Relative Calabi–Yau structures and perverse schobers on surfaces

Let $R$ be the base ${𝔼}_{\mathrm{\infty }}\text{-}$ring spectrum. Our setting for the definition of relative Calabi–Yau structures is that of stable, presentable, dualizable, $R\text{-}$linear $\mathrm{\infty }\text{-}$categories and dualizable (in particular colimit-preserving), $R\text{-}$linear functors between them. In the following, we sketch the definition of relative Calabi–Yau structures and describe the gluing properties. The definition makes use of the functoriality of $R\text{-}$linear Hochschild homology, as well as its $S^1\text{-}$action, which we obtain from the formalism of traces [23, 44].

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

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

The above definition is thus analogous to the definition of relative Calabi–Yau structures given in the setting of dg categories by Brav–Dyckerhoff [2]. We will show in Lemma 3.6 that a dg functor $f:A\to B$ admits a weak Calabi–Yau structure if and only if the $k\text{-}$linear functor $𝒟(f):𝒟(A)\to 𝒟(B)$ between the compactly generated derived $\mathrm{\infty }\text{-}$categories admits a weak Calabi–Yau structure. The possibility to consider relative Calabi–Yau structures on dualizable $k\text{-}$linear $\mathrm{\infty }\text{-}$categories thus makes the $\mathrm{\infty }\text{-}$categorical setting slightly more general than the dg categorical setting. The definition of weak relative left and right Calabi–Yau structures for compactly generated categories linear over a commutative ring spectrum was also previously described in a model categorical framework in the unpublished Master’s thesis [29].

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

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

Gluing Calabi–Yau structures. Relative Calabi–Yau structure can be glued together along pushouts or pullbacks of $\mathrm{\infty }\text{-}$categories. There are also variants of this for $(\mathrm{\infty },2)\text{-}$categorical lax pushouts and pullbacks (see [9, Lem. 6.3.3]).

For the gluing of left Calabi–Yau structures, consider a pushout diagram of smooth, dualizable $R\text{-}$linear $\mathrm{\infty }\text{-}$categories and dualizable functors.

For the gluing of right Calabi–Yau structures, we consider a pullback diagram of proper dualizable $R\text{-}$linear $\mathrm{\infty }\text{-}$categories and dualizable functors as follows.

We note that the analogue of Theorem 2 was not previously known for right Calabi–Yau structures on dg categories [2].

Perverse schobers are, in general, a conjectural categorification of perverse sheaves, proposed by Kapranov–Schechtman [26]. In this paper, we use the framework of perverse schobers parametrized by ribbon graphs of [5]. This describes perverse schobers on marked surfaces with boundary in terms of constructible sheaves valued in stable $\mathrm{\infty }\text{-}$categories defined on a spanning ribbon graph embedded in the surfaces. Concretely, a perverse schober $ℱ$ parametrized by a graph $\mathbf{G}$ is encoded as a functor $ℱ:\mathrm{Exit}(\mathbf{G})\to \mathrm{St}\text{.}$ Here $\mathrm{St}$ denotes the $\mathrm{\infty }\text{-}$category of stable $\mathrm{\infty }\text{-}$categories, and the domain denotes the exit path category of $\mathbf{G}\text{,}$ whose objects are the vertices and edges of $\mathbf{G}$ and whose morphisms describe the incidence between vertices and edges. The limit of this functor, denoted by $\mathrm{\Gamma }(\mathbf{G},ℱ)\text{,}$ is called the $\mathrm{\infty }\text{-}$category of global sections of $ℱ\text{.}$

We remark that the usage of enhanced triangulated categories (such as the stable $\mathrm{\infty }\text{-}$categories we employ in this paper) in our treatment of perverse schobers is essential, since there is no sensible theory of homotopy (co)limits of non-enhanced triangulated categories, which is needed for such a sheaf theory.

Monodromy of perverse schobers. A perverse sheaf on a topological surface restricts to a cochain complex of locally constant sheaves on the top-dimensional stratum, which is the complement of the discrete set of singularities. The cohomology sheaves of this cochain complex are trivial except in degree $-1$ (in the typical convention), thus defining a local system of vector spaces. We discuss in Section 4.3, and sketch in the following, how to associate a similar local system of stable $\mathrm{\infty }\text{-}$categories to parametrized perverse schobers.

