1 Introduction
Cluster algebras were introduced by Fomin and Zelevinsky [Reference Fomin and ZelevinskyFZ02] as a class of commutative algebras equipped with a combinatorial structure relating different subsets of the algebra called clusters. Since then, there has been a great interest in cluster algebras and their relation to other subjects, including Teichmüller theory, polyhedral surfaces, representation theory of quivers and aspects of noncommutative algebraic geometry such as Calabi–Yau algebras, Calabi–Yau categories and stability conditions. A survey with many references can be found in [Reference KellerKel08]; we also refer to the clusteralgebra portal [Reference FominFom] for further surveys and information on cluster algebras.
Relevant for this work is a particular class of cluster algebras associated to oriented marked surfaces equipped with an ideal triangulation, introduced in [Reference Gekhtman, Shapiro and VainshteinGSV05, Reference Fock and GoncharovFG06b, Reference Fock and GoncharovFG09] and further studied in [Reference Fomin, Shapiro and ThurstonFST08, Reference Fomin and ThurstonFT18]. These cluster algebras can be described in two different ways. The first perspective is geometric and provides a description in terms of the decorated Teichmüller spaces of the surfaces. The cluster variables arise as lambda lengths, which form the coordinates of the Teichmüller space. These lambda lengths satisfy an analogue of the classical Ptolemy relations, which gives rise to the cluster exchange relations. The second perspective makes direct use of the combinatorics of the ideal triangulation. The mutation matrix used to define the cluster algebra arises as the signed adjacency matrix of the ideal triangulation, which counts the number of incidences of the ideal triangles. The resulting algebra does not depend on the choice of ideal triangulation but only on the underlying marked surface.
This second perspective in particular shows that cluster algebras of marked surfaces can be considered as cluster algebras associated to quivers, which can be categorified via $2$ Calabi–Yau (CY) triangulated categories, called cluster categories, and $3$ CY triangulated categories. To describe the $3$ CY categorification of the cluster algebra associated to a quiver Q, one chooses a nondegenerate potential W. The $3$ CY categorification is then given by the derived category of the Ginzburg algebra $\mathscr {G}(Q,W)$ associated to the quiver with potential $(Q,W)$ . The $2$ CY cluster category can be obtained from the derived category of the Ginzburg algebra via the Verdier quotient $\mathcal {D}(\mathscr {G}(Q,W))^{\operatorname {perf}}/\mathcal {D}(\mathscr {G}(Q,W))^{\operatorname {fin}}$ [Reference AmiotAmi09]. There is also a direct link between the Ginzburg algebras and the combinatorics of the cluster algebras; we refer to [Reference KellerKel12] for a survey.
To describe the results of this work, we first recall the construction of the quiver $Q_{\mathcal {T}}^{\circ }$ , and a choice of nondegenerate potential $W_{\mathcal {T}}$ , associated to an ideal triangulation $\mathcal {T}$ of a marked surface $\mathbf {S}$ [Reference LabardiniFragosoLF09, Reference Geiß, LabardiniFragoso and SchröerGLFS16]. We assume for simplicity that $\mathcal {T}$ has no selffolded triangles. The quiver $Q^{\circ }_{\mathcal {T}}$ has as vertices the internal edges of $\mathcal {T}$ and an arrow $a:i\rightarrow j$ for each ideal triangle containing the edges $i,j$ , where the edge j follows the edge i in the clockwise order of the edges of the ideal triangle induced by the orientation of the surface. The nondegenerate potential $W_{\mathcal {T}}=W^{\prime }_{\mathcal {T}}+W^{\prime \prime }_{\mathcal {T}}\in kQ^{\circ }_{\mathcal {T}}$ consists of a part $W^{\prime }_{\mathcal {T}}$ which is the sum of the clockwise $3$ cycles inscribed in the interior ideal triangles of $\mathcal {T}$ and a part $W^{\prime \prime }_{\mathcal {T}}$ which is a sum of anticlockwise cycles, one for each interior marked point of $\mathbf {S}$ .
The $2$ CY and $3$ CY categorifications can be described in terms of the combinatorial geometry of $\mathcal {T}$ – see [Reference Qiu and ZhouQZ17] and the references therein for the $2$ CY cluster category and [Reference QiuQiu18, Reference Qiu and ZhouQZ19] for the finite part of the derived category of the Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W^{\prime }_{\mathcal {T}}\right )$ . Most relevant for us is Ivan Smith’s realisation of the finite part of the derived category of $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ as a full subcategory of the Fukaya category of a Calabi–Yau $3$ fold $Y^{\circ }$ equipped with a Lefschetz fibration $\pi :Y^{\circ }\rightarrow \Sigma $ [Reference SmithSmi15]. The surface $\Sigma $ is obtained from $\mathbf {S}$ by removing all interior marked points – that is, $\Sigma = \mathbf {S}\backslash (M\cap \mathbf {S}^{\circ })$ , where $\mathbf {S}^{\circ }=\mathbf {S}\backslash \partial \mathbf {S}$ denotes the interior of $\mathbf {S}$ and M denotes the set of marked points. Inspired by the geometry of $\pi $ , we give in this paper a description of the entire unbounded derived category of the Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ in terms of the global sections of a perverse schober.
Before we describe our model for $\mathcal {D}\left (\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )\right )$ , we highlight the relation to a model for the partially wrapped Fukaya categories of graded surfaces or, equivalently, the derived categories of gentle algebras [Reference Haiden, Katzarkov and KontsevichHKK17, Reference Lekili and PolishchukLP20]. Consider an ideal triangulation of a graded marked surface $\mathbf {S}$ and the dual ribbon graph $\Gamma $ . The Fukaya category of the surface $\mathbf {S}$ is equivalent to the dgcategory of global sections of a constructible cosheaf of dgcategories on the ribbon graph $\Gamma $ [Reference Dyckerhoff and KapranovDK15, Reference Haiden, Katzarkov and KontsevichHKK17]. The cosheaf description of the Fukaya category categorifies the statement that the middle cohomology $\operatorname {H}_{\Gamma }(\Sigma ,\mathbb {Z}[1])$ of the surface $\Sigma $ with support on $\Gamma $ is equivalent to the abelian group of global sections of a constructible cosheaf $\underline {\operatorname {H}}_{\Gamma }(\mathbb {Z}[1])$ on $\Gamma $ whose stalk at a point x is the homology $\operatorname {H}_{\Gamma \cap U}(U,\mathbb {Z}[1])$ of a small neighbourhood $x\in U\subset \Sigma $ with support on $\Gamma \cap U$ . Our model describes the derived category of the Ginzburg algebra in terms of the global sections of a different constructible cosheaf of dgcategories on $\Gamma $ . Denote by $\Gamma ^{\circ }$ the ribbon graph obtained by removing all exterior edges of $\Gamma $ . Decategorified, the idea behind our model is to express the middle cohomology of the $3$ fold $Y^{\circ }$ with support on $\pi ^{1}(\Gamma ^{\circ })$ in terms of the abelian group of global sections with support on $\Gamma ^{\circ }$ of the perverse pushforward $\pi _{*}(\mathbb {Z}[3])$ to $\Sigma $ , which in turn is equivalent to the global sections with support on $\Gamma ^{\circ }$ of a constructible cosheaf $\underline {\operatorname {H}}_{\Gamma }(\pi _{*}\mathbb {Z}[3])$ on $\Gamma $ . We will not provide a systematic categorification of the perverse pushforward functor $\pi _{*}$ , but rather provide an explicit description of the categorification of the constructible cosheaf $\underline {\operatorname {H}}_{\Gamma }(\pi _{*}\mathbb {Z}[3])$ . This will be achieved by constructing a perverse schober on the surface that is classified locally, at every critical value of Smith’s Lefschetz fibration, by the $\operatorname {Ind}$ complete version of the spherical adjunction
The explicit computability of our model then arises from a concrete algebraic description of this adjunction, as well as the resulting categorification of $\underline {\operatorname {H}}_{\Gamma }(\pi _{*}\mathbb {Z}[3]))$ in terms of variants of Waldhausen’s $\operatorname {S}_{\bullet }$ construction. A full definition of the notion of a perverse schober on a surface is not yet documented in the literature; we thus introduce a framework for the treatment of perverse schobers on surfaces which are parametrised by ribbon graphs. Our definition of a parametrised perverse schober can be seen as a generalisation of the approach to topological Fukaya categories of surfaces of [Reference Dyckerhoff and KapranovDK18, Reference Dyckerhoff and KapranovDK15], allowing for the treatment of nonconstant coefficients. The main result of this paper is the following:
Theorem 1. Let $\mathcal {T}$ be an ideal triangulation of an oriented marked surface $\mathbf {S}$ and consider the dual ribbon graph $\Gamma $ . There exists a $\Gamma $ parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ whose stable $\infty $ category of global sections with support on $\Gamma ^{\circ }$ satisfies
That is, it is equivalent to the unbounded derived $\infty $ category of the Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W^{\prime }_{\mathcal {T}}\right )$ .
Note that if $\mathcal {T}$ contains no interior marked points, the potential $W^{\prime }_{\mathcal {T}}=W_{\mathcal {T}}$ is nondegenerate. Given an ideal triangulation $\mathcal {T}$ with interior marked points, the potential $W^{\prime }_{\mathcal {T}}$ is in general degenerate. In this case, the Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ is not expected to fully capture the cluster combinatorics.
Informally, Theorem 1 can be summarised as the statement that the derived $\infty $ category $\mathcal {D}\left (\mathscr {G}\left (Q^{\circ }_{\mathcal {T}},W^{\prime }_{\mathcal {T}}\right )\right )$ arises via the gluing of simpler $\infty $ categories. The pieces used in the gluing construction are the derived $\infty $ categories of certain relative Ginzburg algebras of ngons. This terminology was suggested by Bernhard Keller in his ICRA 2020 lecture series on relative Calabi–Yau structures. The derived $\infty $ category of a relative Ginzburg algebra also appears as the $\infty $ category of global sections $\mathcal {H}(\Gamma ,\mathcal {F}_{\mathcal {T}})$ of the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ (without any restrictions on the support). The $\infty $ category $\mathcal {H}(\Gamma ,\mathcal {F}_{\mathcal {T}})$ contains $\mathcal {H}_{\Gamma ^{\circ }}(\Gamma ,\mathcal {F}_{\mathcal {T}})\simeq \mathcal {D}\left (\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )\right )$ as a full subcategory. The passage from all global sections to global sections with support on the interior thus constitutes a loss of information, which explains why the nonrelative Ginzburg algebras cannot directly be glued. In terms of the underlying cluster algebras, our gluing construction seems to be a special case of the procedure of amalgamation and defrosting of cluster algebras of [Reference Fock and GoncharovFG06a].
To make the gluing construction of $\mathcal {D}\left (\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )\right )$ work, we need to determine the correct way to glue the pieces. Making different choices would lead to different signs of the differentials of the Ginzburg algebra. The total choice of signs is equivalent to a choice of spin structure on the surface $\Sigma =\mathbf {S}\backslash (M\cap \mathbf {S}^{\circ })$ ; see Section 7.1.
The formalism used for the description of the perverse schober $\mathcal {F}_{\mathcal {T}}$ works not only in the klinear setting but also over the sphere spectrum. Many of our results naturally extend to this more general setting; see Section 7.2.
In Section 1.1 we recall the full definition of the $3$ CY Ginzburg algebra and continue by introducing relative Ginzburg algebras. Section 1.2 contains a discussion of parametrised perverse schobers and Smith’s results. In Section 1.3 we describe the gluing construction of the Ginzburg algebra.
1.1 Relative Ginzburg algebras of triangulated surfaces
A quiver Q consists of a finite set of vertices, denoted $Q_{0}$ , and a finite set of arrows, denoted $Q_{1}$ , together with source and target maps $s,t:Q_{1}\rightarrow Q_{0}$ . A quiver is called graded if each arrow carries an integer labelling. Given a graded quiver Q, we denote by $kQ$ the graded path algebra over a commutative ring k. A potential W for a quiver Q is an element of the cyclic path algebra $kQ^{\operatorname {cyc}}$ , meaning the algebra of klinear sums of cyclic paths.
For the definition of the Ginzburg algebra, due to [Reference GinzburgGin06], we follow [Reference KellerKel11]. Consider a quiver with potential $(Q,W)$ . We denote by $Q^{\prime }$ the graded quiver with the same set of vertices as Q and graded arrows of the following three kinds:

