.19.

A spherical functor between presentable $\infty$-categories is monadic if and
only if it is comonadic if and only if it is conservative, as follows from the
$\infty$-categorical Barr-Beck theorem [Lur17, 4.7.3.5]. The terminology
introduced in the first two points in Definition 3.18 is hence purely
convention and may be swapped.

##  4 Generalized cluster categories of surfaces and perverse schobers

We fix a field $k$ and a marked surface ${\bf S}$ with a trivalent spanning
graph $\mathcal{T}$. Theorem 6.1 in [Chr22b] constructs a
$\mathcal{T}$-parametrized perverse schober $\mathcal{F}_{\mathcal{T}}$, whose
$\infty$-category of global sections
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$ is equivalent to the
unbounded derived $\infty$-category of some relative Ginzburg algebra
$\mathscr{G}_{\mathcal{T}}$ associated to $\mathcal{T}$. The potential of the
underlying quiver consists of $3$-cycles, one for each triangle of the dual
ideal triangulation of $\mathcal{T}$. In Sections 4.1 and 4.2, we describe the
generalized cluster category associated to $\mathscr{G}_{\mathcal{T}}$ in
terms of the global sections of a component of a semiorthogonal decomposition
of $\mathcal{F}_{\mathcal{T}}$. Using this, we are able to deduce in Section
4.3 that the generalized cluster category is equivalent to the $1$-periodic
topological Fukaya category.

###  4.1 A semiorthogonal decomposition of perverse schobers

The $\mathcal{T}$-parametrized perverse schober $\mathcal{F}_{\mathcal{T}}$
from [Chr22b, Theorem 6.1] is locally at each vertex of $\mathcal{T}$
classified by the spherical adjunction

$f^{*}:\mathcal{D}(k)\longleftrightarrow\operatorname{Fun}(S^{2},\mathcal{D}(k)):f_{*}$

where $f^{*}$ is the pullback functor along $S^{2}\rightarrow\ast$. There is a
$k$-linear equivalence
$\operatorname{Fun}(S^{2},\mathcal{D}(k))\simeq\mathcal{D}(k[t_{1}])$, where
$k[t_{1}]$ is the graded polynomial algebra with $|t_{1}|=1$, under which the
functor $f_{*}$ is identified with the pullback functor $\phi^{*}$ along
$\phi:k[t_{1}]\xrightarrow{t_{1}\mapsto 0}k$, see [Chr22b, Prop. 5.5]. In
particular, the functor $f^{*}$ is conservative and the adjunction
$f^{*}\dashv f_{*}$ thus comonadic. The adjunction $f^{*}\dashv f_{*}$ is
however not monadic. The Eilenberg-Moore $\infty$-category
$\operatorname{Fun}(S^{2},\mathcal{D}(k))^{\operatorname{mnd}}$ of the monad
$f_{*}f^{*}$ can be identified with the full subcategory
$\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}^{\operatorname{perf}}(k))\subset\operatorname{Fun}(S^{2},\mathcal{D}(k))$,
see Lemma 2.10. Using this observation, we can obtain a monadic and comonadic
$\mathcal{T}$-parametrized perverse schober
$\mathcal{F}^{\operatorname{mnd}}_{\mathcal{T}}$ by restricting in the
construction of $\mathcal{F}_{\mathcal{T}}$ in [Chr21] the spherical
adjunction $f^{*}\dashv f_{*}$ to the spherical monadic and comonadic
adjunction

$(f^{*})^{\operatorname{Ind-
fin}}:\mathcal{D}(k)\longleftrightarrow\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}^{\operatorname{perf}}(k)):f_{*}^{\operatorname{Ind-
fin}}\,.$

We remark that to give this definition of
$\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}$, we also use that in the
construction of $\mathcal{F}_{\mathcal{T}}$ all appearing autoequivalences of
$\operatorname{Fun}(S^{2},\mathcal{D}(k))$ preserve the subcategory
$\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}^{\operatorname{perf}}(k))\subset\operatorname{Fun}(S^{2},\mathcal{D}(k))$.

By construction, there is an inclusion of perverse schobers
$\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}\rightarrow\mathcal{F}_{\mathcal{T}}$.

######  Lemma 4.1.

The inclusion
$\mathcal{F}^{\operatorname{mnd}}_{\mathcal{T}}\rightarrow\mathcal{F}_{\mathcal{T}}$
is right admissible.

###### Proof.

By Remark 3.15, we need to show that for $1\leq i\leq 3$, the diagram

${\mathcal{V}^{3}_{(f^{*})^{\operatorname{Ind-
fin}}}}$${\mathcal{V}^{3}_{f^{*}}}$${{\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}^{\operatorname{perf}}(k))}}$${{\operatorname{Fun}(S^{2},\mathcal{D}(k))}}$$\scriptstyle{\varrho_{i}}$$\scriptstyle{\varrho_{i}}$
(9)

is right adjointable. We only consider the case $i=1$, the other cases can be
treated analogously or also follow by [Chr22b, Prop. 3.11] from the $i=1$
case.

The right adjoint of the inclusion functor
$i:\mathcal{V}^{2}_{(f^{*})^{\operatorname{Ind-
fin}}}\hookrightarrow\mathcal{V}^{2}_{f^{*}}$ can be determined from the
following diagram in $\mathcal{P}r^{R}_{\operatorname{St}}$ containing two
pullback squares:

${\mathcal{V}^{3}_{f^{*}}}$${\mathcal{V}^{3}_{(f^{*})^{\operatorname{Ind-
fin}}}}$${{\operatorname{Fun}(\Delta^{2},\operatorname{Fun}(S^{2},\mathcal{D}(k)))}}$${{\operatorname{Fun}(\Delta^{2},\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}^{\operatorname{perf}}(k)))}}$${\mathcal{D}(k)}$${\mathcal{D}(k)}$${{\operatorname{Fun}(S^{2},\mathcal{D}(k))}}$${{\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}^{\operatorname{perf}}(k))}}$$\scriptstyle{\operatorname{radj}(i)}$${\lrcorner}$${\lrcorner}$$\scriptstyle{\operatorname{ev}_{0}}$$\scriptstyle{\operatorname{Fun}(\Delta^{2},\pi_{\operatorname{mnd}})\leavevmode\nobreak\
\leavevmode\nobreak\
}$$\scriptstyle{\operatorname{ev}_{0}}$$\scriptstyle{f^{*}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{f^{*}}$$\scriptstyle{\pi_{\operatorname{mnd}}}$