If we only consider connected surfaces, then all generic, that is, non-singular, stalks of a given perverse sheaf are equivalent. The same is true for a $\mathbf{G}\text{-}$parametrized perverse schober $ℱ$: the value of $ℱ$ at any edge of $\mathbf{G}$ is independent of the chosen edge, up to equivalence, and should be considered as the generic stalk. Denote the generic stalk by $𝒩\text{.}$ Locally at each vertex $v$ of $\mathrm{\Gamma }\text{,}$ a perverse schober is described by a spherical adjunction $𝒱\leftrightarrow 𝒩\text{,}$ with $𝒱$ called the $\mathrm{\infty }\text{-}$category of vanishing cycles at $v\text{.}$ If $𝒱\ne 0\text{,}$ we call the vertex $v$ a singularity of $ℱ\text{.}$

Let ${\mathbf{G}}_0$ be the set of vertices of $\mathbf{G}\text{.}$ Given a perverse schober $ℱ$ with a set of singularities $P\subset {\mathbf{G}}_0\text{,}$ we wish to associate a local system valued in $\mathrm{St}$ on $\mathbf{S}\backslash P\text{,}$ which we describe as a group homomorphism

to the group of equivalence classes of autoequivalences of the generic stalk $𝒩\text{.}$

There is in general no canonical choice of such a local system. We can however canonically define a local system $ℒℱ$ on the total space of the frame bundle $\mathrm{Fr}(\mathbf{S}\backslash {\mathbf{G}}_0)\to \mathbf{S}\backslash {\mathbf{G}}_0\text{.}$ The fiber of the frame bundle has the homotopy type of the circle $S^1\text{,}$ the monodromy of the local system along the fiber is given $[2]\text{.}$ Suppose now that we choose a framing $\xi$ of the surface $\mathbf{S}\backslash P\text{,}$ meaning a section of its frame bundle. We can then pull back to a local system ${\xi }^{\ast }ℒℱ$ on $\mathbf{S}\backslash {\mathbf{G}}_0\text{,}$ and crucially, this local system extends to $\mathbf{S}\backslash P\text{.}$ This defines the desired monodromy local system of $ℱ\text{.}$ We stress that this local system depends on the choice of framing $\xi \text{.}$

Note that in the special case that $𝒩$ is $2\text{-}$periodic, that is, $[2]\simeq {\mathrm{id}}_{𝒩}\text{,}$ the local system on the frame bundle has trivial monodromy on the fiber. It thus already reduces to a local system on $\mathbf{S}\backslash P\text{,}$ and no choice of framing is required as input.

Perverse schobers without singularities are fully determined by their monodromy.

The notion of a non-singular parametrized perverse schober is thus non-canonically equivalent to the notion of a local system of stable $\mathrm{\infty }\text{-}$categories on the surface. Note that what we refer to as the global sections of the non-singular perverse schober is however very different from the global sections of a local system. The former type of global sections describes a generalized topological Fukaya category and categorifies the first cohomology of the surface relative to the complement in the boundary of the marked points.

Our results on the monodromy of perverse schobers relate to the problem of defining the topological Fukaya category of a marked surface over an arbitrary base $\mathrm{\infty }\text{-}$category $𝒩$: without further assumptions on $𝒩\text{,}$ a choice of framing of the surface is required. Then there exists a perverse schober (unique up to equivalence) with generic stalk $𝒩$ and trivial monodromy relative to the chosen framing. Its $\mathrm{\infty }\text{-}$category of global sections describes the desired $𝒩\text{-}$valued topological Fukaya category. In the case $𝒩=𝒟(k)\text{,}$ this $\mathrm{\infty }\text{-}$categorical topological Fukaya category recovers the derived $\mathrm{\infty }\text{-}$category of the dg categorical topological Fukaya category or equivalently of the $A_{\mathrm{\infty }}\text{-}$categorical partially wrapped Fukaya category. If $𝒩$ is $2\text{-}$periodic, then no choice of framing is required, and there is already a perverse schober with a well-defined trivial monodromy, whose global sections give the $𝒩\text{-}$valued topological Fukaya category. In the setting of dg categories, this problem of constructing topological Fukaya categories (up to a contractible space of choices) was fully solved by Dyckerhoff–Kapranov [14, 15] using the formalism of $2\text{-}$Segal objects. Their construction in fact additionally supplies a choice of perverse schober with trivial monodromy for every choice of spanning ribbon graph. Their construction was extended to the case of $𝒩$ the stable $\mathrm{\infty }\text{-}$category of right modules over the $2\text{-}$periodic sphere spectrum by Lurie [33].