• an arrow $a:i\rightarrow j$ in degree $0$ for each $a:i\rightarrow j\in Q_{1}$ ,

• an arrow $a^{*}:j\rightarrow i$ in degree $1$ for each $a:i\rightarrow j\in Q_{1}$ and

• an arrow $l_{i}:i\rightarrow i$ in degree $2$ for each $i\in Q_{0}$ .
The cyclic derivative $\partial _{a}:kQ_{\operatorname {cyc}}\rightarrow kQ$ with respect to $a\in Q_{1}$ is the klinear map taking a cycle c to $\partial _{a}c=\sum _{c=uav}uv$ , where $u,v\in kQ$ are allowed to be lazy paths. We denote the lazy path at a vertex $i\in Q_{0}$ by $p_{i}$ . We define the Ginzburg algebra $\mathscr {G}(Q,W)$ to be the dgalgebra whose underlying graded algebra is given by the graded path algebra $kQ^{\prime }$ and whose differential d is determined by the following action on the generators:
Note that $\mathscr {G}(Q,W)$ is not the completed Ginzburg algebra, as considered, for example, in [Reference Keller and YangKY11, Reference SmithSmi15]. We will not consider completed Ginzburg algebras in this paper. In terms of the associated derived $\infty $ categories of these dgalgebras, this does not mean much of a loss, because the derived $\infty $ category of the completed Ginzburg algebra can be realised as a full subcategory of the derived $\infty $ category of the noncompleted Ginzburg algebra. This perspective, however, neglects the additional topological structure of the completed Ginzburg algebra; see, for example, [Reference Keller and YangKY11 , Appendix].
We now introduce a relative version of the Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}},W^{\prime }_{\mathcal {T}}\right )$ associated to an ideal triangulation $\mathcal {T}$ of an oriented marked surface $\mathbf {S}$ . We define a quiver $Q_{\mathcal {T}}$ by adapting the definition of the quiver $Q^{\circ }_{\mathcal {T}}$ to include the boundary of $\mathbf {S}$ . We let $Q_{\mathcal {T}}$ be the quiver with a vertex for each edge of $\mathcal {T}$ (including boundary edges) and an arrow $a:i\rightarrow j$ for each ideal triangle containing the edges $i,j$ , where the edge j follows the edge i in the clockwise order. If $\mathcal {T}$ contains selffolded triangles, we additionally include an arrow $a:i\rightarrow i$ for each selffolded edge i of $\mathcal {T}$ . The quiver $Q_{\mathcal {T}}$ contains a clockwise $3$ cycle $T(f)$ for each ideal triangle f of $\mathcal {T}$ . We define the potential
We denote by $\tilde {Q}_{\mathcal {T}}$ the graded quiver with the same set of vertices as $Q_{\mathcal {T}}$ and graded arrows of the following three kinds:

• an arrow $a:i\rightarrow j$ in degree $0$ for each $a:i\rightarrow j\in (Q_{\mathcal {T}})_{1}$ ,

• an arrow $a^{*}:j\rightarrow i$ in degree $1$ for each $a:i\rightarrow j\in (Q_{\mathcal {T}})_{1}$ and