The functor $\pi_{\operatorname{mnd}}$ is the right adjoint of the inclusion.
Recall that the functor $\varrho_{1}$ is defined in Section 3.2 by evaluating
at the vertex $2\in\Delta^{2}$. The above diagram shows that evaluating
$\operatorname{radj}(i)$ at $2$ gives $\pi_{\operatorname{mnd}}$. It follows
that the diagram (9) is right adjointable. ∎

######  Definition 4.2.

We define the functor
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}:\operatorname{Exit}(\mathcal{T})\rightarrow\operatorname{St}$
as the cofiber of the inclusion
$\mathcal{F}^{\operatorname{mnd}}_{\mathcal{T}}\hookrightarrow\mathcal{F}_{\mathcal{T}}$
in $\operatorname{Fun}(\operatorname{Exit}({\mathcal{T}}),\operatorname{St})$.

######  Lemma 4.3.

The functor $\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ is a
$\mathcal{T}$-parametrized perverse schober. At each vertex of $\mathcal{T}$,
it is described by the trivial spherical adjunction

$0\leftrightarrow\mathcal{D}(k[t_{1}^{\pm}])\,.$

###### Proof.

Using Lemma 2.11, this is straightforward to check by using that pushouts in
the functor category
$\operatorname{Fun}(\operatorname{Exit}(\mathcal{T}),\operatorname{St})$ are
computed pointwise in $\operatorname{Exit}(\mathcal{T})$. ∎

######  Remark 4.4.

The abbreviation $\operatorname{clst}$ stands for ’cluster’. It is also an
anagram to $\operatorname{lcst}$, which stands for ’locally constant’. Note
that the perverse schober $\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ is
locally constant in the sense of Definition 3.18.

######  Proposition 4.5.

The pair
$\\{\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}},\mathcal{F}^{\operatorname{mnd}}_{\mathcal{T}}\\}$
forms a semiorthogonal decomposition of the $\mathcal{T}$-parametrized
perverse schober $\mathcal{F}_{\mathcal{T}}$.

###### Proof.

The inclusion
$\alpha:\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}\rightarrow\mathcal{F}_{\mathcal{T}}$
is by Lemma 4.1 right admissible. A straightforward adaptation of the argument
from the proof Lemma 4.1 further shows that the cofiber morphism
$\mathcal{F}_{\mathcal{T}}\rightarrow\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$
arising from the definition of
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ is the pointwise left
adjoint of a left admissible inclusion
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}\rightarrow\mathcal{F}_{\mathcal{T}}$.
This shows that
$\\{\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}},\mathcal{F}^{\operatorname{mnd}}_{\mathcal{T}}\\}$
indeed forms a semiorthogonal decomposition of $\mathcal{F}_{\mathcal{T}}$. ∎

###  4.2 The generalized cluster category of a triangulated surface

######  Definition 4.6.

Let $\mathcal{D}$ be a $k$-linear $\infty$-category with a compact generator
$X\in\mathcal{D}$.

  1. (1)

We define the finite subcategory
$\mathcal{D}^{\operatorname{fin}}\subset\mathcal{D}$ as the full subcategory
consisting of objects $Y\in\mathcal{D}$, satisfying that
$\operatorname{Mor}_{\mathcal{D}}(X,Y)\in\mathcal{D}(k)$ is perfect. Note that
the notation $\mathcal{D}^{\operatorname{fin}}$ suppresses the choice of
compact generator $X$.

  2. (2)

Suppose that $\mathcal{D}$ is smooth. Then $\mathcal{D}^{\operatorname{fin}}$
consists of compact objects and
$\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}\subset\mathcal{D}$. We
define the generalized cluster category of $\mathcal{D}$ as the
$\operatorname{Ind}$-complete Verdier quotient
$\mathcal{D}/\operatorname{Ind}\mathcal{D}^{\operatorname{fin}}$ (i.e. cofiber
in $\mathcal{P}r^{L}_{\operatorname{St}}$).

######  Remark 4.7.

Let $\mathcal{D}$ be as in Definition 4.6. Then $\mathcal{D}$ admits a
semiorthogonal decomposition
$(\mathcal{D}/\operatorname{Ind}\mathcal{D}^{\operatorname{fin}},\operatorname{Ind}\mathcal{D}^{\operatorname{fin}})$
into its generalized cluster category and its $\operatorname{Ind}$-finite
part, see Lemma 2.7.

The goal of this section is to describe the generalized cluster category of
the $\infty$-category
$\mathcal{D}(\mathscr{G}_{\mathcal{T}})\simeq\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$
with compact generator $\mathscr{G}_{\mathcal{T}}$. We begin by giving an
intrinsic description of this compact generator in terms of the perverse
schober.

######  Lemma 4.8.

For $e$ an edge of $\mathcal{T}$, consider the left adjoint

$\operatorname{ev}_{e}^{*}:\mathcal{D}(k[t_{1}])\simeq\mathcal{F}_{\mathcal{T}}(e)\rightarrow\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$

of the evaluation functor $\operatorname{ev}_{e}$ at $e$. The object
$\bigoplus_{e\in\mathcal{T}_{1}}\operatorname{ev}_{e}^{*}(k[t_{1}])\in\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$
is a compact generator.