Local Calabi–Yau structures. Let $ℱ$ be an $R\text{-}$linear $\mathbf{G}\text{-}$parametrized perverse schober. Locally at any vertex $v$ of the graph $\mathbf{G}\text{,}$ with incident edges $e_1,\mathrm{\dots },e_m\text{,}$ the perverse schober $ℱ$ is given by a collection of functors $ℱ(v)\to ℱ(e_i)\simeq 𝒩\text{,}$ $1\le i\le m\text{.}$ One can show that these functors arise, up to suitable equivalence, from a single spherical adjunction $F:𝒱\leftrightarrow 𝒩:G$ via an explicit construction based on the relative Waldhausen $S_{•}\text{-}$construction (see Proposition 4.10). We say that the adjunction $F⊣G$ underlies $ℱ$ at $v\text{.}$

We further prove a novel criterion for a spherical functor $F:𝒱\to 𝒩$ between compactly generated, proper $R\text{-}$linear $\mathrm{\infty }\text{-}$categories, where $𝒩$ is weakly right $(n-1)\text{-}$Calabi–Yau, to admit a weak right $n\text{-}$Calabi–Yau structure: this is the case if and only if its twist functor $T_{𝒱}$ is equivalent to the shifted Serre functor ${\mathrm{id}}_{𝒱}^{\ast }[1-n]$ (see Proposition 5.9).

Global Calabi–Yau structures. Given a $\mathbf{G}\text{-}$parametrized perverse schober $ℱ\text{,}$ we can evaluate global sections at the external (i.e., boundary) edges of $\mathbf{G}\text{,}$ whose set is denoted by ${\mathbf{G}}_1^{\partial }\text{.}$ This yields a functor

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

Typically, a relative Calabi–Yau structure on the $\mathrm{\infty }\text{-}$category of global sections $\mathrm{\Gamma }(\mathbf{G},ℱ)$ arises in the smooth setting as a left Calabi–Yau structure on the functor $\partial ℱ$ and in the proper setting as right Calabi–Yau structure on the functor ${\prod }_{e\in {\mathbf{G}}_1^{\partial }}{\mathrm{ev}}_e\text{.}$

Note that finite limits of proper dualizable $R\text{-}$linear $\mathrm{\infty }\text{-}$categories in $\mathrm{St}$ (or equivalently in the $\mathrm{\infty }\text{-}$category ${\mathrm{LinCat}}_R$ of $R\text{-}$linear $\mathrm{\infty }\text{-}$categories) are not necessarily again dualizable and proper. We can fix this issue by forming the limits in the $\mathrm{\infty }\text{-}$category ${\mathrm{LinCat}}_R^{\mathrm{dual}}$ of dualizable $R\text{-}$linear $\mathrm{\infty }\text{-}$categories (see Corollary 3.14). The arising notion of $\mathrm{\infty }\text{-}$category of global sections is denoted by ${\mathrm{\Gamma }}^{\mathrm{dual}}(\mathbf{G},ℱ)\text{;}$ we call these the dualizable global sections. In the proper setting, we should thus ask for the restriction of ${\prod }_{e\in {\mathbf{G}}_1^{\partial }}{\mathrm{ev}}_e$ to ${\mathrm{\Gamma }}^{\mathrm{dual}}(\mathbf{G},ℱ)$ to be relative right Calabi–Yau.