• an arrow $l_{i}:i\rightarrow i$ in degree $2$ for each vertex $i\in (Q_{\mathcal {T}})_{0}$ given by an internal edge of $\mathcal {T}$ .
We define the relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ to be the dgalgebra whose underlying graded algebra is given by the graded path algebra $k\tilde {Q}_{\mathcal {T}}$ and whose differential is determined by the following action on the generators:
The relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ is an example of the more general relative Ginzburg algebras associated to ice quivers with potential [Reference WuWu21]. An ice quiver is a quiver equipped with the further datum of a subquiver, whose vertices and arrows are called frozen. The ice quiver underlying $\mathscr {G}_{\mathcal {T}}~$ is given by $Q_{\mathcal {T}}$ , with frozen vertices given by the boundary edges of $\mathcal {T}$ and no frozen arrows. The potential is $\overline {W}_{\mathcal {T}}^{\prime }$ .
The quiver $Q_{\mathcal {T}}^{\circ }$ is the full subquiver of $Q_{\mathcal {T}}$ spanned by the vertices corresponding to internal edges. The potential $W_{\mathcal {T}}^{\prime }=\sum _{f}T(f)\in \left (kQ^{\circ }_{\mathcal {T}}\right )^{\operatorname {cyc}}$ consists of all $3$ cycles inscribed into internal ideal triangles of $\mathcal {T}$ . Note that if the boundary of $\mathbf {S}$ is empty, then $\left (Q_{\mathcal {T}},\overline {W}_{\mathcal {T}}^{\prime }\right )=\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ and the relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}~$ is equivalent to $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ . As an example, let $\mathbf {S}$ be the $3$ gon and $\mathcal {T}$ a triangle. The relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ is then given by the graded path algebra of the graded quiver
with differential d mapping each arrow of degree $1$ to the composite of the two opposite arrows of degree $0$ . The Ginzburg algebra $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ of the triangle $\mathcal {T}$ is, however, zero.
Theorem 1 extends to relative Ginzburg algebras in the following way:
Theorem 2. Let $\mathcal {T}$ be an ideal triangulation of an oriented marked surface $\mathbf {S}$ with dual ribbon graph $\Gamma $ . The $\infty $ category of global sections of the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ satisfies
In [Reference KellerKel11, Section 7.6] it shown that mutation of quivers with potential induces derived equivalences between the respective Ginzburg algebras. In [Reference LabardiniFragosoLF09] it is shown that if two ideal triangulations $\mathcal {T},\mathcal {T}^{\;\prime }$ are related by a flip of an edge, the associated quivers with potentials $\left (Q^{\circ }_{\mathcal {T}},W_{\mathcal {T}}\right )$ and $\left (Q^{\circ }_{\mathcal {T}^{\;\prime }},W_{\mathcal {T}^{\;\prime }}\right )$ are related by quiver mutation. In combination, these two results show that flips of ideal triangulations induce derived equivalences of the associated Ginzburg algebras. We extend the derived equivalences to the relative Ginzburg algebras.
Theorem 3. Let $\mathbf {S}$ be an oriented marked surface with two ideal triangulations $\mathcal {T},\mathcal {T}^{\;\prime }$ related by a flip of an edge e of $\mathcal {T}$ . Then there exists an equivalence of $\infty $ categories
We will prove Theorem 3 in Section 6.4 using an intrinsic feature of the theory of parametrised perverse schobers, namely equivalences of global sections induced from contractions of the underlying ribbon graphs.
We thank Bernhard Keller for informing us about an alternative approach to Theorem 3. A result of Yilin Wu [Reference WuWu21] extends the argument from [Reference KellerKel12, Section 7.6] to relative Ginzburg algebras, showing that the mutations of ice quivers with potential of [Reference PresslandPre20] induce derived equivalences between the associated relative Ginzburg algebras. Theorem 3 may then be recovered by additionally extending the results of [Reference LabardiniFragosoLF09] relating flips of the ideal triangulation and mutations of quivers with potentials to ice quivers.
1.2 Perverse schobers and Fukaya categories
Perverse schobers are a conjectured categorification of the notion of perverse sheaves [Reference Kapranov and SchechtmanKS14]. An approach to the categorification of a perverse sheaf on a disc was suggested in [Reference Kapranov and SchechtmanKS14]. The datum of a perverse sheaf on a disc with a single singularity in the centre is equivalent to the datum of a certain quiver diagram; the proposed ad hoc categorification of the quiver description is a spherical adjunction. In this paper, we extend this ad hoc categorification to perverse schobers on oriented marked surfaces. We combinatorially describe perverse schobers using ribbon graphs. Such a ribbon graph arises as the dual to an ideal triangulation of the marked surface. Given a ribbon graph $\Gamma $ , we define a poset $\operatorname {Exit}(\Gamma )$ with

• objects the vertices and edges of $\Gamma $ and

• morphisms of the form $v\rightarrow e$ , with v a vertex and e an incident edge.
For each nvalent vertex v of $\Gamma $ , there exists a subposet $\operatorname {Exit}(\Gamma )_{v/}\subset \operatorname {Exit}(\Gamma )$ consisting of the vertex v and the n incident edges. We define a perverse schober $\mathcal {F}$ parametrised by $\Gamma $ to be a functor $\mathcal {F}:\operatorname {Exit}(\Gamma )\rightarrow \operatorname {St}$ into the $\infty $ category of stable $\infty $ categories such that the restriction to $\operatorname {Exit}(\Gamma )_{v/}$ is for every vertex v equivalent to a particular diagram obtained from a spherical adjunction. The exact definition is based on the categorified Dold–Kan correspondence of [Reference DyckerhoffDyc21] and categorifies the ‘fractional spin’ description of perverse sheaves on a disc of [Reference Kapranov and SchechtmannKS16a]. The definition of a parametrised perverse schober captures the idea that a perverse schober on a surface is a collection of suitably gluedtogether spherical adjunctions, categorifying the description of perverse sheaves on surfaces given in [Reference Kapranov and SchechtmannKS16a]. The $\infty $ category of global sections $\mathcal {H}(\Gamma ,\mathcal {F})$ of a parametrised perverse schober $\mathcal {F}$ is defined as the limit of $\mathcal {F}$ in $\operatorname {St}$ . Under mild technical assumptions, the global sections of $\mathcal {F}$ are equivalent to a suitable colimit of the dual to $\mathcal {F}$ (left adjoint diagram), which describes a constructible cosheaf, see Section 4.3.
Given an ideal triangulation $\mathcal {T}$ without selffolded triangles of an oriented marked surface $\mathbf {S}$ , Smith [Reference SmithSmi15] defines a Calabi–Yau $3$ fold Y with an affine conic fibration $\pi :Y\rightarrow \mathbf {S}$ . The relation to Ginzburg algebras is as follows:

• The derived category of finite modules over $\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )$ arises as a full subcategory of the derived Fukaya category $\operatorname {Fuk}(Y)$ of Y, where $W_{\mathcal {T}}^{\prime }$ is the potential of $Q_{\mathcal {T}}^{\circ }$ consisting of clockwise $3$ cycles.

• The derived category of finite modules over $\mathscr {G}\left (Q^{\circ }_{\mathcal {T}},W_{\mathcal {T}}\right )$ arises as a full subcategory of the derived Fukaya category $\operatorname {Fuk}(Y,b)$ of Y with a twisting background class $b \in H^{2}(Y,\mathbb {Z}_{2})$ . Here, $W_{\mathcal {T}}=W_{\mathcal {T}}^{\prime }+W_{\mathcal {T}}^{\prime \prime }$ is the potential consisting of clockwise $3$ cycles and anticlockwise cycles.
The geometry of $\pi $ becomes clear when considering its fibres, which are given as follows:

• The generic fibre of $\pi $ is diffeomorphic to $T^{*}S^{2}$ .

• In the interior of each ideal triangle of $\mathcal {T}$ , there exists exactly one singular value with singular fibre given by the $2$ dimensional $A_{1}$ singularity.

• The fibres of the interior marked points in $\mathbf {S}$ are given by $\mathbb {C}^{2}\amalg \mathbb {C}^{2}$ .
We denote by $\Sigma := \mathbf {S}\backslash (M\cap \mathbf {S}^{\circ })$ , with $\mathbf {S}^{\circ }$ the interior of $\mathbf {S}$ , the surface without the interior marked points and $Y^{\circ } := \pi ^{1}(\Sigma )$ . Note that the restriction $\pi \rvert _{Y^{\circ }}:Y^{\circ }\rightarrow \Sigma $ of $\pi $ is a Lefschetz fibration.
The twist by the background class $b\in H^{2}(Y,\mathbb {Z}_{2})$ changes signs in the signed count of pseudoholomorphic curves passing through the fibres of the interior marked points. Without the background class, the signed count of such pseudoholomorphic curves always vanishes, so that the derived Fukaya category of $Y^{\circ }$ is equivalent to the derived Fukaya category of Y. The change in the $A_{\infty }$ structure of the derived Fukaya category of Y induced by the background class b accounts exactly for the difference between the potentials $W^{\prime }_{\mathcal {T}}$ and $W_{\mathcal {T}}$ .
We expect the $\infty $ category of global sections of the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ of Theorem 1 to describe (the $\operatorname {Ind}$ completion of) a partially wrapped Fukaya category of $Y^{\circ }$ . We further expect the global sections with support on $\Gamma ^{\circ }$ (the graph obtained from $\Gamma $ by removing boundary edges) to then correspond to (the $\operatorname {Ind}$ completion of) the wrapped Fukaya category of $Y^{\circ }$ . In the case of the unpunctured ngon, where $Y^{\circ }=Y$ is the $3$ dimensional $A_{n3}$ singularity and $Q_{\mathcal {T}}^{\circ }$ the $A_{n3}$ quiver, it is shown in [Reference Lekili and UedaLU21] that $\mathcal {W}(Y^{\circ })\simeq \mathcal {D}\left (\mathscr {G}\left (Q_{\mathcal {T}}^{\circ },W_{\mathcal {T}}^{\prime }\right )\right )^{\operatorname {perf}}$ , meaning that $\mathcal {H}_{\Gamma ^{\circ }}(\Gamma ,\mathcal {F}_{\mathcal {T}})$ is equivalent to the $\operatorname {Ind}$ completion of the wrapped Fukaya category of $Y^{\circ }$ .
We describe in Section 1.3 how the geometry of the Lefschetz fibration manifests itself in the definition of $\mathcal {F}_{\mathcal {T}}$ . We expect that the twisting by the background class b can be described as a deformation of the wrapped Fukaya category. It would be interesting to study the relation between such a deformation and the description in terms of parametrised perverse schobers.
1.3 The gluing construction of Ginzburg algebras
We now describe the construction of the perverse schober $\mathcal {F}_{\mathcal {T}}$ appearing in Theorem 1 and Theorem 2. We assume for simplicity that all ideal triangles of $\mathcal {T}$ are not selffolded. The ribbon graph $\Gamma $ dual to $\mathcal {T}$ parametrising $\mathcal {F}_{\mathcal {T}}$ consists of a vertex for each ideal triangle and an edge for each edge of $\mathcal {T}$ . Boundary edges of $\mathcal {T}$ correspond to external edges of the ribbon graph. Parametrised perverse schobers can, as can sheaves, be glued. To define $\mathcal {F}_{\mathcal {T}}$ , it thus suffices to define $\mathcal {F}_{\mathcal {T}}$ locally at each vertex of $\Gamma $ . The local datum at each vertex is a spherical adjunction, which we choose to be
where $\operatorname {Fun}\left (S^{2},\mathcal {D}(k)\right )$ is the $\infty $ category of local systems on the $2$ sphere with values in $\mathcal {D}(k)$ and $f^{*}$ is the pullback functor along $S^{2}\rightarrow \ast $ . This adjunction is shown in [Reference ChristChr20] to be spherical.
The $\infty $ category $\operatorname {Fun}\left (S^{2},\mathcal {D}(k)\right )$ is equivalent to the derived $\infty $ category of the polynomial algebra $k[t_{1}]$ with generator $t_{1}$ in degree $1$ (see Proposition 5.5). This derived $\infty $ category is, by a result of [Reference AbouzaidAbo11], equivalent to the Indcompletion of the wrapped Fukaya of the cotangent bundle $T^{*}S^{2}$ , which is the generic fibre of the Lefschetz fibration $\pi \rvert _{Y^{\circ }}$ . Under these equivalences, the image $f^{*}(k)$ corresponds to the Lagrangian zero section of $T^{*}S^{2}$ . The fibration $\pi \rvert _{Y^{\circ }}$ has exactly one singular value in each ideal triangle of $\mathcal {T}$ , so that up to homotopy of $\Gamma $ , the vertices of $\Gamma $ lie at the singular values of $\pi \rvert _{Y^{\circ }}$ . The singular fibres are given by the $A_{1}$ singularity. The relation between the geometry of $\pi \rvert _{Y^{\circ }}$ and the definition of $\mathcal {F}_{\mathcal {T}}$ can thus be summarised as follows:

• The wrapped Fukaya category of the generic fibre $T^{*}S^{2}$ of $\pi \rvert _{Y^{\circ }}$ gives rise to the $\infty $ category on the right of formula (2). This $\infty $ category describes the generic stalk of $\mathcal {F}_{\mathcal {T}}$ .

• Each vertex of $\Gamma $ corresponds to a singular value of $\pi \rvert _{Y^{\circ }}$ . The $\infty $ category on the left of formula (2) describes the categorification of the vector space of vanishing cycles at that singularity of $\pi \rvert _{Y^{\circ }}$ . Since the $A_{1}$ singularity has a unique vanishing cycle, this $\infty $ category is given by $\mathcal {D}(k)$ .

• The spherical adjunction $f^{*}\dashv f_{*}$ arises from a spherical object, the Lagrangian zero section, in the wrapped Fukaya category of $T^{*}S^{2}$ describing the vanishing cycle.
We further note that the perverse schober only models the Lefschetz fibration $\pi \rvert _{Y^{\circ }}$ , not the full fibration $\pi $ . The fibres $\mathbb {C}^{2}\amalg \mathbb {C}^{2}$ of $\pi $ over the interior marked points of $\mathbf {S}$ are not encoded in $\mathcal {F}_{\mathcal {T}}$ .
The parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ in total corresponds to the datum of a diagram
in the $\infty $ category $\operatorname {St}$ of stable $\infty $ categories indexed by the poset $\operatorname {Exit}(\Gamma )$ (see Section 1.2). The computations in Section 5 show that the parametrised perverse schober $\mathcal {F}_{\mathcal {T}}$ assigns the following:

• To each vertex of $\Gamma _{\mathcal {T}}$ , a stable $\infty $ category equivalent to the derived $\infty $ category of the relative Ginzburg algebra of the $3$ gon, depicted in diagram (1); this uses the fact that each vertex of $\Gamma _{\mathcal {T}}$ is trivalent.

• To each edge of $\Gamma _{\mathcal {T}}$ , a stable $\infty $ category equivalent to the derived $\infty $ category of the polynomial algebra $k[t_{1}]$ with generator $t_{1}$ in degree $1$ . Note that $k[t_{1}]$ is equivalent to the $2$ Calabi–Yau completion of k in the sense of [Reference KellerKel11] – that is, a $2$ dimensional Ginzburg algebra.
The equivalence $\mathcal {H}(\Gamma ,\mathcal {F}_{\mathcal {T}})\simeq \mathcal {D}(\mathscr {G}_{\mathcal {T}})$ of Theorem 2 thus expresses that the derived $\infty $ category of the relative Ginzburg algebra $\mathscr {G}_{\mathcal {T}}$ is glued from relative Ginzburg algebras of $3$ gons along $2$ dimensional Ginzburg algebras. We further illustrate the gluing construction of $\mathscr {G}_{\mathcal {T}}$ in two examples in Section 6.2.
Notation and conventions
We follows the notation and conventions of [Reference LurieLur09] and [Reference LurieLur17]. In particular, we always use the homological grading.
2 Preliminaries
This paper is formulated using the language of stable $\infty $ categories. It would in principle be possible to formulate most results in the framework of dgcategories. Our reason for using stable $\infty $ categories is to gain access to the powerful framework developed in [Reference LurieLur09, Reference LurieLur17]. As a side effect, we also profit in Section 7.2 from the added generality of stable $\infty $ categories over dgcategories. The essential computations in the gluing construction of the Ginzburg algebras are, however, performed using the category of dgcategories, with its quasiequivalence model structure.
The goal of this section is to review background material on the relation between, on the one hand, ring spectra, stable $\infty $ categories and their colimits, and on the other hand, dgalgebras, dgcategories and their homotopy colimits. All material appearing in this section for which we could not find references in the literature is well known to experts. In Sections 2.1 and 2.2 we discuss some generalities on limits and colimits in $\infty $ categories of $\infty $ categories and on $\infty $ categories of modules associated to ring spectra. In Sections 2.3 to 2.5 we relate dgcategories with $\infty $ categories. In Section 2.6 we discuss semiorthogonal decompositions.
For an extensive treatment of the theory of $\infty $ categories and stable $\infty $ categories, we refer to [Reference LurieLur09] and [Reference LurieLur17], respectively.
2.1 Limits and colimits in $\infty $ categories of $\infty $ categories
We begin by introducing the following $\infty $ categories of $\infty $ categories:
Definition 2.1. We denote

1. by $\operatorname {Cat}_{\infty }$ the $\infty $ category of $\infty $ categories,

2. by $\operatorname {St}\subset \operatorname {Cat}_{\infty }$ the subcategory spanned by stable $\infty $ categories and exact functors and

3. by $\operatorname {St}^{\operatorname {idem}}\subset \operatorname {St}$ the full subcategory spanned by idempotent complete stable $\infty $ categories.
An $\infty $ category is called presentable if it is equivalent to the Indcompletion of a small $\infty $ category and admits all colimitsFootnote ^{1} [Reference LurieLur09, Section 5.5]. We further denote

4. by $\mathcal {P}r^{L}\subset \operatorname {Cat}_{\infty }$ the subcategory spanned by presentable $\infty $ categories and colimitpreserving functors,

5. by $\mathcal {P}r^{R}\subset \operatorname {Cat}_{\infty }$ the subcategory spanned by presentable $\infty $ categories and accessible and limitpreserving functors and

6. 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 spanned by stable $\infty $ categories.
We are further interested in Rlinear $\infty $ categories, where R is an $\mathbb {E}_{\infty }$ ring spectrum – that is, a commutative algebra object in the symmetric monoidal $\infty $ category $\operatorname {Sp}$ of spectra. The $\infty $ category $\mathcal {P}r^{L}$ also admits the structure of a symmetric monoidal $\infty $ category [Reference LurieLur17, Section 4.8.1]. Given an $\mathbb {E}_{\infty }$ ring spectrum R, the $\infty $ category $\operatorname {LMod}_{R}\in \mathcal {P}r^{L}$ of left modulespectra over R is an algebra object of $\mathcal {P}r^{L}$ .
Definition 2.2.