###### Proof.

This follows using the adjunction
$\operatorname{ev}_{e}^{*}\dashv\operatorname{ev}_{e}$ and the observation
that global sections which vanish on all edges of $\mathcal{T}$ are zero,
since $f^{*}$ is a conservative functor. ∎

As shown in [Chr22b, Prop. 6.7], the compact generator from Lemma 4.8 is given
by the relative Ginzburg algebra
$\mathscr{G}_{\mathcal{T}}\in\mathcal{D}(\mathscr{G}_{\mathcal{T}})\simeq\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$

######  Theorem 4.9.

The generalized cluster category of $\mathcal{D}(\mathscr{G}_{\mathcal{T}})$
is equivalent to the $\infty$-category
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}})$ of
global sections of the perverse schober
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$.

######  Remark 4.10.

A shown in [Chr22b, Section 6.4], the $\infty$-category
$\mathcal{D}(\mathscr{G}_{\mathcal{T}})\simeq\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$
does not depend on the choice of trivalent spanning graph $\mathcal{T}$ of the
marked surface ${\bf S}$, up to equivalence of $\infty$-categories. The same
thus also holds for its associated generalized cluster category. We thus write
$\mathcal{C}_{\bf
S}\coloneqq\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}})$
and call $\mathcal{C}_{\bf S}$ the generalized cluster category of the marked
surface ${\bf S}$.

To prove Theorem 4.9, we show that the $\infty$-category of global sections of
$\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}$ is equivalent to
$\operatorname{Ind}\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{fin}}$
and then make use of the semiorthogonal decomposition
$\\{\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}},\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}\\}$
of $\mathcal{F}_{\mathcal{T}}$. Showing the former involves an explicit
argument using compact generators arising from curves described in [Chr21].

######  Definition 4.11.

We denote by
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{Ind-
fin}}\subset\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$ the full
subcategory of global sections
$X\in\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$ satisfying that
$\operatorname{ev}_{e}(X)\in\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}(k)^{\operatorname{perf}})$
for all edges $e$ of $\mathcal{T}$.

######  Lemma 4.12.

The $\infty$-category $\mathcal{H}^{\operatorname{Ind-
fin}}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$ is compactly generated by the
objects $M_{\gamma}^{L}$ associated to finite matching data
$(\gamma,f^{*}(k))$ with $\gamma$ pure and
$f^{*}(k)\in\operatorname{Fun}(S^{2},\mathcal{D}(k))^{\operatorname{fin}}\subset\operatorname{Fun}(S^{2},\mathcal{D}(k))$,
as defined in [Chr21].

###### Proof.

Using that $f^{*}(k)$ is a compact generator of
$\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}(k))^{\operatorname{fin}}$,
[Chr21, Prop. 5.19] shows that a certain collection of maching data with local
value $f^{*}(k)$ and pure, finite underlying matching curves compactly
generate $\mathcal{H}^{\operatorname{Ind-
fin}}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$. We remark that for the fact
that the matching curves giving rise to this compact generator are finite, it
is crucial that the marked surface has a marked point on each boundary
component. ∎

######  Proposition 4.13.

There exist equivalences of $\infty$-categories

$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}})\simeq\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{Ind-
fin}}\simeq\operatorname{Ind}\mathcal{D}(\mathscr{G}_{\mathcal{T}})^{\operatorname{fin}}\,.$

###### Proof.

It follows from the definition of
$\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}$, that the image on global
sections of the inclusion
$\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}}\rightarrow\mathcal{F}_{\mathcal{T}}$
consists of global sections which evaluate on each edge of $\mathcal{T}$ to an
object in
$\operatorname{Ind}\operatorname{Fun}(S^{2},\mathcal{D}(k)^{\operatorname{perf}})\subset\operatorname{Fun}(S^{2},\mathcal{D}(k))$.
We thus see that
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}}^{\operatorname{mnd}})\simeq\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{Ind-
fin}}$.

We proceed with showing the equivalence
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{Ind-
fin}}\simeq\operatorname{Ind}\mathcal{D}(\mathscr{G}_{\mathcal{T}})^{\operatorname{fin}}$.
We can realize both of these $\infty$-categories as presentable, stable
subcategories of $\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$. An
object of $\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$ is finite with
respect to the compact generator of Lemma 4.8 if and only if its evaluations
to the edges of $\mathcal{T}$ are finite in
$\operatorname{Fun}(S^{2},\mathcal{D}(k))$ (with respect to the compact
generator $k[t_{1}]$), which is equivalent to lying in
$\operatorname{Fun}(S^{2},\mathcal{D}(k)^{\operatorname{perf}})\subset\operatorname{Fun}(S^{2},\mathcal{D}(k))$.
It follows that
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{fin}}\subset\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{Ind-
fin}}$. Using that the objects in
$\mathcal{H}({\mathcal{T}},\mathcal{F}_{\mathcal{T}})^{\operatorname{fin}}$
are compact in $\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$, we
further get
$\operatorname{Ind}\mathcal{H}({\mathcal{T}},\mathcal{F}_{\mathcal{T}})^{\operatorname{fin}}\subset\mathcal{H}^{\operatorname{Ind-
fin}}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$. Lemma 4.12, in combination with
the fact that the global sections associated to finite, pure matching data
with finite local value, in the sense of [Chr21], lie by construction in
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{fin}}\subset\mathcal{H}^{\operatorname{Ind-
fin}}(\mathcal{T},\mathcal{F}_{\mathcal{T}})$, now implies that both
$\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}})^{\operatorname{Ind-fin}}$
and
$\operatorname{Ind}\mathcal{D}(\mathscr{G}_{\mathcal{T}})^{\operatorname{fin}}$
are compactly generated by the same set of objects and thus equivalent. ∎

###### Proof of Theorem 4.9..

Combine Proposition 4.5, Remark 3.17.2 and Proposition 4.13. ∎

The dg-algebra $k[t_{2}^{\pm}]$ is commutative and the $\infty$-category
$\mathcal{D}(k[t_{2}^{\pm}])$ comes with an apparent $k[t_{2}^{\pm}]$-linear
structure. This can be used to construct a factorization of the perverse
schober $\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ through
$\operatorname{Lin}_{k[t_{2}^{\pm}]}\rightarrow\operatorname{St}$. In
particular, its limit $\mathcal{C}_{\bf
S}=\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}})$
inherits a $k[t_{2}^{\pm}]$-linear structure

######  Proposition 4.14.

The $k[t_{2}^{\pm}]$-linear $\infty$-category $\mathcal{C}_{\bf S}$ is smooth
and proper.