When the global sections describe the partially wrapped Fukaya category of a surface, the difference between $\mathrm{\Gamma }(\mathbf{G},ℱ)$ and ${\mathrm{\Gamma }}^{\mathrm{dual}}(\mathbf{G},ℱ)$ can be explained as follows: the category $\mathrm{\Gamma }(\mathbf{G},ℱ)$ describes the usual smooth partially wrapped Fukaya category of the marked surface. The proper full subcategory ${\mathrm{\Gamma }}^{\mathrm{dual}}(\mathbf{G},ℱ)\subset \mathrm{\Gamma }(\mathbf{G},ℱ)$ consists of those Lagrangians which do not end at the boundary components containing no marked points.

For perverse schobers without singularities, we prove the following.

Theorem 3 generalizes Brav–Dyckerhoff’s result [2] on relative Calabi–Yau structures on topological Fukaya categories of framed marked surfaces (corresponding to the case $𝒩=𝒟(k)$ and a perverse schober with trivial monodromy relative to the chosen framing).

There is no direct analogue of Theorem 3 for general perverse schobers with singularities. Essentially, this is because a perverse schober is not determined up to equivalence by the separate records of monodromy data and local singularity data (see Example 4.35). There is however an almost immediate consequence of the gluing property of relative Calabi–Yau structures for global sections of perverse schobers (see Theorem 5.7), which can be applied in practice by using the local Calabi–Yau structures from Proposition 2.

Fukaya–Seidel categories. Let $X$ be an exact symplectic manifold of dimension $2n$ and $\pi :X\to 𝔻$ a Lefschetz fibration with base the disc. Let $F$ be the regular fiber of $\pi$ and $\mathrm{Fuk}(F)$ the proper Fukaya $A_{\mathrm{\infty }}\text{-}$category of compact Lagrangians in $F\text{.}$ The Fukaya–Seidel $A_{\mathrm{\infty }}\text{-}$category $\mathrm{FS}(\pi )$ is equivalent to the directed $A_{\mathrm{\infty }}\text{-}$subcategory of $\mathrm{Fuk}(F)$ on the vanishing cycles of the Lefschetz fibrations [37]. The corresponding derived Fukaya–Seidel category is a triangulated category and admits a canonical enhancement to a $k\text{-}$linear stable $\mathrm{\infty }\text{-}$category $𝒟(\mathrm{FS}(\pi ))\text{.}$

The formalism of parametrized perverse schobers on the disc $𝔻\text{,}$ considered as a marked surface with a single marked point, realizes the derived Fukaya–Seidel category $𝒟(\mathrm{FS}(\pi ))$ as the global sections of a perverse schober. The generic stalk of the schober is the derived Fukaya category of the fiber $𝒟(\mathrm{Fuk}(F))\text{.}$ The singularities of the perverse schober lie at the singular values of the Lefschetz fibration; the corresponding spherical adjunctions arise from the spherical objects in $\mathrm{Fuk}(F)$ given by the vanishing cycles. The ribbon graph parametrizing $ℱ$ is chosen so that the singular values all lie at $1\text{-}$valent vertices. There is a further non-singular $(m+1)\text{-}$valent vertex $v\text{,}$ with $m$ the number of vanishing cycles. The value of $ℱ$ at $v$ is the directed $\mathrm{\infty }\text{-}$category $ℱ(v)\simeq \mathrm{Fun}({\mathrm{\Delta }}^{m-1},𝒟(\mathrm{Fuk}(F)))\text{.}$

Any spherical object in a weak $(n-1)\text{-}$Calabi–Yau category gives rise to a weak $n\text{-}$Calabi–Yau spherical functor (see Lemma 6.4). The gluing properties of right Calabi–Yau structures thus yield a weak relative right $n\text{-}$Calabi–Yau structure on $𝒟(\mathrm{FS}(\pi ))\text{.}$ This induces the known natural transformation [38] from the Serre functor ${\mathrm{id}}_{𝒟(\mathrm{FS}(\pi ))}^{\ast }\to {\mathrm{id}}_{𝒟(\mathrm{FS}(\pi ))}[n+1]\text{.}$

In Section 6.1, we will give a more detailed account of the above construction in the alternative framework of [20] for the definition of Fukaya–Seidel categories. In this framework, we furthermore prove that the smooth and proper Fukaya–Seidel category admits not only a weak relative right $n\text{-}$Calabi–Yau structure but also a weak relative left $n\text{-}$Calabi–Yau structure.