7. Let R be an $\mathbb {E}_{\infty }$ ring spectrum. The $\infty $ category of $\operatorname {LinCat}_{R}=\operatorname {LMod}_{\operatorname {LMod}_{R}}\left (\mathcal {P}r^{L}\right )$ of left modules in $\mathcal {P}r^{L}$ over $\operatorname {LMod}_{R}$ is called the $\infty $ category of Rlinear $\infty $ categories.
Remark 2.3. Though not directly contained in the definition, it can be shown that any Rlinear $\infty $ category is automatically stable; see [Reference LurieLur18, D.1.5] for a discussion.
Remark 2.4. A lefttensoring of an $\infty $ category $\mathcal {M}$ over a monoidal $\infty $ category $\mathcal {C}^{\otimes }$ is a coCartesian fibration of $\infty $ operads $\mathcal {O}^{\otimes } \rightarrow \mathcal {LM}^{\otimes }$ over the leftmodule $\infty $ operad $\mathcal {LM}^{\otimes }$ , such that there are equivalences of fibres $\mathcal {O}^{\otimes }_{\langle m\rangle }\simeq \mathcal {M}$ and $\mathcal {O}^{\otimes }_{\langle a\rangle }\simeq \mathcal {C}^{\otimes }$ . We refer to [Reference LurieLur17, Section 4.2.1] for more details. Objects of $\operatorname {LinCat}_{R}$ can be identified with stable and presentable $\infty $ categories $\mathcal {C}$ equipped with the datum of a lefttensoring over the symmetric monoidal $\infty $ category $\operatorname {LMod}_{R}$ , such that the tensor product preserves colimits separately in each variable [Reference LurieLur18, Appendix D]. Let $\mathcal {M}_{1},\mathcal {M}_{2}$ be Rlinear $\infty $ categories as witnessed by the coCartesian fibrations $\mathcal {O}^{\otimes }_{1},\mathcal {O}^{\otimes }_{2}\rightarrow \mathcal {LM}^{\otimes }$ . An Rlinear functor $\mathcal {M}_{1}\rightarrow \mathcal {M}_{2}$ thus corresponds to a morphism of $\infty $ operads $\mathcal {O}^{\otimes }_{1}\rightarrow \mathcal {O}^{\otimes }_{2}$ over $\mathcal {LM}^{\otimes }$ .
We now recall (in order of appearance) results on

i) how to compute limits in $\operatorname {Cat}_{\infty }$ ,

ii) how to compute limits and colimits in $\mathcal {P}r^{L}$ , $\mathcal {P}r^{L}_{\operatorname {St}}$ and $\mathcal {P}r^{R}$ , $\mathcal {P}r^{R}_{\operatorname {St}}$ ,

iii) how to compute limits and colimits in $\operatorname {LinCat}_{R}$ and

iv) how to compute limits and colimits in $\operatorname {St}^{\operatorname {idem}}$ .
i) There is a general formula for limits in $\operatorname {Cat}_{\infty }$ . Let $D:Z\rightarrow \operatorname {Set}_{\Delta }$ be a diagram taking values in $\infty $ categories. Consider the coCartesian fibration $p:X\rightarrow Z$ classified by D. The limit $\infty $ category $\operatorname {lim}D$ is equivalent to the $\infty $ category of coCartesian sectionsFootnote ^{2} of p [Reference LurieLur09, 3.3.3.2]. If Z is the nerve of a $1$ category, the model for computing limits in $\operatorname {Cat}_{\infty }$ can be described more explicitly. We can use the relative nerve construction [Reference LurieLur09, 3.2.5.2] for the coCartesian fibration classified by D, which is very explicitly defined. We denote this model for the coCartesian fibration by $p:\Gamma (D)\rightarrow K$ and call it the (covariant) Grothendieck construction. A more detailed introduction to the relative nerve construction can be found in [Reference ChristChr20 , Section 1.2].
ii) One of the nice features of presentable $\infty $ categories is that there is an $\infty $ categorical adjoint functor theorem, which states that a functor between presentable $\infty $ categories admits a right adjoint if and only if it preserves all colimits, and admits a left adjoint if and only if it is accessibleFootnote ^{3} and preserves all limits. There thus exists an adjoint equivalence of $\infty $ categories
with the functors $\operatorname {radj},\operatorname {ladj}$ acting as the identity on objects. The functor $\operatorname {radj}$ maps a colimitpreserving functor to its right adjoint and the functor $\operatorname {ladj}$ maps an accessible and limitpreserving functor to its left adjoint. The adjoint equivalence $\operatorname {radj}\dashv \operatorname {ladj}$ also restricts to an adjoint equivalence between $\mathcal {P}r^{L}_{\operatorname {St}}$ and $\left (\mathcal {P}r_{\operatorname {St}}^{R}\right )^{op}$ . The equivalences $\operatorname {radj},\operatorname {ladj}$ preserve all limits and colimits, so that we can exchange the computations of limits and colimits of diagrams of (stable) presentable $\infty $ categories. For the computation of limits, we can use i) and the fact that that the inclusions $\mathcal {P}^{L}_{\operatorname {St}}\subset \mathcal {P}r^{L}\subset \operatorname {Cat}_{\infty }$ and $\mathcal {P}r^{R}_{\operatorname {St}}\subset \mathcal {P}r^{R}\subset \operatorname {Cat}_{\infty }$ preserve all limits.
iii) The computation of limits and colimits of Rlinear $\infty $ categories reduces to the computation of limits and colimits in $\mathcal {P}r^{L}$ , because the forgetful functor $\operatorname {LMod}_{\operatorname {LMod}_{R}}\left (\mathcal {P}r^{L}\right )\rightarrow \mathcal {P}r^{L}$ preserves all limits and colimits [Reference LurieLur17, 4.2.3.1, 4.2.3.5].
iv) The inclusion functor $\operatorname {St}^{\operatorname {idem}}\subset \operatorname {Cat}_{\infty }$ preserves all limits. The computation of colimits of idempotent stable $\infty $ categories can be related to the computation of colimits of presentable stable $\infty $ categories via the colimitpreserving Indcompletion functor $\operatorname {Ind}:\operatorname {St}^{\operatorname {idem}}\rightarrow \mathcal {P}r^{L}_{\operatorname {St}}$ . Given an $\infty $ category $\mathcal{C}\in \mathcal{P}r^L_{\operatorname {St}}$ , we denote by $\mathcal{C}^c\in \operatorname {St}^{\operatorname {idem}}$ its full subcategory of compact objects. Note that for $\mathcal{C}\in \operatorname {St}^{\operatorname {idem}}$ , there exists an equivalence $\operatorname {Ind}(\mathcal{C})^c\simeq \mathcal{C}$ .
2.2 Modules over ring spectra
Consider the symmetric monoidal $\infty $ category $\operatorname {Sp}$ of spectra. $\operatorname {Sp}$ is a stable and presentable $\infty $ category. An $\mathbb {E}_{1}$ ring spectrum is an object of $\operatorname {Alg}(\operatorname {Sp})$ , the $\infty $ category of (coherently associative) algebra objects in $\operatorname {Sp}$ . For every such $\mathbb {E}_{1}$ ring spectrum R, there is a stable and presentable $\infty $ category $\operatorname {RMod}_{R}$ of right Rmodules in $\operatorname {Sp}$ . If R can be enhanced to a commutative algebra object of $\operatorname {Sp}$ – that is, an $\mathbb {E}_{\infty }$ ring spectrum – then $\operatorname {RMod}_{R}$ inherits the structure of a symmetric monoidal $\infty $ category. In this case, we can form the $\infty $ category $\operatorname {Alg}(\operatorname {RMod}_{R})$ of algebra objects in $\operatorname {RMod}_{R}$ . Given $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ , we can again form the $\infty $ category $\operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ of right Amodules in $\operatorname {RMod}_{R}$ . Alternatively, we can also consider the $\mathbb {E}_{1}$ ring spectrum $\xi (A)\in \operatorname {Alg}(\operatorname {Sp})$ underlying A, obtained as follows. We consider the forgetful functor $\operatorname {RMod}_{R}\rightarrow \operatorname {Sp}$ , mapping a right Rmodule to the underlying spectrum. This functor extends to a functor $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(\operatorname {Sp})$ , which we apply to A. We can form the $\infty $ category of right modules $\operatorname {RMod}_{\xi (A)}$ over $\xi (A)$ . We will show in Corollary 2.7 that this does not yield a further $\infty $ category; there exists an equivalence of $\infty $ categories
Let $\mathcal {D}$ be a stable $\infty $ category and consider any object $X\in \mathcal {D}$ . We can find an $\mathbb {E}_{1}$ ring spectrum $\operatorname {End}(X)\in \operatorname {Alg}(\operatorname {Sp})$ , called the endomorphism algebra, with the following properties [Reference LurieLur17, 7.1.2.2]:

• $\pi _{n}\operatorname {End}(X) \simeq \pi _{0}\operatorname {Map}_{\mathcal {D}}(X[n],X)$ for all $n\in \mathbb {Z}$ .