###### Proof.

Using the duality
$\mathcal{P}r^{L}_{\operatorname{St}}\simeq(\mathcal{P}r^{R}_{\operatorname{St}})^{\operatorname{op}}$,
we find that $\mathcal{C}_{\bf S}$ is equivalent to the colimit of the right
adjoint diagram of $\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$. The
smoothness thus follows from the fact that finite colimits of smooth
$\infty$-categories along compact objects preserving functors are again
smooth.

To see that $\mathcal{C}_{\bf S}$ is proper, consider the compact generator
$\bigoplus_{e\in\mathcal{T}_{1}}\operatorname{ev}^{*}(k[t_{1}^{\pm}])$, given
by the sum of the images of the left adjoints of the evaluation functors
$\operatorname{ev}_{e}:\mathcal{H}(\mathcal{T},\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}})\rightarrow\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}(e)\simeq\mathcal{D}(k[t_{1}^{\pm}])$.
Proposition 5.19 in [Chr21] shows that
$\operatorname{ev}^{*}_{e}(k[t_{1}^{\pm}])\simeq M_{c_{e}}^{k[t_{1}^{\pm}]}$
for some matching datum $(c_{e},k[t_{1}^{\pm}])$, where $c_{e}$ is a finite,
pure matching curve (or equivalently
$\operatorname{ev}^{*}_{e}(k[t_{1}^{\pm}])\simeq M_{c_{e}}$ in the notation of
Section 6.2 for some matching curve $c_{e}$). Using Theorems 6.7 and 6.8, we
see that
$\operatorname{End}(\bigoplus_{e\in\mathcal{T}_{1}}\operatorname{ev}^{*}(k[t_{1}^{\pm}]))$
is perfect in $\mathcal{D}(k[t_{2}^{\pm}])$, showing that $\mathcal{C}_{\bf
S}$ is proper. ∎

We will prove the following theorem in a separate work on relative Calabi-Yau
structures for perverse schobers [Chr22c].

######  Theorem 4.15.

The $k[t_{2}^{\pm}]$-linear functor

$\prod_{e\in\mathcal{T}_{1}^{\partial}}\operatorname{ev}_{e}:\mathcal{C}_{\bf
S}\longrightarrow\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}(e)$

admits a $2$-Calabi-Yau structure in the sense of Definition 2.14

######  Corollary 4.16.

The adjunction

$\partial\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}:\prod_{e\in\mathcal{T}_{1}^{\partial}}\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}(e)\longleftrightarrow\mathcal{C}_{\bf
S}:\prod_{e\in\mathcal{T}_{1}^{\partial}}\operatorname{ev}_{e}$

is spherical.

###### Proof.

The twist functor of the adjunction
$\partial\mathcal{F}^{\operatorname{clst}}_{\mathcal{T}}\dashv\prod_{e\in\mathcal{T}_{1}^{\partial}}\operatorname{ev}_{e}$
is by Theorem 4.15 a suspension of the inverse Serre functor
$\operatorname{id}_{\mathcal{C}_{\bf S}}^{!}$ and thus invertible.
Furthermore, a Serre functor commutes with any natural transformation, so that
$\operatorname{id}_{\mathcal{C}_{\bf S}}^{!}$ commutes with the counit of the
adjunction. To prove the sphericalness of the adjunction, it thus suffices by
[Chr22d, Thm. 4.1] applied to opposite categories to prove that the functor
$\partial\mathcal{F}^{\operatorname{clst}}_{\mathcal{T}}$ is comonadic. By the
$\infty$-categorical Barr-Beck theorem [Lur17, 4.7.3.5], we only need to show
that $\partial\mathcal{F}^{\operatorname{clst}}_{\mathcal{T}}$ is
conservative. We denote the left adjoint of $\operatorname{ev}_{e}$ by
$\operatorname{ev}_{e}^{*}$. As noted in the proof of Proposition 4.14 above,
one has $\operatorname{ev}_{e}^{*}(k[t_{1}^{\pm}])\simeq M_{c_{e}}$, in the
notation from Section 6.2, for some matching curve $c_{e}$ in ${\bf S}$. This
shows that $\operatorname{ev}_{e}^{*}$ is conservative. It follows that
$\partial\mathcal{F}^{\operatorname{clst}}_{\mathcal{T}}=\prod_{e\in\mathcal{T}_{1}^{\partial}}\operatorname{ev}_{e}^{*}$
is also conservative, concluding the proof. ∎

###  4.3 Relation to the 1-periodic topological Fukaya category

The perverse schober $\mathcal{F}_{\mathcal{T}}$ is locally constant, i.e. has
no singularities, and its generic stalk is the derived $\infty$-category
$\mathcal{D}(k[t_{1}^{\pm}])$ of $1$-periodic chain complexes. It is thus
locally described by the $1$-periodic derived category of the $A_{2}$-quiver.
We hence refer to its $\infty$-category of global sections $\mathcal{C}_{\bf
S}=\mathcal{H}({\mathcal{T}},\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}})$
as the topological Fukaya category of ${\bf S}$ with values in the derived
$\infty$-category of $1$-periodic chain complexes, or the $1$-periodic
topological Fukaya category for short. There is a reasonable notion of
monodromy for parametrized perverse schobers and this monodromy of
$\mathcal{F}_{\mathcal{T}}^{\operatorname{clst}}$ along any closed curve in
${\bf S}^{\circ}$ is trivial; the monodromy is defined and shown to be trivial
in the proof of Theorem 6.5 in [Chr21]. We are hence justified in calling the
generalized cluster category $\mathcal{C}_{\bf S}$ the $1$-periodic
topological Fukaya category of ${\bf S}$.