The Fukaya–Seidel category is defined in [20] as a partially wrapped Fukaya category with a stop in the fiber over $\mathrm{\infty }\text{.}$ Part (i) of Theorem 6.1 should readily generalize to the partially wrapped Fukaya categories arising from Lefschetz fibrations over an arbitrary marked surface. Up to technicalities, this follows from the cosheaf properties of such a partially wrapped Fukaya category shown in [20]. The statement about the relative left Calabi–Yau property of part (ii) of Theorem 6.1 may be generalized to this setting, given an understanding of the action of the monodromy of the Lefschetz fibration on the non-degenerate Hochschild class of the wrapped Fukaya category of the fiber.

Periodic topological Fukaya categories. The author’s initial motivating example for treating relative Calabi–Yau structures over an arbitrary base was the construction of relative right $2\text{-}$Calabi–Yau structures on $1\text{-}$periodic topological Fukaya categories of marked surfaces. These can be considered as $\mathbb{Z}/1\mathbb{Z}\text{-}$graded versions of the partially wrapped Fukaya categories. Their construction is the topic of Section 6.2. Relative right $2\text{-}$Calabi–Yau structure induces $2\text{-}$Calabi–Yau Frobenius exact $\mathrm{\infty }\text{-}$structures (see [8]), which in turn induce $2\text{-}$Calabi–Yau Frobenius extriangulated structures on the homotopy $1\text{-}$categories. In the case of $1\text{-}$periodic topological Fukaya categories, this exact/extriangulated structure allows for the additive categorification of cluster algebras with coefficients associated with surfaces (see [8]).

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

Considering the $n\text{-}$periodic topological Fukaya category over the base $k[t_n^{\pm }]\text{,}$ or $k[t_{2n}^{\pm }]$ if $n$ is odd, Theorem 3 yields the desired relative $(n+1)\text{-}$Calabi–Yau structure on it.

Relative Ginzburg algebras over any base ring spectrum. The derived $\mathrm{\infty }\text{-}$categories of relative Ginzburg algebras of $n\text{-}$angulated surfaces arise as the global sections of parametrized perverse schobers (see [5, 6]). In Section 6.3, we construct relative left $n\text{-}$Calabi–Yau structures on these derived $\mathrm{\infty }\text{-}$categories. In the case $n=3\text{,}$ this result is a special case of results [45, 46] on relative left $3\text{-}$Calabi–Yau structures on relative Ginzburg algebras of ice quivers with potentials or equivalently relative Calabi–Yau completions. In the case $n=3\text{,}$ these $\mathrm{\infty }\text{-}$categories can further be expected to describe the derived $\mathrm{\infty }\text{-}$categories of the partially wrapped Fukaya categories of the 3-folds studied in [41].

The relevant perverse schobers are locally near each vertex described by the spherical adjunction $f^{\ast }:𝒟(k)\leftrightarrow \mathrm{Fun}(S^{n-1},𝒟(k)):f_{\ast }\text{,}$ where $f:S^{n-1}\to \ast$ is the map from the singular simplicial set of the $(n-1)\text{-}$sphere to the point. The functor $f_{\ast }$ is left $n\text{-}$Calabi–Yau. Furthermore, the functor ${\overline{f}}^{\ast }\text{,}$ obtained by restricting $f^{\ast }$ to a functor $𝒟(k)\to \mathrm{Ind}\mathrm{Fun}(S^{n-1},{𝒟}^{\mathrm{perf}}(k))\text{,}$ is right $n\text{-}$Calabi–Yau.