• The induced ring structure of $\pi _{*}\operatorname {End}(X)$ is determined by the composition of endomorphisms in the homotopy category $\operatorname {Ho}(\mathcal {D})$ .
The algebra object $\operatorname {End}(X)$ is an endomorphism object of X in the sense of [Reference LurieLur17, Section 4.7.1], and its existence expresses the enrichment of the stable $\infty $ category $\mathcal {D}$ in spectra.
Assume that the stable $\infty $ category $\mathcal {D}$ is also presentable. An object $X\in \mathcal {D}$ is called a compact generator if

• X is compact – that is, commutes with filtered colimits – and

• an object $Y\in \mathcal {D}$ is zero if and only if $\operatorname {Map}_{\mathcal {D}}(X,Y[i])\simeq \ast $ for all $i\in \mathbb {Z}$ .
The importance of this notion is that if X is a compact generator, there exists an equivalence of $\infty $ categories $\mathcal {D}\simeq \operatorname {RMod}_{\operatorname {End}(X)}$ [Reference LurieLur17, 7.1.2.1].
We now restrict to Rlinear $\infty $ categories where R is an $\mathbb {E}_{\infty }$ ring spectrum. The most important case will be where $R=k$ is a commutative ring. Suppose that $\mathcal {D}$ is an Rlinear $\infty $ category and $X\in \mathcal {D}$ a compact generator. Lemma 2.5 shows that we can lift $\operatorname {End}(X)$ along the forgetful functor $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(\operatorname {Sp})$ to an algebra object in $\operatorname {RMod}_{R}$ :
Lemma 2.5. Let R be an $\mathbb {E}_{\infty }$ ring spectrum. Let $\mathcal {C}$ be a stable and presentable Rlinear $\infty $ category with a compact generator X. Then there exist an algebra object $\operatorname {End}_{R}(X)\in \operatorname {Alg}(\operatorname {RMod}_{R})$ and an equivalence of Rlinear $\infty $ categories
The algebra object $\operatorname {End}_{R}(X)$ is mapped under the functor $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(\operatorname {Sp})$ to the endomorphism algebra $\operatorname {End}(X)\in \operatorname {Alg}(\operatorname {Sp})$ .
Proof. The lefttensoring of $\mathcal {C}$ over R determines an Rlinear functor
. By the adjoint functor theorem, the functor admits a right adjoint G. We denote $\operatorname {End}_{R}(X):=G(X)\in \operatorname {RMod}_{R}$ . The existence of a lift of $\operatorname {End}_{R}(X)$ to $\operatorname {Alg}(\operatorname {RMod}_{R})$ and of the equivalence (3) follow from [Reference LurieLur17, 4.8.5.8] (compare also the proof of [Reference LurieLur17, 7.1.2.1]). The right adjoint of the composite functor
maps X to the endomorphism object $\operatorname {End}(X)$ . By the universal property of $\operatorname {End}(X)$ and $X\in \mathcal {C}\simeq \operatorname {RMod}_{\xi \left (\operatorname {End}_{R}(X)\right )}(\operatorname {Sp})$ , there exists a morphism $\xi (\operatorname {End}_{R}(X))\rightarrow \operatorname {End}(X)$ in $\operatorname {Alg}(\operatorname {Sp})$ , which is an equivalence on underlying spectra and thus an equivalence of $\mathbb {E}_{1}$ ring spectra.
Remark 2.6. In the setting of Lemma 2.5, the algebra object $\operatorname {End}_{R}(X)$ is an endomorphism object of X in the $\infty $ category $\mathcal {C}$ considered as lefttensored over $\operatorname {RMod}_{R}$ . We call $\operatorname {End}_{R}(X)$ the Rlinear endomorphism algebra of X.
Corollary 2.7. Let R be an $\mathbb {E}_{\infty }$ ring spectrum and set $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ . Then there exists an equivalence of $\infty $ categories
where $\xi :\operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {Alg}(Sp)$ denotes the forgetful functor.
Proof. The $\infty $ category $\operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ is presentable by [Reference LurieLur17, 4.2.3.7], stable by [Reference LurieLur17, 7.1.1.4] and lefttensored over $\operatorname {RMod}_{R}$ by [Reference LurieLur17, Section 4.3.2]. Consider the monadic adjunction . The adjunction and the fact that G is conservative and accessible imply that A is a compact generator. The Rlinear endomorphism algebra of $A\in \operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ is given by $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ . The statement thus follows from the second part of Lemma 2.5 and [Reference LurieLur17, 7.1.2.1].
Let R be an $\mathbb {E}_{\infty }$ ring spectrum. We end this section with a brief discussion of the relation between colimits of algebra objects in $\operatorname {RMod}_{R}$ and the colimits of the corresponding $\infty $ categories of right modules in $\operatorname {LinCat}_{R}$ . There is a functor $\theta : \operatorname {Alg}(\operatorname {RMod}_{R})\rightarrow \operatorname {LinCat}_{R}$ that assigns to an algebra object $A\in \operatorname {Alg}(\operatorname {RMod}_{R})$ the $\infty $ category $\operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ [Reference LurieLur17, section 4.8.3]. The functor $\theta $ assigns to an edge $\phi :A\rightarrow B$ in $\operatorname {Alg}(\operatorname {RMod}_{R})$ the relative tensor product
using the right Amodule structure on B provided by $\phi $ . For all $\phi :A\rightarrow B$ , the functor $\theta (\phi )$ admits a right adjoint, given by the pullback functor $\phi ^{*}:\operatorname {RMod}_{B}(\operatorname {RMod}_{R})\rightarrow \operatorname {RMod}_{A}(\operatorname {RMod}_{R})$ along $\phi $ [Reference LurieLur17, 4.6.2.17]. The functor $\theta $ preserves colimits indexed by contractible simplicial sets (that is, simplicial sets whose geometric realisation is a contractible space), most notably pushouts.
2.3 Differential graded categories and their modules
Let k be a commutative ring. A klinear dgcategory is a $1$ category enriched in the $1$ category $\operatorname {Ch}(k)$ of chain complexes of kmodules. Given a dgcategory C and two objects $x,y\in C$ , we write $\operatorname {Hom}_{C}(x,y)$ or $\operatorname {Hom}(x,y)$ for the mapping complex. We consider dgalgebras as dgcategories with a single object.
Definition 2.8. Let A and B be klinear dgalgebras.

• A left Amodule M is a graded left module over the graded algebra underlying A equipped with a differential $d_{M}$ such that
$$ \begin{align*} d_{M}(a.m)=d_{A}(a).m+(1)^{\operatorname{deg}(a)}a.d_{M}(m) \end{align*} $$for all $a\in A$ and $m\in M$ . 
• A right Amodule M is a graded right module over the graded algebra underlying A equipped with a differential $d_{M}$ such that
$$ \begin{align*} d_{M}(m.a)=d_{M}(m).a+(1)^{\operatorname{deg}(m)}m.d_{A}(a) \end{align*} $$for all $a\in A$ and $m\in M$ . We also refer to right Amodules simply as Amodules. 
• An ABbimodule M is a graded bimodule over the graded algebras underlying A and B equipped with a differential $d_{M}$ , which exhibits M as a left Amodule and a right Bmodule. If $A=B$ , we call M an Abimodule.
Remark 2.9. Let M be an ABbimodule with differential $d_{M}$ . The shifted ABbimodule $M[1]$ can be described as follows:

• The differential is $d_{M}$ .

• The left action $.{}_{[1]}$ of $a\in A$ on $m\in M[1]$ is given by $a.{}_{[1]}m=(1)^{\operatorname {deg}(a)}a.m$ , where $a.m$ denotes the left action of $a\in A$ on $m\in M$ .