The $\infty$-category $\mathcal{C}_{\bf S}^{\operatorname{c}}$ can further be
considered as an $\infty$-categorical avatar of the topological Fukaya
(dg-)category in the sense of [DK18] with coefficients in the cyclic $2$-Segal
object arising from the Waldhausen $S_{\bullet}$-construction of the
$2$-periodic dg-category $\operatorname{dgMod}_{k[t_{1}^{\pm}]}$ of dg
$k[t_{1}^{\pm}]$-modules. In particular, Corollary 3.4.7 of [DK18] shows the
following.

######  Theorem 4.17.

The stable $\infty$-category $\mathcal{C}_{\bf S}^{\operatorname{c}}$ is acted
upon by automorphisms in $\operatorname{ho}\operatorname{St}$ by the mapping
class group of the marked surface $({\bf S},M)$ of isotopy classes of
orientation preserving diffeomorphisms ${\bf S}\rightarrow{\bf S}$ restricting
to the identity on $\partial{\bf S}$.

Finally, we also record a description of the generalized cluster category
$\mathcal{C}_{\bf S}$ in terms of a gentle algebra.

######  Proposition 4.18.

There exists a(n ungraded) gentle algebra $\operatorname{gtl}$ and an
equivalence of $\infty$-categories $\mathcal{C}_{\bf
S}\simeq\mathcal{D}(\operatorname{gtl})\otimes_{k}\mathcal{D}(k[t_{1}^{\pm}])$.

###### Proof.

Choose a trivalent spanning graph $\mathcal{T}$ of ${\bf S}$. As observed in
[HKK17], the $\infty$-category of global sections of a locally constant
perverse schober with generic stalk $\mathcal{D}(k)$ on ${\bf S}$ parametrized
by $\mathcal{T}$ admits a formal generator whose endomorphism algebra is a
finite dimensional, and in general graded, gentle algebra, denoted
$\operatorname{gtl}^{\prime}$. Using that tensoring with
$\mathcal{D}(k[t_{1}^{\pm}])$ (with respect to the symmetric monoidal
structure of $\operatorname{Lin}_{k}$) preserves colimits in
$\operatorname{Lin}_{k}$, we find an equivalence $\mathcal{C}_{\bf
S}\simeq\mathcal{D}(\operatorname{gtl}^{\prime})\otimes_{k}\mathcal{D}(k[t_{1}^{\pm}])$
in $\operatorname{Lin}_{k}$. Let $\operatorname{gtl}$ be the ungraded gentle
algebra obtained from $\operatorname{gtl}^{\prime}$ by discarding the grading.
It is easy to see that
$\mathcal{D}(\operatorname{gtl}^{\prime})\otimes_{k}\mathcal{D}(k[t_{1}^{\pm}])\simeq\mathcal{D}(\operatorname{gtl})\otimes_{k}\mathcal{D}(k[t_{1}^{\pm}])$
in $\operatorname{Lin}_{k}$, showing the claim. ∎

##  5 Exact structures from relative Calabi-Yau structures

Consider an exact functor $G:\mathcal{B}\rightarrow\mathcal{A}$, meaning a
functor which preserves finite limits and colimits, where $\mathcal{B}$ is a
stable $\infty$-category and $\mathcal{A}$ an additive $\infty$-category. We
can pull back the the split-exact structure on $\mathcal{A}$ to an exact
structure on the $\infty$-category $\mathcal{B}$, such that a sequence in
$\mathcal{B}$ is exact if and only if its image under $G$ is a split-exact
sequence in $\mathcal{A}$, see Example 5.9. The goal of this section is to
show that if $G$ carries a right $2$-Calabi-Yau structure and is spherical,
then the exact structure on $\mathcal{B}$ is Frobenius and its extriangulated
homotopy category is $2$-Calabi-Yau.

In Section 5.1, we recall the notion of an exact $\infty$-category and
describe pullbacks of exact structures. In Section 5.2, we recall what is an
extriangulated category and why it is natural to consider the extriangulated
homotopy categories of exact $\infty$-categories. In Section 5.3, we describe
the Frobenius exact structure arising from a functor carrying a $2$-Calabi-Yau
structure.

###  5.1 Exact $\infty$-categories

######  Definition 5.1 ([Bar15]).

An exact $\infty$-category is a triple
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$, where
$\mathcal{C}$ is an additive $\infty$-category and
$\mathcal{C}_{\dagger},\mathcal{C}^{\dagger}\subset\mathcal{C}$ are
subcategories (called subcategories of inflations and deflations), satisfying
that

  1. (1)

every morphism $0\rightarrow X$ in $\mathcal{C}$ lies in
$\mathcal{C}_{\dagger}$ and every morphism $X\rightarrow 0$ in $\mathcal{C}$
lies in $\mathcal{C}^{\dagger}$.

  2. (2)

pushouts in $\mathcal{C}$ along morphisms in $\mathcal{C}_{\dagger}$ exist and
lie in $\mathcal{C}_{\dagger}$. Dually, pullbacks in $\mathcal{C}$ along
morphisms in $\mathcal{C}^{\dagger}$ exist and lie in $\mathcal{C}^{\dagger}$.

  3. (3)

Given a commutative square in $\mathcal{C}$ of the form

${X}$${Y}$${X^{\prime}}$${Y^{\prime}}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{c}$$\scriptstyle{d}$

the following are equivalent.

     * •

The square is pullback, $c\in\mathcal{C}_{\dagger}$ and
$d\in\mathcal{C}^{\dagger}$.

     * •

The square is pushout, $b\in\mathcal{C}_{\dagger}$ and
$a\in\mathcal{C}^{\dagger}$.

When the exact structure is clear from the context, we also simply refer to
$\mathcal{C}$ as an exact $\infty$-category.

######  Definition 5.2.

An exact sequence $X\rightarrow Y\rightarrow Z$ in an exact $\infty$-category
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ consists of a
fiber and cofiber sequence in $\mathcal{C}$

${X}$${Y}$${0}$${Z}$$\scriptstyle{a}$${\square}$$\scriptstyle{b}$

with $a\in\mathcal{C}_{\dagger}$ and $b\in\mathcal{C}^{\dagger}$.

######  Example 5.3.

  1. 1.