We can replace $𝒟(k)$ by ${\mathrm{RMod}}_R\text{,}$ with $R$ an ${𝔼}_{\mathrm{\infty }}\text{-}$ring spectrum, to obtain an $R\text{-}$linear version of this adjunction. We expect that both the Calabi–Yau structures of $f_{\ast }$ and ${\overline{f}}^{\ast }$ can be lifted to the $R\text{-}$linear setting, but we only prove that we have a weak right Calabi–Yau structure on ${\overline{f}}^{\ast }\text{.}$ Its existence is proven using the criterion for the existence of weak right Calabi–Yau structures on spherical functors of Proposition 5.9. Via gluing, the Calabi–Yau structure on ${\overline{f}}^{\ast }$ yields weak relative right $n\text{-}$Calabi–Yau structures on the locally compact global sections of $R\text{-}$linear perverse schobers that generalize ($\mathrm{Ind}\text{-}$finite) derived categories of relative Ginzburg algebras.

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

We generally follow the notation and conventions of [31, 34]. In particular, we use the homological grading convention. Given an $\mathrm{\infty }\text{-}$category $𝒞$ and two objects $X,Y\in 𝒞\text{,}$ we denote by ${\mathrm{Map}}_{𝒞}(X,Y)$ the mapping space. We denote the $\mathrm{Ind}\text{-}$completion of $𝒞$ by $\mathrm{Ind}(𝒞)={\mathrm{Ind}}_{\omega }(𝒞)$ and the subcategory of ($\omega \text{-}$)compact objects of $𝒞$ by ${𝒞}^{\mathrm{c}}\text{.}$ Given a functor $F:𝒞\to 𝒟\text{,}$ we denote its left and right adjoints, if existent, by $\mathrm{ladj}(F)$ and $\mathrm{radj}(F)\text{,}$ respectively.

Let ${\mathrm{Cat}}_{\mathrm{\infty }}$ be the $\mathrm{\infty }\text{-}$category of $\mathrm{\infty }\text{-}$categories and $𝒮$ the $\mathrm{\infty }\text{-}$category of spaces. We denote by $𝒫r^L\subset {\mathrm{Cat}}_{\mathrm{\infty }}$ the subcategory of presentable $\mathrm{\infty }\text{-}$categories and left adjoint functors and by $𝒫r^R\subset {\mathrm{Cat}}_{\mathrm{\infty }}$ the subcategory of presentable $\mathrm{\infty }\text{-}$categories and right adjoint functors. The $\mathrm{\infty }\text{-}$category $𝒫r^L$ admits a symmetric monoidal structure such that a commutative algebra object in $𝒫r^L$ amounts to a symmetric monoidal presentable $\mathrm{\infty }\text{-}$category $𝒞\text{,}$ satisfying that its tensor product $\text{-}\otimes \text{-}:𝒞\times 𝒞\to 𝒞$ preserves colimits in both entries (see [34, § 4.8]). An example of a commutative algebra object in $𝒫r^L$ is the $\mathrm{\infty }\text{-}$category ${\mathrm{RMod}}_R$ of right module spectra over an ${𝔼}_{\mathrm{\infty }}\text{-}$ring spectrum $R\text{.}$ Note that if $R=k$ is a commutative ring, then ${\mathrm{RMod}}_k$ is equivalent as a symmetric monoidal $\mathrm{\infty }\text{-}$category to the (unbounded) derived $\mathrm{\infty }\text{-}$category $𝒟(k)$ (see [34, Thm. 7.1.2.13]).

As noted in [35, § D.1.5], $R\text{-}$linear $\mathrm{\infty }\text{-}$categories in the above sense are automatically stable. Given $𝒞\in {\mathrm{LinCat}}_R\text{,}$ we denote the result of the action of an element $C\in {\mathrm{RMod}}_R$ on $X\in 𝒞$ by $C\otimes X\in 𝒞\text{.}$

We thus have ${\pi }_i{\mathrm{Mor}}_{𝒞}(X,Y)\simeq {\pi }_0{\mathrm{Map}}_{𝒞}(X[i],Y)$ for all $i\in \mathbb{Z}\text{.}$