• The right action $.{}_{[1]}$ of $b\in B$ on $m\in M[1]$ is given by $m.{}_{[1]}b=m.b$ , where $m.b$ denotes the right action of $b\in B$ on $m\in M$ .
We can identify left Amodules with dgfunctors $A\rightarrow \operatorname {Ch}(k)$ , right Amodules with dgfunctors $A^{op}\rightarrow \operatorname {Ch}(k)$ and ABbimodules with dgfunctors $A\otimes B^{\operatorname {op}}\rightarrow \operatorname {Ch}(k)$ . The following definition is thus consistent with Definition 2.8:
Definition 2.10. Let C be a dgcategory. We call a dgfunctor $C^{op}\rightarrow \operatorname {Ch}(k)$ a right Cmodule. We denote by $\operatorname {dgMod}(C)$ the dgcategory of right Cmodules.
Remark 2.11. Given any dgcategory C, the dgcategory $\operatorname {dgMod}(C)$ is pretriangulated, with distinguished triangles of the form .
Given a dgcategory C and an object $~x\in C$ , we denote by $\operatorname {End}^{\operatorname {dg}}(x)$ the endomorphism dgalgebra with underlying chain complex given by $\operatorname {Hom}_{C}(x,x)$ and algebra structure determined by the composition of morphisms in C.
Lemma 2.12. Let C be a dgcategory with finitely many objects $x_{1},\dotsc ,x_{n}$ . Then there exists an equivalence of dgcategories $\operatorname {dgMod}(C)\simeq \operatorname {dgMod}\left (\operatorname {End}^{\operatorname {dg}}\left (\bigoplus _{i=1}^{n} x_{i}\right )\right )$ , where $\operatorname {End}^{\operatorname {dg}}\left (\bigoplus _{i=1}^{n}x_{i}\right )$ is the endomorphism dgalgebra of $\bigoplus _{i=1}^{n} x_{i}$ in $\operatorname {dgMod}(C)$ .
Proof. This follows directly from spelling out the datum of a right module over C and over $\operatorname {End}^{\operatorname {dg}}\left (\bigoplus _{i=1}^{n}x_{i}\right )$ .
2.4 A model for the derived $\infty $ category of a dgalgebra
Let A be a klinear dgalgebra. The $1$ category $\operatorname {dgMod}(A)_{0}$ underlying the dgcategory $\operatorname {dgMod}(A)$ is the $1$ category with the same objects and with mapping sets given by the $0$ cycles. This $1$ category admits the projective model structure, where the weak equivalences are given by quasiisomorphisms and the fibrations are given by degreewise surjections. All objects of $\operatorname {dgMod}(A)_{0}$ are fibrant. A description of the cofibrant objects in $\operatorname {dgMod}(A)_{0}$ can be found, for example, in [Reference Barthel, May and RiehlBMR14], where they are called qsemiprojective objects. A right Amodule M is cofibrant if and only if

• the ungraded module $\bigoplus _{i\in \mathbb {Z}}M_{i}$ is a projective right module over the ungraded algebra $\bigoplus _{i\in \mathbb {Z}}A_{i}$ and

• for all acyclic right Amodules N, the mapping complex $\operatorname {Hom}_{A}(M,N)$ is acyclic.
If $A=k$ is a commutative ring, the cofibrant objects are the complexes of projective kmodules. We denote by $\operatorname {dgMod}(A)^{\circ }\subset \operatorname {dgMod}(A)$ the full dgsubcategory spanned by fibrantcofibrant objects. We call the dgnerve $\mathcal {D}(A):= N_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })$ the (unbounded) derived $\infty $ category of A.
Before we can further discuss the properties of $\mathcal {D}(A)$ , we need to briefly discuss localisations of $\infty $ categories.
Definition 2.13. A functor $f:\mathcal {C}\rightarrow \mathcal {C}^{\prime }$ between $\infty $ categories is a reflective localisation if f has a fully faithful right adjoint.
In [Reference LurieLur09], localisations in the sense of Definition 2.13 are simply called localisations. We are, however, interested in a more general class of localisations, which can be characterised by the following universal property:
Definition 2.14. Let $\mathcal {C}$ be an $\infty $ category and let W be a collection of morphisms in $\mathcal {C}$ . We call an $\infty $ category $\mathcal {C}^{\prime }$ the $\infty $ categorical localisation of $\mathcal {C}$ at W if there exists a functor $f:\mathcal {C}\rightarrow \mathcal {C}^{\prime }$ such that for every $\infty $ category $\mathcal {D}$ , composition with f induces a fully faithful functor
whose essential image consists of those functors $F:\mathcal {C}\rightarrow \mathcal {D}$ for which $F(\alpha )$ is an equivalence in $\mathcal {D}$ for all $\alpha \in W$ . In that case, we also write $\mathcal {C}^{\prime }=\mathcal {C}\left [W^{1}\right ]$ .
It is shown in [Reference LurieLur09, 5.2.7.12] that reflective localisations are localisations in the sense of Definition 2.14. If the collection of morphisms W is closed under homotopy and composition and contains all equivalences in $\mathcal {C}$ , we can regard $\mathcal {C}\left [W^{1}\right ]$ as a fibrant replacement of $(\mathcal {C},W)$ in the model category of marked simplicial sets (see also the discussion in the beginning of [Reference LurieLur17, Section 4.1.7]).
Our first goal in this section is to prove the following analogue of [Reference LurieLur17, 1.3.5.15], relating the derived $\infty $ category of A with the $\infty $ categorical localisation of $\operatorname {dgMod}(A)_{0}$ at the collection of quasiisomorphisms:
Proposition 2.15. Let A be a dgalgebra and let W denote the collection of quasiisomorphisms. There exists an equivalence of $\infty $ categories
Given a model category C, the $\infty $ categorical localisation of $N(C)$ at the collection of weak equivalences is called the $\infty $ category underlying C. We refer to [Reference HinchHin16] for general background. Proposition 2.15 thus shows that the derived $\infty $ category of A is the $\infty $ category underlying the model category $\operatorname {dgMod}(A)_{0}$ .
For the proof of Proposition 2.15, we need the following two lemmas.
Lemma 2.16. Let A be a dgalgebra. The inclusion functor $N(\operatorname {dgMod}(A)_{0})\rightarrow N_{\operatorname {dg}}(\operatorname {dgMod}(A))$ induces an equivalence of $\infty $ categories
where H is the collection of chain homotopy equivalences.
Proof. The proof of [Reference LurieLur17, 1.3.4.5] applies verbatim.
Lemma 2.17. Let A be a dgalgebra. There exists an equivalence of $\infty $ categories
Proof. We adapt the proofs of [Reference LurieLur17, 1.3.4.6, 1.3.5.12]. We show that the inclusion functor $i:\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })^{\operatorname {op}}\rightarrow \operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{\operatorname {op}}$ admits a left adjoint which exhibits $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })^{\operatorname {op}}$ as a reflective localisation at the collection of quasiisomorphisms. Note that any functor is a localisation if and only if the opposite functor is a localisation. We thus conclude that $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })$ is equivalent as an $\infty $ category to the localisation of $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))$ at the collection of quasiisomorphisms.
To verify that $i^{\operatorname {op}}$ is a reflective localisation, we need to show that it admits a left adjoint $G:\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{\operatorname {op}}\rightarrow \operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A)^{\circ })^{\operatorname {op}}$ . To show that W is the collection of quasiisomorphisms, we need to show by [Reference LurieLur09, 5.2.7.12] that any edge $e:M\rightarrow N$ in $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{op}$ is a quasiisomorphism if and only if $G(e)$ is an equivalence. Consider a trivial fibration $f:Q^{\prime }\rightarrow Q$ in $\operatorname {dgMod}(A)$ given a cofibrant replacement and any $P\in \operatorname {dgMod}(A)^{\circ }$ . [Reference LurieLur09, 5.2.7.8] shows the existence of G, provided that the composition with f induces an isomorphism of spaces
We deduce this from the assertion that composition with f induces a quasiisomorphism
The surjectivity of $\alpha $ follows from the lifting property of the cofibration $0\rightarrow P$ with respect to trivial fibrations. The kernel of $\alpha $ is given by $\operatorname {Hom}_{\operatorname {dgMod}(A)}(P,\operatorname {ker}(f))$ . Using the fact that f is a quasiisomorphism, we deduce that $\operatorname {ker}(f)$ is acyclic. The contractibility of the kernel of $\alpha $ thus follows from the property of P being cofibrant. We can thus deduce the existence of G. We note that G is pointwise given by choosing a cofibrant replacement. Consider an edge $e:M\rightarrow N$ in $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))^{op}$ . If e is a quasiisomorphism, it follows from Whitehead’s theorem for model categories that $G(e)$ is an equivalence. If $G(e)$ is an equivalence, we have the following commutative diagram in $\operatorname {N}_{\operatorname {dg}}(\operatorname {dgMod}(A))$ :
The vertical edges and the upper horizontal edge are quasiisomorphisms. It follows that e is also a quasiisomorphism.
Proof of Proposition 2.15.
By Lemmas 2.16 and 2.17, there exists an equivalence of $\infty $ categories
Using the fact that $H\subset W$ , the statement follows.
Let k be a commutative ring. The symmetric monoidal structure of the $1$ category $\operatorname {Ch}(k)$ can be used to also endow the $\infty $ category $\mathcal {D}(k)$ with a symmetric monoidal structure. As shown in [Reference LurieLur17, 7.1.4.6], there exists an equivalence of $\infty $ categories
The left side of formula (6) is the $\infty $ categorical localisation of the nerve of the $1$ category of dgalgebras at the collection of quasiisomorphisms. The right side is the $\infty $ category of algebra objects in $\mathcal {D}(k)$ . The equivalence (6) expresses that every dgalgebra can be considered as an algebra object in $\mathcal {D}(k)$ and that every algebra object in $\mathcal {D}(k)$ can be obtained this way (meaning it can be rectified). Unless stated otherwise, we will omit this identification and consider dgalgebras as algebra objects in the symmetric monoidal $\infty $ category $\mathcal {D}(k)$ .
We can consider k also as an $\mathbb {E}_{\infty }$ ring spectrum. The $\infty $ category $\operatorname {RMod}_{k}$ of right modules over k thus inherits a symmetric monoidal structure. The $\infty $ categories $\mathcal {D}(k)$ and $\operatorname {RMod}_{k}$ are equivalent as symmetric monoidal $\infty $ categories [Reference LurieLur17, 7.1.2.13].
Let A be a klinear dgalgebra and X a cofibrant Amodule. Consider the Quillen adjunction
between the tensor functor on the level of chain complexes and the internal Hom functor composed with the forgetful functor $\operatorname {dgMod}(A)\rightarrow \operatorname {dgMod}(k)$ . Given a Quillen adjunction between model categories, there is an associated adjunction between the underlying $\infty $ categories [Reference MazelGeeMG16]. We denote the adjunction of $\infty $ categories underlying the Quillen adjunction (7) by
Lemma 2.18. Let A be a klinear dgalgebra. The $\infty $ category $\mathcal {D}(A)$ admits the structure of a klinear $\infty $ category such that for any $X\in \mathcal {D}(A)$ , the functor is klinear.
Proof. The $\infty $ category $\mathcal {D}(A)$ is stable and presentable by [Reference LurieLur17, 1.3.5.9, 1.3.5.21]. We now show that $\mathcal {D}(A)$ is lefttensored over $\mathcal {D}(k)$ . Note that $\operatorname {dgMod}(k)_{0}\simeq \operatorname {Ch}(k)$ is a symmetric monoidal model category with respect to the tensor product, which we denote in the following by $\otimes $ [Reference LurieLur17, 7.1.2.11]. We further denote the Quillen bifunctor $\operatorname {dgMod}(k)\times \operatorname {dgMod}(A)\rightarrow \operatorname {dgMod}(A)$ given by the relative tensor product by . Recall that $\mathcal {LM}^{\otimes }$ denotes the leftmodule $\infty $ operad [Reference LurieLur17, 4.2.1.7]. We define a $1$ category $O_{A}^{\otimes }$ as follows:

• An object of $O_{A}^{\otimes }$ consists of an object $(\underbrace {a,\dotsc ,a}_{i\operatorname {many}},\underbrace {m,\dotsc ,m}_{j\operatorname {many}})\in \mathcal {LM}^{\otimes }$ and objects
$$ \begin{align*} (x_{1},\dotsc,x_{i})\in (\operatorname{dgMod}(k)^{\circ})^{\times i}, \qquad \left(m_{1},\dotsc,m_{j}\right)\in (\operatorname{dgMod}(A)^{\circ})^{\times j}. \end{align*} $$ 
• For $n=1,2$ , consider the object $X_{n}$ of $O_{A}^{\otimes }$ given by $l_{n}=(\underbrace {a,\dotsc ,a}_{i_{n}\operatorname {many}},\underbrace {m,\dotsc ,m}_{j_{n}\operatorname {many}})\in \mathcal {LM}^{\otimes }$ and
$$ \begin{align*} \left(x^{n}_{1},\dotsc,x^{n}_{i_{n}}\right)\in (\operatorname{dgMod}(k)^{\circ})^{\times {i_{n}}},\qquad \left(m_{1}^{n},\dotsc,m^{n}_{j_{n}}\right)\in (\operatorname{dgMod}(A)^{\circ})^{\times {j_{n}}}. \end{align*} $$A morphism $X_{1}\rightarrow X_{2}$ consists of a morphism $\alpha :l_{1}\rightarrow l_{2}$ in $\mathcal {LM}^{\otimes }$ , which we also consider as a morphism of sets $\tilde {\alpha }:\{1,\dotsc ,i_{1}+j_{1}\}\rightarrow \{1,\dotsc ,i_{2}+j_{2}\}$ , morphisms$$ \begin{align*} \bigotimes_{e\in \tilde{\alpha}^{1}(i)} a^{1}_{e}\rightarrow a_{i}^{2} \end{align*} $$in $\operatorname {dgMod}(k)^{\circ }$ for $1\leq i \leq i_{2}$ and morphisms$$ \begin{align*} \left( \bigotimes_{e\in \tilde{\alpha}^{1}\left(j\right)\backslash \operatorname{max}\left(\tilde{\alpha}^{1}\left(j\right)\right)} a^{1}_{e} \right) \otimes_{k} m_{\operatorname{max}\left(\tilde{\alpha}^{1}\left(j\right)\right)i_{1}}^{1}\rightarrow m^{2}_{ji_{2}} \end{align*} $$in $\operatorname {dgMod}(A)^{\circ }$ for $i_{1}+1\leq j \leq i_{2}+j_{2}$ .
The forgetful functor $N\left (O_{A}^{\otimes }\right )\rightarrow \mathcal {LM}^{\otimes }$ is a coCartesian fibration of $\infty $ operads, exhibiting $N((\operatorname {dgMod}(A)^{\circ })_{0})$ as lefttensored over the symmetric monoidal $\infty $ category $N((\operatorname {dgMod}(k)^{\circ })_{0})$ . By the discussion following [Reference LurieLur17, 4.1.7.3] and the fact that preserves weak equivalences in both entries, it follows that the lefttensoring passes to the $\infty $ categorical localisations at the chain homotopy equivalences, meaning that we obtain that $\mathcal {D}(A)$ is lefttensored over $\mathcal {D}(k)$ . The action of $\mathcal {D}(k)$ on $\mathcal {D}(A)$ preserves colimits in both variables, as follows from the monoidal product being a Quillen bifunctor. To see that is a klinear functor, we need to describe an extension of it to a map $\alpha :N\left (O^{\otimes }_{k}\right )\rightarrow N\left (O^{\otimes }_{A}\right )$ of $\infty $ operads over $\mathcal {LM}^{\otimes }$ . We leave to the reader the details of the description of a functor of $1$ categories $\alpha ^{\prime }:O^{\otimes }_{k}\rightarrow O^{\otimes }_{A}$ whose nerve $N(\alpha ^{\prime })$ defines the desired functor $\alpha $ .
Proposition 2.19. Let A be a klinear dgalgebra. Using the symmetric monoidal equivalence $\mathcal {D}(k)\simeq \operatorname {RMod}_{k}$ , we can consider $\operatorname {RMod}_{A}\overset {\text {formula~}(4)}{\simeq } \operatorname {RMod}_{A}(\operatorname {RMod}_{k})$ as lefttensored over $\mathcal {D}(k)$ . There exists an equivalence
of $\infty $ categories lefttensored over $\mathcal {D}(k)$ .
Proof. Consider the adjunction of $\infty $ categories underlying the Quillen adjunction . Using the adjunction, it can be directly checked that A is a compact generator of $\mathcal {D}(A)$ . It follows from [Reference LurieLur17, 4.8.5.8] that there exists an equivalence
of $\infty $ categories lefttensored over $\mathcal {D}(k)$ , where $\operatorname {End}_{k}(A)\in \operatorname {Alg}(\mathcal {D}(k))$ is the klinear endomorphism algebra of A (see Remark 2.6). We note that the underlying chain complex satisfies $\operatorname {End}_{k}(A)\simeq \operatorname {RHom}_{A}(A,A)\simeq A$ . By the universal property of $\operatorname {End}_{k}(A)$ , there exists a morphism of dgalgebras $\chi :A\rightarrow \operatorname {End}_{k}(A)$ , whose underlying morphism of chain complexes is induced by the actions $A\otimes _{k}A\rightarrow A$ and $A\otimes _{k} \operatorname {End}_{k}(A)\rightarrow A$ . The latter is induced by the counit of the adjunction and is thus equivalent to the former. It follows that $\chi $ induces a quasiisomorphism $\operatorname {End}_{k}(A)=\operatorname {RHom}_{A}(A,A)\simeq A$ on underlying chain complexes and is hence a quasiisomorphism of dgalgebras. In total, we obtain that there also exists an equivalence of klinear $\infty $ categories $\operatorname {RMod}_{\operatorname {End}_{k}(A)}(\mathcal {D}(k))\simeq \operatorname {RMod}_{A}(\mathcal {D}(k))\simeq \operatorname {RMod}_{A}(\operatorname {RMod}_{k})$ , which combined with formula (10) shows the statement.
Let $A,B\in \operatorname {Alg}(\mathcal {D}(k))$ be dgalgebras and $F:\operatorname {RMod}_{A}\rightarrow \operatorname {RMod}_{B}$ a klinear functor. Clearly $F(A)\in \operatorname {RMod}_{B}$ carries the structure of a right Bmodule. Let $m:A\otimes _{k} A\rightarrow A$ be the multiplication map of A. Using the klinearity of F, we find an action map