Let $\mathcal{C}$ be an additive $\infty$-category. Then there is an exact
$\infty$-category $(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$
with inflations consisting of inclusions of direct summands and deflations
consisting of projections onto direct summands, called the split-exact
structure. The exact sequences are the split-exact sequences $X\hookrightarrow
X\oplus Y\twoheadrightarrow Y$.

  2. 2.

Let $\mathcal{C}$ be a stable $\infty$-category. Then
$(\mathcal{C},\mathcal{C},\mathcal{C})$ is an exact $\infty$-category.

######  Definition 5.4.

Let $(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ be an exact
$\infty$-category.

  1. 1)

An object $P\in\mathcal{C}$ is called projective if every exact sequence
$X\rightarrow Y\rightarrow P$ is split-exact. An object $I\in\mathcal{C}$ is
called injective if every exact sequence $I\rightarrow Y\rightarrow Z$ is
split-exact.

  2. 2)

We say that $\mathcal{C}$ has enough projectives if for each object
$X\in\mathcal{C}$ there exists an exact sequence $X\rightarrow P\rightarrow Y$
with $P$ projective. Similarly, we say that $\mathcal{C}$ has enough
injectives if for each object $Y\in\mathcal{C}$ there exists an exact sequence
$Y\rightarrow I\rightarrow X$ with $I$ injective.

  3. 3)

We say that $\mathcal{C}$ is Frobenius if $\mathcal{C}$ has enough projectives
and injectives and the classes of projective and injective objects coincide.

The $\infty$-categorical version of the stable $1$-category of a Frobenius
exact $1$-category is the following.

######  Proposition 5.5 ([JKPW22]).

Let $(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ be a Frobenius
exact $\infty$-category and $W$ the class of morphisms $f:X\rightarrow Y$
which fit into an exact sequence

$X\xlongrightarrow{(f,g)}Y\oplus I\longrightarrow J$

with $I$ and $J$ injective and $g$ arbitrary. Then, the $\infty$-categorical
localisation $\bar{\mathcal{C}}\coloneqq\mathcal{C}[W^{-1}]$ of $\mathcal{C}$
at $W$ is a stable $\infty$-category.

###### Proof idea.

Applying [Cis19, Prop. 7.5.6] and its dual version, one finds that
$\bar{\mathcal{C}}$ admits finite limits and colimits and pushouts and
pullbacks coincide. ∎

######  Lemma 5.6.

Suppose that $\mathcal{C}$ is a $k$-linear $\infty$-category and
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ a Frobenius exact
$\infty$-category. Then the stable $\infty$-category $\bar{\mathcal{C}}$ from
Proposition 5.5 inherits the structure of a $k$-linear $\infty$-category in
the sense of [Lur18, D.1.1.1].

###### Proof.

As argued in [Lur18, Section D.1.2], the datum of a $k$-linear structure on a
presentable $\infty$-category $\mathcal{D}$ is equivalent to the datum of a
monoidal functor
$\operatorname{LMod}_{k}^{\operatorname{ff}}\rightarrow\operatorname{Fun}(\mathcal{D},\mathcal{D})$
with domain the symmetric monoidal $\infty$-category of (automatically free)
$k$-modules of finite rank. In turn, we can identify this datum with the datum
of an object of
$\operatorname{LMod}_{k}^{\operatorname{ff}}\times_{\operatorname{Alg}(\operatorname{Cat}_{\infty})}\operatorname{LMod}(\operatorname{Cat}_{\infty})\times_{\operatorname{Cat}_{\infty}}\mathcal{D}$,
see [Lur17, Section 4.7.1], where $\operatorname{Cat}_{\infty}$ is considered
as a monoidal $\infty$-category with the Cartesian monoidal structure.

Let $Z=\coprod_{W}\Delta^{1}$ and $\bar{Z}=\coprod_{W}\Delta^{0}$. We have
apparent functors $\pi:Z\rightarrow\bar{Z}$ and $Z\rightarrow\mathcal{C}$.
Using the module structure of $\mathcal{C}$, the latter functor gives rise to
a functor
$f:Z\times\operatorname{LMod}_{k}^{\operatorname{ff}}\rightarrow\mathcal{C}\times\operatorname{LMod}_{k}^{\operatorname{ff}}\rightarrow\mathcal{C}$.
We define the $k$-linear $\infty$-category $\mathcal{C}^{\prime}$ as the
pushout

${W\times\operatorname{LMod}_{k}^{\operatorname{ff}}}$${\mathcal{C}}$${\bar{W}\times\operatorname{LMod}_{k}^{\operatorname{ff}}}$${\mathcal{C}^{\prime}}$${\ulcorner}$

in
$\operatorname{LMod}_{k}^{\operatorname{ff}}\times_{\operatorname{Alg}(\operatorname{Cat}_{\infty})}\operatorname{LMod}(\operatorname{Cat}_{\infty}^{\times})$.
By [Lur17, 4.2.3.5], the underlying diagram in $\operatorname{Cat}_{\infty}$
is also pushout. For any $\infty$-category $\mathcal{A}$, we obtain a pullback
square in $\operatorname{Cat}_{\infty}$ as follows.

${{\operatorname{Fun}(\mathcal{C}^{\prime},\mathcal{A})}}$${{\operatorname{Fun}(\mathcal{C},\mathcal{A})}}$${{\operatorname{Fun}(\bar{W}\times\operatorname{LMod}_{k}^{\operatorname{ff}},\mathcal{A})}}$${{\operatorname{Fun}(W\times\operatorname{LMod}_{k}^{\operatorname{ff}},\mathcal{A})}}$${\lrcorner}$

Since $W$ is closed under the action by
$\operatorname{LMod}_{k}^{\operatorname{ff}}$, we find that
$\mathcal{C}^{\prime}$ is the $\infty$-categorical localization of
$\mathcal{C}$ at $W$, see [Cis19, Def. 7.1.2]. This concludes the argument,
showing that $\bar{\mathcal{C}}$ inherits a $k$-linear structure. ∎

######  Definition 5.7.

  1. 1.