The $\mathrm{\infty }\text{-}$category ${\mathrm{LinCat}}_R$ inherits a symmetric monoidal structure, as the module category over a commutative algebra object. We will often make use of this monoidal product and denote it by $\otimes \text{.}$ The tensor product of $𝒞,𝒟\in {\mathrm{LinCat}}_R$ arises as the geometric realization of the two-sided bar construction $\mathrm{Bar}{(𝒞,𝒟)}_{\ast }:{\mathrm{\Delta }}^{\mathrm{op}}\to 𝒫r^L\text{,}$ which is given informally by the formula $\mathrm{Bar}{(𝒞,𝒟)}_n=𝒞{\otimes }_{𝒫r^L}{\mathrm{RMod}}_R^{{\otimes }_{𝒫r^L}n}{\otimes }_{𝒫r^L}𝒟\text{,}$ where ${\otimes }_{𝒫r^L}$ denotes the symmetric monoidal product of $𝒫r^L\text{.}$ The symmetric monoidal $\mathrm{\infty }\text{-}$category ${\mathrm{LinCat}}_R$ is closed. As observed in [23, § 4.1], using that $𝒫r^L$ is closed, the internal Hom in ${\mathrm{LinCat}}_R\text{,}$ denoted ${\mathrm{Lin}}_R(𝒞,𝒟)\text{,}$ can be obtained as the limit of a cosimplicial object obtained from replacing the tensor products in the two-sided bar resolution by the right adjoint internal Homs in $𝒫r^L\text{.}$ We record the following functoriality of the internal Hom of ${\mathrm{LinCat}}_R\text{.}$

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

and

satisfying the triangle identities.

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

An $R\text{-}$linear $\mathrm{\infty }\text{-}$category $𝒞\in {\mathrm{LinCat}}_R$ is dualizable if and only if it is compactly assembled (see [35, Thm. D.7.0.7]), which is equivalent to $𝒞$ being a retract of a compactly generated, presentable and stable $\mathrm{\infty }\text{-}$category in the $\mathrm{\infty }\text{-}$category $𝒫{𝓇}_{\mathrm{St}}^L\subset 𝒫r^L$ of stable, presentable $\mathrm{\infty }\text{-}$categories (see [35, Prop. D.7.3.1]).

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

to the restriction of the morphism object functor ${\mathrm{Mor}}_{𝒞}(\text{-},\text{-})$ (see [35, Prop. D.7.2.3, Rem. D.7.7.6]). Under the assumption that $𝒞$ is compactly generated, an $R\text{-}$linear functor is dualizable if and only if it preserves compact objects (see [31, Prop. 5.5.7.2]). We thus have a fully faithful inclusion ${\mathrm{LinCat}}_R^{\mathrm{cpt}-\mathrm{gen}}\subset {\mathrm{LinCat}}_R^{\mathrm{dual}}\text{.}$ We remark that this inclusion preserves both limits and colimits, as follows from the results of [17].

We again fix a base ${𝔼}_{\mathrm{\infty }}\text{-}$ring spectrum $R\text{.}$ Suppose we are given two $R\text{-}$linear ring spectra $A,A^{\prime }\text{.}$ The $\mathrm{\infty }\text{-}$category of $A\text{-}$ $A^{\prime }\text{-}$bimodules ${}_A{}^{}\mathrm{BMod}_{A^{\prime }}^{}({\mathrm{RMod}}_R)$ is equivalent to the $\mathrm{\infty }\text{-}$category ${\mathrm{Lin}}_R({\mathrm{RMod}}_A,{\mathrm{RMod}}_{A^{\prime }})$ of $R\text{-}$linear functors between the respective right module $\mathrm{\infty }\text{-}$categories (see [34, Thms. 4.3.2.7 and 4.8.4.1]). In terms of functors, left and right duals of bimodules, if they exist, correspond to left and right adjoints of the corresponding functors. In the following, we will work with functors instead of bimodules.

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

We can use the equivalences ${\mathrm{\Theta }}_{𝒞}$ and ${\mathrm{\Xi }}_{𝒞}$ to define the dual of an $R\text{-}$linear endofunctor $𝒞\to 𝒞\text{,}$ considered as a functor ${𝒞}^{\vee }\otimes 𝒞\to {\mathrm{RMod}}_R$ or ${\mathrm{RMod}}_R\to 𝒞\otimes {𝒞}^{\vee }\text{.}$