An exact functor
$G:(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})\rightarrow(\mathcal{D},\mathcal{D}_{\dagger},\mathcal{D}^{\dagger})$
between exact $\infty$-categories consists of a functor
$G:\mathcal{C}\rightarrow\mathcal{D}$ which preserves zero objects,
inflations, deflations, pushouts along inflations and pullbacks along
deflations.

  2. 2.

A sub-exact structure of an exact $\infty$-category
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ consists of an
exact $\infty$-category
$(\mathcal{C},\mathcal{C}_{\ddagger},\mathcal{C}^{\ddagger})$, satisfying that
$\mathcal{C}_{\ddagger}\subset\mathcal{C}_{\dagger}$ and
$\mathcal{C}^{\ddagger}\subset\mathcal{C}^{\dagger}$.

Sub-exact structures can be pulled back along exact functors as follows.

######  Lemma 5.8.

Let
$G:(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})\rightarrow(\mathcal{D},\mathcal{D}_{\dagger},\mathcal{D}^{\dagger})$
be an exact functor between exact $\infty$-categories and
$(\mathcal{D},\mathcal{D}_{\ddagger},\mathcal{D}^{\ddagger})$ a sub-exact
structure of $(\mathcal{D},\mathcal{D}_{\dagger},\mathcal{D}^{\dagger})$. Then
there exists a sub-exact structure
$(\mathcal{C},\mathcal{C}_{\ddagger},\mathcal{C}^{\ddagger})$ of
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$, such that $G$
defines an exact functor
$G:(\mathcal{C},\mathcal{C}_{\ddagger},\mathcal{C}^{\ddagger})\rightarrow(\mathcal{D},\mathcal{D}_{\ddagger},\mathcal{D}^{\ddagger})$.

###### Proof.

We set $\mathcal{C}_{\ddagger}\subset\mathcal{C}_{\dagger}$ to be the
subcategory of morphisms whose image under $G$ lies in
$\mathcal{D}_{\ddagger}$. We define $\mathcal{C}^{\ddagger}$ similarly. It is
straightforward to verify that
$(\mathcal{C},\mathcal{C}_{\ddagger},\mathcal{C}^{\ddagger})$ is an exact
$\infty$-category and that
$G:(\mathcal{C},\mathcal{C}_{\ddagger},\mathcal{C}^{\ddagger})\rightarrow(\mathcal{D},\mathcal{D}_{\ddagger},\mathcal{D}^{\ddagger})$
is an exact functor. ∎

######  Example 5.9.

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are stable $\infty$-categories
and $G:\mathcal{C}\rightarrow\mathcal{D}$ is an exact functor in the usual
sense, i.e. a functor that preserves finite limits and colimits. Then $G$
defines an exact functor
$G:(\mathcal{C},\mathcal{C},\mathcal{C})\rightarrow(\mathcal{D},\mathcal{D},\mathcal{D})$
between exact $\infty$-categories. Applying Lemma 5.8, we obtain a non-trivial
additional exact structure
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ on $\mathcal{C}$
by pulling back the split-exact structure
$(\mathcal{D},\mathcal{D}_{\dagger},\mathcal{D}^{\dagger})$ on $\mathcal{D}$.
A fiber and cofiber sequence in $\mathcal{C}$ is exact in
$(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ if and only if its
image under $G$ is split-exact.

######  Proposition 5.10.

Let $G:\mathcal{C}\rightarrow\mathcal{D}$ be an exact functor between stable
$\infty$-categories. If $G$ is a spherical functor, then the exact
$\infty$-category $(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$
from Example 5.9 is Frobenius. The subcategory of injective and projective
objects of $\mathcal{C}$ consists of the additive closure of the essential
image of the right adjoint $H$ of $G$.

######  Remark 5.11.

A result similar to Proposition 5.10 appears for triangulated categories in
[BS21], see Theorem 4.23 and Remark 4.24 in loc. cit.

###### Proof of Proposition 5.10.

Let $F$ be the left adjoint of $G$ and $H$ the right adjoint of $G$. By the
sphericalness of $G$, we have that $H\simeq F\circ T_{\mathcal{D}}$, with
$T_{\mathcal{D}}$ the cotwist functor of the adjunction $G\dashv H$, see
[DKSS21, Cor. 2.5.16]. In particular, it follows that the essential images of
$F$ and $H$, and hence also their additive closures in $\mathcal{C}$, agree.

We begin by showing that every object of the form $F(d)\in\mathcal{C}$ with
$d\in\mathcal{D}$ is injective. A dual argument shows that every object of the
form $H(d)\in\mathcal{C}$ is projective. Let $c\in\mathcal{C}$ and consider an
extension $\alpha\in\operatorname{Ext}^{1}_{\mathcal{C}}(F(d),c)$. The
extension $\alpha$ corresponds to a fiber and cofiber sequence $c\rightarrow
z\rightarrow F(d)$ in $\mathcal{C}$. The image of this fiber and cofiber
sequence under $G$ splits (i.e. defines an exact sequence) if and only if
$G(\alpha)\simeq 0$. The commutative diagram,

${{\operatorname{Ext}^{1}_{\mathcal{D}}(GF(d),G(c))}}$${{\operatorname{Ext}^{1}_{\mathcal{C}}(F(d),c)}}$${{\operatorname{Ext}^{1}_{\mathcal{D}}(d,G(c))}}$$\scriptstyle{\operatorname{u}}$$\scriptstyle{\simeq}$$\scriptstyle{G}$

with $\operatorname{u}$ the unit of $F\dashv G$, shows that $G(\alpha)\simeq
0$ if and only if $\alpha\simeq 0$. This is the case, if and only if the fiber
and cofiber sequence $c\rightarrow z\rightarrow F(d)$ already splits. Since
$c$ is chosen arbitrarily, it follows that $F(d)$ is injective.

Next, we show that $\mathcal{C}$ has enough projective objects. Let
$c\in\mathcal{C}$. We have a fiber and cofiber sequence
$c\xrightarrow{\operatorname{u}_{c}}HG(x)\rightarrow T_{\mathcal{C}}(c)$,
where $\operatorname{u}_{c}$ is a unit map and $T_{\mathcal{C}}$ the twist
functor of $G\dashv H$. We apply $G$ and extend to the diagram

${T_{\mathcal{D}}G(c)}$${0}$${G(c)}$${GHG(c)}$${G(c)}$${0}$${GT_{\mathcal{C}}(c)}$${0}$${\square}$$\scriptstyle{G\operatorname{u}_{c}}$${\square}$$\scriptstyle{\operatorname{cu}_{G(c)}}$${\square}$

with $\operatorname{cu}_{G(c)}$ a counit map and all squares biCartesian. This
shows that the sequence
$G(c)\xrightarrow{G\operatorname{u}_{c}}GHG(x)\rightarrow GT_{\mathcal{C}}(c)$
in $\mathcal{D}$ splits and hence that that
$c\xrightarrow{\operatorname{u}_{c}}HG(x)\rightarrow T_{\mathcal{C}}(c)$ is
exact. This shows that $\mathcal{C}$ has enough projective objects.

Finally, we show that any injective or projective object lies in the additive
closure (closure under direct summands) of the essential image of $H$. Let $I$
be injective. Then we have an exact sequence $I\rightarrow HG(I)\rightarrow
T_{\mathcal{C}}(I)$ in $\mathcal{C}$. Since $I$ is injective, this sequence
splits, showing that $I$ is a direct summand of $HG(I)$, as desired. If $P$ is
projective, we similarly find an exact sequence
$T_{\mathcal{C}}^{-1}\rightarrow FG(P)\rightarrow P$, showing that $P$ is a
direct summand of $FG(P)\simeq HT_{\mathcal{D}}^{-1}G(P)$, as desired,
concluding the proof. ∎

###  5.2 Extriangulated categories

Extriangulated categories were introduced by Nakaoka-Palu in [NP19] as a
simultanious generalization of triangulated and exact $1$-categories. An
extriangulated category $(C,\mathbb{E},\mathfrak{s})$ consists of

  * •

an additive $1$-category $C$,

  * •

an additive bifunctor $\mathbb{E}:C^{\operatorname{op}}\times
C\rightarrow\operatorname{Ab}$ to the additive $1$-category of abelian groups,
of which we think as describing an interesting class of extensions in $C$ and

  * •

a sequence $\left(Y\rightarrow Z\rightarrow X\right)=\mathfrak{s}(\alpha)$,
called realization, associated to each $\alpha\in\mathbb{E}(X,Y)$,

subject to number of conditions, see [NP19].

Examples of extriangulated categories are, besides triangulated and exact
categories, extension closed subcategories of triangulated categories. A
further natural source of extriangulated categories are the homotopy
$1$-categories of exact $\infty$-categories. To see this, note that by [Kle20]
any exact $\infty$-category admits a stable hull, meaning that it can be
embedded as an extension closed subcategory in a stable $\infty$-category.
Passing to homotopy categories, one obtains an extriangulated structure on the
homotopy category, as it is an extension closed subcategory of the
triangulated homotopy category of the stable hull. An independent and more
direct proof that the homotopy category is extriangulated also appears in
[NP20].

Given an exact $\infty$-category $\mathcal{C}$, the additive functor of
extensions
$\mathbb{E}:\operatorname{ho}\mathcal{C}^{\operatorname{op}}\times\operatorname{ho}\mathcal{C}\rightarrow\operatorname{Ab}$
in the extriangulated structure of $\operatorname{ho}\mathcal{C}$ describes
isomorphism classes of exact sequences in $\mathcal{C}$. The extriangulated
structure on $\operatorname{ho}\mathcal{C}$ can be useful when studying the
exact $\infty$-category $\mathcal{C}$. For instance, one can use it to
formulate a notion of cluster-tilting object in the exact $\infty$-category
$\mathcal{C}$, which is simply an object which is cluster-tilting when
considered as an object in the extriangulated homotopy category
$\operatorname{ho}\mathcal{C}$ in sense of Definition 5.17 below.

The definition of a Frobenius extriangulated category is analogous to the
definition of a Frobenius exact $\infty$-category, see Definition 5.4.

######  Definition 5.12.

Let $(C,\mathbb{E},\mathfrak{s})$ be an extriangualated category.

  1. 1)

An object $P\in C$ is called projective if $\mathbb{E}(P,X)\simeq 0$ for all
$X\in C$ and injective if $\mathbb{E}(X,P)\simeq 0$ for all $X\in C$.

  2. 2)

We say that $C$ has enough projectives if for each object $X\in C$ there
exists an exact sequence $X\rightarrow P\rightarrow Y$ with $P$ projective.
Similarly, we say that $C$ has enough injectives if for each object $Y\in C$
there exists an exact sequence $Y\rightarrow I\rightarrow X$ with $I$
injective.

  3. 3)

We say that $C$ is Frobenius if $C$ has enough projectives and injectives and
the classes of projective and injective objects coincide.

The condition of being a projective or injective object in an exact
$\infty$-category can be tested in the extriangulated homotopy category. We
also get the following.

######  Lemma 5.13.

An exact $\infty$-category is Frobenius if and only if its extriangulated
homotopy category is Frobenius.

######  Remark 5.14.

Given a Frobenius extriangulated category $C$, its stable category $\bar{C}$
is defined as the quotient category by the ideal of morphisms which factor
through an injective object. The stable category $\bar{C}$ inherits the
structure of a triangulated category, see Corollary 7.4 and Remark 7.5 in
[NP19].

If $(\mathcal{C},\mathcal{C}_{\dagger},\mathcal{C}^{\dagger})$ is a Frobenius
exact $\infty$-category, the triangulated homotopy category of its associated
stable $\infty$-category $\bar{\mathcal{C}}$ can be identified with the
triangulated stable category $\bar{\operatorname{ho}\mathcal{C}}$ of the
Frobenius extriangulated homotopy category $\operatorname{ho}\mathcal{C}$.